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Micro Raman Study of Filled Double Walled Carbon Nanotubes By Tom Hartsfield A Thesis In Partial Fulfillment of the Requirements for the D egree Bachelor of Arts, Concentration in Physics Written Under the S ponsorshi p of Professor Mariana Sendova S arasota, Florida, 2010
ii Table of Contents: Chapter I: Introduction Carbon Nanotubes ................................ ................................ ............. 1 1. Nanotechnology ................................ ................................ ................................ ............................... 1 2. Why do we study Carbon nanotubes? ................................ ................................ ........................... 4 3. Structure of carbon nanotubes ................................ ................................ ................................ ...... 6 Chapter II: Theory of Phonon s and Raman Spectroscopy ................................ ..................... 10 1. Phonons ................................ ................................ ................................ ................................ .......... 10 2. Theoretical phonon models ................................ ................................ ................................ .......... 13 3. ................................ ................................ ................................ ........... 15 4. Resonance Raman spectroscopy ................................ ................................ ................................ .. 17 5. Raman spectrum of carbon nanotubes ................................ ................................ ....................... 20 Chapter III: Experimental Techniques ................................ ................................ .................... 23 1. Creation and filling of DWCNT ................................ ................................ ................................ .. 23 2. Acquiring R aman spectra ................................ ................................ ................................ ............ 27 Chapter IV: A Theoretical Model of the Nanotube System ................................ .................... 30 1. An introduction to our model ................................ ................................ ................................ ...... 30 2. Nanotube coupled oscillator model ................................ ................................ ............................. 35 Chapter V: Data Analysis and Results ................................ ................................ ...................... 38 1. Extraction of peak data from Raman spectra ................................ ................................ ............ 38 2. Fitting of peak data to phonon models ................................ ................................ ........................ 41 3. Composition of the nanotube samples ................................ ................................ ......................... 43 4. Results of temperature dependent phonon frequency fits ................................ ......................... 46 5. Results from the coupled oscillator model ................................ ................................ .................. 54 6. Brief numerical analysis of the coupled oscillator model ................................ .......................... 55 7. Conclusions ................................ ................................ ................................ ................................ .... 59 References: ................................ ................................ ................................ ................................ ... 62
iii List of Figures: 1.1 Three types of nanoparticles 1.2 Computer rendering of a carbon nanotube 1.3 Graphene lattice and unit cell 1.4 CNT Unit cell 1.5 CNT 2D electron density of states, 1D DOS 2.1 Phonons: transverse and longitu dinal 2.2 Stokes scattering 2.3 Conceptual Schematic of a Raman spectrometer 2.4 Energy level transitions 2.5 Raman spectrum of an empty DWCNT sample 2.6 Radial Breathing Mode phonons of a 10,10 CNT 2.7 Three G band phonon modes 3.1 HRTEM of DWCNT sample, diameter distributions of sample 3.2 SEM of DWCNT sample before acid wash 3.3 Se filled DWCNT experimental sample image with laser spot marker 4.1 Three dimensional CNT model translating to 1D nearest neighbor model 4.2 Single mass and spring oscillator sy stem 4.3 Three spring two mass system with identical masses and springs 4.4 Three spring two mass system with different masses and springs 5.1 Typical D band and Lorentzian fit with linear baseline 5.2 Typical G band four Lorentzian fit 5.3 Kataura plot fo r nanotube diameter 5.4 Unfilled nanotube sample D band data and anharmonic phonon decay fitting
iv 5.5 Plot of D band center location vs temperature, anharmonic phonon decay fit and resulting fit parameters for each filling material 5.6 Unfilled nanotube sam 5.7 resulting fit parameters for each filling material 5.8 ecay fit and resulting fit parameters for each filling material 5.9 Unfilled nanotube sample G band data and anharmonic phonon decay fitting 5.10 Plot of G band center location vs temperature, anharmonic phonon decay fit and resulting fit parameters for ea ch filling material 5.11 Oscillator model parameter a 2 vs parameter b 5.12 Oscillator model parameter a 2 vs parameter a 3 5.13 Variation in a 2 with error in
v List of Tables: 1.1 Summary of some of the properties of CNTs 5.1 RBM analysis for nanot ube chirality and diameter results 5.2 Parameters and results from the coupled oscillator model 5.3 Error analysis of a 2 5.4 Comparison of Raman band center location fit anharmonicity parameters
1 Chapter I: Introduction Carbon Nanotubes 1. Nanotechnology Ov er the past few years, a new phrase has entered the vernacular of scientists and followers of scientific advances: nanotechnology. To futurists, nanotechnology is a panacea of engineering marvels; new medical miracles, chemical processes, semiconductor de vices, energy concepts, aerospace materials and consumer applications all spring forth. A more precise definition our purposes let it be defined as by N. Tanaguchi 1 The processing of, separation, consolidation, and deformation of materials by one atom or by one molecule." described by t he same laws we are familiar with in everyday life; their important properties are heavily dependent upon the behavior and position of every constituent atom. Our basic objects become individual molecules. Quantum mechanical effects become apparent and u navoidable in methods must be used which rely on smaller wavel ength incident materials such as electrons pushed by the electron cloud around the sensed atom. To move the same structure, clever methods of minute electric currents and single molecules that move like pincers in response to them are employed.
2 comprehend, much like the enormous size scales of space. If the width of a single human hair was the diameter of a football field, one of these tiny objects would still only be the size of a coarse grain of sand or a small pebble. If humans were nanoparticles, all the human beings living on Earth could stand shoulder to shoulder on the head of a nail. Figure 1.1: Three types of nanoparticles: Niobium nanoclusters , Silicon nanowires, buckminsterfullerene Within this size range, several new classes of nanoparticles have been discovered and intensively studied in recent y ears. Some of the first were the clusters, groups of between three and several million atoms of the same element, which increasingly demonstrate quantum effects as the number of atoms decreases. Buckminsterfullerenes, for example, are hollow polyhedrons of 60 or more carbon atoms. Other nanoparticles are similarly geometrically arranged simple groups of atoms. In the nanoscience realm, very often the boundary between nanoparticle and molecule is blurry. Quantum dots (or other spherical nanoparticles) a re zero dimensional objects of semiconducting material which have applications in optics and quantum computing, while nanowires are one dimensional objects of conducting, semi conducting or insulating material
3 which are heavily researched for electronics u ses. Various organic molecules and polymers can usefully be manipulated at the atomic level. Cylindrical nanotube structures including carbon nanotubes are another type of one dimensional nanoparticle, probably the most commonly researched of all of thes e materials. In this work, we focus our study on carbon nanotubes (CNTs) and their properties. Figure 1.2: Computer rendering of the atomic structure of a carbon nanotube 
4 2. Why do we study Carbon nanotubes? Carbon Nanotubes (CNTs) are hollow cyl inders of hexagonal lattice carbon atoms, which can be thought of as rolled up sheets of graphene planar hexagonal lattice carbon. First famously reported by Iijima 4 in 1991 (though observed earlier), these new structures have rapidly become a subject of strong interest to physicists and engineers. T hey have extraordinary material properties (see Table 1.1), including stiffness similar to that of diamond and the greatest tensile strength of any known material. They have extraordinary heat conduction cap ability as well as rich electronic structure. CNTs exist in both metallic and semiconducting varieties. Semiconducting tubes have an extraordinary variety of band gap widths from 0 to 1.2 eV 3 and conducting nanotubes can carry very high currents. Experi mentally Measured Property Carbon Nanotube Comparable material 1.0 1.5 5,6 Diamond 1.2 Tensile Strength (MPa) 150,000 7 Titanium 900 Specific Strength (kNm/kg) 110,000 Kevlar 2,500 Bulk Metallic Current Carrying Ca p. (A/m 2 ) (10 9 10 10 ) 8 Copper (5*10 6 ) Table 1.1: Summary of some of the properties of CNTs Carbon nanotubes have found applications in an ever widening array of places ranging from advanced technical scientific equipment to relatively inexpensive cons umer products. They have found electronics applications such as field effect transistors and printed flexible circuits with nanotube based inks 9 for greater charge carrying capacity; the same ability to be put into a printable ink has allowed their use in prototype printable solar cells. They have also been used to
5 create simple photodiode cells and as wires in more conventional photovoltaic cells. CNTs have been used as atomic force microscope (AFM) tips. The consumer industry is starting to see some c arbon nanotube products as well, although at this time these are predominantly performance and luxury sporting goods such as baseball bats and cycling components. Striking proposals exist for future uses of carbon nanotubes. Researchers have developed prototype paper thin flexible battery cells 10 composed of cellulose paper with nanotubes embedded as electrode and electrolyte components. The Japanese government and NASA are working on designs for a space elevator, an enormous cable over thirty five tho usand kilometers long with the capability to withstand enough tensi on to be anchored and held tau t with a counterweight in space and have cargo brought into orbit up its length 11 Nanotubes have been found to be effective as a reinforcing material, to inc rease the strength of solid materials such as concrete, polyethylene and other carbon composites when imbedded in them. Ultra strong fiber for textile applications can also be made with nanotubes, clothing and bullet proof vests as examples being designed
6 3. Structure of carbon nanotubes Carbon nanotubes can be categorized into distinct groups by their physical properties. A given nanotube structure consists of a certain number of coaxial hollow cylindrical shells. These shells have the molecular s tructure of a single planar sheet of graphene (hexagonal lattice carbon atoms) rolled up and bonded edge to edge (See Fig. 1.2). A single walled carbon nanotube has only one cylinder; multi walled carbon nanotubes may have two, three or more tubular layer s. Multi walled nanotubes are most commonly created on the order of three to thirty nanometers in diameter, single walled nanotubes tend to be smaller, around one to two nanometers. Both are commonly found in lengths up to 1 (or 10000 times their diame ter), nanotubes up to the range of centimeters long (roughly one million times their diameter!) have recently been created. At the end of the tube a cap is generally formed. Caps vary in structure; they can be fullerene in shape, consisting of a symmetri cal C 60 like sphere section, a smaller roughly hemispheric shape with higher curvature or a less symmetrical group of atoms which bridge across the open end of the tube. Figure 1.3: Graphene lattice and unit cell. The lattice vectors are a 1 = and a 2 = (1) a c c a 1 a 1 a 2 C a = 2.46 C C = 1.42
7 The unit cell (smallest area of lattice that can be continuously tiled without rotation to form the entire lattice) for the graphene lattice, contains two atoms. However, once the graphene sheet is rolled to form a CNT, th e unit cell becomes much larger. Figure 1.4: CNT unit cell The electronic properties of a given nanotube can be determined entirely by its chirality. The chiral vector C h is a vector which stretches across the graphene plane from an origin at one carb on atom to a terminus at another atom which would occupy the exact same position in the lattice structure once rolled up into the tu be. The CNT is described by the components of C h in the a 1 and a 2 basis. The components are a pair of integers called the n and m indices. Nanotubes which satisfy the relation (2n+m)mod3 = 0 are metallic conductive and the rest are
8 semiconducting. Nanotubes with chirality (n,0) are known as zig zag nanotubes, while those with chirality (n,n) are known as armchair nanotubes Armchair nanotubes are always metallic. All other CNTs fall into a range between these two and are called chiral. A second vector, called the translation vector, T can be created as the shortest vector perpendicular to the chiral vector that ends at a lattice point (Carbon atom nucleus location). It is then described in terms of indices of the same lattice vectors a 1 and a 2 The translation vector C h an d T and the endpoint of C h +T is the unit cell of the nanotube, the smallest area which can be repeatedly tiled to construct the entirety of the lattice of the CNT. If a rectangular strip with a long side along the translation vector and with width C h (Fi g. 2.3 above) is cut out of the graphene and rolled into a cylinder the nanotube can then be constructed (excepting the endcaps). The electronic structure of carbon nanotubes is somewhat more complicated to calculate 12 Nanotubes with diameters of above roughly one nanometer are considered to effectively possess the same electronic configuration as bulk graphene, namely purely sp 2 bonds (very small nanotubes may start to have some sp 3 bond components). Using an approximation known as the tight binding me thod, the electron energy bands (and resulting band gap for semiconducting nanotubes) can be calculated. The approximation made is to unroll the curved lattice of the nanotube (as it holds the cylinder shape) into a flat two dimensional surface. For nano tubes of roughly .6 nm and larger this gives an increasingly good approximation as the diameter increases
9 and the curvature correspondingly decreases. The final function that describes the electron density of states achieved from these calculations is: (1.1) Where k x and k y are the wave numbers of the electrons (see Figure 1.5 below). Figure 1.5: Left: A plot of the two dimensional electron DOS . Right: One dimension DOS plots for three nanotubes demonstrating a conducting nanotube (cen ter) and two semiconducting nanotubes . Using this density of states model, the gap between the top of the lower energy valence and the bottom of the higher energy conduction bands can then be determined.
10 Chapter II: Theory of Phonons and Raman Spec troscopy 1. Phonons The materials we investigate in this work are crystalline solids. That is, they are rigid, periodic lattices of atoms chemically bonded together. We will make the approximation that the lattices are perfectly free of defects. Waves may travel through this lattice by periodic displacements of the atoms away from their equilibrium lattice points. These waves may be either transverse or longitudinal. Figure 2.1 : Phonons. Left  transverse phonon modes. Right  longitudinal phonon. Phonons are collective wave excitations of the lattice. The vibrational state of the lattice is characterized as the sum of all phonons in each individual oscillating state. Allowable phonons span the possible set of periodic resonances in a lat tice. This is what allows all lattice waves to
11 then be constructed by combination of individual normal mode phonons. The phonon solution below is derived entirely using classical mechanics. While we can describe simple models for the waves themselves wi thout requiring quantum mechanical explanation, a phonon actually must be a quantum mechanical entity. This is because on the scale of atoms, quantum mechanical effects are significant; the energy of a lattice vibration must be quantized. Thus a phonon i s defined as a quantum of energy of a lattice vibration mode. To create a model for phonons, we can describe the interaction between any two atoms as a potential energy function V(r). As a first approximation we can simplify this potential as a simple har monic potential: V(r) = Ar 2 where r is the distance of the atom from its undisturbed position in the structure (x x 0 ). This potential can be derived by modeling the forces as simple F = k ( x x 0 ) and considering Inter estingly, the same potential approximation is also arrived at by considering the first two terms of the Taylor series expansion about x 0 of any potential with a minimum at point x 0 (a central force potential): (2.1) w here primes indicate derivatives. The linear term drops because the derivative at the minimum must be 0. The second derivative evaluated at x 0 and constant form A. The equation of motion for this potential can be found by solving the differential equation:
12 (2.2) This system has a resonant frequency of oscillation, which is equal to
13 2. Theoretical phonon models In more complex realistic situations, the simple harmonic potential model of the phonon described above loses its abilit y to describe observed events. Harmonic theory does not allow for interaction between phonons or the decay or change in time of a single wave 16 To analyze decay and interaction effects, we consider a new anharmonic potential with higher order terms. (2.3) Anharmonic models based off of the classical harmonic oscillator can be used to describe quantum mechanical effects. The Debye temperature (T D ) of the G band optical mode frequency (T) from 80 K to 720 K. This makes t he occupation number (N) for the harmonic oscillator model phonon energy states of the lattice very small: (2.4) We are in the quantum mechanical limit of the classical harmonic oscillator. The model that we use i n this work is a theoretical model for cubic anharmonic decay of phonons by Klemens 17 and its extension to quartic decay by Balkanski, Wallis and Haro 18 It is beyond the scope of this work to fully quantitatively derive the expressions used, but some com ments would be worthwhile. The physical process corresponding to the cubic anharmonicity term described by Klemens is the decay of the original phonon to two phonons of opposite momentum; a three phonon total process. An approximation (suggested by Kleme ns) is made
14 that the two final phonons are of identical frequency, each having one half the frequency of the initial phonon. The final result of calculating the frequency shift including the third order term is a temperature dependent function for the fre quency of the initial excited phonon : (2.5) Where is the observed frequency of the phonon, is a fitting parameter corresponding to the phonon frequency, C is fitting parameter corresponding to the cubic anharmonicity contrib ution to the phonon decay, the Boltzmann constant, the reduced Planck constant and T the temperature. An extension of the model to include quartic anharmonicity with the cubic anharmonicity works similarly to the cubic model. Here the initial phonon decays into three phonons. Here we again use equal to The relation that follows from calculating the frequency shift includ ing the third and fourth order anharmonic terms (where D is the new coefficient for the fourth order term ) is: (2.6) where A final equation to describe observed phonon frequencies as a function of temperature from their observed Raman spectroscopic fe atures will be introduced in chapter V.
15 3. scattering theory Nearly all light that is reflected by a surface is the result of an elastic collision where no kinetic energy is lost into vibration of the surface. A ray strikes the surfac e and reflects at an angle equal to the incoming angle reflected over the normal to the surface The reflected light has the same frequency as the light that came in and impacted. Before the advent of quantum theories this was all that anyone thought to look for. various substances 19 investigating the specific origins of Raleigh scattering in different liquids. Through years of careful observation he noti ced that always a very tiny fraction (roughly one out of a million photons) of the light reflected from a surface were of a lower frequency than the incident light. This frequency change was large enough to be caused by the loss of energies necessary to s tretch atomic bonds. Concluding that this was a new phenomenon, Raman published first his discovery of a single frequency shift made by analyzing only reflected sunlight. He began to use a Mercury lamp and observed that different liquids reflected severa l different shifted frequencies, representative of the individual structure of the molecules present. Realizing that this scattering must not be elastic or there would be no reflected light of differing frequency, models that correctly described how the energy of the light changed were investigated. Raman did not contribute much to the further work on vibrationally shifted scattering beyond the cataloging of different spectra, and others discovered and explained most of the details of its workings
16 There are two basic types of Raman scattering: Stokes and anti Stokes. In Stokes scattering, the incoming photon loses a certain amount of energy which generates phonons. Anti Stokes scattering occurs when the photon gains energy from phonons already in the m aterial. Since the phonon energy is quantized, incoming photons gain or lose only discrete amounts of energy. Thus these two processes occur with the same frequency change (Stokes scattering is red shifted by that frequency and anti Stokes blue shifted th e same amount). The anti Stokes scattering however is less likely to occur so we only record and analyze the Stokes scatters in our work. Figure 2.2: Stokes scattering process a redshifted photon is produced
17 4. Resonance Raman spectroscopy In the year focusing optics were used to take measurements of the Raman spectra of materials. It was not until the invention of the laser in 1960 however that Raman spectrometers became wi despread commercially. The very simplest conceptual schematic of a Raman backscattering spectrometer works something like this. A monochromatic beam is directed towards sample material. This beam interacts with the surface and a fraction of the incoming photons are scattered. The optics above the sample filter and record the photons that come back out of the sample. Filtering is done to remove the Rayleigh scattered photons, the blackbody radiation photons that naturally radiate from the sample due to its temperature and the general noise photons (from environmental light and internal electronic noise). A count is maintained of the number of photons emitted across the spectrum of frequencies. Modern micro Raman spectrometers have many additional featu res. They are generally built onto existing optical microscopes, sending light into and out of the objective lenses of this host microscope and including features such as the variable focusing and optics settings of the host microscope and additional opt ics equipment.
18 Figure 2.3 : Conceptual schematic of a Raman spectrometer. Traditional Raman spectroscopy excites the material into a virtual state a s hort lived unobserv able state that lies below any electronic transition and is extremely short lived. Resonance Raman spec troscopy is a particular type of Raman spectroscopy which gives much stronger response peaks. In this technique, incoming photon energies are chosen to excite a virtual state above an electronic excited state in the material 20
19 Fig ure 2.4 : Energy level transitions. Compiling a count of all the Raman scattered photons and sorting them by the Raman Shift the change in wavelength is to gather a Raman spectrum. This spectrum is usually plotted as a function: number of p hotons counted as a function of Raman shift. The spectral resolution of the spectrometer used in this experiment is less than one nanometer.
20 5. Raman s pectrum of carbon nanotube s 21 A band in the frequency spectrum occurs in an area around a photon fr equency redshift that matches exactly the correct energy to excite a certain phonon process (or multiple processes with very similar to tal energies) in the material. For carbon nanotubes, the Raman spectrum is relatively complex, containing multiple bands In our work, we focus our analysis on the four most intense bands in the Raman spectrum. These are (see Figure 2.5 below) the Radial Breathing Mode (RBM) band, located from the low frequency cutoff (110 cm 1 for our system) up to roughly 300 cm 1 the D band, which is located around 1300 cm 1 the G band, centered around 1580 cm 1 and finally the G Prime or 2D band at around 2600 cm 1 Figure 2.5 : Raman Spectrum of an empty DWCNT sample. Note the frequency shift increases to the left. The four mai n band areas which we investigate are shown here.
21 The RBM band spans more than 200 cm 1 because each different nanotube diameter has a slightly different resonant frequency in this range. The phonons which give rise to the RBM shift can be visualized as r adial stretching and contraction of the nanotube (altering the diameter of the cylinder). Figure 2.6 : RBM phonons for a (10,10) nanotube (adapted from ). The individual peaks in the RBM band are hard to individually resolve due to the convoluted na ture of this band in any sample consisting of more than one type of nanotubes. The D (for disorder) band is simpler in composition, consisting of only one component. This band is a result of defects present in the lattice structure of the nanotubes. Def ects include missing or substituted atoms and grain boundaries. Finite size effects (the area of phonon excitation being near to the end of a nanotube, where the infinite size approximation of the lattice fails) also contribute. Because of this property it can be used to detect and compare defect
22 despite being a secon d order effect, still occurs in samples lacking in D band excitations. This is because it is also an allowed mode in systems with correct symmetry. In the case of CNTs the Finally the G ( for graphite) band is the dominant spectral feature of carbon nanotubes. This mode is associated with tangential stretching phonons in the nanotube (increasing and decreasing the length of the nanotube). Figure 2.7 : Three G band phonon modes demonstrat ing tangential stretching (adapted from ). This band is a composite of several lines with between 1500 cm 1 and 1660 cm 1 The G band is intrinsic to all sp 2 bonded carbon materials and is the the most intense band. It consists of a number of differ ent theoretical lines The qualitative features of each band will be looked at in greater detail in chapter V.
23 Chap ter III: Experimental Techniques 1. Creation and filling of DWCNT Three main methods 22 are commonly used for creating carbon nanotubes. Nano tubes were first discovered as an unintentional byproduct of the arc plasma jet method for creating fullerenes. In this method, a large arc of current in a helium atmosphere is passed between two graphite electrodes, one of the electrodes is doped with ni ckel and yttrium. The doped electrode evaporates and carbon soot containing carbon nanotubes forms on the other electrode, producing nanotubes at the rate of several g/min. A problem here however is that a large amount of other allotrope forms of carbon is also present in the soot and further purification methods are necessary. A second method for creation of nanotubes is laser ablation. A laser beam is directed onto a graphite surface mixed with a catalytic metal. The atmosphere is heated to a tempera ture of 1200 C and argon gas is pumped along the direction of the beam. As the ablation occurs carbon particles and nanotubes are formed and carried along the argon flow to a cooled copper collector. This process however also forms roughly five to ten percent other structures such as fullerenes which must then be removed. The carbon nanotubes used for the work described in this thesis are created via a variant of the third method chemical vapor deposition (CVD), known as catalyzed ch emical vapor deposition (CCVD). Chemical vapor deposition methods have been known and used in various semiconductor fabrication processes for decades. The basic technique is to flow a gas across a substrate and cause some of the gas atoms/molecules to be deposited on the surface of th e substrate which is
24 heated to around 700 C. A gas mixture containing both the volatile material to be deposited and other inert gases is flowed into the deposition area or chamber, usually at a high temperature. The component of the targeted gas intera cts with the substrate in the deposition chamber and is deposited onto the surface out of the gas flow. The non interacting gas and other byproducts of the main deposition process are then carried out of the chamber by the flow. Nanotubes are nucleated s tarting from particular sites on the substrate where the catalyst particles are located. The catalyst particles either remain on the substrate at the base of the forming nanotube or are carried along the tip of the nanotube as it grows. This process is o ften used to grow thin films and layers of both amorphous and crystalline metals, semiconductors and other compounds. Catalyzed C VD utilizes an additional chemical on the surface of the substrate with the metal nanoparticles to further catalyze the growth of the desired deposition product. In the case of the carbon nanotubes in this work 23 a catalyst of Mg 0.99 Co 0.075 Mo 0.025 O was used, where the subscripts indicate the relative percentages of the elements in the catalyst material. After the growth is com pleted, the nanotubes are washed in a weak acid to remove any remaining catalyst material. The final product of the CCVD growth process is a batch of carbon nanotubes which are 77% double walled, have a median inner wall diameter of 1.35 nm and a median o uter wall diameter of 2.05 nm and have extreme diameter ranges of roughly one nanometer greater and less for each wall. The average surface area is an amazing 800 m 2 /g (23)
25 Figure 3.1 (Reproduced from ) : (a) HRTEM image of DWCNTs, (b) distribution o f the numbers of walls for the whole population (established from 96 individual CNTs). Distribution of inner (di) and outer (do) diameter for the whole population of CNTs (c) and for DWNTs only (d). Figure 3 2 : Scanning electron microscope image  o f grown carbon nanotubes before the acid wash (still some catalyst visible)
26 In order to fill the carbon nanotubes, they are subjected to a second process 24,25 The guest materials we worked with are the elements Se and Te and the diatomic compound PbI 2 The process of the filling begins with physically grinding together the empty nanotubes and the filling material. This ground mixture is then vacuum sealed in a quartz ampoule and heated in a furnace at the rate of five K/min to a temperature well above the melting point of the filler (all fillers have melting points far below that of the DWCNT). The mixture is maintained at this temperature for 24 hours and then cooled to a temperature below the melting point at a rate of .1K/min. Finally the mixture i s cooled back to ambient temperature at the rate of 1 K/min. The resulting combination of nanotubes and excess filler material is then washed in a mild solution, differing for each material, to remove that extraneous filler remaining outside of the nanotu bes.
27 2. Acquiring Raman s pectra The next step in our work is acquiring Raman spectra in the range of 110 cm 1 to 2850 cm 1 of each type of carbon nanotubes filled and unfilled. We begin with empty reference nanotubes and all varieties of filled na notubes in fine powder form. From each sample a miniscule sample portion is drawn out and placed onto a quartz slide. This slide is then placed into a temperature controlled chamber with a glass viewing window. The chamber is placed with its viewing win dow under the objective lens of the Raman spectrometer microscope and interfaced with a heater and cooling system connected to a liquid nitrogen Dewar, allowing for temperatures between 80 K and 730 K to be achieved. The Raman spectrometer is turned on, t he x axis (Raman shift) and y axis (intensity) are calibrated. The x axis is calibrated by measuring known emission lines of neon; the y axis is calibrated using an NIST measured spectrum of a white light source. The wavelength of the laser is also calib rated to a tolerance of less than .1 nm using the Raman shift of the most intense line of Toluene. The software receiving the photon counts from the CCD then modifies t he counts to negate the differences between the observed spectra and the known ideal ca libration spectra. The sample is set initially at 80 K and allowed fifteen minutes to reach thermal equilibrium. A particular spot 16 in size (the size of the laser spot) is selected based on maximizing the Raman signal for the temperature study and an image saved of this spot (Fig. 2.3). Special care was taken to insure that Raman spectra were all taken from exactly the same spot at each temperature.
28 Figu re 3 .3 : The Selenium filled sample and the on it The entire laser spot itself e xtends roughly one quarter of the way to the first mark from the center on the reticule in each direction A spectrum is then taken using an eighty second standard sequence. Forty seconds of machine noise collection (laser off, detector shutter closed), twenty seconds of dark sample noise (laser off, detector shutter open) and concluding with a twenty second exposure (laser on, detector shutter open) time at a given temperature. The raw light input data entering the spectrometer from the sample during th e laser exposure is recorded by a CCD. This measured data then has the previously recorded noise subtracted out and is translated into a computer file of intensity vs. Raman shift pairs. After the each spectrum is taken, the temperature is raised twenty or thirty
29 Kelvin and fifteen minutes again allowed for thermal equilibrium. The exact same spot location is again manually found and focused upo n using for reference an image of the spot produced in the first spectrum capture and a new spectrum is taken. At roughly 200 K the temperature increments are increased to thirty and forty Kelvin respectively. Great care is taken to acquire each spectrum under the same time cycle of heating procedures. Spectra are taken up to 730 K for all samples except Seleniu m which reaches melting point at only 494 K. At the conclusion of this work a set of between sixteen and twenty five complete Raman spectra are obtained of each spot. These temperature dependent sets of Raman spectra are then mathematically analyzed and compared.
30 Chapter IV: A Theoretical Model of the Nanotube System 1. An introduction to a new model In this work, we are looking to determine the strength of the interaction between the inner wall of the carbon nanotube and the filling material. T his system allows us to make two simple approximations that we can harness to build a model. The first is an interaction potential between atoms that is dependent only upon distance, and in a simple way. The second is to ignore all interactions beyond th e nearest neighbors. Figure 4.1: Three dimensional nanotube system translates to one dimensional nearest neighbor interaction model. The rings are the nanotube walls, blue circles are c atoms, red circles guest atoms and double arrows are interaction s. Using a harmonic potential for our simple interaction potential, we look at spring block models. to consider a physical spring and mass system.
31 Figure 4 .2 : Single mass and spring oscillator system The strength of the harmonic interaction is represented by the spring constant k and the atom by the mass m. The equation of motion for this system is: (4.1) where a dot indicates a time derivative. We ar rive at the same equation of motion as previously described for the harmonic oscillator. Adding more complexity to this system, we consider two identical masses connected to each other and to external motionless endpoints by three springs, each of identic al spring constant k. Fig. 4.3 : Three spring two mass system with identical masses and springs For this system, we observe two degrees of freedom: the position of each mass in the system. In the Lagrangian for this system, there are two kinetic energ y terms (one for each mass) and three potential energy terms (one for each spring).
32 (4.2) (4.3) Solving the Euler Lagrange equations for each generalized coordinate of this system: (4.4) (4. 5 ) (4. 6 ) We see that there is a second term in both equations containing the coordinate of both masses, which couples the motions of the two masses. Considering this we look to find the normal modes of the system. Normal modes are the patterns of motion where all the masses in the system oscillat e sinusoidally at the same frequency. To do this, we set up our equations of motion as eigenvalue problems. Defining two matrices M and K which contain the masses and spring constants, we write out the equations of motion in matrix form: (4. 7 ) (4 8 )
33 (4. 9 ) (4. 10 ) Eigenvalue equation for matrix A with eigenvalue and eigenvectors : (4. 11 ) Since our equations of motion are homogeneous second order linear differential equations, we know the solutions will be o f the form: (4. 1 2 ) W here and are constants. Taking A and B to be unity, the eigenvalue equation becomes (4. 13 ) We then solve the characteristic equation (determinant) for the eigenvalue 2 This equation is a quartic, but fortunately it can be easily separated into two q uadratic terms which can be solved individually. (4. 14 ) (4. 15 )
34 W e can see that the roots of Equation 4.15 are (4. 1 6 ) The lower frequency solution physically represents both masses moving in phase. They are at the same displacement from equil ibrium, moving in the same direction with no compression or extension of the spring between them at any time. The higher frequency solution describes the case when the two masses oscillate out of phase opposite and identically about the center. The oute r springs compress and expand mirror symmetrically about the center and the center spring compresses and expands at twice the rate of the outer springs.
35 2. Nanotube coupled oscillator model To estimate the carbon guest interaction in the nanotube sy stem we are investigating, we propose this model. We consider one mass to be a carbon atom in the inner wall of the nanotube and one mass to be a guest atom in the filling material. The three springs constants (from left to right) represent the strength of the carbon carbon interaction (k 1 ), the strength of the carbon guest atom interaction (k 2 ) and the strength of the guest atom guest atom interaction (k 3 ). In this model we consider the masses to be unequal and the springs of uneven strengths. Figure 4.4 : Three spring two mass model with d guest in the nanotube model. As before we set up the Lagrangian for the system: (4. 17 ) Applying the Euler Lagrange equation to each coordina te: (4. 18 ) (4. 19 )
36 (4. 20 ) We use the matrix formulation and solve for the eigenvalues considering the same differential equation solutions as before This time, the K and M matrices are more complicated: (4. 21 ) (4. 22 ) Expanding 4.2 2: (4. 2 3 ) We make several substitutions before we find the eigenvalues. (4. 24 ) Solving, we have four real roots, pos itive and negative values of two root s : (4. 25 ) This expression gives us the normal mode frequen cies of the sys tem. We solve the expression with a computer algebra system for the variable we are interested in:
37 (4. 26 ) Fur ther simplifying (4.26) we can write: (4. 27 ) And finally: (4. 28 ) Equation 4.28 estimates the carbon guest atom interactio n in terms of the variables a 3 and b. Using this model we will then take the values experimentally measured for the resonance frequencies from the Raman spectroscopy data, the known interaction masses and carbon carbon and guest guest inter actions strengths and solve for the interaction strength between carbon and 0 are experimentally determined from the temperature data fits of the D a 3 is determined from the known bond st rengths between carbon nanotube walls and between guest atoms. b is estimated with the masses of the atoms. When the guest material is a compound, the average mass is used for b.
38 Chapter V : Data Analysis and Results 1. Extraction of peak data from Raman spectra Once taken, the temperature dependent set of spectra for a given filling material is imported into GRAMS AI, a spectrum analysis software. A consistent baseline for the spectra (zeroing the intensity value at a predetermined fixed point) was esta blished and the spectra were normalized to its highest intensity value (the maximum height of the G band). R ecursive mathematical fitting tools were then used to fit each spectral feature with mathematical line shapes known as Lorentzian functions, of th e form: ( 5.1 ) Where a is a constant C a constant magnitude coefficient and x 0 is the center of the peak. The Lorentzian line shape turns out to be most appropriate in this case because it describes the homogeneous broadening of a spectral line due to phonon electron coupling 26 We begin the fitting of each Raman band with exactly identical initial conditions at all temperatures for all spectra to insure consistency between results. Initial conditions for the fitting of a band consist of the left and right limits of the entire fit range on the Raman shift axis and the center location, width (FWHM) and height of each Lorentzian line shape in the fit. The initial conditions are first defined by starting with a random guess generated by the software whi ch is fitted recursively to the spectrum of the reference sample at 80 K. Starting from the initial condition of this first converged fit, we use the software to perform a mathematical
39 recursion algorithm until a stable fit is reached. We fit the D band Lorentzian. Figure 5.1 : (b ) A typical D band and its Lorentz ian fit with linear baseline; (a ) a typical band and its Lorentzian fit. The red line is the experimentally me asured spectrum and the black is the peak fit. Th e G band is fit with a group of four Lorentzians due to its complex shape. Four was found to be the minimum number of components which consistently converge and have spectral widths above 10 cm 1 for every sample type from the same initial condition in or der to produce the most comparable overall fits. For this band we concentrated on relative similarity and consistency in final fit over theoretical precision to facilitate comparison between the spectra, our main goal.
40 Figure 5.2 : Typical G band four Lo rentzian fit. Here the re d line is the experimental data and the black line mirroring it the sum fit of the four individual peaks which are grey This fitting is performed for the spectrum taken at each temperature for a given material. The center locat ion, height and width of the final line shape fit are then recorded for the fit at each temperature. The result of this work is a new set of temperature dependent data, showing peak features as a function of temperature for a given sample. This process i s performed on all four filling materials and the empty nanotubes.
41 2. Fitting of peak data to phonon models For our phonon frequency analysis, we look at several specific peak properties as a function of temperature. These are the band center locations of th peak in the G band fit and the width of the band. These sets of data are now input to a second mathematical software, ORIGIN. This software is used to fit the parameters from the temperature dependent phono n frequency model equations to the peak parameter vs temperature data points. ORIGIN uses a non linear regressio n method known as the Levenberg Marquardt algorithm to achieve function fits. It arrives at a solu tion by successive iterations, combining the Gauss Newton algorithm and the gradient descent algorithm. The Gauss Newton algorithm is the minimizing of the sum of the squares of the difference at each point. The gradient descent method takes the negative of the gradient of a function and moves in the direction of greatest change. To fit the relation between peak center locations determined by the pr evious Lorentzian fitting and temperature with our phonon equations, we vary the parameter and the coefficients A, B, C and D: D band thr ee phonon anharmonic decay peak center location with E quation 2.5 : peak center location with E quation 2.6 : where and
42 band width: ( 5. 2 ) G band center location with Equation 2 .5 : Our initial conditions for this fitting are guesses at the correct parameters. These guesses are simple, rough and need not be the same in every case; the algorithm converges identically to a given fit for any guess to which it conv erges acceptably at all. From these fits we obtain for each type of nanotube sample both a function which can be plotted with and compared to the data points, and the set of function parameters which have been converged upon by the iteration process to de scribe the phonon interactions within the system. Our fits cover specific frequency ranges in the Raman spectrum and our initial guesses for the fit parameters are the correct fit parameters for the unfilled D WNT sample (Figures 5.4, 5.6 and 5.9 ). For th e D band, the Raman shift range is 1270 cm 1 to 1350 cm 1 Raman shift range is 2450 cm 1 to 2700 cm 1 Finally, the G band is fit from 1500 cm 1 to 1660 cm 1
43 3. Composition of the nanotube samples Factors which al low for characteri zation of the nanotube sample s were examined. Manual analysis of scanning electron microscope images of the sample of empty CNTs showed that on average there was .025 nm of nanotube length per nm 2 of sample area. The measured diameter of the laser spot o n t (this was the smallest spot size that could be consistently be refocused upon for each spectrum) Multiplying the total spot size by the length of nanotube per nm 2 gives 2.0x 10 6 nm of nanotube length under the laser. If the average nan otube length is nearly 5x10 3 nm w e can estimate that 400 nanotubes are sampled by the laser Analyzing the RBM band of the nanotubes, we can estimate the different diameters sampled as well. Using the Kataura plot 27 (Figure 5.3), each RBM frequency can be associated with a specific chirality and diameter of nanotube. Finding the intersection on the plot of the photon energy ( for our laser, 1.58 eV 0.01 eV ) on one axis and the Raman shift on the other axis, a certain nanotube is found.
44 Figure 5.3: Ka taura plot for nanotube diameter. Specific chirality versions also exist but require much higher resolution to display. Energy Seperation refers to the energy of the Raman scattered photon. Analyzing the RBM of our spectra is difficult due to its compli cated nature. Several peak locations were resolved with a combination of manual and algorithmic Lorentzian peak fitting methods. It is important to note that these results are calculated for single walled CNT, not DWCNT.
45 Table 5.1 : RBM analysis for nan otube chirality and diameter composition r esults Here we can see that of the 18 possible theoretical peaks in this region, we observed eight. Of these nanotube types, one is armchair metallic (11,11) and the remain ing seven are semi conducting.
46 4. Resul ts of temperature dependent phonon frequency fits temperature. This is in agreement with previous findings for empty and Selenium filled DWCNT 28 All three bands exhibi ted this characteristic for the unfilled nanotubes and all filling materials. The anharmonic decay fits and their parameters can be compared quantitatively. The D band of the unfilled nanotubes is centered at 1299.6 cm 1 at 80 K and red shifts down to be low 1296 cm 1 as the temperature nears 700 K. The parameters from the anharmonic decay equation for these empty nanotubes are 0 = 1305 cm 1 and C = 6.0 .5 cm 1 Figure 5.4 : Unfilled nanotube sample D band, D band temperature dependence and anharmonic phonon fit.
47 Looking at the center locations of the D band for the filled nanotube samples, in each case we see a greater Rama n shift at temperatures below 270 K. The Selenium sample consistently shows much larger Raman shifts while the other samples red shift towards the unfilled 0 cm 1 C, cm 1 ( T= 0), cm 1 Se @ DWNT 131 4 9 1305 Te @ DWNT 1312 11 1301 PbI 2 @ DWNT 1310 8 1302 HgTe @ DWNT 1307 6 1301 DWNT 1305 5 1300 Figure 5.5 : Top: Plot of D band center location, three phonon anharmonic decay fit (Equation 2.5 ); Bottom: Table of fit parameters Filled circle s indicate Se filled sample, triangles indicate Te, Squares PbI2 and empty circles the unfilled DWNT.
48 The fit parameters agree with a qualitative assessment that the unfilled samples show greater Raman shift red shift with temperature increase. The Se anh armonicity coefficient is slightly higher and the PbI2 and Te anharmonicity coefficients are much higher. The 0 harmonic frequency is greater for each of the unfilled nanotubes as well. s red shift with temperature as well, from 2577 cm 1 at 80 K to 2569 cm 1 at 720K. The width of the band increases with temperature from 69 cm 1 to 81 cm 1 Figure 5.6 : n fit
49 1 and 9 cm 1 upshifted relative to the empty DWNTs and shows stronger temperature dependent anharmoni city with temperature increase. 0 cm 1 C, cm 1 D, cm 1 T= 0), cm 1 Se @ DWNT 1314 28 5 1291 Te @ DWNT 1312 26.5 4.5 1290 PbI 2 @ DWNT 1310 21.5 3 1291.5 HgTe @ DWNT 1307 19.5 3.5 1291 DWNT 1305 19.5 3 1288.5 Figure 5.7 : Top: Plot of anharmon ic phonon deca y fit (Equation 2.6 ); Bottom: Table of fit parameters. Filled circles indicate Se filled sample, triangles indicate Te, Squares PbI2 and empty circles the unfilled DWNT.
50 re, with larger anharmonicity coefficients as well. 0 cm 1 A, cm 1 B, cm 1 Se @ DWNT 1314 45 8.5 Te @ DWNT 1312 43.5 7.5 PbI 2 @ DWNT 1310 44.5 7 HgTe @ DWNT 1307 42 6 DWNT 1305 42 7 Figure 5.8 : Top: Plot of nharmonic phonon decay fit (Equation 5.2 ); Bottom: Table of fit param eters. Filled circles indicate Se filled sample, triangles indicate Te, Squares PbI2 and empty circles the unfilled DWNT.
51 For the G band, the second highest Raman shifted Lorentzian component in the four component fit was chosen as the point of center loc ation comparison between the different samples. This component consistently had the greatest intensity and height. It also corresponded the most closely to the maxima of the band. For the unfilled nanotube sample the G band center shifted from a peak va lue of 1585 cm 1 at 80 K to 1580 cm 1 at 720K. Figure 5.9 : Unfilled nanotube sample G band showing four Lorentzian fit G band temperature dependence and anharmonic phonon fit. The filled nanotube also showed downshift in Raman shift with increase in tem perature. The Se and Te filled samples however consistently showed higher G band center frequencies at all temperatures, while the PbI 2 filled sample showed slightly higher frequencies at most temperatures.
52 0 cm 1 C, cm 1 T= 0), cm 1 Se @ DWNT 1605 17 1588 Te @ DWNT 1600 13 1587 PbI 2 @ DWNT 1598 12 1586 HgTe @ DWNT 1601 15 1586 DWNT 1596 11 1585 HOPG 1593 12 1581 Figure 5.1 0 : Top: Plot of G band peak center location, an harmonic phonon dec ay fit (Equation 2.5) ; Bottom: Table of fit parameters Fille d diamonds indicate HOPG sample, filled circles indicate Se filled sample, triangles indicate Te, Squares PbI2 and empty circles the unfilled DWNT. The filled nanotube samples had greater base 0 values and greater anharmonicity parameters in all cases than the unfilled sample. For comparison, a temperature dependent set of Raman
53 spectra were taken on a sample of Highly Oriented Poly Graphite (HOPG), a material which consists of many graphene s heets (Figure 5.10 ). HOPG lacks some of the nanotube specific spectral features, but retains a G band due to the Carbon sp 2 bond. The band center was determined at each temperature and fit with the same phonon anharmonicity equation as the nanotube sampl es. The harmonic frequency was lower than that observed for the unfilled nanotubes and much lower than those of the filled nanotubes. The anharmonicity was nearly the same as the empty DWNT, and similar than or lower to all the filled nanotube samples.
54 5. Results from the coupled oscillator model Having extracted corresponding harmonic phonon energies of the different nanotube systems through the application of the temperature dependant anharmonic phonon fits, we can then introduce them into the oscillator model equation for a 2 We take the resonant frequency from equation as described in chapter IV, we then calculate the ratio of the guest carbon interac tion to the carbon carbon interaction. 0 cm 1 b a 3 a 2 Se @ DWNT 1314 6.6 1.58 1.40% Te @ DWNT 1312 10.6 0.75 1.10% PbI 2 @ DWNT 1310 12.8 0.56 0.80% HgTe @ DWNT 1307 13.7 0.41 0.30% DWNT 1305 1 1 Table 5. 2: Parameters and results from the couple d oscillator model Final column displays the ratio of the guest CNT Van der Waals interaction to the C C bond in the nanotube
55 6. Brief numerical analysis of the coupled oscillator model Numerical modeling can help show the properties of the coupled os cillator model for different situations. The ratio of the mass of the filling atom over the mass of a carbon atom of the system, b, was varied while keeping the parameters a 3 and constant. Varying from .01, which would be a Hydrogen atom in corporated inside the CNT, up to 20, which would be the heaviest atoms on the periodic table, an interesting picture emerges. Figure 5.11 : a 2 vs b. There is a singularity in a 2 when the denominator of its equation is near zero namely at the value of b for whic h The measured bond strength ratio a 2 reaches the same stable value to any desired number of digits shortly before and after this singularity.
56 Next the change in a 2 with change in a 3 (the ratio of the guest guest interaction of t he carbon carbon interaction) was analyzed. This allows an analysis of how filling materials with stronger and weaker guest guest bonds would effect the strength of the bond between the guest and the nanotube inner wall. The analysis was run from bond st rength of nearly zero to bond strength ten times higher than that of the carbon carbon sp 2 bond (2.9 eV), greater than any known interatomic bond strength. Results were similar to the previous case. Figure 5.12 : a 2 vs a 3 There is again a singularity at the value for which the denominator of a 2 is zero. Increase or decrease in either direction very quickly stabilizes the value of a 2 far beyond experimental error. To calculate the possible error introduced into a 2 by experimental error in the measurem ent of parameter another numerical simulation was performed. The value of a 2 was calculated while
57 varying by increments of .001 from +1 cm 1 to 1 cm 1 1 cm 1 was chosen because this is the experimental error in the measurement and fitting process. Calculations were performed for all four filled nanotube systems. Figure 5.13 : Variation in a 2 The error in is linear, resulting in the same error for values equally above and below the calculated value, represented by horizontal the line. This line intercepts the function at the measured value. Each sample showed a linear growth in a 2 error with error in For 1, the total error was .002 for each sample.
58 Table 5.3 : Error analysis of a 2 The uncertainty is identical in every case. Th e relative uncertainty is therefore much higher in the materials with weaker interactions.
59 7. Conclusions Utilizing Raman spectroscopy between the temperatures of 80 K and 720 K, valuable insight into the physics of filled carbon nanotubes was gai ned. The dependence of the Raman spectrum characteristics upon the phonon processes of the nanotubes allowed an elucidation of the internal vibrational and structural properties of the unfilled and filled DWNT. The independently variable range of tempera tures allowed the phonon information extracted from the Raman shift data taken to be considered in models of phonon anharmonicity and nanotube physics. These models offer new results about the effect of filling materials on DWCNT. The data and analysis s uggest a clear and consistent ordering of the strength of the effect of the filling materials on the Raman shift of the D and G bands. The filling material also strongly effects the Raman shift dependence on temperature. The Se filled DWCNT showed the gre atest effect, followed by the Te filled sample, The PbI 2 sample and lastly the HgTe filled sample. The anharmonicity parameters increase in the same order (Table 5.4 ).
60 D band G' band G band C, cm 1 C, cm 1 D, cm 1 C, cm 1 Se @ DWNT 9 28.0 5. 0 14 Te @ DWNT 11 26.5 4.5 13 PbI 2 @ DWNT 8 21.5 3.0 12 HgTe @ DWNT 6 19.5 3.5 15 DWNT 6 19.5 3.0 11 Table 5.4 : Comparison of Raman band center location fit anharmonicity parameters. The 0 parameters (Table 5. 2 ) corresponding to the phonon frequency were initially found in fitting the D band anharmonic phonon decay. The accuracy of these parameters was supported by their strong fit to the sec ). Interpreti ng the coupled oscillator model results, we first note that the inner outer nanotube wall Van der Waals interaction strength is .9% of that of the sp 2 C C bond. This allows comparison of the strength of the Van der Waals interaction between the filling ma terial and the inner nanotube wall with the outer wall to inner wall interaction. The Se @ DWNT sample shows significantly stronger bonding between the guest material and the inner tube than between the tube walls. The Te @ DWNT sample shows slightly str onger interaction, while the PbI 2 @
61 DWNT show slightly weaker interaction. The HgTe @ DWNT sample shows a much weaker interaction. The comparison of the G band fits for the nanotubes with those for the HOPG gives another reference for the filling materia l effect. The pa rameters of the fit (Figure 5.10 ) show that the unfilled DWCNT G band phonon frequency parameter 0 is upshifted by 3 cm 1 relative to that of the HOPG. The upshift of 0 for the Se@DWNT relative to the unfilled DWCNT is 9 cm 1 The other filled nanotube samples range from 2 cm 1 to 5 cm 1 in difference from the unfilled DWCNT.
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