structural,electronicandmechanical propertiesofone-andtwo

Structural, Electronic and MechanicalProperties of One- and Two-Dimensional

Transition Metal Dichalcogenide Materialsby

Nourdine Zibouche

A thesis submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

in Physics

Approved, Thesis Committee

Chair: Prof. Dr. Thomas Heine (Jacobs University)

Prof. Dr. Veit Wagner (Jacobs University)

Prof. Dr. Thomas Niehaus (University of Regensburg)

Dr. Agnieszka Kuc (Jacobs University)

Date of Defense: October 9, 2014

Engineering and Science

Statutory Declaration

I, Nourdine Zibouche, hereby declare that I have written this PhD thesis independently,

unless where clearly stated otherwise. I have used only the sources, the data and the

support that I have clearly mentioned. This PhD thesis has not been submitted for

conferral of degree elsewhere.

Nourdine Zibouche

Bremen, Germany

December 17, 2014

Abstract

The successful isolation of single sheet of graphene and the considerable progress in

miniaturizing electronic devices have prompted researchers to explore alternative materi-

als other than silicon, particularly two-dimensional (2D) materials. This has led to the

renaissance of layered Transition-Metal diChalchogenide (TMC) materials, which have

recently received special considerations due to their fundamentally and technologically

intriguing properties. In fact, bulk layered TMC compounds have been studied for many

years and have mainly been exploited as lubricants and intercalation materials. Recent

development in the exfoliation and synthesis techniques o�ered the opportunity of ex-

ploring their properties at low dimensions, in particular 1D and 2D. Consequently, many

applications of low dimension TMCs have been developed and proposed for the next gener-

ation of nanoelectronic devices, including �eld-e�ect transistors, photodetectors, sensors,

light-emitting diodes, solar cell and so on.

In this Ph.D. dissertation, the physical properties of one- and two-dimension semicon-

ducting TMC materials have been studied via �rst-principles approach based on density

functional theory. The main focus is about the electronic structure of nanotubular, mono-

and few-layer TMC systems. The role of quantum con�nement and the e�ect of spin-orbit

coupling are examined. The results show the thickness dependence of the electronic prop-

erties, when the bulk systems are thinned down to the monolayer level. A giant spin-orbit

splitting is revealed in the monolayered systems due to the inversion symmetry breaking.

The electronic properties of TMC nanotubes are also investigated and compared to that

of the layered counterparts. Moreover, the response of these TMC materials to external

factors, in particular tensile strain and electric �eld, is explored. The electronic band

structures, band gaps and charge carrier mobilities with respect to the applied tensile

strain or the electric �eld are strongly a�ected. This shows the possibility of controlling

and tuning the properties of TMC materials, which may provide new functionalities and

hence eminent applications in nanoelectronics, optoelectronics and �exible devices.

Keywords: Transition-metal dichalchogenides, DFT, nanotubes, quantum con-

finement, spin-orbit coupling, tensile strain, electric field.

i

Outline

This thesis is written in a cumulative form, as permitted by Jacobs University Bremen.

The major part of the achieved work has been published in peer-reviewed journals. The

remaining part is submitted or at the �nal manuscript revision before submission. The

list of the articles is given below. Permissions to the copyrighted �gures that have been

adopted in this thesis have been received.

This thesis is organized and structured into three di�erent parts. The �rst part is

of an introductory aspect and presents a general overview about layered transition metal

dichalcogenide materials. This part is divided into two main sections; the �rst section deals

with bulk and monolayer systems and the second one describes nanotubular materials.

Fundamental and essential characteristics of these materials are highlighted including their

structural, electronic and mechanical properties, a brief description of methods of synthesis

and fabrication as well as their recent and major applications in electronics, optoelectronics

and other �elds in nanotechnology. Note that the principal emphasis will be put on to

TMCs with a transition metal belonging to the group 6 of the periodic table, namely Mo

and W, and the chalcogens S, Se and Te.

The second part presents the key concepts of the density functional theory method,

which is used to carry out calculations in this work. Here, we recall the fundamental

theorems of Hohenberg and Kohn, the Kohn�Sham approach and the main important

density functionals, from pure to hybrid, that are largely employed in computational

chemistry and physics. A general outline about existing basis sets is also given as well as

the choice of di�erent computational ingredients that have been adopted and used in the

calculations.

In the third part, the accomplished research work and the contribution to the �eld of

one- and two-dimensional transition metal dichalcogenide materials are summarized and

assembled into various extended abstracts of articles, in which the obtained major results

and conclusions are highlighted. The full-text of the articles describing the complemented

work of this Ph.D. project on TMC materials are given as appendices at the end of this

dissertation.

iii

March 29, 2017

1 List of articles

1. Influence of quantum confinement on the electronic structure of the transi-tion metal sulfide TS2, Agnieszka Kuc, Nourdine Zibouche, Thomas Heine;Phys. Rev. B 83 (2011) 245213.

2. Transition-metal dichalcogenides for spintronic applications;Nourdine Zi-bouche, Agnieszka Kuc, Janice Musfeld, Thomas Heine. Ann. Phys.(Berlin), (2014). Accepted for publication

3. Electron transport in MoWSeS monolayers in the presence of an exter-nal electric field; Nourdine Zibouche, Pier Philipsen, Thomas Heine, Ag-nieszka Kuc; Phys. Chem. Chem. Phys. (2014) accepted for publication.

4. Transition-metal dichalcogenide bilayers: switching materials for spin- andvalleytronic applications; Nourdine Zibouche, Pier Philipsen, AgnieszkaKuc, Thomas Heine; Accepted for publication in Phys. Rev. B. (PRB).

5. From layers to nanotubes: Transition metal disulfides TMS2; NourdineZibouche, Agnieszka Kuc, Thomas Heine; Eur. Phys. J. B. (2012) 49.

6. Electromechanical Properties of Small Transition-Metal DichalcogenideNanotubes; Nourdine Zibouche, M. Ghorbani-Asl, Thomas Heine, Ag-nieszka Kuc; Inorganics 2014, 2, 155-167.

7. Tunable Electronic Properties and Transport of MoS2 andWS2 NanotubesUnder External Electric Field; Nourdine Zibouche, Pier Philipsen, Ag-nieszka Kuc, Thomas Heine; To be submitted to Nanoscale.

8. Noble-Metal Chalcogenide Nanotubes; Nourdine Zibouche, Agnieszka Kuc,Pere Miro, Thomas Heine; submitted to Inorganics.

9. Mahdi Ghorbani-Asl, Nourdine Zibouche, Mohammad Wahiduzzaman,Augusto F. Oliveira, Agnieszka Kuc, Thomas Heine; Electromechanicsin MoS2 and WS2 nanotubes and Monolayers; Scientific Reports 3 (2013)2961. (not included in the thesis)

10. Double Walled Transition-Metal Dichalcogenide Nanotubes; Nourdine Zi-bouche, Thomas Heine and Agnieszka Kuc (in preparation)

1

Acknowledgements

First and foremost, I would like to express my gratitude and thanks to my supervisor

professor Thomas Heine for his generous guidance, support and discussions that made this

Ph.D. work constructive and productive. His invaluable scienti�c intuition and experience

have inspired me and made me eager to learn more, to deepen my knowledge, and to grow

as research scientist.

I gratefully thank professor Veit Wagner, professor Thomas Niehaus, and Dr. Agnieszka

Kuc for being the my dissertation committee members. Their insightful comments, sug-

gestions, advice, and the time spent on the evaluation of this dissertation are sincerely

appreciated.

Special thanks to Dr. Agnieszka Kuc for her tremendous help and patience. This work

would not have been achieved without her support, attention, and crucial contribution

and involvement.

I thank Dr. Lyuben Zhechkov for his assistance in using the computational resources

and Dr. Augusto Oliveria, Dr. Andreas Mavrantonakis, and Dr. Patrice Donfack for

proofreading and correcting my thesis.

Our project assistant, Mrs. Britta Berninghausen, deserves special thanks for her assis-

tance and support in all the administrative a�airs and administration.

I would like to thank the Scienti�c Computing & Modelling (SCM) company, the CEO

Dr. Stan van Gisbergen, and the other sta� at Vrije University in Amsterdam for the

valuable experience and the pleasant chats that I have acquired during my one-year em-

ployment and stay. Spacial thanks to the o�ce manager Mrs. Frieda Vansina for all her

support and to Dr. Pier Philipsen with whom I have mainly worked. His instructions and

brilliant scienti�c discussions have enormously contributed to achieve part of this Ph.D.

work.

I also want to thank professor Hélio Anderson Duarte and his group members at the

University of UFMG in Brazil for the scienti�c collaboration and for the enjoyable time

that I have spent there during my exchange program.

I am deeply grateful to the European research Council (ERC) and the Quasinano project

for the generous �nancial support.

My former and present colleagues deserve my thanks and regards for the academic

discussions and the unforgettable funny moments and chats that we have shared since I

joined the group. They have been source of friendship, support and inspiration.

I am deeply and forever indebted to my parents and siblings for their love, support and

encouragement throughout my entire life and studies. I am also very grateful to all of my

vii

other family members and friends.

Lastly, I o�er my regards to all those who supported me for the completion of this Ph.D.

thesis.

viii

Contents

Abstract i

Outline iii

List of articles v

Acknowledgements vii

1. Introduction 11.1. Layered transition metal dichalcogenides . . . . . . . . . . . . . . . . . . . 3

1.1.1. Crystal structure and composition . . . . . . . . . . . . . . . . . . 3

1.1.2. Electronic and optical properties . . . . . . . . . . . . . . . . . . . 4

1.1.3. Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2. Transition-metal dichalcogenide nanotubes . . . . . . . . . . . . . . . . . . 11

1.2.1. Synthesis of TMC nanotubes . . . . . . . . . . . . . . . . . . . . . 12

1.2.2. Properties of TMC nanotubes . . . . . . . . . . . . . . . . . . . . . 12

1.2.3. Applications of TMC nanotubes . . . . . . . . . . . . . . . . . . . 13

1.2.4. Geometry of a nanotube: an example of CNT . . . . . . . . . . . . 14

1.2.5. Geometry of a TMC nanotube . . . . . . . . . . . . . . . . . . . . 16

2. Methods 192.1. Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2. Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3. Relativistic e�ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4. Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3. Results and discussion 273.1. Layered transition metal dichalcogenides . . . . . . . . . . . . . . . . . . . 27

3.2. Transition-metal dichalcogenide nanotubes . . . . . . . . . . . . . . . . . . 34

4. Summary and concluding remarks 43

5. References 45

Appendices 57

ix

Contents

A. Influence of quantum confinement on the electronic structure of the transitionmetal sulfide TS2 59

B. Transition-metal dichalcogenides for spintronic applications 69

C. Electron transport in MoWSeS monolayers in the presence of an external elec-tric field 77

D. Transition-metal dichalcogenide bilayers: switching materials for spin- and val-leytronic applications 83

E. From layers to nanotubes: Transition metal disulfides TMS2 89

F. Electromechanical Properties of Small Transition-Metal Dichalcogenide Nan-otubes 97

G. Tunable Electronic Properties and Transport of MoS2 and WS2 NanotubesUnder External Electric Field 111

H. Noble-Metal Chalcogenide Nanotubes 119

x

1. Introduction

The progress in nanotechnology and material science requires miniaturization of the

new generation of electronic devices, including transistors and integrated systems. Over

the past �fty years, the silicon-based technology has dominated the semiconductor and

the integrated circuit industry due to its low production cost, silicon thermal stability,

large-area substrates, easy doping, etc. and the size of the devices has become smaller

and smaller. This was supported by the trends in the Moore's law, which states that the

amount of transistors that can be squeezed into an integrated circuit will approximately

double every two years. However, the practical performance and size limitation of silicon

e�ciency and capability have almost been reached. Therefore, alternative solutions are

strongly required and have yet to be achieved. This implies �nding novel materials with

excellent and unique properties, which is one of the driving forces that can serve as building

blocks to manufacture, assemble and implement these devices and components at the

nanoscale size.

In 2004, graphene, which is one-atom-thick layer of 𝑠𝑝2 bonded carbon atoms in a hexag-

onal arrangement on a honeycomb lattice, has successfully been isolated.1 Graphene has

been considered as a potential substitute for silicon and hence the material of the future.

This is due to its excellent electronic and mechanical properties such as a very high charge

carrier mobility at room temperature that is hundred times greater than that of silicon.

However, graphene is a zero gap material, which drastically prevents its utilization in

electronic applications, where the electronic band gap is a fundamental characteristic of

semiconducting devices that governs their e�ciency, performance and properties.2 Never-

theless, the advent of graphene has prompted and motivated researchers to explore new

two dimensional (2D) materials, which can ful�ll the desired properties for the next gen-

eration of electronic devices. Layered Transition Metal diChalcogenides (TMC) are one

class of materials that are promising to take over silicon and to complement graphene

or even to compete with the two. These TMC materials such as MoS2, WS2, and TiS2have been studied for decades in their bulk forms and were mainly used as lubricants

and intercalation materials. However new attention to these materials has been drawn at

lower dimensionality, since Radisavljevic and co-workers3 made a breakthrough in 2011

with the fabrication of a �eld-e�ect transistor (FET) using MoS2 monolayer as a semi-

conducting channel. This FET has shown auspicious characteristics.3 MoS2 monolayer is

a semiconductor with a direct band gap of about 1.9 eV3,4 and with a signi�cant photo-

luminescence5,6 in comparison to bulk MoS2. This features is suitable, for example, for

light-emitting diodes. In addition, this material possesses a valley degeneracy,7–12 which

1

1. Introduction

occurs at nonequivalent high symmetric 𝐾 and 𝐾 ′ points of the Brillouin zone, and a

large spin-orbit splitting due to the lack of inversion symmetry.13 These characteristics

have led to the renaissance of TMC compounds, including those that are isoelectronic to

MoS2, such as MoSe2, WS2, and WSe2, and suggests potential applications in valleytron-

ics, spintronics and optoelectronics.

Like carbon, which has di�erent allotropes, TMC compounds, in addition to multi-

layered systems, also appear to have several polymorphs with di�erent dimensionalities

(from 3D to 0D) and shapes such as nanotubes (NTs), nanoribbons, �akes and inorganic

fullerenes (IF) particles as shown in Fig. 1.1. Due to their low dimensionality, these ma-

terials may exhibit interesting and exceptional physical and chemical properties that arise

from quantum con�nement e�ects and other quantum phenomena. TMC nanotubes are

also of great importance and may o�er new possibilities in nanoelectronics and other ap-

plications in nanotechnology due to their 1D nanostructure. The �rst TMC-NTs, namely

WS2 and MoS2, were synthesized and proposed in the seminal work of Tenne et 𝑎𝑙.14,15 in

1992, simultaneously with the discovery of carbon nanotubes (CNTs).16 These NTs have

shown excellent tribological and mechanical properties. Subsequently, various methods

of synthesis and growth have been developed and many other TMC-NTs have been ob-

tained such as NbS2, ReS2, TiS2, ZrS2,17,18 etc. Nevertheless, these nanotubes are less

explored in comparison to their carbon counterparts, and much work has to be done on

the investigation of their properties from both experimental and theoretical aspects.

In this perspective, the main motivations and objectives of this Ph.D. thesis is to inves-

tigate the physical properties of these inorganic transition metal dichalcogenide materials,

including mono-, few-layer and nanotubular systems. Particular e�ort will be devoted to

understanding how the quantum con�nement e�ects in�uence the properties of these ma-

terials at the nanoscale, how they are coupled to their geometries, and how do they change

Figure 1.1.: Polymorphs of transition metal dichalcogenide compounds.

2

1.1. Layered transition metal dichalcogenides

with the dimensionality, as well as their response to external factors such as strain and

electric �eld. Therefore, we have used in this work, the �rst-principles electronic struc-

ture approach based on density functional theory (DFT). This method is a powerful and

rigorous tool to determine the ground-state and linear response properties of molecules

and solids. DFT has also shown its capability to complement experimental observations

and even predict some other properties of materials with a high level of accuracy, where

experimental investigations are not accessible.


1.1.1. Crystal structure and composition

Layered transition metal dichalcogenides (TMCs) have a general chemical formula MX2,

where M stands for the transition metal atom, such as Mo, W, Nb, Ti, V, Re, Zr, Hf or

Ta and X refers to the chalcogens (S, Se or Te). A hexagonal transition metal layer is

sandwiched between two hexagonal dichalcogenide layers, where the atoms M and X are

covalently bonded in-plane. The adjacent X-M-X sheets are packed one on top of the

other by relatively weak interactions. These interactions are usually referred to as van der

Waals (vdW) type, but they can also be of Coulombic character. The metal atoms have a

six-fold coordination and depending on the number of 𝑑 electrons in the X-M-X trilayer,

they can be arranged in either octahedral (Oℎ) or trigonal prismatic (D3ℎ) manners (see

Fig. 1-a, b), which results in di�erent polytypes, namely trigonal 1T with space group

P-3m1, hexagonal 2H (1H in case of a monolayer) and rhombohedral 3R both with space

group P63/mmc, as shown in Fig. 1.c-e. The digits 1, 2 and 3 refer to the number of

layers in the unit cell. For example, the prototypical TMC material MoS2 can be found

in either 2H or 3R con�gurations, whilst, TiS2 crystallizes in the 1T polytype.19–24 In our

study, the main focus will be on the TMCs of the group 6 with 2H (1H for monolayers)

polytype, unless otherwise stated.

Figure 1.2.: Polytypes of layered TMC materials.

3

1. Introduction

1.1.2. Electronic and optical properties

Layered TMC materials exhibit interesting intrinsic electronic properties due to the

highly anisotropic nature of their structures. The 𝑑 orbitals of the transition metal and

the 𝑝𝑧 orbitals of the chalcogen atom mainly determine the electronic character of the

TMC materials. When the 𝑑 orbitals are partially �lled, TMCs exhibit a metallic behav-

ior, whereas when they are fully occupied, the materials have a semiconducting character.

Consequently, bulk TMC compounds can be insulators such as HfS2, semiconductors such

as MoS2, WS2, TiS2, PtS2, etc. or metals such as VS2, NbS2, ReS2 and TaS2. The

semiconducting TMC materials have an indirect band gap in their bulk structure, which

decreases as the atomic number of the chalcogen element increases (i.e., X = S, Se, Te).

The TMCs of group 6, namely MoX2 and WX2 within the 2H polytype have an indirect

band gap, with the valence band maximum (VBM) situated at the Γ-point and the con-

duction band minimum (CBM) located halfway between the Γ and the K points. When

the bulk is thinned to a monolayer, these materials undergo an indirect-direct interband

transition due the quantum con�nement, where the direct band gap is found at the K-

point of the Brillouin zone. Particularly, bulk MoS2 has an indirect band gap of 1.3 eV,

whereas the MoS2 monolayer has a direct band gap of 1.9 eV.4,5 From photoluminescence

(PL) characterization, it has been found that the magnitude of the peaks is up to 4 orders

larger for MoS2 monolayer than for the bulk.7–10

It has also been shown that the TMC optical properties are dominated by excitonic

transitions instead of band-to-band transitions. In fact, the absorption spectra of MoS2monolayer and bilayer (see Fig. 1.3) show two pronounced peaks known as A and B

excitons, which correspond to the transitions from the two spin-split subbands of the

highest valence band to the lowest conduction band.5,25–27 In these transitions, the exci-

ton binding energy is predicted to be 0.9 eV and 0.4 eV for the monolayer and the bilayer,

Figure 1.3.: (a) Trigonal prismatic structure of monolayer MoS2. (b) Honeycomb latticestructure (b), (c) The lowest-energy conduction bands and the highest-energyvalence bands near the K and K' points of the Brillouin zone. (d) Absorptionspectrum of MoS2 monolayer with two prominent resonances, known as theA and B excitons.26

4


respectively.28–30 The binding energy of the monolayer is high due to the low dielectric

screening in such a system. Moreover, gate-dependent PL measurements on semiconduct-

ing TMC monolayers have indicated that negative trions can be formed by charging an

exciton with an extra electron or hole, having binding energies in the range of 20�40 meV

for di�erent TMC monolayers.5,6,25–27

The symmetry and properties of the semiconducting TMCs provide the control of val-

ley degrees of freedom using circularly polarized light, which makes them suitable for

valleytronic devices. In fact, the hexagonal honeycomb structure of a TMC monolayer

has two distinct valleys K and K'. The degeneracy of these two points is lifted and the

split valleys have charge carriers with opposite spin due to the time reversal symmetry.

The combination of the spin and valley degrees of freedom results in the con�nement

of the charge carriers in a given valley.7,9 The valley population selection by edge elec-

trons excitation using circularly polarized light has been reported for MoS2 and WSe2monolayers.7,8,10–12

Using Raman spectroscopy, it has been shown that TMC monolayers have two most

predominant peaks, A1𝑔 and E12𝑔, corresponding to the out-of-plane and the in-plane

phonon vibrational modes, respectively.31–36 In the �rst one, i.e. A1𝑔, the chalcogen atoms

in the monolayer move perpendicularly to the plane in opposite directions, while the

transition metal M is static. In the second mode, i.e. E12𝑔, all the M and X atoms move in

the in-plane direction, however, the chalcogen atoms X of the two di�erent monoatomic

layers of the sandwich structure move in the opposite direction to the M atoms. As the

number of the layers in the TMC system increases from monolayer to bulk, the out-of-plane

phonon mode becomes sti� and the in-plane bonding is relaxed, which leads the a blue and

red shifts of the A1𝑔 and E12𝑔 modes, respectively.31,37 The increase in the temperature

also results in spectral broadening and a red shift of both A1𝑔 and E12𝑔 modes; this can

be assigned to anharmonic contributions to the interatomic potentials.38–40 It has been

found that the E12𝑔 mode is not a�ected by electron doping, while the A1𝑔 mode, as in the

case of temperature increase, undergoes a red shift and an increase in the peak width.41

The application of a uniaxial strain severely a�ects the E12𝑔 mode, which yields a red shift

and a splitting into two distinct peaks for even small values around 1% of strain.42

1.1.3. Synthesis

Single or multiple layer TMC materials can be obtained by several methods, such as me-

chanical exfoliation, liquid exfoliation and chemical vapor deposition (CVD) techniques.

The weak interactions between layers enable their separation by micromechanical cleav-

age using, for example, simple Scotch tapes.3,43–47 This approach is the most used one to

isolate the layers, since it induces fewer defects and modi�cations in the structure. How-

ever, the low yield and the di�culties with size control limit its usage, particularly for

commercial production. The liquid exfoliation of TMC layers can be achieved by interca-

5

1. Introduction

lation of ions, such as lithium.48–54 The strategy consists of immersing a powder of the

bulk TMC material in an ionic solution and then exfoliating the sheets in water. Elec-

trochemical intercalation of lithium was also used for the synthesis of thin �lms of TMC

nanomaterials.48,49 Other chemical exfoliations in liquid phases, such as organic solvents,

polymers and surfactant solutions, have been demonstrated,55–61 allowing large quantity

production. However, some changes in the electronic structures may occur, which lower

the yield of obtaining single layer nanosheets.

The CVD methods, also called bottom-up methods, involve a direct synthesis of the

TMC �lms from initial solid precursors heated to high temperatures.62–76 They can allow

growth of large-area, uniform and well-controlled atomically thin �lms of layered TMC

compounds. For example, MoS2 �akes have been grown on insulating substrates and

on a CVD-grown template of graphene-covered copper foil.63 However, the usage of the

CVD approaches is still in the early stage for the synthesis and growth of layered TMCs

other than MoS2 and the control of the layer thickness remains one of the challenges that

has to be met in order to produce large-area TMC thin-�lms for the next generation of

optoelectronic and nanophotonic devices.

group V

1.1.4. Applications

Mainly used as bulk materials for decades, layered transition metal dichalcogenides

have shown an early technological interest in various areas such as hydrodesulfurization

and denitrogenation catalysis, photovoltaic cells, photocatalysis, tribology, and lithium

batteries due to their distinctive electronic, optical, and catalytic properties.51,77–94 For

example, TiS2 has been used as an active cathode material in lithium batteries.92 MoS2and WS2 have been used as catalysts for hydrodesulfurization and denitrogenation in

petroleum industry as well as high-temperature solid-state lubricants or as additives in

liquid lubricants.91 TMCs of group 5, such as TaS2, NbSe2, and NbSe2 exhibit high charge

density waves and superconducting properties, which makes them potential candidates as

intercalation compounds.95

The recent growing interest in the layered TMC materials, in particular MoS2 monolayer

and few layers due to their reduced dimensionality and symmetry, has resulted in new

prominent applications at the nanoscale level in optoelectronics, nanophotonic, and �exible

devices, including �eld-e�ect transistors, photovoltaics, photodetectors and sensors.

Field-e�ect devices form one of the most signi�cant areas, where potential applications

of 2D TMC materials have been developed and reported. We recall hereafter some exam-

ples. Radisavljevic and co-workers3 have fabricated the �rst �eld-e�ect transistor based

on MoS2 monolayer using HfO2 as a top-gate dielectric. The FET showed a very high

room-temperature on/o� current ratio of 1 × 108 and mobility of at least 200 cm2 V−1 s−1,

as shown in Fig 1.4. This mobility is found to be comparable to that of graphene nanorib-

6


Figure 1.4.: Top-gated MoS2 monolayer transistor.3

Figure 1.5.: Ambipolar transistor of a MoS2 �ake (left), and change of sheet conductivity𝜎2𝐷 (4-probe), as a function of gate voltage 𝑉𝐺 (right).96

bons but much lower than that of graphene and silicon-based transistors. An ambipolar

double-layer �ake MoS2 transistor was also fabricated,96 having characteristics of a high

on/o� current ratio larger than 102 and carrier mobilities of up to 44 and 86 cm2 V−1 s−1

for electrons and holes, respectively, with an accumulated carrier density of 1 x 1014 cm−2

(see Fig. 1.5). A p-type FET based on WSe2 monolayer as the active channel assembled

together with the chemically doped source and drain terminals shows a high e�ective hole

mobility of about 250 cm2 V−1 s−1, a current on/o� greater than 106 at room tempera-

ture.97

The excellent mechanical properties of TMC materials, especially their malleability,

have made them potential candidates for �exible electronic devices. In fact, �exible FETs

have been developed. Pu et 𝑎𝑙.98 have presented a MoS2 thin-�lm transistor using an ion

gel as a dielectric gate, which operates at low voltage. Its characteristics exhibit a high

on/o� current ratio of 105 and high mobility of 12.5 cm2 V−1 s−1. It has been shown that

such a FET is electrically stable even under mechanical bending with a curvature radius of

0.75 mm. Using conventional solid-state high-𝜅 dielectrics on �exible substrates, such as

Al2O3 and HfO2, Chang et 𝑎𝑙.99 have reported a high performance MoS2 �exible FET with

on/o� current ratio greater than 107, a subthreshold slope of about 82 mV per decade and

a mobility of 30 cm2 V−1 s−1. This MoS2 device can function under mechanical bending

up to a radius of 1 mm.

7

1. Introduction

Figure 1.6.: Integrated circuit based on MoS2 (left), and dependence of the inverter gain(negative value of 𝑑𝑉𝑜𝑢𝑡/𝑑𝑉𝑖𝑛) on the input voltage (right).43

Figure 1.7.: Integrated multistage circuits on MoS2.100

Other important applications in electronic devices based on these 2D TMC materials

have also been proposed, such as in integrated circuits and memory storage. Radisavljevic

et 𝑎𝑙.43 have constructed an integrated circuit composed of two MoS2 monolayer transistors

to perform logic operations (see Fig. 1.6). A fully integrated multistage circuit has

been entirely manufactured for the �rst time on a 2D material, which consists of few

layers of MoS2 for high-performance low-power applications,100 as shown in Fig. 1.7.

Combining multilayer graphene together with a MoS2 monolayer, Bertolazzi et 𝑎𝑙.101 have

demonstrated that these 2D materials can be used as memory devices and information

storage (see Fig. 1.8). Chen et 𝑎𝑙.102 have also fabricated a multibit memory FET device

based on a multilayer MoS2 treated with highly energetic plasmas for memory and data

storage. This memory FET has exhibited a high data retention with binary and multibit

data storage capabilities as well as a fast programming speed (see Fig. 1.9).

8


Figure 1.8.: Memory device based on MoS2.101

Figure 1.9.: Multibit memory device based on MoS2.102

Due to the direct optical band gap observed in monolayered TMC materials, many

applications in optoelectronic devices have also emerged, such in photovoltaic cells, pho-

todetectors, light emitting diodes, etc. The �rst phototransistor has been fabricated using

a mechanically exfoliated MoS2 monolayer. Its photoresponsiveness is found to be much

better than that of the graphene-based devices, in addition to its stable characteristics

such as incident-light control and prompt photoswitching behavior.47 Lee et 𝑎𝑙.103 have

employed di�erent MoS2 layers to construct top-gated nanosheet photodetectors. They

have observed that these photodetectors with monolayer and double-layer depicted in Fig.

1.8 can be used to detect green light, while the three-layer counterpart is suitable for red

light detection.

Heterostructures consisting of graphene and MoS2 layers have also been suggested as

photodetectors. Zhang and co-workers104 have constructed a photodetector based on

graphene�MoS2 bilayer with photoresponsiveness of at least 107 A W−1 and a high pho-

togain greater than 108. Another photodetector device assembled with vertically stacking

graphene�MoS2�graphene structures has been shown to exhibit highly e�cient photocur-

rent generation and photodetection, where the amplitude and the photocurrent polarity

can be modulated by an external electric �eld.105

9

1. Introduction

Sensing devices fall in the applicable �elds, where 2D TMC materials can be relevant

for the implementation and the development of the new generation of highly sensitive

and low-cost FETs-based sensors. Several experimental and theoretical studies have been

reported on the use of TMCs, particularly MoS2, as sensing materials for chemical and bio-

molecules, including H2, O2, H2O, CO, NO, NO2, NH3, etc.45,106–112 In the following, we

recall some of the FETs-based sensors that have recently been proposed, where TMCs have

been used as channels in the transistors. Using mechanical exfoliation method, Dattatray

et 𝑎𝑙.113 have deposited monolayer and multilayers of MoS2 on SiO2/Si substrates and

exposed them to NO2, NH3 and humidity under gate bias and light irradiation. The

fabricated FETs have exhibited remarkable sensing properties, especially in the case of

transistors with a few MoS2 layers. He et 𝑎𝑙.112 have proposed a �exible thin-�lm transistor

(TFT) array based on MoS2, which shows high sensitivity and �exibility as well as an

excellent reproducibility. The �rst MoS2-based FET biosensor was reported by Sarkar et

𝑎𝑙.114 which exhibited ultrasensitive and speci�c detection of biomolecules and excellent

sensitivity for pH sensing (see Fig. 1.10). Other monolayer MoS2 sensors with Schottky

contacts and grown using the CVD technique have shown highly sensitive detection of

NO2 and NH3 at room temperature.115

Figure 1.10.: Photodetector based on MoS2 (right). The schematic band diagrams of ITO(gate)/Al2O3 (dielectric)/single- , double-, triple-layer MoS2 (n-channel) un-der the light (𝐸𝑙𝑖𝑔ℎ𝑡 = ℎ𝜈) illustrate the photoelectric e�ects for band gapmeasurements (left) .103

Figure 1.11.: FET-based Biosensor based on MoS2.114

10

1.2. Transition-metal dichalcogenide nanotubes


Layered transition-metal dichalcogenide compounds can also form nanotubes (NTs),

similar to carbon nanostructures. The TMC-NT can be thought of as a monolayer or

few layers rolled up into cylindrical shape to produce a single-wall (SW) or a multi-

wall (MW) tubular structure, respectively. This can be explained by the fact that the

chalcogen atoms at the layer's edges (or rims) are two-fold bonded to the tetrahedrally

coordinated metal atoms, while in the basal plane, the chalcogen and metal atoms are

three-fold bonded and trigonal prismatic coordinated, respectively.116 Such unsaturated

bonds at the edges render the layers energetically unstable against bending as their number

increases in comparison to the internally bonded atoms in the layers, which consequently

yields to the formation of curved shapes or tubular structures. This, of course, requires

a certain energy so-called strain energy (𝐸𝑆𝑡𝑟𝑎𝑖𝑛) to roll up a planar layer into a tube,

which is de�ned as the di�erence between the total energy of the tube (𝐸𝑇 ) and that of

the respective monolayer (𝐸𝑀𝐿) per atom. In general, the strain energy of nanotubes is

correlated to the tube diameter (𝑑) through the relation 𝐸𝑆𝑡𝑟𝑎𝑖𝑛 ∼ 1/𝑑2. The stability of

a MoS2 SW-NT can be described by folding a rectangular planar stripe (nanoribbon) of

𝑝 atoms with a width 𝑙 and energy 𝐸𝑅 into a tube, where the energies of the ribbon and

the tube are obtained by the relations:

𝐸𝑅 = 𝑝𝑖𝐸𝑀𝐿 + 𝑝𝑒𝐸𝑒 (1.1)

𝐸𝑇 = 𝑝𝐸𝑀𝐿 + 𝑝𝐸𝑆𝑡𝑟𝑎𝑖𝑛, (1.2)

where 𝑝𝑖 and 𝑝𝑒 (𝑝 = 𝑝𝑖 + 𝑝𝑒) are the numbers of the internal and the edge atoms of the

nanoribbon, respectively, and 𝐸𝑒 is the energy per atom of the edge atoms. It is found

that the nanotube is more stable than the corresponding stripe beyond a critical diameter

(𝑑 = 6.2 nm) and the total number of atoms should exceed 223 (see Fig. 1.11).116

Figure 1.12.: Energy as a function of number of atoms of MoS2 NT and stripe.116

11

1. Introduction

1.2.1. Synthesis of TMC nanotubes

There have been signi�cant e�orts on the growth of di�erent TMC-NTs since the �rst

synthesis of WS2 by Tenne et al.14 Generally, TMC-NTs can be obtained by employ-

ing techniques far from equilibrium, such as arc discharge and laser ablation, or by using

chemical reactions routes close to equilibrium conditions. The prototypical WS2 and MoS2nanotubes are produced using gas-solid reaction at high temperatures by the reduction

of their respective oxides WO3 and MoO3 in the presence of a mixture of H2, N2 and

H2S gases.14,15 This method has been modi�ed so that the oxide particles, which have

whiskers or needle-like structures, are used as precursors and thermally treated in a H2S

or H2Se atmosphere.117 MoS2 and WS2 nanotubes have also been prepared by the decom-

position of the respective trisul�de and triselenide or the ammonium chalcometallate at

high temperature under a �ow of H2 gas. Other TMC-NTs have been prepared from their

respective trichalcogenide precursors, such as disul�des of groups 4 and 5. In fact, the

nanotubes of TiS2, ZrS2, and HfS2 can be grown by the hydrogen reduction of TiS3, ZrS3and HfS3, respectively, in an argon atmosphere.17 Similar procedure has also been used

for the dichalcogenide of the group 5. It has been shown that the decomposition of NbS3and TaS3 in a hydrogen atmosphere lead to the formation of NbS2 and TaS2 nanotubes.118

NbSe2 NTs have been prepared by the decomposition of the triselenide under a gas �ow of

argon, though, they can also be obtained using intense electron irradiation. More details

on the growth and synthesis of TMC nanotubes, one can be found in many reports and

reviews.14,15,17,117–125

1.2.2. Properties of TMC nanotubes

TMC nanotubes have interesting physical and chemical properties due to their low di-

mensionality, where quantum e�ects play an important role. The mechanical properties of

TMC-NTs are the most investigated from both experimental and theoretical aspects, with

more emphasis on MoS2 and WS2 NTs. Several measurements by in situ scanning and

transmission electron microscopies (SEM, TEM) have shown that WS2-NTs exhibit ultra-

high strength and elasticity under uniaxial tensile tests. In fact, AFM tips of WS2-NTs

were attached to a cantilever and pushed against a surface of a silicon wafer. The deter-

mined Young's modulus was 171 GPa, which is comparable to that of the bulk material

(150 GPa).126 Furthermore, applied tensile strain measurements on individual WS2-NTs

have evaluated the Young's modulus, strength and elongation to failure to be 152 GPa,

3.7-16.3 GPa, and 5-14%, respectively.127 This was supported by theoretical investiga-

tions on MoS2 SW-NTs, where the nanotubes can be stretched up to 16% with a Young's

modulus of 230 GPa.128 Single-wall MoS2-NT ropes were also investigated and the lowest

Young's modulus value was found to be 120 GPa, which is much smaller than that of bulk

2H-MoS2 (238 GPa).129 An atomic-scale torsional stick-slip behavior was also observed by

twisting single WS2-NTs using external torque.130 It has also been shown that WS2-NTs

12


are good resistants to shock waves and are able to bear a shear stress induced by shock

waves up to 21 GPa.131 Using in situ TEM images, a tensile test of a WS2 MW-NT reveals

that the strain is taken by the outermost layer of the nanotube and the bending sti�ness

obtain by electric-�eld induced resonance measurements was 217 GPa.132 Recently, it has

been found that the thickness of the tubes plays an important role in increasing the re-

sistance to the tensile strain. In fact, it has been shown that MW-NTs bear much higher

strain than the free of defects SW-NTs. There was no degradation observed when the tube

diameter increases from 20 to 60 nm.133

The investigation of the electronic properties of TMC-NTs has shown that these ma-

terials preserve the electronic character of their layered counterparts. Early work, using

density functional tight-binding (DFTB) calculations, has shown that MoS2 NTs exhibit

a semiconducting behaviour.134 Depending on chirality, the band gap is either direct and

similar to that of the monolayered system for zigzag NTs, where the top of the valence

band (VBM) and the bottom of the conduction band (CBM) are located at Γ point, or in-

direct and resembling that of bulk materials for armchair NTs, where the transition occurs

between the VBM at Γ and CBM situated at𝐾 point.134,135 Unlike semiconducting CNTs,

for which the band gap decreases with the tube diameter, the band gap of semiconducting

TMC-NTs increases with the diameter and converges to the monolayer value.134,135 This

was veri�ed by experimental observations using Raman spectroscopy and optical measure-

ments of MoS2 and WS2 fullerene (IF) nanoparticles and nanotubes, where the excitonic

bandgap of these nanoparticles was found to shift to lower energies when their diameter

shrinks.136,137 The electronic properties of TiS2 nanotubes have also been investigated by

means of �rst-principles and are found to be semiconductors irrespective of their chirality

and geometry. However, the band gap vanishes for very small tube diameters.128,138,139

On the other hand, NbS2 and NbSe2 nanotubes exhibit a metallic character independent

of chirality and diameter.140,141

1.2.3. Applications of TMC nanotubes

The intriguing properties of TMC nanotubes render them promising and potential candi-

dates in many areas of nanotechnology. However, to date, only a few e�ective applications

have emerged employing these materials. On the other hand, the advancement and devel-

opment of synthesis and characterization techniques will likely lead to new applications.

Tribology is one of the most prosperous applications �elds of TMC-NTs, particularly MoS2and WS2 NTs, in which, they are used as solid lubricants or as additives to other �uid

lubricants and greases.143,144 The excellent tribological properties of these nanotubes can

be attributed to rolling and sliding friction provided by the cylindrical shapes of these

nanostructures. MoS2 and WS2 NTs have also been suggested as tips for scanning probe

microscopy.145 In fact, the sti�ness, the inertness and the strong absorption of light in the

visible spectrum are the relevant characteristics of these materials for their application

13

1. Introduction

Figure 1.13.: FET based on WS2 nanotube.142

in nanolithography and optical imaging. The excellent mechanical properties TMC-NTs,

that have been reviewed in the previous section, show that these materials are capa-

ble of being used as reinforcement ingredients together with other compounds to from

high-strength nanocomposites.146 TMC-NTs have also been proposed as intercalation and

sorption materials, since the interlayer spacing and the hollow vacuum in the center of

the tubes can serve as hosts for guest molecules and atoms. Hydrogen adsorption and

desorption in TiS2 and MoS2 NTs was demonstrated at room temperature.147,148 It has

been shown that high-purity uncapped TiS2 NTs can e�ciently store up to 2.5 wt% of hy-

drogen at 298 𝐾 under a pressure of 4 MPa.148 The highest gaseous storage capacity and

electrochemical discharge capacity are found to be 1.2 wt% hydrogen and 262 mA h g−1,

respectively, at 298 𝐾 and a for current density of 50 mA g−1 for MoS2 NTs.147 The inser-

tion of organic molecules and the reversible copper intercalation were also demonstrated in

VS2 NTs.149 These characteristics can �nd interesting applications in the use of TMC-NTs

as host materials for electrodes of rechargeable batteries. A nanocomposite material of

MoS2 NTs and Ni nanoparticles was shown to be very e�ective for hydrodesulfurization

of thiophene and its derivatives at low temperatures, thus suggesting this material as a

catalyst for sulfur depolluting in the petroleum production.150 The electronic properties

of these TMC-NTs, though less explored, may also �nd relevant applications in electronic

and optoelectronic devices. Recently, the �rst FET based on individual WS2 NTs has

been reported, exhibiting a �eld-e�ect mobility of 50 cm2 V−1 s−1 and a current density

of 1019 cm−3(See Fig. 1.13).142

1.2.4. Geometry of a nanotube: an example of CNT

A nanotube can be perceived as rolling up a planar sheet of the corresponding material

into a cylindrical or tubular shape. To illustrate the geometry and the symmetry features

of a nanotube, we consider here the simplest case, namely, carbon nanotube (CNT), which

can be obtained by bending a stripe of a graphene sheet into a tube-like form.151–155 In

Fig. 1.14, 𝑋 and 𝑌 are the Cartesian axes. The vectors �� and �� are the primitive vectors

of the unit cell with two atoms at coordinate positions 𝑝1 and 𝑝2:

14


𝑝1 =(𝑎+ ��)

3𝑎𝑛𝑑 𝑝2 =

2(𝑎+ ��)

3. (1.3)

We de�ne the vectors �� and �� in the Cartesian coordinate system as follow

�� = 𝑎

(��1 +

1

2��2

)𝑎𝑛𝑑 �� = 𝑎

(√3

2��1 −

1

2��2

), (1.4)

where ��1 and ��2 are the unit vectors along 𝑋 and 𝑌 , respectively, and 𝑎 = 2.46 Å is the

lattice constant of graphite, which is related to the carbon-carbon bond length 𝑎𝐶−𝐶 by :

𝑎 =√

3𝑎𝐶−𝐶 .

To obtain a nanotube one needs to roll up the graphene stripe following the so-called

chiral vector �� (also called helical vector) de�ned between the origin point 𝑂 and the

equivalent point 𝐴. The angle 𝜃 = 𝐴𝑂𝐾, called the chiral angle, de�nes the direction of

the chiral vector �� (�� = 𝑂𝐾) and has a maximum value of 𝜋6 (i.e. 0 ≤ 𝜃 ≤ 𝜋

6 ). The chiral

vector can be written in the basis vector set as :

�� = 𝑛��+𝑚�� (1.5)

or in the Cartesian coordinates:

�� =

√3

2𝑎(𝑛+𝑚)��1 +

1

2𝑎(𝑛−𝑚)��2, (1.6)

Figure 1.14.: Graphene sheet, ��, �� are lattice vectors of the graphene unit cell (uc); ��, ��are the Cartesian coordinates; 𝑂𝐴 is the helical vector, 𝐴𝑂𝐾 is the helicalangle; 𝑂𝐿 and 𝑂𝐾 are the armchair and zigzag directions, respectively.

15

1. Introduction

where (n, m) are a pair of integers that characterize the chiral vector, and are referred to

as the chiral indices, which de�ne the nanotube. The length 𝐻 of the chiral vector and

the chiral angle 𝜃 are determined as :

𝐻 = |��| = 𝑎(𝑛2 +𝑚2 + 𝑛𝑚)1/2 (1.7)

𝑐𝑜𝑠𝜃 =𝑎1.��

|𝑎1|.|��|(1.8)

or

𝑐𝑜𝑠𝜃 =2𝑛+𝑚

2(𝑛2 +𝑚2 + 𝑛𝑚)1/2(1.9)

The length 𝐿 of the chiral vector �� also constitutes the circumference of a (n,m) nan-

otube; thus, the tube diameter 𝑑 can be determined by :

𝑑 =𝐻

𝜋=𝑎(𝑛2 +𝑚2 + 𝑛𝑚)1/2

𝜋(1.10)

In Fig. 1.14, the zigzag axis of the graphene sheet corresponds to 𝜃 = 0, which means

𝑚 = 0, then if the rolling chiral vector is along this axis (i.e. 𝑂𝐾), a zigzag nanotube is

generated, and hence a zigzag NT is an (n,0) nanotube. On the other hand, if the direction

of the rolling chiral vector is along 𝑂𝐿, i.e. the armchair direction, the nanotube is called

armchair NT, which corresponds to 𝜃 = 𝜋6 or to 𝑛 = 𝑚. Consequently, the armchair NT

is an (n,n) nanotube. A nanotube generated for any other direction of the helical vector

between 𝜃 = 0 and 𝜃 = 𝜋6 is referred to as a general chiral (n,m) nanotube. By de�nition,

the zigzag and armchair directions are called achiral directions.

The point-group symmetry of the hexagonal lattice makes many of these nanotubes

equivalent. So all the possible individual NTs are generated by using only a 112 irreducible

wedge of the Bravais lattice, i.e. the wedge that is contained in the interval 𝜃 ∈ [0, 𝜋6 ].156,157

1.2.5. Geometry of a TMC nanotube

In analogy to CNTs, a TMC single-wall nanotube is also constructed by folding a TMC

monolayer into a tube form. The TMC monolayer consists of three monoatomic planes

X-M-X. The pair of the parallel sulfur planes at the distance of 𝛿 ≈ 3.1 Å are separated

by the transition metal plane. All three planes belong to the same trigonal lattice, with

the basis vectors �� and �� of equal length 𝑎 = 3.16 Å. The unit cell contains two sulfur

and one metal atoms. The sulfur atoms in the di�erent planes are exactly one on top of

the other and separated by 𝛿, and the metal atoms are between the centers of the sulfur

triangles at a height 𝛿2 . This TMC monolayer has a trigonal symmetry of the space group

P6m2. Similar to CNT, the TMC-SW nanotube (n,m) is obtained by rolling up the helical

vector in a given direction, where the chiral angle is also in the range 𝜃 ∈ [0, 𝜋6 ], as shown

16


in Fig. 1.15.158

The TMC tube wall, which consists of three coaxial cylinders (X-M-X) of the thickness

𝛿, endures di�erent distortions when the monolayer is folded into a tube. Considering

that the tube's radius corresponds to the distance between the axis of the tube and the

metal cylinder, the interior and exterior sulfurs are additionally shrunken and stretched,

respectively.158

Figure 1.15.: TMC nanotubes; ��, �� are lattice vectors, �� is the rolling direction; �� is theunit cell vector and �� is the helical vector.

17

2. Methods

2.1. Density functional theory

The basic ideas of Density Functional Theory (DFT) are contained in the two original

papers of Hohenberg-Kohn and Kohn-Sham.159,160 This theory has had a tremendous im-

pact on realistic calculations of the properties of molecules and solids, and its applications

to di�erent problems continue to expand. The fundamental concept is that instead of

dealing with the many-body Schrödinger equation

𝐻Ψ = 𝐸Ψ, (2.1)

which involves the many-body wavefunction Ψ, one deals with the formulation of the

problem that involves the total density of electrons, where 𝐻 is the Hamiltonian of the

system expressed by

𝐻 = 𝑇𝑒 + 𝑇𝑁 + 𝑉𝑒𝑁 + 𝑉𝑁𝑁 + 𝑉𝑒𝑒 (2.2)

The terms 𝑇𝑒 + 𝑇𝑁 are the kinetic energies of electrons and nuclei, respectively. 𝑉𝑒𝑁

corresponds to the attractive electrostatic interaction between electrons and nuclei. 𝑉𝑁𝑁 +

𝑉𝑒𝑒 are repulsive potentials due to the nuclei-nuclei and electron-electron interactions,

respectively.

The expression of the many-body wavefunction in terms of the electronic density is

a huge simpli�cation, since there is no need to explicitly specify Ψ, as it is the case in

the Hartree-Fock approximation. Thus, instead of starting with a drastic approximation

for the behavior of the system, which is what the Hartree-Fock wavefunctions represent,

one can develop the appropriate single-particle equations in an exact manner, and then

introduce approximations as needed. This was �rst simpli�ed by the so-called 𝐵𝑜𝑟𝑛 −𝑂𝑝𝑝𝑒𝑛ℎ𝑒𝑖𝑚𝑒𝑟 approximation, which considers that the nuclei are much heavier than the

electrons. Therefore, the nuclei move much slower that the electrons. In this case, the

electrons are treated as they are moving in a �eld of �xed nuclei. Consequently, the term

𝑇𝑁 is neglected and the repulsive potential between nuclei 𝑉𝑁𝑁 is considered as a constant

in Eq. 2.2. Hence, the Hamiltonian becomes

𝐻 = 𝑇𝑒 + 𝑉𝑒𝑁 + 𝑉𝑒𝑒 (2.3)

and the solution to the Eq. 2.1 is the electronic wavefunction

19

2. Methods

Ψ =𝑁∑𝑖=1

𝜓𝑖, (2.4)

where 𝜓𝑖 is the one-electron wavefunction.

The Hohenberg-Kohn theorem159 states that the groundstate density 𝜌(��) of a system

of interacting electrons in an external potential 𝑉𝑒𝑥𝑡(��) determines the total energy of the

system uniquely. In others words, if 𝑁 interacting electrons move in an external potential

𝑉𝑒𝑥𝑡(��), the ground-state electron density 𝜌0(��) minimizes the total energy

𝐸[𝜌(��)] = 𝐹 [𝜌(��)] +

∫𝜌(��)𝑉𝑒𝑥𝑡(��) 𝑑�� (2.5)

where 𝐹 is a universal functional of 𝜌(��) and the minimum value of the energy functional 𝐸

is 𝐸0, the exact ground-state electronic energy, which can be obtained using the variational

principle

𝐸(Ψ) =⟨Ψ|𝐻|Ψ⟩⟨Ψ|Ψ⟩

(2.6)

Assuming a non-interacting system, Kohn and Sham160 have derived a set of di�erential

equations enabling the ground state density 𝜌(��) to be found. They have separated 𝐹 [𝜌(��)]

into three distinct parts, so that the total energy 𝐸 becomes

𝐸[𝜌(��)] = 𝑇 [𝜌(��)] +1

2

∫∫𝜌(��) 𝜌(𝑟′)

|�� − 𝑟′|𝑑�� 𝑑𝑟′ + 𝐸𝑥𝑐[𝜌(��)] +

∫𝜌(��)𝑉𝑒𝑥𝑡(��) 𝑑�� (2.7)

where 𝑇 [𝜌(��)] de�nes the kinetic energy of a non-interacting electron gas with density 𝜌(��)

𝑇 [𝜌(��)] = −1

2

𝑁∑𝑖=1

∫𝜓*𝑖 (��)∇2 𝜓𝑖(��) 𝑑��. (2.8)

𝐸𝑥𝑐[𝜌(��)] represents the exchange-correlation energy functional in the Eq. 2.7. Introducing

a normalization constraint by a Lagrange multiplier on the electron density,∫𝜌(��)𝑉 (��)𝑑�� =

𝑁 , one obtains

𝜕

𝜕𝜌(��)[𝐸[𝜌(��)] − 𝜇

∫𝜌(��)𝑑��] = 0 ⇒ 𝜕𝐸[𝜌(��)]

𝜕𝜌(��)= 𝜇 (2.9)

Eq. 2.17 may now be written in terms of an e�ective potential, 𝑉𝑒𝑓𝑓 (��), as follows:

𝜕 𝑇 [𝜌(��)]

𝜕𝜌(��)+ 𝑉𝑒𝑓𝑓 (��) = 𝜇 (2.10)

20

2.1. Density functional theory

where

𝑉𝑒𝑓𝑓 (��) = 𝑉𝑒𝑥𝑡(��) +

∫𝜌(𝑟′)

|�� − 𝑟′|𝑑𝑟′ + 𝑉𝑥𝑐(��) (2.11)

and

𝑉𝑥𝑐(��) =𝜕𝐸𝑥𝑐[𝜌(��)]

𝜕𝜌(��), (2.12)

𝑉𝑥𝑐 is the exchange-correlation potential. In order to �nd the ground-state energy 𝐸0 and

the ground-state density 𝜌0, the one-electron Schrödinger equations

[−1

2∇2

𝑖 + 𝑉𝑒𝑓𝑓 (��) − 𝜀𝑖]𝜓𝑖(��) = 0 (2.13)

should be solved self-consistently along with Eq. 2.9 and Eq. 2.10, where

𝜌(��) =𝑁∑𝑖=1

|𝜓𝑖(��)|2. (2.14)

A self-consistent solution is required due to the dependence of 𝑉𝑒𝑓𝑓 (��) on 𝜌(��). The above

equations provide a theoretically exact method to �nd the ground-state energy of an inter-

acting system, by giving the form of 𝐸𝑥𝑐. Unfortunately, the form of 𝐸𝑥𝑐 is unknown and

its exact value has been calculated for only a few very simple systems, such as H and He.

In electronic structure calculations, 𝐸𝑥𝑐 is most commonly approximated within the local

density approximation (LDA)161–165 or generalized-gradient approximation (GGA)166,167

with the exact exchange part or by hybrid functionals such as B3LYP,168 that we brie�y

review in the following paragraphs.

The local density approximation (LDA, or LSDA counting the spin) is the simplest

approximation to the true Kohn-Sham functional, where the local exchange-correlation

potential in the Kohn-Sham equations is de�ned as the exchange potential for the spatially

uniform electron gas with the same density as the local electron density

𝑉 𝐿𝐷𝐴𝑥𝑐 (��) = 𝑉 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛−𝑔𝑎𝑠

𝑥𝑐 [𝜌(��)]. (2.15)

The exchange-correlation functional for the uniform electron gas is known to high preci-

sion for all values of the electron density 𝜌(��). Albeit its simplicity, LDA has proven to be

remarkably successful to describe many properties, such as structure (lattice constants),

vibrational frequencies, elastic moduli and phase stability (of similar structures) for many

homogeneous systems. However, in computing energy di�erences between rather di�erent

structures, the LDA can have signi�cant errors. For instance, binding energies of many

systems are overbinding and activation energies in di�usion or chemical reactions may be

too small or absent.

Another approximation to the Kohn-Sham functional is the generalized gradient approx-

imation (GGA),166,167 which is considered as an improvement of LDA. The physical idea

behind the GGA is that real electron densities are not uniform for most of the systems, so

21

2. Methods

by including information on the spatial variation in the electron density can allow better

�exibility in describing real materials. This is valid for slowly varying densities, where

the exchange-correlation functional is expressed using both the local electron density and

gradient of the electron density

𝑉 𝐺𝐺𝐴𝑥𝑐 (��) = 𝑉𝑥𝑐[𝜌(��),∇𝜌(��)]. (2.16)

The enhancement of LDA by GGA can be seen in the most cases mentioned above,

where LDA fails, such as the description of the binding energies of molecules and solids,

energy barriers in di�usion or chemical reactions, the relative stability of bulk phases. It

is also more realistic for magnetic solids and useful for electrostatic hydrogen bonds.

Currently, there are two main GGA functionals well established and widely used, which

are LYP169 (Lee�Yang�Parr) and PBE167 (the Perdew�Burke�Ernzerhof). Many other

GGA functionals have been developed and described in the literature and can be classi�ed

into two types; functionals that contain empirical parameters whose values have been �tted

to reproduce experiments or more accurate calculations, such as B88,170 LYP,169 etc. The

second type corresponds to functionals with no empirically determined parameters, such as

PBE,167 B86,171 PW91,172 etc. In our calculations, we have mainly used PBE functional,

the exchange-correlation energy is given by the formula

𝐸𝑃𝐵𝐸𝑋𝐶 [𝜌(��)] =

∫𝑑3𝑟𝜌(��)𝜖𝑢𝑛𝑖𝑓𝑋 (𝜌(��))𝐹𝑋𝐶(𝑟𝑠, 𝜉, 𝑠), (2.17)

where 𝜖𝑢𝑛𝑖𝑓𝑋 is the exact exchange energy per electron of a uniform electron gas, 𝐹𝑋𝐶 is

the enhancement factor over the local exchange, 𝑟𝑠 is the Seitz radius, 𝜉 is the relative

spin polarization and 𝑠 is the scaled density gradient variable. The philosophy behind

the construction of PBE is to avoid empirical parameterizations as much as possible in

the correction and enhancement of the LDA and PW91 densities, which obey certain

fundamental physical constraints such as obeying the limit of uniform electron gas density,

recovering the proper linear response and satisfying the Lieb�Oxford bound.173

The progress from LDA to GGA was a signi�cant accomplishment in 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 methods,

resulting in the success of DFT among the chemistry community, nevertheless the expected

level of accuracy could not be reached yet as well as the description of some properties

was not successful; for example the bonds are softened which overestimates the lattice

constants and decreases the bulk moduli, the van der Waals interactions are omitted and

some GGA workfunctions turn out to be smaller than those of LDA for several metals.

The band gaps are also underestimated by both LDA and GGA.

More recently, other functionals have been developed to reach higher accuracy and better

describe properties, where LDA and GGA fail. Semi-local information was included by

adding the laplacian of the orbital spin density ∇2𝜌𝑖𝜎(��) or by the orbital kinetic energy

density to the electronic density 𝜏𝜎(��) and the gradient of the density (𝜎 =↑, ↓ ; 𝜌𝑖𝜎(��) =

|𝜓𝑖𝜎(��)|2). These functionals are called meta-GGAs174 and have the form

22

2.2. Basis sets

𝑉 𝑚𝑒𝑡−𝐺𝐺𝐴𝑥𝑐 (��) = 𝑉𝑥𝑐[𝜌(��),∇𝜌(��),∇2𝜌𝑖𝜎(��), 𝜏𝜎(��)], (2.18)

where 𝜏𝜎(��), given by the relation

𝜏𝜎(��) =1

2

𝑜𝑐𝑐∑𝑖

∇𝜓𝑖𝜎(��)

𝑚𝑒

(2.19)

can be used to determine whether or not the localized model of the exchange-correlation

hole is a good approximation to the true exchange-correlation energy functional of the

considered electronic system. These functionals are semi-local in a sense they only de-

pend on the density and the Kohn-Sham orbitals at a given point (��) and an in�nitesimal

interval around this point. Examples of meta-GGA functionals are TPSS,175 M06L,176

revTPSS,177 MBJ178 TB-mBJ.179 The latter one (TB-mBJ) was developed to correct the

di�erence between the averaged exchange potential and the exact exchange potential, ob-

tained by applying the optimized e�ective potential method to the Hartree-Fock method.

The TB-mBJ potential is designed to reproduce and predict band gaps of semiconduc-

tors and solids with high level of accuracy and computational costs comparable to regular

GGA.179,180

The non-locality character can only be achieved by considering hybrid functionals.168

The term hybrid refers to the combination of the exact exchange energy of the Hartree-

Fock model and the exchange-correlation energy obtained from DFT. The exact exchange

energy can be derived from the exchange electron density, which can be written in terms

of the Kohn-Sham orbitals as

𝐸𝑋(��) =1

2𝜌(��)

∫ |∑𝑜𝑐𝑐

𝑖 𝜓*𝑖 (𝑟′)𝜓𝑖(��)|2

|�� − 𝑟′|𝑑3��. (2.20)

This means that a functional, based on a non-local quantity, cannot be evaluated at one

particular spatial location unless the electronic density is known for all spatial locations.

These functionals are known as hybrid-GGAs and the most widely used ones are the

B3LYP181,182 PBE0183,184 and HSE185,186 functionals.

2.2. Basis sets

In order to solve the one electron Kohn-Sham equations, the wavefunctions should be

expanded in terms of atomic basis functions. A basis set is a mathematical representation

of wavefunctions (or molecular orbitals) of any multi-electron system. The molecular

orbitals can be described as a linear combination of basis functions as follow

23

2. Methods

Ψ𝑗(��) =𝑀∑𝑖=1

𝑎𝑗𝑖𝜑𝑖(��), (2.21)

where 𝑗 labels the orbitals and 𝑖 the basis functions. The sum runs over all the basis

functions up to the size of the basis set 𝑀 , 𝑎𝑗𝑖 are the expansion coe�cients and 𝜑𝑖(��)

the basis functions. There are several types of basis sets that are used in computational

chemistry, materials science and solid state physics, such as plane waves, augmented waves,

hydrogen-like atomic orbitals, numerical basis functions, Slater�type orbitals (STO) and

Gaussian type orbitals (GTO). The two latter types of basis sets are used in all our studies

and will be discussed in the following paragraphs.

The choice of localized basis set is of a crucial importance in the linear combination of

atomic orbitals (LCAO) approach or 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 calculations in general. Unlike plane wave

methods, the orbitals must be chosen for a given system to be accurate and e�cient. There

is also problem of overcompleteness if one attempts to go to convergence. Nevertheless,

there is a great experience in constructing appropriate localized orbitals, which provides

crucial understanding and calculation procedures that can be both fast and accurate with

a careful choice of orbitals.

Starting from the hydrogen-like atomic orbitals, which are exact solutions for only mono-

electronic systems, Slater had the idea of splitting the basis functions into two analytical

forms; a radial part which depends only on the principal quantum number 𝑛 and hence

independent of the angular quantum number 𝑙; and an angular part described by spherical

harmonics. The radial part can be written as:

𝑅𝑛(��) = 𝑁𝑟𝑛−1𝑒−𝜁 ��. (2.22)

The parameter 𝜁 corresponds to 𝑍*

𝑛 , where 𝑍* represents the e�ective nuclear charge. The

energy level associated to this orbital is

𝐸𝑛 = −1

2

(𝑍*

𝑛

)2

. (2.23)

The general form of Slater type orbitals (STO)187 is given by the following relation

Ψ𝑛 𝑙𝑚 𝜁(��) = 𝑁 𝑌𝑙 𝑚(𝜃, 𝜙) 𝑟𝑛−1 𝑒−𝜁 ��. (2.24)

These basis functions have the advantage of describing the eigenfunctions of systems

that are physically related to the one of interest. However, to generate STOs, one should

choose a sequence of basis functions with variable exponents for each atomic orbital, the

reason is that STOs are nodeless, so that, linear combination of several STOs are needed

to reproduce the real nodal atomic orbitals in order to reach the required �exibility and

the level of accuracy. Moreover, three and four center two-electron integrals cannot be

done analytically. Also, the speci�c kind of an exponential function involved makes STOs

24

2.3. Relativistic effects

slow in computing since integrals over STOs need considerable computational e�ort in the

case of electron-repulsion integrals.

Gaussian-type orbitals (GTO)188,189 are one of the widely used basis sets, especially in

the chemists community. They are generally expressed as:

Ψ𝑛 𝑙𝑚 𝜁(��) = 𝑁 𝑌𝑙 𝑚(𝜃, 𝜙) 𝑟2𝑛−2−𝑙 𝑒−𝜁 ��2 . (2.25)

These functions depend on ��2 and they are numerically easy to handle because the product

of two Gaussians is also a Gaussian, which is not the case for Slater-type basis functions,

and the calculation of the electronic integrals can simply be obtained with ��2. However,

the dependence on ��2 does not adequately describe the electronic behavior in the region

close to or very far from the nuclei; this is because GTOs have a zero slope when ��2 → 0

and decrease faster comparing to Slater functions, which have a �nite slope. Accordingly,

Slater basis functions describe well the atomic orbitals, particularly hydrogenoide orbitals.

Nevertheless, there exists several possibilities to represent the atomic electron density using

GTOs such as :

1. by considering minimal basis functions representation; thus, for example, the 1𝑠 type

orbital of hydrogen can be reproduced by at least one Gaussian function. In practice the

minimal number basis functions that are required is three.

2. by doubling or tripling the number of initial functions to represent the atomic orbitals,

with di�erent values of 𝜁 in the augmentation. This is de�ned as double-zeta or triple-zeta.

3. by a linear combination of initial Gaussian functions; they are de�ned as primitive

and contracted Gaussian basis functions.

Some of the most commonly used basis functions are for example STO-3G, used to

reproduce one STO and split-valence basis sets suh as 3-21G, 4-31G and 6-31G.

Two other types of functions are also added to better describe the atomic, molecular or

charged systems. They are called polarization and di�usive functions. The former type

(polarization functions) are added to reproduce a modi�ed form of the electron density

close to the nuclei; they are represented by a star sign (*), such as 6-31G*. The later type

(di�usive functions) are added to describe the modi�ed electron density at distances far

from the nuclei, particularly charged atoms; they are represented by a plus sign (+), for

example 6-31+G. The above mentioned GTO are called Pople basis functions.

2.3. Relativistic effects

Relativistic e�ects including spin-orbit coupling, mass-velocity and Darwin corrections

play a signi�cant role in determining the electronic structure and properties of materials

in general, as the electrons velocity approaches the velocity of light. Spin-orbit coupling

(SOC), which is the interaction of the electron's spin and its orbital motion, has a strong

e�ect, especially in semiconducting materials. This interaction causes a shift in degenerate

25

2. Methods

atomic energy levels of the electrons in the band structures and thus induces a bands

splitting. These e�ects can be described by the Dirac equation, which is written as

𝐻𝑠𝑜 = − ℎ

4𝑚20𝑐

2𝜎 · 𝑝×∇𝑉𝑐, (2.26)

where ℎ is Planck's constant, 𝑚0 is the mass of a free electron, 𝑐 is the velocity of light,

𝑝 is the momentum operator, 𝑉𝑐 is the Coulomb potential of the atomic core, and 𝜎 =

(𝜎1, 𝜎2, 𝜎3) is the vector of Pauli-spin matrices.

Adding this term to the Eq. 1, this becomes

𝐻𝜓𝑖 = 𝐸𝜓𝑖 +

(− ℎ

4𝑚20𝑐

2𝜎 · 𝑝×∇𝑉𝑐

)𝜓𝑖, (2.27)

where

𝐻 = 𝐻0 +𝐻𝑠𝑜 (2.28)

and 𝐻0 represents the Hamiltonian of a non-relativistic system. For light elements in the

periodic table up to the fourth row, these e�ects can be considered as perturbations to be

added in the Hamiltonian of a given system. However, when going down in the periodic

table, the atoms become heavier and thus the relativistic e�ects also become stronger and

apparent, which can eventually dominate the physical or chemical interactions.

2.4. Computational details

As this thesis is dealing with TMC materials of mainly one- and two-dimensional peri-

odicity, i.e. nanotubes, monolayers and bilayers, one needs to use appropriate codes that

explicitly represent these periodic boundary conditions. The convenient choice is the use

of localized basis sets-based codes, as they properly describe the wavefunctions along the

required periodicity in comparison to, for example, plane waves codes, where the use of

supercell approaches are necessary and unavoidable in order to represent such systems.

Therefore, in our study, we have used two di�erent DFT codes based on localized basis sets,

namely, Gaussian type and Slater type orbitals, which are implemented in Crystal09190

and ADF-BAND,191 respectively. The reason to use two di�erent codes is that, from one

hand, Crystal09 provides the use of the helical periodic boundaries features to perform

calculations on nanotubes and other one-dimensional systems, which drastically reduces

the computational costs. On the other hand, in ADF-BAND, the treatment of relativistic

e�ects is included, which are expressed by the Zero Order Regular Approximation (ZORA

and SO).192–194 Exhaustive computational details can be found in each method section of

every speci�c research work that is included as appendices in the last part of the thesis.

26

3. Results and discussion


The design of materials, particularly semiconductors, at the nanometer size is of great

interest, due to quantum phenomena that play major roles in 1D and 2D materials, while

they are not present in their bulk counterparts. Quantum con�nement is a very important

one and it can be described, in a simple way, as when electrons are spatially con�ned in

a given dimension. This causes an increase in the minimum energy of the electrons in

the con�ned direction. A direct consequence of the quantum con�nement can be seen

in the densities of states (DOS) of the con�ned systems when scaling down from three

to zero dimension (see Fig. 3.1), which a�ects the electronic behaviour of the systems.

The optical band gap in a semiconductor, which is experimentally measurable, increases

as the system size decreases. In fact, decreasing the size of the semiconducting system,

the lowest energy level in the conduction band region goes upward and the highest en-

ergy level in the valence band region shifts downward. This was established in layered

semiconducting TMC materials. Bulk MoS2 is an indirect gap semiconductor. When this

Figure 3.1.: Densities of states of 3D, 2D, 1D and 0D systems; The upper and middleparts describe TMC polymorphs and their corresponding quantum systems,respectively.

27


material is thinned to a monolayer (ML), a transition from indirect to direct band gap

(∆) is observed. This has been demonstrated and reported by several experimental works

using photoluminescence measurements.3–5,7–10

Analogously, we have investigated the dependence of electronic structure on the thick-

ness of several TMC materials, including MoS2, WS2, NbS2 and ReS2. The study consisted

of examing bulk, few layers and monolayers of these systems. Fig. 3.2. shows the band

structures (BS) of MoS2 materials, given here as an example. Bulk MoS2 exhibits an

indirect ∆ of 1.2 eV, which correspends to the transition from the valence band maximum

(VBM) situated at Γ point to the conduction band minimum (CBM) located halfway be-

tween Γ and 𝐾 points. Decreasing the number of layers, the energy gap between VBM and

CBM becomes larger and reaches the maximum value of 1.9 eV at the ML limit, and the

material becomes a direct semiconductor, where both band extrema are located at the 𝐾

point. Note that the aforementioned band gap values are obtained using PBE functional.

It is well known that PBE generally underestimates the band gap. Our �ndings are in a

good agreement with the experimental ones, where the di�erence is maximum 0.1 eV for

ML.3,5

-4

-2

0

2

E-E

F

/ e

V

MoS2 bulk MoS

2 8-layer

-4

-2

0

2

MoS2 6-layer

Γ M K Γ-4

-2

0

2

MoS2 quadrilayer

Γ M K Γ

MoS2 bilayer

Γ M K Γ-4

-2

0

2

MoS2 monolayer

= 1.2 eV∆

∆= 1.9 eV

Figure 3.2.: BS of bulk MoS2, its monolayer, as well as, polylayers calculated at thePBE level. The horizontal dashed lines indicate the Fermi level. The arrowsindicate the fundamental band gap (direct or indirect) for a given system.The top of valence band (blue) and bottom of conduction band (green) arehighlighted.

28


12345678

N

1

1.5

2

2.5

3

3.5

∆ /

eV

PBE0 indirectPBE indirectPBE0 direct (k=K)

PBE direct (k=K)

MoS2

12345678

N

1

1.5

2

2.5

3

3.5

WS2

Figure 3.3.: Calculated direct and indirect band gap values of MoS2 and WS2 N-layerslabs. The horizontal solid lines indicate the band gaps of bulk structures.

Similar observations also hold for WS2. When WS2 is thinned to the ML, it undergoes

an indirect to a direct band gap transition, with values of 1.3 and 2.1 eV for both bulk

and ML, respectively. The dependence of band gap size on the material thickness is shown

in Fig. 3.3. However, in contrast to MoS2 and WS2, the electronic properties of metallic

NbS2 and ReS2 materials remain independent and insensitive to the change of the system

thickness.

Furthermore, we have carried out the calculations using another level of theory, namely

the hybrid PBE0, for comparison. The obtained results show that PBE0 overestimates

the experimental band gaps by about 1 eV. Hence, PBE performs better than PBE0 in

describing the electronic structure, particularly the band gap energies, of TMC materi-

als. Accordingly, this shows that it is possible to tune the electronic properties of TMC

materials, particularly the band gap o�sets, so-called band gap engineering, which makes

them suitable for applications in electronics and optoelectronics. (For more details, see

Appendix 1 for the full article).4

Another consequence of the quantum con�nement and scaling down the bulk TMC

materials to 2D, in particular to the monolayered systems, is the gigantic e�ect of the spin-

orbit coupling (SOC). This is not present, for example, in the bulk and bilayer materials

due the spatial inversion symmetry of the crystal lattice and the time reversal symmetry.

These two symmetry operations convert the wave vector �� into −𝑘 and result in a spin

degeneracy of the bands, which means that the dispersion relation implies: 𝐸↑(��) = 𝐸↓(��).

However, the inversion symmetry is suppressed in the monolayered systems, and the spin-

orbit interactions lift the band degeneracy, resulting in a considerable spin-orbit (SO)

splitting of the bands in the valence band region. We have explored, using �rst-principles

calculations, the e�ect of spin-orbit interactions in the TMC-MLs by including relativistic

e�ects corrections. We have examined the modi�cation in the electronic structure of

the MLs when including the SOC together with an applied tensile strain. We have also

discussed the possibility of inducing spin-splitting in TMC-BLs due to Rashba e�ect by

the formation of heterostructures.

29


Fig. 3.4. shows the calculated BS of WX2 systems using non-relativistic (NR), scalar

relativistic (SR) and SR+SOC treatment. The BS show that the materials are direct band

gap semiconductors as it has been established before, where CBM and VBM are located

at 𝐾 point. The ∆ values are, however, larger in the case of SR than those of SR+SOC

by 60-280 meV, for all ML systems. In general, ∆ decreases as the nuclear charge of the

chalcogen atoms becomes larger and increases when going from MoX2 to WX2, except for

WTe2 that has larger ∆ than MoTe2 by about 100 meV.

The SO split-o� bands (∆SO) obtained at the VBM are very signi�cant and become

larger for heavier atoms in the MLs, ranging from 147 meV to 480 meV, as shown

in the Tab. 1. These ∆SO values are in a good agreement with previously reported

data.28,29,195,196

-2

0

2

4

EF

NR SR

-2

0

2

4

WS

2

SR+SO

-2

0

2

4

E-E

F / e

V

-2

0

2

4

WS

e2

Γ K M-2

-1

0

1

2

Γ K M Γ K M-2

-1

0

1

2

WT

e2

∆ = 1.98 eV ∆ = 1.94 eV

∆ = 1.74 eV

∆ = 1.80 eV ∆ = 1.71 eV

∆ = 1.43 eV

∆ = 1.30 eV ∆ = 1.14 eV

∆ = 0.86 eV

∆SO

= 395 meV

∆SO

= 428 meV

∆SO

= 480 meV

Figure 3.4.: Band structures of WX2 monolayers. Fundamental band gaps (∆) and spin-orbit splittings (∆SO) are given. NR - non relativistic, SR - scalar relativistic,SR+SO - scalar relativistic with spin-orbit interactions.

Table 3.1.: Energy band gaps (∆), spin-orbit splitting (∆SO) of VB band and the e�ectivecarrier masses at the 𝐾 point (for both spin polarizations) of TMC monolay-ers. The SO splitting are given from TB-mBJ potential and PBE functionalcalculations. Note: data from the SR+SO simulations.

SystemTB-mBJ PBE

∆ ∆𝑉 𝐵𝑀SO ∆𝐶𝐵𝑀

SO m*𝑒/m0 m*

ℎ/m0 ∆𝑉 𝐵𝑀SO ∆𝐶𝐵𝑀

SO(eV) (meV) (meV) ↓ ↑ ↓ ↑ (meV) (meV)

MoS2 1.62 147 26 0.449 0.520 -0.624 -0.537 147 12MoSe2 1.40 176 34 0.557 0.657 -0.730 -0.616 180 29MoTe2 0.97 190 46 0.541 0.655 -0.773 -0.618 209 46WS2 1.74 395 17 0.276 0.380 -0.491 -0.351 419 10WSe2 1.43 428 3 0.439 0.308 -0.540 -0.369 449 24WTe2 0.86 480 4 0.398 0.246 -0.526 -0.300 476 26

30


The SO splittings in the CBM are found to be much weaker than those in the VBM.

The calculated e�ective charge carriers masses are much smaller for WX2 MLs than that

of MoX2 ones (see Tab. 3.1.).

The e�ect of strain on the electronic structure, particularly on the band gap, of TMC-

MLs has been reported in the literature.42,197–206 It has been shown that ∆ decreases with

the tensile strain and the materials undergo a semiconductor-metal transition at about

10 % of elongation for MoS2. However the e�ect of strain on the SOC has not been yet

addressed. Therefore, we have investigated how the evolution of the SOC is coupled to

tensile stain and how the SO splitting is a�ected. As shown in Fig. 3.5., in contrast to

the band gap, ∆SO increases with tensile strain; the VBM and CBM, respectively, shift

upward and downward and form a Dirac-like point at a critical elongation.

As mentioned above, the SOC e�ect is hampered in the TMC-BL materials and no spin

splitting is observed due the to the inversion and time reversal symmetries. However, by

forming BL heterostructures, SOC can be restored, inducing SO splittings of values as

large as that of the heavier MLs that form the hetero-bilayers, as depicted in Fig. 3.6.

(For more details, see Appendix 2 for the full article).207

The electronic properties of 2D TMC materials can also be optimized and combined

to ful�ll speci�c functionalities and applications by tuning their electronic structures in

a reversible way. This can be achieved by introducing intrinsic modi�cations, such as

doping and defects or by employing external factors, such as strain, temperature, electric

and magnetic �elds or light. One of the common strategies is the application of a gate

voltage, which is employed, for example, to open up a band gap in graphene. This involves

the exposition of the 2D system to an external perpendicular electric �eld, which induces

polarization and shifts in the energy levels of the electronic structure. Accordingly, we

have used this approach to study the response of MoX2 to WX2 MLs and BLs, where

X = S, Se, to an external electric �eld applied vertically to the basal planes of the 2D

systems. We have performed DFT-PBE calculations, using explicit 2D periodic boundary

conditions, and taking into account the relativistic e�ects and SO interactions.

Γ M K

-6

-4

-2

0

2

4

6

E-E

F / e

V

EF

ε = 0.0%

Γ M K

ε = 4.7%

Γ M K Γ

ε = 9.5%

Figure 3.5.: Band structure of WS2 monolayer under tensile strain (𝜀) calculated withSR+SO treatment. 𝜀 of 9.5% mixes the bands di�erently and closes the bandgap, leading to the semiconductor-metal transition.

31


Γ M K-3

-2

-1

0

1

2

3

E-E

F / e

V

MoS2-MoSe

2 BL

Γ M K Γ

MoS2-WSe

2 BL

Γ M K Γ

WS2-WSe

2 BL

∆ = 0.843 ∆ = 0.695 ∆ = 0.851

∆SO

= 202 meV ∆SO

= 423 meV ∆SO

= 423 meV

Figure 3.6.: Band structures of MoS2 and WSe2 monolayers versus bilayers calculated withthe SR+SO method. Fundamental band gaps (∆) and spin-orbit splittings(∆SO) are given.

0 2 4 6 8

EField

/ V Å-1

0

0.5

1

1.5

2

∆

/ e

V

MoS2

WS2

MoSe2

WSe2

a

0 2 4 6 8

EField

/ V Å-1

0

5

10

15

µ

/ D

eb

ye

MoS2

WS2

MoSe2

WSe2

b

Figure 3.7.: Bandgap (a) and total dipole moment (b) versus electric �eld of MoX2 andWX2 MLs.

Fig. 3.7. depicts ∆ and the dipole moment (𝜇) evolution with respect to the applied

electric �eld of MX2-MLs. ∆ remains una�ected and the dipole moment increases linearly

as the electric �eld strength increases in the range of 0.0-3.5 and 0.0-2.0 for disul�de and

diselenide MLs, respectively. Beyond these ranges, the polarization increases with the �eld

strength and 𝜇 deviates from the linear progression, resulting in a rapid decrease of the

band gaps, which close at critical electric �eld strengths. The e�ective masses of electrons

and holes and the SO splittings in the VBM at the high symmetry K point of all ML

systems are found to be insensitive to the applied electric �eld.

The BS of WX2 BL systems are represented in Fig. 3.8 for di�erent electric �eld

strengths, as an example. The electric �eld induces a polarization of the electronic den-

sity, which lifts the spin degeneracy of the energy levels due to Stark e�ect. This results

in an inversion symmetry breaking in the BL structures and causes SO splittings in both

conduction and valence band regions. As the electric �eld strength increases, the conduc-

tion and valence bands, respectively, shift downward and upward, cross the Fermi level

and close the band gap at critical �elds. consequently, the BLs become metallic. The SO

32


splittings in the valence band at the K point appear at rather weak �elds and saturate at

values comparable to those of their respective monolayers.

Fig. 3.9. shows ∆ and 𝜇 evolution with the electric �eld of the BL systems. Unlike

the monolayers, ∆ and 𝜇, respectively, decreases and increases linearly until the critical

electric �eld strength, at which the materials undergo a semiconductor-metal transition.

However, this transition occurs at relatively smaller values than that of the corresponding

MLs. Moreover, the WX2 BL materials are found to be more responsive to the applied

electric �eld than the MoX2 counterparts. In consequence, applying gate voltages is one of

the possibilities to control and tune the electronic properties of 2D TMC materials, which

render them potential candidates for applications in nanoelectronic devices, particularly

in spintronics and valleytronics. (For more details, see Appendices 3 and 4 for the full

articles)208,209

0.0 VÅ-1

-2

-1

0

1

2

E-E

F /

eV

0.6 VÅ-1

1.25 VÅ-1

-2

-1

0

1

2

WS2

Γ M K Γ

-2

-1

0

1

2

Γ M K Γ M K Γ

-2

-1

0

1

2

WSe2

EF

EF

Figure 3.8.: Band structures versus electric �eld of WX2 BLs.

0 0.5 1 1.5

Efield

/ VÅ-1

0.0

0.5

1.0

1.5

∆ / e

V

0 0.5 1 1.50

2

4

6

µ / D

ebye

MoS2

MoSe2

WS2

WSe2

a b

Figure 3.9.: Bandgap (a) and total dipole moment (b) versus electric �eld of MoX2 andWX2 BLs.

33



As discussed in the Introduction, several layered TMC materials are able to form

nanotube-like structures. These materials are indeed unstable towards folding and have

a high tendency to roll up into cylindrical shapes. Substantial advancement has been

made for the growth and the synthesis of TMC nanotubes since the �rst report of WS2and MoS2 NTs in 1992.14,15 However, their physical and chemical properties are yet to

be explored. Therefore, we are interested to investigate, by means of �rst-principles, the

properties of several TMC-NTs, including the systems of groups 4�7 and 10. First, we

have reassessed the study of the energetic stability and the electronic properties of MoS2and WS2 NTs. This has already been investigated using semi-empirical method, namely

density functional tight binding (DFTB).134,210–212

The strain energy versus diameter of MoS2 and WS2 NTs in shown in Fig. 3.10. We

have considered diameters in the range 12�43 Å, corresponding to chiral indices 11-24 and

both zigzag and armchair chiralities. The strain energy decreases with the tube diameter

𝑑 and follows the 1/𝑑2 dependence. It is found that the energy values are one order

of magnitude larger than that of CNTs with similar diameters. The armchair NTs are

slightly more favorable than zigzag NTs for a given diameter and for both MoS2 and WS2NT systems. However, MoS2-NTs become more stable than WS2-NTs, particularly for

diameters larger than 15 Å, irrespective of the chirality.

Fig. 3.11. depicts the band structures of (24,0) and (24,24) MS2-NTs, as an example,

where M = Mo and W. Both MoS2 and WS2 NT materials exhibit a semiconducting

character. However, zigzag NTs have a direct band gap corresponding to that of their re-

spective monolayered systems and the armchair NT show an indirect band gap, resembling

that of the bulk counterparts. The band gaps increase with the tube diameter approach-

ing those of the corresponding monolayer limit for both zigzag and armchair chiralities, as

shown in Fig. 3.12. Our �ndings are also in a good agreement with those obtained using

DFTB.134,210,212 (For more details, see Appendix 5 for the full article).135

5 10 15 20 25 30 35 40 45d [Å]

0

0.1

0.2

0.3

0.4

Estr

ain [

eV

]

MoS2 (n,0)

WS2 (n,0)

MoS2 (n,n)

WS2 (n,n)

Figure 3.10.: Strain energy versus diameter (𝑑) of MoS2 and WS2 nanotubes.

34


-1

0

1

E0 [

eV

]

(24,0)

-1

0

1

(24,24)

K Γ K

-1

0

1

K Γ K

-1

0

1

MoS2

WS2

Figure 3.11.: Band structure of (24,0) and (24,24) of MoS2 and WS2 nanotubes.

5 10 15 20 25 30 35 40 45d [Å]

1.2

1.4

1.6

1.8

2

∆ [e

V]

MoS2 (n,0)

WS2 (n,0)

MoS2 (n,n)

WS2 (n,n)

Figure 3.12.: Band gap of (n,0) and (n,n) MoS2 and WS2 NTs with respect to the tubediameter (d). The corresponding layered structures are given as horizontallines: solid-bulk, dashed-monolayer.

We have also examined the response of TMC-NTs to an applied tensile strain and how

their electronic properties and the e�ective carrier masses are modi�ed when undergoing

the elongation. We have considered single wall MX2-NTs, with M = Mo, W and X =

S, Se, and Te. Fig. 3.13. shows the stress-strain curves for all studied NTs. For small

elongations, the stress-strain relation follows a linear progression, which corresponds to an

elastic regime associated to Hooke's law; for large strains, the trends deviate from linearity

and thus exhibit plastic deformations. The calculated Young's moduli are consistent with

previous experimental and theoretical �ndings.126–128,130,132,213,214 The electronic proper-

ties of these materials are also altered; the band gaps decrease linearly with the mechanical

deformations and eventually a semiconductor-metal transition is observed for large elon-

gations, as shown in Fig. 3.14. The band gaps of armchair NTs appear to be independent

of the tube diameter under strain. However, in contrast to the corresponding layered

systems, the nanotubes preserve the direct and the indirect characters when the materials

are subject to the tensile stain, as depicted in Fig. 3.15, for zigzag and armchair chirali-

35


ties, respectively. The calculated quantum conductance (𝒢) becomes apparent close to theFermi level (𝐸𝐹 ) under the mechanical deformations. 𝒢 decreases with the strain below

𝐸𝐹 for all NTs, and remains una�ected above 𝐸𝐹 . Consequently, transport channels start

to open close to 𝐸𝐹 . The charge carrier mobilities are notably sensitive to tensile strain,

since the band dispersions in both CBM and VBM are a�ected by the elongations, and

thus, the e�ective electron and hole masses are signi�cantly changed.

Accordingly, this shows that the electronic properties of TMC-NTs can be tailored ap-

plying mechanical deformations, in particular the tensile strain, which may result in poten-

tial applications and may provide new functionalities in optoelectronic and nanoelectronic

devices. (For more details, see Appendix 6 for the full article).215

MoS2

0

50

100

150

200

σ =

F/A

/ G

Pa

MoSe2

MoTe2

0

15

30

45

60

0 0.05 0.10

WS2

0

50

100

150

0 0.05 0.10ε = ∆L/L

0

WSe2

0 0.02 0.04 0.06

WTe3

0

5

10

15

(21,21)

(24,24)

(21,0)

(24,0)

Quartic function

Harmonic approx.

Figure 3.13.: Stress-strain relation of MoX2 and WX2 NTs under applied tensile strainalong the tube axis.

0.0

1.0

2.0

∆ /

eV

(24,0) MoS2

(21,0) MoS2

(24,0) MoSe2

(21,0) MoSe2

(24,0) MoTe2

(21,0) MoTe2

0.0

1.0

2.0(24,24) MoS

2

(21,21) MoS2

(24,24) MoSe2

(21,21) MoSe2

(24,24) MoTe2

(21,21) MoTe2

0.0 0.1 0.2

ε = ∆L/L0

0.0

1.0

2.0(24,0) WS

2

(21,0) WS2

(24,0) WSe2

(21,0) WSe2

(24,0) WTe2

(21,0) WTe2

0.0 0.1 0.20.0

1.0

2.0(24,24) WS

2

(21,21) WS2

(24,24) WSe2

(21,21) WSe2

(24,24) WTe2

(21,21) WTe2

Figure 3.14.: Band gap evolution with the applied tensile strain of zigzag and armchairMoX2 and WX2 NTs.

36


-1

0

1

2

E-E

F /

eV

EF

ε = 0% ε = 10% ε = 17%

Γ X|Γ-1

0

1

2

E-E

F /

eV

X|Γ X

∆ = 1.4 eV

∆ = 0.7 eV∆ = 0.4 eV

∆ = 1.1 eV

∆ = 0.4 eV

∆ = 0.1 eV

(24,0) MoS2

(24,0) MoSe2

-1

0

1

2

E-E

F /

eV

EF

ε = 0% ε = 10% ε = 17%

Γ X|Γ-1

0

1

2

E-E

F /

eV

X|Γ X

∆ = 1.7 eV∆ = 0.9 eV

∆ = 0.5 eV

∆ = 1.4 eV∆ = 0.6 eV

∆ = 0.3 eV

(24,24) MoS2

(24,24) MoSe2

Figure 3.15.: Band structure response to the applied tensile strain of zigzag (left) andarmchair (right) of MoS2 NTs.

0 1 2 3 4

MoS2

0

0.5

1

1.5

∆ / e

V

0 1 2 3 4

EField

/ V nm-1

0

0.5

1

1.5

WS2

(18,18)

(21,21)

(24,24)

(18,0)

(21,0)

(24,0)

Figure 3.16.: Band gap versus electric �eld of MoS2 and WS2 nanotubes.

In the case of layered TMC materials, we have shown, that it is possible to tune their

electronic properties by applying an external electric �eld. This provides an insightful

motivation to further explore the response of TMC nanotubes to an electric �eld applied

perpendicularly to the periodic direction of the tubes and how their electronic properties

are correlated to the tube curvature under gate voltages. Therefore, correspondingly to

the layered systems, we have used DFT method at the PBE level and explicit 1D periodic

boundary conditions to investigate the electronic structure of disul�de Mo and W NTs in

the presence of a perpendicular electric �eld. Fig. 3.16 shows the band gap evolution with

respect to the electric �eld of zigzag and armchair MoS2 and WS2 NTs. The band gaps of

all NTs decrease linearly as the �eld strength increases and close at critical �eld strengths,

leading to a semiconductor-metal transition. However, the band gaps of armchair tubes

decrease faster than that of zigzag ones and the values of the band gap closure of armchair

NTs are twice as small than those of the corresponding zigzag NTs for a given chiral index

𝑛. The band gap closure is also faster as the tube diameter becomes larger, irrespective

of the chirality. Note that the ranges of the �eld strength at which the band gaps close

are at least one order of magnitude smaller than that of ML and BL counterparts.

37


0 1 2 3 4

MoS2

0

10

20

30

40

µ

/ D

eb

ye

0 1 2 3 4

EField

/ V nm-1

0

10

20

30

40WS2

(18,18)(21,21)(24,24)(18,0)(21,0)(24,0)

Figure 3.17.: Dipole moment versus electric �eld of MoS2 and WS2 nanotubes.

The induced dipole moments versus the electric �eld strength of MoS2 and WS2 NTs

are presented in Fig. 3.17. As in the case of TMC MLs and BLs, 𝜇 increases linearly with

the �eld strength. The polarization appears to be stronger for armchair NTs than zigzag

ones and increases with the tube diameter, which explains that the band gaps of armchair

NTs close faster than those of zigzag NTs.

An exemplary band structures of (24,0) and (24,24) NTs are plotted in Fig. 3.18 for

both MoS2 and WS2 materials. The conduction and the valence bands, respectively, shift

downward and upward, resulting in a band gap closure at critical �elds. The zigzag and

armchair systems, respectively, preserve their direct and indirect band gap characters until

the semiconductor-metal transition. In addition, the degeneracy of the energy bands is

altered by the Stark e�ect, causing band splittings, and thus bands mixture, in both

conduction and valence regions. The e�ective masses of electrons and holes increase and

decrease with the electric �eld strength, respectively. At zero electric �eld, the e�ective

electron masses of MoS2-NTs are larger than those of WS2-NTs and for both zigzag and

armchair chiralities. However, as the �eld strength increases, the changes in the e�ective

electron mass values of MoS2 and WS2 NTs can reach a maximum of about 30% and 12%

for zigzag and armchair NTs, respectively.

Essentially, it has been shown that the electronic structure of TMC nanotubes is signif-

icantly modi�ed by the external electric �eld. TMC-NTs appear to be more sensitive to

gate voltages than their respective layered forms. This implies that the electronic proper-

ties of these materials can be controlled by employing external factors such as tensile strain

or gate voltages or even the combination of the two, which lead to new functional charac-

teristics that can be used in the new generation of �exible and nanoelectronic devices.(For

more details, see Appendix 7 for the full article).

38


0.0 V nm-1

-1

0

1

2

E-E

F / e

V

2.5 V nm-1

-1

0

1

20.0 V nm

-1

-1

0

1

21.2 V nm

-1

-1

0

1

2

MoS2

Γ X

(24,0)

-1

0

1

2

Γ X-1

0

1

2

Γ X

(24,24)

-1

0

1

2

Γ X-1

0

1

2

WS2

EF

Figure 3.18.: Band structures versus electric �eld of (24,0) and (24,24) MoS2 and WS2nanotubes.

The scope of TMC nanotubes is also extended to dichalcogenide materials other than

MoS2 and WS2 and their isoelectronic systems. Our attention is particularly drawn to

the TMC materials that include the noble metal elements, namely Pt and Pd. Recently,

it has been reported that the layered noble TMC materials, i.e. PtX2 and PdX2, exhibit

interesting electronic properties and that quantum con�nement is the key factor that

governs these properties when scaling down from bulk to monolayered systems.216,217 In

fact, it has been shown that bulk and monolayers are semiconductors with indirect band

gaps, while the bilayer systems are found to be metallic.216,217 Therefore, we are also

interested to examine how do the electronic properties of these noble TMC materials

change when going from the 2D to 1D systems, i.e. from monolayer to nanotubular forms.

The materials considered here are PtS2, PtSe2, PdS2 and PdSe2 in the 1T polytype. It

has been shown that these monolayered materials are more stable in this phase.216,217

Note that the noble TMC nanotubes are not yet experimentally observed. However, as

for most TMC materials, they are more likely able to form. The energetic stability of

these nanotubes is examined. The strain energy (𝜀𝑆) with respect to the tube diameter is

depicted in Fig. 3.19. The plots show that 𝜀𝑆 is correlated to 1/𝑑2, which is typical for

nanotubular materials. PdX2 are found to be sti�er than PtX2 for small diameters, since

their strain energy is smaller than that of PtX2 in the corresponding range of diameters.

In addition, the strain energy of these noble TMC-NTs is chirality independent and the 𝜀𝑆values are smaller than that of prototypical MoS2 and WS2 NTs. This means that noble

TMC-NTs are more favorable and easier to form than MoS2 and WS2 NTs.

The electronic structure of these noble TMC-NTs is also investigated. Fig. 3.20 shows

39


10 15 20 25 30 35 40 45 50

d / Å

0.00

0.05

0.10

0.15

0.20

0.25

εs / e

V

(n,0) PtSe2

(n,n) PtSe2

(n,0) PdSe2

(n,n) PdSe2

10 15 20 25 30 35 40 45 50

d / Å

0.00

0.05

0.10

0.15

0.20

0.25

εs / e

V

(n,0) PtS2

(n,n) PtS2

(n,0) PdS2

(n,n) PdS2

Figure 3.19.: Strain energy (𝜀𝑆) versus diameter (𝑑) of PtX2 and PdX2 nanotubes.

the band structures of PtS2 and PdS2 NTs and their respective monolayers, as an example.

Layered PtX2 and PdX2 are semiconductors with indirect band gaps; this is in agreement

with early reports.216,217 The noble TMC-NTs are also found to be semiconducting with

indirect band gaps for both zigzag and armchair chiralities, while in the case of MoX2 and

WX2, the zigzag NTs have direct band gaps and the armchair ones have indirect band

gaps. This is probably due to the electronic con�guration of the transition metals and the

di�erence between 1T and 1H symmetries. Nevertheless, similar to MoX2 and WX2 NTs,

the band gaps of noble TMC-NTs increase with the diameter and rapidly converge to that

of their respective monolayered systems, as shown in Fig. 3.21.

In conclusion, these noble TMC-NTs exhibit interesting electronic properties that may

expand and broaden the capabilities of TMC materials, particularly 1D systems, and

yield to new applications in nanodevices. However, the the growth and synthesis of these

materials yet has to be demonstrated. (For more details, see Appendix 8 for the full

article)

1L

0.0

1.4

2.8

E-E

F / e

V

(14,0)

0.0

0.8

1.6

PtS2

(8,8)

0.0

0.8

1.6

E-E

F / e

V

M Γ K M

0.0

1.4

2.8

X Γ

0.0

0.8

PdS2

X Γ

0.0

0.8

1.75 eV

1.26 eV

1.18 eV 1.28 eV

0.79 eV 0.92 eV

Figure 3.20.: Band structures of monolayers and (14,0), (8,8) nanotubes of PtS2 and PdS2systems.

40


10 15 20 25 30 35 40 45 50

d / Å

0.2

0.5

0.8

1.0

1.2

1.5

1.8

2.0

∆

/ e

V

(n,0) PtSe2

(n,n) PtSe(n,0) PdSe(n,n) PdSe

2

10 15 20 25 30 35 40 45 50

d / Å

0.2

0.5

0.8

1.0

1.2

1.5

1.8

2.0

∆ /

eV

(n,0) PtS2

(n,n) PtS2

(n,0) PdS2

(n,n) PdS2

Figure 3.21.: Band gap (∆) versus diameter (𝑑) of PtX2 and PdX2 nanotubes.

41

4. Summary and concluding remarks

Scaling down semiconductor devices, that are presently based on silicon nanotechnology,

will ultimately reach the size limits, which may hamper their performance and utilization

for the next generation of nanoelectronic devices. Therefore, alternatives to silicon based

devices are required to reach the desired goals and objectives. Layered transition metal

dichalcogenide materials constitute potential candidates to complement or take over sili-

con. Recently, extensive research has been carried out on these TMC materials from both

experiment and theory. Considerable e�ort has been made for the development and the

improvement of synthesis and growth methods to obtain large-area samples with desirable

properties. Preeminent and attractive innovations and applications, based on TMC mate-

rials, have been accomplished and suggested for nanoelectronic devices, including FETs,

LEDs, photodetectors, sensors, memory devices, solar cells and so on.

This dissertation presents some theoretical works that have been contributed to the �eld

of TMC materials, using �rst-principles approaches based on density functional theory.

Hereafter, we recall the major obtained results and conclusions. The work has mainly

dealt with one- and two-dimensional TMC materials, including monolayers, bilayers and

nanotubes.

It has been shown that quantum con�nement e�ects represent the key role of tuning

and tailoring the electronic properties of these materials. In fact, thinning TMC materials

from bulk to the monolayer level causes a change in the band gap size and induces a

transition from indirect to direct band gap, which is relevant for band gap engineering.

The e�ect of spin-orbit coupling has been investigated in the monolayered TMC systems

and reveals considerable spin-splittings due to the lack of inversion symmetry in these

structures. The size of the spin-splittings becomes signi�cant for heavier atoms and when

the materials are subject to tensile strain. It has also been shown that spin-splittings can

be induced in the bilayer materials by forming heterobilayers, which initially do not exist

due to inversion and time reversal symmetries.

The exposure of TMC materials to external perpendicular electric �elds shows that the

monolayer systems are very stable and their electronic properties remain almost una�ected.

On the other hand, in the case of the bilayers, the band gap decreases linearly and a

semiconductor-metal transition occurs for practical electric �elds. Moreover, spin-orbit

splittings can be induced in these systems due to the inversion symmetry breaking caused

by the electric �eld.

The investigation of MoS2 and WS2 nanotubes shows that their strain energies are cor-

related to the typical 1/𝑑2 relation, where 𝑑 de�nes the tube diameter. The armchair

43

4. Summary and concluding remarks

tubes are slightly more favorable than zigzag ones for similar diameters and MoS2 NTs

are more stable than WS2 NTs, for large diameters. Furthermore, the range of the strain

energies is found to be one order of magnitude higher than that of carbon nanotubes with

comparable diameters. The examination of the electronic structure of these TMC nan-

otubes indicates that these materials are all semiconductors. The armchair and zigzag NTs

exhibit indirect and direct gaps similar to that of their corresponding bulk and monolayer

systems, respectively. Moreover, the band gaps increase with the diameter, irrespective of

the chirality.

The electromechanical properties of TMC nanotubes under an applied tensile strain have

been examined. The stress-strain diagram has exhibited elastic regime for small elonga-

tions and thereafter the materials endure a plastic deformation. The obtained Young's

moduli thoroughly agree with available experimental and theoretical data. The band gaps

linearly decrease with the strain and the materials undergo a semiconductor-metal tran-

sition for mechanical strain larger than those of the monolayered forms. Nevertheless,

the band gaps remain indirect and direct when the materials are subject to tensile strain

for both armchair and zigzag nanotubes, respectively. The charge carrier mobilities are

strongly a�ected and the transport channels open close to the Fermi level particularly for

large strain.

The response of TMC nanotubes to an external perpendicular electric �eld is also evalu-

ated. It is found that the band gaps decrease linearly with the �eld strength and ultimately

close at critical �elds; hence, a semiconductor-metal transition is observed. The band gap

closure is reached much faster for armchair NTs than zigzag ones along with the increase

of tube diameter. Split-o� bands are also observed in the conduction and valence bands

due to Stark e�ect; however, the indirect and direct characters of the band gap are con-

served for both armchair and zigzag NTs, respectively. The charge carrier mobilities are

slightly sensitive to gate voltages, since the e�ective carrier masses linearly change under

an electric �eld.

The investigation of TMC nanotubes is also extended to materials beyond MoX2 and

WX2, namely noble metal dichalcogenide PtX2 and PdX2 materials. The energetic sta-

bility expressed by the strain energy indicates that these materials are able to form, since

their strain energies are lower than that of MoX2 and WX2 counterparts. The strain ener-

gies are also chirality independent for a given material and exhibit the 1/𝑑2 characteristic.

The nanotubes are semiconductors; they all exhibit indirect band gaps, irrespective of the

chirality, which rapidly approach that of the corresponding monolayers.

44

5. References

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The following article has been removed from the online version because of copyright restrictions.

Influence of quantum confinement on the electronic structure of the transition metal sulfide TS2

A. Kuc, N. Zibouche, and T. Heine

Bulk MoS2, a prototypical layered transition-metal dichalcogenide, is an indirect band gap

semiconductor. Reducing its slab thickness to a monolayer, MoS2 undergoes a transition to the direct

band semiconductor. We support this experimental observation by first-principle calculations and

show that quantum confinement in layered d-electron dichalcogenides results in tuning the

electronic structure. We further studied the properties of related TS2 nanolayers (T = W, Nb, Re) and

show that the isotopological WS2 exhibits similar electronic properties, while NbS2 and ReS2 remain

metallic independent of the slab thickness.

Physical Review B, Volume 83, Issue 24, June 2011

DOI: 10.1103/PhysRevB.83.245213

http://dx.doi.org/10.1103/PhysRevB.83.245213

How quantum confinement influences the electronic

structure of transition metal sulfides TMS2

A. Kuc, N. Zibouche and T. Heine

School of Engineering and Science, Jacobs University Bremen,

Campus Ring 1, 28759 Bremen, Germany

April 29, 2011

The following results are obtained with PBE0 functional and are complementary to the

results presented in the main paper.

163

-0.2

0

0.2

EF

[Har

tree

]

MoS2 bulk MoS2 8-layer

-0.2

0

0.2

MoS2 6-layer

Γ M K Γ

-0.2

0

0.2

MoS2 quadrilayer

Γ M K Γ

MoS2 bilayer

Γ M K Γ

-0.2

0

0.2

MoS2 monolayer

= 2.3 eV∆

∆= 2.9 eV

Figure 1: Band structures of bulk MoS2, its monolayer, as well as, polylayers calculated as the DFT/PBE0.

The horizontal dashed lines indicate the Fermi level. The arrowas indicate the fundamental band gap (direct

or indirect) for a given system.

264

-0.2

0

0.2

EF

[Har

tree

]

WS2 bulk WS2 8-layer

-0.2

0

0.2

WS2 6-layer

Γ M K Γ

-0.2

0

0.2

WS2 quadrilayer

Γ M K Γ

WS2 bilayer

Γ M K Γ

-0.2

0

0.2

WS2 monolayer

= 2.4 eV∆

∆= 3.0 eV

Figure 2: Band structures of bulk WS2, its monolayer, as well as, polylayers calculated as the DFT/PBE0.

The horizontal dashed lines indicate the Fermi level. The arrowas indicate the fundamental band gap (direct

or indirect) for a given system.

365

Γ M K Γ

-0.2

0

0.2

EF

[Har

tree

]

NbS2 bulk

Γ M K Γ

NbS2 monolayer

Γ M K Γ

NbS2 bulk

Γ M K Γ

-0.2

0

0.2

NbS2 monolayer

PBE0 PBE

Figure 3: Band structures of bulk NbS2 and its monolayer calculated as the DFT/PBE0 and the DFT/PBE.

The horizontal dashed lines indicate the Fermi level.

Γ M K Γ

-0.2

0

0.2

EF

[Har

tree

]

ReS2 bulk

Γ M K Γ

ReS2 monolayer

Γ M K Γ

ReS2 bulk

Γ M K Γ

-0.2

0

0.2

ReS2 monolayer

PBE0 PBE

Figure 4: Band structures of bulk ReS2 and its monolayer calculated as the DFT/PBE0 and the DFT/PBE.

The horizontal dashed lines indicate the Fermi level.

466

0

50

100

DO

S [a

rb. u

nits

]

Mo PDOS

MoS2 bulk

0

50

100

Mo 4d-orbitals

MoS2 monolayer

-0.2 0 0.2

EF [Hartree]

0

50

100

S PDOS

-0.2 0 0.2

EF [Hartree]

0

50

100

S 3p-orbitals

Figure 5: Partial density of states of bulk MoS2 and its monolayer calculated as the

DFT/PBE0. The projections of Mo and S atoms are given together with the contribu-

tions from 4d and 3p orbitals of Mo and S, respectively. The vertical dashed lines indicate

the Fermi level. (Online color).

567

The following articles have been removed from the online version because of copyright restrictions.

Transition-metal dichalcogenides for spintronic applications

Nourdine Zibouche, Agnieszka Kuc, Janice Musfeldt, Thomas Heine

Spin-orbit (SO) splitting in transition-metal diChalcogenide (TMC) monolayers is investigated by

means of density functional theory within explicit two-dimensional periodic boundary condition. The

SO splitting reaches few hundred meV and increases with atomic number of the metal and chalcogen

atoms, resulting in nearly 500 meV for WTe 2. Furthermore, we find that similar to the band gap, SO

splitting changes drastically under tensile strain. In centrosymmetric TMC bilayers, SO splitting is

suppressed by the inversion symmetry. However, this could be induced if the inversion symmetry is

explicitly broken, e.g. by a potential gradient normal to the plane, as it is present in heterobilayers

(Rashba-splitting). In such systems, the SO splitting could be as large as for the heavier monolayer

that forms heterobilayer. These properties of TMC materials suggest them for potential applications

in opto-, spin- and straintronics.

annalen der physik, Volume 526, Issue 9-10, October 2014

DOI: 10.1002/andp.201400137

Electron transport in MoWSeS monolayers in the presence of an external electric field

Nourdine Zibouche, Pier Philipsen, Thomas Heine and Agnieszka Kuc

The influence of an external electric field on single-layer transition-metal dichalcogenides TX2 with

T = Mo, W and X = S, Se (MoWSeS) has been investigated by means of density-functional theory

within two-dimensional periodic boundary conditions under consideration of relativistic effects

including the spin–orbit interactions. Our results show that the external field modifies the band

structure of the monolayers, in particular, the conduction band. This modification has, however, very

little influence on the band gap and effective masses of holes and electrons at the K point, and also

the spin–orbit splitting of these monolayers is almost unaffected. Our results indicate a remarkable

stability of the electronic properties of TX2 monolayers with respect to gate voltages. A reduction of

the electronic band gap is observed starting only from field strengths of 2.0 V Å−1 (3.5 V Å−1) for

selenides (sulphides), and the transition to a metallic phase would occur at fields of 4.5 V Å−1 (6.5 V

Å−1).

Phys.Chem.Chem.Phys., 2014, 16, 11251

DOI: 10.1039/c4cp00966e

http://dx.doi.org/10.1002/andp.201400137

http://dx.doi.org/10.1039/c4cp00966e

Transition-metal dichalcogenide bilayers: Switching materials for spintronic and valleytronic

applications

Nourdine Zibouche, Pier Philipsen, Agnieszka Kuc, and Thomas Heine

We report that an external electric field applied normal to bilayers of transition-metal

dichalcogenides TX2 (T = Mo, W, X = S, Se) creates significant spin-orbit splittings and reduces the

electronic band gap linearly with the field strength. Contrary to the TX2 monolayers, spin-orbit

splittings and valley polarization are absent in bilayers due to the presence of inversion symmetry.

This symmetry can be broken by an electric field, and the spin-orbit splittings in the valence band

quickly reach values similar to those in the monolayers (145 meV for MoS2,..., 418 meV for WSe2) at

saturation fields less than 500 mV Å−1. The band gap closure results in a semiconductor-metal

transition at field strength between 1.25 (WX2) and 1.50 (MoX2) V Å−1. Thus, by using a gate voltage,

the spin polarization can be switched on and off in TX2 bilayers, thus activating them for spintronic

and valleytronic applications.

DOI: 10.1103/PhysRevB.90.125440

From layers to nanotubes: Transition metal disulfides TMS2

N. Zibouche, A. Kuc, and T. Heine

MoS2 and WS2 layered transition-metal dichalcogenides are indirect band gap semiconductors in

their bulk forms. Thinned to a monolayer, they undergo a transition and become direct band gap

materials. Layered structures of that kind can be folded to form nanotubes. We present here the

electronic structure comparison between bulk, monolayered and tubular forms of transition metal

disulfides using first-principle calculations. Our results show that armchair nanotubes remain indirect

gap semiconductors, similar to the bulk system, while the zigzag nanotubes, like monolayers, are

direct gap materials, what suggests interesting potential applications in optoelectronics.

The European Physical Journal B, Volume 85, Issue 49, January 2012

DOI: 10.1140/epjb/e2011-20442-1

Electromechanical Properties of Small Transition-Metal Dichalcogenide Nanotubes

Nourdine Zibouche, Mahdi Ghorbani-Asl, Thomas Heine and Agnieszka Kuc

Transition-metal dichalcogenide nanotubes (TMC-NTs) are investigated for their electromechanical

properties under applied tensile strain using density functional-based methods. For small

elongations, linear strain-stress relations according to Hooke’s law have been obtained, while for

larger strains, plastic behavior is observed. Similar to their 2D counterparts, TMC-NTs show nearly a

linear change of band gaps with applied strain. This change is, however, nearly diameter-

independent in case of armchair forms. The semiconductor-metal transition occurs for much larger

deformations compared to the layered tube equivalents. This transition is faster for heavier

chalcogen elements, due to their smaller intrinsic band gaps. Unlike in the 2D forms, the top of

valence and the bottom of conduction bands stay unchanged with strain, and the zigzag NTs are

direct band gap materials until the semiconductor-metal transition. Meanwhile, the applied strain

causes modification in band curvature, affecting the effective masses of electrons and holes. The

quantum conductance of TMC-NTs starts to occur close to the Fermi level when tensile strain is

applied. Inorganics, Volume 2, Issue 2, April 2014. DOI: 10.3390/inorganics2020155

http://dx.doi.org/10.1103/PhysRevB.90.125440

https://doi.org/10.1140/epjb/e2011-20442-1

https://doi.org/10.3390/inorganics2020155

Tunable Electronic and Transport Properties of MoS2 and WS2

Nanotubes Under External Electric Field

Nourdine Zibouche,1 Pier Philipsen,2 Agnieszka Kuc,1 and Thomas Heine1

December 17, 2014

1School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany2Scientific Computing & Modelling NV, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands

Abstract

We have investigated the effect of a perpendicular electric field on the electronic properties and charge

carrier mobilities of MoS2 and WS2 nanotubes (NTs) by means of density functional theory. The ap-

plied electric field induces a strong band gap modulation. In fact, the band gap rapidly closes causing a

semiconductor-metal transition, particularly for large tube diameters and for armchair forms. The band-

structures and the effective masses of electrons and holes are modified even by very weak field strengths.

Therefore, the effet of gate voltage on the TMC nanotubes may lead to electrically tunable switching

devices and other nanoelectronic applications.

1 Introduction

Molybdenum and tungsten disulfides (MoS2 and WS2) nanotubes (NTs) have been the first synthesized

and characterized inorganic NTs.1,2 MoS2 and WS2 NTs belong to the family of layered transition metal

chalcogenides (TMCs). They are semiconducting materials with direct and indirect band gaps for zigzag and

armchair NTs, respectively. They have also demonstrated excellent mechanical properties.3–10 For example,

it has been shown that WS2-NTs are good resistants to shock waves3 and exhibit ultrahigh strength and

elasticity under uniaxial tensile tests.7 These materials have been mainly used as solid lubricants11 because

of their excellent tribological properties. They are also suggested as scanning probe tips,12 catalysts,13

reinforcements for composite materials,14 photo-transistors,15 storage or as host materials.16,17 However,

MoS2 and WS2 nanotubes are still less explored in comparison to carbon NTs (CNTs) or to their layered

counterparts. Therefore, many more fundamental issues need to be addressed, particularly, the control of

their electronic properties by means of doping, mechanical strain or electric field. For example, recently, we

have investigated the effect of an external tensile strain on the electronic and transport properties of these

TMC-NTs.10,18 We have shown that Raman spectroscopy is suitable for monitoring the strain of individual

tubes. Moreover, the band gap reduction, the electronic structure and the effective masses of the charge

carriers can be modulated by an applied tensile strain.18

There have also been reports on the influence of external electric field on various layered and tubular

nanomaterials, such as CNTs,19–30 BN31,32 and ZnO33 NTs, where significant modifications have been ob-

served in the electronic and mechanical properties of these nanostructures, suggesting them as promising

materials for applications in nanoelectronic devices.

1

111

However, the effect of an electric field applied on MoS2 and WS2 NTs has not yet been explored. In this

perspective, we report in this paper the response of zigzag and armchair MoS2 and WS2 NTs to an external

perpendicular electric field using density functional theory (DFT). We show that the band gaps of these

materials drastically reduce in the presence of weak electric fields, especially for armchair forms, involving

a degenerate band states separation colse to the Fermi level due to the Stark effect. The transition from

semiconductor to metal is rapidly attained for large NT diameters and the effective masses of electrons and

holes are also modified with the field strengths.

Methods

The electronic structure and charge carrier mobilities of single-wall (SW) MoS2 and WS2 nanotubes sub-

jected to an expernal perpendicular electric field have been calculated using density functional theory (DFT),

as implemented in the ADF-BAND software.34,35The PBE36 exchange-correlation functional is adopted for

all calculations. Valence triple-zeta polarized (TZP) basis sets composed of Slater-type and numerical or-

bitals with a small frozen core have been employed. Both zigzag (n,0) and armchair (n,n) chiralities and

different diameters have been examined in this study by considering NTs with n = 18, 21, and 24. The struc-

tures were fully optimized (atomic positions and lattice vectors), with a the maximum gradient threshold of

10−3 hartree A−1. Relativistic effects were taken into account by employing the scalar Zero Order Regular

Approximation (ZORA).37 The k-points mesh over the first Brillouin zone was sampled according to the

Wiesenekker-Baerends scheme,38 where the k-space integration parameter is set to 5 to ensure a uniform

k-point spacing in the irreducible wedge.

2 Results and discussion

The band gap evolution of (n,0) and (n,n) MoS2 and WS2 NTs under a perpendicular electric field is

shown in Fig. 1. We can see that the band gap of all tubes decreases linearly with the electric field strength.

The critical values of the electric field, at which the band gap of MoS2 and WS2 NTs closes, are found to

be in the range of 1.2–1.9 V nm−1 and 2.5–3.8 V nm−1 for both armchair and zigzag NTs, respectively. It

is clear that these critical field strengths are as twice small as for armchair NTs than for the zigzag NTs.

This means that armchair NTs are more sensitive to external electric fields than their zigzag counterparts,

although armchair NTs at zero electric field. It is also interesting to mention that the larger the diameter,

the faster the band gap closure for both zigzag and armchair NTs. Note that the planar monolayers of the

MoS2 and WS2 are more stable than the corresponding SW-NTs in the presence of the electric fields.39

The studied diameters of MoS2 and WS2 NTs are among the smallest experimentally observed values. In

addition, the band gaps of TMC bilayers close faster that those of the respective monolayers.39,40 Therefore,

one can expect that the band gaps of multiwall (MW) NTs close faster that those of SW-NTs.

Fig. 2 depicts the dipole moments of MoS2 and WS2 NTs induced by the external electric field. We

observe that, as the electric field strength increases, the dipole moments increase linearly for all tubes. The

results indicate that the polarization is higher for armchair NTs and becomes larger as the diamter increases,

which explains why the band gap closure is faster for this type of tubes. Therefore, the band gap of MoS2

and WS2 nanotubes with large diameters can be modulated using very small electric fields.

2

112

0 1 2 3 4

MoS2

0

0.5

1

1.5

∆ /

eV

0 1 2 3 4

EField

/ V nm-1

0

0.5

1

1.5

WS2(18,18)

(21,21)

(24,24)

(18,0)

(21,0)

(24,0)

Figure 1: Band gaps of (n,0) and (n,n) MoS2 and WS2 nanotubes versus external electric field.

0 1 2 3 4

MoS2

0

10

20

30

40

µ

/ D

eb

ye

0 1 2 3 4

EField

/ V nm-1

0

10

20

30

40WS2

(18,18)(21,21)(24,24)(18,0)(21,0)(24,0)

Figure 2: Dipole moments of (n,0) and (n,n) MoS2 and WS2 nanotubes versus external electric field.

The band structures of examplary (24,0) and (24,24) MoS2/WS2 nanotubes in the presense of electric

field are presented in Fig. 3. The band gaps are direct and indirect for zigzag (24,0) and armchair (24,24)

tubes of both MoS2/WS2, respectively, what was already reported in previous works.41,42 We can see that

at critical fields, the direct/indirect character of the band gap remains unchanged for zigzag/armchair NTs.

However, the energy bands are modified by the so-called Stark effect, which alters the degenerate energy

levels due to the applied electric field, inducing a mixure of the neighboring subband states in both valence

and conduction regions. When the electric field is applied on zigzag nanotubes we observe that the highest

valence band is split into two subbands, whereas the lowest band of the conduction region stays unperturbed

at Γ point, at which the valence band maximum (VBM) and the conduction band minimum (CVM) are

situated. Armchair nanotubes have VBM and CBM located, respectively, at Γ and at K point, which is

situated at 23 between Γ and X. In this case, the degenerate states of both the highest valence band and the

lowest conduction band are split and shifted apart. The highest valence band is split at Γ and K, whereas

the lowest conduction band is split only at K. The splitting ∆split values of VBM and CBM at critical field

strengths are reported in Table 1 for all studied nanotubes. For the highest valence band, the values of ∆split

at Γ and K points decrease with the diameter for all tubes, except for MoS2 at K, where they are almost

unaffected. This is also the case for lowest conduction band at K for armchair NTs, whereas at Γ, no ∆split

is observed for all tubes.

3

113

0.0 V nm-1

-1

0

1

2

E-E

F

/ e

V

2.5 V nm-1

-1

0

1

20.0 V nm

-1

-1

0

1

21.2 V nm

-1

-1

0

1

2

MoS2

Γ X

(24,0)

-1

0

1

2

Γ X-1

0

1

2

Γ X

(24,24)

-1

0

1

2

Γ X-1

0

1

2

WS2

EF

Figure 3: Band structures of (24,0) and (24,24) MoS2 and WS2 nanotubes versus external electric field.

Table 1: Splitting values (∆split) of conduction band minimum (CBM) and valence band maximum (VBM)

at Γ and K of (n,0) and (n,n) MoS2 and WS2 nanotubes.

System Chirality ∆split of VBM (meV) ∆split of CBM (meV)

Γ K Γ K

MoS2 (18,0) 22 0 0.0 0

(21,0) 194 0 0.0 0

(24,0) 170 0 0.0 0

(18,18) 135 114 0.0 167

(21,21) 97 114 0.0 158

(24,24) 85 117 0.0 152

WS2 (18,0) 255 0 0.0 0

(21,0) 220 0 0.0 0

(24,0) 180 0 0.0 0

(18,18) 147 160 0.0 217

(21,21) 115 144 0.0 206

(24,24) 92 150 0.0 199

The energy changes of VBM and CBM with respect to the electric field of all tubes are plotted in Fig. 4.

Both VBM and CBM energies shift linearly upward and downward, respectively, when increasing the electric

field, and hence leading to the semiconductor-metal transition at critical field strengths. Analogous to the

band gap, the VBM and CBM energies change faster for larger diameters of both zigzag and armchair NTs.

The effective masses of electrons and holes of the studied nanotubes with respect to the applied electric

4

114

field are shown in Fig. 5. In general, the effective masses of electrons/holes increase/decrease with the electric

field strength, respectively. Note that the effective masses of holes are considerably large due to the flatness

of the bands at the VBM, situated at Γ point for both zigzag and armchair tubes. At zero electric field,

the electrons effective masses of MoS2 NTs are larger than those of WS2 NTs by at least 0.11/0.17 m0 for

zigzag/armchair chiralities. However the changes in the electrons effective masses with increasing the field

strength are almost linear for both MoS2 and WS2 systems, except for zigzag NTs at weak fields, where the

maximum changes are of 28/26% for zigzag MoS2/WS2 NTs and 10/12% for armchair MoS2/WS2 NTs.

(n,0)

-6.0

-5.5

-5.0

-4.5

E / e

V

VBM

CBM

(n,n)

-6.0

-5.5

-5.0

-4.5

MoS2

VBM

CBM

0 1 2 3 4

EField

/ V nm-1

-6.0

-5.5

-5.0

-4.5

-4.0

(18,0)

(21,0)

(24,0)

0 0.5 1 1.5 2-6.0

-5.5

-5.0

-4.5

-4.0

WS2

(18,18)

(21,21)

(24,24)

Figure 4: CBM and VBM of (n,0) and (n,n) MoS2 and WS2 nanotubes versus external electric field.

(n,0)

0.3

0.4

0.5

me* \

m0

(18,0) MoS2

(21,0) MoS2

(24,0) MoS2

(18,0) WS2

(21,0) WS2

(24,0) WS2

(n,n)

0.3

0.4

0.5

0 1 2 3 4

-10

-8

-6

-4

mh* \

m0

0 0.5 1 1.5 2

EField

/ V/nm

-10

-8

-6

-4

(18,18) MoS2

(21,21) MoS2

(24,24) MoS2

(18,18) WS2

(21,21) WS2

(24,24) WS2

Figure 5: Electrons and holes effective masses of (n,0) and (n,n) MoS2 and WS2 nanotubes versus external

electric field.

5

115

3 Conclusions

We have studied the response of MoS2 and WS2 nanotubes to an external electric field applied perpen-

dicularly to the tube periodicity using density functional theory. It is clearly shown that potential gradient

causes linear reduction and increase in the band gaps and dipole moments, respectively, what results in a

semiconductor-metal transition. Moreover, the polarization is stronger for armchair than zigzag tubes and

for large diameters. The band structure analysis shows that the indirect and direct band gap characters are

preserved under the applied electric field until the band gap closure for armchair and zigzag chiralities, respec-

tively. However, the degeneracy of energy band states is altered by the Stark effect, causing a modification

in the charge carrier mobilities.

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116

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The following article has been removed from the online version because of copyright restrictions.

Noble-Metal Chalcogenide Nanotubes

Nourdine Zibouche, Agnieszka Kuc, Pere Miró and Thomas Heine

We explore the stability and the electronic properties of hypothetical noble-metal chalcogenide

nanotubes PtS2, PtSe2, PdS2 and PdSe2 by means of density functional theory calculations. Our

findings show that the strain energy decreases inverse quadratically with the tube diameter, as is

typical for other nanotubes. Moreover, the strain energy is independent of the tube chirality and

converges towards the same value for large diameters. The band-structure calculations show that all

noble-metal chalcogenide nanotubes are indirect band gap semiconductors. The corresponding band

gaps increase with the nanotube diameter rapidly approaching the respective pristine 2D monolayer

limit.

Inorganics, Volume 2, Issue 4, October 2014

DOI: 10.3390/inorganics2040556

http://doi.org/10.3390/inorganics2040556

structural,electronicandmechanical propertiesofone-andtwo

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