Electronic structure, transport and optical
properties of MoS
2
monolayers and
nanoribbons
Niterói 2017
Electronic structure, transport and optical
properties of MoS
2
monolayers and
nanoribbons
Tese apresentada ao Curso de Pós-Graduação em Física da Universidade Federal Fluminense, como requisito parcial para obtenção do Título de Doutor em Física.
Orientador:
Prof. Dr. CAIO HENRIQUE LEWENKOPF
Universidade Federal Fluminense Instituto de Física
Niterói 2017
Caio Henrique Lewenkopf. –- Niterói, 2017. 208 p. : il.
Tese (Doutorado) – Universidade Federal Fluminense, Instituto de Física, Niterói, 2017.
Bibliografia: p. 201-208.
1.MODELO DE LIGAÇÃO FORTE. 2.MOLIBDÊNIO DE ENXOFRE. 3.TRANSPORTE ELETRÔNICO. 4.DEFEITO. 5.NANOFITA. 6.EXCITON. 7.CONDUTIVIDADE ÓPTICA I. Lewenkopf, Caio Henrique,
orientador. II.Universidade Federal Fluminense. Instituto de Física, Instituição responsável. III. Título.
Acknowledgements
First and foremost, I would like to express my gratitude to my supervisor, professor Caio H. Lewenkopf, for his incredible support, advice and guidance throughout my entire PhD program.
I also would like to thank all the people of the nano group for the highly formative meetings, conversations and debates during these years, and also for the invaluable com-panionship, respect and camaradrie during conferences or working day life. The Instituto de Física at UFF has been a very hospitable environment, where I believe that I found not only a dynamical research environment, but also great people. I take the opportunity to express my gratitude to the UFF sta, both administrative and technical, for the kind support and assistance provided along all these years.
I would like to thank the hospitality and support from the University of Central Florida, (USA). Words come short to express my gratitude to Prof. Eduardo R. Mucciolo, for his dedicated mentoring during the short but very productive period, as well for the very kind reception.
I would like to thank the hospitality and support from the NUS Centre for Advanced 2D Materials, Singapore. A special acknowledgement to professor Vitor M. Pereira, who expertly guided me throughout the last part of my PhD run.
We would like to thank the hospitality of Prof. Fanyao Qu. and the helpful conver-sations and Prof. Nuno Peres and Prof. Marcus Moutinho for helpful discussions. This work was supported by the Brazilian funding agencies CNPq, CAPES, FAPERJ.
I must thank the poor soul that is not from Physics, but nearly every week in this last period was earing my research progress in a child friendly explanation and helped me to have good insights.
de nitly as important as the little advance in understanding of the MoS2 properties. Life
is made of encounters of souls, it does not matter if we think about them as spirits or ensembles of atoms and wave-frequencies. Every soul changes you and is changed by you. Four years have meant lots of souls I came across and I want to thanks all of them.
Rio has a wonderful landscape, colours and music but also strong contrasts. So dierent to give occasion to grow and change, so hard to make me hope in the slowly path to a better society.
Among all people that took part in the advertures of these years, I do not know who to acknowledgement most. It is a big list:
the people that welcomed me when I rst arrived and were my family there; the enlarged family that welcomed me across the south;
the mixed transformable gang composed by the people always ready to explore and enjoy a beach, a trek, a lunch, an event or carnival;
the partner in crime for 2 years;
the capoeira crews around the world, the mestres that I met and the one that I always look forward to see again;
the pole dance crew in Niteroi that I loved;
the ones who came from far far away to visit me and Rio;
some people with a very special energy, and in particularly one that entered my life in lots of possible ways;
and the angel that appered in this last year.
Finally, I thank the friends far far away, that in an available or necessary occasion can make the magic of getting as close as they were everytime here.
Last, my large large family, where everyone is special. I hope the distance will reduce to be able to join more family conferences!
Abstract
First, we propose an accurate tight-binding parametrization for the band structure of MoS2 monolayers near the main energy gap. We introduce a generic and straightforward
derivation for the band energies equations based on the Slater-Koster model that can be employed for other monolayer dichalcogenides. The parametrization includes spin-orbit coupling. The proposed set of model parameters reproduces both the correct orbital compositions and location of valence and conductance band in comparison with ab initio calculations. The model gives a suitable starting point for realistic large-scale atomistic electronic transport calculations.
Next, we study the electronic structure and transport properties of zigzag and arm-chair monolayer molybdenum disul de nanoribbons using the tight-binding model men-tined above, that accurately reproduces their bulk band structure near the band gap. We study the electronic properties of pristine zigzag and armchair nanoribbons, paying particular attention to the edge states that appear within the MoS2 bulk gap. By
an-alyzing both their orbital composition and their local density of states, we nd that in zigzag-terminated nanoribbons, in distinction to graphene, edge states can be localized at a single edge for certain energies independent of the nanoribbon width. We also study the eects of disorder in these systems using the recursive Green's function technique. We show that for the zigzag nanoribbons, the conductance due to the edge states is strongly suppressed by short-range disorder such as vacancies. In contrast, the local density of states still shows edge localization. We also show that long-range disorder has a small eect on the transport properties of nanoribbons within the bulk gap energy window.
Finally, we study the excitonic spectrum of MoS2monolayers. Our approach takes into
account the anomalous screening in two dimensions and the presence of a substrate by the eective Keldish potential. The Bethe-Salpeter equation is solved for a multi-band tight-binding description of the single particle spectrum. We obtain the main features of the optical conductivity spectrum analyzing the localization in k-space of the excitonic peaks. We study the eect of dierent TB descriptions on the main features of the experimental optical absorption, namely, the contributions of higher energy bands, and the validity of reduced TB models with and without spin- ipping terms in the spin-orbit contribution to the Hamiltonian. We compare the absolute magnitude of the linear optical conductivity, one-particle optical conductivity and one of the most recent experimental measurements of this quantity. Other related TB and eective-mass calculations fail to reproduce the actual experimental magnitude of the optical conductivity.
Resumo
Propomos uma parametrização do modelo de ligação fortes (tight-binding) que de-screve com precisão a estrutura de bandas de monocamadas de MoS2 na vizinhança do
gap de energia. Introduzimos uma derivação genérica e simples baseada no modelo de Slater-Koster que pode ser também utilizada para outros sistemas de dicalcogenetos mono-camada. A parametrização inclue o acoplamento spin-órbita. O conjunto de parâmetros reproduz tanto a composição orbital, como a relação de dispersão das bandas de condução e valência obtidas por cálculos de primeiros princípios. O modelo fornece um ponto de partida convinciente para cálculos atomísticos de larga escala de transporte electrônico de sistemas de dimensões realísticas.
Em seguida estudamos a estrutura eletrônica e as propriedades de transporte de nano tas de MoS2 com terminações ziguezague e poltrona usando o modelo de
tight-binding mencionado acima, o qual reproduz com precisão sua estrutura de bandas próx-imas ao gap principal. Estudamos as propriedades eletrônicas de tas ziguezague e poltrona perfeitas, com particular atenção nos estados de borda que aparecem na região de énergia correspondente ao gap do bulk. Analisando sua composição orbital e a densi-dade local de estados identi camos que em nano tas de MoS2, ao contrário do grafeno, os
estados lcalizados podem aparecer em uma única borda, independente da largura da ta. Estudamos também efeitos de desordem nestes sistemas usando a técnica das funções de Green recursivas. Mostramos que para nano tas ziguezague a condutância devida a esta-dos de borda é fortemente suprimida por centros espalhadores de curto alcance, tais como vacâncias. Em contraste, a densidade local de estados continua a mostrar localização de borda. Mostramos ainda que desordem de longo alcance tem um efeito muito pequeno nas propriedades de transporte para os estados dentro do bulk gap em nano tas de MoS2.
Finalmente, estudamos o espectro de excitons de monocamadas de MoS2. Nossa
abordagem leva em consideração efeitos anômalos de blindagem em duas dimensões e a presença de substrato usando um potencial de Keldysh efetivo. A equação de Bethe-Salpeter é resolvida para uma descrição de partícula independente baseado num modelo de tight-binding de multibandas. Obtemos as principais características da condutividade óptica e analizamos a localização dos picos excitônicos no espaço k. Estudamos o efeito de diferentes descrições TB nas principais características da absorção óptica, mais es-peci camente as contribuções das bandas de mais alta energia, a validade do modelo TB reduzido com e sem termos de spin- ip na interação spin órbita do Hamiltoniano. Estu-damos magnitude absoluta da condutividade óptica linear, comparando nossos resultados com a condutividade óptica de particula única e com dados experimentais recentes. Out-ros modelos TB e cálculos usando massas efetivas falham em reproduzir a magnitude da condutividade óptica medida experimentalmente.
List of Figures
1 Building van der Waals heterostructures. If one considers 2D crystals to be analogous to Lego blocks (right panel), the construction of a huge
variety of layered structures becomes possible. Extracted from [3] . . . p. 3 2 Monolayers proved to be stable under ambient conditions (room
temper-ature in air) are shaded blue; those probably stable in air are shaded green; and those unstable in air but that may be stable in inert atmo-sphere are shaded pink. Grey shading indicates 3D compounds that have been successfully exfoliated down to monolayers, as is clear from atomic force microscopy, for example, but for which there is little further
information. Extracted from [3] . . . p. 4 3 Elements forming layered sul des or selenides with the metal in
octahe-dral or trigonal prismatic coordination (niobium and tantalum are found
in both). Adapted from Ref.[7] . . . p. 10 4 Schematics of the structural polytypes of MoS2: 2H (hexagonal
sym-metry, two layers per repeat unit, trigonal prismatic coordination), 3R (rhombohedral symmetry, three layers per repeat unit, trigonal prismatic coordination) and 1T (tetragonal symmetry, one layer per repeat unit, octahedral coordination). The chalcogen atoms (X) are yellow and the metal atoms (M) are grey. The lattice constants a are in the range 3.1 to 3.7 Å for dierent materials. The stacking index c indicates the number of layers in each stacking order, and the interlayer spacing is ∼ 6.5 Å. Adjacent layers in the 2H polytype are rotated by 60°, whereas those in the 3R polytype can be superimposed with a translation only. Extracted
from Ref. [7]. . . p. 11 5 Atomic force microscope (AFM) image of a single layer of MoS2 deposited
on a silicon substrate, obtained with scotch tape-based micromechanical
Monolayer ake of MoS2. (b) Optical microscopy image and (c) atomic
force microscopy (AFM) image. Extracted from [4] . . . p. 12 7 Carrier mobility in monolayer MoS2 as a function of temperature (a) and
carrier density (b), calculated from rst-principles DFT for the electronic band structure, phonon dispersion and electronphonon interactions. In panel a, the grey band shows the uncertainty in calculated mobility values due to a 10% uncertainty in computed deformation potentials associated
with phonons. Adapted from Ref. [78]. . . p. 17 8 Three dimensional schematic structure of MoS2, with the chalcogen atom
S in yellow and the metal atom Mo in grey. . . p. 19 9 Top view of the MoS2 lattice structure. Dark (light) circles represent Mo
(S) atoms. Notice that in this view two S atoms sit on top of each other. The unit cell is shown in the highlighted hexagon. The lattice constant in the Mo plane is a. The two Bravais lattice vectors (R1 and R2) are
indicated. Six other auxiliary vectors that connect a Mo atom with its
nearest S atoms, δ1±, δ2±, and δ3±, are indicated. . . p. 20
10 Top view (left side) and tridimensional view (central) of the rst neigh-bors of a Mo atom involved in the tight binding model. The reference trig-onal prism coordination unit and other useful quantities are also shown. On the right side, Schematic view of the MoS2 unit cell. The bottom and
top S atoms are laterally displaced to facilitate the visualization. The
solid lines indicate nearest-neighbor hopping amplitudes. . . p. 21 11 Brillouin zone for the MoS2 lattice. K1 and K2 are the reciprocal
lat-tice basis vectors, and Γ, K, K0, and M are the high-symmetry points
considered in this study. . . p. 22 12 Band structures calculated using DFT for bulk and MoS2 monolayer.
The horizontal dashed lines indicate the Fermi level. The arrows indicate the fundamental bandgap (direct or indirect). The top of the valence band (blue) and bottom of the conduction band (green) are highlighted.
Extracted from [4, 5] . . . p. 23 13 The DFT-HSE06 band structure of MoS2 near the gap region. . . p. 25
tracted from [42]. . . p. 28 15 Left) 2D pavimentation. Right) 1D pavimentation . . . p. 33 16 Behavior of ptop
z under z symmetry. The ptz goes in −pbz. . . p. 36
17 Scheme of the hopping amplitudes. Each symbol v, u, and t represents a number of amplitudes. The v's involve the Mo d orbitals; the u's involve the S p orbitals; the t's involve the Mo d and S p orbitals. Notice the
direction of the hoppings. . . p. 40 18 Reference DFT-HSE06 band structure calculation with the constraint
points indicated. Blue circles: analytical constraints. Orange circles: numerical constraints. Predominantly even (odd) bands with respect to
z inversion are shown in blue (red). . . p. 52 19 Comparison between the band structures obtained with the DFT-HSE06
(blue) and with the optimized tight-binding model using the parameters
from the CB-VB optimization (red) near the gap region. . . p. 56 20 Comparison between the band structures obtained with the DFT-HSE06
(blue) and with the optimized tight-binding model using the parameters from the VB optimization (red) near the gap region. The VB
optimiza-tion focuses on reproducing accurately the valence band. . . p. 57 21 Comparison between the DFT-HSE06 band structure (blue points) and
the best t to the simpli ed tight-binding model (red points). . . p. 59 22 Comparison between the DFT-HSE06 spin-resolved band structure (blue
points) and the best- t tight-binding model (red points). The spin
split-ting is due to inclusion of spin-orbit coupling. . . p. 63 23 Transverse unit cells of zigzag (a) and armchair (b) MoS2 ribbons. The
arrows indicate the lattice parameters aZ and aA. The red dashed boxes
mark zigzag lines (a) or armchair dimers (b) with its respective value n indicated to the left. Dark (light) circles represent Mo (S) atoms. Note that in the zigzag nanoribbon one edge is S-terminated while the opposite
one is Mo-terminated. . . p. 71 24 Sketch of the primitive unit cell (PUC) of a zigzag MoS2 nanoribbon. . p. 73
(PUC) of a armchair MoS2 nanoribbon. . . p. 77
26 Spin-unpolarized electronic band structure of nanoribbons with dierent edge orientations: (a) Band structure of 90-Z-MoS2-NR for the
even-band model with 540 orbitals, with the inset showing the level crossing that takes place near the Brillouin zone edge. (b) Band structure for the all-band model with 990 orbitals, where we identify zigzag midgap states as the four bands (from 0 through 3). The bottom gures show the band structure of the 90-A-MoS2-NR using the even-band model with
540 orbitals (c) and the all-band model with 990 orbitals (d). The odd parity bands are plotted in red. The band structures in (c) and (d) are
doubly degenerate. The red dashed line represents the Fermi level. . . p. 81 27 Comparison between the spin-unpolarized and the spin-polarized band
structures of zigzag and armchair MoS2 nanoribbons. The top panels
correspond to the band structure of the 10-Z-MoS2-NR with (b) and
without (a) spin resolution within the all band model with 110 orbitals. Analogously, the bottom panels show the band structure of the 10-A-MoS2-NR with (d) and without (c) spin resolution. The blue dashed line
represents the Fermi level. . . p. 84 28 Transmission of pristine ribbons as a function of the Fermi energy. The
solid blue curve is the transmission for the 10-Z-MoS2-NR and the dashed
red curve corresponds to the 10-A-MoS2-NR. . . p. 87
29 LDOS at the energies chosen out of the main gap: (a) E = −0.04 eV, and (e) E = 2.46 eV for the pristine 10-Z-MoS2-NR. The ribbon edge
at y = 0 is S-terminated while the opposite one is Mo-terminated. The image in each plane corresponds to a single atomic nature, S or Mo, as indicated by the labels in each plane. In (a) E = −0.04 bulk bands and bands 2 and 3 contribute with available states, while in (e) E = 2.46 bulk bands and band 0 contribute with available states. In both (a) and
2.05 eV, (d) E = 2.19 eV for the pristine 10-Z-MoS2-NR. The ribbon
edge at y = 0 is S-terminated while the opposite one is Mo-terminated. The image in each plane corresponds to a single atomic nature, S or Mo, as indicated by the labels in each plane. For (b) E = 1.73 only states from band 1 contribute, while for E = 2.19 eV, only states from band 0
contribute. Note for E = 2.05 both band 1 and band 0 contribute. . . . p. 89 31 Band structure of a topological insulator. . . p. 89 32 Electronic spin-resolved dispersion between two boundary Kramers
de-generate points Γa = 0 and Γb = πa. In (a) the number of surface states
crossing the Fermi energy is even, whereas in (b) it is odd. An odd number of crossings leads to topologically protected metallic boundary
states.[125] . . . p. 90 33 Scheme to compute Landauer conductance with RGF method for MoS2. p. 92
34 Electronic transmission in a 10-Z-MoS2-NR as a function of the Fermi
energy for the clean case (black curve) and in the presence of a vacancy at the edge (red curve). The nanoribbon contains M = 100 cells and the vacancy is present in the middle cell that is equally distant to the contacts. The ribbon contains 20 atoms in each cell. The insets indicate the vacancy position. The panels on the left show results for vacancies in the top edge (Mo-terminated edge): (a) one Mo vacancy, (b) two S vacancies, one in bottom plane and one in the top plane, and (c) one S vacancy in the upper or bottom plane. The panels on the right show results for vacancies in the bottom edge (S-terminated edge): (d) one Mo vacancy, (e) two S vacancies, one in bottom plane and one in the top plane, and (f) one S vacancy in the upper or bottom plane. The gray areas mark the perfect transmission regions due to the states at the
opposite edge. . . p. 93 35 Electronic transmission in a 10-Z-MoS2-NR as a function of the Fermi
energy in the presence of random vacancies. The dashed black lines indicate perfect transmission of the pristine ribbon. The nanoribbon contains M = 100 cells with 30 atoms in each cell. The inset indicates the vacancies positions (red balls) distributed in both S (black squares)
MoS2-NR with added random vacancies, as shown in Fig. 35. The ribbon
edge at y = 0 is S-terminated while the opposite one is Mo-terminated. The image in each plane corresponds to a single atomic nature, S or Mo,
as indicated by the labels in each plane. . . p. 95 37 Electronic transmission in a 30-Z-MoS2-NR as a function of the Fermi
energy. We compare the transmissions due to vacancies and the on-site impurity for ranges d = 3 Å, 12 Å, and 26 Å. The nanoribbon contains N = 100cells and the vacancy is present in the middle cell that is equally distant from the contacts. The ribbon contains 20 atoms in each cell. The insets indicate the vacancy and impurity position. In (a) we show the results for defects in the top edge at the S atom, while in (b) the impurity is on the Mo atom. The gray areas mark the energy regions where the states are characterized by a propagation at the edge where the defect is
placed. . . p. 96 38 Schematic diagram showing the optical transitions at the K-point in the
monolayer MoS2. A and B-exciton peak are both signals of the 1st exciton
level. The dierence of their excitation energy is due to the spin-orbit splitting of the valence band (VB). The background peak corresponds to the energy dierence between the conduction band and the VB. Adapted
from [149] . . . p. 98 39 (a) Comparison of the re ectance spectra of two dierent exfoliated MoS2
monolayers. (b) Comparison of the refctance spectra leof exfoliated (red) and CVD-grown (blue) MoS2 monolayers. (c) Real part of the sheet
conductivity (in units of G0 = 2e2/hfor monolayer MoS2. Adapted from
Ref. [148] . . . p. 98 40 MoS2 monolayer sandwiched between two dielectrics. Adapted from [52] p. 106
41 Lowest exciton level (A) as a function of Nks, where Nk = Nks2 is the
number of k points, for dierent regularizations. Top: TB model of
Ref.[53] (Wu et al.), bottom: our TB model [69] (Ridol et al.). . . p. 109 42 Sample states ( rst states) that contribute to the A-B peaks. Left: Wu
et al. TB model [53]. Right: Our TB model [69] (Ridol et al.). The
dol et al. model [69]. Top: one-particle joint density of states, bottom:
one-particle optical conductivity. . . p. 111 44 Linear optical conductivity as a function of k-sampling mesh size Nks.
Left: Wu et al. model [53]. Right: Ridol et al. model [69]. . . p. 112 45 Sample states that contributes to the C peak. Left: Wu et al. model
[53]. Right: Ridol et al. model [69]. The vertices of the rhombus are
the Γ points. . . p. 113 46 Linear optical conductivity by varying the number of conduction bands.
Left: Wu et al. model [53], right: Ridol et al. [69]. . . p. 114 47 Linear optical conductivity for an all-band model (Nc = 8, Nv = 2,
Nks = 60) and an even-band model (Nc = 4, Nv = 2, Nks = 60). Left:
Wu et al. [53]. Right: Ridol et al. [69] . . . p. 115 48 Linear optical conductivity for the even-band model (Nc= 4, 2, Nv = 2,
Nks = 60). Left: Wu et al. model and right: Ridol et al. model. . . . p. 116
49 Linear optical conductivity of a MoS2 monolayer calculated with and
without particle-hole interactions and compared with the experimental results from Ref. [148] (all are presented in the same units). Linear optical conductivity and one-particle optical conductivity for both Wu
et al. model and Ridol et al. model (Nc= 8, Nv = 2, Nks = 60). . . . p. 117
50 Rotations . . . p. 135 51 First: Next-to-nearest neighbor hoppings between Mo atoms.
Second:Next-to-nearest neighbor hoppings between S atoms. . . p. 139 52 Left: scheme of a real system. Each slice of the two leads is described by
H. Right: each lead can be represented by one single slice: the left lead
described by H0, the right lead described by HN +1 . . . p. 167
Contents
1 Introduction p. 1
1.1 Motivation and purpose . . . p. 1 1.2 Techniques, results and structure of the thesis . . . p. 4
2 Theoretical background p. 8
2.1 Metal dichalcogenides . . . p. 8 2.2 Molybdenum disul de MoS2 (molybdenite) . . . p. 12
2.2.1 Applications: optoelectronic . . . p. 13 2.2.2 Electronic transport and devices . . . p. 13 2.2.3 Transport and scattering mechanism background . . . p. 15
3 Review of the Electronic Structure of MoS2 p. 19
3.1 Crystal structure . . . p. 19 3.1.1 Lattice parameters . . . p. 21 3.1.2 Brillouin zone . . . p. 22 3.2 Electronic band structure . . . p. 22 3.2.1 Eective masses . . . p. 26 3.2.2 Orbital composition . . . p. 27 3.3 Tight-binding method . . . p. 29 3.3.1 Standard formulation [70] . . . p. 30 3.3.2 Optimized notation for MoS2 bulk . . . p. 31
3.4.1 All orbital model . . . p. 35 3.4.2 Three-orbitals model . . . p. 35 3.4.3 7x2 orbitals model . . . p. 36 3.4.4 11x2 orbital model: our choice . . . p. 36
4 Tight-binding model for bulk monolayer of molybdenum disul de p. 38
4.1 Model fundamentals . . . p. 38 4.2 Overlap amplitudes in terms of SK integrals . . . p. 41 4.3 Energy bands . . . p. 44 4.3.1 Unsymmetrized matrix formulation . . . p. 45 4.3.2 Symmetrized matrix formulation . . . p. 46 4.4 Expansion around symmetry points . . . p. 47 4.4.1 Γ point . . . p. 47 4.4.2 K point . . . p. 49 4.5 Optimization of the parameters . . . p. 51 4.5.1 K point . . . p. 53 4.5.2 Γ point . . . p. 53 4.6 Eleven band model . . . p. 54 4.7 Simpli ed model . . . p. 58 4.8 Spin-orbit interaction . . . p. 59 4.9 Comparison with the tight-binding model of Cappelluti et al. . . p. 64
5 Electronic structure of MoS2 nanostructures p. 67
5.1 Introduction . . . p. 67 5.2 Band structure of MoS2 pristine nanoribbons . . . p. 71
5.2.3 Results . . . p. 80
6 Transport in pristine and disordered nanoribbons p. 85
6.1 Experimental motivation . . . p. 85 6.2 Pristine nanoribbon: zigzag and armchair cases . . . p. 86 6.3 Topological aspects of MoS2 . . . p. 88
6.4 Zigzag nanoribbons with defects . . . p. 91 6.4.1 Short-ranged scattering . . . p. 92 6.4.2 Study of dierent sources of defects: long ranged scattering . . . p. 95
7 Optical properties of the MoS2 monolayer p. 97
7.1 State of the art . . . p. 98 7.2 Theory and methods . . . p. 101 7.2.1 Excitons: the Bethe-Salpeter equation . . . p. 101 7.2.2 Eective screened potential . . . p. 106 7.2.3 Optical response . . . p. 106 7.3 Implementation and Results . . . p. 108 7.3.1 Convergence of the Bethe-Salpeter spectrum . . . p. 108 7.3.2 One-particle optical conductivity . . . p. 110 7.3.3 Linear optical conductivity . . . p. 112 7.4 Discussion and further analysis . . . p. 116
8 Conclusions p. 120
A Simpli cations and reduction of parameter number p. 124
A.1 Hopping amplitudes . . . p. 124 A.2 Slater-Koster parameters . . . p. 129
A.4 Reduction of hopping amplitudes parameters by symmetry considerations p. 133 A.4.1 Hopping amplitudes . . . p. 134 A.4.2 On-site energies . . . p. 146
B Explicit tight-binding derivations p. 147
B.1 Tight binding energy bands . . . p. 147 B.2 Symmetrized equations . . . p. 151 B.2.1 First step . . . p. 151 B.2.2 Second step . . . p. 155 B.3 Dispersion around the K and M points . . . p. 157 B.3.1 K point . . . p. 157 B.3.2 M point . . . p. 164
C The recursive Green function method for transport p. 166
C.1 Elements of linear mesoscopic transport . . . p. 166 C.2 Contact Green's function . . . p. 167 C.2.1 Self-consistency relation for the leads Green's functions . . . p. 169 C.3 Full Green's function . . . p. 170 C.3.1 Left Green Function . . . p. 170 C.3.2 Right Green Function . . . p. 172 C.3.3 Connecting right and left Green's functions . . . p. 173 C.4 Summary of the basic expressions for the transport recursive calculations p. 175 C.5 Why to use RGF method . . . p. 176
D State of the art for the optical conductivity of MoS2 p. 179
1 Introduction
1.1 Motivation and purpose
The synthesis of graphene in 2004 [1, 2], the rst single-atom thick material, has boosted the research in atomically thin two-dimensional (2D) materials raising the interest in their fundamental properties and potential applications. 2D materials are very unique because they lead to the appearance of phenomena that are very dierent from those of their bulk material counterparts. They are inherently exible, strong, and extremely thin. Additionally since they are entirely made up of their surface, the interface between the surface and the substrate and the presence of adatoms and defects can dramatically alter their material's inherent properties. The development of 2D materials is expected to improve current device technology, and to provide additional opportunities for spintronic devices and quantum computing[3]. The past years of graphene research has yielded many methods for synthesizing, transferring, detecting, characterizing, and manipulating the properties of 2D materials [22, 3]. Research on graphene and other two-dimensional atomic crystals is intense and is likely to remain one of the leading topics in condensed matter physics and materials science for many years [22, 3]. The huge electronic mobility of graphene at room temperature has motivated many proposals of graphene-based electronic devices [9]. However, a drawback in engineering graphene-based electronic device is the absence of a gap in the monolayer samples, and the di culty in opening a gap in multilayer systems without aecting the mobility. As an alternative route, isolated atomic planes can also be reassembled into design heterostructures made layer by layer in a precisely chosen sequence using dierent materials [3]. For instance devices made by stacking dierent 2D crystals on top of each other like LEGO panels, see Fig. 1. The rst such heterostructures (often referred to as `van der Waals') have recently been fabricated and investigated, revealing unusual properties and new phenomena. The ability to manipulate isolated single atomic layers and reassemble them to form heterostructures layer-by-layer in a precise sequence, opens enormous possibilities for applications [3, 4, 6, 12].
As a consequence, researchers have now started paying more attention to other two-dimensional (2D) atomic crystals such as isolated monolayers and few-layer crystals. Un-fortunately, many 2D crystals that are stable in theory are unlikely to survive in reality because they would corrode, decompose, segregate and suer contamination and oxi-dization. A known exceptions are, for instance, semiconducting dichalcogenides that are promising compounds since they can be easily exfoliated and present a suitable small gap both in bulk and as a single layer. Moreover, layers of hexagonal boron nitride (hBN), molybdenum disulphide (MoS2), and other dichalcogenides have been proved to be stable
under ambient conditions [3]; see Fig. 2.
In this category of 2D dichalcogenides systems, monolayers (ML) of molybdenum disul de (MoS2) have recently gained attention for combining an electron mobility
compa-rable to that of graphene devices with a nite energy gap [13]. Unlike its bulk form, which is an indirect gap semiconductor, monolayer MoS2 has a direct gap [3, 14]. Monolayer
molybdenum disulphide (ML-MoS2), a prototypical transition layered metal
dichalco-genide, shows a transition from an indirect band gap of 1.3 eV in a bulk structure to a direct band gap of 1.8 eV in the monolayer structure. The direct band gap has the maxi-mum of the valence and minimaxi-mum of the conduction band (VBMAX/CBMIN) located at the K point of the Brillouin zone (BZ). The direct band gap is in the visible frequency range, most favorable for optoelectronic applications [14].
Further, the materials with electrically controllable optical properties nd uses in diverse applications ranging from electroptical modulators to display screens. Unfortu-nately, the optical constants of most bulk semiconducting materials do not vary signi -cantly with electric eld. Very recently, two- dimensional (2D) atomic crystals emerged as a potential alternative to bulk semiconductors to modulate optical properties for photonic applications[10, 26]. In graphene, the most widely studied 2D material, changes in optical absorption larger than 100% produced by the electric eld eect have been used to demon-strate nanoscale electro-optical modulators in the infrared range [11]. However, the lack of a band gap in graphene makes its practical application at visible frequencies infeasible. It has been recently demonstrated electrical control of photoluminescence quantum yield and absorption coe cient in the visible range for MoS2, because of the interaction of the
excitons with charge carriers [27]. Another important aspect of 2D semiconducting ML-dichalcogenides such as MoS2 is the important role of many-body electronic interactions.
In particular the optical properties are dominated by excitonic transitions [49].
Figure 1: Building van der Waals het-erostructures. If one considers 2D crys-tals to be analogous to Lego blocks (right panel), the construction of a huge variety of layered structures be-comes possible. Extracted from [3] Because of absence of interlayer phonon scattering, the ideal thermal conductivity, is expected to be higher than the values of the bulk stacks. Another interesting feature is that the electronic properties appear to be highly sensitive to external pressure [16], strain [17, 18, 19], and temperature [20], which aect the gap and, under certain conditions, can also induce an insulator/metal transition.
In addition, the lack of lattice inversion symmetry together with spin-orbit coupling (SOC) leads to coupled spin and valley physics in monolayers of MoS2 and other group-VI
dichalcogenides [26, 28], making it possible to control spin and valley in these materials [6, 29].
The electronic structure of ML-MoS2 exhibits a valley degree of freedom indicating
that the valence and conduction bands consist of two degenerate valleys (K, K0
) located at the corners of hexagonal Brillouin zone. The strong spin-orbit coupling, the coupling between the spin and valley degrees of freedom, and their eect on the exciton photolumi-nescense have sparked strong interest [29]. Due to their peculiar band structure, a variety of nanoelectronics applications [4, 6] including valleytronics, spintronics, optoelectronics, and room temperature transistor devices [13] have been suggested for monolayers of MoS2
[4, 13].
In light of the growing interest in this material, an accurate and yet reasonably simple model describing the band structure and electronic properties of bulk MoS2 is highly
desirable. In this thesis we put forward a model of such kind.
The great interest in the electronic and mechanical properties of graphene and other 2D materials, has also triggered an intense investigation and production eorts of nanowires and nanoribbons with sub-nanometer width. As in graphene, the presence of edges dra-matically modi es the low energy spectrum of TMDs. Thus it is straightforward the necessity to extend a simple model to study the electronic properties of pristine
nanorib-Figure 2: Monolayers proved to be stable under ambient conditions (room temperature in air) are shaded blue; those probably stable in air are shaded green; and those unstable in air but that may be stable in inert atmosphere are shaded pink. Grey shading indicates 3D compounds that have been successfully exfoliated down to monolayers, as is clear from atomic force microscopy, for example, but for which there is little further information. Extracted from [3]
bons of MoS2. This thesis proposes an accurate simple model for the band structure of
MoS2 nanoribbons near the Fermi energy at the charge neutrality point.
The poor mobility of the nowadays state of the art of TMD samples restrict their applications in electronic devices. This issue a subject of intense theoretical and experi-mental debate [50, 21, 20, 22, 23, 24, 25] and a challenge for future practical uses of these materials in devices. This thesis contributes to the understanding of the low conductance by analyzing transport properties of disordered nanoribbons.
Another key quantity to study the behavior of the material is the linear optical con-ductivity. In this thesis, this optical property will be explored on the basis of the band structure model we put forward including the eects of two-body interactions such as excitons. Our work gives an accurature overview of both theoretical and experimental ndings, and even cast new light on the nature of some features of the optical properties.
1.2 Techniques, results and structure of the thesis
While much of the theoretical work of graphenic materials has been based on tight-binding-like approaches, so far the electronic properties of single-layer, few-layer and nanoribbons dichalcogenides have been mainly investigated by means of ab initio cal-culations, based on Density Functional Theory (DFT) [29, 17]. Such methods provide valuable information about electronic properties of pristine dichalcogenide crystals, but
are computationally prohibitive to treat disordered systems with a large number of atoms. To address the latter, one needs to resort to a simple eective model, such as the k · p Hamiltonian (see, for instance, Ref. [30] for a review the k · p theory for two-dimensional transition metal dichalcogenides) or the tight-binding approximation. In this thesis we choose the latter route, which provides a more accurate description for the entire band structure than the k · p method. From the computational point of view, tight-binding models are highly scalable and therefore suited to address large structures, both ordered and disordered. Hence, the tight-binding model applied to a single-layer MoS2 as well as
to similar transition metal dichalcogenides, constitutes a key tool for further studies of the low-energy electronic transport properties of these materials, such as the description of the conductivity in diusive samples, as well as for evaluating the conductance of ballistic samples as a function of carrier concentration. Moreover the tight-binding model applied for single-layer MoS2 as well as for similar transition metal dichalcogenides.
We emphasize that it is not our goal to achieve the best quantitative agreement between the numerical DFT (Density functional theory) calculations and analytical cal-culations. The main goal of our method is to describe and investigate the low-energy band structure which plays a central role for transport, and in MoS2 for the
optoelec-tronic properties as well. We develop a qualitative and intuitive understanding of the relative importance of competing eects and of the physical mechanisms that give rise to the eects, which, on the other hand, can be quanti ed by means of accurate numerical calculations.
In recent years, a variety of tight-binding models have been proposed for MoS2
mono-layers [31, 32, 33]. Unfortunately, they are neither practical nor su ciently accurate for transport calculations. For that purpose, one needs a tight-binding model with a man-ageable number of parameters and interactions that accurately reproduces the ab initio electronic properties of the conduction band (CB) near its minimum energy and the va-lence band (VB) near its maximal energy.
Another model presented in Refs. [41, 42] yields to two bands that look very similar to the conduction and the valence bands obtained by standard DFT calculations. However, we observe that the tight-binding orbital compositions of Refs. [41, 42] have actually no relation with those calculated using DFT. Hence, a new tight-binding parametrization, reproducing both energies and orbital composition is badly needed. This is the main goal of the rst part of the thesis.
the electronic properties of monolayer MoS2 nanoribbons. Our calculations reproduce
qualitatively the band structure of narrow nanoribbons obtained by DFT. We show the necessity of considering the full Hamiltonian, with even and odd parities with respect to z-axis re ection, for an accurate description of nanoribbon electronic states within the bulk gap energy window. By doing so, we observe the appearance of an odd-parity band close to the Fermi level for both kinds of edge terminations. In the zigzag case, we analyze the edge nature of states inside the bulk gap. Interestingly, we nd that the metallic bands correspond to states localized at a single edge independent of nanoribbon width and disorder.
We also study the conductance and local density of states (LDOS) of both pristine and disordered MoS2zigzag and pristine armchair nanoribbons using the recursive Green's
function (RGF) method. We focus our attention on the eect of short and long-range disorder on the conductance of zigzag nanoribbons. The results are interpreted in terms of topological invariants and their robustness protection against disorder. We nd that even a modest concentration of vacancies close to the edges can cause a large transmission suppression, particularly within the bulk gap energy window. In contrast, long-range scattering does not have a signi cant eect on the conductance.
Finally, we address the optical properties of MoS2 monolayers. Recent experimental
results have studied the absorption, re ectance and optical conductivity spectra of few layers and a single layer of MoS2. Few theoretical calculations exist with which to interpret
these ndings [52, 154, 156, 53]. In particular, it is of interest to understand the intensity and the nature of the excitonic absorbption peaks. Here, we present calculations of the linear optical conductivity based on the tight-binding band structure put forward. We implement the calculations in a Bethe-Salpeter framework that includes excitons features. We compare directly our results with recent experimental ndings demonstrating a good agreement with the excitonic theory. Furthermore, we analyse and discuss the nature of high-energy peaks in the optical conductivy spectra, comparing our result with another very recent proposed tight-binding in the literature.
The thesis is structured as follow:
Chapter 2 contains an introductory presentation of the transition metal dichalco-genides (TMDs) properties. In provides all necessary elements to develop a model to study transport and opto-electronic properties.
Chapter 3 describes the crystal and electronic structure of a MoS2 monolayer. We
band structure calculation, done in collaboration with Duy Lee and Talat Raman (Uni-versity of Central Florida), that serves as a reference for our tight-binding model. We also recall the tight binding method formulation and we review the tight-binding models for dichalcogenides found so far in the literature.
Chapter 4 presents the main model. We analyze the band equations for a few high-symmetry k-point, allowing us to obtain simple analytical expressions for the bands. These are used to nd the best set of tight-binding parameters that t the DFT band structure. We nd the best parameter set and we present the corresponding band structure for the full case and in the case we are focused in the valence band. We also consider a simpli ed model with a reduced number of parameters and we add spin-orbit interaction. We also present a comparison between our 11-band tight-binding formulation and that of Refs. [41, 42].
Chapter 5 presents the theoretical model and the band structure calculations for pristine MoS2 nanoribbons with zigzag and armchair edges.
Chapter 6 discusses the electronic transport in clean and disordered nanoribbons with Fermi energy in the vicinity of the bulk gap. First, we analyze the conductance of pristine zigzag and armchair nanoribbons, supplemented by a discussion of the LDOS of the zigzag case. Next, we study the eect of both short- and long-ranged disorder in the transmission and the LDOS of zigzag nanoribbons.
In Chapter 7 we investigate the optical properties of MoS2 in the light of recent
exper-imental ndings. The calculations of this chapter were performed at the HPC facilities of the NUS Centre for Advanced 2D Materials, under the supervision of Professor V. Pereira. Finally we draw our conclusions in Chapter 8. The main text is supplemented by a number of appendices containing technical aspects of the calculations. In Appendix A we present in detail all the elements required to construct the tight-binding band equations. In Appendix B we show how to implement the band equations in the unsymmetrized and symmetrized ones. We make the analysis of the band structure at symmetry points useful in the optimization process. Appendix C explains into detail the equations of the Recursive Green Function method used for the transport calculations. Finally Appendix D includes a review of the state of art of ML-MoS2 optical conductivity and excitons.
2 Theoretical background
2.1 Metal dichalcogenides
Many two-dimensional (2D) materials exist in bulk form as stacks of strongly bonded layers with weak interlayer attraction, allowing exfoliation into individual, atomically thin layers. These crystal structures feature single-atom-thick or polyhedral-thick layers of atoms that are covalently or ionically connected with their neighbors within each layer, whereas the layers are held together via van der Waals bonding along the third axis [6].
The material receiving the most attention today is graphene, the monolayer counter-part of graphite. The electronic band structure of graphene has a linear dispersion near the K point, and its charge carriers can be described as massless Dirac fermions, pro-viding scientists with an abundance of new physics [111]. Graphene is a unique example of an extremely thin electrical and thermal conductor, with high carrier mobility, and surprising properties.
Many other 2D materials are known, such as the transition metal dichalcogenides TMDs, transition metal oxides and graphene analogues such as boron nitride (BN). In particular, TMDs show a wide range of electronic, optical, mechanical, chemical and thermal properties that have been studied by researchers for decades [6, 8]. There is at present a resurgence of scienti c and engineering interest in the atomically thin 2D forms of TMDs because of recent advances in sample preparation, optical detection, transfer and manipulation of 2D materials, and physical understanding of 2D materials learned from graphene. The 2D exfoliated versions of TMDs oer properties that are complemen-tary or even distinct from those in graphene. Graphene displays an exceptionally high carrier mobility exceeding 106cm2V−1s−1 at 2K and exceeding 105cm2V−1s−1 at room
temperature for devices encapsulated in BN dielectric layers. However, because pristine graphene lacks a bandgap, eld-eect transistors (FETs) made from graphene cannot be eectively switched o and have low on/o switching ratios. Bandgaps can be engineered in graphene using a variety of methods, but these methods add complexity and diminish
mobility. In contrast, several 2D TMDs possess sizable bandgaps around 12 eV [7, 4, 6]. In few words, transition metal dichalcogenides (TMDs) are layered materials with strong in-plane bonding and weak out-of-plane interactions enabling exfoliation into two-dimensional layers of single unit cell thickness. TMDs such as MoS2, MoSe2, WS2 and
WSe2 have sizable bandgaps that change from indirect to direct in single layers, allowing
applications such as transistors, photodetectors and electroluminescent devices.
Let us give a more precise de nition of TMDs. It is common that the three heaviest elements of the sulfur sub-group, namely selenium Se, tellurium Te, and polonium Po, be collectively referred to as the chalcogens, but the term chalcogen, according to the o cial guides to inorganic nomenclature, applies equally to all the elements in Group 16 of the Periodic Table, thus being proper also for oxygen O and sulfur S. It becomes internationally accepted that the elements oxygen, sulfur, selenium, and tellurium will be called chalcogens and their compounds chalcogenides. The term derives from the Greek terms χαλκoζ meaning 'copper' and γννω meaning 'giving birth', and it was meant in the sense of ore 1-forming element. In particular the inorganic compounds of sulfur,
selenium, and tellurium with metals and semimetals, which collectively may be termed as metal chalcogenide (MCh) systems [7].
Metal chalcogenides have played a major role in the eld of low-dimensional solids. Metal clustering and low-dimensional structures are frequently found among TMDs. These compounds tend to form covalent structures, so that the reduced relative charge on the metal favors metalmetal bonding. In layered chalcogenides, which have enough d-electrons for signi cant MM bonding in two dimensions, the dimensionality of MM interactions is increased to two [7].
Most transition elements M from group IV (Ti, Zr..), group V (V, Nb..) or group VI (Mo, W...) react with chalcogen atoms to give dichalcogenides MX2 with a precise
1:2 stoichiometry, crystallizing in either 2D or 3D structures, as originating from the competition between cationic d levels and anionic sp levels. The 2D layered structures, which can be formulated as M4+(X2−)
2, consist of sandwiched sheets of the XMX form,
separated by a van der Waals gap between the X layers of adjacent sheets. Inside the sheets, the coordination of the metal ions is sixfold, either octahedral or a body-centered trigonal prism (see Fig. 3).
Molybdenum and tungsten dichalcogenides MX2 (M = Mo, W) occur in the
(hexago-nal) 2H stacking polytype, while MoS2, MoSe2, WS2 occur in addition also in the
Figure 3: Elements forming layered sul des or selenides with the metal in octahedral or trigonal prismatic coordination (niobium and tantalum are found in both). Adapted from Ref.[7]
bohedral) 3R polytype. Also, tetragonal 1T-MoS2 has been observed for single layers
exfoliated from 2H- or 3R-MoS2 crystals. The bonding in X-M-X is predominantly
cova-lent. The XMX layers are stacked in van der Waals contact with one another along the c-axis of the material (see Fig. 4).
The MoX2 (X = S, Se, Te) and WX2 (X = S, Se) are all semiconductors with band
gaps well matched to the solar spectrum and inherent corrosion resistance, since their energy gaps are derived from non-bonding molecular orbitals. These properties suggest applications as solid-state and photoelectrochemical solar cells [8].
Atomically thin, 2D TMDs of high quality properties are essential. There is a variety of available reliable production methods that vary from top-down exfoliation from bulk and bottom-up synthesis[4, 8].
The methods of top-down synthesis are [4]: a) adhesive tape, see Fig.5, (produces single crystal akes of high purity but does not allow control of ake thikness and size), b) thermal ablation with laser (challenging for scale-up), c) liquid-phase preparations, it uses MoS2 suspensions, (very promising), d)chemical exfolation: intercalation of TMDs
by ionic species allows the layers to be exfoliated in liquid, ex. lithium-based intercalation, (but can change the electronic structure), e)exfoliation by ultrasonication in appropriate liquids.
Alternatively, the typical bottom-up methods are [4]: a) chemical vapour deposition (CVD) on metal substrates (use dierent solid precursors heated at high temperature). It has been shown that the electrodeposition of molybdenum chalcogenides from high-temperature molten salts can give large, well-de ned crystals of these compounds. b) Chemical bath deposition (CBD)2
2Chemical bath deposition (CBD) can be regarded as an analogue to chemical vapor deposition (CVD),
since they rely on the same principle to utilize chemical reaction of the precursor molecules or ions, except that CBD growth takes place in low-temperature (soft-processing) conditions in liquid solution.
Mechanical exfoliation remains the most powerful approach for studying layer prop-erties, since it is less destructive than the other methods and has succesfully been used to create large, 10µm single- akes on a variety of substrates [4, 8].
Figure 4: Schematics of the structural polytypes of MoS2: 2H (hexagonal symmetry, two
layers per repeat unit, trigonal prismatic coordination), 3R (rhombohedral symmetry, three layers per repeat unit, trigonal prismatic coordination) and 1T (tetragonal symme-try, one layer per repeat unit, octahedral coordination). The chalcogen atoms (X) are yellow and the metal atoms (M) are grey. The lattice constants a are in the range 3.1 to 3.7 Å for dierent materials. The stacking index c indicates the number of layers in each stacking order, and the interlayer spacing is ∼ 6.5 Å. Adjacent layers in the 2H polytype are rotated by 60°, whereas those in the 3R polytype can be superimposed with a translation only. Extracted from Ref. [7].
Characterization of 2D materials is inherently challenging due to small sample size, the necessity of minimizing substrate signals and destructive eects. However, since graphene, the number of methods that have been developed to enable their identi cation and char-acterization, has been enhanced [4, 8]:
a) Optical microscopy is initially the most powerful method for identifying single and multi-layer akes. b) Fluorescence microscopy: same methal dichalcogenides have direct gaps only as single layers, enabling their direct visualization increasing uorescence signal. c) Atomic Force Microscope AFM is a powerful technique to determine layer thickness
Figure 5: Atomic force microscope
(AFM) image of a single layer of MoS2
deposited on a silicon substrate, obtained with scotch tape-based micromechanical cleavage. Extracted from Ref[13]
with a precision of 5%. d) Raman Spectroscopy is a useful method for ngerprinting a material and layer-dependent changes of the vibrational structure. Raman spectra can also detect vibrational modes that are active due to symmetry breaking in single-layer lms and enhanced vibrations in out-of-plane modes when the single layer is suspended. e)Transmission Electron Microscopy (TEM) and Scanning Tunneling Microscope (STM) can provide detailed topographic information, layer sizes, interlayer stacking and elemen-tal composition. Moreover STM can also manipulate atoms at speci c points. 3 f) X-ray
Diraction (XRD), small-angle X-ray scattering (SAXS) and wide-angle XRD, can supply information about the unit cell structure, and dimensions.
2.2 Molybdenum disul de MoS
2(molybdenite)
Mo: [Kr]5s14d5 group 6 (or group VI B) in the periodic table. Transition metal.
S: [Ne]3s23p4group 16 (or group VI A) in the period table. Chalcogens.
a) b) c)
Figure 6: (a)Photograph of a bulk MoS2 crystal (approximately 1 cm in size). Monolayer
ake of MoS2. (b) Optical microscopy image and (c) atomic force microscopy (AFM)
image. Extracted from [4]
Among 2D-TMDs, Molybdenum disul de, MoS2, has attracted considerable interest
in connection with its electronic and optoelectronic properties [14, 13, 17].
In what follows we discuss the main aspects of the current research on this material and their relation to some of the speci c motivations of this thesis.
3Recent focus has shifted toward studies of adsorbates and defects, including metal adatoms and
2.2.1 Applications: optoelectronic
Because single-layer TMDs have primarily direct semiconducting bandgaps, they are of great interest for applications in optoelectronics, and because they are atomically thin and processable, they have great potential for exible and transparent optoelectronics [4]. Optoelectronic devices are electronic devices that can generate, detect, interact with or control light. Nanomaterials such as carbon nanotubes, semiconductor quantum dots and nanowires have been much studied for use in optoelectronic applications such as:
lasers, LEDs, optical switches, photodetectors, displays. The realization of exible and transparent optoelectronics in applications such as displays and wearable elec-tronics will require a wide variety of transparent and exible components such as conductors, semiconductors, optical absorbers, light emitters and dielectrics. These diverse functionalities will require integrating dierent classes of 2D materials with dierent properties. For transparent conductors, graphene is a promising exi-ble and earth-abundant candidate. For semiconducting components with tuna exi-ble bandgaps, 2D TMDs are a promising choice. For 2D dielectrics, materials such as layered perovskites and BN are promising.
solar cells. The relatively high Earth abundance of TMDs and their direct bandgaps in the visible range make them attractive as the light-absorbing material in alterna-tive thin- lm solar cells, including exible photovoltaics that could coat buildings and curved structures. Moreover the band gap may be tuned with various inter-calants or using layers of dierent thickness.
2.2.2 Electronic transport and devices
Electronic and spintronic devices use electronic charge and spin, respectively, to carry signals. The valley index is another property of charge carriers that can be exploited, and it refers to the con nement of electrons or holes in distinct conduction-band minima or valence-band maxima at the same energies but dierent positions in momentum space, leading to potential `valleytronic' devices. Materials that have strong spin splitting, which can be due to various eects that push the system out of equilibrium or due to symmetry breaking, allow spin-polarized carrier populations to be maintained, and are needed for spintronic devices.
In the group-IV semiconducting dichalcogenides (MoS2, MoSe2, WS2 and WSe2), a
unique set of conditions gives rise to both strong spinorbit-induced electronic band split-ting and spinvalley coupling. Monolayer TMDs such as MoS2 lack inversion symmetry,
unlike graphene or bilayer MoS2, which are centrosymmetric. The lack of inversion
sym-metry, con nement of electron motion in plane and high mass of Mo lead to a very strong spinorbit splitting, with the valence-band around 0.15 eV. This is in contrast to graphene, which has very weak spinorbit interaction primarily owing to the low mass of carbon. The lack of inversion symmetry along with strong spinorbit coupling also leads to the coupling of spin and valley physics [4].
One of the most important applications of semiconductors are transistors in digital electronics. For digital logic transistors, desirable properties are high charge-carrier mo-bilities for fast operation, a high on/o ratio (that is, the ratio of on-state to o-state conductance) for eective switching, and high conductivity (that is, the product of charge density and mobility) and low o-state conductance for low power consumption during operation. In most semiconductors, doping can be used to increase the charge density, but can also lead to decreased mobility owing to scattering. For digital logic, on/o ratios of 104 − 107 are generally required for use as switches. Much interest in graphene has
centered on electronic device applications because it is two-dimensional, it has exception-ally high carrier mobilities, and an external gate voltage can readily modulate its current ow. But the lack of bandgap in graphene means that it cannot achieve a low o-state current, limiting its use as a digital logic transistor. A bandgap is essential for transistors: it oers the possibility for the realization of an interband tunnel FET4. A conventional
FET consists of a semiconducting channel that is connected to source and drain electrodes and can be switched on and o by the application of a eld via a gate electrode that is separated by a dielectric material. Engineering a graphene bandgap is possible, but in-crease fabrication complexity and either reduces mobilities ( 200cm2V−1s−1 for 150meV
bandgap) or required high voltages (for 250meV requires a voltage exceeding 100V). Thus researchers are now turning to TMDs as ultrathin materials with tunable bandgaps that can be made into FETs with high on/o ratios [4]. In the basic FET structure, which has been adapted to 2D TMDs, a semiconducting channel region is connected to the source and drain electrodes, and separated by a dielectric layer from a gate electrode. The cur-rent owing between the source and drain electrodes is controlled by the gate electrode modulating the conductivity of the channel.
Moreover, excellent mechanical exibility of MoS2 also makes it a compelling
semi-conducting material for exible electronics [122]. Mechanical measurements performed on single-layer MoS2show that it is 30 times as strong as steel and can be deformed up to 11%
before breaking. Most existing studies and device demonstrations were performed on ex-4Field eect transistor
foliated MoS2 akes. In particular, eld-eect-transistors based on monolayer MoS2 were
found to exhibit high on/o ratios of ∼ 108, steep subthreshold swing of ∼ 70mV/dec,
with reported electron eective mobility ranging from 1 to 480cm2/Vs depending on the
device structures, dielectric environment and processing [122].
The rst implementation of a top-gated transistor based on monolayer MoS2 was
reported by Kis and co-workers [13]. This device showed excellent on/o current ratio, n-type conduction and room-temperature mobility of ≥ 200cm2V−1s−1. Although MoS
2 will
not compete with conventional IIIV transistors on mobility values alone, its attractive electrical performance characteristics, relatively high earth abundance and high degree of electrostatic control could make MoS2 a viable candidate for low-power electronics.
In Ref. [80] we see a speci c interest in tunnel eld-eect transistor in 2D transition metal dichalcogenide materials. They show that reducing the channel thickness reduced the natural scaling length of the potential which results in higher electric eld and ON-currents. Another advantage of 2D materials is that thinning the material does not increase the band gap as much as it does in 3D materials (larger band gap results in a lower ON-current). Moreover mono-layer usually have small dielectric constants, which can also increase the ON-current.
In 2013 Hone, and colleagues demonstrated that they could dramatically improve the performance of graphene by encapsulating it in boron nitride (BN), an insulating material with a similar layered structure [138]. Recently they have also shown that the performance of molybdenum disul de can be similarly improved by BN-encapsulation. Combination of BN and graphene electrodes is like a 'socket' into which we can place many other materials and study them in an extremely clean environment to understand their true properties and potential. They found that the room-temperature mobility was improved by a factor of about 2, approaching the intrinsic limit. Upon cooling to low temperature, the mobility increased dramatically, reaching values 5-50Ö that those measured previously (depending on the number of atomic layers) [83].
2.2.3 Transport and scattering mechanism background
The low mobility and the main sources of disorder in MoS2 are topics of intense
current investigation. The carriers are con ned to the plane. The degree to which scat-tering mechanisms aect the carrier mobility is also in uenced by layer thickness, carrier density, temperature, eective carrier mass, electronic band structure and phonon band structure. Many of these scattering mechanisms are also seen in other semiconductors
and in graphene. We identify the main scattering mechanisms:
(i) short range scattering (vacancies and impurities or other structural defects). Short-range scattering plays a dominant role in the highly conducting regime at low temperature[22, 21, 24, 122]. Experiments on CVD samples reported values of the electron mobility smaller than the highest measured mobility in exfoliated samples, possibly due to a large amount of localized states[122, 24]. TMDs can contain a number of dierent structural defects in their crystal lattices which signi cantly al-ter their physico-chemical properties. One of the main cause that aects mobility is the presence of atomistic vacancies. Single and double sulfur (top and bottom) vacancies have the lowest energy formation and are ubiquitous in MoS2 [132, 50].
(ii) Acoustic and optical phonon scattering: Ref. [79] reports the temperature dependence of the mobility for T > 200 K indicating acoustic phonon scattering as the dominant mobility-limiting mechanism. The room temperature mobility of 410cm2V−1s−1,
calculated from rst principles studies for temperatures T >100 K, is found to be mainly limited by optical phonon scattering [78]. Carrier mobility is increasingly aected by phonon scattering with increasing temperature. Figure (7-a) shows the temperature dependence of electron mobility in single-layer MoS2 calculated from
rst principles. The mobility due to acoustic phonon scattering alone is shown along with the total eect of acoustic and optical phonon scattering. At low temperatures (T < 100 K), the acoustic component dominates, but at higher temperatures the optical component dominates.
(iii) Coulomb scattering at charged impurities. Scattering of charge carriers in electrical transport are the central performance limiting factors in electronic devices[25, 21, 24]. Coulomb scattering in 2D TMDs is caused by random charged impurities lo-cated within the 2D TMDC layer or on its surfaces, and is the dominant scattering eect at low temperatures. This Coulomb eect also generally limits the mobility of single-layer graphene to values around ∼ 10000cm2V−1s−1when it is placed on
mildly dielectric materials such as SiO2 [77]. The carrier concentration and bandgap
can be tuned by adding ionic impurities, but the mobility is also decreased through scattering, so the choice of doping level in a particular device can strongly in u-ence its performance. The eect of carrier concentration and temperature on MoS2
mobility is shown in Fig. (7-b). By considering the mean free path of carriers, the impurity concentration required to make impurity scattering dominant over phonon scattering is calculated to be ∼ 5 · 1011cm−2, which corresponds to heavy doping.
Ref.[78]
disor-der). The eect of surface phonon scattering and roughness scattering can be very important in extremely thin 2D materials.
In general for nearly all materials, we can follow a typical behavior of the mobility in function of the temperature, and we can identify two dierent regions.
At very low temperature, where impurity contribution dominate, and the main mechanism is elastic scattering. The potential to consider, is that of the impurities. To kill phonon contribution and to fall into this region we typically have to be under the order of ∼ 10K. This value is quite lower than the typically Debye temperatures that range around 200− 300K.
At high temperature (100−300K as seen in Fig.7), where phonon contribution dominate, and the main mechanism is inelastic scattering. The potential to consider, is that of phonons. Note that inside this region: for low temperature, acoustic phonons dominate; while for higher temperature optical phonons dominate.
Figure 7: Carrier mobility in monolayer MoS2as a function of temperature (a) and carrier
density (b), calculated from rst-principles DFT for the electronic band structure, phonon dispersion and electronphonon interactions. In panel a, the grey band shows the uncer-tainty in calculated mobility values due to a 10% unceruncer-tainty in computed deformation potentials associated with phonons. Adapted from Ref. [78].
In this thesis we focus on the study of disorder mechanisms that limit the mobility mainly at low temperature, thus excluding the analysis of phonons. Structural defects in 2D TMDs can be classi ed according to the dimensionality [132]:
0D: Point defects. Among them, the typical ones are: vacancies (Sulfur vacancy has the lower formation energy), substituting elements and adatoms (they can be adsorbed resonant scatterers or adsorbed charged impurities). Note that they also
can be not strictly zero-dimensional as it is possible to meet adsorbed molecules or charge inhomogenities.
1D: Single and double line vacancies, grain boundaries and instability of terminated edges.
2D: Ripples, tilted stacking (when 2D materials are placed one on top pf another), ring defects.
All these factors remain yet poorly understood: it is not clear the role of each one of these source of disorder. With our atomistic model, on the contrary of DFT, we are able to describe samples of micrometer size treating intrinsic defects at the atomistic level. We will discuss in particular: vacancies and charge inhonogenities (represented by uctuations in the local scalar potential)
3 Review of the Electronic
Structure of MoS
2
This chapter presents a short review of the electronic structure and transport proper-ties of monolayer MoS2 covering both experimental and theoretical aspects. It highlights
the main current research challenge controversies and contextualizes the research devel-oped in this thesis.
3.1 Crystal structure
The MoS2 lattice structure is shown in Figs. 8 and 9. Figure 9 shows the top view
of a single-layer, while Fig. 8 illustrates the multi layered structure (lateral view). Like other TMDS, a single MoS2 layer is a 2D rhombic lattice with a three-atom basis (one
Mo and two S atoms). ML- MoS2 consists of one plane of Mo atoms surrounded by two
planes of S atoms in a way that each Mo atom is coordinated by six S atoms in a trigonal prismatic geometry and each S atom is coordinated by three Mo atoms.
The two Bravais lattice vectors are
R1 = (a, 0, 0) and R2 = a 2, √ 3 2 a, 0 ! , (3.1)
Figure 8: Three dimensional
schematic structure of MoS2, with
the chalcogen atom S in yellow and the metal atom Mo in grey.
where a is the lattice constant. The S atoms are located in planes above and below the Mo plane. The distance between the S planes is indicated by dS−S = c. While the distance
between neighboring Mo and S atoms is dM o−S = d. The angle between the Mo-S bond
and the Mo plane is θb.
Figure 9: Top view of the MoS2 lattice structure. Dark (light) circles represent Mo (S)
atoms. Notice that in this view two S atoms sit on top of each other. The unit cell is shown in the highlighted hexagon. The lattice constant in the Mo plane is a. The two Bravais lattice vectors (R1 and R2) are indicated. Six other auxiliary vectors that connect
a Mo atom with its nearest S atoms, δ1±, δ2±, and δ3±, are indicated.
To complete the description of the crystal structure, we introduce the nearest neighbor vectors δn±. Their meaning is depicted in Fig. 10. We denote by b ( or -) and t ( or
+) the S atoms at the bottom and top layers of the unit cell, respectively. The vectors connecting the Mo atom to these atoms are δn− and δn+, respectively. The coordinates
of these vectors are:
δ1± = d (0, cos θB,± sin θB) = (x±, y±, z±), (3.2) δ2± = d − √ 3 2 cos θB,− 1 2cos θB,± sin θB ! , (3.3) δ3± = d + √ 3 2 cos θB,− 1 2cos θB,± sin θB ! . (3.4)
Note that the distance between the two S layers is c = 2d sin θB = a/
√
3and d cos θB = √a3.
The symmetry space group of ML-MDS is D3hwhich contains the discrete symmetries
C3 (trigonal rotation), σv (re ection by the yz plane), σh (re ection by the xy plane) and
