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A Deformation Induced Quantum Dot Woodsworth, Daniel James 2008-05-31

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A Deformation Induced Quantum DotInvestigating the Electronic Properties of a CarbonNanotube CrossbyDaniel James WoodsworthA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFBachelor of Science (Honours)inThe Faculty of Science(Physics)The University Of British Columbia(Vancouver)April, 2008c Daniel James Woodsworth 2008AbstractDue to their extraordinary electronic properties, Quantum Dots (QDs) are potentially very usefulnanoscale devices and research tools. As their electrons are con ned in all three dimensions, theenergy spectra of QDs is descrete, similar to atoms and molecules. Because the gaps betweenthese energy levels is inversely related to the size of the QD, very small QDs are desirable.Carbon nanotubes have long been touted as fundamental units of nanotechnology, due totheir structural, optical and electronic properties, many of which are a result of the con nementof electrons in the trans-axial plane of the nanotube. It is known that their band gap structureis altered under deformation of their cross section.It is proposed that one way to fabricate a very small quantum dot is by con ning electronsin the nanotube so that they may not freely move along its length. A structure to produce thiscon nement has been described elsewhere, namely the carbon nanotube cross, consisting of twocarbon nanotubes, with the the one draped over the other at ninety degrees. It is thought thatthis structure will induce local physical deformations in the nanotube, resulting in local changesin electronic structure of the top nanotube at the junction of the cross. These band gap shiftsmay cause metal-semiconductor transitions, resulting in tunnel barriers that axially the con neelectrons in the nanotube. This thesis investigates the possibility that the carbon nanotube crossmay exhibit QD behavior at the junction of the cross, due to these local band gap shifts.A device for carbon nanotube growth, using Chemical Vapor Deposition, has been designed,and may be built using microfabrication techniques. This device consists of electrodes (for elec-trical measurements of the nanotubes) and catalyst regions (to initiate nanotube growth), litho-graphically patterned in a con guration that promotes carbon nanotube formation. Unfortu-nately, due to fabrication issues, this e ort is a work in progress, and these devices have not yetbeen constructed. However, an experimental methodolgy has been developed, which provides aframework for eventually building a carbon nanotube cross, and investigating the possibility ofQD behavior at the junction of the cross.This structure has also been investigated computationally. Molecular dynamics simulationswere used to obtain equilibrium geometries of the carbon nanotube cross, and it was foundthat their are many di erent meta stable states, corresponding to di erent types of nanotube,and di erent physical arrangements of these nanotubes. The electronic structure of the carbonnanotube cross was calculated using the density functional theory. Band gap energies similar toexperimental values were obtained. A one-to-one spatial correlation between deformation andband gap and conduction band shifts were observed in the top carbon nanotube of the nanotubecross. Small tunnel barriers, inferred from both the calculated band gap and LUMO energies, areobserved, and could well be su cient to con ne electrons along the axis of the nanotube.The results described in this thesis, while not de nitive, certainly indicate that a QD probablywould form at the junction of a carbon nanotube cross, and that further investigation, bothexperimental and computational, is warranted.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Carbon Nanotube Cross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Investigation of the Carbon Nanotube Cross . . . . . . . . . . . . . . . . . . . . . 52 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Energy States of Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Physical Structure of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . 82.4 Band Structure of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 E ects of Deformation on the Band Structure of Carbon Nanotubes . . . . . . . . 103 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Quantum Chemistry Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 Hartree-Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 The Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . 144 Fabricating the Carbon Nanotube Cross . . . . . . . . . . . . . . . . . . . . . . . 164.1 Design and Fabrication of the Chip . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.1 Microfabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Carbon Nanotube Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Measuring the Current-Voltage Characteristics of the CNT Cross . . . . . . . . . 22iiiTable of Contents5 Modeling the Carbon Nanotube Cross . . . . . . . . . . . . . . . . . . . . . . . . 245.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38ivList of Figures1.1 The Carbon Nanotube Cross. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Physical and Electronic Structure of Graphene and CNTs. . . . . . . . . . . . . . . 94.1 Device and Chip Design Schematics. . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Microfabrication Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Current-Voltage (I-V) Measurements of Quantum Dots. . . . . . . . . . . . . . . . 215.1 Comparison of Gaussian Pro les of CNT Cross . . . . . . . . . . . . . . . . . . . . 265.2 MD Relaxation of Carbon Nanotube Crosses . . . . . . . . . . . . . . . . . . . . . 285.3 Investigating the Inter-Molecular Interactions in NanoHive. . . . . . . . . . . . . . 295.4 Comparison of GAUSSIAN methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5 Electronic Structure of CNT Crosses: Band Gap Energy. . . . . . . . . . . . . . . . 325.6 Electronic Structure of CNT Crosses: LUMO Energy. . . . . . . . . . . . . . . . . 34vAcknowledgementsI would like to thank my supervisor, Dr. Alireza Nojeh, for his extensive and continued supportand patience throughout this project. Dr. Nojeh has been extremely generous with his timeand knowledge, for which I am very grateful. Under his guidance I have gained a far deeperunderstanding of this  eld than I otherwise would have. He has my sincere appreciation for hisinvolvement in all aspects of this work.viChapter 1IntroductionAs the semiconductor industry continually strives to construct ever smaller structures for micro-electronics applications, it has provided the scienti c community with a wealth of nanofabriactiontechniques, as well as the ability to shape matter on the atomic scale. This capacity has beenone of the main factors in the explosion of research and development at the nanoscale, which hasalready had a deep impact on both fundamental science, as well commercial technology. Oneof the objects which is perhaps most representative of this union of technological promise andscienti c insight is the so called arti cial atom, or Quantum Dot.A novel structure has been proposed[28], consisting of a cross of Carbon Nanotubes, with onenanotube draped over the other at ninety degree angles, as in Fig. 1.1(a). This thesis describesthe investigation of this structure, speci cally examining its potential to exhibit Quantum Dotbehavior.1.1 Quantum DotsStarting around twenty years ago semiconductor fabrication technology had progressed su cientlysuch that it was possible to construct objects which could con ne countable numbers of electronsto three dimensional regions small enough that their discrete energy spectrum could be resolved[2,17]. These so-called Quantum Dots (QDs) are often described as zero-dimensional structures,because, as opposed to a bulk three dimensional solid, the electrons are con ned in all threedimensions[8]. Because of this con nement, instead of the virtually continuous energy levelsof electrons in a bulk solid, the electrons in a QD, having no translational elements in theirHamiltonian, are con ned to a  nite set of quantized states[8]. In this way these QDs are verysimilar to natural atoms, except that the energy spectra of QDs, which de nes most of theirimportant physical properties, are a function of size, composition and morphology[5]. As theseproperties may be controllably adjusted, it follows that it is possible to engineer nanoscale,`arti cial atoms', which would have a myriad of possible applications.To the researcher, QDs present a controlled environment in which to study all manner ofquantum and atomic phenomena. Examples of the former include time reversal symmetry break-ing under applied magnetic  eld[1], spin and coherence e ects[18] and electron transport and theKondo e ect[32]. The arti cial atom analogy is indeed apt, as QDs provide the opportunity tosystematically study atomic phenomena and electron-electron interactions[22], as well as mimicexperiments in atomic physics with macroscopically tunable QDs[23]. Furthermore, it has beenargued that it may even be possible to see novel physical e ects in QDs that are not found innature[2, 16].The  elds in which QDs have probably made the furthest progress towards widespread tech-nological application are biological analysis[15] and medicine[31]. In both cases, QDs are used1Chapter 1. Introductionas a means of in vivo  uoroscopic imaging: examining cellular mechanisms in the former anddiagnostic imaging in the latter. Here the tunable energy spectrum of QDs is exploited to createparticles that will absorb and emit arbitrary wavelengths of light, enabling extremely accurateimaging.QDs provide one of the most promising routes to a solid state quantum computing platform,in which multiple entangled QDs are combined to form a single qubit[9]. A wide range of op-toelectronic applications of QDs are also envisioned, including lasers and optical detectors[6],as well as nonlinear optics, telecommunications and photovoltaics[8]. Finally, there are ongoingattempts to assemble groups of QDs, `arti cial molecules', and arrays of QDs, `arti cial solids',raising the possibility of fully engineered materials and macro-molecules[2, 5].1.2 Carbon NanotubesAlthough they had been observed, and subsequently misclassi ed, as early as the 1970s[35],Carbon Nanotubes (CNTs) were  rst reported by Iijima in 1991[13], and can divided into twomain types: Multi Walled Carbon Nanotubes and Single Walled Carbon Nanotubes. A SingleWalled Carbon Nanotube, an example of which is shown in Fig. 1.1(d), may be thought of asa hollow cylinder made of a hexagonal array of carbon atoms, one atomic layer thick. A MultiWalled Carbon Nanotube is simply many Single Walled Nanotubes, arranged in concentric shells.I will be dealing exclusively with Single Walled Carbon Nanotubes in this thesis, and so willsimply refer to them as Carbon Nanotubes, or nanotubes for brevity.Carbon Nanotubes have garnered a great deal of attention and excitement, both from thepublic, as well as industry and academia. This is in large part due to their extraordinary struc-tural, electronic, thermal and optical properties[3]. Typical diameters for CNTs are on the orderof nanometers, while their length is on the order of hundreds, even thousands, of microns, givingthem very large aspect ratios, which is in many cases responsible for their unique properties[3].Particularly important for this thesis is that CNTs can be either semi-conducting or metallic[25].Due to these characteristics, a wide array of applications incorporating CNTs has been en-visioned, including: \electronic devices and interconnects,  eld emission devices, electrochemicaldevices, such as supercapacitors and batteries, nanoscale sensors, electromechanical actuators,separation membranes,  lled polymer composites, and drug-delivery systems"[10].1.3 The Carbon Nanotube CrossSince they were  rst studied, all manner of QDs have been reported, possesing a wide variety ofcompositions, structures and properties, to the point that there is currently some ambiguity inthe literature as to what exactly a QD is[27]. One unifying theme is the crucial notion of threedimensional con nement of electrons, and how this is achieved is always the central challenge whentrying to design or manufacture a QD. Original means of con nement included lithographicallypatterning electrodes on a silicon wafer, Fig. 1.1(c), and con ning a two dimensional electrongas to a small point in the plane using electric  elds. More recently there has been much workdone on self assembled QDs, which are usually small, semi-hemispherical objects made of somesemiconducting material (GaAs, InGaAs for example)[8]. Here con nement is generally due tostrain induced band gap changes in the semiconductor.2Chapter 1. Introduction(a) Carbon Nanotube Cross. (b) Discretized EnergyStates[27].(c) Two Dimensional Electron Gas QD. (d) Single Walled Carbon Nanotube.Figure 1.1: Motivation for the Carbon Nanotube Cross. The method proposed in this work aimsto exploit the inherently small nature of carbon nanotubes (d), by investigating a cross structureof two CNTs (a). As a QD at the junction of a CNT cross would potentially have a much smallercharacteristic size than traditional QDs (c), it would have much larger energy level spacing (b).Images (c) and (d) are taken from www.unibas.ch and www.almaden.ibm.com respectively.Many of these methods rely heavily on microfabrication techniques, and despite their unde-niable success, as science and technology strives to work at ever smaller scales, and researcherstry to achieve con nement in ever smaller volumes, they try to thread a needle through a hole ofever decreasing size. Inevitably this e ort is subject to a law of diminishing returns: researchersmust spend more time and resources in an e ort to achieve ever smaller feature sizes on siliconwafers. While they continue to succeed, it is likely that eventually this progress will slow to apainful, perhaps prohibitive rate.Alternatively, rather than forcing macroscale technology into the nanoscale, it may be possibleto exploit the unique features of this regime, and adapt existing mechanisms, structures and3Chapter 1. Introductionobjects to the purposes of research and technology at the nanoscale. This is the philosophyadopted throughout this thesis, and is manifested in the basic structure of this project: thecarbon nanotube cross (Fig. 1.1(a)).Consider attempting to con ne electrons starting with a macroscopic piece of a three dimen-sional material, perhaps a chunk of silicon. If one could manage to machine a cube with lengthof perhaps, a nanometer, then con nement would have been achieved. Unfortunately this wouldbe a very di cult prospect, and would involve very re ned versions of those techniques describedabove. A somewhat easier route, is if one was given a material in which electrons were alreadycon ned in one direction, and all that remained was to achieve con nement in the remaining twodimensions. This is indeed how many QDs are fabricated, but again still relies on microfabrica-tion techniques, and results in relatively bulky QDs, as seen in Fig. 1.1(c). Finally, imagine beinggiven an object whose electrons were con ned to a single axis. All that would remain would beto somehow stop a certain number of these electrons from moving along this one axis.Perhaps somewhat surprisingly, these last objects do indeed exist. Both carbon nanotubesand silicon nanowires, at least to a very good approximation, are one dimensional objects, withtheir electrons e ectively con ned to a line. However, the problem remains to con ne theseelectrons along this line. There is of course the possibility of manufacturing some kind of moldinto which one could press a nanotube, and so bend it or shear it, which, as will be described,would probably con ne the electrons. Alternatively one could attempt to directly interact with it,via Scanning Tunneling Microscopy (STM) or Atomic Force Microscopy (AFM) probes. Howeverall these methods have the same limitations and looming dead end as described above.One potential way to achieve con nement is to grow two nanotubes in a crossed formation,the one draped across the other, as is shown in Fig. 1.1(a). The resulting deformation causes thenecessary change in electronic structure to con ne the electrons from moving freely along the axisof the nanotube. In this way a QD can be constructed in a manner that harnesses the featuresof the nanoscale world, rather than forces an outside pattern onto it. Carbon nanotubes maysimply be grown, and provided appropriate stimulus, it should be possible to grow them in thiscrossed formation[28].At this point, it is reasonable to wonder why it is necessary to avoid using conventionalmicrofabrication techniques. Certainly some esthetic desire to exploit the uniqueness of the nanoworld is not enough. However, there is one fundamental reason for attempting to do this, andthat is size. As will be shown later, the energy gap ( E in Fig. 1.1(b)) between two energystates in a QD's spectrum is inversely related to its characteristic size. Thus, given a relativelylarge QD, with a size in the hundreds of nanometers, the energy states will have a very small,or  ne separation. In many cases this separation will be much less than the thermal energy ofelectrons1, and so these QDs must be operated at very cold temperatures, or else the discretenature of their energy spectra is completely washed out. On the other hand, small QDs, withsizes on the order of a nanometer, should have energy spectra that would be measurable at roomtemperatures.Aside from the pure convenience of room temperature operation, for many of the eventualapplications envisioned for QDs, this would be absolutely essential. This small characteristic sizerequirement is one of the fundamental reasons for studying the carbon nanotube cross structure.If one were to try and construct a QD on the order of 1 nanometer by con ning a two dimensional1At room temperature kBT   26meV .4Chapter 1. Introductionelectron gas using electrodes, one would have to pattern these electrodes to around one nanometer,which is very di cult: notice the size of the electrodes in Fig. 1.1(c). However, the diameter of(single walled) nanotubes are usually   0:8   1:5 nm. A QD formed at the junction of a carbonnanotube cross would naturally have a characteristic size of around one nanometer, and so itpresents a unique way of obtaining an extremely small QD, which could possibly be operable atroom temperatures.This structure displays several other desirable features, for the most part in terms of potentialapplication. First, there is the synergistic combination of QDs and CNTs: two objects that aretouted as some of the fundamental building blocks of nano-electronics and nano-electromechanicalsystems (NEMS). For example it has already been proposed that CNTs would make excellent eld emitters, while QDs are potentially very good single electron transistors. Combined in asingle package, and therefore avoiding the enormous di culty of interfacing at the nanoscale, onemight have a means of a nanoscale  eld emitter, whose emission is regulated to an extraordinarily ne degree by the QD.As most QDs are fabricated from semiconductor materials, it is often necessary to coat themwith organic materials for biological compatibility, and even with these additional measures, tox-icity is a concern[14]. Furthermore, for cellular imaging applications, the necessary size of thebiologically suitable QDs that have been produced is such that the QDs cannot be introducedto the cells via the natural transmembrane di usion processes. Rather, QDs must be introducedusing far more invasive approaches such as microinjection, which has obvious detrimental conse-quences for in vivo imaging[14]. Due to its carbon composition and extremely small size, it ispossible that QDs in a nanotube cross might remedy these problems.1.4 Investigation of the Carbon Nanotube CrossRecognizing its potential, this thesis describes the ongoing study of the carbon nanotube cross.The ultimate goal of this work is to achieve three dimensional electron con nement in a carbonnanotube, speci cally at the junction of the cross, and investigate the properties of this QD.Following previous experiments by my supervisor, Dr. Alireza Nojeh, [28], two main avenuesof study were undertaken. First, a methodology and design for growing carbon nanotubes ina crossed formation was pursued, consisting of growing CNT crosses, using Chemical VaporDeposition (CVD), on lithographically patterned chips and then performing Current-Voltagemeasurements on the these crosses. Second, in order to gain more insight into the physicalprocesses involved in the CNT cross, it was modeled by generating relaxed structures usingMolecular Dynamics (MD), whose electronic structure were then calculated using ComputationalQuantum Chemistry (QC) algorithms.Through these two complimentary approaches, it is hoped that this work will lay the foun-dations for a better understanding of a structure so rich in scienti c interest and technologicalutility: the Quantum Dot in a Carbon Nanotube Cross.5Chapter 2Theory2.1 Energy States of Quantum DotsMuch of the interest in QDs is due to their controllable, discretized, electronic structure, and therehas been an enormous amount of theoretical and experimental study of the energy states in QDs.Although the details of these works are quite complicated, I would like to at least heuristicallyshow the dependence of a QD's energy level spacing on its size, in order to give some physicalgrounding to the claim put forward in the introduction, namely that a very small QD should haveeasily resolvable discrete energy levels at higher temperatures.The Schr odinger equation for the wave function, j (r)i, of a particle in a time independentpotential, V (r), is:Hj (r)i = Ej (r)i (2.1)H =    h22mr2 + V (r) (2.2)For our purposes here, the important feature of a QD, namely the three dimensional con- nement of the electrons, can be modeled using the potential, V. Two suitable potentials are:i) a three dimensional box of dimensions Lx;Ly;Lz, where V = 0 inside the box, and is in niteelsewhere and ii) a sphere of radius Ro where V = 0 inside the sphere and is in nite elsewhere.Substituting these potentials into the Schr odinger equation and performing the straightforwardseparation of variables, and invoking appropriate boundary conditions gives:Enx;ny;nz =  2 h22mn2x + n2y + n2zL2x + L2y + L2z ; ni = 1;2;3;::: (2.3)Enl =  h22mk2nlR2o (2.4)where Enx;ny;nz are the energy states of the box potential and Enl are those of the sphericalpotential, ni = nx;ny;nz and knl is the nth zero of the Bessel function of order l. The importantresult, con rmed by experiment, is that the energy states of a QD are inversely proportional to(in this case) the square of its length. Therefore to achieve a large separation of energy states,and create a viable room temperature QD, its characteristic size should be very small.2.2 Band StructureAs most of the theory described here lies within the realm of condensed matter physics, andrelies heavily on the concepts of band structure and band gap energy (Eg), I will brie y review6Chapter 2. Theorythese concepts here. However, this is a very large  eld, and for more complete treatments see,for example, Kittel[19], whom I have followed closely here.As a  rst approximation, we may model a solid crystal as a collection of electrons con nedto a cube of side length L. The valence electrons of each atom are completely delocalized, andare free to move throughout the solid. As we are considering a bulk crystal, we assume periodicboundary conditions, so that the solid `appears' the same from all points throughout the solid(neglecting edges). We further assume that the individual electrons are non-interacting. Recallthat the state of an electron is given by the electron's wave function  (r), where  (r) solvesthe Schr odinger equation (Eq. 2.1). As the electrons are free to move, the potential, V (r), isde ned to be zero inside the cube. Through separation of variables we arrive at solutions that aretravelling plane waves:   = eik r with E =  h2k22m . Here k is the wave vector and has componentski = 2n L for i = x;y;z and n is any integer.While this so-called free electron model provides a good understanding of many properties ofmetals, such as thermal and electrical conductivity, it fails to take into account the periodicityof the crystal lattice, and the associated potentials of each atom, and so fails to give a gooddescription of most optical and electronic properties[19]. To account for this periodicity, weassume that electrons in the above model are only weakly perturbed by the periodic potentialof the atoms. Thus the travelling wave solutions now `see' periodic crystal planes, as opposedto the uniform zero potential of the free electron model. The travelling waves undergo Braggre ection o  these planes, and this process is described by Bragg's law, 2dSin  = n , where d isthe lattice spacing of the crystal and   = 2 k is the wavelength of the travelling wave. It followsthat waves that have k = n d , regardless of the direction they are travelling in, are completelyre ected, according to the Bragg Law, resulting in a standing wave for this energy eigen state.The two possibilities for the form of this standing wave are[19]: (+) = ei x=d + e i x=d = 2cos( x=d) (2.5) ( ) = ei x=d   e i x=d = 2icos( x=d) (2.6)These standing waves produce localized regions of elevated electron density (as can be seenby taking the modulus squared of either wave function), and the important thing to note is thateach wave produces a di erent distribution. In particular j (+)j2 has high charge density at thelocation of the ion cores, and due to the attraction between the negative electrons and positiveion cores, electrons with this wave function have lower potential energy. Conversely, j ( )j2 hashigh charge density away from the ion cores: a state of higher potential energy. This di erence inpotential energies is essentially the origin of the band gap in crystal solids, because, as we havejust seen, electrons with wave vectors that satisfy k = n d are con ned to have wave functionsof either  (+) or  ( ). If we take the expectation values of the energies for  (+) or  ( ), we nd that they are, in general, separated by some energy Eg, which is to be expected, as the twowave functions correspond to electron distributions with di erent potential energies. Therefore,there is a so-called band gap in the set of allowed energy states for electrons. If we continuallyadd energy to an electron, it will be excited to higher and higher states, until it meets the Braggcondition, and then it must have su cient energy, namely Eg, in order to reach the next band ofallowed states.7Chapter 2. TheoryThe set of states with lower potential energy, namely those of  (+), is called the valence band,while the set with higher potential energy,  ( ), is called the conduction band, and the energydi erence between the highest energy state of the valence band and the lowest energy state of theconduction band is the band gap energy. In reality, the band structure of any material is extremelycomplicated, with many di erent allowed bands and gaps, and is usually measured using someform of di raction technique (neutron, electron or x-ray). However, the important point is thatthe relative positioning of the valence and conduction bands, and the size of Eg generally de nesthe nature of a crystalline solid. Metals have zero band gap energy and overlapping valenceand conduction bands, so there are many electrons in the conduction band available to moveabout freely, as is characteristic of metals, which are excellent conductors. Insulators are just theopposite, with a very large band gap, meaning that a large amount of energy must be supplied toexcite an electron into the conduction band, which results in relatively few free charge carriers,and low conductivity. Semi-conductors are in between these two cases, with a moderate bandgap. Thus by supplying reasonable amounts of energy, electrons can be promoted or preventedfrom reaching the conduction band, giving semiconductors their extremely useful controllable ordiode-like characteristics.This work essentially exploits the fact that deformations and strains in the physical structureof carbon nanotubes cause spatially correlated local changes in the electronic band structurecausing local transitions from semi-conducting to metallic nanotubes, or vice versa. I will  rstgive a brief overview of the relevant physical and electronic properties of CNTs, before describinghow the carbon nanotube cross produces these shifts in band gap energy, thereby con ning theelectrons.2.3 Physical Structure of Carbon NanotubesCarbon has several allotropes, one of which is graphite, a common structure found in lead pencilsand lubricants. Graphite consists of planar sheets of carbon atoms stacked upon each other, whereeach sheet is only one atomic layer thick and is known as graphene. These graphene sheets consistexclusively of carbon atoms, sp2 bonded in a hexagonal network. A CNT is essentially a sheet ofgraphene rolled into a tube, as depicted in Fig. 2.1(a). As such many of the properties of CNTscan be understood in the context of the lattice structure of graphene, which can be described2by making a few de nitions with reference to Fig. 2.1(a). The chirality vector, C = na1 + ma2,connects two crystallographically equivalent sites in the two dimensional graphene sheet. Heren and m are integers and a1 and a2 are the graphene lattice vectors, which de ne a unit cell inthe hexagonal graphene structure. In other words, for any location r in the graphene sheet, anyother location r0 = r+C, will be completely equivalent for all m;n. This set of points de nes thegraphene lattice[19]. The translation vector, T, is de ned to be orthogonal to C, and is extendeduntil the  rst crystallographically equivalent site is encountered.The area spanned by C and T (shaded gray in Fig. 2.1(a)) de nes the unit cell of the CNTwhen it is constructed by rolling up the graphene sheet. When the sheet of graphene is rolledup the lattice points de ned by the two ends of the chirality vector coincide, resulting in thecontinuous hexagonal structure of CNTs, seen in Fig 2.1(a). Note that T points along the axis2The discussion here closely follows that of Leiber[29].8Chapter 2. Theory(a) Hexagonal structure of graphene. Takenfrom [29].(b) Band structure of graphene and CNTs.Taken from [26].Figure 2.1: Physical and Electronic Structure of Graphene and CNTs. (a) The grey regionspanned by C and T de nes the unit cell of a carbon nanotube (CNT). The free parameters inthis unit cell are (n;m), which de ne the chirality and diameter of the CNT. (b) The valenceand conduction bands meet at K1 and K2. The quantization of kc gives a discrete set of allowedstates (horizontal lines at left and parabolas at right). The 1-D band structure of CNTs are slicesof the 2-D band structure of graphene at points de ned by the quantization of kc.of the CNT and C is its circumference.As a CNT is simply the unit cell repeated along T, and the unit cell is purely a function ofC, it follows that a CNT is completely de ned by the integer pair (n;m). On the other hand,upon considering the honeycombed structure in Fig. 2.1(a), it is clear that there are only twophysical degrees of freedom of the rolled up CNT: the diameter of the CNT (which is just C  )and the chirality of the CNT. To understand the latter, consider the two dashed lines in Fig2.1(a), each of which corresponds to a di erent chirality vector. All that is meant by chirality isthat for the (n;n) case, C is parallel to one side of the hexagons (which would appear `twisted'about the CNT axis), while for the (n;0) case, C runs straight across the hexagons (which wouldappear `straight'). It is indeed con rmed, by both theoretical and experimental studies[29], thatthe two physical degrees of freedom of a CNT, chirality and diameter, are completely de ned bythe integer pair.2.4 Band Structure of Carbon NanotubesAs with the physical properties of CNTs, their electronic structure can also be understood bystarting with the electronic structure of graphene, and then accounting for the radial con nementof electrons in the CNT caused by the rolling up of the graphene sheet. The explicit calculation9Chapter 2. Theoryof the band structure of graphene is far beyond the scope of this paper, and here I will onlyqualitatively discuss some of the structure's relevant characteristics. Using a tight binding model(see, for example, [33]) it is found that graphene is a zero-band gap semiconductor in which thevalence and conduction bands overlap at two points in k-space3, K1 and K2 in Fig. 2.1(b)[26].The dispersion function around these points is linear, which implies a conical band gap energysurface in k-space as in Fig 2.1(b).When a  nite piece of graphene is rolled up to form a CNT, there must be a perfect overlapof the two lattice points de ned by the ends of C, which gives the periodic boundary condition:C   k = 2 q q = 1;2;3 ::: (2.7)where k is the electron wave vector. This gives kcD = 2 q q = 1;2;3 ::: (2.8)where kc is the circumferential component of the electron wave vector and D is the diameter ofthe CNT. As the radial con nement of the electrons has quantized kc, k is also quantized, whichmeans that, in general, there will not necessarily be an allowed state with kj = Ki; i = 1;2and j denotes the discrete nature of kj due to this new quantization. Rather the closest statesto Ki will be kj = Ki    kc. In other words the quantization of k only allows certain planesbisecting the cones at kc = kc;j, as is shown in Fig. 2.1(b). Note that the cones are the bandstructure of graphene, and so these bisected curves de ne the band structure of the associatedCNT, with the upper and lower curves de ning the conduction and valence bands respectively.Thus unless there is some j such that kj = Ki there will be a band gap of Eg =  h F  kc[26],between the conduction and valence bands of the CNT. The size of  kc is a function of chiralityand any physical perturbations, such as curvature, axial strain and twist[26].In summary CNTs can exhibit metallic or semiconducting behavior depending on the quan-tization of k, which is dependent on C, which is in turn a function (n;m), and it is indeed foundthat a CNT is metallic when n = m, has a small gap when n m = 3i and is truly semiconductingwhen n   m 6= 3i, where i is an integer[3].2.5 E ects of Deformation on the Band Structure of CarbonNanotubesCNTs are often termed `quantum wires' or one-dimensional structures. These terms refer tothe fact that any electrons in the conduction band of a CNT are con ned in two dimensions,namely the trans-axial plane, and are only free to move in one single dimension: along the axisof the CNT. In this section I brie y show how deforming the CNTs from their relaxed state cancause signi cant shifts in their electronic structure, which can in turn cause axial con nement ofelectrons.Deformation, or the transition from a circular to elliptical cross section of a CNT, can beachieved via AFM/STM manipulation, or other more creative techniques[28]. Using the densityfunctional theory to simulate various types of semiconducting nanotubes, Mazzonni et. al.[25]3k-space is related to momentum space by k = p= h.10Chapter 2. Theoryreport notable band gap closure due to AFM deformation. Notably, the larger the degree ofdeformation, greater the decrease in band gap energy. They conclude that this band gap closureis due to the hybridization of bonds due to the increase in the curvature of the CNT as a result ofthe applied deformation. Cho[34] reports DFT calculations that support these results, con rmingthat the band gaps in semiconducting CNTs close under deformation of their cross sections. Incontrast, deforming a metallic CNT causes its band gap to open, as shown by Chang[30]. Cho[34]also reports band gap opening in metallic CNTs as a result of deformation, noting that themagnitude of these shifts is less than those in semiconducting CNTs. There are many otherresults in the literature, all of which show that under deformation, semiconducting nanotubesshow band gap closure, while the band gap of metallic nanotubes opens.If these deformations are introduced locally, over a short axial distance, then the change inband gap energy can be used to con ne electrons to a very small region of a CNT. For example,suppose that a deformation is introduced in a CNT by laying it across a ridge. Due to the  nitewidth of the ridge, the deformation will not be discontinuous. Rather there will be a gradualtransition from its normal relaxed, nearly circular cross section, to a more severe elliptical crosssection. Due to the complex, nonlinear behavior of dEgd  (where Eg is the band gap energyand   characterizes the deformation of the CNT), the band gap energy will oscillate creatingnumerous, alternating regions of semiconducting and metallic behavior. This results in variousregions of con nement, isolated from the rest of the CNT by tunnel barriers consisting of localsemiconducting regions: electrons in a one dimensional quantum wire have been further con nedto create a zero dimensional QD.The rest of this work will be dedicated to investigating the veracity of these theoretical predic-tions as applied to the carbon nanotube cross, both by attempting to fabricate and grow CNTsin this con guration, as well as by modelling the structure using various classical and quantumsimulation methods, to which I now turn.11Chapter 3Simulation MethodsI would now like to give brief descriptions of each of the theories used to model the carbonnanotube cross. As these methods are all standard within the  eld, I will try to avoid replicatingpages of equations, leaving this to the many excellent references4, instead trying to provide somequalitative insight into the physical basis of these methods.3.1 Molecular DynamicsConsider a collection of interacting atoms. The basic idea of molecular dynamics (MD), is toconsider the motion of each atom in a potential landscape, or mean  eld, that results from theforces of all the other atoms. These calculations are made using purely classical physics, and so,in a nutshell, MD consists of integrating Newton's Second Law, F = ma. The general idea of MDis to choose a particular atom, note its current position, sum the forces of all the other atomsacting on this chosen atom, and then move the atom in a direction and distance calculated fromNewton's Second Law. This motion is calculated for a brief period of time,  t, short enough thatwe assume that the external force  eld due to the other atoms does not change, and therefore theacceleration remains constant. We then do this for all other atoms, therefore evolving the entiresystem from a time t to a time t +  t, and then repeat this for the duration of the simulation.From mechanics we know that F(r) =  rU(r) and a = d2r=dt2, so for a given atom, we haveF(r) = ma (3.1) rU(r) = d2rdt2 (3.2)Depending on the form of the potential U, it may be possible to solve this analytically.However, usually U involves the positions of every other atom, and so all of the di erentialequations are coupled, and it is usually necessary to solve the system numerically. Mathematicallyit is straightforward to discretize both space and time by choosing an appropriate time step andspatial grid, and then apply this discretization to the above set of di erential equations, to obtaina set of  nite di erence equations which can be used to step the system forward from its initialco-ordinates. The system energy may easily be obtained simply by adding the kinetic energy,obtained from the time derivatives of the position coordinates, and the sum of the potentialenergies of each atom.Given the initial co-ordinates, the only thing that remains is to choose a way of calculating thepotential, U. Probably the simplest technique, for small scale objects where gravity is negligible,4For example, see Balbuena[4] for Molecular Dynamics, Lundqvist[24] for Density Functional Theory andOstlund[37] for Hartree-Fock, as well as Nojeh[27] for excellent introductions to all three theories.12Chapter 3. Simulation Methodswould be to apply a Coulomb interaction to each pair of atoms. However, this ignores all mannerof electromagnetic interactions such as van der Waals and dipole-dipole interactions, as well as thebonds between individual atoms. There are many di erent potentials in the literature, and theone used in this study is the AIREBO potential[36], an extension of the REBO[7] potential, whichincorporates the Terso -Brenner potentials, the standard for modelling hydrocarbon atoms. TheAIREBO potential has the form U = UREBO + ULJ + Utors, where UREBO is a Terso  potentialwhich models the bonds between atoms, and is essentially a sum of short range Coulomb typeattractive and repulsive terms, with some modi cation using empirical constants. To model theinter-molecular interactions between non-bonded atoms two potentials are used. Utors is a singleminimum potential to model torsional interactions between atoms, while ULJ is a classic 6-12Lennard Jones potential used to model the van der Waals interactions between atoms. Thereader is referred to the references for the mathematical details of these terms, but the importantthing to realize is that having chosen this potential, one may discretize it, and insert it into thediscretized form of Eq. 3.1. Then at each time step, the position of all atoms are  xed, U iscalculated for all atoms, and then used to step forward the positions and velocities of all atoms.This process is then repeated for the duration of the simulation.3.2 Quantum Chemistry TheoriesWhile the MD model described above is adequate for calculating the geometrical con gurationof atoms, as well as an estimate of the total system energy, it is purely classical, and so cannotgive any information about many crucial properties of a nanoscale system, for example its bandstructure, and in order to examine these properties we must turn to models that incorporatequantum mechanics. Due to the complexity of nanoscale systems, it is necessary to developapproximation schemes to make the problem tractable, while still accounting for the small scalequantum e ects that are crucial to the system's behavior. I will describe two so called ab initiomethods that start by seeking the exact solution to the Schr odinger equation, and make a seriesof approximations to render the problem workable.3.2.1 Hartree-Fock MethodThe Hartree-Fock (HF) method starts by seeking the exact solution,  , to the many body prob-lem. We  rst assume that the nuclei of each atom is  xed, the so called Born Oppenheimerapproximation. Dynamic interactions between electrons are neglected, and are replaced by amean  eld, which is an average over space and time of the e ects of all the electrons in thesystem. In this so called Hartree approximation, we assume that the many body wave functionis separable, and can therefore be written as a product of single electron wave functions, each ofwhich is a solution of the Schr odinger equation with a Hamiltonian consisting of only the electron'skinetic energy and the mean  eld potential. Recall that, due to the indistinguishability of elec-trons and the Pauli exclusion principle, any wave function for fermions must be anti-symmetricwith respect to electron exchange. Therefore the many body wave function may be written as aSlater determinant of single particle wave functions:13Chapter 3. Simulation Methods  = det0BBBBBB@ 1(r1  1(r2)  1(r3) :::  1(rN) 2(r1)  2(r2)  2(r3) :::  2(rN) 3(r1)  3(r2)  3(r3) :::  3(rN)... ... ... ... ... N(r1)  N(r2)  N(r3) :::  N(rN)1CCCCCCAHere  i(rj) is the ith single electron wave function, applied to the jth electron. This expansionof   is really the key of the HF method, as it seperates the many body wave function into a productof single body wave functions.The variational principle is invoked, which states that for any wave function  , its energyexpectation value is greater than or equal to the true ground state energy of the system, usingthe above Slater determinant as a trial wave function. Thus the (HF) method gives an upperbound on the system's energy. This variational principle is then applied iteratively, `tweaking'the wave function at each iteration, in order to minimize the system energy. This results in a setof N single electron equations, the so called Fock equations:f n =  n nf = hcore +n=2Xi=1[Ji + Ki] (3.3)Here hcore =  12r2n   PA ZArAn is the single electron Hamiltonian for an electron movingin the mean  eld electronic potential due to all the atoms, and the sum is over all electrons.Ji is the Coulomb operator, which accounts for electron-electron repulsion, by replacing thetrue two electron instantaneous Coulomb operator (proportional to r 1ij ) with a one electronoperator consisting of the average of this interaction over all space and spin. Ki is the so calledexchange operator, which arises from the anti-symmetry requirement (ie. the indistinguishabilityof electrons), and gives the exchange energy of the electrons. Notice that the Fock operator, f, isessentially a single electron hamiltonian, with some extra energy terms to account for electron-electron interactions. However, as opposed to an exact treatment, which would involve doublesums over all electrons, the Fock operator is only dependant on a single electron. This separabilityinto N coupled equations is the crucial feature of the HF approximation that makes it possible tosolve. In practice these N equations are recast as a matrix eigenvalue problem, and then the basissets are iteratively adjusted using the variational principle subject to Eq. 3.3 until an energyminimum is reached. The set of  i and  i are then recombined to  nd the Hartree-Fock estimateof the wave function and ground state energy, respectively.3.2.2 The Density Functional TheoryRather than treating the many body problem and seeking the many body wave function,  ,which is a function of all the coordinates of all particles, the Density Functional Theory (DFT)seeks the particle density,   over all space. This transformation from wave function to particledensity is rigorously justi ed by two theorems published by Kohn and Hoenberg[12]:1. There is a one-to-one correspondence between the ground state particle density and theground state many body wave function. This is justi ed by proving that the external14Chapter 3. Simulation Methodspotential acting on the exact many body system is uniquely determined by the particledensity.2. There is an analog to the wave function variational principle, described above, that statesthat the true particle density minimizes ground state energy. In other words the groundstate energy of the correct particle density is less than the ground state energy of any otherparticle density.Notice that this transformation results in a huge simpli cation. While   is dependant on3N coordinates,   is dependant on only 3. Essentially Kohn and Hoenberg proved that the wavefunction's dependence on the particle density may be inverted. That is given  (r) = R k (r)j2drwe can calculate a unique   =  [ (r)]. It can be shown that all other observables are alsofunctionals of  , speci cally the total energy: E = E[ ].The actual method for  nding the solution was layed out by Kohn and Shan[20], and involvesthe introduction of an e ective potential. With this modi cation, the problem of  nding themany body solution can be transformed to  nding a set of non-interacting single body solutions,the Kohn-Sham equations. Each Kohn-Sham equation is an eigenvalue equation for one of thesystem's orbitals and its associated energy. Using the above variational principle in Theorem 2the system energy is minimized using Lagrange's method of multipliers. This set of solutions tothe Kohn-Sham equations can be combined to give the global particle density.Practically this is implemented as follows. An initial density is chosen, and is used to calculatethe e ective potential. Note that this e ective potential incorporates all the many body e ects ofthe other electrons and atoms (such as electrostatics and exchange energy) into a single, lumpedterm. Next the Kohn-Sham equations are solved, from which a new particle density is calculated.This process is continued until the particle density converges.While the details of DFT can be rather complex, the important point is that by seeking theparticle density rather than the many body wave function, one can greatly simplify the problem.Furthermore, due to the above two theorems, all other relevant observables can be obtained fromthis particle density.The methods described in this chapter, namely MD, HF and DFT, will be employed laterto investigate the geometric and electronic structure of the carbon nanotube cross. However, asnoted in the introduction, the ultimate goal of this work is to observe QD behavior at the junctionof a real, physical CNT cross, and so I would like to describe the progress made to that end.15Chapter 4Fabricating the Carbon NanotubeCrossWhile the merits of the CNT cross have been discussed, the question remains how to actuallyphysically realize a CNT cross. Furthermore, having built one, how do we actually see it, interactwith it, and measure it. Clearly the very property which is its strength, its size, will also be anissue in light of the above challenges. In brief, the requirements for any experimental method areas follows. CNTs must be fabricated, and preferably in a manner that promotes the formation ofa CNT cross. After growth it is necessary to make electric contact with the CNT cross, so thatmeasurements can be made.CNTs will be grown using Chemical Vapor Deposition (CVD) on a chip fabricated usingmicrofabrication techniques. CNT crosses will be located using Scanning Electron Microscopy(SEM) imaging. Electrical contact can be made using both electrodes on the chip and an AtomicForce Microscope (AFM) tip. In order to investigate the possibility of QD behavior at the CNTcross junction, its Current-Voltage (I-V) characteristics can be measured. In this chapter I willdescribe this process in more detail, as well as the progress made and remaining challenges toobserving a Quantum Dot in the CNT cross.4.1 Design and Fabrication of the Chip4.1.1 DesignRecalling that the diameter of CNTs is on the order of a nanometer, it is clear that makingelectrical contact with them will not be as easy as simply attaching wires at either end of theCNT using alligator clips. What is needed is a sort of `Nano Lab Bench': a step down transformer,something that I can interact with, which in turn can interact with the CNTs. A rectangular`chip' shown in Fig. 4.1(b) was designed to meet these requirements and is at the macro end ofthe step down transformer: having dimensions of   1 cm square, it is large enough to handle andput in a CVD furnace for CNT growth. Each chip consists of an array of 27 30 smaller `devices',with equal numbers of four di erent variations of the same device design, shown in Fig. 4.1(c).The design of this chip was intentionally kept simple, to reduce potential fabrication compli-cations. The electrodes (red in Figs. 4.1(b) and 4.1(c)) are 100   200  m, and are large enoughto make ohmic contact with using a probe station. Along the inside edge of the electrodes arestrips of catalyst (black in Fig. 4.1(c)). The catalyst will be described later; for now consider itsimply a place that initiates CNT growth. These strips are between  ve and ten microns wide,and are between 2 and 4 microns away from the edge of the electrode. This placement is crucial,as it ensures that any CNT that grows from these catalyst strips towards the gap between theelectrodes will  rst travel over the electrode, thus ensuring good electrical contact between the16Chapter 4. Fabricating the Carbon Nanotube Cross(a) Wafer Design. (b) Layout of Chip for CVDGrowth.(c) Device for CNT CrossGrowthFigure 4.1: Fabrication Designs. An array of chips (b) is patterned over the entire four inch wafer(a), along with alignment marks for aligning the two layers (crosses in (a)). Within each chip(b) is an array of devices (c), consisting of two opposing electrodes (red), with strips of catalyst(black) along the edges of the electrode. In between the electrodes are catalyst islands (blue),and it is hoped that CNT cross growth will occur in the region de ned by the catalyst strips andtwo adjacent islands.CNT and the electrode. The blue circles are catalyst islands, of chemical composition identicalto the catalyst strips.The rational for the design is as follows. Each rectangular area de ned by two adjacentcatalyst islands and two opposing electrodes is a potential region for CNT cross growth. CNTswill grow randomly from all catalyst regions, but it is hoped that the diamond type array of thefour catalyst regions (two strips and two islands) will result in the occasional formation of CNTcrosses at the center of this region. Unfortunately, it is probable that this will not occur with anyregularity. To increase the yield of CNT crosses per device, there are between  ve and  fteen ofthese regions on each, depending on which the variant of the device. These variations are simplydi erent combinations of: i) catalyst island size (5   10  m); ii) catalyst strip width (5   10 m);iii) distance between catalyst islands (10   20  m) and iv) distance between catalyst island andelectrode edge (5   10  m). The reason for incorporating all of these variations is a priori thereis no way of knowing what are the optimal growth conditions for CNTs. As a further means ofo setting the probable low yield of CNT crosses, there are 810 devices per chip, giving on theorder of thousands of potential regions per chip for CNT cross growth. Finally, in order to obtainmany chips from one fabrication process, the chip was replicated over the entire four inch wafer(see below), as is shown in Fig. 4.1(a). The spacing between these chips (  1 mm) was chosento ensure the chips were not damaged when the whole wafer was sliced into individual chips (seebelow).4.1.2 FabricationThe chips and devices can be manufactured using standard microfabrication techniques, whichare the very well established methods that make all of micro electronics possible. While there aremany excellent resources on the subject (see for example Franssila[11]), I will brie y outline the17Chapter 4. Fabricating the Carbon Nanotube Crossprocess I have designed, and describe the challenges encountered.The fundamental technique employed in this process is photolithography. In a nutshell, thisis a resistive technique, similar to the plate lithography of the middle ages, and many wax resisttechniques used in the arts. Starting with your substrate (`blank canvas'), in this case a siliconwafer, a layer of protective material is applied to the wafer. Then a template or stencil (aphotomask) is used to selectively expose various parts of the protective material (photoresist)to light, which changes the chemical structure of the exposed photoresist, making it susceptibleto certain chemical etching agents (developers). The entire wafer, with photoresist on top, isthen washed in the developer, removing the parts of the photoresist that were exposed. Then,whatever material is desired is deposited over the entire wafer, in this case either metal, for theelectrodes, or catalyst. Finally, in a procedure known as lifto , the entire assembly is washedin acetone. This removes the remaining unexposed photoresist, and anything that sits on topof it, while not a ecting the last deposited layer (metal or catalyst). Thus one is left with alayer of `lithographically' patterned objects. This procedure may be repeated as many times asis required, each time resulting in a new layer of material.For each di erent set of patterns, or layer, a di erent photomask is required and in this work,two were necessary: one for the electrodes and one for the catalyst strips and islands. Thedrawings of the chips and devices in Fig. 4.1 are directly taken from my photomask designs,which were fabricated at the University of Alberta Nanofabrication Facility. These photomasksare 5 inch square glass plates, and the exposure pattern is controlled by a thin layer of chrome,which has been etched away in certain places (wherever one wishes to have material in any givenlayer). In order to facilitate good alignment of the two layers, crucial to ensure that, say thecatalyst strip, ends up being deposited where it is supposed to be, several alignment marks wereincluded on each mask. These alignment marks consist of nested crosses, with the size of thecross on the catalyst mask (second layer) slightly smaller than on the electrode mask ( rst layer).By adjusting the mask until the smaller cross is centered in the larger cross, one can ensurealignment.Fabrication was carried out in the AMPEL cleanroom, in order to avoid contaminating therelatively sensitive process. While simple in concept, microfabrication is in general tedious andtime consuming. One signi cant challenge encountered was that, during the lifto  process, theunexposed photoresist5 was not completely removed in the acetone bath. This was a problem,as it is probably an organic polymer residue, which could have various, uncontrollable, negativee ects, including contaminating the chip during CVD growth and invalidating the assumptionthat the substrate is an insulating equipotential. Various methods were tried to remove thisresidue, including washing the wafer in hot acetone, as well as sonicating the wafer in hot acetone,neither of which were successful. The one possibility that would work, an oxygen plasma etch toremove the residue, was not possible, as this would oxidize the metal electrode, hampering ohmiccontact between the CNT and electrode, as well as quite possibly damaging the catalyst.As a result of this residue issue, a di erent process was designed, involving a double resistmethod for each layer. Speci cally, a layer of Poly(methyl methacrylate) (PMMA) resist was  rstdeposited, followed by conventional photoresist (PR). The PR was then patterned photolitho-5That it was the unexposed resist was con rmed by performing a blanket exposure on a wafer of patternedphotoresist, and then washing it in the developer, which resulted in a clean wafer. Unfortunately this was not aviable solution to the problem, as the developer could have unforeseen e ects on both the electrode and catalyst.18Chapter 4. Fabricating the Carbon Nanotube Crossgraphically, leaving exposed PMMA, which was then etched using an oxygen plasma. However,as the etch rates for the two resists are di erent, and there is no available data for these valuesfor the machine in AMPEL, it has been very di cult to reliably etch all of the exposed PMMA,without completely etching away the PR/PMMA patterns (in other words completely etchingall the resist, resulting in a clean, blank wafer). Furthermore, it is important to obtain accurateetch rates for both resists, in order to avoid oxidizing the electrode while patterning the secondcatalyst layer.6Due to these challenges, the experimental side of this thesis is a work in progress and is still inthe fabrication stages. However, before describing the work done modelling this structure, whichis su ciently encouraging to provide motivation to continue fabrication, I would like to outline aprocedure that can serve as a basis for ultimately fabricating and measuring the characteristicsof the CNT cross.4.2 Experimental DesignThis section is intended to be a brief illustration of the experimental design that has been devised.It is included because I believe that it is a relatively promising route to growing and measuring aCNT cross. However, as much of it has not been realized, I will be fairly terse. For example I willnot explain the necessity of the PMMA layer, as that has already been discussed in the previoussection.4.2.1 MicrofabricationAs described above, this is essentially a two layer photolithographic process, which is schematicallydepicted in Fig. 4.2.Molybdenum Electrode LayerStarting with a four inch silicon wafer, coated with a thin layer of thermal oxide, a layer of AZP4110 photoresist (  1:4  m thick) is spun on the wafer, followed by a layer of MicroChem 950PMMA in Anisole (4% solids) (  350 nm thick). Following each spin step, the resist is bakedon a hot plate and allowed to cool. The whole wafer is then exposed in a mask aligner, usingthe  rst layer mask (with electrode patterns). The wafer is then immersed in a bath of 1:4, (AZ400K Developer: De-Ionized Water), removing the exposed PR. The wafer is etched with anoxygen plasma, removing the PMMA7. Following this 50nm of Molybdenum is evaporated ontothe wafer in a vacuum chamber. Finally the remaining resists (as well as any Mo on top of theresists) is removed using an acetone bath.Catalyst LayerThis layer is almost identical to the Mo layer, with a few notable exceptions. The PR and PMMAare applied in exactly the same manner, and the photolithography is identical, except that the6In this case, the electrode layer is already deposited, with the PR/PMMA layer on top. We wish to etch downto the electrode, so that we may deposit catalyst. However, if the oxygen plasma etch is continued for too long,then the metal electrode will be oxidized, which is undesirable as described above.7Much of the remaining PR will also be removed19Chapter 4. Fabricating the Carbon Nanotube CrossFigure 4.2: Schematic of Microfabrication Process of Chip for CNT Growth in CVD. a) Spin PRand PMMA resist layers; b) Expose to UV light, using mask with electrode patterns; c) DevelopPR; d) Etch PMMA using O2 plasma; e) Evaporate Mo; f) Lifto  of remaining PR/PMMA(and Mo on top of resists) using acetone; g) Spin PR and PMMA resist layers again (over bothwafer and Mo); h) Repeat photolithography (steps b)-d)) using mask with catalyst patterns; i)Spin catalyst over entire assembly and j) Lifto  of remaining PR/PMMA (and catalyst on topof resists) using acetone, leaving completed device. Note that the pro le in j) corresponds to thedevice in Fig. 4.1(c).20Chapter 4. Fabricating the Carbon Nanotube Cross(a) Schematic of I-V Charac-teristics of QDs.(b) Tunneling Across a QD:Unbiased.(c) Tunneling Across a QD:Biased.Figure 4.3: Current-Voltage (I-V) Measurements of Quantum Dots. The quantum well formedby two potential barriers, (b), is a model for a QD. Electrons cannot tunnel through the entireassembly but may use the states inside the QD (dashed lines) as stepping stones. As a voltageis applied across the QD, the chemical potential shifts, (c), and more and more states becomeavailable as stepping stones, subject to  1 > Ei >  2. Each new state that becomes availablewith increasing voltage results in a current spike, giving rise to the characteristic staircase-likeI-V curve, (a).second layer mask (with catalyst strip and island patterns) is used. The developing of the PRand etching of the PMMA is also identical. Then the catalyst, which initiates CNT growth, isspun on to the entire wafer, after which it is baked, to remove the methanol solvent (see below).Finally lifto  is performed with acetone, leaving the  nished wafer. Note that the pro le in Fig.4.2j) corresponds to the device shown in Fig. 4.1(c).Finally the four inch wafer is cut into the individual chips using a dicing saw.4.3 Carbon Nanotube GrowthThe general methodology of growing CNTs on micro-fabricated devices using chemical vapordeposition (CVD) was initially developed by Dai and colleagues[21], and is essentially followedhere. The broad strokes of this scheme are as follows. A chip is placed in a CVD furnace, whichis heated to around 700oC, at which point methane (CH4) is injected into the furnace. It is herethat the catalyst8, which consists of iron and molybdenum nanoparticles in an alumina supportmatrix, is crucial. Due to the high temperature, the gas molecules break down at the surface ofthe catalyst nanoparticles, and the now free carbon atoms spontaneously form CNTs. While thephysical mechanism of this CNT growth at the catalyst particle surface is still under debate, thesalient feature is that it is spontaneous. Intervention is not required: as long as the chemicals areinside the CVD in the right concentrations, at the right temperature, CNTs will grow.21Chapter 4. Fabricating the Carbon Nanotube Cross4.4 Measuring the Current-Voltage Characteristics of the CNTCrossWhile it is known that CNTs will grow in the above conditions, and that, given enough chances,there will be a few CNT crosses, these last still need to be located on the chip. While thelithographically patterned features can be resolved with an optical microscope, CNTs are toosmall for this, and so the chip must be imaged using a Scanning Electron Microscope (SEM)9.Due to the regular array the devices form on the chip, it should be fairly easy to quickly cyclethrough the chip, visually searching for CNT crosses row by row.Having located a CNT cross, in order to make measurments it is necessary to make electricalcontact with this structure, and there are two cases to consider. Ideally one of the CNTs inthe cross will be in electrical contact with both of the electrodes. In other words, one end ofthe CNT, on one side of the junction, will be touching one electrode, and the other end, on theother side of the junction, will be touching the opposing electrode. In this case electrical contactmay be made with each electrode separately. More probably, only one of the ends of one of theCNTs will be in contact with a single electrode. In this case, in order to complete the circuitfor measurement, contact with the other, free, end of the CNT must be made either using directcontact via an Atomic Force Microscope (AFM) tip or via a tunneling current, using a ScanningTunneling Microscope (STM). As both of these microscopes are also imaging modalities, it shouldbe possible to see the CNT cross, and then `zoom in' and make contact with the appropriate regionof the CNT. The circuit is completed by making contact with a probe station to the electrodethat is in contact with the other end of the CNT.Once contact has been made, Current-Voltage (I-V) measurements can be taken. A DCvoltage bias is applied across the CNT cross (the possible QD), and the voltage is graduallyincreased, all the while measuring the current  owing through the CNT cross. The current isthen plotted versus the voltage10. What shows the presence of a QD is the staircase type behaviorshown in Fig. 4.3(a): each `step' is characteristic of one of the quantized energy levels in theQD.11How this staircase-like current con rms the presence of a QD can be heuristically understoodas follows. Consider, as a toy model of a QD, a uniform potential, with two very high, closelyspaced potential barriers, as depicted in Fig. 4.3(b). The region in between the two barriers isthe model QD, with its set of discrete states (dashed lines). Initially the chemical potential ofthe electrons is the same (Fig. 4.3(b)) on either side of the barriers (in other words on either sideof the CNT junction). However, when the voltage bias is applied, the chemical potentials will8The catalyst consists of 0:05mmol of Fe(NO3)3   9H2O, 0:015mmol of MoO2 (acac)2 and 15mg aluminananoparticles (aluminum oxide C) in 15ml of methanol. The entire mixture was sonicated for twenty four hoursfollowing intial mixing, and then for one hour prior to each use. See [21] for details.9Alternatively a Scanning Tunnelling Microscope (STM) or Atomic Force Microscope (AFM) could be used10This is not crucial, V vs. I is also possible- the important feature is the staircase behavior11In reality this behavior is a combination of both the quantum e ect of the discrete energy states in the QD aswell as the purely classical electrostatic interaction between electrons. As the QD has a very small area, it thereforehas a very small capacitance, and so each additional electron that is added to the QD has an energy greater thanthe previous by a value of e2C , where e is the fundamental charge, and C is the capacitance of the QD. Since C issmall, this is an appreciable energy. This e ect is known as the Coulomb blockade. Thus the simple capacitivecharging of the QD will also show a discrete behavior, and it is necessary to decouple the two e ects, which whilesometimes di cult, is possible.22Chapter 4. Fabricating the Carbon Nanotube Crossshift on both sides, as shown in Fig. 4.3(c). From statistical mechanics, particles tend to  owfrom high to low chemical potential. Now if, instead of two seperate barriers, there was simplya large, single barrier of equal height and as wide as both barriers and the well, then this tunnelgap would be too large for any kind of appreciable tunnel current. However, the discrete statesof the QD can act as `stepping stones' through the QD: an electron may  rst tunnel throughone barrier, into one of the allowed states of the QD, and then through the second barrier, andaround the circuit, which will register as a current. Recall that, without extra energy, a particlecan never move from a state of lower chemical potential to higher chemical potential. Thus theonly states that are able to act as stepping stones are those with  1 > Ei >  2, where  1 and  2are the chemical potentials on either side of the QD and Ei is the energy of the ith allowed statein the QD. Initially as  1 =  2 no current  ows. As the voltage increases, still no current  ows,until the voltage is large enough so that the above inequality is satis ed for at least one state, atwhich point some electrons will tunnel through the QD, as described above, and so there will bea spike in the current. However this state will quickly saturate, and so the current will plateau.As the voltage is further increased,  1 keeps rising, and eventually another state will satisfy theinequality, leading to another spike in current. This leads to the staircase current shown in Fig.4.3(a), indicating the presence of the QD.This experiment is still in the fabrication stages, and so the above discussion is somewhatspeculative. However, it should provide at least a framework for eventually performing I-Vmeasurements on a CNT cross. There is good reason for believing that one may indeed seeQD behavior in these measurments, in large part due to the simulation studies described in thefollowing sections.23Chapter 5Modeling the Carbon NanotubeCrossTo gain another perspective of, and more insight into, the physical processes that govern itsgeometrical and electronic structure, the CNT cross structure was modeled using a variety ofmethods. Broadly these simulations fall into two categories: Molecular Dynamics studies ofthe system's geometry and Quantum Chemistry calculations (using Hartree-Fock and DensityFunctional Theory) of its electronic structure.Modelling nanoscale devices presents unique challenges, as their length and energy scales tendto be right on the cusp, or overlap, of two di erent regimes. They are small enough to exhibit allmanner of inherently quantum phenomena, for example the very discretization of energy levelsthat is integral to this work, and therefore any e orts to model systems on this scale must at leastconsider quantum e ects. However, they are large enough, and have enough protons, electronsand neutrons so as to be virtually impossible to model using the full, unaltered machinery ofquantum mechanics. In other words one cannot simply write down the Hamiltonian for all theparticles of the system and solve for the wave function. That being said, one cannot go too far inthe other direction, and treat the system in a continuum fashion, using say the classical theoriesof electrodynamics, mechanics, and electronic transport equations.Thus any attempt at modeling devices at this scale usually must incorporate a variety ofdi erent methods for di erent parts of the simulation. In this section I will describe how I havemodelled the CNT cross structure, for the most part avoiding technical discussion, as this hasalready been done in Sec. 3, and rather focusing on the methodology adopted, and the variousresults that have been obtained and seem to support the possibility of QD behavior in the CNTcross.5.1 GeometryTo determine the geometry of the system, that is the physical con guration and orientation ofall the atoms, the Molecular Dynamics methods described in Sec. 3.1 were employed. This iscertainly somewhat of an approximation, as in theory the atoms will align themselves and createand break bonds according to the laws of quantum mechanics. As there were on the order ofseven thousand atoms in the system it would be impossible to model using quantum methods,and the AIREBO potential is an excellent approximation.As a starting point for any simulation an initial con guration should be chosen that mimicsexperiment as closely as possible. This was chosen to be a  at sheet of graphene (used as anapproximation to the silicon substrate), with one CNT on top of it, and the other draped over thesubstrate and the  rst CNT, as shown in Fig. 1.1(a). A Matlab script was written to generate24Chapter 5. Modeling the Carbon Nanotube Crossa hexagonal array of carbon atoms in a plane, which is exactly the structure of graphene. Aprogram called Wrapping12, developed by Dr. Shigeo Maruyama from The University of Tokyo,was used to generate co-ordinates of  at CNTs of arbitrary length and chiral indices (m;n). TheCNTs were 10 nm in length and the graphene sheet was 11   11nm.Recall from statistical mechanics that, within its constraints, a system will evolve to the statewith the lowest possible energy, corresponding to its equilibrium state. This is a very convenientway of obtaining the physical, or `true', geometrical state of a system: let it evolve in a MDsimulation until its energy saturates (approaches some value asymptotically). Ideally it would benice to simply use two  at CNTs, the one over top of the other, both over top of the graphenesubstrate, as seen in Fig. 5.3(a), as the starting con guration for the MD simulation, and thenlet the top CNT slowly drift downward to the substrate, eventually conforming itself to thesubstrate and the bottom CNT. Unfortunately this does not happen, for two reasons. First,the inter-molecular interactions in the AIREBO potential are relatively short range. Therefore,as will be discussed below, the top CNT, being around 1nm above the substrate, simply doesnot `see' the substrate, and behaves as though it is not there, and therefore does not `feel' theattractive van der Waals forces that physically should drag the CNT downward. Secondly, evenif this interaction were realized mathematically, it would be very weak, requiring the simulationto be run weeks or even months.Because of this, an initial con guration needed to be developed that was as close as possibleto the suspected physical con guration, such that the MD simulation is `closer to its  nal goal',and need only minorly re-arrange atoms to reach an equilibrium state. Noting that the SEMimage of the CNT cross shown in Fig. 5.3(c) has a bell curve shape, I decided to map the topCNT co-ordinates to a Gaussian pro le according to the transformation (x;y) ! (x;y0) wherey0 = h(1   e x24ln(2)w2 ) (5.1)Here h is the height above the substrate of the centre (peak) of the top CNT, and w is the FullWidth at Half Maximum (FWHM) of the Gaussian function. Having completed this mapping,the results of which are shown in Fig. 1.1(a), another question immediately arises, namely whatvalues to use for the free parameters in the simulation. These are i) the type of CNT, which, fromSec. 2.3, is a function of the integers (m;n) and ii) the width of the Gaussian pro le (w), that iswhether the CNT has a very gradual rise from the substrate to the junction, or rather remains at on the substrate, and then closely contours the bottom CNT at the junction, and then fallsquickly back to the substrate and is  at on the substrate for the rest its length.There is no way to know a priori what combination of these parameters one would obtainfrom CNT growth, nor is there any way to ascertain this information from imaging or mea-surements. Thus, it is appropriate to investigate a wide array of these parameters, in variouspairings, examples of which are shown in Fig 5.1, and see the e ect this has on the geomet-rical and electronic structure of the CNT cross. The two types of CNT investigated were thesemi-conducting (10;0) and the metallic (5;5), and for each of these a range of `widths' wereinvestigated (w = 15;20;:::;45;50) as well as a ` at' CNT cross (Fig. 5.3(a), identical to otherCNT crosses, but without the Gaussian mapping, for comparison. Because in an actual physicalexperiment the CNTs are growing on macroscopic silicon wafers, which are structurally rigid, the12Freely available at http://www.photon.t.u-tokyo.ac.jp/%7Emaruyama/wrapping3/wrapping.html.25Chapter 5. Modeling the Carbon Nanotube Cross(a) (10;0) (! = 20) (b) (10;0) (! = 35)(c) (10;0) (! = 45) (d) (5;5) (! = 15)(e) (5;5) (! = 30) (f) (5;5) (! = 50)Figure 5.1: Comparison of Gaussian Pro les of CNT Cross. Here (m;n) are the integer indicesof the given CNT, and ! is the Full Width at Half Maximum of the Gaussian function used togenerate the co-ordinates. These are examples of the pro les which were obtained using MDrelaxation.26Chapter 5. Modeling the Carbon Nanotube Crossgraphene sheet was  xed throughout the simulation, as would be the wafer.The MD simulations were done in NanoHive13 and the results are shown in Fig. 5.2. At  rstglance, these results are extremely worrying. As the width of the Gaussian pro le increases, thepotential energy of the system decreases, both initially, and throughout the simulation. Further-more, the ` at' CNT cross (Fig. 5.3(a)) has the lowest energy of all. As it is known that CNTcrosses do indeed grow (see Fig. 1.1(d) for example), and are therefore physically realistic states,it would seem that the MD implementation is  awed, casting doubt on these results.It turns out that the MD method is  awed, but in a controllable and non-catastrophic man-ner. To investigate this issue, a series of simulations were performed, consisting of a straight,  atCNT, placed at various heights h above the substrate, depicted in Fig. 5.3(b). These systemswere then relaxed via MD in exactly the same manner as the CNT crosses, and the results ofthese simulations are shown in Fig. 5.2(c). Notice that only the nanotube that is 1 A above thesubstrate, exhibits any kind of potential energy minimization. Furthermore the potential energycurves for the two higher nanotubes (5:5 A and 10 A above the substrate) are identical. Theseresults are explained by the fact that the e ective range of the inter-molecular interactions, mod-elled using the Lennard-Jones potential (see Sec. 3.1), is relatively short, and can be interpretedas follows.First, the only nanotube which changed its con guration in any way, and evolved to a lowerenergy state, is that which was initially closest to the substrate. Thus, the e ective range of theinter-molecular interactions is su cient such that this close CNT interacts with the moleculesof the substrate, but small enough that the two more distant CNTs do not. Since they are notinteracting with the substrate, they behave as if they were isolated. As the initial co-ordinatesof a simple CNT are already essentially in a relaxed state, there is no energy minimizationpossible. This observation is reinforced by the identical energy curves of both of the more distantnanotubes. While the geometric con guration of these two systems is di erent- one is twice asfar as the other from the substrate- they show identical behavior. This is only possible if neitherof the CNTs interact with the substrate, implying that the e ective range of the inter-molecularinteractions in the AIREBO potential is smaller than 5:5 A.Returning to the relaxation of the various Gaussian pro les, shown in Figs. 5.2(a) and 5.2(b),this also explains the decreasing energy of the CNT crosses with increasing spread. For a verynarrow spread, for example w = 15 A in Fig 5.1(d), the CNT is interacting with the substratefor most of its length, forming covalent bonds. Thus the CNT is slightly deformed, having a anoval cross-section, and is not free to relax to its circular, minimum energy, pro le. Conversely,for very large spreads, say w = 50 A, Fig 5.1(f), much of the CNT is well above the substrate,and so a greater portion of the CNT is free to evolve to its lowest possible energy state, withoutany `interference' from the graphene, as it is out of the e ective range of interaction. Carryingalong in this vein, the  at CNT cross is out of range of the substrate, and so is virtually free toevolve to its global minimum energy.14 Therefore this decrease in energy with increasing spreadis an artifact of the MD simulation, speci cally the AIREBO potential.13Freely available from http://nanoengineer-1.com/nh1/14That the potential energy curve does show a minimization is also due to the interactions of the two CNTs aswell as the interactions between the bottom CNT and the substrate. Also recall that the substrate is  xed in space,and so has no e ect on these arguments, as it cannot rearrange to any other con guration to reach a lower energystate.27Chapter 5. Modeling the Carbon Nanotube Cross(a) MD Relaxation of various (5;5) CNT Crosses(b) MD Relaxation of various (10;0) CNT Crosses(c) MD Relaxation of Flat CNT above SubstrateFigure 5.2: MD Relaxation of Carbon Nanotube Crosses. For both types of nanotube, thepotential energy of the structure, for all values of w, saturates, implying a set of equilibrium,meta-stable states, which are physically realistic. Note that, although the  at CNT has thelowest energy, this is an artifact of the relatively short range of the inter-molecular term in theAIREBO potential. This is con rmed by (c): notice that only the CNT that is 1 A above thesubstrate evolves to a lower energy state, by interacting with the substrate. The other two CNTsdo not `see' the substrate, and so behave as if they were isolated. 28Chapter 5. Modeling the Carbon Nanotube Cross(a) Flat Carbon Nanotube Cross. (b) Single Flat CNT Above Sub-strate.(c) SEM Image of a CNT Cross.Figure 5.3: Investigating the Inter-Molecular Interactions in NanoHive. While the MD relaxationresults (Figs. 5.2(a) and 5.2(b)) indicate that a CNT cross would not form in nature, insteadstaying in a  at con guration (a), they have been observed (c). To resolve this issue MD simula-tions were run with a single CNT above the substrate (b), at varying heights. These results areshown in Fig. 5.2(c).Having eliminated this one concern, the other obvious feature of these results is that thepotential energy of all the various CNT crosses saturates with time. This implies that eachof these CNT crosses, both (10;0) and (5;5) CNTs, and for all values of w, are meta-stablestates. That is, once the carbon atoms are in this initial con guration, barring some fairlysevere perturbation, these CNT crosses will not spontaneously switch to some other physicalarrangement. Therefore all of these structures are potentially viable, and could possibly occurexperimentally. As such, a selection of these structures was studied, to see if their electronicstructures' varied. At this point, Molecular Dynamics has ful lled its role: it has providedequilibrium CNT cross con gurations and demonstrated that viable CNT crosses may occur in avariety of combinations of both nanotube type, and width of Gaussian pro le.5.2 Electronic StructureOne might wonder why it was ever advisable to use a classical theory such as Molecular Dynamicsto model a QD dot device, whose very name suggests a quantum mechanical description of theCNT cross structure is required. While the MD package used in this work does an excellentjob of empirically accounting for quantum e ects such as chemical bonds, it is certainly anapproximation, and MD was chosen out of necessity due to the large size of the CNT structure,and because for geometry calculations, quantum e ects are less crucialWhen calculating the electronic structure of the system, however, it is imperative to turn tothe quantum chemistry methods described in Sec. 3.2. Unfortunately, due to the vastly increasedcomplexity of these methods, the total system cannot be simulated at the same time. Indeedthese methods are so computationally expensive, that even the single top CNT, namely thatwhich is draped over the bottom CNT and substrate, cannot be simulated in its entirety. This isbecause for each electron that is added to the system, the overlap and exchange integrals of thisnew electron must be calculated with every other electron already present. It follows that the29Chapter 5. Modeling the Carbon Nanotube CrossFigure 5.4: Comparison of di erent electronic structure calculation methods in the GAUSSIANpackage. While the STO-3G basis sets are faster, they greatly overestimate the band gap en-ergy. The method used to perform electronic structure calculations on the CNT cross was theDFT method B3LYP using a 6-311G(d) basis set, as electronic structure calculations using thiscombination gave a bandgap of 0:57 eV , similar to the empirical values of Eg   0:5   1:5eV .number of integrals that must be computed, and therefore the computation time of the simulation,exhibits a power law type dependance on the number of electrons in the system.The only option is to split the CNT into more manageable sections, and calculate the elec-tronic structure of each of these sections, and then re-assemble these results into a picture ofthe electronic structure of the CNT as a whole. The top CNT was split into 5 A slices, centredabout the CNT cross junction. In order to approximate these slices as though they were part of acontinuous CNT, the atoms at the ends of each slice were terminated with hydrogen atoms, thusavoiding any dangling bonds, which are notorious for causing problems in electronic structurecalculations.Each of these slices contained forty carbon atoms, a far more manageable number than theseven thousand odd atoms that make up the complete system that was relaxed using MD. Elec-tronic structure calculations were performed in GAUSSIAN15, using the hydrogen terminated CNT15Not freely available anywhere. See www.gaussian.com. Calculations were performed on Abacus, a cluster30Chapter 5. Modeling the Carbon Nanotube Crossslices as input. GAUSSIAN has a vast array of di erent methods available, most based on theHartree-Fock (HF) and Density Functional Theory (DFT) methods described in Sec. 3.2. Whilethere are certain trends in these methods, for example using more basis functions in the expansionof a wave function often produces more accurate results, there is no de nitive best choice for anyparticular system.A series of calibration simulations were performed, to determine which method to use for theelectronic structure calculations. A straight CNT, 7:5 A long, was generated in Wrapping, andits electronic structure calculated using HF, DFT and so-called Semi Empirical Methods16, thelast of which was used for calculations not described here. Within the two former methods, onemust make a choice of basis set. These basis functions are used in the expansion of the wavefunction in both algorithms. The mathematical details of these bases are fairly complicated, andare described elsewhere17. Su ce to say that from the STO-3G basis set towards 6-311++(d,p)basis set, there is an increasing number and variety of basis functions assigned to each atom,which enables more accurate modeling of higher level molecular orbitals. The tradeo  for thisincreased accuracy is, not surprisingly, drastic penalties in CPU time. Within DFT there are twocommon variations, namely BLYP and B3LYP, with the latter an extension of the former, as thelatter also solves the HF problem.18Essentially, what GAUSSIAN returns as output from these electronic structure calculations, inthis case using DFT, is a hierarchical arrangement of molecular orbitals19, and their associatedenergies, along with other values calculated from these two fundamental quantities. Among manyother things, GAUSSIAN outputs the energy (EH) of the Highest Occupied Molecular Orbital(HOMO) and the energy (EL) of the Lowest Unoccupied Molecular Orbital (LUMO). Thesetwo orbitals have a straightforward physical interpretation. Consider the discrete set of allowedmolecular orbitals of a molecule, ordered from lowest to highest energy. Then consider taking allof the electrons from all of the atoms, and  lling these discrete electronic states with electrons,from lowest to increasing energy, thus constructing the molecule in its global ground state. Themolecular orbital state in which the  nal electron is placed is the HOMO state, while the nexthighest energy state is the LUMO state. In this work the band gap energy was taken to beEg = EL   EH, as this is the energy required to excite an electron across the `band gap' from the`valence band', or HOMO, which is full and so electrons cannot move to conduct charge, to the`conduction band', or LUMO, which is, in general, empty and so electrons are free to move andmaintained by UBC Chemistry.16These are essentially hybrid methods which use many  rst principles results, but also use empirically obtainedparameters to simplify the calculations.17The Gaussian online manual (http://www.gaussian.com/g ur/g03mantop.htm) has basic descriptions of thesebases, and references to all of the original papers, in which each basis (and for that matter, exchange functional,simulation method etc.) was  rst described.18In general the two main choices to be made when running a DFT simulation are the exchange functional andthe basis sets. The former is used when making the transformation to particle density from the wave function,while the latter is used to expand the single particle wave function (in an e ective potential which accounts for theother particles), which is solved as a  nal step in DFT.19These molecular orbitals have a somewhat convoluted relationship to other, more mathematical and less physicalso-called spin orbitals, which are the actual orbitals that are used in the wave function expansions. This relationshipis somewhat complicated, and far beyond the scope of this work, especially as they are simply a means to an end,namely the band structure of the CNT cross, and furthermore they are employed in a very large, well validatedcommercial package.31Chapter 5. Modeling the Carbon Nanotube Cross(a) (5;5), w = 20 (b) (5;5), w = 45(c) (10;0), w = 20 (d) (10;0), w = 45Figure 5.5: Band Gap Energy of CNT Crosses. The actual CNT that was used as the physicalstructure for the electronic structure calculations is shown above each plot. Notice that there is aone-to-one correlation between local deformation of the CNT and local band gap change, whichsupports the possibility of a QD at the CNT cross junction.conduct charge. The validity of this choice of de nition of band gap energy will be discussed inSec. 6; for now the important point is that it represents the energy necessary to move an electroninto a region in which it is free to move throughout the structure.Returning to the results of the calibration simulations, shown in Fig. 5.4, it is immediatelyevident how the more complex basis functions result in more \computational complexity": foreven an array of forty atoms, all methods fail after around  ve days, having failed to converge.The most signi cant result of these simulations is that, as is generally accepted, the HF methodsgreatly overestimate EL, compared to DFT methods, which also results in greatly in ated bandgap energies. A similar e ect is observed with the relatively inaccurate STO-3G basis set, makingall HF methods and any DFT methods using this basis set undesirable. As all of the simulationsusing the highly accurate 6-311++(d,p) basis set crashed, along with the DFT method(B3LYP,basis set 6-31G) there were really only two choices: 1) B3LYP with a 6-311G(d) basis set or 2) BLYPwith a 6-31G basis set. In this case, the added accuracy provided by the extra basis functions waswarranted, as the simulation using the B3LYP/6-311G(d) combination gave Eg = 0:47eV , muchcloser to the empirical values of Eg   0:5   1:5[38] than those obtained using the BLYP/6-31Gcombination (Eg = 0:59eV ). Therefore the B3LYP method using a 6-311G(d) basis set was usedfor all subsequent electronic structure calculations.32Chapter 5. Modeling the Carbon Nanotube CrossHaving chosen a DFT method, and adopted a de nition for the band gap energy, it is possibleto calculate the electronic structure for various CNT cross structures, speci cally those withw = 20;45 A for both (10;0) and (5;5) CNTs. The band gap energy for each slice was calculatedin GAUSSIAN and then plotted as a function of the location of the center of the slice on the `parent'CNT. These plots are shown in Fig. 5.5, and on the whole are promising. Probably the mostimportant result is the global observation that there is a one-to-one correlation between localdeformation of the top CNT and local band gap change of the CNT, across all di erent CNTcross structures. This is an encouraging  rst step towards showing QD behavior at the junctionof the CNT cross, as the original stated goal was to create a QD in a CNT by introducing localphysical deformations, in the hopes of inducing local band gap change, to create tunnel barrierswhich would axially con ne electrons in the CNT. This observation theoretically con rms the rst half of this goal.There are several other important features of these plots. Consider the case of the metallic(5;5) CNT, with a wide Gaussian pro le (w = 45), shown in Fig. 5.5(b). As discussed in Sec. 2.5,deforming a metallic CNT causes its band gap to open, and this is observed here. Moving fromleft to right initially the CNT is  at on the substrate, and as such the CNT is slightly deformed,`squished' from a circular to oval cross section, similar to an in atable ball which has lost muchof its air resting on the ground. This causes an opening of the band gap. As the CNT rises o the substrate, it is free to be in a more relaxed shape, circular in cross section, and so has a muchsmaller band gap. At the junction, the CNT is once again deformed, this time by the bottomCNT, and so the band gap rises again. Then, symmetrically, the band gap lowers once again(as the CNT is in a free, non-deformed state), before rising again, as it hits the substrate and isdeformed.Now examine the semi-conducting (10;0) tight pro le (w = 20) cross, shown in Fig. 5.5(c).For semi-conducting CNTs, deformation causes a band gap closure (Sec. 2.5), and so the bandgap pro le should be just the opposite of that discussed in the previous paragraph, and it is.Starting in a mildly deformed state resting on the substrate, it has a decreased band gap. As theGaussian pro le is quite narrow, the rise o  the substrate is quite sharp, leading to a signi cantdeformation, and a further decrease in the band gap at around  15 A, as the CNT comes o  thesurface. Once it is free of the surface, it is in a non-deformed state, and so the band gap increases,until the CNT hits the other, bottom CNT, causing deformation and a decreased band gap onceagain. As above, this process is mirrored as the CNT falls back to the substrate.While the other two plots do not have so clean an interpretation, they do exhibit similarfeatures. The question then becomes what is the e ect of the type of CNT (m;n) and thecon guration of the CNT cross (w). About the only valid observation that can be made is tonote that the two di erent types of CNTs behave di erently in response to deformation, as theyshould as one is metallic and one is semiconducting. It is not possible from Fig. 5.5 to make anyconclusions as to which Gaussian pro le, or which type of CNT, is optimal for QD behavior, noris it possible to state that these parameters have no e ect.As a  nal note, observe that for all of the CNT crosses, the calculated band gaps (  0:5  1:5 eV ) are in a normal range, as compared to the experimentally observed range of   0:5  2 eV [38]. This indicates that these computational investigations of CNT crosses, from the MDcalculation of an equilibrium geometry to the choice of GAUSSIAN method to the de nition of bandgap energy, while not necessarily quantitatively accurate, are certainly producing very physically33Chapter 5. Modeling the Carbon Nanotube Cross(a) (5;5), w = 20 (b) (5;5), w = 45(c) (10;0), w = 20 (d) (10;0), w = 45Figure 5.6: LUMO Energy of CNT Crosses. Notice, as in Fig. 5.5 that there is a one-to-onecorrelation between local deformation of the CNT and local band gap change, which supports thepossibility of a QD at the CNT cross junction. The LUMO energy is interpreted as the `conductionband' energy. Notice the local minimums, probably a result of the spatial deformation, in theLUMO energy, which could well be su cient tunnel barriers to produce electron con nement.reasonable results. Therefore the methods adopted here may be considered viable for futurestudy.In terms of visualization and physical interpretation,20 it may be easier to see the regions ofelectron con nement from plots of the LUMO energy (EL), shown in Fig. 5.6. Recall that theLUMO (Lowest Unoccupied Molecular Orbital) is the  rst electronic state that is not occupiedin the system's ground state. This leads to the natural interpretation that the LUMO, and allstates above it, are the `conduction band' and the HOMO, and all states below it, are the valenceband21. Thus, heuristically, EL gives a rough image of the `resistance to movement' that animaginary electron near the HOMO and LUMO states would `see' as it is traveled down the axisof the CNT. This can be seen as follows. Suppose that an electron is in the HOMO of a slicesomewhere along the CNT and is going to move to the left. Further suppose that, as in this slice,the adjacent slice, where the electron is going, is in its ground state, and so all electronic states20It should be noted that both Figs. 5.5 and 5.6 are derived from the same calculations. This section and theseplots are included as they add a di erent perspective and interpretation of how the deformation of the CNT con neselectrons21Recall this motivated the de nition of the band gap energy as Eg = EL  EH34Chapter 5. Modeling the Carbon Nanotube Crossup to and including the HOMO are occupied. In order for this electron to move to the adjacentslice, it must move to the LUMO of that slice, as this is the lowest available state. The lowerthe energy of this LUMO, the easier it will be for the electron to reach this LUMO, and so moveto the adjacent slice. Extending this argument to large numbers of electrons, it follows that,statistically, for sections with larger values of EL fewer electrons will be able to pass throughthese sections.With this interpretation of the LUMO energy, it is almost self evident that the plots in Fig.5.6 show intriguing behavior that appears to be local regions of con nement at the junction ofthe CNT cross, due to the local physical deformation of the CNT. For example, consider Fig.5.6(d), and imagine an electron travelling from left to right in the conduction band (skipping fromLUMO state to LUMO state). When it encounters the  rst rise in EL at x =  20 A, statisticallyit is less probable that it will continue in this direction, as it needs more energy to remain tothe LUMO state (`conduction band'). Now consider an electron in one of the two minima atx =  5 A. If the height of the three triangular peaks (at x =  10;0;10 A) are high enough, theymay be considered as tunnel barriers, trapping this electron in a very small region of around1 nm. This would be a Quantum Dot.These minima in EL are present in all plots in Fig. 5.6, and are all potential regions ofcon nement. However, it is impossible to know if the height of these tunnel barriers is su cientto con ne any appreciable number of electrons. To resolve this question, the transport propertiesof the CNT must be calculated.Returning to whether a QD can be induced in a CNT cross formation, it must be said thatwhile promising, these results do not de nitively provide con rmation. However, they do providea sketch of how a QD might arise in a CNT cross. Examining Fig. 5.5(b) once again, the regionof the potential QD would essentially be the region over which local band gap changes occur:around 20 A on either side, for a total width of 4nm. The EL minima in Fig. 5.6, which representupward conduction band shifts, which can, if large enough, produce con nement. In particularthe `W' shape at the centre of Fig. 5.6(d), could well represent a QD with a width of around 1 nm.Thus, although I cannot conclusively infer the presence of a QD in these CNT crosses (whichwould require electronic transport calculations), the electronic structure change over extremelyshort ranges, on the order of nanometers, which is clearly caused by the deformation induced byCNT crosses, is certainly a positive indication that QDs may well occur in these structures.35Chapter 6Conclusions and Future WorkQuantum Dots have very intriguing and promising properties resulting from the con nement ofelectrons in all three dimensions. One important aspect of QDs is they have discretized energylevels, the spacing of which is inversely proportional to the size of the QD. Therefore, in order toobserve and exploit this discretization, very small QDs, on the order of nanometers are desired.Rather than attempt to use traditional microfabrication techniques at ever smaller, more chal-lenging length scales, this work proposed to exploit the inherent small size of carbon nanotubes.As CNTs are already `one dimensional structures', it is only needed to axially con ne the electronsin the CNT, to achieve three dimensional con nement. A structure has been proposed, namelythe carbon nanotube cross, which is expected to display excellent QD behavior at the junction ofthe CNT cross, and will have an extremely small characteristic size.As may be evident from the phrasing of the above paragraph, I am not able to report afunctioning QD in a CNT cross structure. However, much progress has been made, and thegroundwork for further work has also been well established. An experimental methodology hasbeen laid out, and this aspect of the project is currently a work in progress and is in the fabricationstages.The CNT cross was also investigated computationally, using classical mechanics (MolecularDynamics) to calculate the geometrical con guration of the CNT cross and quantum mechanics(Density Functional Theory) to calculate its electronic structure. It was found that there aremany meta stable con gurations of the CNT cross, corresponding to di erent types of CNT, aswell as di erent manners in which the CNT cross was arranged. As each of these states showedstable equilibrium energy behavior, it was concluded that all of these states are physically realistic,and therefore worth investigating.Using electronic structure calculations, I was able to demonstrate a one-to-one spatial corre-lation between local physical deformation, and local electronic structure change. The calculatedband gaps are very similar to experimentally measured values, and the band gap changed sig-ni cantly at the junction of the CNT cross, over a region as small as 1   2nm. Furthermore,signi cant conduction band shifts (based on the LUMO energy) were observed, over regions ofsimilar sizes. Potentially these local electronic structure shifts could be adequate potential bar-riers to locally con ne electrons, resulting in a QD.The rich potential of the CNT cross that has been demonstrated in this work certainly justi esfurther research. I was not able to report any de nitive results as to whether the type of CNT,or the width of the CNT cross, has any bearing on its electronic structure. To further investigatethis I would propose to collect more data points along the CNT axis, gaining  ner resolution,as well as run simulations of a wider variety of Gaussian pro les, and other types of CNTs.Furthermore, it would be worthwhile to investigate using some less computationally expensive36Chapter 6. Conclusions and Future Workmethods to calculate the electronic structure of the entire CNT22, examine the resultant localdensity of states, and extract the band structure of the whole system from this information.Finally, the de nition of the band gap energy adopted in this work is fairly primitive, and is verysensitive to local errors or irregularities in the band structure, either physical or computational.It would be bene cial to develop a more physical and robust de nition of the band gap energy.All of these suggestions are fairly straightforward, but would quite probably yield some veryinteresting results, and some of this work has already been started.In order to truly validate the CNT cross as a QD device, two things would be required:simulating the electronic transport properties of the CNT cross, and growing and measuringCNT crosses. In both cases, staircase-like current-voltage curves would have to be observed tocon rm that the CNT cross displays QD behavior. And while this work has not provided thesede nitive curves, it hopefully has provided enough evidence to suggest that it is probable thatCNT crosses do exhibit QD behavior at their junction, and that further investigation of thesestructures, both experimental and computational, is merited.22This was tried using semi-empirical methods, but convergence could not be achieved. I have been working onmodifying the simulation to remedy this.37Bibliography[1] Y. Alhassid. The statistical theory of quantum dots. Rev. Mod. Phys, 72:895, 2000.[2] R.C. Ashoori. Electrons in arti cial atoms. Nature, 379:413, 1996.[3] P. Avouris, J. Appenzeller, R. Martel, and S.J. Wind. Carbon nanotube electronics. Proc.IEEE, 104:1772, 2003.[4] P. B. Balbuena and J. M. Seminario. Molecular Dynamics: From Classical to QuantumMethods. Elsevier, Amsterdam, 1999.[5] J.V. Barth, G. Costantini, and K. Kern. Engineering atomic and molecular nanostructuresat surfaces. Nature, 437:671, 2005.[6] P. Bhattacharya, S. 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