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Electronic Structure of MoS2 NanotubesLingyun XuClemson University, neilxu@gmail.com

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Electronic Structure of MoS2 Nanotubes

A DissertationPresented to

the Graduate School ofClemson University

In Partial Fulfillmentof the Requirements for the Degree

Doctor of PhilosophyPhysics

byLingyun XuAugust 2007

Accepted by:Dr. Murray S. Daw, Committee Chair

Dr. Apparao M. RaoDr. D. Catalina Marinescu

Dr. Pu-Chun Ke

Abstract

First-principles methods enable one to study the electronic structure of solids, sur-

faces, or clusters as accurately as possible with moderate computational effort.

So we used a first-principles electronic structure method to calculate the electronic

structure of free-standing layer of MoS2 with ABA and ABC stacking. Our results suggest

MoS2 with ABA stacking which appears as an insulator has an energy gap of 1.64 eV.

The covalent bonds between molybdenum and sulfur atoms are strong enough to form this

gap. The ABC stacking will break the symmetry and becomes metallic. The valance and

impurities calculations show the rigid-band picture of MoS2 with ABA stacking.

For treating larger systems, one can also use the tight-binding method. We applied

this method to fit the band structure of single layer of S to the result from the first-principles

calculation.

The electronic structure of MoS2 nanotubes has been studied using a first-principles

electronic structure method. We investigated MoS2 zigzag (n, 0) nanotubes as well as

armchair (n, n) structures. We constructed MoS2 nanotubes with ABA and ABC stack-

ing. The structures have been completely optimized. We compare our results to previous

tight-binding calculations by Seifert et al.[29] and find significant differences in configura-

tion, bond lengths and resulting electronic structure in several MoS2 nanotubes. For zigzag

structures, almost all the nanotubes with ABA stacking and small tubes with ABC stack-

ing are semiconducting. For armchair structures, all (n, n) tubes with ABA stacking are

semiconducting and with ABC stacking are metallic. For armchair and zigzag tubes of a

given n, the lowest energy structure is semiconducting.

ii

Dedication

This thesis is dedicated to my mother and in loving memory of my grandparents.

iii

Acknowledgments

I would like to express my gratitude to my advisor, Murray S. Daw, for his support,

patience, and encouragement throughout my graduate studies. His academic advice was

essential to the completion of this dissertation and has taught me innumerable lessons and

insights on the workings of academic research in general. I also like to thank professors Terry

Tritt and Apparao Rao, and Dr. Xing Gao, all of Clemson University for their support.

The work is supported by the DOE under grant DE-FG02-04ER-46139. I also acknowledge

the use of VASP and DOE support through time on NERSC.

iv

Table of Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 First-Principles Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Free-Standing Layer of MoS2 . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Tight-binding Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Tight Binding Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Slater-Koster Matrix of Single Layer of Sulfur . . . . . . . . . . . . . . . . . 273.3 Slater-Koster Matrix of Single Layer of Mo . . . . . . . . . . . . . . . . . . 36

4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Configurations of nanotubes of MoS2 . . . . . . . . . . . . . . . . . . . . . . 454.2 Armchair MoS2 nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Zigzag MoS2 nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Comparison of zigzag and armchair . . . . . . . . . . . . . . . . . . . . . . . 56

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

v

List of Tables

2.1 Optimized bond distances for MoS2 layers with two types of stacking. Allthe distances are in A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Impurities in free-standing layer of (MoS2)4 with ABA stacking . . . . . . . 17

4.1 The band gap energies of armchair structures. All the band gaps are in eV. 464.2 Bond lengths for (n, n) tubes with ABA stacking MoS2 nanotubes. . . . . . 504.3 The band gap energies of zigzag structures. All the band gaps are in eV. . . 554.4 Bond lengths for (n, 0) tubes with ABA stacking MoS2 nanotubes. . . . . . 554.5 Comparison of band gap and total energy. All the band gaps are in eV. All

the energies are in eV/unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

vi

List of Figures

2.1 Full all-electronic wavefunction and electronic potential and the correspond-ing pseudo wavefunction and potential . . . . . . . . . . . . . . . . . . . . . 11

2.2 (a) Top view of free-standing layer of MoS2 with ABA stacking. (b) Sideview of same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Electronic density of states of free-standing layer of MoS2 with ABA stacking.The bandgap results from mirror-plane symmetry. . . . . . . . . . . . . . . 14

2.4 (a) Top view of free-standing layer of MoS2 with ABC stacking. (b) Sideview of same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Electronic density of states of a free-standing layer of MoS2 with ABC stack-ing. The loss of mirror symmetry results in a metallic system. . . . . . . . . 17

2.6 Electronic Density of States of (MoS2)12 with one S vacancy . . . . . . . . . 182.7 Electronic Density of States of (MoS2)12 with one Mo vacancy . . . . . . . 192.8 Electronic Density of States of Mo4S7P . . . . . . . . . . . . . . . . . . . . 202.9 Electronic Density of States of Mo4S7Cl . . . . . . . . . . . . . . . . . . . . 212.10 Electronic Density of States of Mo3S8Nb . . . . . . . . . . . . . . . . . . . . 222.11 Electronic Density of States of Mo3S8Tc . . . . . . . . . . . . . . . . . . . . 232.12 Electronic Density of States of Mo3S8Ti . . . . . . . . . . . . . . . . . . . . 24

3.1 Single Layer of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 The band structures of single layer of S from two different calculation. . . . 37

4.1 Cross-section of zigzag (10, 0) of MoS2 nanotube. Larger atoms are Mo. . . 474.2 Cross-section of armchair (14, 14) of MoS2 nanotube. Larger atoms are Mo. 484.3 Cross-section of supercell of (12, 12) MoS2 nanotube. larger atoms are Mo.

This structures contains 6-fold symmetry. . . . . . . . . . . . . . . . . . . . 494.4 Electronic density of state of (6, 6) ABA stacking MoS2 nanotube . . . . . 514.5 Band structure of a (6, 6) tube with ABA stacking . . . . . . . . . . . . . . 524.6 Electronic density of state of (6, 6) ABC stacking MoS2 nanotube . . . . . 534.7 Calculated bond distances of (n, n) tubes with ABA stacking tubes as func-

tion of n. The bond distance of free-standing layer is shown as a reference. . 544.8 Electronic density of state of (6, 0) ABA stacking MoS2 nanotube . . . . . 574.9 Band structure of a (6, 0) tube with ABA stacking . . . . . . . . . . . . . . 584.10 Electronic density of state of (10, 0) ABA stacking MoS2 nanotube . . . . . 594.11 Band structure of a (10, 0) tube with ABA stacking . . . . . . . . . . . . . 604.12 Electronic density of state of (18, 0) ABA stacking MoS2 nanotube . . . . . 614.13 Band structure of a (18, 0) tube with ABA stacking . . . . . . . . . . . . . 62

vii

4.14 Electronic density of state of (6, 0) ABC stacking MoS2 nanotube . . . . . 634.15 Band structure of a (6, 0) tube with ABC stacking . . . . . . . . . . . . . . 644.16 Band structure of a (6, 0) tube with ABC stacking . . . . . . . . . . . . . . 654.17 Electronic density of state of (18, 0) ABC stacking MoS2 nanotube . . . . . 664.18 Calculated bond distances of (n, 0) tubes with ABA stacking tubes as func-

tion of n. The bond distance of free-standing layer is shown as a reference. . 674.19 Calculated band gap energies of MoS2 nanotubes with ABA stacking as func-

tion of n. The band gap of free-standing layer is shown as a reference. Allband gap energies are in eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.20 Calculated strain energies per MoS2 unit per unit length of the tube as func-tion of n. All energies are in eV. . . . . . . . . . . . . . . . . . . . . . . . . 69

viii

Chapter 1

Introduction

Molybdenum disulfide (MoS2) is a very interesting material with numerous appli-

cations [22, 6]. Its structure and appearance are similar to graphite. Due to the weak

interactions between the sheets of sulfide atoms, MoS2 has a low coefficient of friction re-

sulting in a lubricating effect. So it is often used as lubricant [21]. Finely powdered MoS2

is also often mixed into various oils and greases, which allows the mechanisms lubricated

by it to keep running for a while longer, even in cases of almost complete oil loss - finding

an important use in aircraft engines. It is often used in motorcycle engines, especially in

areas of two-stroke engines which are not otherwise well lubricated.

Recent applications involved thin films of fullerene-like MoS2 nanoparticles [5].

Single-wall subnanometer-diameter MoS2 nanotubes were synthesized in 2001, with sig-

nificant amounts of intercalated iodine [28].

Though there are some electronic measurements and ab initio calculations published

for bulk MoS2 [7, 3] and an ab initio study of MoS2I1/3 nanotube bundles [32], no first-

principles calculation of electronic structures has been reported for nanotubes of MoS2.

In July 2000, Seifert, et al. reported the electronic structure of MoS2 nanotubes

using density-functional-based-tight-binding (DFTB) [27, 9]. Their results found that both

MoS2 zigzag (n, 0) and armchair (n, n) nanotubes are semiconducting.

MoS2 forms in sheets composed of three triangularly packed layers, such that a layer

1

of Mo is sandwiched by S layers. Normally, MoS2 is observed to form such that the three

layers are stacked in ABA fashion so that the Mo atom lies at the centers of a trigonal

prism. In the present work, we observe conditions - in nanotubes - where the layers are

stacked according to ABC fashion.

We have, therefore, performed first-principles calculations for MoS2 nanotubes with

ABA and ABC stacking. The configurations are fully optimized. For armchair structures,

all (n, n) tubes with ABA stacking are semiconducting and (n, n) with ABC stacking are

metallic. For zigzag structures, almost all the nanotubes with ABA stacking and small

tubes with ABC stacking are semiconducting. For a given n, the lowest energy structure is

semiconducting.

In the results of Seifert, et al.[29], only tubes with ABA stacking were calculated.

The prediction they made, that all nanotubes are semiconducting, is not confirmed by our

calculation.

2

Chapter 2

Approach

2.1 First-Principles Calculation

Beginning with Schrodinger’s equation without making assumptions such as fitting

parameters, the first-principles method (a.k.a. ab initio) is used for calculation of the

complete many electron system. This section will give a brief description of theories and

approximations made to solve this many-body problem.

2.1.1 Hartree-Fock Approximation

To solve a many-body system with interactions, we start from (2.1):

HΨ = EΨ (2.1)

where Ψ(~r1, ~r2, . . . , ~rN ) is the N -electron wavefunction, E is the system energy and H is

the Hamiltonian of system.

A first approximation, Born-Oppenheimer approximation[4], is to decouple the nu-

clear and electronic degrees of motion. Because nuclei are thousands of times more massive

than the electrons, they move very slowly. So they may be considered to be stationary on

the electronic timescale. It is possible to neglect the nuclear kinetic energy contribution to

the system energy.

3

The Hamiltonian in Equation 2.1 describing the interaction of electrons and nuclei

becomes:

H =N∑

i=1

(− h2

2m∇2

i − Ze2∑~R

1|~ri − ~R|

) +12

∑i6=j

e2

|~ri − ~rj |(2.2)

Here ~ri is the position of electron i and ~R is the position of nucleus. The first

term is the many-body kinetic energy operator which yields the electronic kinetic energies

and the second is the interaction of the electrons with the nuclei. The third describes the

interactions between electrons. The total energy of the system will also include the Coulomb

repulsion between the ions.

Usually, it is impossible to solve this many-body equation analytically because there

are so many electrons (N ∼ 1028 in one mole of a solid) and each electron contains 3N de-

grees of freedom. Moreover, the correlation between electrons which prevents a separation

of 3N degrees into N single-body problems has to be taken account of. Further, the inter-

action can not be treated as a perturbation. Consequently other approximations have to

be applied.

In the Hartree approximation [12], all electrons are treated independently and Ψ

can be written as a product of N one-electron wavefunctions:

Ψ(~r1, ~r2, . . . , ~rN ) = ψ1(~r1)ψ2(~r2) . . . ψN (~rN ) (2.3)

So the one-electron Schrodinger equation is now:

− h2

2m∇2ψi(~r) + [Vion(~r) + Ve(~r)]ψi(~r) = εiψi(~r) (2.4)

where the potential that the electron would feel from the ions:

Vion(~r) = −Ze2∑~R

1|~ri − ~R|

(2.5)

and Ve is the potential that the electron would feel from other electrons.

However, the product of N one-electron wavefunctions is incompatible with the

4

Pauli exclusion principle which requires the many-body wavefunctions to be antisymmetric

under the interchange of two electrons, that is:

Ψ(~r1, ~r2, . . . , ~rN ) = −Ψ(~r2, ~r1, . . . , ~rN ) (2.6)

The form of the wavefunction can be generalised to incorporate asymmetry by re-

placing the Hartree wavefunction by a Slater determinant of one electron wavefunctions.

Ψ(~r1σ1 . . . ~rNσN ) =1√N

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ψ1(~r1σ1) ψ1~r2σ2) · · · ψ1(~rNσN )

· · ·

· · ·

· · ·

ψN (~r1σ1) ψN (~r2σ2) · · · ψN (~rNσN )

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(2.7)

Under the Hartree-Fock approximation[8, 30], the equation 2.5 can be written as:

− h2

2m∇2ψi(~r)+Vion(~r)ψi(~r)+Ve(~r)ψi(~r)−

∑j

∫ d~rψ∗j (~r′)ψi(~r′)ψj(~r)ψ∗i (~r)|~r − ~r′|

= εiψi(~r) (2.8)

The last term on the left-hand side is the exchange term because of Pauli exclusion

principle. Although the exchange energy is included in Hartree-Fock equation 2.8, it neglects

the correlations due to many-body interactions and Density Functional Theory includes

exchange and correlation energy.

2.1.2 Density Functional Theory

Density functional theory (DFT) is a quantum mechanical method used in physics

and chemistry to investigate the electronic structure of many-body systems, in particular

molecules and the condensed phases. The electron density only has three spatial variables

rather than 3N variables as the many-body wavefuntion is. This difference significantly

5

simplifies the problem. In the Kohn-Sham DFT, the theory is a one-electron theory and

replaces the many-body electronic wavefunction with the electronic density. In practice,

approximations are required to implement this theory.

Hohenberg and Kohn [13] stated that if N interacting electrons move in an external

potential Vext(~r), the minimum value of the total energy functional is the ground state

energy of the system:

E[n] =∫n(~r)Vext(~r)d~r + F [n] (2.9)

where F is a universal functional of electronic density n, independent of Vext(~r).

It was then shown by Kohn and Sham [14] that it is possible to replace the many

electron problem by an exactly equivalent set of self consistent one electron equations. Then,

they separated F [n(~r)] into three distinct parts, so that the E becomes:

E[n] =∫n(~r)Vext(~r)d~r +

12

∫ ∫n(~r)n(~r′)|~r − ~r′|

d~rd~r′ + T [n(~r)] + EXC [n(~r)] (2.10)

The first two terms are the classical Coulomb interaction between the electrons

and ions and between electrons and other electrons respectively, both of which are simply

functions of the electronic charge density n(~r).

T [n(~r)] is the kinetic energy of a system of non-interacting electrons with density

n(~r) and EXC is the energy of exchange and correlation of an interacting system. Unfortu-

nately there is no known exact expression for either.

The electronic density n would be:

n(~r) =∫

BZd3k|ψ(~r)|2 (2.11)

6

The VXC(~r) can be derived from:

VXC(~r) =δEXC [n(~r)]

δn(~r)(2.12)

The equation 2.8 would be rewritten as:

[− h2

2m∇2

i + Veff (~r)

]ψi(~r) = εiψi(~r) (2.13)

where the effective potential would be:

Veff (~r) = Vext(~r) +∫

n(~r′)|~r − ~r′|

d~r′ + VXC(~r) (2.14)

2.1.3 Local Density Approximation and Generalized Gradient Approxi-

mation

If the exchange and correlation functional is known exactly, it is possible to find the

solutions to the ground state energy of an interacting system 2.13. Unfortunately, the form

of EXC is in general unknown so an approximation has to be employed.

The local-density approximation (LDA) [14] is the simplest approximation for this

functional. The exchange and correlation energy at the coordinate depends only on the

electron density at that point:

EXC [n] =∫εXC(n)n(r)d3r (2.15)

where εXC is equal to the exchange-correlation energy per electron in a homogeneous elec-

tron gas that has the same electron density at the point.

For systems where the density varies slowly, the LDA tends to perform well. In

strongly correlated systems, the LDA is very inaccurate. Also the LDA underestimates the

badgap.

An obvious approach to improving the LDA is to include gradient corrections which

7

is called generalized gradient approximations (GGA), where it not only takes into account

the local density at a point but also the gradient of the density at the same coordinate:

EXC [n] =∫εXC(n, ~∇n)n(r)d3r (2.16)

2.1.4 Bloch’s Theorem and Plane Wave Basis Sets

The ions in a perfect crystal are arranged in a regular periodic way (at 0K). Therefore

the external potential felt by the electrons will also be periodic - the period being the same

as the length of the unit cell l. That is, the external potential on an electron at r can be

expressed as V (~r) = V (~r +~l).

Bloch’s theorem[1] uses the periodicity of a crystal to reduce the infinite number of

one-electron wavefunctions to the number of electrons in the unit cell of the crystal. The

wavefunction is written as the product of a cell periodic part and a wavelike part:

ψki(~r) = ei~k·~ruki(~r) (2.17)

The ~k is a wavevector confined to the first Brillouin zone. The second term is also a periodic

function,

uki(~r +~l) = uki(~r) (2.18)

which can be expended to:

uki(~r) =∑G

Cki(~G)ei ~G·~r (2.19)

where ~G is the reciprocal lattice vectors which are defined by ~G ·~l = 2πn where ~l is a lattice

vector of the crystal and n is an integer. Combined Equation 2.17 and Equation 2.19, the

electronic wavefunction is written as a sum of plane waves:

ψi(~r) =∑G

Ci(~G)ei(~k+ ~G)·~r (2.20)

By the use of Bloch’s theorem, the problem of differential equation in ψ(~r) has now been

8

mapped onto the algebraic equation involving discrete C(~G) in terms of an infinite number

of reciprocal space vectors.

The electronic wavefunctions at each k-point are now expressed in terms of a discrete

plane wave basis set. In principle this Fourier series is infinite. However, the plane waves

with a smaller kinetic energy typically are more important than those with a very high

kinetic energy. The introduction of a plane wave energy cutoff h2

2m |~k+ ~G|2 reduces the basis

set to a finite size.

2.1.5 k-point summation

In the first Brillouin zone, the occupied states at each k-point contribute to the

electronic potential. If a continuum of plane wave basis sets was required, the basis set for

any calculation would still be infinite, no matter how small the plane wave energy cut-off

was chosen.

For this reason electronic states are only calculated at a set of k-points determined

by the shape of the Brillouin zone compared to that of its irreducible part. The reason

is that the electronic wavefunctions at k-points that are very close together will almost be

identical. It is therefore possible to represent the electronic wavefunctions over a region of

reciprocal space at a single k-point. This approximation allows the electronic potential to

be calculated at a finite number of k-points. The Bloch’s Theorem with k-point summation

therefore have changed the problem of an infinite number of electrons to the number of

electrons in the unit cell at a finite number of k-points chosen so as to appropriately sample

the Brillouin zone.

2.1.6 The Pseudopotential Approximation

It is now tractable to solve Kohn-Sham equation for solid state systems with Bloch’s

theorem, that a plane wave expansion of the wavefunction and k-point sampling. Unfor-

tunately a plane wave basis set is usually very poorly suited to expanding the electronic

wavefunctions because it is difficult to accurately describe the rapidly oscillating wavefunc-

9

tions of electrons in the core region.

Usually, the valence electrons which surround the core region determine most phys-

ical properties of solids instead of the core electrons. This is the reason that the pseudopo-

tential approximation is introduced [11]. This approximation removes the core electrons

and the strong nuclear potential and replace them with a weaker pseudopotential which

acts on a set of pseudo wavefunctions rather than the true valence wavefunctions.

As shown in Fig. 2.1, the valence wavefunctions oscillate rapidly in the region oc-

cupied by the core electrons because of the strong ionic potential. The pseudopotential

is constructed in such a way that the pseudo wavefunction in the core region is smooth

and that the pseudo wavefunctions and pseudopotential are identical to the all electron

wavefunction and potential outside a cut-off radius rc.

A pseudopotential is not unique, therefore several methods of generation exist.

Ultra-soft pseudopotentials [31] and PAW [2] pseudopotentials are amongst the most widely

used.

Generally the PAW potentials are more accurate than the ultra-soft pseudopoten-

tials. There are two reasons for this: first, the radial cutoffs (core radii) are smaller than

the radii used for the US pseudopotentials, and second the PAW potentials reconstruct the

exact valence wave function with all nodes in the core region. Since the core radii of the

PAW potentials are smaller, the required energy cutoffs and basis sets are also somewhat

larger. If such a high precession is not required, the older US-PP can be used. In practice,

however, the increase in the basis set size will be anyway small, since the energy cutoffs

have not changed appreciably for C, N and O, so that calculations for models, which include

any of these elements, are not more expensive with PAW than with US-PP.[20]

2.2 Software

We performed all calculations with Vienna Ab-initio Simulation Package (a.k.a.

VASP) [15, 17, 18, 16] which is based on the projector augmented wave (PAW) [2, 19]

10

Figure 2.1: An illustration of the real wavefunction and electronic potential (solid lines)plotted against distance r, from the atomic nucleus. The corresponding pseudo wavefunctionand potential is plotted (dashed lines). Outside a a certain cutoff radius rc, the real electronand pseudo electron values are identical.

11

method employing the generalized gradient approximation (GGA) [23, 24, 25, 26]. Each

configuration was optimized by minimizing the energy. For nanotube calculations, k-point

convergence was achieved with a Γ-centered grid.

2.3 Free-Standing Layer of MoS2

2.3.1 Free-standing Layer of MoS2 with ABA stacking

Before investigating MoS2 nanotubes, we examined as a reference flat, free-standing

sheets of MoS2, which is shown in Fig 2.2.

During the calculation, we set the number of irreducible k-points in the 2D Bril-

louin zone to 72. The unit cell in our calculation contains one Mo and two S atoms. We

determined the optimum distance and lattice constant by minimizing total energy. The

optimized a (S-S intralayer distance) is 3.20A, and 2z (S-S interlayer distance) is 3.13A.

These results are similar to the experimental results [3], which are a = 3.159A, 2z = 3.172A.

Fig. 2.3 shows the density of states of a free-standing layer of MoS2. In that figure,

we can clearly see a band gap which is 1.64 eV. The free-standing layer of MoS2 has a mirror

plane symmetry about the Mo layer. Thus all states have even or odd symmetry under this

operation. The Mo d-states are all even. The S p-states on opposite layers can form even

and odd combinations, of which only the even combinations interact with the Mo d-states.

This arrangement leads to the formation of a gap.

2.3.2 Free-standing Layer of MoS2 with ABC stacking

To illustrate this, we have altered the ABA sequence to ABC and re-calculated the

electronic structure. Fig. 2.4 shows the ABC stacking. In both ABA and ABC stacking,

the layers are triangular. If S in third layer is directly below that in first layer, this is

called “ABA”. If it is directly below the middle center of the triangle, this is called “ABC”

stacking. In Fig. 2.4, two different colors were used to illustrate S atoms in first and third

layers (Mo atoms in the middle).

12

(a)

(b)

SMo

a

2z

Figure 2.2: (a) Top view of free-standing layer of MoS2 with ABA stacking. (b) Side viewof same.

13

0

2

4

6

8

10

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.3: Electronic density of states of free-standing layer of MoS2 with ABA stacking.The bandgap results from mirror-plane symmetry.

14

The loss of the mirror plane symmetry causes the disappearance of the band gap,

as shown in Fig. 2.5.

The energy for ABC stacking is 0.54 eV per MoS2 unit higher than that for ABA

stacking, as is generally expected.

Table 2.3.2 shows the differences in bond distances between these two structures.

ABA stacking ABC stackingS-S (intralayer) 3.201 3.227S-S (interlayer) 3.131 3.155Mo-S 2.422 2.441

Table 2.1: Optimized bond distances for MoS2 layers with two types of stacking. All thedistances are in A.

2.3.3 Point Defects and Impurities

We investigate the effects of various point defects: valance in Mo, valance in S and

substituions on both sublattices.

Fig. 2.6 shows density of states of (MoS2)12 with one S removed. Compared to the

plot of MoS2, there is a peak in the gap area. The localized states on valance in S will trap

carriers resulting in a poor conductivity.

Fig. 2.7 shows density of states of (MoS2)12 with one Mo removed.

We also tried substitutional impurities by substituting Ti, Nb and Tc for Mo and P,

Cl for S. The fermi level is controlled by these impurities, for example, the density of states

of Mo4S7P in Fig. 2.8 clearly illustrates that the fermi level is shifted toward p-type. Our

results are consistent with a rigid-band picture of dopant in the MoS2. The all calculation

results are listed in table 2.3.3. The electronic density of states of other impurities are

shown in Fig. 2.12, Fig. 2.10, Fig. 2.11, Fig. 2.9.

15

(a)

(b)

S in first

Layer

Mo S in third Layer

a

2z

Figure 2.4: (a) Top view of free-standing layer of MoS2 with ABC stacking. (b) Side viewof same.

16

0

1

2

3

4

5

6

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.5: Electronic density of states of a free-standing layer of MoS2 with ABC stacking.The loss of mirror symmetry results in a metallic system.

Impurities TypesMo3S8Nb p-typeMo3S8Ti p-typeMo3S8Tc n-typeMo4S7P p-typeMo4S7Cl n-type

Table 2.2: Impurities in free-standing layer of (MoS2)4 with ABA stacking

17

0

10

20

%0

40

50

(0

)0

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.6: Electronic Density of States of (MoS2)12 with one S vacancy

18

0

10

20

%0

40

50

(0

)0

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.7: Electronic Density of States of (MoS2)12 with one Mo vacancy

19

0

5

10

15

20

25

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.8: Electronic Density of States of Mo4S7P

20

0

5

10

15

20

25

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.9: Electronic Density of States of Mo4S7Cl

21

0

5

10

15

20

25

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.10: Electronic Density of States of Mo3S8Nb

22

0

5

10

15

20

25

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.11: Electronic Density of States of Mo3S8Tc

23

0

5

10

15

20

25

!4 !2 0 2 4

Num

ber o

f Sta

tes

Energy (eV)

Figure 2.12: Electronic Density of States of Mo3S8Ti

24

Chapter 3

Tight-binding Method

In this chapter, we will discuss the tight binding method and our calculation results.

The reason we decided to use tight-binding method instead of first-principles is the size of

MoS2 nanotubes. The tubes were initially constructed with a very large diameter. The

computation was beyond the software. With tight-binding method, it is possible to deal

with these larger systems. Later it was found that the nanotubes can be constructed with

much smaller diameters in Section 4.1, we decided to switch back to the first-principles

method.

3.1 Tight Binding Theory

In order to investigate the properties of larger model systems a simpler and less

computationally demanding method is required. One of the simplifications over the first-

principles calculation is the tight-binding (TB) Hamiltonian method, also referred to as

Linear Combination of Atomic Orbitals (LCAO).

In the tight biding theory, it is assumed that the orbitals that are very similar

to atomic states (i.e. wavefunctions tightly bound to the atoms, hence the term “tight-

binding”) can be used as a basis for expanding the wavefunction.

So the Schrodinger equation 2.1 needs to be rewritten. First of all, the eigenvector

25

ψ is expended out in terms of the basis functions:

ψ =∑iα

ciαφiα (3.1)

where the index i refers to the atoms and the Greek letter α to the orbitals on these atoms.

φiα is an orbital on atom i. The Hamiltonian can then be written as a matrix as:

Hiα,jβ = 〈iα|H|jβ〉 (3.2)

The on-site integrals and represent the energies of the orbitals:

〈φs(r)|H|φs(r)〉 = εs (3.3)

〈φpx(r)|H|φpx(r)〉 = εp (3.4)

For example, the off-site interaction is: (if the bond is assumed to be along the

x-axis):

〈φs(r)|H|φs(r± b)〉 = Vss (3.5)

It is assumed that the interaction between orbitals on different atoms to be inde-

pendent of the position of other atoms, known as the two-center approximation.

One simplifying assumption is that the orbitals on different atoms are orthogonal

to one another.

〈φpx(r)|φpy(r′)〉 = 0 (3.6)

Tight-binding method has major disadvantages compared to first-principles method.

It uses a minimal basis and ignores three-center and higher order effects. The other dis-

advantage is that the magnitude and distance variation of the matrix elements must be

26

determined empirically. There is also the issue of transferability, since it is difficult to prove

that just because a particular model with a particular set of parameters reproduced the

fitting database accurately it will do so for other configurations.

3.2 Slater-Koster Matrix of Single Layer of Sulfur

We start to construct Slater-Koster matrix of single layer of S. Fig. 3.1 shows this

structure.

3.2.1 Hamiltonian Matrix Elements

3.2.1.1 On Site

〈φs(r)|H|φs(r)〉 = εs

〈φs(r)|H|φpx(r)〉 = 0

〈φpx(r)|H|φpx(r)〉 = εp

〈φpx(r)|H|φpy(r)〉 = 0

3.2.1.2 off site x± ax

〈φs(r)|H|φs(r± ax)〉 = Vss

〈φs(r)|H|φpx(r± ax)〉 = ∓Vsp

27

a

Figure 3.1: Single Layer of S. Each S atom has 6 nearest-neighbors.

28

〈φs(r)|H|φpy(r± ax)〉 = 0

〈φs(r)|H|φpz(r± ax)〉 = 0

〈φpx(r)|H|φpx(r± ax)〉 = Vppσ

〈φpx(r)|H|φpy(r± ax)〉 = 0

〈φpx(r)|H|φpz(r± ax)〉 = 0

〈φpy(r)|H|φpx(r± ax)〉 = 0

〈φpy(r)|H|φpy(r± ax)〉 = Vppπ

〈φpy(r)|H|φpz(r± ax)〉 = 0

〈φpz(r)|H|φpx(r± ax)〉 = 0

〈φpz(r)|H|φpy(r± ax)〉 = 0

29

〈φpz(r)|H|φpz(r± ax)〉 = Vppπ

3.2.1.3 off site r± b, where b = (12 ,

√3

2 )a

〈φs(r)|H|φs(r± b)〉 = Vss

〈φs(r)|H|φpx(r± b)〉 = ∓Vsp cos θ

〈φs(r)|H|φpy(r± b)〉 = ∓Vsp sin θ

〈φs(r)|H|φpz(r± b)〉 = 0

〈φpx(r)|H|φpx(r± b)〉 = Vppσ cos2 θ + Vppπ sin2 θ

〈φpx(r)|H|φpy(r± b)〉 = (Vppσ − Vppπ) sin θ cos θ

〈φpx(r)|H|φpz(r± b)〉 = 0

〈φpy(r)|H|φpx(r± b)〉 = (Vppσ − Vppπ) sin θ cos θ

〈φpy(r)|H|φpy(r± b)〉 = Vppσ sin2 θ + Vppπ cos2 θ

30

〈φpy(r)|H|φpz(r± b)〉 = 0

〈φpz(r)|H|φpx(r± b)〉 = 0

〈φpz(r)|H|φpy(r± b)〉 = 0

〈φpz(r)|H|φpz(r± b)〉 = Vppπ

3.2.1.4 off site r± c, where c = (−12 ,

√3

2 )a

〈φs(r)|H|φs(r± c)〉 = Vss

〈φs(r)|H|φpx(r± c)〉 = ±Vsp cos θ

〈φs(r)|H|φpy(r± c)〉 = ±Vsp sin θ

〈φs(r)|H|φpz(r± c)〉 = 0

〈φpx(r)|H|φpx(r± c)〉 = Vppσ cos2 θ + Vppπ sin2 θ

〈φpx(r)|H|φpy(r± c)〉 = (Vppπ − Vppσ) sin θ cos θ

31

〈φpx(r)|H|φpz(r± c)〉 = 0

〈φpy(r)|H|φpx(r± c)〉 = (Vppπ − Vppσ) sin θ cos θ

〈φpy(r)|H|φpy(r± c)〉 = Vppσ sin2 θ + Vppπ cos2 θ

〈φpy(r)|H|φpz(r± c)〉 = 0

〈φpz(r)|H|φpx(r± c)〉 = 0

〈φpz(r)|H|φpy(r± c)〉 = 0

〈φpz(r)|H|φpz(r± c)〉 = Vppπ

3.2.2 Hamiltonian Matrix for k

3.2.2.1 General Form

A B C D

B∗ E F G

C∗ F ∗ H I

D∗ G∗ I∗ J

A = 〈φs(r)|H|φs(r)〉+ 〈φs(r)|H|φs(r± ax)〉e∓ik·ax

32

+〈φs(r)|H|φs(r± b)〉e∓ik·b + 〈φs(r)|H|φs(r± c)〉e∓ik·c

= εs + 2Vss cos (akx) + 4Vss cos (a

2kx) cos (

√3

2ky)

B = 〈φs(r)|H|φpx(r)〉+ 〈φs(r)|H|φpx(r± ax)〉e∓ik·ax

+〈φs(r)|H|φpx(r± b)〉e∓ik·b + 〈φs(r)|H|φpx(r± c)〉e∓ik·c

= 2iVsp[sin(akx) + sin(a

2kx) cos(

√3

2ky)]

C = 〈φs(r)|H|φpy(r)〉+ 〈φs(r)|H|φpy(r± ax)〉e∓ik·ax

+〈φs(r)|H|φpy(r± b)〉e∓ik·b + 〈φs(r)|H|φpy(r± c)〉e∓ik·c

= 2√

3iVsp sin(a

2kx) cos(

√3

2ky)

D = 〈φs(r)|H|φpz(r)〉+ 〈φs(r)|H|φpz(r± ax)〉e∓ik·ax

+〈φs(r)|H|φpz(r± b)〉e∓ik·b + 〈φs(r)|H|φpz(r± c)〉e∓ik·c

= 0

E = 〈φpx(r)|H|φpx(r)〉+ 〈φpx(r)|H|φpx(r± ax)〉e∓ik·ax

+〈φpx(r)|H|φpx(r± b)〉e∓ik·b + 〈φpx(r)|H|φpx(r± c)〉e∓ik·c

= εp + 2Vppσ cos(akx)

+(Vppσ + 3Vppπ) cos(a

2kx) cos(

√3

2ky)

33

F = 〈φpx(r)|H|φpy(r)〉+ 〈φpx(r)|H|φpy(r± ax)〉e∓ik·ax

+〈φpx(r)|H|φpy(r± b)〉e∓ik·b + 〈φpx(r)|H|φpy(r± c)〉e∓ik·c

=√

3(Vppπ − Vppσ) sin(a

2kx) sin(

√3

2ky)

G = 〈φpx(r)|H|φpz(r)〉+ 〈φpx(r)|H|φpz(r± ax)〉e∓ik·ax

+〈φpx(r)|H|φpz(r± b)〉e∓ik·b + 〈φpx(r)|H|φpz(r± c)〉e∓ik·c

= 0

H = 〈φpy(r)|H|φpy(r)〉+ 〈φpy(r)|H|φpy(r± ax)〉e∓ik·ax

+〈φpy(r)|H|φpy(r± b)〉e∓ik·b + 〈φpy(r)|H|φpy(r± c)〉e∓ik·c

= εp + 2Vppπ cos(akx)

+(3Vppσ + Vppπ) cos(a

2kx) cos(

√3

2ky)

I = 〈φpy(r)|H|φpz(r)〉+ 〈φpy(r)|H|φpz(r± ax)〉e∓ik·ax

+〈φpy(r)|H|φpz(r± b)〉e∓ik·b + 〈φpy(r)|H|φpz(r± c)〉e∓ik·c

= 0

J = 〈φpz(r)|H|φpz(r)〉+ 〈φpz(r)|H|φpz(r± ax)〉e∓ik·ax

+〈φpz(r)|H|φpz(r± b)〉e∓ik·b + 〈φpz(r)|H|φpz(r± c)〉e∓ik·c

= εp + 2Vssπ cos (akx) + 4Vssπ cos (a

2kx) cos (

√3

2ky)

34

3.2.2.2 matrix at high symmetry points

Γ = (0, 0, 0)2πa

εs + 6Vss 0 0 0

0 εp + 3Vppσ + 3Vppπ 0 0

0 0 εp + 3Vppσ + 3Vppπ 0

0 0 0 εp + 6Vppπ

Q = (0,2√3, 0)

2πa

εs + 6Vss 0 0 0

0 εp + 3Vppσ + 3Vppπ 0 0

0 0 εp + 3Vppσ + 3Vppπ 0

0 0 0 εp + 6Vppπ

P = (12,

√32, 0)

2πa

εs − 2Vss 0 0 0

0 εp − 2Vppσ

√3(Vppσ−Vppπ) 0

0√

3(Vppσ−Vppπ) εp − 2Vppπ 0

0 0 0 εp − 2Vppπ

35

3.2.3 Parameter

We have the band structure of single layer of S using the first-principles method.

Compared to these results, εs, εp, Vss, Vppσ and Vppπ can be calculated based on the band

structure. Fig. 3.2 shows the band structures with these two methods.

εs = −11.03115 eV

εp = −0.42405 eV

Vss = 0.190525 eV

Vppσ = −0.181675 eV

Vppπ = 0.8997583 eV

3.3 Slater-Koster Matrix of Single Layer of Mo

3.3.1 Hamiltonian Matrix Elements

3.3.1.1 On Site

〈φdxy(r)|H|φdxy(r)〉 = εd

〈φdxy(r)|H|φdyz(r)〉 = 0

36

!5

!4

!3

!2

!1

0

1

2

3

Q!

Ener

gy (e

V)

k!points

!5

!4

!3

!2

!1

0

1

2

3

Q!

Ener

gy (e

V)

k!points

!5

!4

!3

!2

!1

0

1

2

3

Q!

Ener

gy (e

V)

k!points

!5

!4

!3

!2

!1

0

1

2

3

Q!

Ener

gy (e

V)

k!points

!5

!4

!3

!2

!1

0

1

2

3

Q!

Ener

gy (e

V)

k!points

Figure 3.2: The band structures of single layer of S from two different calculation. Thefull line is the band structure from first-principles method. The dashed line is the bandstructure from tight-binding method.

37

〈φdxy(r)|H|φdzx(r)〉 = 0

〈φdxy(r)|H|φdx2−y2 (r)〉 = 0

〈φdxy(r)|H|φd3z2−r2 (r)〉 = 0

3.3.2 off site x± ax

〈φdxy(r)|H|φdxy(r± ax)〉 = Vddπ

〈φdyz(r)|H|φdyz(r± ax)〉 = Vddδ

〈φdzx(r)|H|φdzx(r± ax)〉 = Vddπ

〈φdx2−y2 (r)|H|φdx2−y2 (r± ax)〉 =34Vddσ +

14Vddδ

〈φdx2−y2 (r)|H|φd3z2−r2 (r± ax)〉 = −√

34Vddσ +

√3

4Vddδ

〈φd3z2−r2 (r)|H|φd3z2−r2 (r± ax)〉 = −14Vddσ +

34Vddδ

38

3.3.2.1 off site r± b, where b = (12 ,

√3

2 )a

〈φdxy(r)|H|φdxy(r± ax)〉 =916Vddσ +

14Vddπ +

316Vddδ

〈φdxy(r)|H|φdx2−y2 (r± ax)〉 = −3√

316

Vddσ +√

34Vddπ −

√3

16Vddδ

〈φdxy(r)|H|φd3z2−r2 (r± ax)〉 = −38Vddσ −

38Vddδ

〈φdyz(r)|H|φdyz(r± ax)〉 =34Vddπ +

14Vddδ

〈φdyz(r)|H|φdzx(r± ax)〉 =√

34Vddπ −

√3

4Vddδ

〈φdzx(r)|H|φdzx(r± ax)〉 =14Vddπ +

34Vddδ

〈φdx2−y2 (r)|H|φdx2−y2 (r± ax)〉 =316Vddσ +

34Vddπ +

116Vddδ

〈φdx2−y2 (r)|H|φd3z2−r2 (r± ax)〉 =√

38Vddσ −

√3

8Vddδ

〈φd3z2−r2 (r)|H|φd3z2−r2 (r± ax)〉 =14Vddσ +

34Vddδ

39

3.3.2.2 off site r± c, where c = (−12 ,

√3

2 )a

〈φdxy(r)|H|φdxy(r± ax)〉 =916Vddσ +

14Vddπ +

316Vddδ

〈φdxy(r)|H|φdx2−y2 (r± ax)〉 =3√

316

Vddσ −√

34Vddπ +

√3

16Vddδ

〈φdxy(r)|H|φd3z2−r2 (r± ax)〉 =38Vddσ +

38Vddδ

〈φdyz(r)|H|φdyz(r± ax)〉 =34Vddπ +

14Vddδ

〈φdyz(r)|H|φdzx(r± ax)〉 = −√

34Vddπ +

√3

4Vddδ

〈φdzx(r)|H|φdzx(r± ax)〉 =14Vddπ +

34Vddδ

〈φdx2−y2 (r)|H|φdx2−y2 (r± ax)〉 =316Vddσ +

34Vddπ +

116Vddδ

〈φdx2−y2 (r)|H|φd3z2−r2 (r± ax)〉 =√

38Vddσ −

√3

8Vddδ

〈φd3z2−r2 (r)|H|φd3z2−r2 (r± ax)〉 =14Vddσ +

34Vddδ

40

3.3.3 Hamiltonian Matrix for k

3.3.3.1 General Form

A B C D E

B∗ F G H I

C∗ G∗ J K L

D∗ H∗ I∗ M N

E∗ K∗ L∗ N∗ O

A = 〈φdxy(r)|H|φdxy(r)〉+ 〈φdxy(r)|H|φdxy(r± ax)〉e∓ik·ax

+〈φdxy(r)|H|φdxy(r± b)〉e∓ik·b + 〈φdxy(r)|H|φdxy(r± c)〉e∓ik·c

= εd + 2Vddπ cos (akx) + 4 cos (a

2kx) cos (

√3

2ky)(

916Vddσ +

14Vddπ +

316Vddδ)

B = 0

C = 0

D = 〈φdxy(r)|H|φdx2−y2 (r)〉+ 〈φdxy(r)|H|φdx2−y2 (r± ax)〉e∓ik·ax

+〈φdxy(r)|H|φdx2−y2 (r± b)〉e∓ik·b + 〈φdxy(r)|H|φdx2−y2 (r± c)〉e∓ik·c

= 4 sin(a

2kx) sin(

√3

2ky)

(3√

316

Vddσ −√

34Vddπ +

√3

16Vddδ)

41

E = 〈φdxy(r)|H|φd3z2−r2 (r)〉+ 〈φdxy(r)|H|φd3z2−r2 (r± ax)〉e∓ik·ax

+〈φdxy(r)|H|φd3z2−r2 (r± b)〉e∓ik·b + 〈φdxy(r)|H|φd3z2−r2 (r± c)〉e∓ik·c

= 4 sin(a

2kx) sin(

√3

2ky)(

38Vddσ +

38Vddδ)

F = 〈φdyz(r)|H|φdyz(r)〉+ 〈φdyz(r)|H|φdyz(r± ax)〉e∓ik·ax

+〈φdyz(r)|H|φdyz(r± b)〉e∓ik·b + 〈φdyz(r)|H|φdyz(r± c)〉e∓ik·c

= εd + 2Vddδ cos (akx) + 4 cos (a

2kx) cos (

√3

2ky)(

34Vddπ +

14Vddδ)

G = 〈φdyz(r)|H|φdzx(r)〉+ 〈φdyz(r)|H|φdzx(r± ax)〉e∓ik·ax

+〈φdyz(r)|H|φdzx(r± b)〉e∓ik·b + 〈φdyz(r)|H|φdzx(r± c)〉e∓ik·c

= −4 sin(a

2kx) sin(

√3

2ky)(

√3

4Vddπ −

√3

4Vddδ)

H = 〈φdyz(r)|H|φdx2−y2 (r)〉+ 〈φdyz(r)|H|φdx2−y2 (r± ax)〉e∓ik·ax

+〈φdyz(r)|H|φdx2−y2 (r± b)〉e∓ik·b + 〈φdyz(r)|H|φdx2−y2 (r± c)〉e∓ik·c

= 0

I = 〈φdyz(r)|H|φd3z2−r2 (r)〉+ 〈φdyz(r)|H|φd3z2−r2 (r± ax)〉e∓ik·ax

+〈φdyz(r)|H|φd3z2−r2 (r± b)〉e∓ik·b + 〈φdyz(r)|H|φd3z2−r2 (r± c)〉e∓ik·c

= 0

42

J = 〈φdxz(r)|H|φdzx(r)〉+ 〈φdzx(r)|H|φdzx(r± ax)〉e∓ik·ax

+〈φdzx(r)|H|φdzx(r± b)〉e∓ik·b + 〈φdzx(r)|H|φdzx(r± c)〉e∓ik·c

= εd + 2Vddπ cos (akx) + 4 cos (a

2kx) cos (

√3

2ky)(

14Vddπ +

34Vddδ)

K = 〈φdzx(r)|H|φdx2−y2 (r)〉+ 〈φdzx(r)|H|φdx2−y2 (r± ax)〉e∓ik·ax

+〈φdzx(r)|H|φdx2−y2 (r± b)〉e∓ik·b + 〈φdzx(r)|H|φdx2−y2 (r± c)〉e∓ik·c

= 0

L = 〈φdzx(r)|H|φd3z2−r2 (r)〉+ 〈φdzx(r)|H|φd3z2−r2 (r± ax)〉e∓ik·ax

+〈φdzx(r)|H|φd3z2−r2 (r± b)〉e∓ik·b + 〈φdzx(r)|H|φd3z2−r2 (r± c)〉e∓ik·c

= 0

M = 〈φdx2−y2 (r)|H|φdx2−y2 (r)〉+ 〈φdx2−y2 (r)|H|φdx2−y2 (r± ax)〉e∓ik·ax

+〈φdx2−y2 (r)|H|φdx2−y2 (r± b)〉e∓ik·b + 〈φdx2−y2 (r)|H|φdx2−y2 (r± c)〉e∓ik·c

= εd + 2(34Vddσ +

14Vddδ) cos (akx) + 4 cos (

a

2kx) cos (

√3

2ky)(

316Vddσ +

34Vddπ +

116Vddδ)

N = 〈φdx2−y2 (r)|H|φd3z2−r2 (r)〉+ 〈φdx2−y2 (r)|H|φd3z2−r2 (r± ax)〉e∓ik·ax

+〈φdx2−y2 (r)|H|φd3z2−r2 (r± b)〉e∓ik·b + 〈φdx2−y2 (r)|H|φd3z2−r2 (r± c)〉e∓ik·c

= 2(−√

34Vddσ +

√3

4Vddδ) cos (akx) + 4 cos (

a

2kx) cos (

√3

2ky)(

√3

8Vddσ −

√3

8Vddδ)

43

O = 〈φd3z2−r2 (r)|H|φd3z2−r2 (r)〉+ 〈φd3z2−r2 (r)|H|φd3z2−r2 (r± ax)〉e∓ik·ax

+〈φd3z2−r2 (r)|H|φd3z2−r2 (r± b)〉e∓ik·b + 〈φd3z2−r2 (r)|H|φd3z2−r2 (r± c)〉e∓ik·c

= εd + 2(14Vddσ +

34Vddδ) cos (akx) + 4 cos (

a

2kx) cos (

√3

2ky)(

14Vddσ +

34Vddδ)

44

Chapter 4

Results

4.1 Configurations of nanotubes of MoS2

Like carbon nanotubes [10], MoS2 nanotubes can be constructed by wrapping the

free-standing layer along a chiral vector described by two integer indices. Fig. 4.1 and

Fig. 4.2 show the cross-section of (10, 0) and (14, 14) MoS2 tube, respectively.

For carbon nanotubes, the bonds between C of a (n, 0) tube have a zigzag appear-

ance, and of a (n, n) tube look like armchairs. For MoS2 nanotubes, the bonds between

Mo and S have the same appearance, so we continue using “zigzag” for (n, 0) tubes and

“armchair” for (n, n) tubes.

The smallest zigzag (n, 0) structure we consider is (6, 0), the largest (18, 0). Among

the armchair structures, (6, 6) is the smallest, the largest is (14, 14). The full list is:

• Zigzag

1. ABA stacking (6, 0) (10, 0) (18, 0)

2. ABC stacking (6, 0) (12, 0) (18, 0)

• Armchair

1. ABA stacking (6, 6) (12, 12) (14, 14)

45

ABA stacking ABC stacking(6, 6) 0.24 Metallic(12, 12) 1.10 Metallic(14, 14) 1.22

Table 4.1: The band gap energies of armchair structures. All the band gaps are in eV.

2. ABC stacking (6, 6) (12, 12)

The supercells are constructed as in Fig. 4.3 by arranging a two-dimensional array

of parallel nanotubes; the 2D array of nanotubes is triangular to optimize packing. The

separation between each nanotube is about 12A which we found has negligible intertube

interactions. The triangular supercell itself has 6-fold rotational symmetry around the

nanotube axis. The fullest use of symmetry is obtained when the nanotubes themselves

have also 6-fold symmetry. Those tubes with n which is a multiple of 6 have that 6-fold

symmetry, and those are the most efficient for computation. We have also done a few

nanotubes with other values of n to check that our results are not sensitive to our selection.

During relaxation, all the Mo and S atoms are allowed to move along the axis of the

nanotube, as well as along the radius direction, which keeps the symmetry of tubes.

4.2 Armchair MoS2 nanotubes

The results of armchair (n, n) MoS2 nanotubes are simpler than those of zigzag

tubes. All tubes with ABA stacking are semiconducting. For example, Fig. 4.4 shows the

electronic density of states and Fig. 4.5 shows the one-dimensional band structure of a (6,

6) nanotube with ABA stacking. The band gap energy is 0.24 eV. With increase of n, the

band gap is approaching the band gap of free-standing layer of MoS2 with ABA stacking.

All armchair tubes with ABC stacking are metallic. Fig. 4.6 illustrates the electronic

density of states of (6, 6) nanotube with ABC stacking.

The results for band gaps of armchair tubes are shown in Table 4.2.

For (n, n) tubes, there are two different Mo-Mo bond distances. One is parallel to

46

Figure 4.1: Cross-section of zigzag (10, 0) of MoS2 nanotube. Larger atoms are Mo.

47

Figure 4.2: Cross-section of armchair (14, 14) of MoS2 nanotube. Larger atoms are Mo.

48

Figure 4.3: Cross-section of supercell of (12, 12) MoS2 nanotube. larger atoms are Mo.This structures contains 6-fold symmetry.

49

Free-standing Layer (14, 14) (12, 12) (6, 6)Mo-Mo> 3.201 3.272 3.239 3.418S-S (Inner)> 3.201 2.980 2.886 2.732S-S (Outer)> 3.201 3.565 3.591 4.080S(Inner)-Mo⊥ 2.422 2.400 2.389 2.377S(Inner)-Mo> 2.422 2.397 2.377 2.368S(Outer)-Mo⊥ 2.422 2.496 2.491 2.635S(Outer)-Mo> 2.422 2.434 2.443 2.457Lz(Mo||,S||) 3.201 3.189 3.211 3.220

Table 4.2: Bond lengths for (n, n) tubes with ABA stacking MoS2 nanotubes. || indicatesthe bond is parallel to the tube axis. ⊥ indicates the bond is perpendicular to the tubeaxis. > indicates the bond is 60 degrees to the tube axis. Lz(Mo||,S||) is the bond distanceof Mo-Mo and S-S parallel to the axis of tube. All the distances are in A.

the tube axis and the other is at 60 degrees before wrapping the free-standing layer. S-S

and S-Mo in inner and outer layers have the similar difference. Fig. 4.7 demonstrates the

changes of these bond distances for (n, n) tubes with ABA stacking nanotube. All bond

lengths are in Table 4.2.

4.3 Zigzag MoS2 nanotubes

Fig. 4.8 shows the electronic density of states of a (6, 0) tube which is constructed

from a MoS2 free-standing layer with ABA stacking. It clearly shows this nanotube is

metallic. Fig. 4.9 is the one-dimensional band structure.

This result contradicts the prediction made by Seifert et al.[29] who claimed that

all MoS2 nanotubes with ABA stacking are semiconducting.

Except the smallest (6, 0), all zigzag tubes with ABA stacking are semiconducting.

The band gap increases with n. Fig. 4.10, Fig. 4.10, Fig. 4.12 and Fig. 4.12 illustrate the

electronic density of states and one-dimensional band structure of (10, 0) and (18, 0) with

ABA stacking.

For ABC stacking, the trend of band gap with size is opposite. With increasing n,

the band gap energy decreases and finally it disappears. Fig. 4.14, Fig. 4.15, Fig. 4.16 and

50

0

10

20

30

40

50

60

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 4.4: Electronic density of state of (6, 6) ABA stacking MoS2 nanotube

51

!1

!0.5

0

0.5

1

1.5

0 !/l

Ener

gy (e

V)

k

Figure 4.5: Band structure of a (6, 6) tube with ABA stacking

52

0

10

20

30

40

50

60

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (e:)

Figure 4.6: Electronic density of state of (6, 6) ABC stacking MoS2 nanotube

53

2.35

2.4

2.45

2.5

2.55

2.6

2.65

5 10 15 20n

Bo

nd

Dsit

an

ce Å

Mo-S (Inner)

Mo-S (Inner)

Mo-S (Outer)

Mo-S (Outer)

Free-standing Layer

(a)

(b)

2

2.5

3

3.5

4

4.5

5 10 15 20n

Bo

nd

Dis

tan

ces (

Å)

S-S (Inner)

S-S (Outer)

Mo-Mo

Lz

Free-standing Layer

>

>

>

(Mo ,S )|| ||

⊥

⊥

Figure 4.7: Calculated bond distances of (n, n) tubes with ABA stacking tubes as functionof n. The bond distance of free-standing layer is shown as a reference. || indicates thebond is parallel to the tube axis. > indicates the bond is 60 degrees to the tube axis. Allbond distances are in A. (a) The bond distances of Mo-Mo, S-S in inner and outer layersas function of n. (b) The bond distances of Mo-S in inner and outer layers as function of n.

54

ABA stacking ABC stacking(6, 0) Metallic 0.13(10, 0) 0.31(12, 0) 0.11(18, 0) 0.99 Metallic

Table 4.3: The band gap energies of zigzag structures. All the band gaps are in eV.

Free-standing Layer (18, 0) (10, 0) (6, 0)Mo-Mo⊥ 3.201 3.339 3.621 4.271Mo-Mo< 3.201 3.213 3.240 3.248S-S (Inner)< 3.201 2.794 2.655 2.675S-S (Inner)⊥ 3.201 3.073 2.988 2.702S-S (Outer)⊥ 3.201 3.803 4.486 5.416S-S (Outer)< 3.201 3.352 3.506 3.609S(Inner)-Mo|| 2.422 2.372 2.326 2.320S(Inner)-Mo< 2.422 2.392 2.397 2.471S(Outer)-Mo|| 2.422 2.420 2.402 2.455S(Outer)-Mo< 2.422 2.489 2.587 2.745Lz(Mo||,S||) 5.544 5.472 5.341 4.639

Table 4.4: Bond lengths for (n, 0) tubes with ABA stacking MoS2 nanotubes. || indicatesthe bond is parallel to the tube axis. ⊥ indicates the bond is perpendicular to the tubeaxis. > indicates the bond is 60 degrees to the tube axis.Lz(Mo||,S||) is the bond distanceof Mo-Mo and S-S parallel to the axis of tube. All the distances are in A.

Fig. 4.17 show the electronic density of states and one-dimesional band structures of (n, 0)

tubes with ABC stacking.

All results of zigzag nanotubes are shown in Table 4.3.

Fig. 4.18 shows the bond distances which are Mo-Mo, S-S and Mo-S in inner and

outer layers. All the bond distances for (n, 0) tubes with ABA stacking are in Table 4.3.

In the paper by Seifert, et al.[29], only tubes with ABA stacking were calculated

and only Mo-Mo, Mo-S bond distances were mentioned. However, they failed to clarify the

Mo-Mo and Mo-S with different angles. Some bonds are parallel to the tube axis, some

perpendicular and the others at 30 or 60 degrees. Also they failed to discuss how the bond

distance depends on n. According to Ref.[29], the optimized Mo-Mo and Mo-S bond lengths

in tubes are larger than those of the planar sheet. Our results show, for armchair tubes,

55

(6, 0)ABA (6, 0)ABC (6, 6)ABA (6, 6)ABCNumber of Atoms 36 36 36 36Band Gap Metallic 0.13 0.24 MetallicEnergy -230 -234 -246 -220

Table 4.5: Comparison of band gap and total energy. All the band gaps are in eV. All theenergies are in eV/unit.

S-S in inner layer and Mo-S bond are smaller than those of free-standing layer; for zigzag

tubes, S-S in inner layer is also smaller.

4.4 Comparison of zigzag and armchair

We plot the band gap energies vs. n in Fig. 4.19. With increasing n, the band gaps

in both armchair and zigzag nanotubes with ABA stacking approach the band gap of a

free-standing layer.

Because the (6,0) and (6,6) structures contain the same number atoms, we can

directly compare their total energies. Table 4.4 summarizes the results of 4 different nan-

otubes. The (6, 0) with ABC and (6, 6) with ABA stacking have the lower energy than the

other two. These two tubes with lower total energy are semiconducting, while (6, 0) with

ABA and (6, 6) with ABC stacking structures with higher total energy are metallic.

We also compare (12, 0) and (12, 12) structures. (12, 12) tube with ABC stacking

which is metallic has higher total energy than the rest tubes. So for a given n, the tube

with lowest energy is always semiconducting.

We also plot the strain energy vs. n in Fig. 4.20. With increasing n, all strain

energies approach zero. The armchair tubes with ABC stacking which are all metallic have

the highest strain energy. The strain energy of zigzag nanotubes with ABA stacking is

higher than zigzag with ABC stacking. The armchair structures with ABA stacking which

are all semiconducting have the lowest strain energy.

56

0

10

20

30

40

50

60

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 4.8: Electronic density of state of (6, 0) ABA stacking MoS2 nanotube

57

!1

!0.5

0

0.5

1

0 !/l

Ener

gy (e

V)

k

Figure 4.9: Band structure of a (6, 0) tube with ABA stacking

58

0

10

20

30

40

50

60

70

80

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 4.10: Electronic density of state of (10, 0) ABA stacking MoS2 nanotube

59

!1

!0.5

0

0.5

1

0 !/l

Ener

gy (e

V)

k

Figure 4.11: Band structure of a (10, 0) tube with ABA stacking

60

0

200

400

600

800

1000

1200

1400

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 4.12: Electronic density of state of a (18, 0) ABA stacking MoS2 nanotube. Thedash line is the density of states and full line is the total density of states

61

!1

!0#5

0

0#5

1

1#5

0

Ener

gy (e

V)

k

Figure 4.13: Band structure of a (18, 0) tube with ABA stacking

62

0

10

20

30

40

50

60

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 4.14: Electronic density of state of (6, 0) ABC stacking MoS2 nanotube

63

!1

!0.5

0

0.5

1

0 !/l

Ener

gy (e

V)

k

Figure 4.15: Band structure of a (6, 0) tube with ABC stacking

64

!1

!0.5

0

0.5

1

0 !/l

Ener

gy (e

V)

k

Figure 4.16: Band structure of a (6, 0) tube with ABC stacking

65

0

20

40

60

80

100

120

140

!5 !4 !3 !2 !1 0 1 2 3

Num

ber o

f Sta

tes

Energy (eV)

Figure 4.17: Electronic density of state of (18, 0) ABC stacking MoS2 nanotube

66

2

3

4

5

6

5 10 15 20 25n

Bo

nd

Dis

tan

ce Å

Mo-Mo

Mo-Mo

S-S (Inner)

S-S (Inner)

S-S (Outer)

S-S (Outer)

Free-standing Layer

2

2.2

2.4

2.6

2.8

5 10 15 20 25

n

Bo

nd

Dis

tan

ce (

Å)

S(Inner)-Mo S(Inner)-Mo

S(Outer)-Mo S(Outer)-Mo

Free-standing Layer

(a)

(b)

⊥

⊥⊥

<

<

<

<

<

||

||

Figure 4.18: Calculated bond distances of (n, 0) tubes with ABA stacking tubes as functionof n. The bond distance of free-standing layer is shown as a reference. The S atom ininner layer connects 4 Mo atoms. Because of symmetry, we only show two of Mo-S(Inner)distances. Same reason for the Mo-S in outer layer. || indicates the bond is parallel to thetube axis. ⊥ indicates the bond is perpendicular to the tube axis. < indicates the bond is30 degrees to the tube axis. All bond distances are in A. (a) The bond distances of Mo-Mo,S-S in inner and outer layers as function of n. (b) The bond distances of Mo-S in inner andouter layers as function of n.

67

0

0.5

1

1.5

2

5 10 15 20n

Ban

d G

ap

(eV

)

(n, n) tubes

(n, 0) tubes

Free-standing Layer

Figure 4.19: Calculated band gap energies of MoS2 nanotubes with ABA stacking as func-tion of n. The band gap of free-standing layer is shown as a reference. All band gap energiesare in eV.

68

0

2

4

6

8

10

12

14

16

5 10 15n

Str

ain

En

erg

y (

eV

)

(n, 0) w/ABA

(n, n) w/ABA

(n, 0) w/ABC

(n, n) w/ABC

Figure 4.20: Calculated strain energies per MoS2 unit per unit length of the tube as functionof n. All energies are in eV.

69

Chapter 5

Conclusion

We use a first-principles electronic structure method to investigate the electronic

structure of MoS2 nanotubes. Both zigzag (n, 0) and armchair (n, n) nanotubes are studied.

Also the nanotubes are constructed with ABA and ABC stacking. For armchair structures,

all (n, n) tubes with ABA stacking are semiconducting and with ABC stacking are metal-

lic. For zigzag structures, almost all the nanotubes with ABA stacking and small tubes

with ABC stacking are semiconducting. With increasing n, zigzag and armchair tubes will

demonstrate similar characteristics to free-standing layer system. For armchair and zigzag

tubes of a given n, the tube with lowest energy is semiconducting.

70

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