Electrical properties of Carbon Nanotubes

8/6/2019 Electrical properties of Carbon Nanotubes

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Electrical properties of Carbon Nanotubes

Kasper Grove-Rasmussen

Thomas Jrgensen

August 28, 2000

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Contents

1 Preface 3

2 Introduction to Carbon Nanotubes 4

3 Single wall Carbon Nanotubes 5

4 Reciprocal Lattice 74.1 The Brillouin zone of the graphene lattice . . . . . . . . . . . . . . . . . . 94.2 The 2D Brillouin zone of the nanotube unit cell . . . . . . . . . . . . . . . 94.3 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 1D Brillouin zone of the nanotube . . . . . . . . . . . . . . . . . . . . . . . 12

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 33Appendix

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1 Preface

This project describes the outcome of the Bachelorproject, we have been occupied with inthe 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 inspiresomeone else to look further into it. We will!The project has come into existence in cooperation with the Department of Solid StatePhysics of the University of Copenhagen, in particular our advisors Jesper Nygaard, DavidCobden and Poul Erik Lindelof.We will in this project try to describe some of the knowledge about the electrical propertiesof carbon nanotubes and investigate several different tubes of a broad scale of range topoint out the details. Among other things, you will see that some tubes are metallic andsome are semiconductors. Further more it is our intention to write this project in sucha way that it is easily comprehensible to the reader. We will try to explain some of themethods used in the published articles more carefully, because often it is not obvious ornot explained what exactly happens. The project is purely theoretical and all results arebased on a computer program.

Good luck with the reading!

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2 Introduction to Carbon Nanotubes

As we are moving rapidly towards the twenty first century, the development within theworld 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 thedoor closing behind you, it occurs to you that the value of your loving computer alreadyhas decreased.This is caused by the tremendous research in the microelectronics, which has changed a lotof 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 amountof transistors placed on a micro chip doubles, but it is limited and soon a new techniquehas to be developed to carry on building faster computers. The silicon is the very heart ofthe microelectronics and will probably still be of great importance as we pass the changeof millennium. However, experiments indicate a growfield for a new technique based onmolecules. This is due to the scientists, who try to bring us from the micro scale to thenano scale, though it is another matter in the periodic table that is used, Carbon. Thisleads us to the main issue of this project.Carbon nanotubes describes a specific topic within solid state physics, but is also of inter-est 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 yearsbecome more and more popular to the scientists. Initially, it was the spectacularly elec-tronic properties, that was the basis for the great interest, but eventually other remarkableproperties 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, withdiameters from about 1 to 20 nanometres. They were discovered by the work of the ball-like fullerenes, which are much alike in structure. It turned out that adding a few percentof other atoms, nickel and cobalt, in the creating process of fullerenes, it was possible tolet the carbon chain grow, and thereby making a cylindric tube. Because of their smalldiameter 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.

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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 sheetwith different kinds of symmetries in structure. These structures are shown in figure 1.

All the nanotubes are labelled with

Figure 1: Three kinds of nanotubes. (a) An armchair nan-otube. The pattern in the vertical direction has the shape

of an armchair. (b) A zigzag nanotube. The pattern in thevertical direction is a zigzag line. (c) A chiralnanotube(n,m), (from [2] p. 758).

indices (n,m), - a simple way to tellthe name and the size of a specifictube. Other observations from pic-tures of the TEM are that the carbonnanotubes occur with more than onewall. These are composed of sev-eral concentric cylinders within eachother. In this article we just presentthe properties of the single wall nan-otube. If we return to the indices,we have already mentioned that thesedescribe the size of a tube. To seethis we have to investigate how weroll up the sheet to a cylinder (singlewall nanotube). The structure of thegraphene sheet is carbon atoms boundtogether in a honeycomb lattice, see

figure 2.The first thing we have to do is tochoose our primitive lattice vectors a1and a2 of the graphene sheet. These vectors define a parallellogram, which is called theprimitive unit cell, and fill all space by the repetition of suitable crystal translations op-erations (the graphene sheet). The unit cell also defines the minimum area of the paral-lellogram. So we can always make sure that we have chosen the right ones. There aremany ways of choosing the lattice vectors, but some are more comfortable to work withthan others. In figure 2 we have chosen a1 and a2 to be two upright secants of a isoscelestriangle within two hexagons. In Cartesian coordinates the vectors are

a1 =

a

2,3a

2

, a2 =

a

2,3a

2

(1)

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 Chiralvector) labeled

Ch = na1 + ma2 (2)

which determines the circumference of the carbon nanotube. Here n and m are the indicesof the tube and for that reason it is obvious that (n.m) directly describes the size of the

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tube.In figure 2 we can see that the arbitrary Chiral vector (n,m) for n m 0 lies between = 0 degrees (zigzag axis) and theta = 30 degrees (armchair axis). If we choose avector beyond this area, the symmetry will give us an equivalent vector within the area byreflection.

Also shown is the Translation vector which is perpendicular to Ch and given by1. 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

vectors Ch and T span a rectangle, which is called the 2D nanotube unit cell. This is therectangle, we roll up in the Chiral direction, that forms the cylinder. The side OB is puttogether with the side AB. This defines a unit cell, that repeatedly put together formsthe tube. Thereby making a periodicy equal the length of the translation vector in thetubuleaxis.The nanotubes just created has no distortion of bond angles other than in the circumferencedirection, caused by the cylindric curvature of the surface. As we will understand later,

this curvature, the chiral angle, the diameter dt =|Ch|

and an applied magnetic field alongthe tubule axes have influence of the electrical properties of the tube.

1[2], equation (19.6)

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4 Reciprocal Lattice

In this chapter we are to find the lattice vectors in the reciprocal space given the real spacelattice vectors. All calculations are done in Cartesian coordinates. The 1. Brillouin zoneis found by Wigner-Zeits primitive unit cell2. In the following this zone is referred to asthe 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 becalculated and the choice of real space lattice vectors is ambiguous. The general equationcan be applied to the different kinds of real space lattice vectors connected with the carbonnanotube. The reciprocal lattice vectors are related to the real space lattice vectors by therelation 3

ai bj = 2ij (3)

Figure 3 shows how the unit

Figure 3: Real space with unit vectors (x, y) and reciprocal spacewith unit vectors (kx, ky) of length 2.

vectors of the 2 dimensional Carte-sian coordinate system in real spacetransform to the unit vectors inthe reciprocal space according toequation (3). The x unit vectoris perpendicular to ky and viceversa. The unit length of thereciprocal vectors are 2. (notshown at figure 3). If the tworeal space vectors are not perpendicular it is not possible to do the transformation toreciprocal space that simple.

In a three dimensional lattice with realspace vectors a1, a2 and a3 the reciprocal latticevectors are found by 4

b1 = 2a2 a3

Vreal, b2 = 2

a3 a1Vreal

, b3 = 2a1 a2

Vreal(4)

where Vreal = a1 a2 a3 is the volume of the box spanned by a1, a2 and a3.These vectors obey equation (3). The length of a reciprocal lattice vector is given by

bi = 2|ai| (5)

where i is 1, 2 or 3.5 In our case we are only in two dimensions. Thus setting

2[1] p. 83[1] p. 334[1] p. 335This equation is only correct in the special case where it is later used. a3 = (0, 0, 1) and a1 a2 in

the plane orthogonal to a3.

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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

b1 =2

a

113

, b2 =

2

a

113

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 a1 and a2 are perpendicular torespectively b2 and b1. The hexagon is shown in k-space, but each pair of (kx, ky) corre-sponds to an energy value. The exact dependence is given in section 5.

As mentioned above the choice of real space lattice vector is ambiguous. Furthermorethe coordinates of these vectors depend on the choice of coordinate system. Our choiceis different from the choice of [2], but similar to the choice of [4]. The program we havemade is based on the definitions given above. It might be confusing when comparingwith other literature, but the results we obtain are similar to what would be found withother choices of lattice vectors and coordinate system. In [2] the two real space lattice vec-tors are switched and the x-direction of the coordinate system is along the zigzag direction.

The area of the Brillouin zone con-

Figure 4: The real space (left) and reciprocal (right) lat-tice vectors. The hexagon to the right is the 1. Brillouin of

the graphene sheet.

sists of 6 triangles (one is sketch in thefigure) each with an area of

1

2(

1

2

b2)( 1cos30

2

a

3) =

42

3

3a2

Hence the area of the Brillouin zone is

ABz =82

3a2(8)

This area is used later to calculatethe number of band needed to get the

dispersion relation of a carbon nan-otube and to normalize the density of state.

4.2 The 2D Brillouin zone of the nanotube unit cell

The two vectors defining the nanotube are the chiral vector Ch (2) and the translation

vector T. According to equation (3) the chiral vector Ch has to be perpendicular tothe second reciprocal lattice vector. This gives a reciprocal lattice vector parallel to thetranslation vector T, denoted GT. Similar the other reciprocal lattice vector denoted GCis parallel to Ch. The length of the reciprocal vectors are respectively

2|T| and

2| Ch|

(5). In

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Figure 5: The real space vectors defining the nanotube (left) and the reciprocal vectors defining the 2DBrillouin 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 byequation (6) and (7). The exact expression is not given, because a more convenient way ofdetermining 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 tworeciprocal lattice vectors. The larger the area of the nanotube unit cell gets the smallerthe area of the Brillouin becomes, because the reciprocal vectors are proportional to thereciprocal length of the unit vectors (5). In the chiral case even for small values of n andm the area of the real space nanotube unit cell becomes large and thereby reducing the2D 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 wavevectorsalong the reciprocal vector GC in the reciprocal space are allowed. This results in a numberof 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 thechoice of coordinate system. Hence we first derive it generally. The expression derived canthen be used with another choice than ours.The periodic boundary condition is a result of the required periodicity along the circum-ference of the Block wavefunction express as

(x + Ch) = (x)

where is the blockwavefunction of the graphene sheet and x is along the circumference.This gives rise to the following equation

Ch k = 2q (9)where Ch = n a1+m a2 is the chiral vector defining the nanotube, k the wavevector and q aninteger. Restrictions are later made on q. We write the relation in Cartesian coordinates

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to express the dependence between kx and ky.

n

a1xa1y

+ m

a2xa2y kxky = 2q

(na1x + ma2x)kx + (na1y + ma2y)ky = 2q (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 subsetof wavevectors in k-space are allowed. Because of the linear dependence we speak about(quantization) lines.Using the defined real space lattice vectors a1 and a2 (1) the relation becomes

(n

3a

2 + m

3a

2 )ky = 2q (na

2 + m(a

2 ))kx

(n + m)ky =4q

3a n m

3kx (11)

ky =4q

3a(n + m) n m

3(n + m)kx (12)

Equation (11) expresses the boundary condition and is true for all n and m. Two particularsimple cases appear for n = m (armchair nanotube) and n = m (zig-zag nanotube). Inthe armchair case the quantization condition is ky =

4q3a(n+n)

= 2q3an

which corresponds

to horizontal lines, while the lines are vertical kx

= 2qan

in the zigzag case. These two casesare 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 moreobvious 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, whereall 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 pairsof wavevectors according to the periodic boundary condition and the rectangle is the 2DBrillouin zone of the nanotube unit cell. Only the lines near the Brillouin zone of thegraphene sheet are depicted. The spacing between the lines is 2

|Ch|

, because the allowed

wavevectors k are the projection on Ch of length 2 times an integer (3). This is exactly

the reciprocal lattice vector GC. The edge of the Brillouin zone of the nanotube unit cellparallel to the quantization lines is half the distance of the vector GC.The vector GT is along the lines perpendicular to GC and defines the edge of the Brillouinzone of the nanotube unit cell in that direction. The angle between the lines and thekx-axis is determined by equation (12).In the (9,0) zigzag case the slope of the lines is 1

3, which is a line with an angle to the

kx-axes of -30 degrees (figure 5(a)). The (5,5) armchair has ky =2q53a

, which is a constant.

Thus the horizontal lines in 5(b) If the tubes are chosen with n m the angle of the chiral

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[(9,0) zig-zag]

kxy

[(5,5) armchair]

kx

ky

[(6,5) c

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. Onlyone line crosses the Brillouinin zone of the tubes unit cell. In the chiral case (c) the Brillouin zone getsvery 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 calledzonefolding. So far we know the 2D Brillouin zone of the nanotube unit cell and the allowedpairs (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 bythe dispersion relation of the graphene sheet (see section 5). In the following it will be moreconvenient to speak about pairs of (kT, kC) defined by the coordinate system spanned byGT and GC. It is actually just a rotation about origo ofkx-ky-coordinate system dependingon the choice of tube. This change of coordinate system is done to have the axes of thecoordinate system along the sides of the rectangular 2D Brillouin zone of the nanotubeunit cell.In the kC-kT coordinate system the kCs 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 schemein two dimensions (k-space) without the energy values shown. The states are not onlyrestricted to the 1. Brillouin zone. Instead we want to depicture the dispersion relation inthe 2D Brillouin zone of the nanotube unit cell. This representation is called the reducedzone scheme. All the kC and kT-vectors are translated by an integer times the reciprocallattice 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 Brillouinzone. This is a one dimensional problem. The kC-axis is shown in figure 7 (right). Thedots indicated the allowed values ofkC, determined by the intersection of the quantization

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Figure 7: (left) The quantization lines showed in the kT-kC coordinate system. They are always parallelto the GT vector and separated by GC. (right) The allowed vectors along the kC-axis, indicated by dots

are separated by the length ofGC. Only one dot (wavevector) is allowed inside the Brillouin zone. Theleft 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 GC. Also shown are the reciprocal lattice vector GC and the Brillouin zone (now onedimensional). 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 GC.Thus every wavevector outside the Brillouin zone is translated to the kC = 0. This reducesthe two dimensional Brillouin zone of the nanotube unit cell to a 1 dimensional Brillouinzone along GT, 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 isdetermined by the fact, that the total length of the 1D Brillouin zone of the nanotubetimes the spacing ( GT) equals the area of the Brillouin zone of the graphene sheet

7. Thatmeans for each translated kC the contribution to the total length of the one dimensionalBrillouin zone is the length of GT. One have to translate enough kC to equal the area ofthe Brillouin zone of the graphene sheet 8. N also equals the number of hexagons in thereal space nanotube unit cell.

In the above approach we implicitly suppose that no new energy band are given byzonefolding in the kT- direction. This actually seem to be the case. It is not clear howmany 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 zonefoldingN2

kC values, but then zonefolding the segment of length GT besides the 1D 1. Brillouinzone of the nanotube.

In short the zonefolding can be expressed as the set of N segments of length GT eachseparated by the vector GC are zonefolded (translated to Brillouin zone) into a 1D Brillouinzone of nanotube.

The dispersion relation of the nanotube has to be plot in the above found 1 dimensional

7[?]

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Brillouin zone i.e. kTT

; T

. Only a finite number of states exist equal to the number of

hexagon in the real space unit cell of nanotube.

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5 Graphene sheet dispersion relation

Before we are able to find the dispersion relation of the nanotube, we need the dispersionrelation 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 dispersionrelation is given by 8

(k) = 0(2cos(kxa

2)e

ikya

23 + e

ikya3 ) (13)

To obtain the 1D dispersion relations

Figure 8: The dispersion relation of the graphene

sheet. The blue hexagonal is the Brillouin zone andthe red dots called K-point are zero-gap points.

that describes the properties of the carbonnanotubes, we first have to consider the 2Ddispersion relation that is given by the lat-

ter equation. The +/- absolute value of thisrelation describes the energy bands in the2D graphene sheet (see figure 8), where thelattice constant is a =

3d and d is the

separation of two carbon atoms, who arenearest neighbours. The two parameters = t

0and 0 = t are hopping strength ele-

ments. They describe the possibility for anelectron to tunnel from one carbon atom toits neighbouring atom in the two directions

and vary with the curvature of the graphenesheet (see section 10).

The 2D dispersion relation makes two surfaces that have a shape like a tent raisedover/under the domain in the (kx, ky)-plan. However it is not defined on a rectangularBrillouin zone but on a hexagonal one. The symmetry is very obvious! The upper tentis 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 labelledthe 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 ofthe 2D graphene sheet

8Physical Review B Volume 55 Number 18 t = 0, t = 0. If using the dispersion relation from [2]kx and ky are to be switched, because of different choice of basis and coordinate system.

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6 Dispersion relation of the 1D nanotube

Now we are nearly ready to plot the dispersion relation of the nanotube along the directionof the vector GT in k-space i.e along the quantization lines of the 1D Brillouin zone of thenanotube. The coordinates kx and ky of the dispersion relation (13) are no longer applicableexcept 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 GT and therebymaking them proper coordinates.

In the general case the coordinate system of kT and kC is ro-

Figure 9: The (kx,ky)and (kT,kC) coordi-nate system.

tated with respect to the (kx,ky)-system. In figure 6(b) and 6(c)the Brillouin zone of the nanotube unit cell (blue rectangle) doesnot have any of its sides parallel to the kx-axis or the ky-axis.

Hence the vectorGT is not in the direction of the coordinate axes.The task is to find the relation between the two coordinate sys-

tem, i.e. an expression of kx and ky as a function of kT and kG.They are related by a rotation matrix in the following way.

kxky

=

cos sinsin cos

kTkC

where is the rotation angle counterclockwise. The angle of rotation is the absolute valueof angle related to the slope of equation (12). Thus = |tan1( nm

3(n+m))|.

Evaluating the above equation gives

kx = kT cos kC sin (14)

ky = kT sin + kCcos (15)

By substituting the expressions of kx and ky into the dispersion relation (13) the energybecomes a function of the wavevectors kT and kC. The boundary condition gives thequantization lines with the spacing | GC|. Thus the allowed values of kC can be expressedas kC =

2| Ch|

q, where q is taken the values from 1 to N. The energy is now found for each

value of q (a band) in the first Brillouin zone of the graphene sheet. The edge of the

Brillouin zone is

T

kTa

T

, 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 casesof the a zigzag and a armchair tube. Actually the above approach is necessary because weare to look at the zigzag case (n,0), which are not along the axes of the ( kx,ky)-coordinatesystem.

Figure 10(a) shows the extended zone in k-space of the (9,0) zigzag. In this casethe (kT,kC)-coordinate system are rotated 30 degrees in respect to the (kx,ky)-coordinatesystem. Plotting the energy along for instance the kx-axis is not the same as plottingthe 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 differentbands at one kx. A line parallel to the long side of the rectangular represents a specific

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[Extended zone]

kx

ky

[Dispersion Relation]-1.5 -1 -0.5 0 0.5 1 1.5

k

-3

-2

-1

0

1

2

3

E

Figure 10: (9,0) zigzag nanotube. (a). The dashed lines are the quantization lines. Each line gives riseto 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 = 18segments that has to be shown in the dispersion relation. (b) The dispersion relation of a (9,0) nanotubewith the energy in units of0. It is metallic because the light blue line intersect at k = 0. The plot showsthe Brillouin zone of the nanotube |ka| 1

2GT =

T

, which in this case is |ka| 3

.

kt. It is clearly seen that the intersections are not the same as in the above case. Thebands achieved using the first method are translated in respect to the correct bands of thedispersion relation.The figure shows the extended zone scheme (a) with the number of quantization lines Ndetermined 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 bandsof the dispersion relation (b). For instance the two light blue lines (degeneracy) in theextended zone scheme give the same energy bands. This is the two light blue coloredbands of the dispersion relation crossing at k = 0. In the extended zone scheme thequantization lines cross a K-point (corner of Brillouin zone) at kG = 0 in agreement withthe above. The dispersion relation just represents the slices the quantization lines makewith the three dimensional dispersion relation of the graphene sheet. A good intuitivelyway to get a feeling of the dispersion relation is to draw the extended zone scheme andimagine 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 quanti-zation lines give the dispersion relation of the nanotube. The (5,5) armchair has four doubledegenerated bands (4 middle bands), because two lines in the extended zone scheme slicethe dispersion relation of the graphene sheet in the same way. The top and the lowest bandare non degenerate. This is a general rule of the armchair. There are two non degeneratebands and 2(n 1) degenerate bands. The bands cross the Fermi level at k = 2

3.

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[Extended zone]

kx

ky

[Dispersion Relation]-3 -2 -1 0 1 2 3

k

-3

-2

-1

0

1

2

3

E

Figure 11: (5,5) nanotube. (a). The dashed lines are the quantization lines. Each line correspond tothe same colored band of the dispersion relation. (b) The dispersion relation plot between |kT| T = .The bands pink bands cross the Fermi energy in agreement with the pink quantization line crossing theK-points of the graphene sheet.

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7 Density of state (DOS)

The density of state (DOS) can be calculated from the dispersion relation of the carbonnanotube. It tells how many states at the energy between and + d.The total contribution to the DOS at the energy E can be expressed as the following

double sum

n(E) =1

bands

i

dk(k ki)

k1

(16)

where ki are the roots of the equation E

-3 -2 -1 0 1 2 3E

0

0.05

0.1

0.15

0.2

0.25

DOS

0 0.5 1 1.5 2 2.5 3k

0.5

1

1.5

2

2.5

3

E

Figure 12: Show the contribution to theDOS (below) from one of the band (above)of a (5,5) armchair.

(ki) = 0 and is the total length of the Bril-louin zone of the carbon nanotube i.e the num-ber of bands (with degeneracy) times the length

of the vector GT. We want to use equation (16)to do numerical calculations of the DOS from thedispersion relation. The interpretation is shownat figure 12. For each intersection of a line ofconstant energy E and a specific energyband thereciprocal derivative is found. If there is morethan one intersection the reciprocal derivativesare added together and the procedure is repeatedwith the rest of the bands. The total DOS is

found by evaluations for every energy value presentin the dispersion relation and normalizing it withthe total length of the 1D Brillouin () zone of thenanotube. For instance the band shown at figure12 has two intersections at a energy E = 0.9. Thereciprocal derivative at these points are addedand constitute the contribution to the DOS from that band (not normalized). The DOSgoes to infinity when the band has zero slope (minimum and maximum). The DOS hasa jump at E = 1, because only one contribution from E > 1 and two contributions forE < 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 pro-gram interpret as an energygap. This is corrected, by setting the energygap of the metallictubes to zero and smooth the DOS near the Fermi level. We are allowed to do so, because ifthe energygap was found manually from the DOS and by looking at the dispersion relationthe 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(k = 2

3T) and find the contribution from both sides. This would not give a numerical error.

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

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energybands and the DOS.

8 Comparison of some nanotubes

In this section we will look at some carbon nabotubes with almost the same diameter andtry to describe the similarities and the differences. The length of the rollup vector is givenby |ch| =

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 thesenanotubes you can find in the appendix. Well at least now we know that any differencesare not caused by the diameter, since it is almost the same.

[Extended zone scheme of (7,4)]

kxky

[Dispersion relation of (7,4)]

-3 -2 -1

0

0.2

0.4

0.6

0.8

DOS

- 0. 4 - 0. 2

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

E

Figure 13: (a) The extended zone scheme of (7,4) nanotube. (b) The dispersion relation and DOSof the (7,4) nanotube. This tube is a metallic chiral tube. The energy is plot between |E| 1 and|ka|

T=

31. (c) (8,2) chiral nanotube. The dispersion relation and the DOS shows that this tube has

a bandgap of E0

= 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 metallicwhile the (8,3), (5,6) and (8,0) are semiconducting. This is due to the chiral angle, whichis the only basic property that changes between these tubes. The chiral angle causes anobliquity between the rollup vector and the honeycomb lattice and apparently the angleinfluences 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 Ch direction passesthrough the K points in the Brillouin zones and therefore only leaves a bandgap in thereduced 1D Brillouin zone, as seen here with the three semiconductors. Otherwise for themetallic the symmetry is more nice and the allowed k s of course intersect with the K

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points. The numbers of bands in the seven carbon nanotubes are quite different. They gofrom 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 Ch is almost the samebut the lengths of T differ. The nanotubes with a lot of bands also have a long T. Thatmeans they span rectangles of different sizes in the real space. The larger the rectangularis the smaller it appears in the reciprocal space. Similar if T is long in real space it appearsproportional smaller in the reciprocal space, GT. Now the GT determines the length of1D Brillouin zone and if this one gets smaller with increasing length of T then we have togo much further out in the GC direction, because the area of this long rectangular, whichcontains all the allowed k s, must be the same as the area of the Brillouin zone. Thereforewe get a larger amount of intersections, which corresponds to the energybands. Thereforesome 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 thetranslation vector T, and there by the area spanned by Ch and T. So in this chapter weconclude that chiralty is very important for electronic properties and it does matter howwe roll up the carbon nanotube.

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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 semi-conducting depending of the choice of (n,m). This is remarkable, since there are neitherdifferences in the bondings between the carbon atoms nor any impurities of donor atomspresent, which normally cause the above mentioned features. This fact arise interestingperspectives for developing new electronic devices. The figure 14 shows the pseudoenergy-gap versus 1/diameter for the armchair nanotubes. We use the terminology pseudogapbecause the armchair nanotubes always are a metallic and therefore have no real energy-gap. The pseudogap describes the difference of the energy between the peak values closestto the Fermi level in the density of state diagram. For a semiconducting nanotube thisdifference is a real energygap.

[Pseudoenergygap vs n ] [Pseudoenergap vs 1dt ]

Figure 14: (n,n) nanotube. The pseudoenergygap is in unit of0, which is the difference of the energybetween 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 reciprocaldiameter, 1/diameter. That means when the diameter increases the pseudogap decreasesand the energys closest to the Fermi energy with the high density of states move closertogether. The equation of the linear dependence is given by

Epseudogap = 7.10A0 1dt

(17)

where A is Angstrom and 0 = 2.5eV.9 and the constant of the 1. degree polynomial is

almost zero ( E0

= 0.08). The pseudogap of the 20 armchair nanotubes plotted at figure 14

is in the range from E0

= 0.33 for the (20,20) armchair to E0

= 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 0 are shown as afunction of n. This result corresponds to the values that are depicted in figure 19.27 in

9[1] p. 213

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[Energygap vs n] [Energygap vs 1d

]

Figure 15: Energygap is in units of0. 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 E0

= 0.09 and

respectively E0

= 0.18, which are similar to the gaps in figure 15(a). Due to fact that theDOS 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 theenergygap is proportional to the reciprocal diameter. When the diameter increases and thecarbon nanotube become more two dimensional, the semiconducting energygap vanishes.The best fit of the calculated energygaps in figure 15(b) gives the equation

Egap = 2.74A01

dt(18)

The slope of the fitted 1. degree polynomial is smaller than the slope of the armchairgiven in equation 14. This indicates an explanation based on more allowed k vectors inthe circumferential direction. The only thing, that can reduce the bandgap in this case, ismoving the The picture (b) of the energygap for the zigzag nanotube manifests that theintersection of the tent dispersion relation with the quantization lines closer to the Kpoints. It is known that when the size of the nanotube increases there are more allowed kvectors in the circumferential direction with a narrowed spacing. From these two facts itis possible to conclude that the narrowed spacing of the more allowed k vectors somehowreduce the distance between the intersection lines and the K points, which is not obviousfrom 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 everythird 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 (figure16) states the same as for the two previous kinds of nanotubes. The bandgap is linearlyrelated to the size of the tube by the reciprocal diameter. A best fit gives a similar slope2.68 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 betweenthe energygap and the reciprocal diameter.

Remark that chiral tubes of this structure (n,n+1) never occur as metallic even though10see figure from page in Dresselhaus

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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, nm = 3pwhere p is an integer.

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the shifted K-points with > 1. If < 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 inthe kx direction i.e. the points shift along the allowed ky values. The 4 double degeneratedbands in the middle are not lifted, because of the symmetry.

The (9.0) zigzag tube develops a narrow bandgap

-3 -2 -1 0 1 2 3

E

0

0.2

0.4

0.6

0.8

1

DOS

-1.5 - 1 - 0.5 0 0.5 1 1.5k

-3

-2

-1

0

1

2

3

E

Figure 18: (9,0) tube with curvature effect(= 1.05). The lowest band develops a ener-gygap, indicated by the vertical lines close tozero in the DOS.

(figure 19), when the curvature effect is consid-ered. It is seen in the density of state, which van-ish at zero energy. The K-points of the dispersionrelation of the graphene sheet are shifted awayfrom the lines containing the allowed wavevectors(figure 20). Every other nanotube than the arm-

chair tube develops a bandgap when curvatureeffect are considered, because the K-points areshifted away from the quantization lines. Thusthe only metallic nanotube are the armchair tubes.The above is in good agreement with the resultspresented in the article [4], but with < 1. Theshift of the K-points are in the other direction.The bandgap is not predicted by Dresselhaus [2],but another approach with four tight binding pa-rameters are used. The references in the [4] aremore recent than the references in [2], which might

indicate more reliability. The relation betweenthe energygap and for various is depicted infigure 20. The gap increases with increasing ,and is approximately linear. Remark the inter-

esting shape of the curve between = 1 and = 1.02, where it has a local minimum.

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[Energygap vs 1dt

] [(9,0) zigzag tube.]

kx

ky

Figure 19: (b) The crosses indicate the shifted K-points. No quantization lines intersectthe new K-points, thereby making the tube nonmetallic.

Consider the curvature of the nanotubes, the results in figure 20 are a bit superflu-ous,because is a quantity that is dependent of the diameter of the tube, and thereforethere exist a specific value of for each tube, which are measured in experiments. Theabove results correspond to the results in the article [4]. The program has no limitationonly to evaluate curvature effects on the simple nanotubes, but it allows an arbitrary choiceof tube, though we only have depicted the case for a (9.0) zigzag carbon nanotube.

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11 Magnetic field along the tubule axes

In this section we examine how the dispersion relation of a nanotube changes when it isplaced in a applied magnetic field parallel the tubule axis. This gives rise to the Aharonov-Bohm effect and changes the periodic boundary condition (9) with the phasefactor e

h,

where is the magnetic flux and e the elementary charge. In the applied magnetic field Bthe flux through a cross section of the nanotube with radius r is given by = r2B.Thus the boundary condition becomes

Ch k = 2q + e r2

hB (19)

The effect on the nanotube is quite simple. It translates the allowed wavevectors in thereciprocal space. The equation determining which new pairs of reciprocal lattice vectors

are allowed is

ky =2

3a(n + m)(2q + e

r2

hB) n m

3(n + m)kx

This equation is very similar to equation (12) except that the constant in the 1. degreepolynomial is changed. Hence translating the quantization lines along ky dependent onB. The extended zone scheme of a (9,0) zigzag tube in a applied magnetic field with fieldstrength B = 6000 Tesla is depicted in figure 21(a).

[Extended zone]

kx

ky

[Dispersion Relation]-3 -2 -1 0 1 2 3

E

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

DOS

- 1. 5 -1 - 0 .5 0 0 .5 1 1 .5k

-3

-2

-1

0

1

2

3

E

Figure 20: (9,0) tube

The black dashed lines are the allowed pairs of wavevectors of the nanotube in zeromagnetic field while the blue lines shows the allowed pairs in a magnetic field of strength6000 T. Clearly the blue lines are shifted and most important there are no lines whichintersect a K-point. Every allowed wavevector of the nanotube in the magnetic field have anonzero energy given by the dispersion relation of the graphene sheet. Thus the (9,0) tubein a magnetic field of 6000 T is a semiconductor as indicated by the dispersion relationand DOS in figure 21. The DOS gives a bandgap of approximately 0.5, which also can be

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seen in the picture below. More bands appear in the dispersion relation indicating that amagnetic field lifts the band degeneracy.

The field strength of 6000 Tesla is very high, but at this value the effect of shifted quantiza-tion lines becomes very clear. But in fact even a small magnetic field will make the (9,0) asemiconductor, because the line through the K-point of the Brillouin zone of the grapheneare shifted as soon as the magnetic field is nonzero. When the bandgap is bigger than thethermal energy at room temperature it is reasonable to speak of a semiconductor. As tothat the two present cases, (9.0) and (5.5) tubes, are very similar. A simple calculationshow large an applied magnetic field has to be to exceed the thermal energy, E = 25meV,at room temperature. In units of0 the thermal energy corresponds to a value of 0.01 andthe slope of the curves in figure 22 is about 0.1. For Eg > kBT the magnetic field has tobe larger than B = 0.01

0.1= 0.09kTesla = 90Tesla, which is a quite heavy field.

Figure 22a shows the energygap of a (9,0) tube versus the magnetic field strengthB

.As the field strength increases the line through the the K-point at zero magnetic energyare 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 totallysymmetric about half the period. The point at about B = 10500T indicate the periodicbehavior of the energygap with the magnetic field strength.Figure 22b shows energy of the (5,5) armchair nanotube in a magnetic field parallel to theaxes versus the magnetic field strength. The behavior is the very same as explained forthe (9,0) tube. The periodicity is found by the equation

2 = er2

hB

B =h

er2

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 twotubes has about the same diameter. Hence the flux through the nanotubes is about thesame. The same behavior is related to all nanotubes in a applied magnetic field. Thus asemiconducting 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 quantizationlines and when the one of the lines in the Brillouin zone of the nanotube intersects a cornerof the Brillouin zone of the graphene sheet the tube becomes metallic.

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[(9,0) zig-zag] [(5,5) armchair]

Figure 21: Energygap vs. magnetic field parallel to the axes of the nanotube. This a periodic effectwith period B = h

e. 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 ofthe single wall carbon nanotubes. We have illustrated the structures of different kinds ofnanotubes and seen how the pattern of the cylinder changes due to the obliquity betweenthe hexagonals and the rollup vector, also called the chirality. Further more we have shownexplicitly that the electrical properties depend on the four parameters: the chirality, thecurvature, 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 andhow large we make it, to succeed building either a metallic nanotube or a semiconductingone.On account of the deadline for this project there were some interesting issues we did notsucceed to investigate. For instance the scaling properties of the density of state functionsand the universal density of states, described in the Mintmire article [3], could be of greatinterest looking further into.Summarizing the results of this report, we achieved the purpose for the project. We havemade a program, that generates the extended zone scheme, the dispersion relation andthe 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 isnot appropriate for better understanding. As concluded above the program calculates theright dispersion relation for any nanotube. In our opinion the extended zone scheme andthe dispersion relation together make a the basic tool to understand the amazing electricalproperties of the carbon nanotubes. The DOS also gives the right result in terms of shapeand energygap after the corrections explained in 7 considered. In the light of the abovementioned we can conclude that we have a tool to examine the basic electrical propertiesof carbon nanotubes.All the data used in this project are stored in a database.

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otube given different parameters. When the program starts it asks for the two parameters(n,m) defining the nanotube, the curvature parameter and the strength of the magnetic

field along the axes of the tube. If no curvature effect is wanted = 1.The first output is a picture of the Brillouin zone in the extended zone scheme. The al-lowed values of the wavevectors are shown by dashed quantization lines. Furthermore theBrillouin zone of the graphene sheet is depicted by a yellow hexagon. The hexagon repre-sents the area of the tent dispersion relation. If a magnetic field is present the quantizationlines shown are the shifted lines. Next the number of bands to calculate are printed. Eachtime the dispersion relation and the DOS are calculated of one band the band number isprinted. Then it is possible to keep track of the time left before final output. In the caseof 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 thecontribution from these bands.There are some errors first time the program is executed, because Mathematica tells youif some words are spelled similar.We are using the interpolation function ofMathematica with the interpolationorder 1. Theinterpolation 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 gen-erated 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 zeroenergy 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 theprogram, but with a earlier one. These graphs do not have a correct normalized DOS. Thegraphs are found in the following order.

The zig-zag nanotubes from (3,0) to (40,0). These are not normalized correct and inthe 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, thatonly 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 versionof 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 . 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 sameas those written below the DOS of a specific tube, because the energygaps in the data arecalculated with a better resolution.

33

Electrical properties of Carbon Nanotubes

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