Electronic Structure of MoS 2 Nanotubes

Transcription

1 Clemson University TigerPrints All Dissertations Dissertations Electronic Structure of MoS 2 Nanotubes Lingyun Xu Clemson University, neilxu@gmail.com Follow this and additional works at: Part of the Condensed Matter Physics Commons Recommended Citation Xu, Lingyun, "Electronic Structure of MoS 2 Nanotubes" (2007). All Dissertations. Paper 116. This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact awesole@clemson.edu.

2 Electronic Structure of MoS 2 Nanotubes A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Physics by Lingyun Xu August 2007 Accepted by: Dr. Murray S. Daw, Committee Chair Dr. Apparao M. Rao Dr. D. Catalina Marinescu Dr. Pu-Chun Ke

3 Abstract First-principles methods enable one to study the electronic structure of solids, surfaces, 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 MoS 2 with ABA and ABC stacking. Our results suggest MoS 2 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 MoS 2 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 MoS 2 nanotubes has been studied using a first-principles electronic structure method. We investigated MoS 2 zigzag (n, 0) nanotubes as well as armchair (n, n) structures. We constructed MoS 2 nanotubes with ABA and ABC stacking. 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 configuration, bond lengths and resulting electronic structure in several MoS 2 nanotubes. For zigzag structures, almost all the nanotubes with ABA stacking and small tubes with ABC stacking 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

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

5 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 I also acknowledge the use of VASP and DOE support through time on NERSC. iv

6 Table of Contents Title Page i Abstract ii Dedication iii Acknowledgments iv List of Tables vi List of Figures vii 1 Introduction Approach First-Principles Calculation Software Free-Standing Layer of MoS Tight-binding Method Tight Binding Theory Slater-Koster Matrix of Single Layer of Sulfur Slater-Koster Matrix of Single Layer of Mo Results Configurations of nanotubes of MoS Armchair MoS 2 nanotubes Zigzag MoS 2 nanotubes Comparison of zigzag and armchair Conclusion Bibliography v

7 List of Tables 2.1 Optimized bond distances for MoS 2 layers with two types of stacking. All the distances are in Å Impurities in free-standing layer of (MoS 2 ) 4 with ABA stacking The band gap energies of armchair structures. All the band gaps are in ev Bond lengths for (n, n) tubes with ABA stacking MoS 2 nanotubes The band gap energies of zigzag structures. All the band gaps are in ev Bond lengths for (n, 0) tubes with ABA stacking MoS 2 nanotubes Comparison of band gap and total energy. All the band gaps are in ev. All the energies are in ev/unit vi

8 List of Figures 2.1 Full all-electronic wavefunction and electronic potential and the corresponding pseudo wavefunction and potential (a) Top view of free-standing layer of MoS 2 with ABA stacking. (b) Side view of same Electronic density of states of free-standing layer of MoS 2 with ABA stacking. The bandgap results from mirror-plane symmetry (a) Top view of free-standing layer of MoS 2 with ABC stacking. (b) Side view of same Electronic density of states of a free-standing layer of MoS 2 with ABC stacking. The loss of mirror symmetry results in a metallic system Electronic Density of States of (MoS 2 ) 12 with one S vacancy Electronic Density of States of (MoS 2 ) 12 with one Mo vacancy Electronic Density of States of Mo 4 S 7 P Electronic Density of States of Mo 4 S 7 Cl Electronic Density of States of Mo 3 S 8 Nb Electronic Density of States of Mo 3 S 8 Tc Electronic Density of States of Mo 3 S 8 Ti Single Layer of S The band structures of single layer of S from two different calculation Cross-section of zigzag (10, 0) of MoS 2 nanotube. Larger atoms are Mo Cross-section of armchair (14, 14) of MoS 2 nanotube. Larger atoms are Mo Cross-section of supercell of (12, 12) MoS 2 nanotube. larger atoms are Mo. This structures contains 6-fold symmetry Electronic density of state of (6, 6) ABA stacking MoS 2 nanotube Band structure of a (6, 6) tube with ABA stacking Electronic density of state of (6, 6) ABC stacking MoS 2 nanotube Calculated bond distances of (n, n) tubes with ABA stacking tubes as function of n. The bond distance of free-standing layer is shown as a reference Electronic density of state of (6, 0) ABA stacking MoS 2 nanotube Band structure of a (6, 0) tube with ABA stacking Electronic density of state of (10, 0) ABA stacking MoS 2 nanotube Band structure of a (10, 0) tube with ABA stacking Electronic density of state of (18, 0) ABA stacking MoS 2 nanotube Band structure of a (18, 0) tube with ABA stacking vii

9 4.14 Electronic density of state of (6, 0) ABC stacking MoS 2 nanotube Band structure of a (6, 0) tube with ABC stacking Band structure of a (6, 0) tube with ABC stacking Electronic density of state of (18, 0) ABC stacking MoS 2 nanotube Calculated bond distances of (n, 0) tubes with ABA stacking tubes as function of n. The bond distance of free-standing layer is shown as a reference Calculated band gap energies of MoS 2 nanotubes with ABA stacking as function of n. The band gap of free-standing layer is shown as a reference. All band gap energies are in ev Calculated strain energies per MoS 2 unit per unit length of the tube as function of n. All energies are in ev viii

10 Chapter 1 Introduction Molybdenum disulfide (MoS 2 ) is a very interesting material with numerous applications [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 resulting 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 MoS 2 nanoparticles [5]. Single-wall subnanometer-diameter MoS 2 nanotubes were synthesized in 2001, with significant amounts of intercalated iodine [28]. Though there are some electronic measurements and ab initio calculations published for bulk MoS 2 [7, 3] and an ab initio study of MoS 2 I 1/3 nanotube bundles [32], no firstprinciples calculation of electronic structures has been reported for nanotubes of MoS 2. In July 2000, Seifert, et al. reported the electronic structure of MoS 2 nanotubes using density-functional-based-tight-binding (DFTB) [27, 9]. Their results found that both MoS 2 zigzag (n, 0) and armchair (n, n) nanotubes are semiconducting. MoS 2 forms in sheets composed of three triangularly packed layers, such that a layer 1

11 of Mo is sandwiched by S layers. Normally, MoS 2 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 MoS 2 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

12 Chapter 2 Approach 2.1 First-Principles Calculation Beginning with Schrödinger 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 Hartree-Fock Approximation To solve a many-body system with interactions, we start from (2.1): HΨ = EΨ (2.1) where Ψ( r 1, r 2,..., r N ) 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 nuclear 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

13 The Hamiltonian in Equation 2.1 describing the interaction of electrons and nuclei becomes: N H = ( h2 2m 2 i Ze 2 1 i=1 R r i R ) + 1 e 2 2 r i j i r j (2.2) Here r i 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 in one mole of a solid) and each electron contains 3N degrees 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 interaction 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( r 1, r 2,..., r N ) = ψ 1 ( r 1 )ψ 2 ( r 2 )... ψ N ( r N ) (2.3) So the one-electron Schrödinger equation is now: h2 2m 2 ψ i ( r) + [V ion ( r) + V e ( r)] ψ i ( r) = ɛ i ψ i ( r) (2.4) where the potential that the electron would feel from the ions: V ion ( r) = Ze 2 R 1 r i R (2.5) and V e is the potential that the electron would feel from other electrons. However, the product of N one-electron wavefunctions is incompatible with the 4

14 Pauli exclusion principle which requires the many-body wavefunctions to be antisymmetric under the interchange of two electrons, that is: Ψ( r 1, r 2,..., r N ) = Ψ( r 2, r 1,..., r N ) (2.6) The form of the wavefunction can be generalised to incorporate asymmetry by replacing the Hartree wavefunction by a Slater determinant of one electron wavefunctions. Ψ( r 1 σ1... r N σ N ) = 1 N ψ 1 ( r 1 σ 1 ) ψ 1 r 2 σ 2 ) ψ 1 ( r N σ N ) ψ N ( r 1 σ 1 ) ψ N ( r 2 σ 2 ) ψ N ( r N σ N ) (2.7) Under the Hartree-Fock approximation[8, 30], the equation 2.5 can be written as: h2 2m 2 ψ i ( r)+v ion ( r)ψ i ( r)+v e ( r)ψ i ( r) j d rψ j ( r )ψ i ( r )ψ j ( r)ψ i ( r) r r = ɛ i ψ i ( 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 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

15 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 V ext ( r), the minimum value of the total energy functional is the ground state energy of the system: E[n] = n( r)v ext ( r)d r + F [n] (2.9) where F is a universal functional of electronic density n, independent of V ext ( 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)v ext ( r)d r n( r)n( r ) r r d rd r + T [n( r)] + E XC [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 E XC is the energy of exchange and correlation of an interacting system. Unfortunately there is no known exact expression for either. The electronic density n would be: n( r) = BZ d 3 k ψ( r) 2 (2.11) 6

16 The V XC ( r) can be derived from: V XC ( r) = δe XC[n( r)] δn( r) (2.12) The equation 2.8 would be rewritten as: [ ] h2 2m 2 i + V eff ( r) ψ i ( r) = ɛ i ψ i ( r) (2.13) where the effective potential would be: V eff ( r) = V ext ( r) + n( r ) r r d r + V XC ( r) (2.14) Local Density Approximation and Generalized Gradient Approximation 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 Unfortunately, the form of E XC 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: E XC [n] = ɛ XC (n)n(r)d 3 r (2.15) where ɛ XC is equal to the exchange-correlation energy per electron in a homogeneous electron 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

17 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: E XC [n] = ɛ XC (n, n)n(r)d 3 r (2.16) 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: ψ ki ( r) = e i k r u ki ( r) (2.17) The k is a wavevector confined to the first Brillouin zone. The second term is also a periodic function, u ki ( r + l) = u ki ( r) (2.18) which can be expended to: u ki ( r) = G C ki ( G)e i 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: ψ i ( r) = G C i ( G)e i( k+ G) r (2.20) By the use of Bloch s theorem, the problem of differential equation in ψ( r) has now been 8

18 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 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 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. Unfortunately 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

19 tions of electrons in the core region. Usually, the valence electrons which surround the core region determine most physical properties of solids instead of the core electrons. This is the reason that the pseudopotential 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 occupied 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 r c. 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 pseudopotentials. 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

20 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 wavefunction and potential is plotted (dashed lines). Outside a a certain cutoff radius r c, the real electron and pseudo electron values are identical. 11

21 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 MoS Free-standing Layer of MoS 2 with ABA stacking Before investigating MoS 2 nanotubes, we examined as a reference flat, free-standing sheets of MoS 2, which is shown in Fig 2.2. During the calculation, we set the number of irreducible k-points in the 2D Brillouin 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 3.20Å, and 2z (S-S interlayer distance) is 3.13Å. These results are similar to the experimental results [3], which are a = 3.159Å, 2z = 3.172Å. Fig. 2.3 shows the density of states of a free-standing layer of MoS 2. In that figure, we can clearly see a band gap which is 1.64 ev. The free-standing layer of MoS 2 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 Free-standing Layer of MoS 2 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

22 Mo S (a) a 2z (b) Figure 2.2: (a) Top view of free-standing layer of MoS 2 with ABA stacking. (b) Side view of same. 13

23 10 8 Number of States !4! Energy (ev) Figure 2.3: Electronic density of states of free-standing layer of MoS 2 with ABA stacking. The bandgap results from mirror-plane symmetry. 14

24 The loss of the mirror plane symmetry causes the disappearance of the band gap, as shown in Fig The energy for ABC stacking is 0.54 ev per MoS 2 unit higher than that for ABA stacking, as is generally expected. Table shows the differences in bond distances between these two structures. ABA stacking ABC stacking S-S (intralayer) S-S (interlayer) Mo-S Table 2.1: Optimized bond distances for MoS 2 layers with two types of stacking. All the distances are in Å 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 (MoS 2 ) 12 with one S removed. Compared to the plot of MoS 2, 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 (MoS 2 ) 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 Mo 4 S 7 P 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 MoS 2. The all calculation results are listed in table The electronic density of states of other impurities are shown in Fig. 2.12, Fig. 2.10, Fig. 2.11, Fig

25 a S in first Layer Mo S in third Layer (a) 2z (b) Figure 2.4: (a) Top view of free-standing layer of MoS 2 with ABC stacking. (b) Side view of same. 16

26 6 5 Number of States !4! Energy (ev) Figure 2.5: Electronic density of states of a free-standing layer of MoS 2 with ABC stacking. The loss of mirror symmetry results in a metallic system. Impurities Mo 3 S 8 Nb Mo 3 S 8 Ti Mo 3 S 8 Tc Mo 4 S 7 P Mo 4 S 7 Cl Types p-type p-type n-type p-type n-type Table 2.2: Impurities in free-standing layer of (MoS 2 ) 4 with ABA stacking 17

27 )0 (0 Number of States % Figure 2.6:!4! Energy (ev) Electronic Density of States of (MoS 2 ) 12 with one S vacancy 18

28 )0 (0 Number of States % Figure 2.7:!4! Energy (ev) Electronic Density of States of (MoS 2 ) 12 with one Mo vacancy 19

29 25 20 Number of States !4! Energy (ev) Figure 2.8: Electronic Density of States of Mo 4 S 7 P 20

30 25 20 Number of States !4! Energy (ev) Figure 2.9: Electronic Density of States of Mo 4 S 7 Cl 21

31 25 20 Number of States !4! Energy (ev) Figure 2.10: Electronic Density of States of Mo 3 S 8 Nb 22

32 25 20 Number of States !5!4!3!2! Energy (ev) Figure 2.11: Electronic Density of States of Mo 3 S 8 Tc 23

33 25 20 Number of States !4! Energy (ev) Figure 2.12: Electronic Density of States of Mo 3 S 8 Ti 24

34 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 MoS 2 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 firstprinciples 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 tightbinding ) can be used as a basis for expanding the wavefunction. So the Schrödinger equation 2.1 needs to be rewritten. First of all, the eigenvector 25

35 ψ is expended out in terms of the basis functions: ψ = iα c iα φ iα (3.1) where the index i refers to the atoms and the Greek letter α to the orbitals on these atoms. φ iα is an orbital on atom i. The Hamiltonian can then be written as a matrix as: H iα,jβ = iα H jβ (3.2) The on-site integrals and represent the energies of the orbitals: φ s (r) H φ s (r) = ɛ s (3.3) φ px (r) H φ px (r) = ɛ p (3.4) x-axis): For example, the off-site interaction is: (if the bond is assumed to be along the φ s (r) H φ s (r ± b) = V ss (3.5) It is assumed that the interaction between orbitals on different atoms to be independent 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. φ px (r) φ py (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 disadvantage is that the magnitude and distance variation of the matrix elements must be 26

36 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 structure. We start to construct Slater-Koster matrix of single layer of S. Fig. 3.1 shows this Hamiltonian Matrix Elements On Site φ s (r) H φ s (r) = ɛ s φ s (r) H φ px (r) = 0 φ px (r) H φ px (r) = ɛ p φ px (r) H φ py (r) = off site x ± a x φ s (r) H φ s (r ± a x) = V ss φ s (r) H φ px (r ± a x) = V sp 27

37 a Figure 3.1: Single Layer of S. Each S atom has 6 nearest-neighbors. 28

38 φ s (r) H φ py (r ± a x) = 0 φ s (r) H φ pz (r ± a x) = 0 φ px (r) H φ px (r ± a x) = V ppσ φ px (r) H φ py (r ± a x) = 0 φ px (r) H φ pz (r ± a x) = 0 φ py (r) H φ px (r ± a x) = 0 φ py (r) H φ py (r ± a x) = V ppπ φ py (r) H φ pz (r ± a x) = 0 φ pz (r) H φ px (r ± a x) = 0 φ pz (r) H φ py (r ± a x) = 0 29

39 φ pz (r) H φ pz (r ± a x) = V ppπ off site r ± b, where b = ( 1 2, 3 2 )a φ s (r) H φ s (r ± b) = V ss φ s (r) H φ px (r ± b) = V sp cos θ φ s (r) H φ py (r ± b) = V sp sin θ φ s (r) H φ pz (r ± b) = 0 φ px (r) H φ px (r ± b) = V ppσ cos 2 θ + V ppπ sin 2 θ φ px (r) H φ py (r ± b) = (V ppσ V ppπ ) sin θ cos θ φ px (r) H φ pz (r ± b) = 0 φ py (r) H φ px (r ± b) = (V ppσ V ppπ ) sin θ cos θ φ py (r) H φ py (r ± b) = V ppσ sin 2 θ + V ppπ cos 2 θ 30

40 φ py (r) H φ pz (r ± b) = 0 φ pz (r) H φ px (r ± b) = 0 φ pz (r) H φ py (r ± b) = 0 φ pz (r) H φ pz (r ± b) = V ppπ off site r ± c, where c = ( 1 2, 3 2 )a φ s (r) H φ s (r ± c) = V ss φ s (r) H φ px (r ± c) = ±V sp cos θ φ s (r) H φ py (r ± c) = ±V sp sin θ φ s (r) H φ pz (r ± c) = 0 φ px (r) H φ px (r ± c) = V ppσ cos 2 θ + V ppπ sin 2 θ φ px (r) H φ py (r ± c) = (V ppπ V ppσ ) sin θ cos θ 31

41 φ px (r) H φ pz (r ± c) = 0 φ py (r) H φ px (r ± c) = (V ppπ V ppσ ) sin θ cos θ φ py (r) H φ py (r ± c) = V ppσ sin 2 θ + V ppπ cos 2 θ φ py (r) H φ pz (r ± c) = 0 φ pz (r) H φ px (r ± c) = 0 φ pz (r) H φ py (r ± c) = 0 φ pz (r) H φ pz (r ± c) = V ppπ Hamiltonian Matrix for k General Form A B C D B E F G C F H I D G I J A = φ s (r) H φ s (r) + φ s (r) H φ s (r ± a x) e ik a x 32

42 + φ s (r) H φ s (r ± b) e ik b + φ s (r) H φ s (r ± c) e ik c = ɛ s + 2V ss cos (ak x ) + 4V ss cos ( a 3 2 k x) cos ( 2 k y) B = φ s (r) H φ px (r) + φ s (r) H φ px (r ± a x) e ik a x + φ s (r) H φ px (r ± b) e ik b + φ s (r) H φ px (r ± c) e ik c = 2iV sp [sin(ak x ) + sin( a 3 2 k x) cos( 2 k y)] C = φ s (r) H φ py (r) + φ s (r) H φ py (r ± a x) e ik a x + φ s (r) H φ py (r ± b) e ik b + φ s (r) H φ py (r ± c) e ik c = 2 3iV sp sin( a 3 2 k x) cos( 2 k y) D = φ s (r) H φ pz (r) + φ s (r) H φ pz (r ± a x) e ik a x = 0 + φ s (r) H φ pz (r ± b) e ik b + φ s (r) H φ pz (r ± c) e ik c E = φ px (r) H φ px (r) + φ px (r) H φ px (r ± a x) e ik a x + φ px (r) H φ px (r ± b) e ik b + φ px (r) H φ px (r ± c) e ik c = ɛ p + 2V ppσ cos(ak x ) +(V ppσ + 3V ppπ ) cos( a 2 k x) cos( 3 2 k y) 33

43 F = φ px (r) H φ py (r) + φ px (r) H φ py (r ± a x) e ik a x + φ px (r) H φ py (r ± b) e ik b + φ px (r) H φ py (r ± c) e ik c = 3(V ppπ V ppσ ) sin( a 3 2 k x) sin( 2 k y) G = φ px (r) H φ pz (r) + φ px (r) H φ pz (r ± a x) e ik a x = 0 + φ px (r) H φ pz (r ± b) e ik b + φ px (r) H φ pz (r ± c) e ik c H = φ py (r) H φ py (r) + φ py (r) H φ py (r ± a x) e ik a x + φ py (r) H φ py (r ± b) e ik b + φ py (r) H φ py (r ± c) e ik c = ɛ p + 2V ppπ cos(ak x ) +(3V ppσ + V ppπ ) cos( a 2 k x) cos( 3 2 k y) I = φ py (r) H φ pz (r) + φ py (r) H φ pz (r ± a x) e ik a x = 0 + φ py (r) H φ pz (r ± b) e ik b + φ py (r) H φ pz (r ± c) e ik c J = φ pz (r) H φ pz (r) + φ pz (r) H φ pz (r ± a x) e ik a x + φ pz (r) H φ pz (r ± b) e ik b + φ pz (r) H φ pz (r ± c) e ik c = ɛ p + 2V ssπ cos (ak x ) + 4V ssπ cos ( a 3 2 k x) cos ( 2 k y) 34

44 matrix at high symmetry points Γ = (0, 0, 0) 2π a ɛ s + 6V ss ɛ p + 3V ppσ + 3V ppπ ɛ p + 3V ppσ + 3V ppπ ɛ p + 6V ppπ Q = (0, 2 3, 0) 2π a ɛ s + 6V ss ɛ p + 3V ppσ + 3V ppπ ɛ p + 3V ppσ + 3V ppπ ɛ p + 6V ppπ P = ( 1 2, 3 2, 0)2π a ɛ s 2V ss ɛ p 2V ppσ 3(Vppσ Vppπ ) 0 0 3(Vppσ Vppπ ) ɛ p 2V ppπ ɛ p 2V ppπ 35

45 3.2.3 Parameter We have the band structure of single layer of S using the first-principles method. Compared to these results, ɛ s, ɛ p, V ss, V ppσ and V ppπ can be calculated based on the band structure. Fig. 3.2 shows the band structures with these two methods. ɛ s = ev ɛ p = ev V ss = ev V ppσ = ev V ppπ = ev 3.3 Slater-Koster Matrix of Single Layer of Mo Hamiltonian Matrix Elements On Site φ dxy (r) H φ dxy (r) = ɛ d φ dxy (r) H φ dyz (r) = 0 36

46 3 2 1 Energy (ev) 0!1!2!3!4!5! k!points Q Figure 3.2: The band structures of single layer of S from two different calculation. The full line is the band structure from first-principles method. The dashed line is the band structure from tight-binding method. 37

47 φ dxy (r) H φ dzx (r) = 0 φ dxy (r) H φ dx 2 y 2 (r) = 0 φ dxy (r) H φ d3z 2 r 2 (r) = off site x ± a x φ dxy (r) H φ dxy (r ± a x) = V ddπ φ dyz (r) H φ dyz (r ± a x) = V ddδ φ dzx (r) H φ dzx (r ± a x) = V ddπ φ dx 2 y 2 (r) H φ d x 2 y 2 (r ± a x) = 3 4 V ddσ V ddδ 3 3 φ (r) H φ dx 2 y 2 d (r ± a x) = 3z 2 r 2 4 V ddσ + 4 V ddδ φ d3z 2 r 2 (r) H φ d 3z 2 r 2 (r ± a x) = 1 4 V ddσ V ddδ 38

48 off site r ± b, where b = ( 1 2, 3 2 )a φ dxy (r) H φ dxy (r ± a x) = 9 16 V ddσ V ddπ V ddδ φ dxy (r) H φ (r ± a x) = dx 2 y 2 16 V ddσ + 4 V ddπ 16 V ddδ φ dxy (r) H φ d3z 2 r 2 (r ± a x) = 3 8 V ddσ 3 8 V ddδ φ dyz (r) H φ dyz (r ± a x) = 3 4 V ddπ V ddδ φ dyz (r) H φ dzx (r ± a x) = V ddπ 4 V ddδ φ dzx (r) H φ dzx (r ± a x) = 1 4 V ddπ V ddδ φ dx 2 y 2 (r) H φ d x 2 y 2 (r ± a x) = 3 16 V ddσ V ddπ V ddδ 3 3 φ (r) H φ dx 2 y 2 d (r ± a x) = 3z 2 r 2 8 V ddσ 8 V ddδ φ d3z 2 r 2 (r) H φ d 3z 2 r 2 (r ± a x) = 1 4 V ddσ V ddδ 39

49 off site r ± c, where c = ( 1 2, 3 2 )a φ dxy (r) H φ dxy (r ± a x) = 9 16 V ddσ V ddπ V ddδ φ dxy (r) H φ (r ± a x) = dx 2 y 2 16 V ddσ 4 V ddπ + 16 V ddδ φ dxy (r) H φ d3z 2 r 2 (r ± a x) = 3 8 V ddσ V ddδ φ dyz (r) H φ dyz (r ± a x) = 3 4 V ddπ V ddδ φ dyz (r) H φ dzx (r ± a x) = V ddπ + 4 V ddδ φ dzx (r) H φ dzx (r ± a x) = 1 4 V ddπ V ddδ φ dx 2 y 2 (r) H φ d x 2 y 2 (r ± a x) = 3 16 V ddσ V ddπ V ddδ 3 3 φ (r) H φ dx 2 y 2 d (r ± a x) = 3z 2 r 2 8 V ddσ 8 V ddδ φ d3z 2 r 2 (r) H φ d 3z 2 r 2 (r ± a x) = 1 4 V ddσ V ddδ 40

50 3.3.3 Hamiltonian Matrix for k 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 = φ dxy (r) H φ dxy (r) + φ dxy (r) H φ dxy (r ± a x) e ik a x + φ dxy (r) H φ dxy (r ± b) e ik b + φ dxy (r) H φ dxy (r ± c) e ik c = ɛ d + 2V ddπ cos (ak x ) + 4 cos ( a 3 2 k x) cos ( 2 k y)( 9 16 V ddσ V ddπ V ddδ) B = 0 C = 0 D = φ dxy (r) H φ (r) + φ dx 2 y 2 d xy (r) H φ dx 2 y 2 (r ± a x) e ik a x + φ dxy (r) H φ (r ± b) e ik b dx 2 y 2 + φ dxy (r) H φ dx 2 y 2 (r ± c) e ik c = 4 sin( a 3 2 k x) sin( 2 k y) ( V ddσ 4 V ddπ + 16 V ddδ) 41

51 E = φ dxy (r) H φ (r) + φ d3z 2 r 2 d xy (r) H φ d3z 2 r 2 (r ± a x) e ik a x + φ dxy (r) H φ (r ± b) e ik b d3z 2 r 2 + φ dxy (r) H φ d3z 2 r 2 (r ± c) e ik c = 4 sin( a 3 2 k x) sin( 2 k y)( 3 8 V ddσ V ddδ) F = φ dyz (r) H φ dyz (r) + φ dyz (r) H φ dyz (r ± a x) e ik a x + φ dyz (r) H φ dyz (r ± b) e ik b + φ dyz (r) H φ dyz (r ± c) e ik c = ɛ d + 2V ddδ cos (ak x ) + 4 cos ( a 3 2 k x) cos ( 2 k y)( 3 4 V ddπ V ddδ) G = φ dyz (r) H φ dzx (r) + φ dyz (r) H φ dzx (r ± a x) e ik a x + φ dyz (r) H φ dzx (r ± b) e ik b + φ dyz (r) H φ dzx (r ± c) e ik c = 4 sin( a k x) sin( 2 k y)( 4 V ddπ 4 V ddδ) H = φ dyz (r) H φ (r) + φ dx 2 y 2 d yz (r) H φ dx 2 y 2 (r ± a x) e ik a x = 0 + φ dyz (r) H φ (r ± b) e ik b dx 2 y 2 + φ dyz (r) H φ dx 2 y 2 (r ± c) e ik c I = φ dyz (r) H φ (r) + φ d3z 2 r 2 d yz (r) H φ d3z 2 r 2 (r ± a x) e ik a x = 0 + φ dyz (r) H φ (r ± b) e ik b d3z 2 r 2 + φ dyz (r) H φ d3z 2 r 2 (r ± c) e ik c 42

52 J = φ dxz (r) H φ dzx (r) + φ dzx (r) H φ dzx (r ± a x) e ik a x + φ dzx (r) H φ dzx (r ± b) e ik b + φ dzx (r) H φ dzx (r ± c) e ik c = ɛ d + 2V ddπ cos (ak x ) + 4 cos ( a 3 2 k x) cos ( 2 k y)( 1 4 V ddπ V ddδ) K = φ dzx (r) H φ (r) + φ dx 2 y 2 d zx (r) H φ dx 2 y 2 (r ± a x) e ik a x = 0 + φ dzx (r) H φ (r ± b) e ik b dx 2 y 2 + φ dzx (r) H φ dx 2 y 2 (r ± c) e ik c L = φ dzx (r) H φ (r) + φ d3z 2 r 2 d zx (r) H φ d3z 2 r 2 (r ± a x) e ik a x = 0 + φ dzx (r) H φ (r ± b) e ik b d3z 2 r 2 + φ dzx (r) H φ d3z 2 r 2 (r ± c) e ik c M = φ (r) H φ dx 2 y 2 d (r) + φ x 2 y 2 d (r) H φ x 2 y 2 d x 2 y 2 (r ± a x) e ik a x + φ (r) H φ dx 2 y 2 d (r ± b) e ik b x 2 y 2 + φ (r) H φ dx 2 y 2 d x 2 y 2 (r ± c) e ik c = ɛ d + 2( 3 4 V ddσ V ddδ) cos (ak x ) + 4 cos ( a 3 2 k x) cos ( 2 k y)( 3 16 V ddσ V ddπ V ddδ) N = φ (r) H φ dx 2 y 2 d (r) + φ 3z 2 r 2 d (r) H φ x 2 y 2 d 3z 2 r 2 (r ± a x) e ik a x + φ (r) H φ dx 2 y 2 d (r ± b) e ik b 3z 2 r 2 + φ (r) H φ dx 2 y 2 d 3z 2 r 2 (r ± c) e ik c 3 3 = 2( 4 V ddσ + 4 V ddδ) cos (ak x ) + 4 cos ( a k x) cos ( 2 k y)( 8 V ddσ 8 V ddδ) 43

53 O = φ (r) H φ d3z 2 r 2 d (r) + φ 3z 2 r 2 d (r) H φ 3z 2 r 2 d 3z 2 r 2 (r ± a x) e ik a x + φ (r) H φ d3z 2 r 2 d (r ± b) e ik b 3z 2 r 2 + φ (r) H φ d3z 2 r 2 d 3z 2 r 2 (r ± c) e ik c = ɛ d + 2( 1 4 V ddσ V ddδ) cos (ak x ) + 4 cos ( a 3 2 k x) cos ( 2 k y)( 1 4 V ddσ V ddδ) 44

54 Chapter 4 Results 4.1 Configurations of nanotubes of MoS 2 Like carbon nanotubes [10], MoS 2 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) MoS 2 tube, respectively. For carbon nanotubes, the bonds between C of a (n, 0) tube have a zigzag appearance, and of a (n, n) tube look like armchairs. For MoS 2 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

55 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 12Å 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 MoS 2 nanotubes The results of armchair (n, n) MoS 2 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 MoS 2 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

56 Figure 4.1: Cross-section of zigzag (10, 0) of MoS 2 nanotube. Larger atoms are Mo. 47

57 Figure 4.2: Cross-section of armchair (14, 14) of MoS 2 nanotube. Larger atoms are Mo. 48

58 Figure 4.3: Cross-section of supercell of (12, 12) MoS 2 nanotube. larger atoms are Mo. This structures contains 6-fold symmetry. 49

59 Free-standing Layer (14, 14) (12, 12) (6, 6) Mo-Mo > S-S (Inner) > S-S (Outer) > S(Inner)-Mo S(Inner)-Mo > S(Outer)-Mo S(Outer)-Mo > L z (Mo,S ) Table 4.2: Bond lengths for (n, n) tubes with ABA stacking MoS 2 nanotubes. indicates the bond is parallel to the tube axis. indicates the bond is perpendicular to the tube axis. > indicates the bond is 60 degrees to the tube axis. L z (Mo,S ) is the bond distance of Mo-Mo and S-S parallel to the axis of tube. All the distances are in Å. 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 Zigzag MoS 2 nanotubes Fig. 4.8 shows the electronic density of states of a (6, 0) tube which is constructed from a MoS 2 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 MoS 2 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 and Fig 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 and 50

60 60 50 Number of States !5!4!3!2! Energy (ev) Figure 4.4: Electronic density of state of (6, 6) ABA stacking MoS 2 nanotube 51

61 1.5 1 Energy (ev) 0.5 0!0.5!1 0 k!/l Figure 4.5: Band structure of a (6, 6) tube with ABA stacking 52

62 60 50 Number of States !5!4!3!2! Energy (e:) Figure 4.6: Electronic density of state of (6, 6) ABC stacking MoS 2 nanotube 53

63 Bond Dsitance Å Bond Distances (Å) n (a) S-S (Inner) S-S (Outer) Mo-Mo > Lz(Mo,S ) Free-standing Layer Mo-S (Inner) Mo-S (Inner) > Mo-S (Outer) Mo-S (Outer) > Free-standing Layer n (b) Figure 4.7: Calculated bond distances of (n, n) tubes with ABA stacking tubes as function of n. The bond distance of free-standing layer is shown as a reference. indicates the bond is parallel to the tube axis. > indicates the bond is 60 degrees to the tube axis. All bond distances are in Å. (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 and outer layers as function of n. 54

64 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 Mo-Mo < S-S (Inner) < S-S (Inner) S-S (Outer) S-S (Outer) < S(Inner)-Mo S(Inner)-Mo < S(Outer)-Mo S(Outer)-Mo < L z (Mo,S ) Table 4.4: Bond lengths for (n, 0) tubes with ABA stacking MoS 2 nanotubes. indicates the bond is parallel to the tube axis. indicates the bond is perpendicular to the tube axis. > indicates the bond is 60 degrees to the tube axis.l z (Mo,S ) is the bond distance of Mo-Mo and S-S parallel to the axis of tube. All the distances are in Å. Fig 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 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

65 (6, 0)ABA (6, 0)ABC (6, 6)ABA (6, 6)ABC Number of Atoms Band Gap Metallic Metallic Energy Table 4.5: Comparison of band gap and total energy. All the band gaps are in ev. All the energies 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 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 nanotubes. 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 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

66 60 50 Number of States !5!4!3!2! Energy (ev) Figure 4.8: Electronic density of state of (6, 0) ABA stacking MoS 2 nanotube 57