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An overview of tight-binding method for two-dimensional carbon structures

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In: Graphene: Properties, Synthesis and Applications ISBN 978-1-61470-949-7
Editor: Zhiping Xu, pp. 1-36 © 2011 Nova Science Publishers, Inc.

Chapter 1

AN OVERVIEW OF TIGHT-BINDING METHOD FOR

TWO-DIMENSIONAL CARBON STRUCTURES

Behnaz Gharekhanlou and Sina Khorasani*

School of Electrical Engineering
Sharif University of Technology
P. O. Box 11365-9363, Tehran
Iran

Abstract

Since the advent of graphene, the two-dimension allotrope of carbon, there has been
a growing interest in calculation of quantum mechanical properties of this crystal and its
derivatives. To this end, the tight-binding model has been in frequent use for a relatively
long period of time. This review, presents an in-depth survey on this method as possible
to graphene and two of the most important related structures, graphene anti-dot lattice,
and graphane. The two latter derivatives of graphene are semiconducting, while graphane
is semimetallic. We present band structure and quantum mechanical orbital calculations
and justify the results with the density function theory.

Key words: Graphene, Graphane, Carbon Nanostructures, Tight-binding Method,

Quantum Mechanics.

I. TIGHT-BINDING METHOD

I.1. Introduction

The tight-binding model (or TB model) is an approach to the calculation of electronic

band structure using an approximate set of wave functions based upon superposition of wave

*
E-mail address: khorasani@sina.sharif.edu
2 Behnaz Gharekhanlou and Sina Khorasani

functions for isolated atoms located at each atomic site. This model describes the properties
of tightly bound electrons in solids. The electrons in this model should be tightly bound to the
atom to which they belong and they should have limited interaction with states and potentials
on surrounding atoms of the solid. As a result the wave function of the electron will be rather
similar to the atomic orbital of the free atom where it belongs to. This method was developed
by Bloch [1] in 1928 considered only the satomic orbital. In 1934 Jones, Mott, and Skinner
[2] considered different atomic orbitals.
Tight-binding models are applied to a wide variety of solids. The model gives good
qualitative results in many cases and can be combined with other models that give better
results where the tight-binding model fails. Though the tight-binding approach is a one-
electron model, it also provides a basis for more advanced calculations like the calculation of
surface states and application to various kinds of many-body problem and quasiparticle
calculations. It is also a common practice to use optimized tight-binding methods, in which
the values of the matrix elements are derived approximately or fitted to experiment or other
theories [3].

I.2. Secular Equation

In a crystalline solid,the translational symmetry along the directions of thelattice vectors

ai , enforces the Bloch’s theorem to be satisfied by any wave function of the lattice.
Mathematically, this can be written as
i k ⋅a i
T ψk (r ) = e ψk (r ) (i = 1,2, 3) , (1)
where T is the translation operator along the lattice vector ai , and k is the Bloch wave
vector [4, 5]. In the tight-binding model of electronic structures, single electron wave
functions are normally expanded in terms of atomic orbitals [6, 7]
ϕnlm (r , θ, Φ ) = Rnl (r )Ylm (θ, Φ ) , (2)
where Rnl and Ylm are radial and spherical-harmonic functions in polar coordinates and n, l,
and m are the principal, angular-momentum, and magnetic quantum numbers, respectively.
The wave function can be defined in variety of models, but the most commonly used
form is a linear combination of atomic orbitals. Thus the wave function φ(k, r) is defined as
a summation on the atomic wavefunction ϕ(r − R j ) on the site j

1 N i k ⋅R j
φj (k, r) = ∑e
N R
ϕ( r − R j ) ( j = 1,..., n ) , (3)
j

where R j is the position of the atom and N is the number of atomic wavefunctions in the
unit cell. It is easy to verify that this function satisfies Bloch condition
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 3

1 N i k⋅ R j
φj (k, r + a) = ∑e
N R
ϕ(r + a − R j )
j

1 N i k⋅(R j −a) 
= e i k⋅ a j ∑e ϕ  r − (R j − a) (4)
N R −a  
j

= ei k⋅a j φj (k, r).

In a solid, eigenfunctions ψj (k, r) ( j = 1,..., n ) are defined as a linear recombination of
Bloch functions
n
ψj (k, r) = ∑ C jj ′ (k) φj (k, r) . (5)
j ′ =1
Here, C jj ′ (k) are coefficients which have to be determined.
The eigenvalues of the system described by the Hamiltonian H is given by [8]
∫ ψj (k, r)H ψj (k, r)dr .
*
ψ j H ψj
E j ( k) = = (6)
*
ψj ψj ∫ ψj (k, r) ψj (k, r)dr
Now, substituting ψj (k, r) as defined in (5) leads to
n n
∑ C ij* C ij ′ φj H φj ′ ∑ H jj ′ (k)C ij*C ij ′
j , j ′ =1 j , j ′ =1
E j (k) = n = n , (7)
∑ C ij* C ij ′ φj φj ′ ∑ S jj ′ (k)C ij*C ij ′
j , j ′ =1 j , j ′ =1
where H jj ′ (k) and S jj ′ (k) are transfer and overlap matrices, respectively, and are defined by

H jj ′ (k) = φj H φj ′ , (8)

S jj ′ (k) = φj φj ′ . (9)
*
For a given k value, the coefficient C jj ′ (k) is optimized so as to minimize Ei (k)
n
∑ H jj ′ (k)C ij ′ (k)
∂Ei (k) j ′ =1
= n
∂C ij * (k)
∑ S ′ (k)C ij * (k)C ′ (k)
jj ij
j , j ′ =1
n
∑ H ′ (k)C ij * (k)C ′ (k)
jj ij n
j , j ′ =1
 n
−
 2 j ′ =1
∑ S jj ′ (k)Cij ′ (k) = 0. (10)
 
 ∑ S jj ′ (k)C ij * (k)C ij ′ (k)
 j, j ′ =1 
 
This further can be rewritten as
4 Behnaz Gharekhanlou and Sina Khorasani

n
∑ H jj ′ (k)C ij ′ (k)
j ′ =1
n
∑ S ′ (k)C ij * (k)C ′ (k)
jj ij
j , j ′ =1
 n  n
 ∑ H (k)C * (k)C (k)  ∑ S jj ′ (k)C ij ′ (k)
 jj ′ ij ij ′ 
 j , j ′ =1  j ′ =1
− n  n = 0. (11)
 
 ∑ jj ′ S ( k )C ij
* ( k )C
ij ′
( k )  ∑ jj ′ S ( k )C ij
* ( k )C
ij ′
( k )
 j , j ′ =1  j , j ′ =1
 
This in turn can be readily simplified as
n n
∑ H jj ′ (k)C ij ′ (k) − Ei (k) ∑ S jj ′ (k)C ij ′ (k) = 0 . (12)
j ′ =1 j ′ =1
Putting in the matrix form we get
{[H] − Ei (k)[S]} {Ci (k)} = 0 . (13)

If the matrix [ H] − Ei (k)[S] has an inverse, the vector {C i (k )} will be identically zero and
this leads to the trivial solution. Thus,non-trivial solutions require
[ H] − Ei (k)[S] = 0 . (14)
This equation is called the secular equation, whose eigenvalues Ei (k) give the energy
bandstructure.
As an example, suppose that the primitive basis contains one atom with the atomic orbital
function ϕ(r) . We get
1 N i k ⋅R j
φj (k, r) = ∑e
N R
ϕ(r − R j ) ( j = 1,..., n ) . (15)
j

Here we use the nearest-neighbor Hamiltonian of the system; it means that only the effect
of the nearest-neighbors of the atom in the unit cell is considered.The transfer and overlap
matrices are now defined respectively as follows
1 N ik⋅(R −R′ )
H jj (k) = φj H φj = ∑ ∑ e j j ϕ(r − R j ) H ϕ(r − R ′j )
N R R′
j j

1 N
= ∑ ϕ(r − R j ) H ϕ(r − R j )
N R =R ′
j j
(16)
N
1
+ ∑ eik⋅a ϕ(r − R j ) H ϕ  r − (R j + a) + ...
N R ′ = R +a
j j

N
=− α − γ0 ∑ eik⋅a
R ′j = R j + a
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 5

1 N ′
S jj (k) = φj φj = ∑ ∑ eik⋅(R j −R j ) ϕ(r − R j ) ϕ(r − R ′j )
N R R′
j jj

1 N
= ∑ ϕ(r − R j ) ϕ(r − R j )
N R =R ′
(17)
j j

N
1
+ ∑ eik⋅a ϕ(r − R j ) ϕ r − (R j + a) + ... = 1.
N R′ =R +a
j j

Here, we assume that the atomic wavefunction is normalized.Then, the secular equation
reduces to
N
E (k) = −α − γ0 ∑ eik⋅a . (18)
R′j =R j +a
The constant parameters α and γ 0 are usually found by fitting this equation to high
symmetric points of experimental data.
We can use an important approximation in the Tight-binding method, when considering
the orbital number l, as well. One can rewrite (3) as
1 N ik⋅R
φli (k, r) = ∑ e ϕ(r − ti − R) .
N R
(19)

The corresponding eigenfunction would be given by

n
ψ(k, r) = ∑ Cli (k) φli (k, r) . (20)
l ,i
Hence, the Schrödinger Hamiltonian equation takes the form
Hψ = E ψ,
(21)
∑  φmj (k, r) H φli (k, r) − E (k) φmj (k, r) φli (k, r)  Cli (k) = 0.

l ,i
In order to solve this linear system we need to be able toevaluate the following parameters:
1
φmj (k, r) φli (k, r) = ∑ e ik⋅R ϕm (r − t j ) ϕl (r − ti − R)
N R
N
(22)
= ∑ eik⋅R ϕm (r − t j ) ϕl (r − ti − R) .
R
In a similar way, we have
1
φmj (k, r) H φli (k, r) = ∑ eik⋅R ϕm (r − t j ) H ϕl (r − ti − R)
N R
(23)
N
ik⋅R
= ∑e ϕm (r − t j ) H ϕl (r − ti − R) .
R
At this point we can use an important approximation [11]: the overlap matrix elements in (9)
is non-zero only for the same orbitals on the same atom, i.e.
6 Behnaz Gharekhanlou and Sina Khorasani

ϕm (r − t j ) ϕl (r − ti − R ) = δlm δij δ(R) . (24)

This is referred to as an “orthogonal basis”, since any overlap between different orbitals on
the same atom or orbitals on different atoms is taken to be zero. So it will be a unit matrix.
Similarly, we will take the Hamiltonian matrix elements in (8) to be non-zero only if the
orbitals are on the same atom which are referred to as the “on-site energies”:
ϕm (r − t j ) H ϕl (r − ti − R) = δlm δij δ(R)εl . (25)
The above relation implies that if the orbitals belong to different atoms, but situated at nearest
neighbor sites, denoted in general as d , one would get the following
ϕm (r − t j ) H ϕl (r − ti − R) = δ (t j − ti − R − d )Vlm,ij . (26)
The Vlm,ij are also referred to as “hopping” matrix elements.
Now, one needs only to find the on-site and hopping matrix elements, from which the
energy eigenvalues of the related Hamiltonian matrix are found.

II. CARBON ATOM

II.1. Introduction

Carbon is present in nature in many allotropic forms likes graphite, diamond, fullerenes,
nanotubes, etc.This is because of the versatility of its bonding.Carbon is the sixth element of
the periodic table. Each carbon atom has six electrons in the configuration1s2, 2s2, and 2p2
atomic orbitals, i.e. 2 electrons fill the inner shell 1s, which is close to the nucleus and which
is irrelevant for chemical reactions, whereas 4 electrons occupy the outer shell of 2s and 2p
orbitals. Because of low difference in energy of 2s and 2p levels in carbon, the electronic
wave functions for these four electrons can be mixed with each other, changing the
occupation of the 2s and three 2p atomic orbitals so as to enhance the binding energy of the C
atom with its neighboring atoms [8, 9]. These orbitals are called hybrid orbitals. In carbon,
mixing of 2s and 2p orbitals leads to three possible hybridizations sp, sp2 and sp3, generally
called spn hybridization with n = 1,2, 3 . Other elements of this group such as Si, Ge exhibit
sp3 hybridization. Carbon differsfrom Si and Ge because it only has the spherical 1s orbitals
as inner atomic orbitals so that the absence of nearby inner orbitals facilitates hybridizations
involving only valence s and p orbitals for carbon. The lack of sp and sp2 hybridization in Si
and Ge might be related to the absence of “organic materials” made of Si and Ge [8].

II.2. Hybridization in Carbon Allotropes

II.2.1. sp Hybrid Orbitals

The sp hybrid atomic orbitals are one of the possible states of electron in an atom,
especially when it is bonded to others. These electron states have half 2s and half 2p
characters. We can see this kind of hybridization in chemical compounds such as ethyne. This
type of hybridization has been also recently observed in free carbon nanoparticles [10].
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 7

Figure II.2.1. sp hybridization.

For illustration, we choose the 2px state. From a mathematical view point, there are
two ways to combine the 2s and 2p atomic orbitals, obtained by the symmetric and anti-
symmetric combinations
sp+ = C1 2s + C 2 2px ,
(27)
sp− = C 3 2s + C 4 2px .

Using the ortho-normality conditions sp+ sp− = 0 , sp+ sp+ = 1 , and sp− sp− = 1 ,
the relationships between the coefficients are obtained as follow
C 1C 3 + C 2C 4 = 0, C 12 + C 22 = 1,
(28)
C 32 + C 42 = 1, C 12 + C 32 = 1.
The result is
sp+ =
1
2
(
2s + 2px , )
(29)
sp− =
1
2
(
2s − 2px . )
These energy states have a region of high electron probability each, and the two atomic
orbitals are located opposite to each other, centered on the atom.

II.2.2. sp2 Hybrid Orbitals

If we mix two of the 2p orbitals with a 2s orbital, we end up with three sp2 hybridized
orbitals. In the case of a superposition of the 2s and two 2p orbitals, which we maychoose to
be the 2px and the 2py states, one obtains the planar sp2 hybridization. These three
orbitals lie on a plane and make equal angles of 120°, pointing to the vertices of an equilateral
triangle. When the central atom makes use of sp2 hybridized orbitals, the compound so
formed has a trigonal shape as shown in Figure II.2.2.
8 Behnaz Gharekhanlou and Sina Khorasani

Figure II.2.2. sp2 hybridization.

From a mathematical view point we have

sp12 = C 1 2s − 1 − C 12 2py ,

 3 1 
sp22 = C 2 2s + 1 − C 22  2px + 2py  , (30)
 2 2 
 3 1 
sp32 = C 3 2s + 1 − C 32 − 2px + 2py  .
 2 2 
Using ortho-normality conditions, we obtain
C12 + C 22 + C 32 = 1
1 1 1
C1C 2 − 1 − C 12 1 − C 22 = 0 ⇒ C1 = C 2 = , C3 = − . (31)
2 3 3
1
C1C 3 − 1 − C12 1 − C 32 = 0
2
Hence, the three possible quantum-mechanical states are given by
1 2
sp12 = 2s − 2py ,
3 3
1 2  3 1 
sp22 = 2s +  2px + 2py , (32)
3 3  2 2 
1 2  3 1 
sp32 = − 2s + − 2px + 2py .
3 3  2 2 
When carbon atoms make use of sp2 hybrid orbitals for σ bonding, the three bonds lie on
the same plane. Such carbon allotropes include the two-dimensional graphite andgraphene,
the one-dimensional carbon nanotube, and the zero-dimensional fullerene, in which, every
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 9

pair of carbon atoms make use of sp2 hybrid orbitals. The remaining p orbitals of carbon
atoms overlap to form a π bond.
π bonds result from the overlap of atomic orbitals that are in contact through two areas
of overlap. π bonds are more diffuse bonds than the σ bonds. Electrons in π bonds are
sometimes referred to as π electrons.
In the case of sp2 hybridization, the carbon atom is a special case. Because the only
orbital bonded to the nucleus is 1s, the size of atoms is small and the resultant band is
considerably strong. Other elements of group IVnormally appear in sp3 hybridization. Going
down the table of periodic element of this group, and with the physical size of the elements
increasing, the bond energy is reduced,and eventually the last element of this group, that is
Pb, becomes a metal rather than being a semiconductor. Since the π bonds are much weaker
than σ bonds, forming of π bonds in the other elements of this group would be highly
unstable. While the bonding energy of π orbitals in Si is only about 25Kcalmol−1, this value
is about 60Kcalmol−1 for carbon.

II.2.3. sp3 Hybrid Orbitals

Carbon atoms in diamond provide a simple example of sp3 hybrid orbitals. Mixing one s
and all three p atomic orbitals produces a set of four equivalent sp3 hybrid atomic orbitals.
When carbon atoms make use of sp3 hybrid orbitals, the four bonds around each carbon atom
point toward the vertices of a regular tetrahedron and make angles of 109.5°.

Figure II.2.3. sp3 hybridization.

Using calculations similar to previous sections, four sp3 hybrid orbitals are given by
10 Behnaz Gharekhanlou and Sina Khorasani

1
sp13 =
2
{
2s + 2px + 2py + 2pz },
sp23 =
1
2
{
2s − 2px − 2py + 2pz },
(33)
sp33 =
1
2
{
2s − 2px + 2py − 2pz } ,

sp43 =
1
2
{
2s + 2px − 2py − 2pz }.
When sp3 hybrid orbitals are used for the central atom in the formation of molecule, the
molecule is said to have the shape of a tetrahedron.
In general for spn hybridization, n + 1 electrons belong to the carbon atom occupied in
the hybridized σ orbital and 4 − (n + 1) electrons remain in the π orbital.

III.3. Vanishing and Non-vanishing Orbital Interactions Based on TB model

To study carbon allotrops, there are four atomic orbitals per atom: one s-type and three p-
type (px , py , pz ) . We want to use Tight-binding method to obtain the electronic band
structure. Therefore we work within the orthogonal basis of orbitals and nearest neighbor
interactions only, as described by the equations of (14).To solve this eigenvalue problem we
need the Hamiltonian matrix elements between these atomic orbitals at different interatomic
distance. For the sp-bonding, there are only four non-zero overlapping integrals as shown in
Figure II.3.1 because of spherical orientation of these atomic orbitals. The remaining cases
like those shown in Figure II.3.2 correspond to the matrix elements which vanish because of
symmetry constraints. σ and π bondings are defined such that the axes of the involved p
orbitals are parallel and normal to the interatomic vector, respectively.

Figure II.3.1. sp-bonding correspond to non-vanishing matrix elements.

 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 11

Figure II.3.2. sp-bonding

bonding correspond to vanishing matrix elements.

In general the p orbitals are notjust parallel or lie on the line that joins the atomic
positions. We can always project each p orbital into two components, s, one of them along the
bonding line, the other one perpendicular to it. These are called σ and π components,
respectively. This then leads to the general description of the interaction between two
porbitals,or one s and one p orbital [11]. Consider the Hamiltonian matrix element between
the s orbital on one atom and one of the p orbitals, which is here denoted by pa , on another
atom in random directions θ relative to the joining line of their centers. Let d be the unit
vector along the bond and a is the unit vector along one of the Cartesian axes (x , y, z ) as
shown in Figure II.3.3.

Figure II.3.3. The p orbital is in random directions θ relative to the joining line to the s orbital.

The p orbitals can be decomposed into two components that are parallel and normal to d
as illustrated below
pa = a ⋅ d pd + a ⋅ n pn = cos θ pd + sin θ pn . (34)
Now the Hamiltonian matrix element s and pa is given by
s H pa = s H pd cos θ + s H pn sin θ = Hspσ cos θ . (35)
Note that the overlap between s and pn orbitals is zero by symmetry.
12 Behnaz Gharekhanlou and Sina Khorasani

Consider the Hamiltonian matrix element between p1 and p2 orbitals bonded along
the unit vector d inn random directions θ1 and θ2 , respectively. This situation is shown in
Figure II.3.4.

Figure II.3.4. Two p orbitals bonding in random directions.

The p orbitals can be decomposed into two components that are parallel and normal to d
as we have done below in (36) and demonstrated in Figure II.3.5.
p1 = a1 ⋅ d pd 1 + a1 ⋅ n pn1 = cos θ1 pd 1 + sin θ1 pn1 ,
(36)
p2 = a 2 ⋅ d pd 2 + a 2 ⋅ n pn 2 = cos θ2 pd 2 + sin θ2 pn 2 .

Figure II.3.5. Decomposition of p orbitals into parallel and normal components with respect to d .

The Hamiltonian matrix element is then given by

p H p = pd 1 H pd 2 cos θ1 cos θ2 + pn 1 H pn 2 sin θ1 sin θ2
1 2 (37)
= H ppσ cos θ1 cos θ2 + H pp π sin θ1 sin θ2 ,
where we have used the fact that overlap between orthogonal p orbitals are zero by symmetry.
 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 13

III. ENERGY BAND CALCULATION OF GRAPHENE

III.1. Introduction

structure is one-atom-thick planar sheets of sp2

Graphene is an allotrope of carbon. Its str
bonded carbon atoms that are densely packed in a honeycomb crystal lattice.The honeycomb
lattice can be described in terms of two triangular sub
sub-lattices, A and B (Figure III.1.1) so
there are two atoms in each unit cell.

Figure III.1.1. Graphene lattice

ttice is composed of two triangular sub
sub-lattices.
lattices. The lattice constant is a .

The carbon-carbon
carbon bond length in graphene is about 1.42Å and its lattice constant a is
about 3.49Å. The unit vectors of the unde
underlying triangular sub-lattices are given by
a
a1 = ( 3, +1),
2
a (38)
a2 = ( 3, −1) .
2
Graphene is a new material, and known to be not only the thinnest ever semimetal, but
also the strongest material, too. As a conductor of electricity, it performs as well as copper.
As a conductor of heat it outperforms all other known materials. It is almost completely
transparent, yet so dense that not even helium, the smallest atomic gas, can pass through it.
Furthermore, graphene is an excellent support of spin
spin-current, and
d provides an ideal support
bed for the future spintronic devices [12].
Graphene is a basic building block for graphitic materials of all other dimensionalities. It
can be wrapped up into 0D fullerenes, rolled into 1D nanotubes or stacked into 3D graphite
[13]. Graphite is obtained by stacking of graphene layers that is stabilized by weak van der
Waalse interactions [14]. Diamond also can be obtained from graphene under extreme
pressure and temperatures. Carbon nanotubes are formed by graphene wrapping, while whi
14 Behnaz Gharekhanlou and Sina Khorasani

fullerenes are obtainable from graphene by modifying the hexagons into pentagons and
heptagons in a systematic way [15]. It is obvious that there has been enormous interest in
understanding physical properties of graphene. Graphene has been the subject of many recent
investigations due to its peculiar transport properties [16, 17]. The 2010 Nobel Prize in
Physics has been awarded to Andre Geim and Konstantin Novoselov for groundbreaking
experiments regarding the two
two-dimensional material graphene [18] (Figure III.1.3).

Figure III.1.2. The unit vectors and unit cell in graphene lattice.

Figure III.1.3. Konstantin Novoselov (on the left) accepts the Nobel Prize in Physics
hysics in December 10,
2010, “for groundbreaking experiments regarding the two
two-dimensional material graphene” [18].
[

III.2. Orbital Hybridization in Graphene

Graphene is a flat monolayer of carbon atoms tightly packed into a two

two-dimensional
dimensional (2D)
2
honeycombb lattice. Because of this,carbon atoms have the sp hybridization. In this case,
case three
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 15

σ bonds lie on the same plane with bond angles 120°. The remaining 2pz orbital for each
carbon overlaps to form a π bond.

III.3. Calculation of σ Bands Using TB

The σ bands are related to σ bonds between carbon atoms. It is should be pointed out
that each unit cell of graphene has two carbon atoms with sp2 hybridization, and every atom
participates with three 2s, 2px and 2py orbitals in the bonding. Thus we deal with six orbitals
in each unit cell yielding six σ bands, three of which appear below the Fermi energy called σ
, and the other three above the Fermi energy being referred to as σ ∗ .
We have to use the secular equation (14). Here the transfer and overlap matrices are
represented by 6 × 6 matrices.
We start with the transfer matrix
2sA 2px A 2pyA 2sB 2px B 2pyB
 H 
 11 ...
2sA ... ... ... H 16 
 


2px A 





2p  
H = yA  
2sB 


 
2pxB 


 
2pyB  H ... ... ... ... H 66 
 61 6×6
 
 
 
 H AA H AB 
 
 
=  .
 
 
 H BA H BB  (39)
 

 
Here, AA and AB terms belong to integrals between orbitals of atom A in the unit cell, and
between A and B atomic orbitals, respectively.
Written out in component form, the H AA is
H AA (k) = φA H φA
1 N i k⋅(R A −R A′ )
= ∑ ∑
N R R′
e ′)
ϕ(r − RA ) H ϕ(r − RA (40)
A A

1 N
= ∑ ϕ(r − RA ) H ϕ(r − RA ) .
N R =R′
A A

Each matrix element is determined using below relations

16 Behnaz Gharekhanlou and Sina Khorasani

1 N
h11 = φ2s H φ2s = ∑ ϕ (r − RA ) H ϕ2s (r − RA ) = ε2s ,
N R = R ′ 2s
A A

1 N
h22 = φ2p H φ2p =
x x
∑ ϕ (r − RA ) H ϕ2px (r − RA ) = ε2p , (41)
N R =R ′ 2px
A A

h33 = φ2p H φ2p = h22 = ε2p .

y y

The remaining terms are zero because of the spherical harmonics Ylm (θ, ϕ) of px and py
orbitals. In the tabular form, H AA is given by
ε 0 
 2s 0
 
H AA =  0 ε2p 0  . (42)
 
 0
 0 ε2p 
Suggested model parameters values fitted tothe electronic band structure along the high-
symmetry lines are ε2s =− 8.7 eV and ε2p = 0 eV .
We define tabular form of H AB as
h 
 11 h12 h13 
 

H AB = h21 h22 h23  (43)
 
h 
h
 31 32 h 33 
which its components have the general form
1 N ik⋅(R A −R B )
H AB (k) = φA H φB = ∑
N R R
∑ e ϕ(r − RA ) H ϕ(r − RB ) . (44)
A B

Here we consider only the nearest-neighbor interaction. There are three such nearest
neighbors, and hence the vectors RAi are given by
a
RA1 = (−1, 0),
3
a
RA2 =
2 3
(1, + 3 ), (45)
a
RA3 =
2 3
(1, − 3 ).
As discussed before, a is the lattice constant of the graphene. Figure III.3.1 shows these
vectors.
 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 17

Figure III.3.1. Three nearest neighbors of the carbon atom shown by RAi vectors.

Because of the different orientations of the nearest

nearest-neighbor atoms Bi correspond to the
atom A , it is necessary to consider general forms for (40) and (42). Hence, one can write
H AB components as

1 N ik⋅(R A −R B )
h11 = φA2s H φB 2s = ∑ ∑ e ϕA2s (r − RA ) H ϕB 2s (r − RB )
N R R
A B (46)
3
ik⋅R Ai
= H ss σ ∑ e .
i =1
RAi are the vectors pointing from A to Bi atoms. As it was discussed above,, these vectors
show the nearest neighbors of A atom. Therefore, we get the following
h12 = φA2s H φB 2 p
x

1 N ik⋅(R A −R B )
= ∑
N R R
∑ e ϕA2s (r − RA ) H ϕB 2p (r − RB )
x
(47)
A B
3
ik⋅R Ai
= H spσ ∑ e cos θ ,
i =1

h13 = φA2s H φB 2 p
y

1 N
= ∑ ∑ eik⋅(RA −RB ) ϕA2s (r − RA ) H ϕB 2py (r − RB )
N R R
(48)
A B
3
ik⋅R Ai
= H spσ ∑ e sin θ ,
i =1
18 Behnaz Gharekhanlou and Sina Khorasani

h22 = φA2 p H φB 2p
x x

1 N ik⋅(R A −R B )
= ∑
N R R
∑ e ϕA2p (r − RA ) H ϕB 2 p (r − RB )
x x
(49)
A B
3

( ik⋅R Ai
= ∑ H ppσ cos2 θ + H pp π 1 − cos2 θ  e

i =1 
 ),

h23 = φA2 p H φB 2 p
x y

1 N
= ∑ ∑ eik⋅(RA −RB ) ϕA2px (r − RA ) H ϕB 2py (r − RB )
N R R
(50)
A B
3
= ∑ (H ppσ − H pp π ) cos θ sin θ eik⋅R Ai
,
i =1

h33 = φA2 p H φB 2p
y y

1 N
= ∑ ∑ eik⋅(RA −RB ) ϕA2py (r − RA ) H ϕB 2py (r − RB )
N R R
(51)
A B
3
∑ H ppσ sin2 θ + H ppπ (1 − sin2 θ) eik⋅R
 
= Ai
.
i =1
Other parameters can be written based on the facts that s Hp =− p Hs ,
p1 H p2 = p2 H p1 and s1 H s2 = s2 H s1 . We choose the model parameters value
to be Hspσ = 5.5 eV , H ss σ = −6.7 eV , H ppσ = 5.1 eV , and H ppπ = −3.1 eV , which are
close to reported values in [8]. These parameters are obtained from a fit to the experimental
electronic band structure along the high-symmetry lines.
The same treatment yields the overlap matrix elements. We assume that the atomic
wavefunctions are normalized as shown in (17), so the SAA matrix is given by
1 0 0
 
 
S AA = 0 1 0. (52)
 
0 0 1
 
Similarly to H AB ,the elements of S AB is described as below
3
s11 = φA2s φB 2s = Sss σ ∑ eik⋅RAi , (53)
i =1
3
s12 = φA2s φB 2p = Sspσ ∑ eik⋅RAi cos θ , (54)
x
i =1
3
s13 = φA2s φB 2p = Sspσ ∑ e ik⋅RAi sin θ , (55)
y
i =1
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 19

3
s22 = φA2p φB 2p = ∑ S ppσ cos2 θ + S ppπ 1 − cos2 θ
x x 
( ) eik⋅R Ai , (56)
i =1
3
s23 = φA2p φB 2p = ∑ e ik⋅RAi (S ppσ − S pp π ) cos θ sin θ , (57)
x y
i =1
3
s33 = φA2p φB 2p = ∑ S ppσ sin2 θ + S ppπ 1 − sin2 θ
y y 
( ) eik⋅R Ai . (58)
i =1
Other parameters can be written based on the facts that s p = − p s , p1 p2 = p2 p1 ,
and s1 s2 = s2 s1 . Suggested tight-binding parameters are Sspσ =− 0.10 eV ,
Sss σ = 0.20 eV , S ppσ =− 0.15 eV , and S pp π = 0.12 eV .
The two carbon atoms in the unit cell of graphene are identical. Hence the transfer matrix
element H AA , which is given by the interaction of an atom at site A with itself an all other
A atoms in the crystal, is exactly the same as H BB . Similarly, H AB is simply the complex
conjugate of H BA . This condition is also satisfied for SAA and SBB , and S AB and SBA .
So we have
*
H BB = H AA, H BA = H AB ,
(59)
*
SBB = SAA, SBA = SAB .
Finally, the secular equation is expressed in the most general form of
H AA(3×3) H AB(3×3)  S SAB(3×3) 
  − E (k)  AA(3×3)  = 0.
H H  S S 
 BA(3×3) BB(3×3)   BA(3×3) BB (3×3)  (60)
Solving this eigenvalue problem yields the electronic band structure of graphene. Since all
possible eigenstates are specified by the wavevector k within any one primitive cell of the
periodic lattice in reciprocal space, the first Brillouin zone is the uniquely defined cell that is
the most compact possible cell. So it is enough to calculate the band structure in this region.
For graphene, the first Brillouin zone is a hexagon. In this case, six symmetry axes exist
which can be deduced from one another by rotations of π 6 with an invariance under rotation
equal to π 3 . Figure III.3.2 shows this region.

Figure III.3.2. First Brillouin zone of the hexagonal lattice. The irreducible zone is shown in red.
20 Behnaz Gharekhanlou and Sina Khorasani

The range over which k is studied can be further reduced by considering the other
symmetries which constitutes the irreducible Brillouin zone of the hexagonal lattice. It can be
demonstrated that the irreducible Brillouin zone is the half of an equilateral triangle with
apexes Γ, K and M as shown in Figure III.3.2. It is generally sufficient to let the wave vector
k describes the edge of the irreducible Brillouin zone, while the inner region is ignored.
Figure III.3.3 shows the σ band structure of the graphene lattice calculated using Tight-
binding model [19].

30

20

10
E(eV)

-10

Γ K M Γ

Figure III.3.3. σ electronic band structure of graphene calculated using tight-binding method.

III.4. Calculation of π Bands Using TB

Band structure of graphene around the Fermi level and for transition energies well above
the optical range is determined by π orbitals; the Fermi level is normally placed at the Dirac
point for unbiased and undoped graphene. A common approximation for the low energy
electronic properties is hence a tight-binding Hamiltonian including only the carbon 2pz
states. Thep orbitals are all parallel to one another and each contains one electron (Figure
III.4.1).
As mentioned before, each unit cell of graphene has two carbon atoms which construct π
bonds using pz orbitals. Using the secular equation (14), the transfer and overlap matrices are
obtained as 2 × 2 matrices.
We start with transfer matrix
 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 21

2 pz A 2 pz B

2 pz A  H H AB  (61)
[H ] = 
 AA  .
2 pz B  H 
 BA H BB 2×2

Figure III.4.1. p orbitals of carbon atoms in the graphene structure.

Written out in component form, the [H ] is

H AA = φ2p H φ2p
z z

1 N (62)
= ∑ ϕ (r − R A ) H ϕ2pz (r − R A ) = ε2p ,
N R = R ′ 2 pz
A A

H AB = φA2 p H φB 2 p
z z

1 N
= ∑ ∑ e ik⋅(R A −R B ) ϕA2pz (r − RA ) H ϕB 2pz (r − RB )
N R R
(63)
A B
3
ik⋅R Ai
= ∑ H pp πe .
i =1
The same treatment yields the overlap matrix elements as follows
2 pz A 2 pz B

2 pz A  S
S AB  (64)
[S ] = 
 AA 
2 pz B  S 
 BA S BB 
2×2
which can be written out in the component form
1 N
S AA = φ2p φ2p =
z z
∑ ϕ (r − RA ) ϕ2pz (r − RA ) = 1,
N R = R ′ 2 pz
(65)
A A
22 Behnaz Gharekhanlou and Sina Khorasani

S AB = φA2 p φB 2 p =
z z

1 N
= ∑ ∑ eik⋅(R A −R B ) ϕA2pz (r − R A ) ϕB 2pz (r − R B )
N R R
(66)
A B

3
= ∑ S ppπeik⋅R Ai .
i =1
Again, in (65) we assume that the atomic wave functions are normalized.
As mentioned before, the transfer matrix element H AA , given by the interaction of an
atom at site A with itself an all other A atoms in the crystal, is exactly the same as H BB .
Similarly, H AB is simply the complex conjugate of H BA . This condition is also satisfied for
S AA , S BB , and S AB and S BA .
The secular equation is constructed as below
H AA(1×1) H AB (1×1) S AA(1×1) S AB (1×1)
− E (k) = 0.
H BA(1×1) H BB (1×1) S BA(1×1) S BB (1×1)
(67)
Solving this eigenvalue problem yields to the π electronic bands of graphene shown in
Figure III.4.2 [19].

10
E(eV)

-5
Γ K M Γ

Figure III.4.2. π electronic band structurecalculated using tight-binding method.

Figure III.4.3 shows σ and π band structures of graphene together. There is a very good
agreement between these results and those reported in [20] obtained from first-principles
calculations shown in Figure III.4.4.
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 23

30

20

10
E(eV)

-10

Γ K M Γ

Figure III.4.3. σ (blue) and π (red) band structures calculated using Tight-binding method.

Figure III.4.4. σ (red) and π (dashed blue) band structures from first-principles calculations [20].

It is remarkable that graphene is a gapless semiconductor, with no gap at the Dirac point
[16]. Many interesting ways have been proposed and investigated to tune the band gap of
graphene. However, it is well known that the presence of defects such as vacancies or
impurities [19, 21, 22], the interaction with a substrate [23], or the edge structure in graphene
nanoribbons [24, 25, 26] can lead to a bandgap opening in graphene.
The idea of introducing a periodic array of vacancies into the graphene lattice was first
mentioned in [19]. Many works have been done with the same idea, since then, on the so-
called anti-dot graphene lattices [27, 28, 29, 30, 31].
Electronic structures of coupled semiconductor quantum dots arranged as graphene hexa-
gonal lattice are studied theoretically using the tight-binding method up to the third-nearest-
24 Behnaz Gharekhanlou and Sina Khorasani

neighbors. The novel two-dimensional Dirac-like electronic excitations in graphene are found
in these artificial planar quantum dot structures which provide the theoretical basis for
searching Dirac fermions in quantum dot materials and have great significance for
investigating and making semiconductor quantum dot devices [32].
Band gap engineering in hexagonal h-BN antidot lattices is another work done in this
field. Although graphene is a gapless semimetal due to the symmetry between two sublattices
of the honeycomb structure, h-BN is a semiconductor with a wide bandgaparound 5.5 eV
[33]. It is found that when the anti-dot lattice is along the zigzag direction, the band gap
opening can be related to the intervalley scattering and does not follow the simple scaling rule
previously proposed in the literature for the anti-dot lattice along the armchair direction. For
h-BN, the calculations show that the anti-dot lattice results in reducing of band gaps. Coupled
with doping of carbon atoms, the bandgap of h-BN antidot lattice can be reduced to below
2 eV, which might have implications in light-emitting devices or photo-electrochemistry [34].
Regular hydrogen adsorption on graphene is another way to open a bandgap in graphene
[35] to be discussed briefly later in the next section.

IV. ENERGY BAND CALCULATION OF GRAPHANE

IV.1. Introduction

The chemical modification of graphene to create derivatives with different structures and
properties has so far been restricted to graphene oxide [36, 37, 38], a disordered structure that
bears an assortment of functional groups.
Another possible modification, which has been predicted theoretically [39], is the
addition of hydrogen atoms to graphene, altering the sp2 hybridization of carbon atoms to sp3,
and thus changing the structure and electronic properties, to form a new two-dimensional
hydrocarbon called graphane. Recently, the Nobel Laureates Andre Geim and Konstantin
Novoselov from the University of Manchester and co-workers from Russia and the
Netherlands have hydrogenated mechanically exfoliated graphene by exposing it to hydrogen
plasma [40]. The electronic properties of the new material change markedly: the highly
conductive graphene is converted from a semimetal into a semiconductor. Because of its
structure and lowdimensionality, it provides a fertile playground for fundamental science
andtechnological applications.
It is reported that hydrogenation of mono, bi, and trilayer graphene samples via exposure
to H2 plasma occurs as a result of electron irradiation of H2O adsorbates on the samples,
rather than H species in the plasma as reported [41]. It is proposed that the hydrogenation
mechanism is electron-impact fragmentation of H2O adsorbates into H+ ions. At the incident
electron energies of 60eV, hydrogenation is observed which is significantly more stable at
temperatures 200°C than previously reported [42].
This hexagonal arrangement of carbons in the sp3 hybridization with C: H = 1 forms two
constructions: if the H atoms connect at opposite sides of the plane of C-C bond, it is called
chairlike, as illustrated in Figure IV.1.1, and if on the same side of the C-C plane, it is called
boatlike. The typical C-C bond length for chair like conformer is 1.52Å which is similar to the
sp3 bond length of 1.53Å in diamond and that for boatlike conformer it is 1.56Å [39].
 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 25

Figure IV.1.1. The chairlike

hairlike structure of graphane.

Graphane was first predicted to exist only in the boat and chair configurations [37].[ But a
new report has just pointed out that the existence of a new conformer, being referred to as the
stirrup, is also plausible [42].
]. Ne
Nevertheless,
vertheless, all these three conformers are expected to exhibit
very close chemical and physical properties. The structural and phonon properties of these
structures,, as well as their relative stability can be established by first-principles
principles calculations.
calculations
It is shown that all these three graphane conformers respond to any arbitrarily oriented
extension with a much smaller lateral contraction than the one calculated for graphene.
Furthermore, the boatlike-graphane
graphane has a small and negative Poisson ratio along the armchair
and zigzag principal directions of the carbon honeycomb lattice. Moreover, the chairlike-
graphane admits both softening and hardening hyperelasticity, depending oon n the direction of
applied load [43]. Lattice thermal properties of graphane latt lattice
ice have been also calculated
[44]. Some further properties of graphene/graphane interfaces are discussed in [45 45, 46]. It is
also anticipated that doped graphane could serve as a high temperature superconductor with a
transition temperature well exceeding 90K [47, 48, 49].

IV.2. Orbital Hybridization in Graphene

Graphane is again an arrangement of carbon atoms with sp3 hybridization saturated by

hydrogen forming a honeycomb shape. Transformation of the sp2 hybridization in graphene
g to
the sp3 hybridization in grapha
graphane
ne results in a change of the bond lengths and angles (Figure
IV.2.1).

Figure IV.2.1. Transformation of the sp2 hybridization of carbon to the sp3.

26 Behnaz Gharekhanlou and Sina Khorasani

A typical bond length for sp2 C-C bondand the standard bond angle for graphene and
graphite is 1.42Å and 120°, respectively. For sp3 hybridization, the standard value of C-C
bond length is 1.54Å, and the corresponding angle is 109.5°. A typical value for the single
C-H bond length is about 1.14Å. But in graphane, depending on the number of the added
hydrogen, the length of C-C bond is between 1.42Å and 1.54Å. Also C-C-H and C-C-C
angles are intermediate between 90° and 109.5° and 120° and 109.5°, respectively. This
implies an intermediate character of the hybridization between sp2 and sp3 [50].

IV.3. Electronic Band Structure Calculated Using TB

As mentioned before, graphane is a honeycomb structure of carbon atoms which their pz

orbitals are saturated with hydrogen atoms. Here we consider chairlike structure. The top
view of this structure is a hexagonal shape, like in graphene. Thus, its unit cell has a rhombus
cross section and there are four atoms in each unit cell: two carbon and two hydrogen atoms
as shown in Figure IV.3.1.
Each carbon atom participates in the bonding in sp3 hybridization using 2s, 2px, 2py and
2pz orbitals. Each hydrogen atom also participates through its 1s orbital. Hence, the
Hamiltonian matrix related to (25) and (26) is sized originally as 10 × 10.
Therefore, Hamiltonian matrix has a general form given as

2sA 2 px A 2 pyA 2 pzA 1sA 2sB 2 px B 2 pyB 2 pzB 1sB

 H 11 
2sA  H 1,10 
 
 
2 px A  
 
2 pyA  
 
2 pzA   
 
 
1sA   
[H ] =  
2sB   

 
2 px B   
 
2 py B   
 
2 pz B  

 

1sB  H 10,1 H 10,10 
 
10×10
 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 27

 
 
 
 H AA H AB 
 
 
=   .
  (68)
 
 H BA H BB 
 

 

A and B terms belong to each carbon and hydrogen connected to it in the unit cell as
labeled in Figure IV.3.1.

Figure IV.3.1. Unit cell of chairlike graphane.

We define tabular form of H AA as

h h12 h13 h14 h15 
 11 
h h22 h23 h24 h25 
 21 
H AA = h31 h32 h33 h34 h35  . (69)
 
h h42 h43 h44 h45 
 41 
h51 h52 h53 h54 h55 
This matrix is referred to on
on-site energies as defined in (25). Writing out H AA in its
components gives
h ji = ϕm (r − t j ) H ϕl (r − ti − R ) = δlm δij δ(R)εl . (70)
Now, one can find the matrix element
elements as
28 Behnaz Gharekhanlou and Sina Khorasani

h11 = ϕ2s ( r − t j ) H ϕ2s (r − t j ) = ε2s ,

hii = ϕ2 p ( r − t j ) H ϕ2 p ( r − t j ) = ε2 p ,
n n

i = 2, 3, 4 i = 2, n = x (71)
i = 3, n = y
i = 4, n = z
h55 = ϕ1s ( r − t j ) H ϕ1s ( r − t j ) = ε1s .
The remaining terms hij with i ≠ j are zero. In tabular form H AA is given by
ε 0 0 0 0 
 2s
 0 
 ε2 p 0 0 0 

H AA =  0 0 ε2 p 0 0  . (72)
 
 0 0 0 ε2 p 0 
 0 
 0 0 0 ε1s 
Suggested model parameters values fitted to the electronic band structure along the high-
symmetry lines are ε2s = − 5.2 eV , ε2 p = 2.3eV , and ε1s = − 4eV .
Similarly, H AB has the tabular form as
h h12 h13 h14 h15 
 11
h 
 21 h22 h23 h24 h25 

H AB = h31 h32 h33 h34 h35  . (73)
 
h h42 h43 h44 h45 
 41 
h51 h52 h53 h54 h55 
The components of the H AB have the general form of
h ji = ϕm ( r − t j ) H ϕl ( r − ti − R ) = δ ( t j − ti − R − d )Vlm ,ij . (74)
Here, d denotes the nearest neighbor atoms position.Thus,what remains is to determine the
rest of four nearest neighbors vectors R Ai of each carbon atom shown in Figure IV.3.2.
We need to project these vectors to Cartesian coordinate. Figure IV.3.3 shows the
tetrahedral structure constructed by the center carbon atom, denoted by the letter A, and its
nearest neighbors.
The bonding length of two carbon atoms aC−C is about 1.52Å and the bonding length of

carbon and hydrogen atoms aC−H is about 1.1Å. Also C-C-H angle θ is about 109.5°. Figure
IV.3.4 shows the triangles constructed on A, B and C points and A, C and D points. Here β is
the complementary of the angle θ, given by β = π − θ . Using the triangular relations, one
can get the distances h and w as well.
 An Overview of Tight
Tight-binding Method for Two-dimensional
dimensional Carbon Structures 29

Figure IV.3.2. Four nearest neighbors of the carbon atom:: three carbon and one hydrogen atoms.

Figure IV.3.3. The tetrahedral structure constructed by atoms in graphane.

Figure IV.3.4. The triangles structure constructed to calculate h and w.

30 Behnaz Gharekhanlou and Sina Khorasani

Now, suppose that the central carbon atom C0 is located at the origin of the Cartesian
coordinate. Using the recent calculated parameters, one ca
cann get the location of the nearest
neighbors of the carbon atom A as
C 0 : (0, 0, 0 ),
w 
C 1 :  , 0, −h ,
 3 
 w w 
C 2 : − , , −h , (75)
 2 3 2 
 w w 
C 3 : − , − , −h ,
 2 3 2 
H : (0, 0, aC −H ).

Figure IV.3.5 shows the projected location of these atoms on xy-plane.

Figure IV.3.5. Top

op view of the atoms specifiedwith (75).

Now, one can write R Ai vectors needed to calculate Hamiltonian parameters H AB as

w 
R A1 :  , 0, −h ,
 3 
 w w 
R A2 : − , , −h ,
 2 3 2  (76)
 w w 
R A3 : − , − , −h ,
 2 3 2 
R A4 : (0, 0, aC −H ).
For each of the R Ai vectors, one can define the corresponding angles made with respect to
the x, y and z axes as θx , θy and θz . Considering these vectors, the components of H AB can
be written out as
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 31

3
ik ⋅R Ai
h11 = ϕA2s H ϕB 2s =Vss σ ∑ e , (77)
i =1
3
ik ⋅R Ai
h12 = ϕA2s H ϕB 2 p = Vsp σ ∑ cos θx e , (78)
x
i =1
3
ik⋅R Ai
h13 = ϕA2s H ϕB 2 p =Vspσ ∑ cos θy e , (79)
y
i =1
3
ik ⋅R Ai
h14 = ϕA2s H ϕB 2 pz =Vspσ ∑ cos θz e , (80)
i =1
ik⋅R A 4
h15 = ϕA2s H ϕH 1s = V1s 1se , (81)
3

x x

h22 = ϕA2 p H ϕB 2 p = ∑ Vpp σ cos2 θx + Vpp π 1 − cos2 θx

i =1
( ) eik⋅R Ai
, (82)

3
∑ (Vppσ −Vpp π ) cos θx cos θy
ik⋅R Ai
h23 = ϕA2 p H ϕB 2 p = e , (83)
x y
i =1
3

x x
(
h24 = ϕA2 p H ϕB 2 p = ∑ Vpp σ −Vpp π cos θx cos θz e ) ik ⋅R Ai
, (84)
i =1
3
∑ Vppσ cos2 θy +Vpp π (1 − cos2 θy ) e
  ik⋅R Ai
h33 = ϕA2 p H ϕB 2 p = , (85)
y y
i =1
3
∑ (Vppσ −Vpp π ) cos θy cos θze
ik⋅R Ai
h34 = ϕA2 p H ϕB 2 p = , (86)
y z
i =1
ik ⋅R A 4
h35 = ϕA2 py H ϕ1s =−V1s 2 p cos θy e , (87)
3
∑ Vppσ cos2 θz +Vpp π (1 − cos2 θz ) e
  ik⋅R Ai
h44 = ϕA2 p H ϕB 2 p = , (88)
z z
i =1
ik⋅ R A 4
h45 = ϕA2 pz H ϕ1s =−V1s 2 p cos θz e , (89)
ik ⋅R A 4
h55 = ϕH 1s H ϕH 1s = V1sse . (90)
Other parameters can be written based on the facts that s H p =− p H s ,
p1 H p2 = p2 H p1 and s1 H s2 = s2 H s1 . We choose the model parameter values,
fitted tothe electronic band structure along the high-symmetry lines, to be Vsp σ = 3.8eV ,
Vss σ = −4.4eV , Vpp σ = 5.6eV , Vpp π = − 1.8eV , V1s 1s = 0eV , V1s 2 p = 4eV , and
V1ss = −3.5eV .As mentioned before, the matrix element H AA is exactly the same as H BB
and H AB is simply the complex conjugate of H BA .
32 Behnaz Gharekhanlou and Sina Khorasani

Figure IV.3.6 shows the resultant electronic band structure of graphane obtained from
direct Tight-binding calculations. The irreducible Brillouin zone is again the half of an
equilateral triangle with apexes ℘, Κ and Μ, exactly like in graphene, as shown in Figure
III.3.2.

10

0
E(eV)

-5

-10

-15

K Γ M K

Figure IV.3.6. The top view of the atoms specified

There is an excellent agreement compared to the reported result in [39], which are
obtained from density-functional theory calculations, as shown in Figure IV.3.7. This
establishes the usefulness and high accuracy of the Tight-binding method as applied to the
planar graphene and graphane structures. We here notice that the former structure is a semi-
metal with a zero-gap at its Dirac points, while the latter is a semiconductor with a direct
energy gap.
Chemical functionalization is a viable route toward bandgap engineering of graphene-
based materials, as first demonstrated in [40], where exposure to a stream of hydrogen atoms
has led to 100% hydrogenation of a graphene sheet,the so-called graphane.
The band gap of periodically doped graphene with hydrogen is investigated. It is found
through a Tight-binding model that for certain periodicities, called nongap periodicities
(NGPs), no gap is opened at the Dirac point. Density-functional-theory calculations show that
a tiny gap is opened for NGPs due to exchange effects, not taken into account in the TB
model. However, this tiny gap is one order of magnitude smaller than the gap opened for
other periodicities different from NGPs. This remarkable reduction in the band-gap opening
for NGPs provokes a crossing of the midgap and the conduction bands and, consequently, the
metallization of the system. This result is also valid for other adsorbates different from atomic
hydrogen [51].
Although graphene is a gapless semimetal due to the symmetry between two sublattices
of the honeycomb structure, graphane is a semiconductor with a wide bandgap around 3.5eV
 An Overview of Tight-binding Method for Two-dimensional Carbon Structures 33

[39]. This gives good idea for opening gap in Dirac points of graphene. Furthermore, it is
possible to control the gap in graphane by removing some of its hydrogen atoms periodically
[52, 53]. The role of hydrogen vacancies is to decrease the energy gap, and at the same time,
to increase the thermoelectric figure of merit [54], as well as significant lattice contraction. It
has been also noticed thatsuch hydrogen vacancy clusters in graphane may appear as quantum
dots and induction of stable ferromagnetism [55]. Furthermore, it has been shown that single-
sided hydrogenation also causes reduction of energy gap to about 1.1eV [52].

Figure IV.3.7. Band structure and density of states of graphane [40].

Obviously, research on functionalized graphane devices is at an embryonic stage. Several

issues must be addressed to introduce graphane in future generations of electron devices.
From this perspective, theoretical simulations have been done to explore possible solutions
for device fabrication and design, in order to provide an early assessment of the opportunities
of functionalized graphane in nanoscale electronics.
Together with the 2-D geometry and unique transport features of graphene, the possibility
of doping graphane makes it possible to define p and n regions so that 2-D p-n junctions
become feasible with small reverse currents. This junction is designed based on patterned
graphane and its current-voltage characteristics is also predicted [52]. Experimental
characteristics of such a junction is observed in [56]. Recent analysis also shows possibilities
of designing bipolar junction transistors based on patterned graphane [57]. For this reason,
donor and acceptor impurities should be introduced to the graphane with controlled
concentrations, the effects of which have been also studied [58].
34 Behnaz Gharekhanlou and Sina Khorasani

A performance analysis of field-effect transistors (FETs) based on recently fabricated

100% hydrogenated graphene (graphane) [55] and theoretically predicted semihydrogenated
graphene (graphone) [59] has been also reported. The approach is based on accurate
calculations of the energy bands by means of GW approximation, subsequently fitted with a
three-nearest neighbor sp3 tight-binding Hamiltonian, and finally used to compute ballistic
transport in transistors based on functionalized graphene. Due to the large energy gap, the
proposed devices have many of the advantages provided by one-dimensional graphene
nanoribbon FETs, such as large Ion and Ion/Ioff ratios, reduced band-to-band tunneling, without
the corresponding disadvantages in terms of prohibitive lithography and patterning
requirements for circuit integration [60]. Recent advaces in the fabrication of graphene
integrated circuits paves the way toward the expected birth of two-dimensional nanoscale
planar electronics based on the graphene and graphane [61].

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