Centro de Estudos Gerais
Instituto de F´ısica
Graduac
¸˜
ao em F´ısica
Matheus Samuel Martins de Sousa
Effects of the intrinsic spin-orbit interaction on the electronic
structure of nanotubes
Niter´
oi-RJ
2019
Effects of the intrinsic spin-orbit interaction on the electronic structure of nanotubes
Trabalho de Conclus˜ao de Curso apresentado ao Programa de Gradua¸c˜ao em F´ısica do Instituto de F´ısica da Universidade Federal Fluminense, como requisito parcial para obten¸c˜ao do grau de Bacharel em F´ısica.
Orientador: Prof. Dr. MARCOS SERGIO FIGUEIRA DA SILVA
Niter´oi-RJ 2019
Ficha catalográfica automática - SDC/BIF Gerada com informações fornecidas pelo autor
Bibliotecário responsável: Mario Henrique de Oliveira Castro - CRB7/6155
S725e Sousa, Matheus Samuel Martins de
Effects of the intrinsic spin-orbit interaction on the electronic structure of nanotubes / Matheus Samuel Martins de Sousa ; Marcos Sérgio Figueira da Silva, orientador. Niterói, 2019.
54 f. : il.
Trabalho de Conclusão de Curso (Graduação em Física)-Universidade Federal Fluminense, Instituto de Física, Niterói, 2019.
1. Grafeno. 2. Interação spin-órbita. 3. Produção
intelectual. I. Silva, Marcos Sérgio Figueira da, orientador. II. Universidade Federal Fluminense. Instituto de Física. III. Título.
Dedico este trabalho `a minha m˜ae Maisa Dias Martins de Sousa e ao meu pai T´acio de Sousa.
Primeiramente gostaria de agradedecer `a minha m˜ae Maisa e meu pai T´acio por sempre me incentivar, aconselhar e de me dar oportunidade e liberdade de fazer o que amo. Agrade¸co por tudo que fizeram por mim durante minha vida e que sem eles jamais poderia ter chegado neste momento. Agrade¸co tamb´em `
a muitos dos meus familiares, minha av´o Maria, meu avˆo Izael, meus tios e primos do Rio de Janeiro. Tamb´em aos meus grandes amigos Rafael, Erik, Victor, Igor e Jo˜ao. Agrade¸co aos meus colegas da UFF e de Niter´oi, lugar onde vivi meus ´ultimos cinco anos e que fiz muitas amizades que almejo levar para a vida. A todos os professores da UFF que me ajudaram nessa trajet´oria culminando neste trabalho, em particular os professores do Instituto de F´ısica da UFF. Em particular agradecer ao professor George Martins com quem tive contato `a partir de 2017 e com quem aprendi in´umeras coisas, pela sua amizade, hospitalidade e por estar sempre disposto a ensinar, ajudar e que certamente marcou este per´ıodo e que almejo muito. Ao professor Marcos S´ergio que me orientou neste trabalho, pela sua amizade, por sempre ter me ajudado e ter me ensinou muito, dando conselhos e com quem aprendi muitas coisas, muitas delas que levarei para minha vida.
RESUMO
Este trabalho tem como finalidade estudar os efeitos da intera¸c˜ao spin-´orbita intr´ınsica nos nan-otubos de carbono. Para isto, um estudo anal´ıtico detalhado da estrutura eletrˆonica do grafeno ´e feito, onde o modelo tight-binding ´e utilizado. Dentro deste modelo s˜ao calculadas as bandas de energia do grafeno e ´e feita uma aproxima¸c˜ao perto dos pontos de alta simetria do grafeno, os chamados pontos de Dirac. O efeito da intera¸c˜ao spin-´orbita intr´ınsica ent˜ao ´e estudada utilizando o modelo de Kane-Mele para o Spin-Hall quˆantico e sua intera¸c˜ao com o grafeno ´e observada. No trabalho foi poss´ıvel observar o que acontece com os cones de Dirac na presen¸ca da intera¸c˜ao spin-´orbita intr´ınsica, a abertura de uma separa¸c˜ao entre as bandas de valˆencia e de condu¸c˜ao.
Um estudo anal´ıtico dos nanotubos de carbono ´e feito, onde as formas de como se construir um nanotubo ´e demonstrada. Os nanotubos s˜ao classificados em trˆes categorias, chiral, poltrona e zigzag a partir de suas simetrias. Um estudo dos nanotubos de carbono poltronas e zigzags s˜ao feito, uma vez tendo suas estruturas eletrˆonica calculadas. ´E observado que os nanotubos poltronas s˜ao mais interessantes por sempre serem met´alicos independente do seu tamanho, j´a os nanotubos zigzag podem ser met´alicos ou isolantes dependendo da sua configura¸c˜ao. Um estudo anal´ıtico sobre as dispers˜oes para estes nanotubos na inclus˜ao da intera¸c˜ao spin-´orbita ´e feito, utilizando o m´etodo de enrolar a folha de grafeno utilizando condi¸c˜oes de contorno apropriadas. Para ambos os tipo de nanotubos o resultado ´e similar, o do aparecimento de um gap entre as bandas de condu¸c˜ao e de valˆencia.
Para complementar o estudo das propriedades eletrˆonicas desses materiais, ´e calculado a den-sidade de estados (DOS), utilizando o formalismo da fun¸c˜ao de Green e a conex˜ao entre os resultados te´oricos e experimentais ´e feita. As DOS s˜ao calculadas para os nanotubos sem a intera¸c˜ao spin-´orbita, sendo observado algo semelhante ao que se havia sido observado com as dispers˜oes. A transi¸c˜ao metal-isolante nos nanotubos poltrona s˜ao observados tamb´em pela DOS, a transi¸c˜ao se d´a pelo desaparecimento da densidade que existia anteriormente pr´oximo `a energia nula dando origem a um gap entre as bandas. Isso cria possibilidade de se manipular as propriedades eletrˆonica destes materials, em particular o gap entre as bandas. ´E poss´ıvel tamb´em observar que o tamanho deste gap ´e proporcional `a intensidade do acoplamento da intera¸c˜ao spin-´orbita.
This work has the goal of studying the effect of the intrinsic spin-orbit interaction in carbon nanotubes. To reach this goal a detailed analytical study of the electronic structure of graphene is done where the tight-binding model is utilized. Using this model the energy bands of graphene are calculated and the approximation for points close to the high symmetries points of graphene, the so-called Dirac points, is done. The intrinsic spin-orbit interaction is studied using the Kane-Mele model for the Quantum Spin Hall effect and its interaction with graphene is observed. In this work is possible to observe what happens with the Dirac cones in the presence of the intrinsic spin-orbit interaction, the opening of a separation between the valence and conduction bands.
An analytical study of the carbon nanotubes is done, where the many ways of constructing a nanotube is showed. In particular, the nanotubes are classified in three categories, chiral, armchair and zigzag due to its symmetries. A study of the armchair and zigzag nanotubes is done, having calculated the electronic structures. It is observed that the armchair nanotubes are more interesting because of the fact that they have a metallic conduction independent of its size, on the other hand, the zigzag nanotubes can be either metallic or insulator depending on its configuration. An analytical study of the dispersion of these nanotubes in the presence of the of the intrinsic spin-orbit interaction is done, using the technique of rolling up a graphene sheet using the appropriated boundary conditions. For both kinds of nanotubes the result is similar, there is a formation of a gap between the conduction and valence bands.
To complement the study of the electronic properties of these materiais, the density of states (DOS) is calculated, using the Green’s function formalism and a connection between theoretical and experimental results is done. The DOS are calculated for the nanotubes without the spin-orbit interaction, and it is observed something similar to what have been observed previously with the dispersions. A metal-insulator transition for the armchair nanotubes are also observed for the DOS, the transition happens when the density of state that previously existed in the null energy point disappear, creating an energy gap between the conduction and valence bands. This allows the possibility of manipulate the electronic properties of theses materials, in particular the energy gap between the bands. It is also possible to observe that the size of this gap created is proportional to the magnitude of the intrinsic spin-orbit coupling.
LIST OF FIGURES
1 Graphene is one-atom thick layer of carbon atoms arranged in a honeycomb lattice.. . . 5
3 Dispersion for graphene in the first Brillouin zone . . . 11
4 Linear behavior for the dispersion of graphene for points close to the high symmetry points K and K0, such cone formation in the energy dispersion is characteristic of a “Dirac Cone” 13
5 Typical crystalline structure for buckled materials such as Silicene, Germanene and Stanene. In (a) we see that the structure is essentially a honeycomb lattice, with the property that the two sublattices not in the the plane, but separated by a finite distance. In (b) we see that, we can account for the buckling by specifying the angle between the nearest neighbors that is denoted in Table I.. . . 14
6 Visualization of the sign of the matrix element νij as explained in [1]. The signs of the
second-neighboring hoppings are given by a right-hand rule for the red sublattice and left-hand rule for the blue sublattice. (+) sign is given by the green arrow while (-) sign is given by the blue arrow.. . . 15
7 In this picture, we can see the effect of the intrinsic spin-orbit interaction in the dispersion of graphene for two values of β. The interaction induces a gap between the conduction and valence bands. . . 19
8 Effect of the intrinsic spin-orbit interaction in the energy dispersion close to the points K and K0.. . . 21
9 Visualization of the cross-sectional ring of the carbon nanotubes in the case of the (a) Zigzag and (b) Armchair nanotubes.. . . 22
10 Classification of three types of carbon nanotubes with the the chiral vector, diameter and chiral angle. . . 23
11 Dispersion for armchair nanotubes for different configurations, reproducing the results of literature [2]. It can be noted that in the three pictures we have a metallic conduction.. . . 25
12 Dispersion for zigzag nanotubes for different configurations. The nanotube can either be a insulator or metal depending on n.. . . 26
13 Dispersion for armchair nanotubes with Ch = (5, 5) and for different values of β. The
opening of the gap can be noticed, and it is proportional to β. . . 28
14 Here we note that the intrinsic spin-orbit interaction (a) opens the gap in the nanotube that was previously a metal and (b) makes the gap even bigger for the nanotube that was already an insulator.. . . 29
15 DOS for armchair nanotubes, with Ch equal to (a) (7, 7) and (b) (12, 12). . . 33
16 DOS for zigzag nanotubes with Ch equal to (a) (9, 0) and (b) (11, 0). . . 34
17 Visualization the functions G±(k) for β = 0.1 in the first Brillouin Zone. The sharp peaks are located near the Dirac points. . . 37
18 DOS for armchair nanotubes with Ch equal to (a) (7, 7) and (b) (9, 9), in the presence of
the intrinsic spin-orbit interaction with λSO = 0.05. It can be noted that the interaction
Ch = (8, 8). It can be seen that the gap increases proportional to the magnitude of the
spin-orbit coupling λSO.. . . 39
20 Size of gap created by the intrinsic spin-orbit interaction for various armchair nanotubes with λSO = 0.2. Although the dependence with the spin-orbit coupling is very small, it
Resumo vii
Abstract viii
List of Figures ix
I. Introduction 1
II. Graphene — A Sheet of Carbon 2
II.1. Introduction to the formalism 2
II.2. Application to Graphene 4
III. Tight Binding Model Applied to Graphene — Second Quantization 6
III.1. First Neighbors Interaction 6
III.2. Expansion for Low Energies — First Neighbors Interaction 10
III.3. Intrinsic Spin-orbit — Second Neighbors Interaction 12
III.4. Expansion for Low Energies — Intrinsic Spin-orbit Interaction 18
IV. Single-Wall Nanotubes 21
IV.1. Classification of carbon nanotubes 21
IV.2. Electronic Properties 22
IV.3. Spin-orbit Interaction for Single-Wall Nanotubes 24
V. Calculating the Density of States 27
V.1. First Neighbors Interaction — Simplest Case 27
V.2. Density of States in the presence of Spin-Orbit interaction 32
VI. Conclusion and Future Works 36
A. Second Quantization 41
1. Fermions 42
B. Green’s Function 43
Graphene is the name given to a flat mono-layer of carbon atoms tightly packed into a two-dimensional honeycomb lattice [3] and has gathered great interest in the last two decades, mainly due to its remarkably properties such as stability, chemically inertness and crystalline under ambient conditions [4]. Theoretically, graphene was first studied by P. R. Wallace [5] and realized experimentally sixty years later by Geim [3] and his collaborators. First evidences for graphene were found by Geim and Novoselov [3] using simply a trace of graphite pencil and adhesive tape and rubbing it off against a silicon surface [4], but to produce actual specimes suitable for experiments, bulk graphite is gently pushed along a Si surface effectively drawing grapnene in to the surface [3]. The pieces of graphene, otherwise invisible, could only then be spotted by eye, with the help of optical microscopes, when placed on top of a Si wafer with a carefully chosen thickness of SiO2substrate [3].
Experiments confirmed what was before thought only an academic results, the electronics prop-erties of isolated graphene were exceptional. In particular, one of the theoretical predictions were that for low-energies electrons in graphene behaved as massless Dirac fermions [4] [6]. In that regime, graphene mimics the physics of quantum electrodynamics (QED) for massless fermions except that with velocity vF 300 times smaller than the speed of light [6]. Structurally, graphene has a sp2 hybridization between
one s orbital and two p orbitals, leading to a trigonal planar structure and forming a σ bond between carbon atoms that are separated by 1.42˚A [6].
Following the success of graphene there is a extensive search for other mono-layer materials. Materials by-products of graphene such as nanoribbons and nanotubes are still being throughly studied. In particular, it can be shown that mono-layer honeycomb systems made of other kinds of atoms such as silicon, germanium and tin are possible, giving rise to silicene, germanene and stanene [7]. These materials have a distinct property from graphene, a buckled structure [8] [9] due to an ionic radii much larger than in graphene. Evidence of a stable structure of silicene, having a buckled structure is found since 2010 [10] and this class of materials are being throughly studied experimentally.
As a consequence of the large ionic radius, these materials are expected to have a large spin-orbit interaction. This interaction leads to a spin-dependent shift of the orbitals with different signs for the sublattices of graphene, acting as a effective mass [6] allowing it to be probed experimentally with the Kane-Mele-type Quantum Spin Hall [1] and showing topological insulator properties [11]. Thus very rich physics arises by replacing carbon atoms with heavier atoms in a honeycomb system.
It is the goal of this monograph to explore part of this physics, in particular to make an analytical study of the role of the spin-orbit interaction in the context of nanotubes. First graphene is introduced and, using a tight-binding model, the energy dispersion is derived, together with the Dirac points. An investigation about the expansion for low energies close to the Dirac points is made, concluding that the dispersion in that regime is linear. After that, the Kane-Mele model for the intrinsic spin-orbit interaction is used resulting in a gap in the energies dispersions of graphene as indicated in the literature. Then the prescription [2] to roll up a graphene sheet into a carbon nanotube is used, and the dispersion for armchair and zigzag nanotubes are obtained. The spin-orbit interaction is turned on in the nanotubes and the resulting energy dispersions are obtained. Following that, there is a section on the density of
states (DOS), where a connection between theory and experimentation is made and the DOS is calculated for all the previous systems.
II. GRAPHENE — A SHEET OF CARBON
II.1. Introduction to the formalism
One of the methods of calculating the electronic structures is the tight-binding (TB). This is an approximation where the solid is taken as a collection of weakly interacting neutral atoms [12]. All electrons wave functions are thought to be very close to their atoms and interact very little with their neighbors. The electrons are said to be “tightly bounded” to the atoms and weakly interacting with the potentials around them. In this approximation, the electron’s wave functions can be computed rather easily as being in atomic levels localized at lattice sites. The wave function for the whole system then would be linear combinations of such wave functions. What differs this configuration from a product of atomic wave function is the possibility of overlap between them. When the wave function spread is comparable to the lattice distance being considered this overlap gains importance in calculating electronic structure. The tight-binding approximation then is good when the spread of the electron wave function is not negligible, but such that we can make corrections from the atom wave functions but also not too big that the picture of the electron “tightly bounded” to an atom would not be valid. We proceed as in [2], considering a lattice with translational symmetry of the unit cells in the direction of the lattice vectors, ai(i = 1, 2, . . .). In this lattice any wave function Ψ should satisfy Bloch’s theorem
TaiΨ = e
ik·aiΨ (i = 1, 2, . . .), (1)
where Tai is the translational operator in the direction of the lattice vector ai and k is the wave vector.
The wave function Ψ in this approximation is related to a linear combination of atomic orbitals, indeed a solution of this kind satisfies equation Eq. (1). We can write a functional Φj based on the j-th atomic
orbital φj located at the lattice site,
Φj(k, r) = 1 √ N N X R eik·Rφj(r − R) (j = 1, . . . , n), (2)
where R is the position of the atom and φj is the atomic wave function in state j and it is
considered to form an orthonormal basis
Z
φ∗(r − Rn)φ(r − Rm)dr = δnm. (3)
This functional is called Bloch orbital. We can show that equation Eq. (2) satisfies Bloch’s Theorem
Φj(k, r + a) = 1 N N X R eik·Rφj(r + a − R), (4) = eik·a 1 √ N N X R eik·(R−a)φj(r + a − R), (5) = eik·a 1 √ N N X R eik·R0φj(r − R0) = eik·aΦj(k, r). (6)
We can see that Eq. (2) is consistent with periodic boundary condition imposed by the translation operator Tai by noting that in three dimensions we can take a number M such that
Φj(k, r + M ai) = Φj(k, r) (i = 1, 2, 3). (7)
Therefore, TM ai = 1 and by using Eq. (4) we find that
eikM ai = 1. (8)
The possible values of k are therefore discretized by the boundary condition. We see that for condition Eq. (8) to hold we require that
k = 2pπ M ai
(i = 1, 2, 3). (9)
Because k is defined in three dimensions with kx, ky and kz and each component is independent
then there is M3 = N values in the first Brillouin zone where each component of the wave vector is
considered a continuum variable. So far, we have considered only wave function of the form of atomic orbitals located at the lattice sites, in this configuration the energies are known to be simply the energy of such atomic orbitals En(k). A more realistic approach is to say that those wave functions become small as
the potential due to its neighbors start to become non-negligible. This suggestion points in the direction that the wave function of the solid Ψj(j = 1, . . . , n) is not necessarily Φj but a linear combination of
theses functions Ψj(k, r) = n X l=1 Cjl(k)Φl(k, r), (10)
where we need to determine Cjl. To do this, we need to solve Schr¨odinger equation
HΨj(k, r) = Ej(k)Ψj(k, r), (11)
where H is the Hamiltonian of the solid. We can find the value for Ej by multiplying Ψ∗j(k, r) from the
Z r Ψ∗j(k, r)HΨj(k, r)dr = Ej(k) Z r Ψ∗j(k, r)Ψj(k, r)dr. (12)
We find the following expression for the energy
Ej(k) = R Ψ∗ jHΨjdr R Ψ∗ jΨjdr . (13)
Substituting Eq. (13) in to Eq. (10), we find a relation between the coefficients Cjl and the ith
energy Ei(k) = Pn jj0=1Hjj0(k)Cij∗Cij0 Pn jj0=1Sjj0(k)Cij∗Cij0 , (14)
where we have defined
Hij = hΦi| H |Φji , (15)
Sij = hΦi|Φji . (16)
Where H and S are integrals over the Bloch orbitals and are called transfer integral matrices and overlap integral matrices, respectively.
When the n × n elements of the matrices Hjj0 and Sjj0 are specified for a given k, the coefficient
C∗
ij is optimized so as to minimize Ei(k). By means extremization of Ei(k), it can be shown that these
conditions are enough to specify the dispersion for the system, and the matrices H and S satisfy the following eigenvalue problem
HCi= Ei(k)SCi, (17)
that allows non-null solution for Ci when
det H − ES = 0. (18)
This equation is called the secular equation and is of degree n and its solutions give all n eigen-values of Ei(k) for a given k
II.2. Application to Graphene
Graphene has two atoms in its unit cell, one in the A sublattice and one in the B sublattice and can be visualized in Fig1a. Therefore, the Hamiltonian is two dimensional and using Eq. (2) and Eq. (15) we find that
(a)
FIG. 1: Graphene is one-atom thick layer of carbon atoms arranged in a honeycomb lattice.
HAB= 1 N N X RARB eik·(RA−RB) Z RA Z RB φ∗B(r − RB)HφA(r − RA), (19) where we define t = Z RA Z RB φ∗B(r − RB)HφA(r − RA), (20)
as the transfer energy t. This is the energy necessary for an electron to jump from one site in sublattice A to another site in sublattice B. In the first-neighbors approximation, the electron at any site can only jump to one of its neighbors, and the transfer energy t is consider to be constant t0One electron in the A
sublattice can only travel to one of its neighbors in the B sublattice and vice-versa. In this approximation,
RA− RB = δi, (21)
where δ is the vector of vectors associated with the first-neighbors in the first Brillouin Zone. Substituting this in Eq. (20), we get the following
HAB= t0
eik·δ1+ eik·δ2+ eik·δ3
= t0g(k), (22)
H = 2p t0g(k) t0g(k) 2p , S = 1 t0g(k) t0g(k) 1 . (23)
Then, solving the secular equation Eq. (18) gives
E(k) =2p± t|g(k|
1 ± s|g(k)|. (24)
To take a better glance at the properties of the energy dispersion, we can set 2p = 0, leveling
the Fermi energy to zero. Further, we make the approximation by saying that the overlap is essentially zero, s = 0. Therefore, the energy dispersion is given as
E±(k) = ±|g(k)|, (25)
where g(k) will depend on the crystalline structure of graphene and will be computed in the next section.
III. TIGHT BINDING MODEL APPLIED TO GRAPHENE — SECOND QUANTIZATION
III.1. First Neighbors Interaction
(a) Unit cell of the graphene sheet, including the location of the first-neighbors.
(b) Reciprocal space for unit cell in the First Brillouin Zone.
Following the last section, we proceed with the same formalism but in the language of second quantization that is introduced in the AppendixA. The calculations here are done in the same way as in [6].
Graphene is a two dimensional mono-layer material made of carbon atoms placed in honeycomb structure. This honeycomb lattice is a combination of two triangular lattices A and B in such way that we can represent as a Bravais lattice in a two atoms basis. The Bravais lattice vectors a1 and a2 are:
a1= a 2 3,√3 a2= a 2 3, −√3 , (26)
transform these vectors to obtain the reciprocal lattice, one way of doing this is by using the relations [12] b1= 2π a2× a3 a1· (a2× a3) , b3= 2π a1× a2 a1· (a2× a3) , b2= 2π a3× a1 a1· (a2× a3) . (27)
We can verify that, these vectors are such that bi· aj = δij as it should. Considering a3 = a 2 0, 0, 1, we obtain b1= 2π 3a 1,√3 , (28) b2= 2π 3a 1, −√3 . (29)
From the picture, we can calculate the first neighbors vectors δ, starting from δ3 we can obtain
the next vector by performing a rotation of 120◦.
δ1= a 2 1,√3 , (30) δ2= a 2 1, −√3 , (31) δ3= −a 1, 0 . (32)
Moreover, the points with high symmetry, at the corner of the Brillouin zone, have a special name: “Dirac points”. They are denoted by K and K0 and will be important later on when we study behavior of electrons in the low energy approximation. Their position in the k space is
K =2π 3a, 2π 3√3a , (33) K0=2π 3a, − 2π 3√3a . (34)
Thus, the unitary cell of graphene has two distinct atoms, one in each sublattice A and B. The most general state in this lattice is then a linear combination of orbitals belonging to each sublattice
|Ψi = αk|σ, kia+ βk|σ, kib, (35)
where αk and βk are lattice coefficients depending in k, |σ, kia is the abstract state representing the
Bloch orbital in the A lattice and |σ, kib the same in the B lattice. These orbitals are such that
|σ, kia= √1 N N X i=1 eik·Ri|σ, ii , (36) |σ, kib= √1 N N X j=1 eik·Rj|σ, ji , (37)
where |σ, ii and |σ, ji are the atomic orbitals at site i/j in each sublattice.
The most general Hamiltonian for the tight-binding approximation is one that accounts for over-lapping between state vectors of electrons in the same sublattice and overover-lapping between different sub-lattices. This Hamiltonian can be written as
H =X σ N X i,j sij a†σ,ibσ,j+ b†σ,jaσ,i + dij a † σ,iaσ,j+ b†σ,jbσ,i, (38)
where c†σ,i is an operator that creates an electron of the c sublattice in the ith position with spin polariza-tion σ and cσ,iannihilates an electron of the c sublattice in the ith sublattice with spin σ. sijis the energy
associated with the overlapping between the wave functions of the i and j sites in the same sublattice and d
ijfor different sublattices. These are fermionic operators and obey the usual anti-commutation relations
discussed in AppendixA
{cσ,i, c†τ,j} = δi,jδσ,τ (c = a, b), (39)
and the operators of the A sublattice anticommute with the operators of the B sublattice. In the first-neighbors tight-binding approximation, there is interaction only between the first first-neighbors, that is, only the adjacent sites will contribute to the Hamiltonian. That means that, in this approximation, the state vectors that are farther away will not overlap at all. The interaction between its neighbors is taken to be constant throughout all the sites, this constant is −t and is named hopping between first-neighbors or overlap integral. Therefore, in this approximation we do not need to consider the inter-sublattice part of the Hamiltonian Eq. (38), we need only to sum over the first neighbors and consider only constant values for the hopping.
H = −t X
hi,ji,σ
a†σ,ibσ,j+ b†σ,jaσ,i. (40)
In second quantization, this Hamiltonian also has the interpretation that the electron has kinetics energy to “jump” between neighboring sites, this is usually called a “hop” between sites or a “hopping”. The lower sum limit hi, ji means that the summation is should be held only on the first neighbors. To find the energies, formally we need to solve the corresponding Schr¨odinger equation
H |Ψi = E |Ψi , (41)
where |Ψi is given by Eq. (35) and H is
H = hσ, k|aH |σ, kia hσ, k|aH |σ, kib hσ, k|bH |σ, kia hσ, k|bH |σ, kib . (42)
operator can be written in the Fourier space as follows c†σ,i= √1 N X k c†σ,ke−ikxi, (43) cσ,i= 1 √ N X k cσ,keikxi. (44)
In such way that the Hamiltonian can be rewritten in this space, noting that
X hi,ji a†σ,ibσ,j= X hi,ji 1 N X k X k0 a†σ,kbσ,ke−i(kxi−k 0x j), (45) X hi,ji b†σ,jaσ,i= X hi,ji 1 N X k X k0 b†σ,kaσ,ke−i(kxj−k 0x i). (46)
We note then that the P
hi,ji imposes that the only summation is to be done across the first
neighbors, in that case xi − xj = δi, where δi is the coordinate of the ith first-neighbor in Eq. (30).
Therefore, we substitute the summation in hi, ji by a summation in δi
X δ 1 N X k X k0 a†σ,kbσ,k0e−ik 0 ·δe−i(k−k0)xi =X k a†σ,kbσ,k X δ e−ik·δ, (47) X δ 1 N X k X k0 b†σ,kaσ,k0eik 0 ·δe−i(k−k0)xi =X k b†σ,kaσ,k X δ eik·δ, (48)
where we use the fact that
1 N X k0 e−i(k−k0)xi = δ k,k0, (49)
is the Kronecker delta, and simplifies the expression above. Next step is to perform the following sum-mation
X
δ
e−ik·δ, (50)
using Eq. (30). The scalar products k · δ is given by
k · δ1= a 2 kx+ √ 3ky , (51) k · δ2= a 2 kx− √ 3ky , (52) k · δ3= −akx. (53)
The sum Eq. (50) then is
X δ e−ik·δ= e−ia2 kx+ √ 3ky + e−ia2 kx− √ 3ky + eiakx (54) = 2 coskya √ 3 2 e−ikxa2 + eikxa. (55)
Identifying
g(k) = 2e−ikxa2 cos(kya
√ 3 2 ) + e
ikxa, (56)
we can rewrite the Hamiltonian in Fourier space as
H = −tX σ,k g(k)a†σ,kbσ,k+ g(k)b†σ,kaσ,k, (57) or in matrix form: H = −tX σ,k a†σ,k b†σ,k 0 g(k) g(k) 0 aσ,k bσ,k . (58)
To find the dispersion E(k) as a function of k and σ for this Hamiltonian, we solve the secular equation
det(H − IE) = 0, (59)
which readily gives ±|g(k)| as eigenvalues. Thus, the dispersion relation for one layer graphene considering only first-neighbors interaction is
E±(k) = ±t s 1 + 4 cos2 √ 3 2 kya + 4 cos √ 3kya 2 cos3kxa 2 , (60)
and is independent of spin polarization.
III.2. Expansion for Low Energies — First Neighbors Interaction
For low energies, we can expand Eq. (50) close to the K and K0points [6]. To do this, we place the momentum at K and make a small vector perturbation q.
X
δ
e−i(K+q)·δ. (61)
Then, we can expand the exponential in a Taylor series, because of the assumption that the vector perturbation is small expanding only to the first order would be enough.
X δ e−i(K+q)·δ≈X δ e−iK·δ1 − iq · δ=X δ e−iK·δ− iX δ eiq·δq · δ. (62)
-3 -2 -1 0 1 2 3 kx -3 -2 -1 0 1 2 3 ky -3 -2 -1 0 1 2 3 E(k) (a)
FIG. 3: Dispersion for graphene in the first Brillouin zone
X δ e−iK·δ= e−i2π3 + 1 + e 2π 3 = 1 + 2 cos 2π 3 = 0, (63)
and the summation left is
−iX
δ
e−ik·δq · δ = −ie−i2π3 a
2(qx+ √ 3qy) + a 2(qx− √ 3qy) − ei 2π 3 aqx (64) = 3a 4 qx(1 − i √ 3) + 3a 4 qy(i + √ 3). (65)
Because we have previously defined Eq. (61) as a function g(k) and written the Hamiltonian with g(k) as one of its parameters, in this approximation we can do the same, defining
g(k) = 3a 4 qx(1 − i √ 3) + 3a 4 qy(i + √ 3), (66)
further we can define an Hamiltonian analogous to Eq. (57), by noting that the fermionic operators are now also a expansion of the previous operators close to the points K and K0. In the reciprocal space these operators have the form
an ≈ e−iK·Rna1,n+ e−iK 0R na 2,n, (67) bn ≈ e−iK·Rnb1,n+ e−iK 0R nb 2,n. (68)
The Hamiltonian then reads
H1ef f = −tX σ,k a†σ,k b†σ,k 0 3a4qx(1 − i √ 3) +3a4qy(i + √ 3) 3a 4qx(1 − i √ 3) +3a4qy(i + √ 3) 0 aσ,k bσ,k . (69)
To make this Hamiltonian cleaner and easier to be interpreted, it is useful to make a rotation of
π/3in the q space such that
1 − i√3 → 2, (70)
i +√3 → 2i, (71)
therefore, the Hamiltonian simplifies to
H1ef f = −tX σ,k a†σ,k b†σ,k 0 3a2qx+ iqy 3a 2 qx− iqy 0 aσ,k bσ,k . (72)
The Hamiltonian Eq. (72) have now the nice interpretation, of being of the form of the Dirac equation for a massless particle
H1ef f = vFσ · q = 0 vF qx+ iqy vF qx− iqy 0 , (73)
where σ are the Pauli matrices. We make the identification
vF =
3at
2 , (74)
as the Fermi velocity and a is the lattice constant. The energies are given by solving the corresponding eigenvalue problem
E(q) = ±vF|q| = ±vF
q q2
x+ qy2. (75)
III.3. Intrinsic Spin-orbit — Second Neighbors Interaction
Spin-orbit coupling occurs in general when a particle with spin moves from one orbital wave function to another. It is a mixing of the spin and the orbital angular momentum, due to the relativistic
FIG. 4: Linear behavior for the dispersion of graphene for points close to the high symmetry points K and K0, such cone formation in the energy dispersion is characteristic of a “Dirac Cone”
motion of the particle and can be derived from Dirac’s theory for the electron. This mixing is bigger for heavier atoms, due to the bigger ionic radius as mentioned before. Therefore, to model different honeycomb systems we have use the appropriated values for the lattice distance a, buckling angle and spin-orbit coupling. This property called “buckling” is a deformation of the honeycomb lattice caused by the ionic radius of the these heavier atoms and can be visualized in Figs5aand 5b. The parameters corresponding to each system are given in TableI.
System a (˚A) θ λSO(meV ) λR(meV )
Graphene 2.46 90◦ 1.3 × 10−3 ≈ 0
Silicene 3.86 101.7◦ 3.973 0.7
Germanene 4.02 106.5◦ 46.3 10.7
Stanene 4.70 107.1◦ 64.4 9.5
TABLE I: Parameters for different honeycomb systems
Carbon is relatively light and therefore the spin-orbit coupling in graphene is expected to be really small. The model used here for the intrinsic spin-orbit interaction is the Kane-Mele [1] that was proposed to model the Quantum Spin Hall effect (QSH) in graphene, although because of the small spin-orbit coupling in graphene this effect could not be observed there. The Kale-Mele model was first introduced [1] by analyzing what kind of Hamiltonian would preserve all symmetries of graphene and also open a gap. When considering Hamiltonians where the mirror symmetry were preserved the intrinsic spin-orbit coupling was used, otherwise the Rashba Hamiltonian. The Kale-Mele Hamiltonian has the following form
(a) (b)
FIG. 5: Typical crystalline structure for buckled materials such as Silicene, Germanene and Stanene. In (a) we see that the structure is essentially a honeycomb lattice, with the property that the two sublattices not in the the plane, but separated by a finite distance. In (b) we see that, we can account
for the buckling by specifying the angle between the nearest neighbors that is denoted in TableI.
HSO= λSO
Z
d2r ˆΨ†(r) ˆs
zσˆzτˆzΨ(r),ˆ (76)
where ˆs, ˆσ and ˆτ are Pauli matrices representing interaction between spins, states in the A(B) sublattices, states in K(K0) points respectively, and λSO is the spin-orbit coupling. The gap generated by this
Hamiltonian is differente from simply applying an electric field in perpendicular direction and such gap is connected with topological properties of materials such as silicene, germanene and stanene. In graphene, this interaction is only of second order while in materials with heavier atoms such as silicene, germanene and stanene this interaction is of first order. For graphene, it is expected that the gapped due to spin-orbit effects is approximally 0.07meV while for materials such as silicene the gap is approximally 1.55meV and planar germanene 23.9meV [8] [13] [14]. In this work, we only use “artifical” gaps, much larger than naturally occurring in nature with the propurse of actually observing what the resulting effects are.
In the language of second quantization, the Hamiltonian for the intrinsic spin-orbit interaction is the following HSO= iλSO 3√3 X hhi,jiiαβ νij c†α,iσ z α,βcβ,j + h.c, (77)
where λSO is the spin-orbit coupling, hhi, jii means that the summation is over the second neighbors.
its second-neighboring sites in a clockwise direction would be different from moving to another site in a anti-clockwise direction and moreover the interaction will be opposite for different polarization of spin. Therefore, the matrix elements of ν for positive spin polarization are +1 if the movement to one of its second-neighbors is clockwise and −1 for anti-clockwise movement. This is an explicit way of finding the effects proposed by Eq. (77), and is due to [? ]. The effective way proposed here to find the sign of νij
is also proposed in [1]. σz is the Pauli matrix and c†σ,i is a fermionic creation operator. The value of νij
formally is given by
νij =
γi× γj
|γi× γj|
= ±1, (78)
and can be visualized in Figs6aand6b. The location of all the second-neighbors can be calculated from the lattice vectors a1 and a2 and are denoted here as γ.
(a) (b)
FIG. 6: Visualization of the sign of the matrix element νij as explained in [1]. The signs of the
second-neighboring hoppings are given by a right-hand rule for the red sublattice and left-hand rule for the blue sublattice. (+) sign is given by the green arrow while (-) sign is given by the blue arrow.
γ1= ±a1 γ2= ±a2 γ3= ± a2− a1. (79)
The Hamiltonian Eq. (77) is written in a compact way and we can take a better glance at it by expanding it. First we deal with the summation in α and β by noting that
σz= 1 0 0 −1 , (80)
HSO= iλSO 3√3 X hhi,jii νij c†↑,ic↑,j− c†↓,ic↓,j , (81)
where we have to sum the contributions from both A and B sublattices. With this, the Hamiltonian reads HSO = iλSO 3√3 X hhi,jii νij
a†↑,ia↑,j− a†↓,ia↓,j− b†↑,ib↑,j+ b†↓,ib↓,j+ h.c
. (82)
To find the dispersion for this Hamiltonian, we can proceed with the same scheme as in the first-neighbors case, by showing that any operator can be written in the Fourier space
c†σ,i =√1 N X k c†σ,ke−ikx1, (83) cσ,i= 1 √ N X k cσ,keikx1, (84) noting that X hhi,jii νija†↑,ia↑,j= X hhi,jii νij 1 N X k X k0 a†↑,ka↑,k0e−i(kxi−k 0x j), (85) X hhi,jii νij− a†↓,ia↓,j= X hhi,jii νij 1 N X k X k0 −a†↓,ka↓,k0e−i(kxi−k 0x j), (86) X hhi,jii νij− b†↑,ib↑,j= X hhi,jii νij 1 N X k X k0 −b†↑,kb↑,k0e−i(kxi−k 0x j), (87) X hhi,jii νijb†↓,ib↓,j= X hhi,jii νij 1 N X k X k0 b†↓,kb↓,k0e−i(kxi−k 0x j). (88)
Because this is a sum only over the second-neighbors, we can use the fact that xi− xj = γi, where
γi is a coordinate of the ith second-neighbor. Substituting the summation in hhi, jii, by a summation in
γi we have X hhi,jii νij 1 N X k X k0 a†↑,ka↑,k0e−i(kxi−k 0x j)=X k a†↑,ka↑,k X γ νγe−ik·γ, (89) X hhi,jii νij 1 N X k X k0 −a†↓,ka↓,k0e−i(kxi−k 0x j)=X k −a†↓,ka↓,k X γ νγe−ik·γ, (90) X hhi,jii νij 1 N X k X k0 −b†↑,kb↑,k0e−i(kxi−k 0x j)=X k −b†↑,kb↑,k X γ νγe−ik·γ, (91) X hhi,jii νij 1 N X k X k0 b†↓,kb↓,k0e−i(kxi−k 0x j)=X k b†↓,kb↓,k X γ νγe−ik·γ, (92)
1 N
X
k0
e−i(k−k0)= δk,k0, (93)
to simplify the expressions. Next, we need to evaluate the following summation
X
γ
νγe−ik·γ. (94)
To do this, we calculate the dot products
k · γ1= ± a 2 3kx+ √ 3ky , (95) k · γ2= ± a 2 3kx− √ 3ky , (96) k · γ3= ∓a √ 3ky. (97)
The best way to evaluate if the value of νγi is either one or minus one is to look at the Figs6b
and6aof the unit cell and seeing if the hopping associated with the second neighbor γi is clockwise or
anti-clockwise. The summation is
X γ νγe−ik·γ = −ei a 2(3kx+ √ 3ky)+ eia2(3kx+ √ 3ky)+ e−ia2(3kx− √ 3ky)− eia2(3kx− √ 3ky)− eia √ 3ky + e−ia √ 3ky (98) =ei3kxa2 − e−i 3kxa 2 ei √ 3ky a 2 + e−i √ 3ky a 2 − 2i sin(√3kya) (99) = 4i sin3kxa 2 cos √ 3kya 2 − 2i sin(√3kya), (100) where we define ih(k) = 4i sin3kxa 2 cos √ 3kya 2 − 2i sin(√3kya). (101)
The Hamiltonian now reads
HSO= iλSO 3√3 X k
ih(k)a†↑,ia↑,j− ih(k)a†↓,ja↓,i− ih(k)b†↑,ib↑,j+ (−i)h(k)b†↓,jb↓,i
, (102) or in matrix form HSO = λSO 3√3 X k a†↑,k b†↑,k a†↓,k b†↓,k −h(k) 0 0 0 0 h(k) 0 0 0 0 h(k) 0 0 0 0 −h(k) a↑,k b↑,k a↓,k b↓,k . (103)
Therefore, we can find the energies by solving the associated eigenvalue problem, because this matrix is diagonal we can solve the energies immediately. Namely,
E±σ(k) = ±h(k) = ± 4 sin3kxa 2 cos √ 3kya 2 − 2 sin(√3kya) ! . (104)
To find the effects of the intrinsic spin-orbit interaction in the Hamiltonian, we can sum the Hamiltonians Eq. (57) and Eq. (103). Therefore we can write the full Hamiltonian H as
H = H + HSO (105) = −t X hi,ji,σ a†σ,ibσ,j+ b†σ,jaσ,i+ iλSO 3√3 X hhi,jii νij
a†↑,ib↑,j− b†↑,ja↑,i− a†↓,ib↓,j+ b†↓,ja↓,i
, (106) or in matrix form H = −βh(k) −tg(k) 0 0 −tg(k) βh(k) 0 0 0 0 βh(k) −tg(k) 0 0 −tg(k) −βh(k) , (107) where we defined β =λSO
3√3. We find the energy dispersion by solving each block separately.
E±σ = ± r t|g(k)| 2 +βh(k) 2 . (108)
III.4. Expansion for Low Energies — Intrinsic Spin-orbit Interaction
In the case of the spin-orbit interaction, we need to expand equation Eq. (50) around the points K and K0. Similarly to the previous section, we have to expand this expression in a Taylor Series of order one. X γ νγe−i(K+q)·γ ≈ X γ νγe−iK·γ 1 − iq · γ (109) =X γ νγe−iK·γ− i X γ νγe−iK·γq · γ. (110)
Then, we can evaluate these summations by first calculating the dot products between K and γ, by using Eq. (33) and Eq. (79)
K · γ1= ± a 2 2π a + 2π 3a = ±4π 3 , (111) K · γ2= ± a 2 2π a − 2π 3a = ±2π 3 , (112) K · γ1= ∓a 0 + 2π 3a = ∓2π 3 . (113)
-3 -2 -1 kx 0 1 2 3 -3-2 -1 0 1 2 3 ky -3 -2 -1 0 1 2 3 E(k) (a) β = 0.05 -3 -2 -1 kx 0 1 2 3 -3-2 -1 0 1 2 3 ky -3 -2 -1 0 1 2 3 E(k) (b) β = 0.1
FIG. 7: In this picture, we can see the effect of the intrinsic spin-orbit interaction in the dispersion of graphene for two values of β. The interaction induces a gap between the conduction and valence bands.
Considering the values of γi, the first summation is given by
X
γ
νγe−iK·γ = (−1)e−i
4π 3 + (+1)ei 4π 3 + (+1)e−i 2π 3 + (−1)ei 2π 3 + (−1)ei 2π 3 + (+1)e−i 2π 3 (114) = 2i sin4π 3 − 4i sin2π 3 = −3i√3. (115)
In the same way, for the second summation
X
γ
νγe−iK·γq · γ = (−1)e−i
4π 3 a 2 3qx+ √ 3qy (116) + (+1)ei4π3 −a 2 3qx+ √ 3qy (117) + (+1)e−i2π3 a 2 3qx− √ 3qy (118) + (−1)ei4π3 −a 2 3qx− √ 3qy (119) + (−1)ei2π3 −a√3qy (120) + (+1)e−i2π3 a√3qy , (121)
which turns out to be exactly zero. Therefore, the effective Hamiltonian for the intrinsic spin-orbit interaction Eq. (103) can be written by making the same assumptions as in the previous section, that the operators here are also approximated to operators near the Dirac points
HSOef f =X k a†↑,k b†↑,k a†↓,k b†↓,k λSO 0 0 0 0 −λSO 0 0 0 0 −λSO 0 0 0 0 λSO a↑,k b↑,k a↓,k b↓,k . (122)
Together with the Hamiltonian for the first-neighbors interaction Eq. (73), we have the following effective Hamiltonian Hef f =X k a†↑,k b†↑,k a†↓,k b†↓,k λSO vF qx+ iqy 0 0 vF qx− iqy −λSO 0 0 0 0 −λSO vF qx+ iqy 0 0 vF qx− iqy λSO a↑,k b↑,k a↓,k b↓,k , (123)
where its eigenvalues are readily obtained by solving each 2 × 2 block diagonal. The eigenvalues for both blocks are the same
E±σ = ± r vF|q| 2 +λSO 2 , (124)
thus, we conclude that the intrinsic spin-orbit interaction creates a “gap” in the energy dispersion between the Dirac cones.
(a) λSO= 0 (b) λSO= 0.05 (c) λSO= 0.1
FIG. 8: Effect of the intrinsic spin-orbit interaction in the energy dispersion close to the points K and K0.
IV. SINGLE-WALL NANOTUBES
A single-wall nanotube is defined by rolling a graphene sheet, if we consider the nanotube long enough its electronic properties are defined by the configuration of the sites in the cross section [2]. In that way, the nanotube can be thought as a one-dimensional structure. The sites in the cross section can take almost arbitrarily shapes, although the most common one are the armchair and zigzag.
IV.1. Classification of carbon nanotubes
Carbon nanotubes are generally classified in three categories, depending on the way that the carbon sheet is “sliced” and “glued” together, armchair, zigzag or chiral. Nonetheless, a more general classification may be done. The primary classification of a carbon nanotube is as either being a achiral or chiral, depending on either the nanotube is the same under a mirror transformation. Under the class of achiral nanotubes, there is the armchair and zigzag, which are the most symmetrical kind of nanotubes. Armchair and zigzag nanotubes have this name because of their cross-sectional ring. Chiral nanotubes do not have this reflection property, rather a spiral symmetry.
A general configuration of a carbon nanotube has a defined chiral vector Ch, which can be
expressed by the real space unit cell vectors ai
Ch= na1+ ma2≡
n, m, (125)
where n and m are integers and m is at most equal to n. The (n, m) notation is useful because a carbon nanotubes can be constructed by slicing two edges of sheet graphene and gluing the edges together. This can be done in many ways, each of them giving different electronic properties. Therefore this notation allows us to classify and study many kinds of nanotubes under the same scope.
The diameter of a nanotube is given in terms of the chiral vector as
L = |Ch| = a
p
(a) Zigzag nanotube
(b) Armchair nanotube
FIG. 9: Visualization of the cross-sectional ring of the carbon nanotubes in the case of the (a) Zigzag and (b) Armchair nanotubes.
We also define the chiral angle, as the angle between the chiral vector Ch and the lattice vector
a1 in the range 0 ≤ |θ| ≤ 30◦. The chiral angle θ denotes the tilt between the hexagons in the direction
of the nanotube ˆz axis. We define the chiral angle as
cos(θ) = Ch· a1 |Ch||a1|
= 2n + m
2√n2+ m2+ nm. (127)
Therefore, we can classify any nanotube in terms of the chiral angle and the chiral vector
Type θ Ch
armchair 30◦ (n, n)
zigzag 0◦ (n, 0)
chiral 0◦< |θ| < 30◦ (n, m)
TABLE II: Classification of nanotubes in terms of the chiral angle and the chiral vector.
IV.2. Electronic Properties
To obtain the dispersion relation for the nanotubes, we use for simplicity the configurations of highest symmetry, namely the armchair and zigzag, then we can use appropriate periodic boundary conditions to roll the graphene sheet into a nanotube. In the case of the armchair, the chiral vector is (n, n) and we roll the tube in the x direction, implying a discretization of the wave vector in that
(a) Chiral nanotube with Ch= (3, 1). (b) Armchair nanotube with Ch= (2, 2)
(c) Zigzag nanotube with Ch= (3, 0)
FIG. 10: Classification of three types of carbon nanotubes with the the chiral vector, diameter and chiral angle.
direction. The boundary condition for this case can be derived from the lattice vector a1 and is simply
the condition that the x-component of this vector must be proportional to π
a1xkx=
3a 2 kx=
m
nπ (m = 1, . . . , 2n), (128)
substituting this condition in to Eq. (60), we obtain
E±m(ky) = ±t s 1 + 4 cos2 √ 3 2 kya + 4 cos √ 3kya 2 cosm nπ (m = 1, . . . , 2n), (129)
for momenta in the first Brillouin Zone, in other words −√π 3 < kya < π √ 3. (130)
In the Figs11a,11band11ceach band is doubly degenerated, the values for m = 0, . . . , n give the same bands as m = n + 1, . . . , 2n. For all armchair nanotube the bands are degenerated at the boundary of the Brillouin Zone, where 3kya
2 = π. In the same way, for all armchair nanotubes there are degeneracy
points between the valence and conduction bands at the points ± 2π
3√3, forming a zero-gap point and the
material has a metallic conduction, where only a infinitesimal energy excitation is necessary to take an electron from the valence to conduction band.
In the case of the zigzag nanotubes, the chiral vector is (n, 0) and the discretization of the wave vector is given by a1yky= √ 3kya 2 = m nπ (m = 1, . . . , 2n). (131)
Imposing this condition on the graphene sheet dispersion Eq. (60) gives
E±m(kx) = ±t r 1 + 4 cos2m nπ + 4 cosm nπ cos3kxa 2 (m = 1, . . . , 2n), (132)
for momenta in the first Brillouin Zone, that is
−π
3 < kxa < π
3. (133)
Here, we have different behavior for different values of n, reproducing the results in the litera-ture [2]. For n = 5 in Fig12awe have a energy gap, where in the n = 6 Figs12bwe have no energy gap. Making the first an insulator and the second a metal at finite temperatures. Indeed, it can be shown that we have a gapless metal when n is a multiple of 3 and an insulator when n is not a multiple of 3.
IV.3. Spin-orbit Interaction for Single-Wall Nanotubes
Following the previous section, to introduce the intrinsic spin-orbit interaction in armchair and zigzag nanotubes we have to simply impose the appropriate boundary conditions to the energy dispersion Eq. (108). In the case of armchair nanotubes, we impose the boundary condition Eq. (128) in the x-component of the wave vector giving
E±m(ky) = ± r t|gm(k y)| 2 +βhm(k y) 2 (m = 1, . . . , 2n), (134) where β =λSO/3√3and
-3 -2 -1 0 1 2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E(k y )/t ky (a) Ch= 5, 5 -3 -2 -1 0 1 2 3 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E(k y )/t ky (b) Ch= 9, 9 -3 -2 -1 0 1 2 3 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E(k y )/t ky (c) Ch= 12, 12
FIG. 11: Dispersion for armchair nanotubes for different configurations, reproducing the results of literature [2]. It can be noted that in the three pictures we have a metallic conduction.
-3 -2 -1 0 1 2 3 -3 -2.8 -2.6 -2.4 -2.2 E(k x )/t kx (a) Ch= 5, 0 -3 -2 -1 0 1 2 3 -3 -2.8 -2.6 -2.4 -2.2 E(k x )/t kx (b) Ch= 6, 0 -3 -2 -1 0 1 2 3 -3 -2.8 -2.6 -2.4 -2.2 E(k x )/t kx (c) Ch= 9, 0
FIG. 12: Dispersion for zigzag nanotubes for different configurations. The nanotube can either be a insulator or metal depending on n.
|gm(k y)| = s 1 + 4 cos2 √ 3 2 kya + 4 cos √ 3kya 2 cosm nπ , (135) hm(ky) = 4 sin m nπ cos √ 3kya 2 − 2 sin(√3kya), (136)
in the interval defined in Eq. (130). The result for a couple of armchair nanotube with spin-orbit inter-action can be seen in Figs13a,13band 13c.
For the zigzag case, we have to impose the conditions Eq. (131) in the y-component of the wave vector E±m(kx) = ± r t|gm(k x)| 2 +βhm(k x) 2 (m = 1, . . . , 2n), (137) and |gm(k x)| = t r 1 + 4 cos2m nπ + 4 cosm nπ cos3kxa 2 , (138) hm(kx) = 4 sin 3kxa 2 cosmπ n − 2 sin2mπ n , (139)
in the interval Eq. (133). The results for a two zigzag nanotubes can be seen in Fig 14a, where the interaction induces a gap and in Fig14b, where the gap increases.
V. CALCULATING THE DENSITY OF STATES
V.1. First Neighbors Interaction — Simplest Case
The Density of States (DOS) are derived from the Green’s function formalism introduced in AppendixB. The Green’s function is defined by
ˆ G(E) =X n |n, σi hn, σ| E − εn , (140)
where n is the eigenstate and εn is the corresponding energy. In the case of the first neighbors interaction,
the Hamiltonian was found to take the form
H =X σ,k 0 −tg(k) −tg(k) 0 . (141)
In diagonalizing this Hamiltonian, we found that the eigenenergies were
E±σ(k) = ±t|g(k)|, (142)
-3 -2 -1 0 1 2 3 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E(k y )/t ky (a) β = 0 -3 -2 -1 0 1 2 3 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E(k y )/t ky (b) β = 0.05 -3 -2 -1 0 1 2 3 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 E(k y )/t ky (c) β = 0.1
FIG. 13: Dispersion for armchair nanotubes with Ch= (5, 5) and for different values of β. The opening
-3 -2 -1 0 1 2 -3 -2.8 -2.6 -2.4 -2.2 E(k x )/t kx (a) Ch= (6, 0) and β = 0.1 -3 -2 -1 0 1 2 3 -3 -2.8 -2.6 -2.4 -2.2 E(k x )/t kx (b) Ch= (11, 0) and β = 0.1
FIG. 14: Here we note that the intrinsic spin-orbit interaction (a) opens the gap in the nanotube that was previously a metal and (b) makes the gap even bigger for the nanotube that was already an
insulator.
|σ, k, ±i = pg(k) |σ, kia∓ q
g(k) |σ, kib
p2|g(k)| . (143)
|σ, k, ±i hσ, k, ±| = pg(k) |σ, kia∓ q g(k) |σ, kibqg(k) hσ, k|a∓pg(k) hσ, k|b 2|g(k)| (144) = |g(k)| |σ, kiahσ, k|a 2|g(k)| + |g(k)| |σ, kibhσ, k|b 2|g(k)| ∓ g(k) |σ, kibhσ, k|a 2|g(k)| ∓ |g(k)| |σ, kiahσ, k|b 2|g(k)| (145) = |σ, kiahσ, k|a 2 + |σ, kibhσ, k|b 2 ∓ g(k) |σ, kibhσ, k|a 2|g(k)| ∓ |g(k)| |σ, kiahσ, k|b 2|g(k)| . (146)
Projecting into the real space, by making the Fourier Transform
|σ, kia =√1 N N X i eik·xi|σ, ii , (147) |σ, kib= √1 N N X j eik·xj|σ, ji . (148)
Further, to get the results we need only to care about the projection of the Green’s function onto one of the sublattices. If we choose to project onto the B sublattice, the matrix elements corresponding would be hσ, j| G |σ, j0i, because of the equivalence between the sublattices this choice is not relevant to the final result. Therefore, to compute the relevant matrix elements we only need to compute the following projector in real space
|σ, k, ±i hσ, k, ±| ≈ |σ, kibhσ, k|b 2 (149) = 1 2N N X jj0 eik·(xj−xj0)|σ, ji hσ, j0| . (150)
Now we can use the use these results to calculate the Green’s function, using the formula Eq. (140). Therefore, the Green’s function reads
gjj0 = hσ, j| G |σ, j0i = 1 2N X k X jj0 eik·(xj−xj0) E − Eσ +(k) +e ik·(xj−xj0) E − Eσ −(k) . (151)
We can simplify this further, by noting that
E+σ(k) = −E−σ(k) ≡ ε(k). (152)
Leading to the following expression, which will allow us to compute the DOS which is done in Appendix ??. gjj0 = 1 N X k X jj0 Eeik·(xj−xj0) E2− ε(k)2 . (153)
the integration in the kx or ky phase space, depending if the nanotube considered is zigzag or armchair,
of the nanotube over the first Brillouin Zone [15]. Further, we need to sum over all 2n energy bands for the nanotube. Thus, in the case of the armchair nanotubes, we substitute the summation in k by
1 N Z k dk = 1 2n 2n X m √ 3a 2π Z √π3 −π √ 3 dky. (154)
Therefore, we have the following expressions for the armchair nanotubes Green’s Function
hσ, j| G |σ, j0i = 1 2n 2n X m √ 3a 2π Z √π3 −π √ 3 dky
Ee2πim3an (xj−xj0)eiky(yj−yj0)
E2− ε m(ky)2
, (155)
where we used the fact that the values of kx are expressed as Eq. (128) and xj = xjx, yˆ j = xjy. Toˆ
perform the integration in Eq. (155) over the variable ky, we need to make use of the residue theorem
I
C
dzf (z) = 2πiXresidues of f (z) inside contour C, (156)
where we define f (ky) = p(ky) q(ky) , (157) where p(ky) = eiky(yj−yj0), (158) q(ky) = E2− εm(ky)2. (159)
We note that p(ky) is an analytic function in ky and q(ky) is periodic function in ky, with singles
poles in the first Brillouin Zone. The residues in this case are formally given by
Residue = p(ky) q0(k y) k y=pole . (160)
In the case of the armchair nanotubes, the poles are given by
cos √ 3 2 kya = −1 2 cos m nπ ± r cos2m nπ +E 2 t2 − 1 ! . (161)
After integrating, we are left with the following expression
hσ, j| G |σ, j0i = i 4t2n
2n
X
m
Ee2πim3an (xj−xj0)eiky(yj−yj0)
sin(√3kya) + sin( √ 3 2 kya) cos( m nπ) , (162)
where ky is given by taking the inverse function of Eq. (161). This last expression could be, in principle,
be evaluated analytically but here it is evaluated numerically. In the case of zigzag nanotubes, we have to make the following substitution
1 N Z k dk = 1 2n 2n X m 3a 2π Z π3 −π 3 dkx. (163)
Then the expression for the Green’s function is
hσ, j| G |σ, j0i = 1 2n 2n X m 3a 2π Z π3 −π 3 dkx Eeikx(xj−xj0)e 2πim √ 3an(yj−yj0) E2− ε m(kx)2 . (164)
By making the same procedures as before, we have that the poles are given by
cos3 2kxa = −4 cosm nπ 2 +Et22 − 1 4 cosmnπ , (165)
and the final expression is
hσ, j| G |σ, j0i = i 2t2n 2n X m Eeikx(xj−xj0)e 2πim √ 3an(yj−yj0) sin(32kxa) cos(mnπ) . (166)
Where kx is given by taking the inverse of Eq. (165). This expression then is also evaluated
numerically. The results for armchair nanotubes are given in Fig 15aand Fig15b, while the results for zigzag nanotubes are given in Fig16aand Fig16b. We can see the for the armchair case, there is a finite density around E = 0 for any value of n, meaning that there is a metallic conduction for that nanotube. As in the case of zigzag nanotubes, this value can be either finite or zero depending on n, this was also noted when analyzing the energy bands. The density of states (DOS) gives us a probe to the electronic structure, it is actually what experimentally a tip of STM would see if it probed the material, this is nice because allows theoretical results to be compared with experiments. We can see that, when comparing with experimental results [16] [17] [18] and others, there is evidence for a DOS of the form that was found, although there is no resolution is such experiments that could distinguish between armchair, zigzag or a more general chiral nanotube. The connection from the experimental results and the theory is made by measurering (V /I)dI/dV , which provides a good measure of the main features of the DOS for metals and semiconductors [18].
V.2. Density of States in the presence of Spin-Orbit interaction
In order to calculate the DOS we need to calculate the Green’s function for the system, with both first neighbors and spin-orbit interactions we need to diagonalize both Hamiltonians together.
0 0.2 0.4 0.6 0.8 1 1.2 -4 -3 -2 -1 0 1 2 3 4 DOS(E) E (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -4 -3 -2 -1 0 1 2 3 4 DOS(E) E (b)
0 0.2 0.4 0.6 0.8 1 1.2 -4 -3 -2 -1 0 1 2 3 4 DOS(E) E (a) 0 0.2 0.4 0.6 0.8 1 1.2 -4 -3 -2 -1 0 1 2 3 4 DOS(E) E (b)
H = iλSO 3√3 X k −βh(k) −tg(k) 0 0 −tg(k) βh(k) 0 0 0 0 βh(k) −tg(k) 0 0 −tg(k) −βh(k) , (167)
where we found that the eigenvalues were given by Eq. (108). The eigenvectors are
|↑, k, ±i = βh(k) ∓ r βh(k) 2 +t|g(k)| 2 |↑, kia+ tg(k) |↑, kib s |tg(k)|2+ βh(k) ∓ r βh(k) 2 +t|g(k)| 2 , (168) |↓, k, ±i = βh(k) ± r βh(k) 2 +t|g(k)| 2 |↓, kia− tg(k) |↓, kib s |tg(k)|2+ βh(k) ± r βh(k) 2 +t|g(k)| 2 . (169)
We proceed to calculate the projectors |σ, k, ±i hσ, k, ±|. In the same way as in the previous section, we only take only the projection of the Green’s function onto one of the sublattices, because of the nature of the interaction the choice of which one we choose must be done carefully and it is expected to give different results, however for these calculations for both sublattices give similar results.
|↑, k, ±i h↑, k, ±| ≈ |tg(k)| 2|↑, ki bh↑, k|b |tg(k)|2+ βh(k) ∓ r βh(k) 2 +t|g(k)| 2, (170) |↓, k, ±i h↓, k, ±| ≈ |tg(k)| 2|↓, ki bh↓, k|b |tg(k)|2+ βh(k) ± r βh(k) 2 +t|g(k)| 2. (171) Denoting G±(k) = |tg(k)|2 |tg(k)|2+ βh(k) ± r βh(k) 2 +t|g(k)| 2, (172)
the Green’s function reads
G(E) = 1 N X k G−|↑, ki bh↑, k|b E − E+↑(k) + G+|↑, ki bh↑, k|b E − E↑−(k) + G+|↓, ki bh↓, k|b E − E+↓(k) + G−|↓, ki bh↓, k|b E − E−↓(k) . (173)
This can be simplified, by noting that
E+σ = −E−σ ≡ ε(k), (174)
G(E) =X k G−+ G+E |↑, kibh↑, k|b E2− ε(k)2 + G−− G+ε(k) |↑, kibh↑, k|b E2− ε(k)2 (175) +G++ G−E |↓, kibh↓, k|b E2− ε(k)2 + G+− G−ε(k) |↓, kibh↓, k|b E2− ε(k)2 . (176)
Making the Fourier Transform back to real space and taking the matrix element hσ, j| G |σ, j0i
hσ, j| G |σ, j0i = 1 N X k N X jj0 G−+ G+Ee ik·(xj−xj0) E2− ε(k)2 + G−− G+ε(k)e ik·(xj−xj0) E2− ε(k)2 (177) +G++ G−Ee ik·(xj−xj0) E2− ε(k)2 + G+− G−ε(k)e ik·(xj−xj0) E2− ε(k)2 (178) = 2 N X k N X jj0 G−+ G+Ee ik·(xj−xj0) E2− ε(k)2 . (179)
The functions G±(k) are visualized in Figs17aand17band have sharp peaks in the vicinities of the points K and K0. In particular, we used that fact that the functions G±(k) have the property that
G+(k) + G−(k) = 1, (180)
(181)
and simplifying the expression to
gjj0 = hσ, j| G |σ, j0i = 2 N X k N X jj0 Eeik·(xj−xj0) E2− ε(k)2 , (182)
which is very similar to Eq. (153), except for the multiplicative factor. The calculations to find the DOS for the spin-orbit interaction are the same as in the previous section, but due to fact that the expression are more complicated, some of theses steps such as find the poles is done by numerical methods and the long expressions are omitted here.
We can see from the results in Figs18a and 18bthat the spin-orbit coupling induces a metallic-insulator transition, opening the gap in the DOS even for small values of λSO. When the size of the
nanotube is varied and the spin-orbit coupling is held fixed, it can be seen in the Fig. ??, that the gap increases as the size of the nanotube increases. In Fig19a, for an armchair nanotube with n = 8 and various values for λSO, the size of the gap created between the bands is proportional to the intensity of
the intrinsic spin-orbit coupling. This results can be compared with experimental data [19], that shows a formation of an gap when studying silicon nanotubes. These properties mean that we are able to control the gap of an system by manipulating the size and intensity of the spin-orbit coupling.
VI. CONCLUSION AND FUTURE WORKS
In this work we were able to calculate analytically the electronic properties of the graphene with the intrinsic spin-orbit coupling and also rolling these graphene sheets in different ways forming
-3 -2 -1 0 1 2 3 kx -3 -2 -1 0 1 2 3 ky 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 G+ (a) -3 -2 -1 0 1 2 3 kx -3 -2 -1 0 1 2 3 ky 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 G -(b)
FIG. 17: Visualization the functions G±(k) for β = 0.1 in the first Brillouin Zone. The sharp peaks are
0
0.2
0.4
0.6
0.8
1
1.2
-4
-3
-2
-1
0
1
2
3
4
DOS(E)
E
0
0.1
-0.3 0 0.3
0
0.1
-0.3 0 0.3
(a) Ch= (7, 7)0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-4
-3
-2
-1
0
1
2
3
4
DOS(E)
E
0
0.1
-0.3 0 0.3
0
0.1
-0.3 0 0.3
(b) Ch= (9, 9)FIG. 18: DOS for armchair nanotubes with Ch equal to (a) (7, 7) and (b) (9, 9), in the presence of the
intrinsic spin-orbit interaction with λSO= 0.05. It can be noted that the interaction caused a formation
0
0.05
0.1
0.15
0.2
0.25
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
DOS(E)
E
λ
SO=0.05
λ
SO=0.1
λ
SO=0.2
λ
SO=0.3
λ
SO=0.4
λ
SO=0.5
(a)FIG. 19: Size of gap created by the intrinsic spin-orbit interaction in a armchair nanotube with Ch= (8, 8). It can be seen that the gap increases proportional to the magnitude of the spin-orbit
coupling λSO.
nanotubes. We were able to find the energy dispersion of graphene analytically and analyze the behavior of electrons with low energies and found that the dispersion is linear in the vicinities of the so-called Dirac points. In that regime we were able to expand the dispersion and find a linearized version near the points K and K0. The spin-orbit interaction is added to the system using the Kane-Mele model [1] of the Quantum Spin Hall Effect (QSH) in the language of second quantization. It was found that the effects of the intrinsic spin-orbit coupling in the graphene was to open gap between the conduction and valence bands and a expansion near the Dirac cones corroborated to this fact.
The carbon nanotubes are studied in this work, the idea of how rolling a sheet of carbon in differ-ent ways may lead to differdiffer-ent electronic properties is explained. The characterization of the nanotubes is made using the chiral vectors, diameter and chiral angles, where the differentiation between achiral and chiral nanotubes is made. Under these categories we are able to classify the nanotubes in three categories: armchair, zigzag and chiral. The nanotubes that we had most interest in this work was the armchair and zigzag ones. For those nanotubes we were able to compute the energy dispersion by noting that for nanotubes with high symmetry we only need to specify the boundary conditions for the wave
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
DOS(E)
E
N=4
N=5
N=6
N=7
N=8
N=9
0
0.05
0.1
-0.066
-0.064
(a)FIG. 20: Size of gap created by the intrinsic spin-orbit interaction for various armchair nanotubes with λSO= 0.2. Although the dependence with the spin-orbit coupling is very small, it can be seen that the
size of the gap is proportional to the size of the nanotube.
vector in the appropriated direction and then could observe their electronic properties. It was observed that the energy dispersion, and thus the electronic properties, for these two kinds of nanotubes is gen-erally different. The energy dispersion for zigzag nanotubes showed that they could be either metallic or insulator depending on their size, for the armchair nanotubes the energy dispersion showed that the material is always metallic independent of their size. The many degeneracies could also be observed on those materials.
The spin-orbit is introduced by the same procedure of rolling a sheet of graphene, the boundary conditions were applied to the spin-orbit Hamiltonian and the energy dispersion could be obtained. In the same way as in graphene the intrinsic spin-orbit coupling created a gap between the conduction and valence bands, this is most interesting in the case of armchair nanotubes where they are always metallic. Using this interaction then we could induce a transition between the metallic and insulator phase of the armchair nanotubes.
To investigate further in the details of the electronic properties of these materials we calculated the density of states, which is the theoretical equivalent of a STM tip probing the material at each site.
