Band Gaps of Single-Walled Carbon Nanotubes: A Computational Study

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1 1 Band Gaps of Single-Walled Carbon Nanotubes: A Computational Study Brittany Chase May 21, 2008 I Introduction Carbon Nanotube (CNT) structures are a recently discovered (1991) phase of carbon first synthesized in the laboratory of Sumio Iijima at Tsukuba[1]. They are known to exhibit a wide variety of electronic properties. They can be purely metallic or semiconducting. Semiconducting nanotubes are of particular interest for their potential industrial applications. In this thesis, results on the electronic properties of carbon nanotubes are presented. CNTs are interesting because their strength and nanoscale size make them ideal for several applications in nanotechnology, electronics, optics, and materials science. The carbon bonds in CNTs are similar to graphite, but their structure provides for bonds stronger than diamonds which give rise to the unique strength of CNTs[1]. The electronic properties of CNTs can also miniaturize the current microscale in electronic devices. For example, CNT based transistors exist that operate at room temperature and can switch using a single electron[2]. Thus, high speed switches for future computers would require a fraction of the current that today s computers utilize and they would take up much less space than electronic components on today s CPU chips. Also, CNTs could result in high density storage for RAM applications since they can be miniaturized to scales far below current technologies. Moreover, it is predicted that the switching time of these electronic components will be around a terahertz, which is 10,000 times faster than today s computers[3]. Future computers will be faster and orders of magnitude smaller with the new technology enabled by CNT-based transistors. CNTs have a low resistance which means they are great thermal conductors. Smaller diameter nanotubes can pack more tightly and carry larger amounts of current without overheating. Since they are great thermal conductors, they can also be used as heat sinks and transfer heat away from CPU chips[3]. Possible applications also include nanoscale electronic motors, mechanical memory elements, and flexible nanotube films for use in computers and cell phones[2]. Tunable band gaps in nanomaterials could lead to full color holographic projectors smaller than a postage stamp[2]. These projectors will offer a highly reliable touch screen, since CNTs are very flexible. Tunable band gaps can also give rise to lasers with a variable spectrum: by altering the band gap, the color of the laser can change. Moreover, it has been proposed that CNTs could possibly be used to transport genes in combination with radiofrequency fields to destroy cancer cells[4]. In addition, CNTs can be used in strain gauges in materials doped with CNTs that can sense tiny variations in internal strain in materials and relay that information optically, resulting in extremely fast failure detection in bridges, buildings, or other public infrastructure settings. Not only would the CNTs relay the stress information, but they would also improve the strength of the

2 2 steel used in the construction. CNTs can even revolutionize everyday items such as clothes and sports gear where low weight and high strength is desired. They can be used as composite fibers in polymers to improve the mechanical, electrical, and thermal properties of the material[2]. II History In 1985, a group led by Rick Smalley discovered the first fullerene, C60. This fullerene is more commonly known as the bucky ball. It is an icosahedral, or soccer ball, shaped molecule composed of sixty carbon atoms bonded together in hexagons and pentagons using sp 2 hybridized bonding. The new arrangement of carbon provided for a new material with different properties. Five years later, Smalley proposed the idea of a bucky tube which could be created by elongating a bucky ball[2]. have six electrons that are evenly shared between their 1s, 2s, and 2p orbitals. They make covalent bonds when the 2s electrons are moved to one or more of the 2p orbitals. When the 2s orbital hybridizes with two 2p orbitals, the result is three sp 2 orbitals and the carbon atoms for a zigzag chain with angles of 120. The three orbitals created in the process are characterized by trigonal bonding[5]. The CNT structure is best described as a single graphene sheet of sp 2 hybridized carbon rolled into a seamless cylindrical tube. In a sense, the graphene model is given periodic boundary conditions. In rolling the tube, the hexagonal lattice is slightly distorted due to large curvature effects in smaller diameter nanotubes[5]. The way the tube is rolled determines its fundamental properties. The tubes can be classified into three categories depending on the geometry of the graphene sheet edge: zig-zag, armchair, and chiral. These are shown in Figure 1. The existence of carbon nanotubes was established through experiment in 1991 when Sumio Iijima imaged carbon nanotubes using high resolution transmission electron microscopy[1]. Since then, carbon nanotube research has become a leading research area of nanotechnology and it is expanding rapidly. The unique properties and numerous potential applications of CNTs has catapulted them into the world s view. III Structure The structure of CNTs resemble a sheet of graphene rolled up into a tube. A graphene sheet is best described as a single layer of carbon atoms arranged in a hexagonal, or honeycomb, pattern. The bonds of this sheet are sp 2 hybridized carbon bonds. Hybridization occurs when the 2s and 2p orbitals mix, so any linear combination of them is characterized by hybridization; the sp 2 bonds are strong because of overlap[2]. Carbon atoms Figure 1: The three different types of CNTs are named based on the n and m index and their edge. Also, the edge of the tube either follows a zig-zag, armchair, or a seemingly random pattern, named chiral. The different structures determine the electronic and mechanical properties of the CNT. Each CNT

3 3 is uniquely identified by the index pair (n,m) that define the translation vector C on the graphene lattice; C maps an origin atom to its periodic partner at the opposite edge of the lattice. When the two ends of C are connected, they coincide which forms the circumference of the CNT s circular cross-section, which is directly proportional to the diameter of the tube. In Figure 2, the vector T would be connected to the opposite side of the shaded quadrilateral and it would be the length of the tube. In addition, the vectors a 1 and a 2 represent the unit vectors of the hexagonal lattice and the integer index pair, (n,m), is a linear combination of the two[1]. Also, a 1 and a 2 define the area of a unit cell on the honeycomb lattice[5]. Defined by the integer index pair, (n,m), the chiral angle ranges from 0 to 30 degrees: zigzag nanotubes have chiral angles of 0 degrees, armchair nanotubes have chiral angles of 30 degrees, and chiral nanotubes have chiral angles between 0 < α < 30[1]. IV Previous Research Since their discovery, CNTs have intrigued the research community. Several groups have confirmed the structure of the CNTs through high resolution microscopy techniques[1]. Noriaki Hamada led a team at the NEC Laboratory in Tsukuba in finding the phase diagram for a variety of CNT geometries[1]. The phase diagram is a representation of which CNTs are conducting and which are semiconducting; it is laid out like the graphene model for illustrative purposes and each point on the graph represents a different diameter CNT. They used the graphene model as a basis for their diagram since the CNT structure is similar to a rolled up sheet of graphene. Using the graphene model, theory predicts that CNTs obey the rule: if n m = 3i(i = 0, 1, 2,...), then the CNT is conducting[1]. Figure 2: Graphene model of a CNT showing the index pair notation used to characterize the CNT geometry. The index pair, (n,m), uniquely identifies the diameter, chiral angle, and type of CNT [6]. The chiral angle, α, is the angle between the translation vector, C, and the (n,0) line as seen in Figure 2. α = arctan( 3n/(2m + n)) (1) Figure 3: The graphene model of carbon predicts that if the difference between n and m is a multiple of three, then the CNT is conducting. This model is appropriate for larger diameter nanotubes. According to the model given in Figure 3, one third of all CNTs should be conducting.

4 4 In addition, research groups have investigated the band gap energies of larger diameter CNTs, and they have been found to be inversely proportional to the diameter as seen in Figure 4[7]. than that of steel. Young s modulus, Y, is the applied tensile stress over the strain: Y = σ ɛ. (2) In Equation 2, σ is the stress (the amount of applied force per cross-sectional area), and ɛ is the strain (the deformation, or change in length, caused by applied stress). The sp 2 bonds can rehybridize as they are strained which make the CNTs more apt to resist breaking when strained[3]. Figure 4: The band gap energies of semiconducting, chiral CNTs are obtained from local-density functional (LDF) calculations. The curve represents E g = V 0 d/r, where d=0.144 nm and V 0 = 2.51eV [7]. V Mechanical Properties One of the properties of CNTs is their unique strength; they are also very resilient. When CNTs are bent, they buckle like straws, but they do not break[3]. In fact, they can easily be straightened out without any damage to the physical structure. Young s modulus is a measure of how stiff or how flexible a material is under tensile strain: how a material would react when bent or stetched[8]. The Young s modulus of CNTs ranges from TPa (terapascals)[3]. One terapascal is a pressure of about seven orders of magnitude greater than atmospheric pressure. The Young s modulus of steel is 0.21 TPa, which means the Young s moduli of CNTs are typically ten times greater Furthermore, the tensile strength of CNTs is Pascals. The tensile strength is a measure of the amount of stress needed to pull a material apart and tensile stress is defined as a pressure, or the force per cross-sectional area, applied to the CNT. In comparison, the tensile strength of high-strength steel alloys is about Pascals. CNTs are twenty times stronger than steel[3]. VI Simple Model of Band Gap Formation Let us consider a simple model of a periodic potential to explore the origin of band gaps in materials. To model a one-dimensional strand of carbon atoms with periodic potential, we employ a simple approximation called the Dirac comb. The Dirac comb is composed of evenly spaced delta function spikes; it is the simplest possible model of a one-dimensional crystal and is used here for illustrative purposes only[9]. The comb in Figure 5 represents a periodic potential that repeats itself after a fixed distance, a. V (x + a) = V (x) (3)

5 5 where N represents the number of periods within the comb. Therefore, using Equation 5 in Equation 6, we get ψ(x) = e inκa ψ(x). (7) Figure 5: The model of a one-dimensional crystal can be represented by a Dirac comb of periodic potential. It is a simple model, but a fairly accurate representation of the way electrons behave. Solving equation 7 for κ, yields the following solution, where n is an integer[11]. κ = 2πn Na (8) This gives the crystal a translational symmetry. Even though the edges of the crystal stop the periodicity of the model, the edges can be thought to wrap around in a circle and create periodic boundaries: for any macroscopic crystal, the edges will not affect the behavior of the electrons deep inside. The strand of carbon atoms is therefore modeled as a ring. Bloch s Theorem[10] states that the solutions to the time-independent Schrödinger equation, Since the exponential is equal to 1, Nκa must be a multiple of 2π. Also, κ must be real. Looking at Figure 5, the region where 0 x a is at zero potential. Therefore, the timeindependent Schrödinger equation in Equation 4 reduces to the following. h 2 2m d 2 ψ = Eψ (9) dx2 h 2 2m d 2 ψ + V (x)ψ = Eψ, (4) dx2 Here, m is the mass of an electron. We define k as follows. satisfy the condition k 2mE h (10) ψ(x + a) = e iκx ψ(x). (5) These solutions are found when solving within a single cell, 0 x a, due to the periodicity of the crystal: this equation can generate the solution in every other cell[11]. Here, κ is a real constant. The periodic boundary condition from Equation 3 can be written as ψ(x + Na) = ψ(x), (6) From this, we get a relationship between E and k. E(k) = k2 h 2 2m (11) The relation defined in Equation 10 is also the dispersion relation for this system. It means that for a given momentum, a variety of energies are possible and some are not available. This collection of energies is defined in the band structure.

6 6 The general solution to the wave equation is as follows. ψ(x) = A sin(kx) + B cos(kx) (12) Using Bloch s theorem from Equation 5 in Equation 12, the wave function for the cell immediately to the left of this wave equation is ψ(x+a) = e iκa [A sin(k(x+a))+b cos(k(x+a))]. (13) cos(κa) = cos(ka) + mα h 2 sin(ka) (17) k Equation 17 determines the possible values of k and the allowed energies since E is proportional to k 2. Determining the values of k creates the allowed energies of the system. To find values for k, one needs to plot the right side of Equation 17 as in Figure 6 for a range of k since there are no other variables in the equation. The two boundary conditions between these wave functions must be applied: the wave function must be continuous at the boundary, and its derivative must be continuous at the boundary[11]. At x = 0, the two wave functions given in Equations 12 and 13 must be continuous, so B = e iκa [A sin(ka) + B cos(ka)]. (14) At x = 0, the derivatives must be continuous; however, there is a discontinuity in the wave function since the potential consists of delta spikes. Thus, the discontinuity is proportional to the strength of the delta function[11]. ( dψ dx ) = 2mα 2 ψ(0) (15) h The condition on the derivatives leads to: ka e iκa k[a cos(ka) B sin(ka)] = 2mα h 2 k B. (16) A combination of Equations 14 and 16 yields the following simplified result. Figure 6: A plot of the right side of Equation 17 against different values of k. Since the cosine function given in the left side of Equation 17 only has values ranging from -1 to 1, only looking at the graph within that region will give the band gap for this particular crystal. This is represented in Figure 7. The gaps are forbidden energies and the bands represent the possible allowed energies. Band structure is the signature of a periodic potential[11]. Within a given band virtually any energy is allowed; hence, the band of values for energy arises as seen in Figure 8. The band structure is similar the graph in Figure 7 rotated 90 degrees counterclockwise since electron energy, E, is proportional to k 2. In each of these bands, there can be at most two electrons due to the Pauli exclusion principle: they can be either spin up or spin down[10]. A plot of E vs. k can also be obtained. The

7 7 Figure 7: The cosine function from the left side of Equation 17 only takes on values between 1 and - 1; thus, band gaps arise because of the boundaries imposed by the cosine function. model of this for the free electron is shown on the left while the model for a solid with periodic potential is shown on the right in Figure 9[9]. On the right graph in Figure 9, it is clear that there is a gap in the energy: at certain values of k, the energy is not allowed as evidenced by the graphs in Figures 6 and 7. Band gaps in real materials are much more complex than this model; however, this model is a relatively accurate portrait of the gaps in energy[12]. Also, this model is a good basis for the twodimensional graphene model since the electrons are nearly free in graphene. Depending on whether or not a band is entirely filled will determine the material s ability to conduct. Consider the band gap diagram given in Figure 10. For example, if a band is completely filled, it will require a large amount of energy to excite an electron to jump across a gap in energy, or forbidden region, because there is not much room to excite it; this results in an insulator. On the other hand, a partially filled band has plenty of room to allow an electron to become excited; hence, it takes a small amount of energy to Figure 8: The band gaps are formed with the boundaries of the cosine function. Electrons can only take on energy values in the shaded regions and the gaps are forbidden energies. excite it and jump across the gap. These are conductors because the electrons can flow freely. The free electron model is an excellent example of a conductor since there are no gaps in the spectrum of allowed energies[9]. Moreover, if there are extra electrons in the next higher band due to doping or if there are holes in a previously filled band, weak electric currents will result, which give rise to semiconductors[11]. The lower of the two bands, which is almost completely filled, is called the valence band because it is composed of primarily valence electrons, while the next level up, which is almost completely empty, is known as the conduction band since those electrons can contribute to the conductivity of the crystal[12]. The electronic band gap of CNTs is of particular interest. CNTs with band gaps of less than a tenth of an ev are considered conducting and those with band gaps greater than that are semi-

8 8 Figure 9: On the left, the energies for a free electron are graphed against k and on the right, the energies for a crystal with a periodic potential and band gaps are graphed[12]. conductors. I have chosen this value as the boundary between conducting and semiconducting nanotubes because at room temperature, it is the value for the thermal energy of molecules: 0.1eV = 4kT and kt = 1/40eV. A significant fraction of electrons at room temperature will have this energy and they will be able to move from the valence band to the conduction band with ease; these types of conducting CNTs are defined as weak conductors. Applying tensile stress can alter the band gaps of CNTs. Different stresses change the band gap in different ways which makes the band gap tunable. The band gap in the stressed CNT will determine whether or not the photon emitted when an electron jumps from a higher energy band to a lower energy band is visible. The energy a photon emits is found by determining the wavelength from the Equation 18, where h is Planck s constant. E = hc λ (18) This equation is applicable because the energy of a photon is relativistic since it approaches the speed of light. It is desirable to tune the band gap in variable lasers, for instance, so that the photon correspondence will be in the visible range of the spectrum. Lasers are created in much the same way; depending on the type of laser desired, the band gap Figure 10: Existance and size of a band gap energy determines whether a material is conducting, semiconducting, or insulating. A band gap exists when there is a gap between the valence and conduction bands; for insulators, the band gap represents an impractical amount of energy that needs to be added to make the material conducting, whereas in semiconductors, adding energy can make the material conducting. For my study, CNTs are either conducting or semiconducting[13]. can be altered so that the photons will give off a certain color of the spectrum when transitioning between energy bands. VII Objective of Study For this thesis, I obtained a phase diagram for the CNTs so I could easily distinguish between conducting and semiconducting CNTs. I compared my data to the accepted phase diagram for larger diameter nanotubes and looked for a correlation. To obtain the phase diagram, I calculated the band structures for a variety of CNTs with different geometries. Using the Fermi-Dirac distribution, I determined whether or not a band gap existed in a variety of CNTs. For the semiconducting CNTs, I looked for a correlation between the band gap energies and the diameters for the nanotubes and I compared the smaller systems to the trends found among the larger systems. Finally, I measured band gap energies as a function of applied tensile stress to observe the changes in the electronic properties of the CNTs.

9 9 VIII Method I use CASTEP and DMol3 Density Functional Theory (DFT) implementations in Materials Studio (Accelrys, Inc). CASTEP and DMol3 permit ab-initio quantum molecular dynamics calculations on small systems. The theory of Molecular Dynamics treats each individual atom in the system as a point. Using Newton s Laws and the interactions between the atoms, the forces can be derived from the effective potential which can also be used to solve the Schrödinger equation for the system. Finding the exact wavefunctions of electrons is difficult to determine[15]. The many-body problem we face would require calculations which is very impractical[16]. Molecular Dynamics simplifies the problem to calculations involving a single particle that depends on spatial coordinates; therefore, the physical electron density is a function of just three variables. It bypasses the complex wavefunction and it expresses energy in terms of the electron density[15]. Local Density Approximation (LDA) takes the effective exchange-correlation potential felt by an electron at any point in space to be a function of only the electron density at that point; it is derived from the exact solution of the uniform electron gas, which is a reasonable model for a free-electron metal[15]. In 1964, Kohn and Hohenberg set forth a description of DFT analysis and they proposed that the electron density is the fundamental variable[15]. Thus, in the Kohn-Sham approach, DFT analysis finds an effective potential. By approximating the exchange-correlation potential as a functional of density, better energy gap calculations can be obtained[16]. DFT anaylsis, shown in Figure 11, approximates effective potential energy by solving for an electron distribution that minimizes the total energy. This effective potential is used in the numerical integration of Schrödinger s Equation for the electron system. In DFT, the nuclei are assumed to be fixed in time while a calculation of the distribution of electrons is made[14]. In fixed nuclear positions, the ground state energy is the lowest possible energy[15]. Iterating this process in time permits the determination of the interactions among the ions in the system and results in a convergence of the total energy for the optimal electron density[17]. Figure 11: DFT analysis essentially uses a specified geometry to compute the location of charges within the system so that the electron density is found. This goes into solving Schrödinger s equation which is needed to solve the energy functional for the system. This iterative process is repeated until the minimum energy functional has been attained. DFT analysis is computationally intensive. Failures and limitations in DFT arise from the deficiencies in the approximation of the exchange and correlation functionals[15]. Therefore, using available computing resources, we are limited to considerations of small-diameter CNTs, d < 0.814nm. Results are validated for band gaps with existing experimental and numerical work. Faster, more accurate calculations are desired. DFT analysis is one of the most powerful computational tools in theoretical physics; it offers the best trade off between accuracy and feasibility. LDA overestimates binding energies by twenty percent, while DFT calculations are accurate to within one percent[15].

10 10 IX Results IX.1 Phase Diagram So far, several CNTs have been successfully modeled and an analysis of their respective band gaps has determined whether they are conducting or semiconducting. I modeled the Fermi-Dirac distribution for each and compared them to theoretical data to examine the band gaps as in Figures 12 and 13. Figure 13: The Fermi distribution for a conducting CNT is modeled. The black curve represents the theoretical model using the Fermi-Dirac distribution equation. The black dots represent allowed energies that were calculated for the electrons in the simulated nanotube. In the CNT defined by (4,0), I found that a virtually no band gap existed between the valence and conduction bands. found and if a significant energy gap exists around the Fermi energy, then the CNT is classified as a semiconductor. Figure 12: The Fermi distribution for a semiconducting CNT is modeled. The black curve represents the theoretical model using the Fermi-Dirac distribution equation. The black dots represent oribital occupancies that were calculated for the electrons in the simulated nanotube. In the CNT defined by (4,3), I found that a band gap energy of ev existed between the valence and conduction bands. The Fermi-Dirac distribution represents the probability of an energy level to be occupied by an electron. After modeling the band gaps for a variety of CNT geometries, I constructed the phase diagram, shown in Figure 13, to illustrate which CNTs are conducting and which are semiconducting. This phase diagram can then be compared to the theoretical predictions based on the graphene model. Smaller geometry CNTs appear not to follow the n m = 3i (where i =0,1,2,...) is conducting rule that the graphene model suggests. In the graphene model, the rule is based on tight-binding calculations of nearest neighbor electron pairs in a flat 2-D geometry and is only appropriate for larger diameter CNTs. f(e) = e (ɛ ɛ F )/kt (19) IX.2 Band Gap vs. Diameter In Equation 19, ɛ F is the Fermi energy; it is the energy between filled and empty energy bands at absolute zero. From this, allowed energies are For the chiral semiconductors, the size of the band gap is calculated and plotted against the diameter

11 11 In smaller geometries, large curvature affects prevent the CNTs from following the patterns that are appropriate for larger geometries. The carbon atoms in the smaller diameter nanotubes are much closer together than in the larger systems, so each atom interacts with every other one. In larger systems, the carbon atoms only interact with their nearest neighbors and the larger system is considered to be more flat like that of the graphene model. IX.3 Applied Tensile Stress Figure 14: This phase diagram of the electronic properties of smaller diameter CNTs shows that smaller systems do not obey the if the difference between the indices is a multiple of three, then the CNT is conducting rule. in a natural log plot in Figure 14 for each CNT to see if a correlation existed. For small geometry CNTs, no correlation exists. It has been experimentally determined that for larger geometries, the band gap is proportional to 1/d [1]. Figure 16: Applying tensile stress to a CNT can either increase or decrease the band gap energy. By applying a pressure uniaxially, the physical structure of the CNT is altered which dramatically changes the electronic properties as well. Applying tensile stress, as in Figure 15, to a conducting or a semiconducting CNT can alter the band gap. As seen in figure 6, tensile stress applied to a conducting CNT can increase a band gap. For the (3,1) CNT, the band gap increased from 0 ev to about 0.06 ev as seen in Figures 16, 17, and 18. Figures 19 and 20 show that tensile stress applied to a semiconducting CNT can decrease the band gap. The band gap of the (3,2) CNT decreased by about fifty percent. Figure 15: The band gap versus diameter shows that no correlation exists between the two properties of the smaller system CNTs. The rules for the larger systems are inappropriate for the smaller system because of large curvature effects. Figure 17: The band gap energy as a function of applied tensile stress for a conducting CNT shows that the band gap for a conductor can increase and become more like that of a semiconductor. Stress affects the bond length of the carbon atoms within the CNT, which also affects the angles between the bonds. This in turn, changes the band

12 12 the tube (approximately 0.20), and α is the chiral angle[2]. Also, the rate of change of the band gap can be used to help determine the difference in the band gap for a CNT that undergoes axial stress. E g = E g0 + de g dɛ ɛ Figure 18: This is an inset of the band gap energy as a function of applied tensile stress for a conducting CNT shown in Figure 17. Using the above equations and the index values for the (3,1) CNT, the change in band gap energy with respect to stress should theoretically be around ev/gpa. Figure 19: This is an inset of the band gap energy as a function of applied tensile stress for a conducting CNT shown in Figure 18, in an attempt to narrow in on the region where the band gap appears to jump radically. gap energies of CNTs. Previous research predicts that the rate of change of the band gap energy with respect to stress is dependent on the chiral angle of the CNT[18]. de g dɛ = sin(2p + 1)3γ(1 + ν) cos(3α) (20) In the equation above, the sign is -1, 0, or 1 depending on the value of p, which is an integer that satisfies n m = 3q + p where q is also an integer, and γ is the tight-binding overlap integral (approximately 2.6 ev), ν is the Poisson ratio of Figure 20: The band gap energy as a function of applied tensile stress for a semiconducting CNT shows that the band gap for a conductor can decrease and become more like that of a conductor. X Discussion Band structures of materials are important because they are directly related to many macroscopic properties of the material. In CNTs, the band gaps determine the conductivity and applying a tensile stress can alter it. Being able to tune the band gap to a desired energy will make CNTs more attractive for a variety of different applications in technology. However, the band gap energies of smaller diameter CNTs do not follow the model predicted from

13 13 orbitals in the carbon atoms is 1 nm which is the typical size of a CNT diameter. For the smaller diameter CNTs that I have modeled, the size of the p orbitals is larger than the diameters; thus, every carbon atom is a nearest neighbor for every other carbon atom around the circumference of the CNT. Figure 21: This is an inset of the band gap energy as a function of applied tensile stress for a semiconducting CNT shown in Figure 20, in an attempt to narrow in on the region where the band gap appears to jump radically. In measuring band gap energies as a function of applied tensile stress, I found that the band gap increased in conductors and decreased in semiconductors. In these measurements, there are no error bars on the graphs because the errors that exist are instrumental; the DFT code is well validated and it is very accurate. Also, the code will yield the same results for the same set of parameters. theory and the graphene phase diagram. The predicted correlation where energy is inversely proportional to the diameter in larger systems does not work for the smaller systems. The large curvature effects play a major role in affecting the electronic properties of smaller diameter CNTs. XI Future Directions In smaller systems, the carbon atoms interact with the atoms across the diameter in addition to their nearest neighbors due to tight binding. The graphene model is appropriate for larger diameter CNTs since the carbon atoms only interact with their nearest neighbors. The de Broglie wavelength can give the approximate size of the p orbitals of the carbon atoms. λ = h p By calculating the de Broglie wavelength as in Equation 20, I can compare the size of the p orbitals in carbon atoms to the diameters of the CNTs I have been able to model. λ = h 2mE (21) Using Equation 20, the approximate size for the p I would like to calculate band structures for a set of larger CNTs and construct a phase diagram to compare to the theoretical graphene model. Also, I would like to find a correlation between the band gap energies and other parameters for the smaller geometry CNTs since there is not a clear correlation with their diameters. With the larger diameter band gaps, I could also look to see if the data matches the predicted relationship where band gap energy is inversely proportional to diameter. Also, I will need to calculate stressstrain curves and compare them with experimental and theoretical data. The next step in research will be to dope these CNTs with an impurity to investigate its effects on the electronic properties. References [1] M. and G. Dresselhaus, Carbon Nanotubes (2006). Retrieved Nov. 3, 2008, from

14 14 [2] M.J. OConnell, Carbon Nanotubes: Properties and Applications. Taylor and Francis Group, Boca Raton, FL, [3] C.P. Poole and F.J. Owens, Introduction to Nanotechnology. Wiley, [4] Carbon Nanotubes Deliver Payload: Cancer Cell-Killing Heat (2005). Retrieved Feb. 20, 2008, from newscenter/nanotechnews a.asp [16] Aulbur, Basics of Density Functional Theory. Retrieved Oct. 26, 2007, from aulbur/dft/bdft.html [17] T. Pang, An Introduction to Computational Physics, Cambridge University Press, Cambridge, [18] E.D. Minot et al, Tuning carbon nanotube bandgaps with strain (2004). [5] R. Saito and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes. Imperial College Press, [6] Carbon Nanotubes. Retrieved Nov. 3, 2008, from Carbonnanotube [7] I. Cabria, J.W. Mintmire, and C.T. White, Mat. Res. Soc. Symp. Proc. 772 (2003). [8] M. Meyyappan, Carbon Nanotubes: Science and Applications. CRC Press, [9] B.K. Tanner, Introduction to the Physics of Electrons in Solids. Cambridge University Press, [10] C. Kittel, Introduction to Solid State Physics, John Wiley and Sons, Inc., New York, [11] D.J. Griffiths, Introduction to Quantum Mechanics. Pearson Education, Upper Saddle River, NJ, [12] B.J. Zeghbroeck, Energy Bands Retrieved Apr. 16, 2007, from bart/book/eband3.htm [13] Retrieved Nov. 3, 2007, from LED/band.htm [14] D. Sanchez-Portal et al, Mat. Sci. (1998). [15] M. Head-Gordon and E. Artacho, Physics Today (2008).