Structural,ElectronicandMechanical PropertiesofOne-andTwo ...
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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.
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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. Layered transition metal dichalcogenides
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
1.1. Layered transition metal dichalcogenides
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
1.1. Layered transition metal dichalcogenides
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
1.1. Layered transition metal dichalcogenides
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
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
1.2. Transition-metal dichalcogenide nanotubes
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
1.2. Transition-metal dichalcogenide nanotubes
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
1.2. Transition-metal dichalcogenide nanotubes
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
3.1. Layered transition metal dichalcogenides
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
3. Results and discussion
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
3.1. Layered transition metal dichalcogenides
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
3. Results and discussion
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
3.1. Layered transition metal dichalcogenides
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
3. Results and discussion
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
3.1. Layered transition metal dichalcogenides
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
3. Results and discussion
3.2. Transition-metal dichalcogenide nanotubes
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
3.2. Transition-metal dichalcogenide nanotubes
-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
3. Results and discussion
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
3.2. Transition-metal dichalcogenide nanotubes
-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
3. Results and discussion
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
3.2. Transition-metal dichalcogenide nanotubes
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
3. Results and discussion
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
3.2. Transition-metal dichalcogenide nanotubes
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.
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chirality, which rapidly approach that of the corresponding monolayers.
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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
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
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
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
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
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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.
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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