Electrical properties of Carbon Nanotubes
Kasper Grove-Rasmussen
Thomas Jørgensen
August 28, 2000
1
Contents
1 Preface
3
2 Introduction to Carbon Nanotubes
4
3 Single wall Carbon Nanotubes
5
4 Reciprocal Lattice
4.1 The Brillouin zone of the graphene lattice . . .
4.2 The 2D Brillouin zone of the nanotube unit cell
4.3 Boundary condition . . . . . . . . . . . . . . . .
4.4 1D Brillouin zone of the nanotube . . . . . . . .
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5 Graphene sheet dispersion relation
15
6 Dispersion relation of the 1D nanotube
16
7 Density of state (DOS)
19
8 Comparison of some nanotubes
20
9 Energygap
22
10 Curvature effect
25
11 Magnetic field along the tubule axes
28
12 Conclusion
30
13 Perspectives of Carbon Nanotubes
31
A The program
32
B Source code
32
C Graphs and data
Appendix
33
2
1
Preface
This project describes the outcome of the Bachelorproject, we have been occupied with in
the autumn semester of 1998, namely ”The electrical properties of Carbon Nanotubes”.
It has been of great interest working with this issue and we hope the result can inspire
someone else to look further into it. We will!
The project has come into existence in cooperation with the Department of Solid State
Physics of the University of Copenhagen, in particular our advisors Jesper Nygaard, David
Cobden and Poul Erik Lindelof.
We will in this project try to describe some of the knowledge about the electrical properties
of carbon nanotubes and investigate several different tubes of a broad scale of range to
point out the details. Among other things, you will see that some tubes are metallic and
some are semiconductors. Further more it is our intention to write this project in such
a way that it is easily comprehensible to the reader. We will try to explain some of the
methods used in the published articles more carefully, because often it is not obvious or
not explained what exactly happens. The project is purely theoretical and all results are
based on a computer program.
Good luck with the reading!
3
2
Introduction to Carbon Nanotubes
As we are moving rapidly towards the twenty first century, the development within the
world of science and technology is moving even faster. Just think of the everyday situation,
where you go to a store to buy yourself a computer, and as you leave the store, noticing the
door closing behind you, it occurs to you that the value of your loving computer already
has decreased.
This is caused by the tremendous research in the microelectronics, which has changed a lot
of things the last twenty years or so. It has opened up a great range of new possibilities,
as the size of the electrical devices has diminished. About every second year, the amount
of transistors placed on a micro chip doubles, but it is limited and soon a new technique
has to be developed to carry on building faster computers. The silicon is the very heart of
the microelectronics and will probably still be of great importance as we pass the change
of millennium. However, experiments indicate a growfield for a new technique based on
molecules. This is due to the scientists, who try to bring us from the micro scale to the
nano scale, though it is another matter in the periodic table that is used, Carbon. This
leads us to the main issue of this project.
Carbon nanotubes describes a specific topic within solid state physics, but is also of interest in other sciences like chemistry or biology, actually the topic has floating boundaries,
because we are on the molecule level. The carbon nanotubes have in the recent years
become more and more popular to the scientists. Initially, it was the spectacularly electronic properties, that was the basis for the great interest, but eventually other remarkable
properties were discovered too.
The carbon nanotubes are long molecular wires that are able to conduct electrical current.
They are constructed by rolling up a specified rectangular piece of a graphene sheet, with
diameters from about 1 to 20 nanometres. They were discovered by the work of the balllike fullerenes, which are much alike in structure. It turned out that adding a few percent
of other atoms, nickel and cobalt, in the creating process of fullerenes, it was possible to
let the carbon chain grow, and thereby making a cylindric tube. Because of their small
diameter and a typical length of a micro metre they are classified as 1D carbon systems,
which electrical properties are to be investigated more detailed by experiments.
The initial research was stimulated through the use of the transmission electron microscopy
(TEM) in the early experiments, which among other things confirmed the existence of the
nanotube.
4
3
Single wall Carbon Nanotubes
Using the transmission electron microscopy (TEM) or scanning tunnelling microscope
(STM) it is possible to see that the carbon nanotubes are cylinders of graphene sheet
with different kinds of symmetries in structure. These structures are shown in figure 1.
All the nanotubes are labelled with
indices (n,m), - a simple way to tell
the name and the size of a specific
tube. Other observations from pictures of the TEM are that the carbon
nanotubes occur with more than one
”wall”. These are composed of several concentric cylinders within each
other. In this article we just present
the properties of the single wall nanotube. If we return to the indices,
we have already mentioned that these
describe the size of a tube. To see
this we have to investigate how we
roll up the sheet to a cylinder (single
wall nanotube). The structure of the
graphene sheet is carbon atoms bound Figure 1: Three kinds of nanotubes. (a) An armchair nantogether in a honeycomb lattice, see otube. The pattern in the vertical direction has the shape
of an armchair. (b) A zigzag nanotube. The pattern in the
figure 2.
vertical direction is a zigzag line. (c) A ”chiralnanotube”
The first thing we have to do is to (n,m), (from [2] p. 758).
choose our primitive lattice vectors a~1
and a~2 of the graphene sheet. These vectors define a parallellogram, which is called the
primitive unit cell, and fill all space by the repetition of suitable crystal translations operations (the graphene sheet). The unit cell also defines the minimum area of the parallellogram. So we can always make sure that we have chosen the right ones. There are
many ways of choosing the lattice vectors, but some are more comfortable to work with
than others. In figure 2 we have chosen a~1 and a~2 to be two upright secants of a isosceles
triangle within two hexagons. In Cartesian coordinates the vectors are
√ !
√ !
Ã
Ã
a 3a
a 3a
a~1 =
,
, a~2 = − ,
(1)
2 2
2 2
Another choice could be with an angle of 120 degrees instead of our 60 degrees.
The two lattice vectors are now used to determine the roll up vector ( also called the Chiral
vector) labeled
~ h = n~a1 + m~a2
C
(2)
which determines the circumference of the carbon nanotube. Here n and m are the indices
of the tube and for that reason it is obvious that (n.m) directly describes the size of the
5
tube.
In figure 2 we can see that the arbitrary Chiral vector (n,m) for n ≥ m ≥ 0 lies between
θ = 0 degrees (zigzag axis) and theta = 30 degrees (armchair axis). If we choose a
vector beyond this area, the symmetry will give us an equivalent vector within the area by
reflection.
Also shown is the Translation vector which is perpendicular to C~h and given by 1 . This
Figure 2: Show the chiral and the translation vectors in the case of a (4,2) nanotube.
translation vector describes the distance between two similar lattice points. Now, the two
~ h and T~ span a rectangle, which is called the 2D nanotube unit cell. This is the
vectors C
rectangle, we roll up in the Chiral direction, that forms the cylinder. The side OB is put
together with the side AB’. This defines a unit cell, that repeatedly put together forms
the tube. Thereby making a periodicy equal the length of the translation vector in the
tubuleaxis.
The nanotubes just created has no distortion of bond angles other than in the circumference
direction, caused by the cylindric curvature of the surface. As we will understand later,
~
this curvature, the chiral angle, the diameter dt = |Cπh | and an applied magnetic field along
the tubule axes have influence of the electrical properties of the tube.
1
[2], equation (19.6)
6
4
Reciprocal Lattice
In this chapter we are to find the lattice vectors in the reciprocal space given the real space
lattice vectors. All calculations are done in Cartesian coordinates. The 1. Brillouin zone
is found by Wigner-Zeits primitive unit cell2 . In the following this zone is referred to as
the Brillouin zone. The derivation of the reciprocal lattice vectors are first given generally.
This approach is chosen because more than one set of reciprocal lattice vectors are to be
calculated and the choice of real space lattice vectors is ambiguous. The general equation
can be applied to the different kinds of real space lattice vectors connected with the carbon
nanotube. The reciprocal lattice vectors are related to the real space lattice vectors by the
relation 3
a~i • b~j = 2πδij
(3)
Figure 3 shows how the unit
vectors of the 2 dimensional Cartesian coordinate system in real space
transform to the unit vectors in
the reciprocal space according to
equation (3). The x̂ unit vector
is perpendicular to k̂y and vice
versa. The unit length of the
reciprocal vectors are 2π. (not Figure 3: Real space with unit vectors (x̂, ŷ) and reciprocal space
shown at figure 3). If the two with unit vectors (k̂x , k̂y ) of length 2π.
real space vectors are not perpendicular it is not possible to do the transformation to
reciprocal space that simple.
In a three dimensional lattice with realspace vectors a~1 , a~2 and a~3 the reciprocal lattice
vectors are found by 4
a~2 × a~3 ~
a~3 × a~1 ~
a~1 × a~2
b~1 = 2π
, b2 = 2π
, b3 = 2π
(4)
Vreal
Vreal
Vreal
where Vreal = a~1 • a~2 × a~3 is the volume of the box spanned by a~1 , a~2 and a~3 .
These vectors obey equation (3). The length of a reciprocal lattice vector is given by
2π
b~i =
|~
ai |
(5)
where i is 1, 2 or 3.5 In our case we are only in two dimensions. Thus setting
2
[1] p. 8
[1] p. 33
4
[1] p. 33
5
This equation is only correct in the special case where it is later used. a~3 = (0, 0, 1) and a~1 ⊥ a~2 in
the plane orthogonal to a~3 .
3
7






a1x
a2x
0






a~1 =  a1y  , a~2 =  a2y  , a~3 =  0 
0
0
1
the reciprocal lattice vectors of the plane are b~1 and b~2 .
The volume spanned by the real space vectors are

Vreal









a1x
a2x
0
a1x
a2y

 
 


 
=  a1y  •  a2y  ×  0  =  a1y  ◦  −a2x 

0
0
1
0
0
= a1x a2y − a1y a2x
Using equation (4) one gets the reciprocal vectors


a2y
2π


−a
b~1 =

2x 
a1x a2y − a1y a2x
0

b~2 =
2π
a1x a2y − a1y a2x
(6)

−a1y


 a1x 
0
(7)
The vector b~3 is not calculated, because it is irrelevant to our purpose. We are only
working in two dimensions. Now for any choice of real space lattice vectors the reciprocal
lattice vectors can be calculated easily by equation (6) and (7). The Brillouin zone is
obtained by finding Wigner-Zeits primitive cell 6 .
The above equations of the reciprocal lattice vectors are used to find the Brillouin zone of
a general nanotube. In this derivation actually three Brillouin zones are in play. These
three are
• The 2D Brillouin zone of the graphene sheet
• The 2D Brillouin zone of the nanotube unit cell
• The 1D Brillouin zone of the nanotube
It is very important to distinguish these three zones. The first is found by the real space
lattice vectors of the graphene sheet (a~1 , a~2 ), equation (6) and (7). The same procedure is
used in the second case, but now with the real space lattice vectors defining the nanotube
(C~h , T~ ). Both cases give a two dimensional Brillouin zone.
The third Brillouin zone is a result of the boundary condition, which arises when the
graphene is made to a tube. By using the zonefolding technique the dimension of the 2D
Brillouin zone of the nanotube unit cell reduces to one dimension. This is the 1D Brillouin
zone of the nanotube.
The exact derivation are given in the following subsection (4.1- 4.4)
6
[1] p. 8
8
4.1
The Brillouin zone of the graphene lattice
The real space lattice vectors of the graphene sheet are given by 1 Thus using equation (6)
and (7) the reciprocal lattice vectors become
2π
b~1 =
a
Ã
1
!
√1
3
2π
, b~2 =
a
Ã
−1
!
√1
3
where a is the lattice constant.
In figure 4 the real lattice vectors and reciprocal lattice vector are shown. They define
a 2D Brillouin zone with the shape of a hexagon. Clearly a~1 and a~2 are perpendicular to
respectively b~2 and b~1 . The hexagon is shown in k-space, but each pair of (kx , ky ) corresponds to an energy value. The exact dependence is given in section 5.
As mentioned above the choice of real space lattice vector is ambiguous. Furthermore
the coordinates of these vectors depend on the choice of coordinate system. Our choice
is different from the choice of [2], but similar to the choice of [4]. The program we have
made is based on the definitions given above. It might be confusing when comparing
with other literature, but the results we obtain are similar to what would be found with
other choices of lattice vectors and coordinate system. In [2] the two real space lattice vectors are switched and the x-direction of the coordinate system is along the zigzag direction.
The area of the Brillouin zone consists of 6 triangles (one is sketch in the
figure) each with an area of
1 1 ¯¯ ~ ¯¯
2π
4π 2
1
√ )= √
( ¯b2 ¯)(
2 2
cos 30 a 3
3 3a2
Hence the area of the Brillouin zone is
8π 2
ABz = √ 2
3a
(8)
This area is used later to calculate
the number of band needed to get the
dispersion relation of a carbon nanotube and to normalize the density of
4.2
Figure 4: The real space (left) and reciprocal (right) lattice vectors. The hexagon to the right is the 1. Brillouin of
the graphene sheet.
state.
The 2D Brillouin zone of the nanotube unit cell
The two vectors defining the nanotube are the chiral vector C~h (2) and the translation
vector T~ . According to equation (3) the chiral vector C~h has to be perpendicular to
the second reciprocal lattice vector. This gives a reciprocal lattice vector parallel to the
translation vector T~ , denoted G~T . Similar the other reciprocal lattice vector denoted G~C
and |C2π
is parallel to C~h . The length of the reciprocal vectors are respectively |2π
~ | (5). In
T~ |
h
9
Figure 5: The real space vectors defining the nanotube (left) and the reciprocal vectors defining the 2D
Brillouin of the nanotube unit cell (right). The Brillouin zone is the rectangle.
figure 5 the four vectors are shown. Here in the case of an unspecified chiral tube. The
reciprocal lattice vectors of the nanotube unit cell in Cartesian coordinates are given by
equation (6) and (7). The exact expression is not given, because a more convenient way of
determining the direction of GT and GC appear in the next section.
The Brillouin zone depicted in figure 5 (the rectangle) has the area spanned by the two
reciprocal lattice vectors. The larger the area of the nanotube unit cell gets the smaller
the area of the Brillouin becomes, because the reciprocal vectors are proportional to the
reciprocal length of the unit vectors (5). In the chiral case even for small values of n and
m the area of the real space nanotube unit cell becomes large and thereby reducing the
2D Brillouin of the nanotube unit cell.
4.3
Boundary condition
When the nanotube 2D unit cell is folded to a cylinder only a discrete set of wavevectors
along the reciprocal vector G~C in the reciprocal space are allowed. This results in a number
of quantization lines in the reciprocal space which represent the allowed pairs of (kx , ky ).
The condition on kx and ky is dependent on the choice of real space lattice vectors and the
choice of coordinate system. Hence we first derive it generally. The expression derived can
then be used with another choice than ours.
The periodic boundary condition is a result of the required periodicity along the circumference of the Block wavefunction express as
ψ(x + Ch ) = ψ(x)
where ψ is the blockwavefunction of the graphene sheet and x is along the circumference.
This gives rise to the following equation
C~h • ~k = 2πq
(9)
where Ch = na~1 +ma~2 is the chiral vector defining the nanotube, ~k the wavevector and q an
integer. Restrictions are later made on q. We write the relation in Cartesian coordinates
10
to express the dependence between kx and ky .
" Ã
n
a1x
a1y
!
Ã
+m
a2x
a2y
!#
Ã
◦
kx
ky
!
= 2πq
(na1x + ma2x )kx + (na1y + ma2y )ky = 2πq
(10)
The last equation shows a linear dependence between kx and ky and gives the pairs of
(kx , ky ), which are allowed in the reciprocal space. When the tube is made only a subset
of wavevectors in k-space are allowed. Because of the linear dependence we speak about
(quantization) lines.
Using the defined real space lattice vectors a~1 and a~2 (1) the relation becomes
√
√
3a
3a
a
a
(n
+m
)ky = 2πq − (n + m(− ))kx
2
2
2
2
n−m
4πq
(11)
(n + m)ky = √ − √ kx
3a
3
ky = √
4πq
n−m
−√
kx
3a(n + m)
3(n + m)
(12)
Equation (11) expresses the boundary condition and is true for all n and m. Two particular
simple cases appear for n = m (armchair nanotube) and n = −m (zig-zag nanotube). In
4πq
the armchair case the quantization condition is ky = √3a(n+n)
= √2πq
which corresponds
3an
to horizontal lines, while the lines are vertical kx = 2πq
in the zigzag case. These two cases
an
are often used as illustrative examples in articles. It is worth a remark that the (n,n) and
(n,-n) nanotubes are not in the same symmetry area i.e. between the 30 degrees. It is more
obvious to look at the (n,0) zigzag nanotubes, which are equivalent to the (n,-n) tubes,
together with the (n,n) armchair nanotubes. They are defining a symmetry area, where
all nanotube can be constructed.
In the following we will use equation (12) and thus not looking at the case, where n = −m.
Three different nanotubes are shown at figure 6. The dashed lines are the allowed pairs
of wavevectors according to the periodic boundary condition and the rectangle is the 2D
Brillouin zone of the nanotube unit cell. Only the lines near the Brillouin zone of the
graphene sheet are depicted. The spacing between the lines is |C2π
~h | , because the allowed
wavevectors ~k are the projection on C~h of length 2π times an integer (3). This is exactly
~ C . The edge of the Brillouin zone of the nanotube unit cell
the reciprocal lattice vector G
parallel to the quantization lines is half the distance of the vector GC .
The vector G~T is along the lines perpendicular to GC and defines the edge of the Brillouin
zone of the nanotube unit cell in that direction. The angle between the lines and the
kx -axis is determined by equation (12).
In the (9,0) zigzag case the slope of the lines is − √13 , which is a line with an angle to the
√ , which is a constant.
kx -axes of -30 degrees (figure 5(a)). The (5,5) armchair has ky = 52πq
3a
Thus the horizontal lines in 5(b) If the tubes are chosen with n ≥ m the angle of the chiral
11
2DBrilzone.nb
2D
ky
ky
kx
kx
[(9,0) zig-zag]
[(5,5) armchair]
Figure 6: The 2D Brillouin zone of the carbon nanotube unit cell (blue rectangular) compared with the
Brillouin zone of the graphene sheet. The dashed lines represent the allowed values of kx and ky . Only
one line crosses the Brillouinin zone of the tubes unit cell. In the chiral case (c) the Brillouin zone gets
very small in agreement with the big nanotube unit cell.
lines is between the two cases above.
The 2D Brillouin zone of the (6,5) nanotube is much smaller than the two other cases,
because the area of the real space unit cell is larger.
4.4
1D Brillouin zone of the nanotube
The 1 dimensional Brillouin zone of the carbon nanotube is achieved by a technique called
zonefolding. So far we know the 2D Brillouin zone of the nanotube unit cell and the allowed
pairs (kx , ky ) determined by the periodic boundary condition.
We have to bear in mind that each (kx , ky ) corresponds to a specific energy value defined by
the dispersion relation of the graphene sheet (see section 5). In the following it will be more
convenient to speak about pairs of (kT , kC ) defined by the coordinate system spanned by
GT and GC . It is actually just a rotation about origo of kx -ky -coordinate system depending
on the choice of tube. This change of coordinate system is done to have the axes of the
coordinate system along the sides of the rectangular 2D Brillouin zone of the nanotube
unit cell.
In the kC -kT coordinate system the kC ’s constitute a discrete set of values, while the kT is
a continuous set of values (figure 7 (left)). The figure is actually an extended zone scheme
in two dimensions (k-space) without the energy values shown. The states are not only
restricted to the 1. Brillouin zone. Instead we want to depicture the dispersion relation in
the 2D Brillouin zone of the nanotube unit cell. This representation is called the reduced
zone scheme. All the kC and kT -vectors are translated by an integer times the reciprocal
lattice vector GC and GT into the 2D Brillouin zone of the nanotube unit cell.
The best approach is to translate the discrete allowed kC values into the Brillouin
zone. This is a one dimensional problem. The kC -axis is shown in figure 7 (right). The
dots indicated the allowed values of kC , determined by the intersection of the quantization
12
[(6,5) ch
Figure 7: (left) The quantization lines showed in the kT -kC coordinate system. They are always parallel
to the G~T vector and separated by G~C . (right) The allowed vectors along the kC -axis, indicated by dots
are separated by the length of G~C . Only one dot (wavevector) is allowed inside the Brillouin zone. The
left figure is just the second axis of the coordinate system shown to the left.
lines and the kC axis. As mentioned above the dots (lines) are separated by the length
of G~C . Also shown are the reciprocal lattice vector G~C and the Brillouin zone (now one
dimensional). Only one wavevector (kC = 0) is allowed inside the Brillouin zone. The
distance between all the other allowed wavevectors and the kC = 0 is a multiplum of G~C .
Thus every wavevector outside the Brillouin zone is translated to the kC = 0. This reduces
the two dimensional Brillouin zone of the nanotube unit cell to a 1 dimensional Brillouin
zone along G~T , because only one value is possible in the kC -direction.
The number of kC that has to be translated (N) to get all the bands of the dispersion is
determined by the fact, that the total length of the 1D Brillouin zone of the nanotube
times the spacing (G~T ) equals the area of the Brillouin zone of the graphene sheet7 . That
means for each translated kC the contribution to the total length of the one dimensional
Brillouin zone is the length of GT . One have to translate enough kC to equal the area of
the Brillouin zone of the graphene sheet 8. N also equals the number of hexagons in the
real space nanotube unit cell.
In the above approach we implicitly suppose that no new energy band are given by
zonefolding in the kT - direction. This actually seem to be the case. It is not clear how
many kC that has to be zonefolded if every kT is zonefolded first.
In the case of the armchair and zigzag case all bands could be obtained by only zonefolding
N
kC values, but then zonefolding the segment of length GT besides the 1D 1. Brillouin
2
zone of the nanotube.
In short the zonefolding can be expressed as the set of N segments of length GT each
separated by the vector G~C are zonefolded (translated to Brillouin zone) into a 1D Brillouin
zone of nanotube.
The dispersion relation of the nanotube has to be plot in the above found 1 dimensional
7
[?]
13
h
i
Brillouin zone i.e. kT ² −π
; π . Only a finite number of states exist equal to the number of
T T
hexagon in the real space unit cell of nanotube.
14
5
Graphene sheet dispersion relation
Before we are able to find the dispersion relation of the nanotube, we need the dispersion
relation of the graphene sheet. This relation is found by the tight binding approximation.
We are not going to derive it, but instead make it our starting point. The dispersion
relation is given by 8
kx a √
√
²(~k) = γ0 (2 cos(
(13)
)e 2 3 + γe 3 )
2
To obtain the 1D dispersion relations
that describes the properties of the carbon
nanotubes, we first have to consider the 2D
dispersion relation that is given by the latter equation. The +/- absolute value of this
relation describes the energy bands in the
2D graphene sheet (see figure
√ 8), where the
lattice constant is a = 3d and d is the
separation of two carbon atoms, who are
nearest neighbours. The two parameters
γ = tγ⊥0 and γ0 = t are hopping strength elements. They describe the possibility for an
electron to tunnel from one carbon atom to
its neighbouring atom in the two directions Figure 8: The dispersion relation of the graphene
and vary with the curvature of the graphene sheet. The blue hexagonal is the Brillouin zone and
the red dots called K-point are zero-gap points.
sheet (see section 10).
The 2D dispersion relation makes two surfaces that have a shape like a tent raised
over/under the domain in the (kx , ky )-plan. However it is not defined on a rectangular
Brillouin zone but on a hexagonal one. The symmetry is very obvious! The upper ”tent”
is the image of the conduction band and the lower one is the image of the valence band.
These two bands touch in the corners of the hexagonal Brillouin zone, which are labelled
the K points. Here the energy E = 0, which corresponds to the Fermi energy level.
The K points also explain the surprising electronic properties that we shall discuss later,
because here, there are no energygap between the valence band and the conduction band.
Furthermore these two bands are degenerated in the K points because of the symmetry of
the 2D graphene sheet
iky a
8
−iky a
Physical Review B Volume 55 Number 18 t = γ0 , t⊥ = γγ0 . If using the dispersion relation from [2]
kx and ky are to be switched, because of different choice of basis and coordinate system.
15
6
Dispersion relation of the 1D nanotube
Now we are nearly ready to plot the dispersion relation of the nanotube along the direction
of the vector G~T in k-space i.e along the quantization lines of the 1D Brillouin zone of the
nanotube. The coordinates kx and ky of the dispersion relation (13) are no longer applicable
except in the case of the armchair (n,n) and the zigzag (n,-n). Here the lines are horizontal
and vertical respectively. Hence kx and ky are in the direction of the vector G~T and thereby
making them proper coordinates.
In the general case the coordinate system of kT and kC is rotated with respect to the (kx ,ky )-system. In figure 6(b) and 6(c)
the Brillouin zone of the nanotube unit cell (blue rectangle) does
not have any of its sides parallel to the kx -axis or the ky -axis.
Hence the vector G~T is not in the direction of the coordinate axes.
Figure 9: The (kx ,ky ) The task is to find the relation between the two coordinate sysand (kT ,kC ) coordi- tem, i.e. an expression of kx and ky as a function of kT and kG .
nate system.
They are related by a rotation matrix in the following way.
"
kx
ky
#
"
=
cos θ −sinθ
sin θ
cosθ
#"
kT
kC
#
where θ is the rotation angle counterclockwise. The angle of rotation is the absolute value
n−m
of angle related to the slope of equation (12). Thus θ = |tan−1 (− √3(n+m)
)|.
Evaluating the above equation gives
kx = kT cos θ − kC sin θ
(14)
ky = kT sin θ + kC cos θ
(15)
By substituting the expressions of kx and ky into the dispersion relation (13) the energy
becomes a function of the wavevectors kT and kC . The boundary condition gives the
~ C |. Thus the allowed values of kC can be expressed
quantization lines with the spacing |G
2π
as kC = |C~ | q, where q is taken the values from 1 to N. The energy is now found for each
h
value of q (a band) in the first Brillouin zone of the graphene sheet. The edge of the
Brillouin zone is − Tπ ≤ kT a ≤ Tπ , where T is the length of the vector T~ .
Before finding the dispersion relation of a chiral tube, we look closer at the two simple cases
of the a zigzag and a armchair tube. Actually the above approach is necessary because we
are to look at the zigzag case (n,0), which are not along the axes of the (kx ,ky )-coordinate
system.
Figure 10(a) shows the extended zone in k-space of the (9,0) zigzag. In this case
the (kT ,kC )-coordinate system are rotated 30 degrees in respect to the (kx ,ky )-coordinate
system. Plotting the energy along for instance the kx -axis is not the same as plotting
the energy along kT (1D Brillouin zone). The vertical line represents a certain kx value.
The intersections of this line and the quantization lines give the energy of the different
bands at one kx . A line parallel to the long side of the rectangular represents a specific
16
3
2
1
E
0
-1
ky
-2
kx
-3
-1.5
[Extended zone]
[Dispersion Relation]
-1
-0.5
0
k
0.5
Figure 10: (9,0) zigzag nanotube. (a). The dashed lines are the quantization lines. Each line gives rise
to a energy band by slicing the tent dispersion relation of the graphene sheet. The bands are depicted in
the same color (b) as the corresponding quantization line (a). The green rectangle coincide the N = 18
segments that has to be shown in the dispersion relation. (b) The dispersion relation of a (9,0) nanotube
with the energy in units of γ0 . It is metallic because the light blue line intersect at k = 0. The plot shows
the Brillouin zone of the nanotube |ka| ≤ 12 GT = Tπ , which in this case is |ka| ≤ √π3 .
kt . It is clearly seen that the intersections are not the same as in the above case. The
bands achieved using the first method are translated in respect to the correct bands of the
dispersion relation.
The figure shows the extended zone scheme (a) with the number of quantization lines N
determined in a previous subsection. The width of the rectangle equals the length of GT .
The colored lines in the extended zone scheme correspond to a the same colored bands
of the dispersion relation (b). For instance the two light blue lines (degeneracy) in the
extended zone scheme give the same energy bands. This is the two light blue colored
bands of the dispersion relation crossing at k = 0. In the extended zone scheme the
quantization lines cross a K-point (corner of Brillouin zone) at kG = 0 in agreement with
the above. The dispersion relation just represents the slices the quantization lines make
with the three dimensional dispersion relation of the graphene sheet. A good intuitively
way to get a feeling of the dispersion relation is to draw the extended zone scheme and
imagine how the lines intersect the 3D tent of the graphene sheet.
Similar the case of the armchair is shown in figure 11 showing how the different quantization lines give the dispersion relation of the nanotube. The (5,5) armchair has four double
degenerated bands (4 middle bands), because two lines in the extended zone scheme slice
the dispersion relation of the graphene sheet in the same way. The top and the lowest band
are non degenerate. This is a general rule of the armchair. There are two non degenerate
.
bands and 2(n − 1) degenerate bands. The bands cross the Fermi level at k = 2π
3
17
1
1.5
3
2
1
E
0
-1
-2
ky
kx
-3
-3
[Extended zone]
[Dispersion Relation]
-2
-1
0
k
1
Figure 11: (5,5) nanotube. (a). The dashed lines are the quantization lines. Each line correspond to
the same colored band of the dispersion relation. (b) The dispersion relation plot between |kT | ≤ Tπ = π.
The bands pink bands cross the Fermi energy in agreement with the pink quantization line crossing the
K-points of the graphene sheet.
18
2
3
7
Density of state (DOS)
The density of state (DOS) can be calculated from the dispersion relation of the carbon
nanotube. It tells how many states at the energy between ² and ² + d².
The total contribution to the DOS at the energy E can be expressed as the following
double sum
¯
¯−1
¯ ∂² ¯
1 X XZ
¯
¯
n(E) =
dkδ(k − ki ) ¯ ¯
¯ ∂k ¯
Save.nb
` bands i
(16)
where ki are the roots of the equation E −
3
²(ki ) = 0 and ` is the total length of the Bril2.5
louin zone of the carbon nanotube i.e the num2
E 1.5
ber of bands (with degeneracy) times the length
1
of the vector G~T . We want to use equation (16)
0.5
to do numerical calculations of the DOS from the
1
1.5
0
0.5
2
2.5
3
dispersion relation. The interpretation is shown
k
at figure 12. For each intersection of a line of
0.25
constant energy E and a specific energyband the
0.2
reciprocal derivative is found. If there is more
0.15
DOS
than one intersection the reciprocal derivatives
0.1
are added together and the procedure is repeated
0.05
with the rest of the bands. The total DOS is
0
-1
1
-3
-2
0
2
3
found by evaluations for every energy value present
E
in the dispersion relation and normalizing it with
the total length of the 1D Brillouin (`) zone of the
nanotube. For instance the band shown at figure Figure 12: Show the contribution to the
DOS (below) from one of the band (above)
12 has two intersections at a energy E = 0.9. The of a (5,5) armchair.
reciprocal derivative at these points are added
and constitute the contribution to the DOS from that band (not normalized). The DOS
goes to infinity when the band has zero slope (minimum and maximum). The DOS has
a jump at E = 1, because only one contribution from E > 1 and two contributions for
E < 0. In the case of nanotube with zerogap between k = 0 and the edge of the Brillouin
the first calculated DOS makes a numerical error close to the Fermi level, which the program interpret as an energygap. This is corrected, by setting the energygap of the metallic
tubes to zero and smooth the DOS near the Fermi level. We are allowed to do so, because if
the energygap was found manually from the DOS and by looking at the dispersion relation
the apparent bandgap would be recognized as a numerical error, not a bandgap.
Another way of correcting the DOS is to start the calculations from the zerogap point
2π
) and find the contribution from both sides. This would not give a numerical error.
(k = 3T
For instance the zigzag tube has zero bandgap at k = 0, where the DOS calculations starts.
The calculations, we carried out, showed great correspondence to similar calculations
[2]. We also think that we have pointed out the details in the steps of calculating the
19
energybands and the DOS.
8
Comparison of some nanotubes
In this section we will look at some carbon nabotubes with almost the same diameter and
try to describe
the similarities and the differences. The length of the rollup vector is given
√
by |c~h | = n2 + m2 + nm. Therefore we have picked out a (5,5), (9,0), (7,4), (8,3), (8,2),
(5,6) and (8,0) which vary only slightly in diameter. All the pictures and numbers of these
nanotubes youSave2.nb
can find in the appendix. Well at least now we know that any differences
are not caused by the diameter, since it is almost the same.
Save2.nb
1
0.75
0.5
0.25
E
0
-0.25
-0.5
-0.75
-0.4
-0.2
0.8
0.6
DOS
0.4
0.2
ky
kx
0
-3
[Extended zone scheme of (7,4)]
[Dispersion relation of (7,4)]
Figure 13: (a) The extended zone scheme of (7,4) nanotube. (b) The dispersion relation and DOS
of the (7,4) nanotube. This tube is a metallic chiral tube. The energy is plot between |E| ≤ 1 and
|ka| ≤ Tπ = √π31 . (c) (8,2) chiral nanotube. The dispersion relation and the DOS shows that this tube has
a bandgap of
E
γ0
= 0.39. No extended zone is shown, because the (8,3) has 194 bands.
First of all the pictures show that the tubes (5,5), (9,0), (7,4) and (8,2) are metallic
while the (8,3), (5,6) and (8,0) are semiconducting. This is due to the chiral angle, which
is the only basic property that changes between these tubes. The chiral angle causes an
obliquity between the rollup vector and the honeycomb lattice and apparently the angle
influences the intersections between the allowed ~k and the ”tent” in the reciprocal space.
Sometimes the chirality causes that none of the allowed ~k s in the C~h direction passes
through the K points in the Brillouin zones and therefore only leaves a bandgap in the
reduced 1D Brillouin zone, as seen here with the three semiconductors. Otherwise for the
metallic the symmetry is more nice and the allowed ~k s of course intersect with the K
20
-2
-1
points. The numbers of bands in the seven carbon nanotubes are quite different. They go
from 10 to 194 bands and that is for the same diameter, - now how does this happen? If
we look at the facts of the nanotubes, we see that the length of the C~h is almost the same
but the lengths of T~ differ. The nanotubes with a lot of bands also have a long T~ . That
means they span rectangles of different sizes in the real space. The larger the rectangular
is the smaller it appears in the reciprocal space. Similar if T~ is long in real space it appears
proportional smaller in the reciprocal space, G~T . Now the G~T determines the length of
1D Brillouin zone and if this one gets smaller with increasing length of T~ then we have to
go much further out in the G~C direction, because the area of this long rectangular, which
contains all the allowed ~k s, must be the same as the area of the Brillouin zone. Therefore
we get a larger amount of intersections, which corresponds to the energybands. Therefore
some carbon nanotubes, independent of the diameter, have more bands than others. All
these arguments are based on the chiralty, which implicitly determines the length of the
translation vector T~ , and there by the area spanned by C~h and T~ . So in this chapter we
conclude that chiralty is very important for electronic properties and it does matter how
we roll up the carbon nanotube.
21
9
Energygap
In this section we will look futher into the diameter and the n dependence of the energygap.
The previous calculations show that the carbon nanotubes can be either metallic or semiconducting depending of the choice of (n,m). This is remarkable, since there are neither
differences in the bondings between the carbon atoms nor any impurities of donor atoms
present, which normally cause the above mentioned features. This fact arise interesting
perspectives for developing new electronic devices. The figure 14 shows the pseudoenergygap versus 1/diameter for the armchair nanotubes. We use the terminology ”pseudogap”
because the armchair nanotubes always are a metallic and therefore have no real energygap. The pseudogap describes the difference of the energy between the peak values closest
to the Fermi level in the density of state diagram. For a semiconducting nanotube this
difference is a real energygap.
[Pseudoenergap vs
[Pseudoenergygap vs n ]
Figure 14: (n,n) nanotube. The pseudoenergygap is in unit of γ0 , which is the difference of the energy
between the peak values closest to the Fermi level in DOS . (a) The pseudoenergygap versus n for a (n,n)
armchair nanotube. (b) The energygap versus the reciprocal diameter. The dependence is clearly linear.
The armchair diagram confirms that the pseudogap is proportional to the reciprocal
diameter, 1/diameter. That means when the diameter increases the pseudogap decreases
and the energys closest to the Fermi energy with the high density of states move closer
together. The equation of the linear dependence is given by
Epseudogap = 7.10Aγ0
1
dt
(17)
where A is Angstrom and γ0 = 2.5eV .9 and the constant of the 1. degree polynomial is
almost zero ( γE0 = 0.08). The pseudogap of the 20 armchair nanotubes plotted at figure 14
is in the range from γE0 = 0.33 for the (20,20) armchair to γE0 = 1.75 for the (3,3) armchair.
In eV the range is between 0.85 eV to 4.38 eV.
In figure 15(a) the energygap of the (n,0) zigzag tube in units of γ0 are shown as a
function of n. This result corresponds to the values that are depicted in figure 19.27 in
9
[1] p. 213
22
1
]
dt
[Energygap vs d1 ]
[Energygap vs n]
Figure 15: Energygap is in units of γ0 . The energygap of the (n,0) zigzag nanotube versus (a) n and (b)
the reciprocal diameter. The latter show linear dependence with slope 2.73
[2]. For instance the (40,0) and (20,0) zigzag tubes have a energygap of γE0 = 0.09 and
respectively γE0 = 0.18, which are similar to the gaps in figure 15(a). Due to fact that the
DOS determines the energygap, the above results indicate that the DOS is correct.
The picture (b) of the energygap for the (nonmetallic) zag nanotube manifests that the
energygap is proportional to the reciprocal diameter. When the diameter increases and the
carbon nanotube become more two dimensional, the semiconducting energygap vanishes.
The best fit of the calculated energygaps in figure 15(b) gives the equation
1
Egap = 2.74Aγ0
(18)
dt
The slope of the fitted 1. degree polynomial is smaller than the slope of the armchair
given in equation 14. This indicates an explanation based on more allowed ~k vectors in
the circumferential direction. The only thing, that can reduce the bandgap in this case, is
moving the The picture (b) of the energygap for the zigzag nanotube manifests that the
intersection of the ”tent” dispersion relation with the quantization lines closer to the K
points. It is known that when the size of the nanotube increases there are more allowed ~k
vectors in the circumferential direction with a narrowed spacing. From these two facts it
is possible to conclude that the narrowed spacing of the more allowed ~k vectors somehow
reduce the distance between the intersection lines and the K points, which is not obvious
from a graphical argument. Futher more we shall notice that some diameters give a zero
bandgap, metallic zigzag nanotubes. If we look at the latter image, it reveals that every
third nanotube has a zero bandgap, which agrees with the statements of Dresselhaus. 10
For the chiral insulating carbon nanotube (here mapped as (n,n+1)) the picture (figure
16) states the same as for the two previous kinds of nanotubes. The bandgap is linearly
related to the size of the tube by the reciprocal diameter. A best fit gives a similar slope
2.68 as in the zigzag case. Summarizing these results, partly shows the relation p.812 [2],
which says that all semiconducting nanotubes have the same linearly dependence between
the energygap and the reciprocal diameter.
Remark that chiral tubes of this structure (n,n+1) never occur as metallic even though
10
see figure from page in Dresselhaus
23
the energygap decreases. But chiral tubes with other structures e.g. (6,3) are metallic.
They all fulfill the condition for conduction properties as mentioned previously, n−m = 3p
where p is an integer.
24
10
Curvature effect
So far all the properties of the nanotubes have been calculated ignoring the effect of the
curvature. The 2D graphene sheet is isotropic, which means that all the overlap integrals
are equivalent and independent of the direction. However, when the rectangular spanned
by C~h and T~ in the graphene sheet is twisted into a tube, it causes variations in the
1
Save.nb integrals between the nearest carbon neighbours around the circumference. Inside
overlap
the cylinder the electron orbitals overlap more than on the outside, while the overlap
integrals in the tubule axis remain unchanged. Unlike in the graphene sheet, it now makes
a difference dependent on which direction you move. The strength of this interaction is
changed by varying the γ in the dispersion relation of the graphene sheet.
Save.nb
ky
1
Save.nbkx
[13]r5cm
The armchair tube is still metallic with the curvature
E
3
3
2
2
1
1
E
0
-1
-1
-2
-2
-3
-3
[(5,5) γ = 1.05]
0
-3
-2
-1
0
k
1
2
3
[(5,5) γ = 1]
-3
-2
-1
Figure 17: (a) The dispersion relation of the (5,5) with curvature effects (γ = 1.05). (b) The dispersion
relation of the (5,5) zero curvature effect. The effect of curvature is to shift the Fermi point toward k = 0.
The energy is in units of γ0 .
effects considered, because the quantization lines still intersect the shifted K-points (crosses). Here shown
in the case of a (5,5). We start by looking at the two simple cases of a (9,0) zigzag and an
armchair tube with γ = 1.05. The dispersion relation of the (5,5) tube is shown at figure
17. It appears that the effect of curvature has shifted the Fermi energy points closer to the
center, but it is still metallic. This shift is understood by looking at figure 18. The tent
formed dispersion relation of the graphene sheet in the Brillouin zone (hexagon), where
the red point are the K-points at no curvature. A deformation of the tent happens when
the curvature gamma parameter is increased. In k-space the effect is sketched at figure 18.
The corners of the hexagon are the K-points without curvature effect and the crosses are
25
0
k
1
the shifted K-points with γ > 1. If γ < 1 the shift of the K-points reverse the direction,
which is illustrated in the article [4].
The armchair (a) does not develop a bandgap, because the K-points are only translated in
the kx direction i.e. the points shift along the allowed ky values. The 4 double degenerated
bands in the middle are not lifted, because of the symmetry.
The (9.0) zigzag tube develops a narrow bandgap
(figure
19), when the curvature effect is consid3
ered. It is seen in the density of state, which van2
ish at zero energy. The K-points of the dispersion
1
E 0
relation of the graphene sheet are shifted away
from the lines containing the allowed wavevectors
-1
(figure 20). Every other nanotube than the arm-2
-3
chair tube develops a bandgap when curvature
-1.5 -1 -0.5 0 0.5 1 1.5
k
effect are considered, because the K-points are
shifted away from the quantization lines. Thus
the only metallic nanotube are the armchair tubes.
1
The above is in good agreement with the results
0.8
presented in the article [4], but with γ < 1. The
DOS 0.6
shift of the K-points are in the other direction.
0.4
The bandgap is not predicted by Dresselhaus [2],
0.2
but another approach with four tight binding pa0
-3 -2 -1 0
1
2
3
rameters are used. The references in the [4] are
E
more recent than the references in [2], which might
Figure 18: (9,0) tube with curvature effect indicate more reliability. The relation between
(γ = 1.05). The lowest band develops a ener- the energygap and γ for various γ is depicted in
gygap, indicated by the vertical lines close to
figure 20. The gap increases with increasing γ,
zero in the DOS.
and is approximately linear. Remark the interesting shape of the curve between γ = 1 and γ = 1.02, where it has a local minimum.
26
Save.nb
ky
kx
[Energygap vs
1
]
dt
[(9,0) zigzag tube.]
Figure 19: (b) The crosses indicate the shifted K-points. No quantization lines intersect
the new K-points, thereby making the tube nonmetallic.
Consider the curvature of the nanotubes, the results in figure 20 are a bit superfluous,because γ is a quantity that is dependent of the diameter of the tube, and therefore
there exist a specific value of γ for each tube, which are measured in experiments. The
above results correspond to the results in the article [4]. The program has no limitation
only to evaluate curvature effects on the simple nanotubes, but it allows an arbitrary choice
of tube, though we only have depicted the case for a (9.0) zigzag carbon nanotube.
27
11
Magnetic field along the tubule axes
In this section we examine how the dispersion relation of a nanotube changes when it is
placed in a applied magnetic field parallel the tubule axis. This gives rise to the AharonovBohm effect and changes the periodic boundary condition (9) with the phasefactor φe
,
h̄
where φ is the magnetic flux and e the elementary charge. In the applied magnetic field B
the flux through a cross section of the nanotube with radius r is given by φ = πr2 B.
Thus the boundary condition becomes
2
πr
C~h • ~k = 2πq + e
B
(19)
h̄
The effect on the nanotube is quite simple. It translates the allowed wavevectors in the
reciprocal space. The equation determining which new pairs of reciprocal lattice vectors
are allowed is
ky = √
2
πr2
n−m
(2πq + e
B) − √
kx
h̄
3a(n + m)
3(n + m)
This equation is very similar to equation (12) except that the constant in the 1. degree
polynomial is changed. Hence translating the quantization lines along ky dependent on
B. The extended zone scheme of a (9,0) zigzag tube in a applied magnetic field with field
strength B = 6000 Tesla is depicted in figure 21(a).
3
2
1
E 0
-1
-2
-3
0.7
0.6
0.5
DOS 0.4
0.3
0.2
0.1
0
-3
ky
kx
[Extended zone]
-1.5-1 -0.5 0 0.5 1 1.5
k
[Dispersion Relation]
-2
-1
0
E
1
2
3
Figure 20: (9,0) tube
The black dashed lines are the allowed pairs of wavevectors of the nanotube in zero
magnetic field while the blue lines shows the allowed pairs in a magnetic field of strength
6000 T. Clearly the blue lines are shifted and most important there are no lines which
intersect a K-point. Every allowed wavevector of the nanotube in the magnetic field have a
nonzero energy given by the dispersion relation of the graphene sheet. Thus the (9,0) tube
in a magnetic field of 6000 T is a semiconductor as indicated by the dispersion relation
and DOS in figure 21. The DOS gives a bandgap of approximately 0.5, which also can be
28
seen in the picture below. More bands appear in the dispersion relation indicating that a
magnetic field lifts the band degeneracy.
The field strength of 6000 Tesla is very high, but at this value the effect of shifted quantization lines becomes very clear. But in fact even a small magnetic field will make the (9,0) a
semiconductor, because the line through the K-point of the Brillouin zone of the graphene
are shifted as soon as the magnetic field is nonzero. When the bandgap is bigger than the
thermal energy at room temperature it is reasonable to speak of a semiconductor. As to
that the two present cases, (9.0) and (5.5) tubes, are very similar. A simple calculation
show large an applied magnetic field has to be to exceed the thermal energy, E = 25meV ,
at room temperature. In units of γ0 the thermal energy corresponds to a value of 0.01 and
the slope of the curves in figure 22 is about 0.1. For Eg > kB T the magnetic field has to
be larger than B = 0.01
= 0.09kT esla = 90T esla, which is a quite heavy field.
0.1
Figure 22a shows the energygap of a (9,0) tube versus the magnetic field strength B.
As the field strength increases the line through the the K-point at zero magnetic energy
are shifted away further away from the K-point thus given rise to an increasing energygap.
At the a magnetic field equal half the period the energygap decreases, because the line (a)
in the extended zone scheme gives the lowest band in the energyband. The graph is totally
symmetric about half the period. The point at about B = 10500T indicate the periodic
behavior of the energygap with the magnetic field strength.
Figure 22b shows energy of the (5,5) armchair nanotube in a magnetic field parallel to the
axes versus the magnetic field strength. The behavior is the very same as explained for
the (9,0) tube. The periodicity is found by the equation
eπr2
B
h̄
2π =
h
eπr2
which gives the periodicity B= 10544 Tesla for the (9.0) tube and B=11431 Tesla for
(5.0). The periodicity of the two carbon nanotubes is almost same, because the two
tubes has about the same diameter. Hence the flux through the nanotubes is about the
same. The same behavior is related to all nanotubes in a applied magnetic field. Thus a
semiconducting nanotube in a applied magnetic field would be metallic for a specific set of
field strength with the period mentioned above. The magnetic field shifts the quantization
lines and when the one of the lines in the Brillouin zone of the nanotube intersects a corner
of the Brillouin zone of the graphene sheet the tube becomes metallic.
B=
29
[(9,0) zig-zag]
[(5,5) armchair]
Figure 21: Energygap vs. magnetic field parallel to the axes of the nanotube. This a periodic effect
h
with period B = eφ
. Both tubes develops bandgaps at field strength different from the period. (a) (9,0)
armchair. (b) (5,5) zig-zag.
12
Conclusion
Through this project we have described some of the basic, but important properties of
the single wall carbon nanotubes. We have illustrated the structures of different kinds of
nanotubes and seen how the pattern of the cylinder changes due to the obliquity between
the hexagonals and the rollup vector, also called the chirality. Further more we have shown
explicitly that the electrical properties depend on the four parameters: the chirality, the
curvature, the diameter and the applied magnetic field along the cylinder axis. Due to these
facts we are able to conclude that it does matter how we roll up the carbon cylinder and
how large we make it, to succeed building either a metallic nanotube or a semiconducting
one.
On account of the deadline for this project there were some interesting issues we did not
succeed to investigate. For instance the scaling properties of the density of state functions
and the universal density of states, described in the Mintmire article [3], could be of great
interest looking further into.
Summarizing the results of this report, we achieved the purpose for the project. We have
made a program, that generates the extended zone scheme, the dispersion relation and
the DOS for a general nanotube. The extended zone scheme is only relevant for nanotube
with limited number of bands. In the case of many bands, the extended zone scheme is
not appropriate for better understanding. As concluded above the program calculates the
right dispersion relation for any nanotube. In our opinion the extended zone scheme and
the dispersion relation together make a the basic tool to understand the amazing electrical
properties of the carbon nanotubes. The DOS also gives the right result in terms of shape
and energygap after the corrections explained in 7 considered. In the light of the above
mentioned we can conclude that we have a tool to examine the basic electrical properties
of carbon nanotubes.
All the data used in this project are stored in a database.
30
13
Perspectives of Carbon Nanotubes
If we zoom out from all the calculations, and try to visualize the perspective of this topic
and what is so amazing about it, we realize that it will be useful in several applications
in the future. Somebody even thinks that creating an elevator into the space should be
possible due to the tremendous strength of the carbon nanotubes.
We only consider possibilities regarding to the electrical properties as we already have
introduced slightly in the beginning. By connecting two leads with a nanotube it is possible
to conduct an electrical current, and thereby creating a electronic device. In fact the
metallic tubes already have been used to build a single-electron transistor that work at the
low temperature level.
In the laboratory of the Department for Solid State Physics here at the University of
Copenhagen, the purpose of research is to build a transistor using the carbon nanotubes.
To do this, they try to develop a technique to place the nanotube appropriately between
two leads on a Silicon surface, because the method used today places the tubes arbitrarily.
Therefore it would be more comfortable to work with and speed up the progress, when
they succeed.
As the size of the electronic devices decrease, they handle transmission of information much
faster. Actually today it is possible to transfer one electron at a time to an electron well,
and eventually this method will be develop to a certain degree, where it can be used in the
microelectronics. We suppose that such a device would increase the speed of a computer
enormously, due to the fact that the electronic information only is given by a shift in a
single electron and not by a flow of many electrons (current).
So when you some day in the future go buying yourself a new computer. You possibly will
not even be able to leave the store, before it has lost in value.
References
[1] Kittel, C. Introduction to solid state physics, Wiley
[2] M.S. Dresselhaus, G. Dresselhaus and P.C. Eklund. Science of Fullerenes and Carbon
Nanotubes, Academic Press, (1996)
[3] J. W. Mintmire and C. T. White. Universal Density of States of Carbon Nanotubes,
Phys. Rev. Lett., Vol. 81, No. 12,(1998).
[4] L. Balents and P. A. Fisher, Correlation effects in carbon nanotubes, Phys. Rev. B,
Vol. 55, No. 18, (1997).
A
The program
All the results presented in the report are calculated by a program made in Mathematica.
It calculates the extended zonescheme, the dispersion relation and the DOS for any nan31
otube given different parameters. When the program starts it asks for the two parameters
(n,m) defining the nanotube, the curvature parameter γ and the strength of the magnetic
field along the axes of the tube. If no curvature effect is wanted γ = 1.
The first output is a picture of the Brillouin zone in the extended zone scheme. The allowed values of the wavevectors are shown by dashed quantization lines. Furthermore the
Brillouin zone of the graphene sheet is depicted by a yellow hexagon. The hexagon represents the area of the tent dispersion relation. If a magnetic field is present the quantization
lines shown are the shifted lines. Next the number of bands to calculate are printed. Each
time the dispersion relation and the DOS are calculated of one band the band number is
printed. Then it is possible to keep track of the time left before final output. In the case
of chiral nanotubes the time calculating the DOS quickly increases as n and m increases,
because of the many bands. It is possible to change the number of band that are shown in
the extended zone scheme and dispersion relation by changing the values qstart and qend.
Only the values of q in between these too values are calculated. The DOS is based on the
contribution from these bands.
There are some errors first time the program is executed, because Mathematica tells you
if some words are spelled similar.
We are using the interpolation function of Mathematica with the interpolationorder 1. The
interpolation order has to be reduced in case of the dispersion less bands (horizontal).
The program saves five files with calculated data.
The source code has comments explaining the syntax. A database is automaticly generated in a directory tubebase.
The energygap is found by looking in the table for the DOS and it is the lowest energy
value, which has a nonzero DOS. In the armchair case the numerical error around zero
energy has to be considered. The value of the energygap
B
Source code
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C
Graphs and data
Some of the graphs presented in this appendix are not made with the final version of the
program, but with a earlier one. These graphs do not have a correct normalized DOS. The
graphs are found in the following order.
The zig-zag nanotubes from (3,0) to (40,0). These are not normalized correct and in
the case of the metallic ones the energygap printed below the graph should be 0 not 0.05.
Clearly the DOS shows no energygap and it is due to a correction of the energygap, that
only applies in the nonmetallic case. It is corrected in the final version of the program.
The armchair nanotubes from (3,3) to (20,20). They are made with the final version
of the program and thus correct.
The nanotubes with about the same diameter. The (5,5), (9,0), (7,4), (8,3), (8,2), (5,6)
and the (8,0). Correct.
The insulating chiral nanotubes (n,n+1). Correct.
The (9,0) nanotube with different γ. Only a part of the energy scale are shown in the
DOS to get a higher resolution.
The (9,0) nanotube in a magnetic field. The magnetic field strength is written in the
upper left corner of the paper.
The (5,5) nanotube in a magnetic field.
Finally the data, which the ”concluding graphs” in the project are based upon. The
energygaps of the (9,0) nanotube in a magnetic field (the data) are not exactly the same
as those written below the DOS of a specific tube, because the energygaps in the data are
calculated with a better resolution.
33