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Raman spectroscopy of carbon nanotubes under axial strain and surface-enhanced Raman spectroscopy of individual carbon nanotubes

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Raman spectroscopy of carbon nanotubes under axial strain and surface-enhanced Raman spectroscopy of individual carbon nanotubes

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Content RAMAN SPECTROSCOPY OF CARBON NANOTUBES UNDER AXIAL STRAIN AND SURFACE-ENHANCED RAMAN SPECTROSCOPY OF INDIVIDUAL CARBON NANOTUBES A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) by Rajay Kumar May 2008 Copyright 2008 Rajay Kumar ii Epigraph When the upper point of a hair is divided into one hundred parts and again each of such parts is further divided into one hundred parts, each such part is the measurement of the dimension of the spirit soul. Svetasvatara Upanishad 5.9 iii Dedication Dedicated to my parents, Satish and Lalita, and my brothers, Sunjay and Vijay who are so integrated into my life, I cannot imagine living without them and who have all taught me so much, given so much, and sacrificed so much out of love. iv Acknowledgements I would like to begin and end this section by mentioning the one man who is almost solely responsible for my graduation, Prof. Stephen B. Cronin. He deserves mention twice because he truly is a remarkable man who has offered much guidance and leadership and taught me so much in the years I have worked with him. Though I consider myself to be a man of science, how we were put together at a trying time in both our lives is indeed almost too good to be true and itself is a microcosm of the test of belief I and all of us are bound to go through. I have more on this, but let me first thank others who have really been there for me on this journey. Thank you mom and papa, Sunjay and Vijay, and Mari and Rika for everything. I’m not sure I would be where I am today if you all were not close by to help me take my mind off troubled experiments and to help me recharge after a week of particularly difficult toil. I’ve been so happy to be here in southern California for Sunjay’s wedding and Vijay’s engagement as well as the birth of the first of the next generation of Kumars, Rika, the first girl born on my father’s side of the family in four generations. Perhaps she will be the next Dr. Kumar of the family. I have also been so happy to be near my mother and father and to just talk and eat home-cooked meals. I also take great pride in being one of the only on my mother’s side in our community of almost 1500 to earn a Ph.D. It just amazes me how much us humans are capable of when I think about how my parents survived the Great Partition of India and Pakistan. This is especially true of my mother who grew up in Palghar, a city with no running water or electricity north of Mumbai who went on to pursue her education, even against her family’s wishes at times, but at the encouragement of her v father, who passed away when I was young, but is a man who I really wish I could meet. She had to move away from home at a young age to pursue her studies and was bright enough to earn a law degree as well as an MBA from Bridgeport University in Hartford, Connecticut. I know she sacrificed time with her family then and I am so proud of the family she has raised here in the United States. I must also mention how my father lived a tough childhood in Amritsar and Chandigarh, but despite all odds, studied very hard and landed a job in Hindustan Motors and later at IBM, after graduating from the University of Michigan in metallurgical engineering. My father was the process engineer for one of IBM’s first lines of integrated circuits and went on to NCR and Rockwell, International where his expertise was regarded so highly, he was awarded US citizenship in only one week from start of file to finish! I am really proud of both of you both. I also want to thank my friends for your discussions, encouragement, and advice. David Farris, you have been a great friend since we both were in Hjalmer’s class for pre-Algebra in the sixth grade. It’s been a crazy, fun road, man and all the times we have hung out have been great, man. David Pratesa, you have really helped me to realize that working does not mean I have to be constantly stressed out and that having fun in life and connecting with people is as important, or even more important than material pursuits. Hanging out with you and Cheryl has at times been a godsend. Brad Mushet, it has been great having you back in LA. Hanging out with you and Jacqui has been fun and I can’t wait to have more good times. Ken and Selene, talking with you two is always a pleasure and the way you both keep really active, open minds is great. Andrew Sutherland, you’ve always been there for me, if vi not in person, then online. I know you’ve had tough deadlines, but you’ve always been available to talk and help me gain perspective on life. And to all my roommates: Will, Themos, Andrew Yick, and Dora, you’ve all been great. Thanks for putting up with all my dirty dishes in the sink and for being there to just hang out and watch movies, go out to dinner, and just talk. Steve Farrell, it’s been fun just hanging out and griping at the world. Jon Kelly, we’ve been through a lot together, man, and I hope we can still hang out after we both graduate! As for my labmates, I’m really proud of all the work you’ve done. Adam, you’ve been here since the beginning and your research and insight never cease to amaze me. I-Kai, we’ve shared a cubicle for a while and it’s been amazing to see you grow as a researcher. I know we’ve talked a lot about research and I’m always glad to see you take my advice to heart. I know you will do great things. Wayne, you too have grown so much here and I know you will do great things as well. Your hard work and good attitude have really helped us all. Memo, you are an amazing person. I am often amazed how you make very difficult things seem so simple. It is truly amazing. Jesse, it’s been good having you around again. Your AFM expertise has really helped us out in here. And David, thanks for being so good-humored all the time. Charles, it’s been great having you in lab. Fernando, seeing you pull those late nights and work so hard has really been inspirational. Mike, our discussions on politics have been fun! Bardia, your positive attitude has really brightened up the lab. Mark Harrison and Brian Luscombe, you guys are really sharp and I wish you all the best in your future engineering careers! vii I would also like to thank those in other labs who have helped with my research. Prof. Chongwu Zhou was one of the first people I met here at USC. When I was starting up the Cronin lab, I took his lab as inspiration for our own as they do amazing research and publish so many papers. In particular, I would like to thank Song Han, Reigh, Koungmin, Daihua, Bo, Kevin, Lewis, Fumi, Thomas, Alex, Hsiao-Kang, Akshay, and the rest of his group for their assistance. I would also like to thank Ronalee Lo and Brian Li from Prof. Ellis Meng’s group. And I would like to thank the members of my Quals and Dissertation Committees, Prof. Thompson, Prof. Levi, Prof. Prata, and of course, Prof. Zhou and Prof. Cronin. I would also like to give thanks to my previous research professor, Aristides Requicha and my group there, including Jonathan Kelly, Dan Arbuckle, Alex Lee, Mike Wang, Mrinal Mahapatro, Babak Mokaberi, Pam Gross, Raymond Glover, and Jesse Theiss. I would also like to mention the USC staff that has been helpful: Kusum Shori, Kim Reid, Mona Gordon, and especially Jaime Zelada, who is a very cool guy and fun guy to talk to at lunch! And thanks to everyone in my life who I haven’t mentioned by name, including my grandmother, Sushila Devi, who is a symbol of strength for our family, and my other relatives who are so many in number, that I unfortunately can’t list them all, but to whom I give thanks and love for their support. I would also like to mention my grandfather who passed away while I was here at USC. He was an amazing engineer and I am proud to follow in his footsteps. We miss you. And a special thanks goes out to all my aunts and uncles, especially Nandu Uncle and Pravin Uncle who have taken a special interest in my research. Thank you. viii Also, I would like to acknowledge support from the Powell Foundation and the James H. Zumberge Research and Innovation Fund for funding my research. And to conclude, I must again thank Steve Cronin. If it were not for him, I probably would have never graduated. Back in April of 2005, after being at USC for nearly two years, I found myself completely unmotivated to work and directionless. During this month, I was let go from my group at the time which though headed by a capable professor, just did not work for me. Though this was bad news, I honestly felt a little relieved since now I was free to figure out what I wanted to do. At the time, I had no interest in graduate school and looked into work positions, though I did not find anything interesting. I quickly realized I wanted to continue doing nanotechnology research. Lo and behold, a few weeks after, my roommate, Andrew Yick, informed me of a new professor coming to USC, Dr. Stephen Cronin. I decided to give it a shot, but knew that I had to make doubly sure that this was going to work. So, as Steve will attest, I really interviewed him, asking him questions about funding, research, etc. I knew going into this meeting that whoever I worked for better be good and be on top of his game. Steve more than exceeded my expectations, as he explained that he obtained his Ph.D. from the esteemed Prof. Mildred Dresselhaus at MIT (who obtained hers from none other than Enrico Fermi!) and had also performed amazing carbon nanotube research. I remember the weekend after this meeting talking to my father on our way to taking him to the airport to tend to my grandfather’s estate after his passing that I would try grad school again. He gave it his blessing and I knew that I would have to really work hard to make this be a success. I really turned a new leaf and worked hard, realizing that taking on ix responsibility was good for me and that treating the lab as my own instead of “someone” else’s would not work. I also knew that a new professor has one of the hardest jobs and did what I could to help out Steve. It will always amaze me how much my life changed in those few weeks after I was let go from my previous group and before I met Steve. Steve, it has been a great experience and I wish you all the best and am glad to be the first in a long line of graduates in this lab! x Table of Contents Epigraph .................................................................................................................... ii Dedication ...................................................................................................................iii Acknowledgements.....................................................................................................iv List of Figures ............................................................................................................xii List of Tables ............................................................................................................xxi Abstract ................................................................................................................xxii Chapter 1 Physical Properties of Carbon Nanotubes ...................................................1 Introduction...................................................................................................1 Electronic Band Structure .............................................................................2 Mechanical Properties...................................................................................7 Chapter 1 Endnotes .....................................................................................12 Chapter 2 Raman Spectroscopy of Carbon Nanotubes..............................................14 Raman Spectroscopy...................................................................................14 Resonance Raman Scattering......................................................................16 Conclusion ..................................................................................................23 Chapter 2 Endnotes .....................................................................................25 Chapter 3 Raman Spectroscopy of Nanotubes under Axial Strain ............................26 Semiconducting Nanotubes under Axial Strain ..........................................30 Strain Dependence of the Raman Intensity.................................................33 Strain Dependence of the Radial Breathing Mode......................................38 Metallic Nanotubes under Axial Strain.......................................................42 Conclusion ..................................................................................................48 Chapter 3 Endnotes .....................................................................................50 Chapter 4 Theoretical Modeling of Nanotubes under Axial Strain ...........................52 Electronic Structure under Axial Strain ......................................................52 Comparison of Theoretical Predictions with Experimental Results ...........56 Calculation of the Vibrational Structure of Carbon Nanotubes..................58 Vibrational Structure of Carbon Nanotubes under Axial Strain .................63 Conclusion ..................................................................................................65 Chapter 4 Endnotes .....................................................................................66 Chapter 5 Surface Enhanced Raman Spectroscopy of Individual Carbon Nanotubes....................................................................................................68 Surface Enhanced Raman Spectroscopy.....................................................68 xi Surface Enhanced Raman Spectroscopy of Individual Carbon Nanotubes 73 Imaging the SERS Nanoparticle Geometry ................................................74 Measuring the SERS Enhancement Factor .................................................79 Plasmonic Heating ......................................................................................84 Conclusion ..................................................................................................86 Chapter 5 Endnotes .....................................................................................88 Chapter 6 Carbon Nanotubes Strained by a Top-Down Microprocessing Method ...91 Experimental Setup .....................................................................................92 Strain-Induced Changes in the Raman Spectra.........................................100 Strain-Induced Changes in the G Band.....................................................102 Strain-Induced Downshifts of the Raman Modes .....................................104 Radial Breathing Mode .............................................................................106 Electrical Measurements ...........................................................................108 Conclusion ................................................................................................110 Chapter 6 Endnotes ...................................................................................111 Chapter 7 Conclusion...............................................................................................112 Summary ...................................................................................................112 Future Outlook ..........................................................................................113 Chapter 7 Endnotes ...................................................................................114 Bibliography.............................................................................................................115 Appendix 1 MATLAB Dispersion Code for the calculation of Electronic Dispersion Relations in Carbon Nanotubes ..............................................123 Appendix 2 Review of Related Literature ...............................................................127 Measurements of Nanotube Composite Materials ....................................129 Individual Nanotube Measurements .........................................................133 Carbon Nanotubes under Axial Strain ......................................................136 Conclusion ................................................................................................137 Appendix 3 Supplemental PDMS Strain Data.........................................................141 Appendix 4 MATLAB Dispersion Code for the calculation of Phonon Dispersion Relations in Carbon Nanotubes under Axial Strain................146 Appendix 5 Supplemental SERS Data.....................................................................152 xii List of Figures Figure 1.1: (left) A 2D graphene sheet with chiral vector OA (or C h ) defined on a honeycomb lattice by chiral angle θ with respect to a 1 . Capped (a) armchair ( θ=30°), (b) zigzag ( θ=0°), and (c) chiral (0 < | θ| < 30°) nanotubes are shown on the right. These particular nanotubes have chiralities of (5,5), (9,0), and (10,5), respectively. 9 ..................................3 Figure 1.2: Electronic dispersion relations in graphene plotted as a 3D projection (left) and as a 2D contour plot (right). ......................................................4 Figure 1.3: (a) Parallel and perpendicular axes of a nanotube. (b) 2D graphene dispersion relation together with zigzag nanotube cutting lines spaced by 2/d t , satisfying the cylindrical quantization of the electron wavevector. (c) Electron conduction (white) and valence (yellow) bands near the Fermi level. 12 ....................................................................4 Figure 1.4: Dispersion relations for a semiconducting (10,0) nanotube (left) and a metallic (9,0) nanotube (right). Note the intersection of the conduction and valence bands in the metallic (9,0) nanotube. ................6 Figure 1.5: Density of states for a (10,0) semiconducting nanotube and a (9,0) metallic nanotube. Note the nonzero density of states near zero energy for the metallic nanotube. The dotted lines represent the dispersion relations for graphene..............................................................7 Figure 2.1: Different possibilities of light scattering: Rayleigh scattering (no Raman effect), Stokes scattering, and anti-Stokes scattering. ................15 Figure 2.2: Electronic excitation processes for resonance Raman scattering. The electron is first excited by light (rightmost arrow), then imparts energy to the creation of a phonon (middle arrow), and then finally emits light at a shifted frequency (leftmost arrow).................................17 Figure 2.3: Relative intensities for Stokes and anti-Stokes Raman intensity plotted as a function of laser energy. The Stokes and anti-Stokes intensities both have peaks at E ii as well as at E ii +E ph for Stokes and E ii -E ph for anti-Stokes. The peaks valued near E ii correspond to the incoming light resonance signal and the other peaks correspond to outgoing resonances...............................................................................................18 Figure 2.4: Atomic force microscope (AFM) image (left) and optical microscope image (right) of a silicon substrate with a metallic grid and sparsely distributed nanotubes. .............................................................................19 xiii Figure 2.5: (a) Radial breathing mode (RBM). The Stokes RBM in (a) appears at 150 cm -1 and the anti-Stokes RBM is at -150 cm -1 . Typical G bands for (b) semiconducting and (c) metallic carbon nanotubes. Note how the G - band shape is broader in the metallic nanotube............................21 Figure 2.6: Illustration showing the atomic motion of the radial breathing mode (RBM) and G band modes of a carbon nanotube. ..................................21 Figure 2.7: Kataura plot calculated using a nearest neighbor tight binding model with overlap integral γ 0 = 2.9 eV and nearest neighbor carbon-carbon distance of a cc = 0.144 nm. The black dots correspond to semiconducting nanotubes while the red circles correspond to metallic nanotubes. ...............................................................................................23 Figure 3.1: Processing steps for fabricating carbon nanotubes pinned to an elastomeric substrate. First, a slab of PDMS is prepared, followed by a deposition of nanotubes in an isopropyl alcohol solution. Gold strips are then patterned on top of the nanotubes using photolithography. Finally, the slab of PDMS is strained, applying strain to the nanotube bundles. An atomic force microscope image of one pinned nanotube bundle is shown in the image on the bottom right.........................................................................................................28 Figure 3.2: Diagram illustrating how strain is determined for a nanotube lying at an angle θ with respect to the applied strain. The length of a nanotube (black angled line) at an angle to the contacts given the distance between contacts (gold rectangles) for an unstrained nanotube (left) and strained nanotube (right). The x value remains constant for all strains since strain is applied in the vertical direction and increases d. The % strain is calculated by dividing the increase in length ( Δ ℓ) by the unstrained length ( ℓ)..........................................................................29 Figure 3.3: (a) AFM image of a carbon nanotube bundle pinned under gold strips with a 0.5 μm diameter laser spot. The scale bar is 2 μm. (b) Raman spectrum from a 532 nm laser taken at 0% strain. The sharp peaks are typical of semiconducting nanotubes and show the high frequency component, G + and lower frequency component G – . (c) G + Raman frequency plotted as a function applied strain. Note the reversible regions of the strain, separated by discrete slips of 1.3 and 4.5%. .........31 Figure 3.4: G + (top), G - (middle), and G’ (bottom) Raman frequencies plotted as a function of applied strain. .......................................................................32 Figure 3.5: G + (top), G - (middle), and G’ (bottom) Raman intensity plotted as a function of strain for a nanotube bundle.................................................34 xiv Figure 3.6: G + band Raman frequency plotted with increasing strain. After an actual strain of 5% (strain point #24), the nanotube downshifted by 24 cm -1 and was held down with ~860 GPa of pressure..............................36 Figure 3.7: Change in D/G band intensity ratio versus strain. An increase in this ratio indicates that defects are forming within the nanotube as strain is applied.....................................................................................................37 Figure 3.8: Radial breathing mode frequency of four metallic nanotubes strained on PDMS and measured with a 633 nm laser. Note the positively sloped trend line for all four nanotubes. .................................................39 Figure 3.9: Radial breathing mode frequencies of four semiconducting nanotubes strained on PDMS and measured with a 532 nm laser. Note the positively sloped trendline observed for all four nanotubes. These nanotubes all have diameters of about 1.3 nm........................................40 Figure 3.10: (a) RBM frequency versus percent strain. (b) RBM intensity versus percent strain. (c) RBM frequency versus number of strain steps. (d) RBM intensity versus number of strain steps. Note how when plotted against the number of strain steps, the frequency increases and the intensity decreases despite decreases in the strain %, indicating the occurrence of irreversible changes to the nanotube................................41 Figure 3.11: (a) G + band and (b) G – band Raman frequency of one of four metallic nanotubes at various degrees of strain. (c) G band Raman spectra taken by a 633 nm laser after subsequent straining and unstraining of the bundle. .......................................................................44 Figure 3.12: G band Raman frequency versus strain for two metallic nanotubes experiencing irreversible upshifts in frequency. The plot in (a) depicts an upshift of 30 cm -1 to a region with a slope of -2.3 cm -1 /% strain. G band spectra before and after the irreversible upshift are shown in (b) for this nanotube. Note the change in lineshape from broad to narrow after the upshift. The plot in (c) depicts an upshift in the G - band of another nanotube of 20 cm -1 to a region with a slope of -1.0 cm -1 /% strain....................................................................................46 Figure 3.13: FWHM linewidths for the (a) G + band and (b) G – band of a metallic nanotube measured with a 633 nm laser and (c) G + band and (d) G – band FWHM linewidths for a semiconducting nanotube measured with a 532 nm laser. Data for other nanotubes can be found in Appendix 3..............................................................................................48 xv Figure 4.1: Chiral angle dependence of the shift in metallic nanotubes’ E 11 subband energies and semiconducting nanotubes’ E 33 subband energies under 1% axial strain. Calculations were made on nanotubes in the diameter range 1.1–1.5 nm............................................................54 Figure 4.2: Joint density of states for a (14,6) semiconducting nanotube and a (16,1) metallic nanotube with and without 1% axial strain. 2 ..................55 Figure 4.3: (a) Phonon dispersion curves, plotted along high symmetry directions, for a two-dimensional graphene sheet. (b) The corresponding density of states vs. phonon energy for phonon modes in units of states/1C- atom/cm -1 . 20 .............................................................................................59 Figure 4.4: Fourth-nearest-neighbors for an (a) A carbon atom and (b) B carbon atom. First nearest neighbors are shown with open circles, second nearest neighbors are shown with solid squares, third nearest neighbors are shown with open squares, and fourth nearest neighbors are shown with open hexagons. The circles connecting atoms are guides for the eye. 20 ................................................................................60 Figure 4.5: Comparison of phonon dispersion relations using a graphene dynamical matrix for a (10,10) nanotube. The phonon dispersion relation was calculated using our MATLAB code as shown in Appendix 4. The y-axis shows the phonon frequency in cm -1 and the x-axis shows the k vector magnitude from 0 to T π . ................................62 Figure 4.6: Phonon dispersion relations for a (10,10) nanotube with (a) 0%, (b) 1%, (c) 2%, and (d) 3% axial strain for Raman resonant phonon frequencies near. One can see the longitudinal optical (upper curve) and transverse optical (lower curve) modes separate and shift in opposite directions. .................................................................................64 Figure 5.1: Comparison of normal Raman Scattering (left) and surface-enhanced Raman Scattering (right). In normal scattering, the conversion of laser light I L into Stokes scattered light I NRS is proportional to the Raman cross section σ R free, the excitation laser intensity I( ν L ), and the number of molecules N that are within the probe volume. The right depicts a SERS experiment where σ R ads describes the increased Raman cross section of the adsorbed molecule, also known as “chemical” enhancement. A( ν L ) and A( ν S ) are the field enhancement factors at the laser and Stokes frequency, respectively. N’ is the number of molecules involved in the SERS process. 6 ............................70 xvi Figure 5.2: SEM image of a numbered fiducial grid, together with gold strips, carbon nanotubes and deposited silver nanoparticles (upper left). Magnified images of regions exhibiting SERS enhancement are also shown. The white circles represent the size and location of the laser spot used to measure SERS enhanced Raman spectra............................75 Figure 5.3: Magnified image of the region indicated in Figure 5.2(a). Note the silver nanoparticles covering the nanotubes near the center of the circle, representing the laser spot size.....................................................76 Figure 5.4: Magnified image of the region indicated in Figure 5.2(b). Note the silver nanoparticles covering the nanotubes near the center of the circle, representing the laser spot size.....................................................77 Figure 5.5: Magnified image of the region indicated in Figure 5.2(c). Note the silver nanoparticles covering the nanotubes near the center of the circle, representing the laser spot size.....................................................78 Figure 5.6: (a) SEM image of nanotubes covered with a film of Ag nanoparticles. The white circle indicates the size and location of the laser spot. (b) Raman spectra of the nanotube from (a) before and after Ag deposition................................................................................................80 Figure 5.7: Raman spectra taken before (top) and after (bottom) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 2 in Table 5.1..................................81 Figure 5.8: Raman spectra taken before (top) and after (bottom) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 3 in Table 5.1..................................82 Figure 5.9: Images of a nanotube before and after Ag nanoparticle deposition and subsequent laser irradiation of a SERS “hot spot.”.................................85 Figure 6.1: Processing steps for HF experiment. (1) First, a piece of silicon with a 500nm oxide is obtained. (2) Aligned nanotubes are transferred to the surface. (3) Using shadowmasking and metal evaporation, 3 μm gaps are created between gold contacts. (4) Finally, the oxide is etched using HF..................................................................................................93 Figure 6.2: SEM image of aligned nanotubes grown using the method of Ryu, et al. 1 ..........................................................................................................94 xvii Figure 6.3: (a) TEM grid used as a shadow-mask for top-down microprocessing shown at 40x optical magnification. (b) SEM image of aligned nanotubes between two gold contacts. The horizontal gap was created by placing a 3 μm tungsten wire over the sample during metal evaporation..............................................................................................94 Figure 6.4: AFM images (left column and top), optical images (middle column), and Raman spectra (right column) from an etched gap containing a resonant carbon nanotube from a prior experiment. The top row of images show the AFM measurements of the gap before etching and after the first etch. Note how the nanotubes are barely visible. The second row of images corresponds to a 34 nm deep gap, the third row corresponds to a 61 nm deep gap, and the bottom corresponds to a 112 nm deep gap. The black oval on the AFM images corresponds to the resonant nanotube. This nanotube’s location is also indicated by the green laser dot in the middle column of optical images for each etch. The G band Raman spectra are taken for each etch as shown in the right column. This data shows that the nanotubes remain unperturbed from their positions as the underlying substrate is etched and dried. Furthermore, the nanotubes appear thicker due to the accumulation of residue. .........................................................................96 Figure 6.5: Plot of nanotube heights versus etch depth. The heights were obtained through section analysis of four nanotubes. The heights were averaged for each depth. Error bars reflect the range of nanotube heights for each etch. ..............................................................................98 Figure 6.6: SEM images of suspended nanotubes dried (a) with and (b) without critical point drying. Note how the nanotube dried in the critical point dryer in (a) is suspended yet the nanotube dried without a critical point dryer in (b) adheres to the substrate due to the high surface tension of the water. The relatively large surface forces pin the nanotube under strain..............................................................................99 Figure 6.7: SEM images of Gaps 2 (upper left), 3 (upper right) and 5 (lower middle) after being etched 582 nm. Note the severely displace nanotubes in all three images and also how NT5 in the lower middle image (upper nanotube) has broken away from the contact. ................100 Figure 6.8: D band intensity divided by the G + band intensity versus trench depth for four out of five nanotubes measured. Note how three nanotubes show consistent upshifts in this ratio for the first three points, indicating a relative increase of the D band intensity with respect to the G + band intensity. Further note how NT5 decreases after the third etch, but not to the initial value.............................................................101 xviii Figure 6.9: Graphs of changes in G + band frequency with etch depth. Note how NT1-NT4 experience upshifts in frequency while NT5 experiences radical downshifts, for the first two etches. After the third etch (not shown), only NT5 spans the gap and downshifts to its pre-strain frequency...............................................................................................103 Figure 6.10: Raman spectra measured from nanotube sample NT5 before and after successive strains induced by the trench etch process. Note how the G band shape changes from two peaks to one broad peak and then to a peak with two shoulders. Also note the change in intensity and downshift ofthe D band.........................................................................105 Figure 6.11: G’ band of NT5 after subsequent etches. Note the downshift by almost 57 cm -1 after the second etch as well as the asymmetry of the peak. Here, the G’ band splits into two peaks. After the final etch, the G’ band upshifts to its original frequency.......................................106 Figure 6.12: Radial breathing mode and anti-Stokes radial breathing mode Raman spectra of a nanotube before etching, after 50 nm of etching, after 500 nm of etching, and after 582 nm of etching. The radial breathing mode in the 500 nm case appears at about 185.5 cm -1 ..........................107 Figure 6.13: Section of a Kataura plot calculated using a tight binding model for a region consisting 1.2-1.4 nm diameter (vertical blue bar) nanotubes resonant with 633 nm laser with a 0.1 eV FWHM (horizontal red bar). The nanotubes falling within this region are all metallic with chiralities of (16,1), (9,9), (15,3), and (13,4)........................................108 Figure 6.14: I-V curve of NT5 after subsequent etches. The calculated resistance decreases by 2 k Ω after the first etch and then increases to 4.0 M Ω after the second etch. No signal was detected after the third etch. ......109 Figure A2.1: Typical Raman spectrum from a semiconducting carbon nanotube...129 Figure A2.2: Left: Schematic diagram of a four-point bending method for inducing tensile strain in a composite material. Right: Strain-induced downshift of the G’ band Raman mode of nanotubes dispersed in epoxy resin. 22 Reprinted with permission from [22], C. A. Cooper, R. J. Young, M. Halsall, Composites: Part A 32, 401 (2001). © 2001, Elsevier. ................................................................................................130 xix Figure A2.3: AFM image showing a carbon nanotube after inducing strain with an AFM tip. 7 Reprinted with permission from [7], S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 94, 167401 (2004). © 2004, American Physical Society....................................................................................134 Figure A2.4: G band Raman spectra for an individual SWNT before and after inducing 1.65% strain, and after breaking the nanotube with an AFM tip. Data taken after the SWNT was broken shows relaxation of the Raman peaks back to their original positions. 7 Reprinted with permission from [7], S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 94, 167401 (2004). © 2004, American Physical Society. .........................135 Figure A3.1: Supplemental strain data of nanotubes on PDMS from a nanotube known as “Slab 2.” Note how the G + shows an irreversible change in nanotube behavior from low frequencies to high frequencies..............141 Figure A3.2: Additional strain data of a nanotube on PDMS known as “Slab3- Grid1-Tube5” which was already shown in Figure 3.11. Note the linear reversible behavior of the G + band and D band compared to the G - band..................................................................................................142 Figure A3.3: Strain data for a nanotube on PDMS known as “Slab3-Grid1- Tube6.” Note the negative curvature trend in the beginning that could not be repeated for the G band..............................................................143 Figure A3.4: Strain data for a nanotube on PDMS known as “Slab3-Grid1- Tube7.” Note the irreversible behavior in the G - band. .......................144 Figure A3.5: Strain data for a nanotube on PDMS known as “Slab3-Grid1- Tube8.” Note the irreversible trend in the G’ spectrum.......................145 Figure A5.1: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 2 in Table 5.1................................152 Figure A5.2: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 3 in Table 5.1................................152 Figure A5.3: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 4 in Table 5.1................................153 xx Figure A5.4: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 5 in Table 5.1................................153 Figure A5.5: Zoomed-out SEM image of nanotubes in Figure 5.2(a).....................154 Figure A5.6: Zoomed-out SEM image of nanotubes in Figure 5.2(b) and (c).........154 xxi List of Tables Table 1.1: Tensile strength to density ratios for high strength materials. 15 .................8 Table 1.2: Reported values of the Young's moduli and breaking stresses and strains for MWNTs and SWNTs..............................................................10 Table 4.1: Force constant parameters for graphite in units of 10 4 dyn/cm. Here, the subscripts r, ti, and to, correspond to radial, transverse in-plane, and transverse out-of-plane, respectively. 20 ............................................61 Table 5.1: SERS related effects of five carbon nanotubes. The Raman data for the first entry is shown in Figure 5.6(b) and the rest are shown in Appendix 5...............................................................................................83 Table A2.1: Strain-induced changes in the Raman spectra using various straining techniques. G ω Δ and G' ω Δ are the shifts observed in the G and G’ bands, respectively. ................................................................................132 xxii Abstract In this thesis, I present resonant Raman spectroscopy of individual carbon nanotube bundles under axial strains up to 17%. The main effect of this strain is to cause nanotube debundling. The G band Raman spectra of metallic and semiconducting nanotubes are found to respond differently to strain and debundling, giving insight into the nature of the broad metallic G - band lineshape. For metallic nanotubes, the G - band upshifts and becomes narrower with strain, making it appear more semiconductor-like. Surprisingly, this metal to semiconductor transition is irreversible with strain, indicating that nanotube-nanotube coupling plays a significant role in the observed G - band of metallic nanotubes. The vibrational and electronic properties of these nanotubes under strain are modeled using tight-binding calculations. This thesis also presents a systematic study of surface enhanced Raman spectroscopy (SERS) of carbon nanotubes. Raman spectra of individual carbon nanotubes are measured by scanning a focused laser spot (0.5 μm diameter) over a large area (100 μm 2 ) before and after depositing silver nanoparticles. Local regions exhibiting SERS enhancement were located relative to a lithographically patterned grid, allowing subsequent scanning electron microscopy to be performed. The uniquely large aspect ratio of carbon nanotubes enables imaging of the nanoparticle geometry together with the SERS active molecule. By measuring the same individual carbon nanotube before and after metal nanoparticle deposition, the SERS enhancement factor is determined unambiguously. SERS enhancement factors up to xxiii 134,000, a consistent upshift in the G band Raman frequency and nanoparticle heating in excess of 600°C are revealed. Nanotubes are also strained through top-down microprocessing. By straining nanotubes to the bottom of a trench made deeper through etching, we induce changes in nanotubes’ resonance with an incident laser. This causes increases in intensity for all the bands, most notably the radial breathing mode. From the radial breathing mode intensity, we calculate the changes to the density of states. Furthermore, we observe two peaks in the G’ band spectrum, which also reflects changes in the density of states. All nanotubes measured experienced irreversible changes in their D bands, indicating the build up of residue on the nanotube surface from etching. 1 Chapter 1 Physical Properties of Carbon Nanotubes Introduction With the torrent of research into nanometer-scale phenomena, devices, and systems, it is important to understand the basic physics of pre-existing, stable nanostructures, especially those that are rigid and possess interesting electronic properties. Among these nanostructures are metallic nanoparticles, semiconducting nanowires, and nanotubes. Of particular interest are carbon nanotubes, which have already found uses in television sets, 1 as light sources, 2 and in transistors. 3 4 Carbon nanotubes are hollow cylinders composed entirely from carbon atoms in a hexagonal lattice structure. This structure gives nanotubes their unique mechanical and electronic properties. Carbon nanotubes have mechanical strengths in their axial direction that are equivalent to diamond, 5 as well as high torsional strength. 6 Under extreme conditions, nanotubes can even adopt superplastic behavior, attaining irreversible strains of 280%. 7 Carbon nanotubes’ electronic structure can be either semiconducting or metallic and represent a quasi-one- dimensional quantum electronic system. 8 These properties make nanotubes interesting from both materials and electronic standpoints. The majority of the work described in this thesis focuses primarily on single- wall carbon nanotubes (SWCNTs or SWNTs). The modifier “single-wall” is used to differentiate between multi-wall carbon nanotubes (MWCNTs or MWNTs). MWNTs consist of a series of nanotubes confined within one another concentrically, like Russian dolls, and have very different properties than SWNTs. 2 Electronic Band Structure To understand carbon nanotubes, it is important to understand their related material, two-dimensional graphene. Graphene is an allotrope of carbon, consisting of a single sheet of graphite. A single wall carbon nanotube, or SWCNT, can be imagined as one of these sheets rolled up into a seamless hollow tubular structure. 8 Graphene consists of atoms of carbon covalently bonded to three neighboring atoms via sp 2 bonding, forming honeycomb lattice structures akin to benzene. This unique structure, featuring trigonal symmetry, gives SWCNTs some of their interesting and unique properties. A nanotube can be thought of as a rolled-up cylinder of graphene with a diameter d t and chiral angle θ, as shown in Figure 1.1. The hexagonal grid has basis vectors a 1 and a 2 . The unit cell of the nanotube is indicated by the box OAB’B, where the circumferential chiral vector, C h , forms an angle θ with the a 1 basis vector. The non-negative integer coefficients (n,m) uniquely identify carbon nanotubes through the relation C h = na 1 + ma 2 . The length (L) of the chiral vector (C h ) is the circumference of the nanotube and is directly related to the diameter (d t ), by the relation L = πd t . Two special classes of nanotubes known as armchair [Figure 1.1(a)] and zigzag [Figure 1.1(b)] nanotubes are achiral and satisfy the condition m = 0 and m = n, respectively. All other nanotubes are chiral nanotubes [Figure 1.1(c)]. 9 3 Figure 1.1: (left) A 2D graphene sheet with chiral vector OA (or C h ) defined on a honeycomb lattice by chiral angle θ with respect to a 1 . Capped (a) armchair ( θ=30°), (b) zigzag ( θ=0°), and (c) chiral (0 < | θ| < 30°) nanotubes are shown on the right. These particular nanotubes have chiralities of (5,5), (9,0), and (10,5), respectively. 9 The electronic energy dispersion relations in graphene can be calculated using a simple tight binding model, given by, 2 / 1 2 2 2 cos 4 2 cos 2 3 cos 4 1 ) , ( ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ± = a k a k a k t k k E y y x y x D g . (1-1) This model considers the energy interaction of a carbon atom with its three nearest neighbors, 10 where the energy overlap integral, γ 0 = 2.9 eV. A plot of this energy dispersion is shown in Figure 1.2 and resembles a turtle shell with a maximum in energy at the Γ point and minima at the K and K’ points in the corners of the hexagonal Brillouin zone. 8 At these corner points of the Brillouin zone, the conduction and valence bands touch, making graphene a semimetal. 11 4 Figure 1.2: Electronic dispersion relations in graphene plotted as a 3D projection (left) and as a 2D contour plot (right). 12 The electronic dispersion relations in nanotubes can be calculated based on those of graphene. When rolled up into a cylindrical tubule, the dispersion relations change from a two-dimensional conduction and valence band to a series of one- dimensional sub-bands, corresponding to a series of slices (or “cutting lines”) of the graphene dispersion relation, as depicted in Figure 1.3(b). These subbands arise from the cylindrical geometry of the nanotube, which results in the quantization of the electron wavevector in the direction perpendicular to the nanotube axis, ⊥ k = 2 πq/L, where q is an integer. 8 Figure 1.3: (a) Parallel and perpendicular axes of a nanotube. (b) 2D graphene dispersion relation together with zigzag nanotube cutting lines spaced by 2/d t , satisfying the cylindrical quantization of the electron wavevector. (c) Electron conduction (white) and valence (yellow) bands near the Fermi level. 12 5 The spacing and angle of these cutting lines is dependent upon the nanotube chirality. The chiral angle is related to the chiral indices (n, m) by ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = − m n m 2 3 tan 1 θ (1-2) and the spacing between the lines is t d 2 , where 2 2 Å 46 . 2 m nm n d t + + = π (1-3) and 2.46 Å is the length of the nanotube unit vector ( 3 times the carbon-carbon bond length). The cutting lines’ spacing are determined by L q k π 2 = ⊥ (1-4) where q is an integer. 12 The implication of this relation is that only some nanotubes will have cutting lines that cross at the K points in the Brillouin zone, where the conduction and valence bands meet, giving them a metallic character. The criterion for metallic nanotubes is that the difference between n and m is a multiple of 3. This means only one-third of all nanotubes have metallic behavior and the remaining two- thirds are semiconducting, possessing a bandgap, such as the one shown in Figure 1.3(c). 9 Appendix 1 contains a computer program in MATLAB that calculates the electronic dispersion relations of any given (n,m) nanotube, using this tight binding approach. Graphs of the dispersion relations of a semiconducting (10,0) nanotube and a metallic (9,0) nanotube, calculated using this program, are shown in Figure 6 1.4. In these graphs, the bandgap is apparent in the dispersion relation of the (10,0) nanotube, whereas none exists for the (9,0) nanotube. Figure 1.4: Dispersion relations for a semiconducting (10,0) nanotube (left) and a metallic (9,0) nanotube (right). Note the intersection of the conduction and valence bands in the metallic (9,0) nanotube. The density of electronic states of carbon nanotubes can be calculated from the dispersion relations by the equation: ∑ = − = N j t t j dk k dE E D 1 1 ) ( 2 1 ) ( π . (1-5) Since nanotubes are quasi-one-dimensional, the density of states scales with the inverse of the square root of the energy, 2 / 1 − E . When the slope of the dispersion relation 0 = dk dE , the density of states diverges about that point, resulting in an 2 / 1 − E dependence, as shown in the peaks in Figure 1.5. 9 Figure 1.5 shows the density of states of a metallic (9,0) and a semiconducting (10,0) nanotube. Notice the nonzero density of states of the metallic nanotube at E = 0, while the semiconducting 7 nanotube has zero density of states at E = 0. These van Hove singularities will become very important in the discussion of resonant Raman spectroscopy of carbon nanotubes in the next chapter. Figure 1.5: Density of states for a (10,0) semiconducting nanotube and a (9,0) metallic nanotube. Note the nonzero density of states near zero energy for the metallic nanotube. The dotted lines represent the dispersion relations for graphene. 13 Mechanical Properties Applying strain to carbon nanotubes provides a method of studying their mechanical strength and limitation. The application of strain simulates real-world use of carbon nanotubes as structural elements. For instance, carbon nanotubes 8 could one day be used for ultra-strong building materials, such as those required for the construction of a space elevator, an Earth-tethered structure for lifting people and materials into space. 14 15 This seemingly impractical application is only made even remotely possible because of nanotubes’ immense strength to density ratio, as shown in Table 1.1. This ratio far exceeds that of any other known material and allows such a colossal structure not to collapse under its own weight. Currently, nanotubes are found in more terrestrial applications, such as improved tennis rackets, strain sensors utilizing Raman spectroscopy, 16 17 and electromechanical oscillators with attoNewton (10 -18 N) sensitivity. 18 Material Ideal tensile strength Density Tensile strength to density ratio Titanium 1.4 GPa 4500 kg m −3 3.1 x 10 -4 GPa/kg m −3 Steel 5 GPa 7900 kg m −3 6.3 x 10 -4 GPa/kg m −3 Kevlar 3.6 GPa 1440 kg m −3 2.5 x 10 -3 GPa/kg m −3 Carbon Nanotube 100 GPa 1300 kg m −3 7.7 x 10 -2 GPa/kg m −3 Table 1.1: Tensile strength to density ratios for high strength materials. 15 The Young’s modulus specifies a material’s stiffness, or ratio of stress to strain. 19 20 21 22 23 24 Although a wide range of values have been reported in the literature, the Young’s modulus of carbon nanotubes, as shown in Table 1.2, is understood to be on the order of 1 TPa for SWNTs, which is roughly the value for diamond. Another key mechanical property of nanotubes is their breaking strain. In order to realize their full potential, as a high strength material and as a strain- sensitive material, it is important to know the maximum amount of axial strain that can be applied to carbon nanotubes before they break. Although nanotubes have a 9 tremendously high Young’s modulus, they are not brittle like most stiff materials and can endure relatively high strains before breaking. At low strains, the C-C bonds in nanotubes elongate elastically. However, at higher strains, these bonds begin to break and re-form, altering the crystalline structure through the creation of Stones-Wales defects. 24 These strain-induced deformities are reversible up to ~5% strain, above which, irreversible defects are formed composed of paired heptagons and pentagons. 24 These so-called 5-7 defects result in an elongation of the nanotubes and help to extend nanotubes to higher breaking strains. 24 In MWNTs, the internal nanotubes may undergo a telescoping effect or sword-in-sheath failure, which also results in irreversible (inelastic) elongation. 24 Elongations of 280% have been reported for SWNTs within a piezo- manipulator under a bias voltage of 2.3 V. 25 These exceptionally high elongations are made possible by the extremely high temperatures (more than 2300 K) produced by the high bias voltage, which enables atomic diffusion and the irreversible coalescence of two concentric nanotubes into one larger nanotube. 25 Table 1.2 lists the Young’s moduli and breaking strains of individual nanotubes as reported by several research groups. While there is some variability in the reported values of the Young’s modulus due to slippage and irreversible failure mechanisms, the numbers converge at ~1 TPa in the absence of these events. The main source of error in these individual nanotube measurements is not in the tiny forces applied, but rather in determining the diameter of the nanotube precisely. Demczyk, et al., have performed a sophisticated set of experiments, where the 10 diameter of the nanotube is measured in situ by transmission electron microscopy (TEM) to minimize this error. 24 Type of Nanotube Authors Method Young’s Modulus Breaking Stress (Strain) Wong, et al. 21 Single AFM tip 1.28 TPa n/a Demczyk, et al. 24 Piezoelectric Manipulator in TEM 0.91 TPa 0.15 TPa (16%) Yu, et al. 20 Pulling NTs with AFM tips in SEM 270-950 GPa 11-63 GPa (12%) Pan, et al. 26 2 mm long aligned MWNTs in stress- strain puller 0.45±0.23 TPa 1.72±0.64 GPa (0.16-1.1%) MWNT Li, et al. 27 10 mm long aligned DWNT strands in stress-strain puller 80 GPa 6 GPa (7.5%) Treacy, et al. 23 Thermal vibration analysis in TEM 1.8 TPa n/a SWNT Yu, et al. 19 Pulling NTs with AFM tips in SEM 320-1470 GPa 30 GPa (5.3%) Table 1.2: Reported values of the Young's moduli and breaking stresses and strains for MWNTs and SWNTs. Nanotube bundles have been observed to have breaking stresses that are more than one order of magnitude weaker than that of individual SWNTs’ Young’s moduli. 19 20 The large reduction in strength arises because most of the nanotubes are much shorter than the length of the bundle. In this case, the strength is limited by the nanotube-nanotube coupling strength, which is far weaker than the C-C bond strength. When strains are exerted on bundles such as these, the nanotubes within the bundle will decouple from each other and become weaker overall than an individual nanotube. The applied strain results in elongation of the bundle through the sliding 11 of its constituent nanotubes along each other rather than by the transfer of strain to the individual nanotubes and their C-C bonds. The optical properties of carbon nanotubes are very sensitive to strain, and provide a multitude of information about their physical and electronic structure. This will be the topic of Chapter 4. 12 Chapter 1 Endnotes 1 M. Kanellos, “Carbon nanotube TV trials on horizon,” c|net, 21 March 2006 <http://www.cnet.com.au/tvs/0,239035250,240061261,00.htm> 2 J.-H. Park, G.-H. Son, J.-S. Moon, J.-H. Han, A. S. Berdinsky, D. G. Kuvshinov, J.- B. Yoo, C.-Y. Park, J.-W. Nam, J. Park, C. G. Lee, and D. H. Choe, J. Vac. Sci. Technol. B 23, 749 (2005). 3 X. Liu, C. Lee, C. Zhou, and J. Han, Appl. Phys. Lett. 79, 3329 (2001). 4 X. Liu, S. Han, and C. Zhou, Nano Lett. 6, 34 (2006). 5 N. M. Pugno, J. Phys.: Condens. Matter 18, S1971 (2006). 6 A. Rochefort, P. Avouris, F. Lesage, and D. R. Salahub, Phys. Rev. B 60, 13824 (1999). 7 J. Y. Huang, S. Chen, Z. Q. Wang, K. Kempa, Y. M. Wang, S. H. Jo, G. Chen, M. S. Dresselhaus, and Z. F. Ren, Nature 439, 281 (2006). 8 M. S. Dresselhaus, G. Dresselhaus, A. Jorio, A. G. Souza Filho, and R. Saito, Carbon 40, 2043 (2002). 9 M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Carbon 33, 883 (1995). 10 R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London (1998). 11 S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 93, 185503 (2004). 12 E. D. Minot, “Tuning the Band Structure of Carbon Nanotubes,” Doctoral Dissertation, Cornell University (2004). 13 R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Appl. Phys. Lett. 60, 2204 (1992). 14 B. I. Yakobson, R. E. Smalley, American Scientist 85, 324 (1997). 15 N. M. Pugno, J. Phys.: Condens. Matter 18, S1971 (2006). 16 M. D. Frogley, Q. Zhao, and H. D. Wagner, Phys. Rev. B 65, 113413 (2002). 13 17 C. Jiang, H. Ko, and V. V. Tsukruk, Adv. Mat. 17, 2127 (2005). 18 V. Sazonova, Y. Yaish, H. Üstünel, D. Roundy, T. A. Arias, and P. L. McEuen, Nature 431, 284 (2004). 19 M.-F. Yu, B. S. Files, S. Arepalli, and R. S. Ruoff, Phys. Rev. Lett. 84, 5552 (2000). 20 M.-F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S. Ruoff, Science 287, 637 (2000). 21 E. W. Wong, P. E. Sheehan, C. M. Lieber, Science 277, 1971 (1997). 22 J. P. Lu, Phys. Rev. Lett. 79, 1297 (1997). 23 M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature 381, 678 (1996). 24 B. G. Demczyk, Y. M. Yang, J. Cumings, M. Hetman, W. Han, A. Zettl, and R. O. Ritchie, Mat. Sci. and Eng. A 334, 173 (2002). 25 J. Y. Huang, S. Chen, Z. Q. Wang, K. Kempa, Y. M. Wang, S. H. Jo, G. Chen, M. S. Dresselhaus, and Z. F. Ren, Nature 439, 281 (2006). 26 Z. W. Pan, S. S. Xie, L. Lu, B. H. Chang, L. F. Sun, W. Y. Zhou, G. Wang, and D. L. Zhang, App. Phys. Lett. 74, 3152 (1999). 27 Y. Li, K. Wang, J. Wei, Z. Gu, Z. Wang, J. Luo, and D. Wu, Carbon 43, 31 (2005). 14 Chapter 2 Raman Spectroscopy of Carbon Nanotubes Raman spectroscopy was one of the first tools used to study carbon nanotubes since the seminal report in 1991 by Sumio Iijima. 1 2 Since Raman spectroscopy is non-contact and provides structural and electronic information about nanotubes, 3 it is more informative and less disruptive compared to alternate methods of characterization. As will be discussed in this chapter, nanotubes’ strong electron- phonon coupling and one-dimensional nature make their Raman signals exceptionally strong. Raman Spectroscopy Raman scattering, discovered in 1928 by Chandrasekhara Venkata Raman, describes the effect by which photons are scattered by a phonon. 4 5 When light is scattered from an atom or molecule, most of the light is scattered elastically with the same energy, frequency, and wavelength as the incident photons. This is known as Rayleigh scattering, as shown in Figure 2.1. About one in every million scattered photons is scattered inelastically by a phonon, resulting in scattered photons having a frequency different from, and usually lower than, the frequency of the incident photons. In Raman scattering, an incident photon with energy ω h is shifted by the energy of a phonon with energy Ω h . There are two Raman scattering processes, known as Stokes and anti-Stokes scattering. In Stokes scattering, the emitted photon is of lower energy than the incident photon, due to the emission of a phonon. In anti- Stokes scattering, the emitted photon is of higher energy than the incident photon, due to the absorption of a phonon. 15 Figure 2.1: Different possibilities of light scattering: Rayleigh scattering (no Raman effect), Stokes scattering, and anti-Stokes scattering. 6 Since a phonon is absorbed in the anti-Stokes process, the anti-Stokes Raman intensity will depend on the phonon occupation number. This is given by the Maxwell-Boltzmann factor: T k E ph B ph e N − = , (2-1) where E ph is the energy of the phonon, k B is the Boltzmann constant, and T is the temperature in Kelvin. Since, in the Stokes process, a phonon is emitted, the Stokes Raman intensity does not depend on the phonon occupation number. Therefore, the ratio of the anti-Stokes to Stokes Raman intensity is given by the Maxwell-Boltzman factor, T k E B ph e − . 16 Resonance Raman Scattering The quantum process of resonance Raman scattering is shown in Figure 2.2. Electrons begin in the ground state (rightmost image) and then absorb the energy of a photon, resonant with a transition between electronic states, promoting an electron from the valence band into the conduction band (second image from right). This electron is now in a virtual state where it decays in energy, generating a phonon (second image from left). Finally, the electron decays back to the valence band, generating a photon, at a frequency shifted from the incident laser light (leftmost image). This process can be described by the following equation, where the Raman intensity is given by, 7 2 ) )( ( 0 0 ∑ − − − ∝ ij i L j phonon L E E Light i i phonon j j Light I ω ω ω h h h . (2-2) The first term in the numerator, j Light 0 , corresponds to the excitation of the electrons, as depicted by the rightmost arrow in Figure 2.2. The middle term, i phonon j , corresponds to the emission (or absorption) of a phonon, and is depicted by the middle arrow in Figure 2.2. The last term, 0 Light i , corresponds to the re-radiation of light, as depicted by the leftmost arrow in Figure 2.2. The first term in the denominator corresponds to resonance with the outgoing light and the second term corresponds to the resonance with the incoming light. 17 Figure 2.2: Electronic excitation processes for resonance Raman scattering. The electron is first excited by light (rightmost arrow), then imparts energy to the creation of a phonon (middle arrow), and then finally emits light at a shifted frequency (leftmost arrow). In practice, the matrix elements in the above intensity relation are not known. They are typically assumed to be constant and the resonance Raman process is described by the equation below. The variation of the intensity of Raman peaks with laser energy is given by the following relation, 2 ) )( ( 1 ) ( Γ − − Γ − − ± ∝ i E E i E E E E I ii laser ii ph laser laser , (2-3) where the first and second terms in the denominator, again, correspond to outgoing and incoming light resonances. The + (–) in the first term applies to the anti-Stokes (Stokes) process for a phonon of energy E ph . E ii is the electronic transition energy between van Hove singularities and Γ gives the inverse lifetime for the resonant scattering process. 8 This relation is plotted in Figure 2.3 with the anti-Stokes process shown in red and the Stokes process shown in blue. 18 Figure 2.3: Relative intensities for Stokes and anti-Stokes Raman intensity plotted as a function of laser energy. The Stokes and anti-Stokes intensities both have peaks at E ii as well as at E ii +E ph for Stokes and E ii -E ph for anti-Stokes. The peaks valued near E ii correspond to the incoming light resonance signal and the other peaks correspond to outgoing resonances. Raman Spectroscopy of Carbon Nanotubes In our Renishaw inVia Raman spectroscopy system, several mirrors guide the incident laser light through a microscope objective lens onto a maneuverable stage holding the sample. The collected light is passed through a notch filter, which filters out light at the laser frequency. The filtered light is reflected off of a grating, which spatially separates different frequencies of light onto the CCD. This automated system is calibrated to determine the frequency of Raman scattered light and displays the phonon count (intensity) versus Raman shift. 19 Since nanotubes’ diameters are almost three orders of magnitude smaller than the wavelength of light, special preparations are required to repeatably measure the Raman spectra of individual nanotubes. By depositing fiducial-micrometer scale metallic grids, using lithography and electron beam metal deposition on silicon substrates, one can identify the general location of an individual carbon nanotube as shown in the atomic force microscope image in Figure 2.4. The corresponding optical microscope image is shown on the right side of Figure 2.4. Repeatable observations of individual strongly resonant nanotubes are necessary to understand the properties of individual nanotubes. It is important to measure individual nanotubes because of the large variance of nanotube properties of different chiralities, that are averaged in our ensemble measurements. Figure 2.4: Atomic force microscope (AFM) image (left) and optical microscope image (right) of a silicon substrate with a metallic grid and sparsely distributed nanotubes. Raman spectra generally consist of a series of Lorentzian peaks that correspond to the phonon modes of the molecule or solid. Selection rules determine which peaks are Raman active. In a carbon nanotube, several peaks are Raman 20 active. One such peak is the radial breathing mode (RBM), as shown in Figure 2.5 and Figure 2.6, which corresponds to radial motion of carbon atoms and typically results in a Raman shift of 100-300 cm -1 . 3 The frequency of the RBM is inversely proportional to the diameter of the nanotube by the relation ω RBM = 223.5 cm -1 nm/d t + 12.5 cm -1 . 9 The G band Raman mode, also shown in Figure 2.5, corresponds to the tangential movement of atoms on the nanotube surface, as illustrated in Figure 2.6. This mode is observed in graphene, where it exists as a single peak. In nanotubes, however, the peak is split into the G + peak and the G - peak, which correspond to axial and circumferential motion of atoms, respectively. The G + peak generally appears at approximately 1590 cm -1 , with the G + band having a higher Raman shift than the G - band. The lineshape of the G - band can be used to distinguish between metallic and semiconducting nanotubes. Metallic nanotubes tend to have broad G - peaks, significantly downshifted with respect to the G + band, whereas semiconducting nanotubes have sharp peaks. 3 While the exact origin of this difference is not fully understood, it is believed to be caused by the phonons coupling strongly to the free electrons in a metallic nanotube. 10 Another important Raman mode is the D band, which is observed around 1350 cm -1 . The D band does not correspond to a simple zero momentum phonon and is only observed in nanotubes with a large amount of defects and disorder, which break the conservation of momentum requirements, that is anything that breaks the sp 2 symmetry of the nanotubes and hence, changes the selection rules. As a result, the intensity of the D band gives a measure of the amount of disorder present in the nanotube. Lastly, the 21 G’ band is the second harmonic of the D band. 3 This is a two-phonon process and does not require defects and disorder in order to be observed. Figure 2.5: (a) Radial breathing mode (RBM). The Stokes RBM in (a) appears at 150 cm -1 and the anti-Stokes RBM is at -150 cm -1 . Typical G bands for (b) semiconducting and (c) metallic carbon nanotubes. Note how the G - band shape is broader in the metallic nanotube. Figure 2.6: Illustration showing the atomic motion of the radial breathing mode (RBM) and G band modes of a carbon nanotube. Resonant Raman Spectroscopy of Carbon Nanotubes The electronic transition energy between the i th valence subband and the corresponding i th conduction subband is denoted as E ii . The transition energies can be seen in the density of states plots in Figure 1.5 as the spaces between the sharp peaks. These peaks arise from the one-dimensional nature of the nanotube and are known as van Hove singularities. Despite the nanotubes’ extremely small geometric 22 cross section, large enhancements in the Raman intensity are possible due to these singularities. Optimal Raman signals are found when the laser energy is resonant with these interband transition energies of nanotube, that is when E Laser = E ii . 3 A Kataura plot, as shown in Figure 2.7, plots the calculated E ii of different chirality nanotubes versus their diameter. Here, each black circle corresponds to a different semiconducting NT and each red open circle corresponds to a different metallic nanotube. Energy bands corresponding to cutting lines away from the K- point in the Brillouin zone of graphene exhibit a trigonal warping effect that is responsible for the unique energy bands for each (n,m) nanotube. The trigonal warping effect also causes the splitting of van Hove singularity peaks in the density of states of metallic nanotubes. This splitting increases with decreasing chiral angle, giving rise to a unique set of electronic transition energies E ii for each (n,m) value. The trigonal warping effect therefore causes a spread in the interband energies for nanotubes with the same diameter, as can be seen in Figure 2.7. The integer i in E ii denotes the index of the singularity in the joint density of states. Therefore, each nanotube (n,m) has a unique set of interband energies E ii denoting the energy differences between the i th valence subband and the i th conduction subband. Given a transition energy E ii and the diameter of a nanotube, the corresponding (n,m) index can be determined uniquely, making the Kataura plot very useful. In practice, these two values, E ii and d t , can narrow down the chirality of the nanotube to a few candidates, using the Kataura plot. The simple tight binding model, described in the previous chapter, predicts the electronic structure of nanotubes remarkably well. 23 However, more accurate models including the effects of nanotube curvature and excitons provide significant corrections to the independent particle approximation. 11 Figure 2.7: Kataura plot calculated using a nearest neighbor tight binding model with overlap integral γ 0 = 2.9 eV and nearest neighbor carbon-carbon distance of a cc = 0.144 nm. The black dots correspond to semiconducting nanotubes while the red circles correspond to metallic nanotubes. Conclusion Raman spectroscopy is a very powerful tool, particularly well-suited for characterizing carbon nanotubes. The Raman spectrum of an individual nanotube provides information about its diameter, metallic/semiconducting nature, atomic structure (n,m), defect concentration, strain, and temperature. Repeatable measurements on individual nanotubes can be performed by making use of a lithographically patterned fiducial grid. These experiments lend insight into the effects of strain, temperature, and defects on the electronic structure of carbon 24 nanotubes of different chiralities, as will be discussed in the remaining chapters of this thesis. 25 Chapter 2 Endnotes 1 S. Iijima, Nature 354, 56 (1991). 2 H. Hiura, T. W. Ebbesen, K. Tanigaki, and H. Takahashi, Chem. Phys. Lett. 202, 509 (1993). 3 M. S. Dresselhaus, G. Dresselhaus, A. Jorio, A. G. Souza Filho, and R. Saito, Carbon 40, 2043 (2002). 4 C. V. Raman, Indian Journal of Physics 2, 387 (1928). 5 C. V. Raman and K. S. Krishnan, Nature 121, 501 (1928). 6 http://en.wikipedia.org/wiki/Image:Ramanscattering.svg. 7 M. Cardona and P. Y. Yu, Fundamentals of Semiconductors: Physics and Material Properties, 3rd Edition (Springer-Verlag, Berlin, Germany, 2001), Chapter 7. 8 A. Jorio, A. G. Souza Filho, G. Dresselhaus, M. S. Dresselhaus, R. Saito, J. H. Hafner, C. M. Lieber, F. M. Matinaga, M. S. S. Dantas, and M. A. Pimenta, Phys. Rev. B 63, 245416 (2001). 9 S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, and R. B. Weisman, Science 298, 2361 (2002). 10 S. D. M. Brown, A. Jorio, P. Corio, M. S. Dresselhaus, G. Dresselhaus, R. Saito, K. Kneipp, Phys. Rev. B 63, 155414 (2001). 11 G. G. Samsonidze, R. Saito, N. Kobayashi, A. Grüneis, J. Jiang, A. Jorio, S. G. Chou, G. Dresselhaus, and M. S. Dresselhaus, App. Phys. Lett. 85, 5703 (2004). 26 Chapter 3 Raman Spectroscopy of Nanotubes under Axial Strain Applying axial strain to a carbon nanotube not only affects its physical structure, but also its electronic and vibrational structure, and hence its optical properties. We present, here, a novel method for measuring the Raman spectra of carbon nanotubes under axial strain. It is important to understand how nanotubes respond to strain, not just as a materials property, but also to gain insight into the behaviors of low-dimensional materials that differ fundamentally from their bulk counterparts. Theoretical ab initio and tight binding calculations predict large changes in the electronic band structure upon application of strain. 1 2 3 4 Conductance measurements on single wall carbon nanotubes (SWCNTs) strained by an atomic-force microscope (AFM) show an increase in resistance with strain due to localized defects, 5 mechanical deformation by the AFM tip, 6 and strain-induced band gaps. 7 While these previous studies focus on the electronic changes in nanotubes under strain, very little work has been done to determine the vibrational characteristics of nanotubes under strain. In previous measurements, large shifts in the vibrational energies of nanotubes were observed under strain, as well as a characteristic difference between metallic and semiconducting nanotubes, which was not well understood theoretically. 8 Resonant Raman spectroscopy was previously performed on bundles of semiconducting and metallic SWNTs using another method of inducing strain. 9 The previous AFM-induced strain technique allowed Raman spectra to be taken before 27 and after inducing strains, limited to 1.65% or less on a single nanotube. 8 Other polymeric nanotube Raman measurements focus on ensemble measurements, whereas our technique focuses on only one nanotube. 10 11 12 13 14 This is made possible by a lithographically defined grid pattern, which provides the high spatial precision required for single nanotube measurements. In this new approach, the strain can be varied in a continuous and reversible fashion up to a maximum strain of 17% on the bundle. By establishing the reversibility of our measurements, we are able to take into account slippage of the SWCNTs on the substrate. Furthermore, we observe both reversible and irreversible changes in the Raman spectra of nanotubes in these bundles, which will be the topic of this chapter. Experimental Measurement of Individual Carbon Nanotubes under Axial Strain By pinning down the ends of nanotube bundles on a silicone polymer, we apply axial strains up to 17%, which results in the debundling of nanotubes. The G band Raman spectra of metallic and semiconducting nanotubes are found to respond differently to strain and debundling, giving insight into the nature of the broad metallic G – band lineshape. For metallic nanotubes, the G – band upshifts and becomes narrower with strain, making it appear more semiconductor-like. Surprisingly, this metal to semiconductor transition is not reversible with strain, indicating that nanotube-nanotube coupling plays a significant role in the observed G – band of metallic nanotubes. 28 In this experiment, samples are prepared through a four step process, shown schematically in Figure 3.1. First, a 1mm thick film of polydimethyl siloxane (PDMS) is created from a Sylgard 184 silicone elastomer kit (Corning, Inc.). Next, SWCNTs synthesized by the laser ablation method 15 are sonicated in an isopropyl alcohol solution for 20 minutes and then deposited onto the PDMS substrate. The PDMS was then coated with S1805 photoresist through spin coating. Although the photoresist does not readily bind to the PDMS, it can usually cover most of the PDMS surface when spun at 4000 RPM. After that, metal strips of Cr-Au are patterned on top of the SWNTs using photolithography, holding the ends of the nanotube bundles fixed to the PDMS substrate. An AFM image of these strips holding a SWNT bundle is also shown in Figure 3.1. Finally, the ends of the PDMS substrate are clamped and the entire substrate is strained along its length. Figure 3.1: Processing steps for fabricating carbon nanotubes pinned to an elastomeric substrate. First, a slab of PDMS is prepared, followed by a deposition of nanotubes in an isopropyl alcohol solution. Gold strips are then patterned on top of the nanotubes using photolithography. Finally, the slab of PDMS is strained, applying strain to the nanotube bundles. An atomic force microscope image of one pinned nanotube bundle is shown in the image on the bottom right. 29 The amount of strain in percent is found by using the Pythagorean Theorem as shown in the following equations and in Figure 3.2: 2 2 d x + = l (3-1) () l l l l − Δ + + = Δ = 2 2 % d d x strain . (3-2) AFM images are taken to ensure that both ends of the nanotube are pinned beneath the metal strips and to determine the angle between the nanotube and the direction of applied strain. Raman spectra are taken with a Renishaw inVia Raman microprobe with a 532 nm Spectra Physics solid state laser and 633 nm HeNe laser, each delivering 1-2 mW of power to the sample. Raman spectra are taken as the PDMS substrate is incrementally strained and unstrained. Figure 3.2: Diagram illustrating how strain is determined for a nanotube lying at an angle θ with respect to the applied strain. The length of a nanotube (black angled line) at an angle to the contacts given the distance between contacts (gold rectangles) for an unstrained nanotube (left) and strained nanotube (right). The x value remains constant for all strains since strain is applied in the vertical direction and increases d. The % strain is calculated by dividing the increase in length ( Δ ℓ) by the unstrained length ( ℓ). 30 Semiconducting Nanotubes under Axial Strain The nanotube bundle shown in Figure 3.3 has a diameter of 13nm, and is expected to contain approximately 60 single wall nanotubes. It is likely the case that several nanotubes contribute to the G band while only one nanotube contributes to the radial breathing mode (RBM). The RBM frequency (ω RBM ) for the resonant nanotubes is 196 cm -1 , implying a diameter (d t ) of 1.3 nm, by the relation ω RBM = A/d t + B, where A = 223.5 cm -1 nm and B = 12.5 cm -1 . 16 Figure 3.3 shows the frequency of the G band Raman mode of a SWNT bundle at various degrees of applied strain. The sharp G band lineshape (upper right inset) indicates that this nanotube is semiconducting in nature. 17 During the initial 3% of strain, there is no change in the Raman frequency, indicating that the nanotube bundle is still slack between the metal strips. Beyond 3%, the G band frequency downshifts at a rate of 2.9 cm -1 /% strain. After reaching a strain of 6.5%, the strain is then reduced to 4.7%, in order to demonstrate that the strain is not relaxing due to slippage underneath the Cr-Au strips. The strain is then increased and decreased twice to ensure that the G band frequency varies reversibly, before increasing to 10% strain and beyond. Above 10% strain, we see a discrete jump in the Raman frequency of this nanotube, indicating a slip in strain of 1.3%. Similarly, we see another discrete jump of 4.5% at 13% strain. In between these discrete “jumps,” the frequency versus strain relationship is found to be reversible, indicating no slippage. Ultimately, the PDMS substrate broke when 17% strain was achieved on the nanotube bundle. Figure 3.3 31 shows typical results that were observed consistently in the four semiconducting nanotubes measured with this technique. Figure 3.3: (a) AFM image of a carbon nanotube bundle pinned under gold strips with a 0.5 μm diameter laser spot. The scale bar is 2 μm. (b) Raman spectrum from a 532 nm laser taken at 0% strain. The sharp peaks are typical of semiconducting nanotubes and show the high frequency component, G + and lower frequency component G – . (c) G + Raman frequency plotted as a function applied strain. Note the reversible regions of the strain, separated by discrete slips of 1.3 and 4.5%. Similar downshifts of the G - and G’ bands of this semiconducting nanotube with applied strain were observed as shown in Figure 3.4. Both the G - and G’ bands exhibited behavior similar to that of the G + band in that after a slack region, there was a downshift in frequency, followed by an upshift to a frequency higher than the initial, pre-strain frequency upon relaxation. This behavior was especially noticeable 32 in the G’ band which downshifts similarly to the G band, in general. 18 The observed downshifts in the G and G’ bands are consistent with strain-induced weakening of carbon-carbon bonds. Figure 3.4: G + (top), G - (middle), and G’ (bottom) Raman frequencies plotted as a function of applied strain. 33 Strain Dependence of the Raman Intensity Significant reduction in the intensities of the G + , G - , and G’ bands were also observed under strain, as shown in Figure 3.5. These decreases in intensity were initially attributed to nanotubes coming off of resonance with increased strain. However, when the strain on the nanotubes was decreased, the intensity continued to drop, indicating that irreversible changes in the nanotube bundle had occurred. For instance, if one compares the variation of the G + band frequency in the top graph of Figure 3.4 to the variation in the G + band intensity in the top graph of Figure 3.5, one can see that there is little correlation. If one also looks at the Raman frequencies at 8% strain in Figure 3.4, one sees that the nanotube takes on values of 1609 cm -1 , 1603 cm -1 , and 1590 cm -1 due to the slipping and straining of the nanotube bundle. However, at 8% strain, the intensity of this nanotube lies within a much smaller range of variation. Though the Raman frequencies surpass the value measured at 0% strain, the intensity never again reaches its initial value. In other words, the G + , G - , and G’ bands change reversibly in frequency as strain is applied, but the intensity of these bands decreases irreversibly. This indicates that nanotube-nanotube coupling contributes to the Raman signal of carbon nanotubes. Or more specifically, the Raman intensity drops as the nanotubes become de-bundled. 34 Figure 3.5: G + (top), G - (middle), and G’ (bottom) Raman intensity plotted as a function of strain for a nanotube bundle. 35 On a separate PDMS slab, the Raman modes of another nanotube were found to downshift by 24 cm -1 under applied strain, as shown in Figure 3.6. Significant downshifts in the G band frequency were achieved after the slab was strained beyond a slack region for the first 3% of strain. After a slip, which accounted for 1% strain, the minimum G band frequency was achieved at a total of 9% applied strain. After accounting for the 1% slippage and 3% slack, we estimate an actual strain value of 5%, for strain point #24. The subsequent additional strain applied to this nanotube bundle (9% to 14%) resulted in upshifts of the G band, indicating continued slippage of the nanotube on the PDMS substrate and/or debundling of the nanotubes within the bundle. This slip indicates a loss of adhesion of the nanotube with the PDMS surface. At maximum strain, we estimate that the nanotube was held down with a stress of ~860 GPa, based on a Young’s modulus of 1 TPa and reports of the G + band downshifting in frequency 28 cm -1 /% axial strain. 19 In this nanotube, both the G + and G - bands downshifted and no D band was observed. 36 Figure 3.6: G + band Raman frequency plotted with increasing strain. After an actual strain of 5% (strain point #24), the nanotube downshifted by 24 cm -1 and was held down with ~860 GPa of pressure. Although we were able to achieve 7.3% strain without slippage of the bundle, twisting and sliding of the nanotubes within the bundle account for a majority of the bundle elongation, 20 resulting in a significantly reduced actual strain on each individual nanotube. By comparing the downshifts observed in this work (2.9 cm -1 /% strain) with the large strain-induced downshifts observed in previous work (14.8 cm -1 downshift at 0.5% strain), we estimate the actual strain on the individual nanotubes within the bundle to be a factor of 10 lower than the nominal strain on the bundle. 27 This implies that the strain-induced changes in the optical transition energies are ten times less than the 100meV predicted theoretically for carbon nanotubes under these high strains, and therefore well within the resonant window. 1 2 The relative intensity of the D band divided by that of the G band (I D /I G ) is a quantity that increases as the sp 2 symmetry of a nanotube is broken. 18 In our measurements, I D /I G only increased under applied strain for one metallic nanotube, 37 as shown in Figure 3.7. This is surprising considering the large amount of strain (7.3%), which is expected to induce a significant distortion of the graphitic hexagons to break the sp 2 symmetry. The relative lack of change in I D /I G , together with the relatively small strain-induced downshifts of the G band (Δω G /Δσ = 2.9%), indicates that there is significant sliding of the nanotubes along each other within the bundle that disperses the nominal strain to the entire bundle. This results in a significantly reduced strain on each individual nanotube. 21 22 Four metallic and four semiconducting nanotubes were measured in this experiment and for each type, all nanotubes showed similar behavior, except for one metallic nanotube, as shown in Figure 3.7. Even for this nanotube, the D/G intensity ratio fluctuates with strain before reaching 13% strain, where strain was likely applied to the nanotube instead of the bundle. Figure 3.7: Change in D/G band intensity ratio versus strain. An increase in this ratio indicates that defects are forming within the nanotube as strain is applied. 38 Strain Dependence of the Radial Breathing Mode Changes in the radial breathing mode (RBM) frequency of several metallic nanotubes, as shown in Figure 3.8, corroborate the strain-induced de-bundling phenomenon. The RBM was observed to shift up irreversibly by 4 cm -1 with applied strain. This irreversible upshift was found consistently in a total of four metallic nanotubes and four semiconducting nanotubes measured using this PDMS strain technique, as shown in Figure 3.8 and Figure 3.9. This result agrees with the previous experimental findings of Rao, et al., who found the RBM frequencies of individual nanotubes to be slightly increased from those of nanotube bundles and attributed it to decoupling rather than a change in diameter. 23 39 Figure 3.8: Radial breathing mode frequency of four metallic nanotubes strained on PDMS and measured with a 633 nm laser. Note the positively sloped trend line for all four nanotubes. 40 Figure 3.9: Radial breathing mode frequencies of four semiconducting nanotubes strained on PDMS and measured with a 532 nm laser. Note the positively sloped trendline observed for all four nanotubes. These nanotubes all have diameters of about 1.3 nm. The quantity strain step (#) is a chronological integer corresponding to how many strain measurements have been taken, whether they resulted in an increase or decrease in strain. As the amount of strain increases, the RBM shifts up in frequency while the intensity decreases. It is tempting to assume that the increase in RBM frequency in Figure 3.10(a) is due to a strain-induced decrease in the nanotube diameter. However, the RBM frequency, on average, tends to increase both when the strain is increased and when it is decreased. This can be seen clearly in Figure 3.10(c), which shows that the RBM frequency correlates better with the number of 41 strain steps (strain #) than with the absolute strain itself (strain %). It is also tempting to assume that the drop in RBM intensity in Figure 3.10(b) is due to strain- induced shifting of the electronic band energies off resonance with the laser energy. However, the RBM intensity does not vary reversibly with strain and, again, correlates better with strain # than with strain %. We attribute these irreversible effects to a change in the nanotube-nanotube coupling within the bundle. The behavior in Figure 3.10 was observed consistently in a total of three semiconducting nanotubes measured in this study. Figure 3.10: (a) RBM frequency versus percent strain. (b) RBM intensity versus percent strain. (c) RBM frequency versus number of strain steps. (d) RBM intensity versus number of strain steps. Note how when plotted against the number of strain steps, the frequency increases and the intensity decreases despite decreases in the strain %, indicating the occurrence of irreversible changes to the nanotube. 42 Inter-nanotube coupling has been found to red-shift the energies of optical transitions by tens of meV. 24 The main effect we observe in Figure 3.10(b) and (d) is most likely due to a change in E ii because of the de-coupling of nanotubes within the bundle. Rao, et al., observed increased RBM frequencies for single nanotubes as compared to bundled nanotubes. 23 In our measurement, there is a single resonant nanotube within the bundle. Thus, we eliminate the ensemble effects in previous measurements, and corroborate that the RBM frequency in fact increases as the nanotube becomes decoupled from its neighbors. This is in contrast to Popov, et al. and Henrard, et al., who calculated the RBM frequency and intensity of isolated nanotubes and nanotubes in bundles. 25 26 27 These calculations predict a decrease in the vibrational frequency as nanotubes are decoupled. Metallic Nanotubes under Axial Strain Figure 3.11 shows the frequency of the upper (G + ) and lower (G – ) components of the G band Raman mode of a different SWCNT bundle measured at various degrees of strain with 633 nm light. The broad and downshifted G – peak indicates that the resonant nanotube in the bundle is metallic. 28 The Raman frequency of the G + mode varies reversibly with strain, while the G – mode varies irreversibly. The data start at 8.4% strain. As the strain was decreased to 7.5%, the G – mode shifted up from 1558 cm -1 to 1567 cm -1 , while the G + mode shifted up by only 2 cm -1 . The strain was subsequently increased to 11.3%, resulting in a downshift of both the G + and G – modes. Here, the G + mode shifts reversibly, and the 43 G – mode shifts irreversibly. After 12%, the strain was reduced. Again, the G + mode varies reversibly along the same slope, while the G – mode increases rapidly in an irreversible fashion. All subsequent straining resulted in reversible shifts of both G + and G – modes, with average slopes of –1.8 cm -1 /% strain and –0.9 cm -1 /% strain, respectively. These reversible downshifts with strain are understood on the basis of the elongation of the C-C bond, which weakens the bond, therefore lowering its vibrational frequency. For the nanotube shown in Figure 3.11, the straining and relaxing resulted in an overall upshift of the G – band by 20 cm -1 , from 1558 cm -1 to 1578 cm -1 . This irreversible upshift can be seen clearly in Figure 3.11(a) and Figure 3.11(b). 44 Figure 3.11: (a) G + band and (b) G – band Raman frequency of one of four metallic nanotubes at various degrees of strain. (c) G band Raman spectra taken by a 633 nm laser after subsequent straining and unstraining of the bundle. The pronounced upshift of the G - band Raman frequency was observed unanimously in all four resonant metallic nanotubes measured in this study, but was not observed in any of the semiconducting nanotubes measured. We attribute this upshift to a change in the nanotube-nanotube coupling within this bundle, which affects this Raman mode through plasmon-phonon coupling. That is, the G – band phonons of the resonant nanotube couple to plasmons in neighboring nanotubes within the bundle. This plasmon-phonon coupling mechanism has been predicted to 45 cause phonon-softening (downshift) of the G band in metallic nanotubes. 6 The irreversible upshifts observed under strain indicate that the plasmonic coupling of neighboring nanotubes is weakened as the bundle is perturbed (strained or relaxed). This debundling causes an upshift of the G – Raman mode. The Raman shift of the G band versus strain can be seen in Figure 3.12 for two other metallic nanotubes. After an initial period of relatively little change in the G band Raman frequency, the frequency increases by about 15 cm -1 . After this jump in Raman frequency, strain is subsequently decreased. However, the G band frequency increases upon relaxation of the PDMS slab and never decreases back to its pre-strain values, as shown in Figure 3.12(a). The G band lineshape changes after this irreversible shift from broad to narrow as shown in Figure 3.12(b). After the significant upshift, the G band frequency decreases reversibly with strain. Another nanotube also experienced an irreversible upshift of 20 cm -1 , as shown in Figure 3.12(c). This upshift is indicative of a transition from metallic to semiconducting behavior. 46 Figure 3.12: G band Raman frequency versus strain for two metallic nanotubes experiencing irreversible upshifts in frequency. The plot in (a) depicts an upshift of 30 cm -1 to a region with a slope of -2.3 cm -1 /% strain. G band spectra before and after the irreversible upshift are shown in (b) for this nanotube. Note the change in lineshape from broad to narrow after the upshift. The plot in (c) depicts an upshift in the G - band of another nanotube of 20 cm -1 to a region with a slope of -1.0 cm -1 /% strain. Reversible changes in the G band linewidths are observed, after the large upshift takes place. This corresponds to the nanotubes ceasing to decouple with strain. In this reversible regime, a difference in the reversible changes of metallic and semiconducting nanotubes is seen. Figure 3.13 shows the G band linewidths (full width at half maximum) plotted as a function of strain for a metallic and a semiconducting nanotube. It is clear from this figure that metallic and 47 semiconducting nanotubes respond very differently to strain. The G band linewidths of semiconducting nanotubes increase with strain [Figure 3.13(c) and Figure 3.13(d)], while that of metallic nanotubes decrease with strain [Figure 3.13(a) and Figure 3.13(b)]. The strain-induced broadening observed in semiconducting nanotubes is attributed to inhomogeneity, which is expected if the strain is not uniformly distributed along the length of the nanotube. The strain-induced narrowing we observe in the case of the metallic nanotubes is very different from this broadening. This narrowing indicates the presence of a fundamentally different phenomenon, namely the formation of a strain-induced bandgap. Unlike nanotube- nanotube coupling, the strain-induced bandgap varies reversibly with strain. The strain-induced bandgap changes the number of free carriers at the Fermi energy, which strongly interact with this phonon mode. The G – band FWHM data shown in Figure 3.13(b) was fit to a slope of –4 cm -1 /% strain. Because the actual strain on each individual nanotube is much less than the nominal strain on the bundle, it is difficult to estimate the change in the bandgap with strain. 48 Figure 3.13: FWHM linewidths for the (a) G + band and (b) G – band of a metallic nanotube measured with a 633 nm laser and (c) G + band and (d) G – band FWHM linewidths for a semiconducting nanotube measured with a 532 nm laser. Data for other nanotubes can be found in Appendix 3. Conclusion We measured the resonant Raman spectra of nanotube bundles under strains up to 17%. The elastic polymer substrate technique allows the strain to be varied in a reversible fashion over a wide range of strains. We find that only a small fraction of the strain applied to the bundle is transferred to individual nanotubes within the bundle. The main effect of this strain is to debundle the nanotubes. The G band Raman frequency is found to decrease with applied strain due to the elongation of the C-C bond. Slippage of the nanotubes on the substrate is found to occur in 49 discrete jumps and can therefore be determined quantitatively from the Raman data. As the nanotube bundles are strained, their RBM frequencies are found to increase, indicating a strain-induced decoupling of nanotubes within the bundle. The G + band Raman frequency decreases by ~15 cm -1 for both metallic and semiconducting nanotubes over the applied strain range. However, the G – band Raman spectra of metallic and semiconducting nanotubes respond differently to strain, giving insight into the nature of the broad metallic G – band lineshape. The G – band frequency decreases by ~15 cm -1 with applied strain for semiconducting nanotubes, while the G – band is found to increase by 20 cm -1 and become more narrow with strain for metallic nanotubes. 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Balkanski, “Lattice Dynamics of Carbon Nanotubes” appearing in Current Topics in Physics: In Honor of Sir Roger J Elliott, World Scientific Publishing Co. (2005). 26 L. Henrard, E. Hernández, P. Bernier, and A. Rubio, Phys. Rev. B 60, R8531 (1999). 27 S. Reich, C. Thomsen, and J. Maultszch, Carbon Nanotubes: Basic Concepts and Physical Properties (Wiley-VCH, Weinheim, Germany, 2004), Vol. 1, p.147. 28 M.S. Dresselhaus, G. Dresselhaus, A. Jorio, A.G. Souza Filho, R. Saito, Carbon 40, 2043 (2002). 52 Chapter 4 Theoretical Modeling of Nanotubes under Axial Strain Our experimental findings demonstrate properties of nanotubes that have been predicted theoretically. The strain-induced changes we observe in the Raman spectra can be explained through an electronic tight binding model and a vibrational force constant based on graphene. These models explain the strain-induced downshift in Raman frequency, changes in nanotube resonance condition and Raman intensity, and metal-to-semiconductor transition. Other models and simulations such as ab initio calculations and density functional theory predict values for nanotubes’ structural properties, including the Young’s modulus. Electronic Structure under Axial Strain Yang, et al., formulated an analytical model for calculating a nanotube’s electronic subband energies under strain. 1 2 Their calculation is based on a simple tight binding model of graphite, and uses zone folding to reduce the 2-dimensional dispersion relation in graphene to the 1-dimensional dispersion in nanotubes. This model was described in detail in Chapter 1 of this thesis. By expressing the three nearest neighbor bond lengths (r i ) in terms of their axial and circumferential components, axial strain is represented by increasing the axial component by ε, as given below. t C m n a c C m n a r h h ˆ ) 1 ( 3 2 ˆ 2 0 0 1 ε + − + + = r (4-1) t C m n a c C m a r h h ˆ ) 1 ( 2 3 2 ˆ 2 0 0 2 ε + + − − = r (4-2) 53 t C m n a c C n a r h h ˆ ) 1 ( 2 3 2 ˆ 2 0 0 3 ε + + − − = r (4-3) Here c ˆ and t ˆ are the circumferential and axial unit vectors, respectively. As strain is applied, the C-C bonds lengthen and the overlap integral decreases with the square of the bond length 1 γ i = γ 0 (r 0 /r i ) 2 , where γ 0 = 2.9 eV and the unstrained bond length r 0 is 0.144 nm. This results in the strain-dependent energy dispersion relation: )] ( cos[ 2 { ) ( 2 1 2 1 2 3 2 2 2 1 r r k k E r r r r − ⋅ + + + ± = γ γ γ γ γ 2 / 1 3 2 3 2 1 3 3 1 )]} ( cos[ 2 )] ( cos[ 2 r r k r r k r r r r r r − ⋅ + − ⋅ + γ γ γ γ . (4-4) In this equation, all terms with subscripts depend independently on the axial strain, ε. 3 4 5 These equations predict strain-induced shifts in the subband transition energies (ΔE ii ). These shifts can be positive or negative depending on the value of (n–m) mod 3. For semiconducting nanotubes, the E 33 subband transition shifts upward in energy with strain if (n–m) mod 3 = 1, and shifts downward if (n–m) mod 3 = 2, as shown in Figure 4.1. In general, the sign of the strain-induced shift can be specified by the expression (-1) i (-1) ν , where i is the subband index,ν = (n-m) mod 3 for semiconducting nanotubes and ν = 1 for − 11 E and ν = 2 for + 11 E in metallic nanotubes. 3 4 5 Figure 4.1 shows the strong chirality dependence of these strain- induced shifts, calculated using this model. 54 Figure 4.1: Chiral angle dependence of the shift in metallic nanotubes’ E 11 subband energies and semiconducting nanotubes’ E 33 subband energies under 1% axial strain. Calculations were made on nanotubes in the diameter range 1.1–1.5 nm. 6 The density of states can be obtained from this dispersion relation by taking its derivative with respect to the axial momentum (k t ) and summing over the subbands: ∑ = − = N j t t j dk k dE E DOS 1 1 ) ( 2 1 ) ( π , (4-5) where N is the total number of subbands. Here, all the momenta in the Brillouin zone are summed at a given energy E. The strain-induced changes in the Raman intensities observed by Lucas, et al. and Cronin, et al., can be explained with this tight binding model. 2 6 Using this model, the effect of strain is calculated for a (14,6) semiconducting nanotube and a (16,1) metallic nanotube. The results are shown in Figure 4.2. It is evident that even small strains can have a large effect on the density of states. For a (14,6) nanotube, a strain of 1% causes a downshift of 57 meV in the position of a van Hove singularity. In metallic nanotubes, the E 11 subband transition splits into two singularities due to 55 the trigonal warping effect. As strain is applied, the energy of the van Hove singularities shift towards each other, as shown in Figure 4.2. The experimentally observed shifting of the RBM on and off of resonance can be explained by these large shifts (57 meV), which exceed the resonant window of nanotubes. Figure 4.2: Joint density of states for a (14,6) semiconducting nanotube and a (16,1) metallic nanotube with and without 1% axial strain. 2 The Raman intensity changes as the resonant subband transition energy shifts toward or away from the laser energy. The intensity of the resonant Raman process in nanotubes can be expressed as: 2 ) )( ( 1 ) ( r ii ph l r ii l l i E E E i E E E I Γ − − ± Γ − − ∝ , (4-6) 56 where E ii is the resonant transition energy, E l is the laser energy, E ph is the phonon energy, and Γ r is the inverse scattering lifetime for the Raman scattering process. 7 In the ±E ph term, the – sign represents Stokes processes (phonon emission) and the + sign represents anti-Stokes processes (phonon absorption). Using an extended tight binging model, Souza Filho, et al., predicted universal shifts in E ii for families of nanotubes with constant (n-m). This model also revealed a strain-induced interference effect in the Raman scattering cross section of metallic nanotubes under strain. 8 Comparison of Theoretical Predictions with Experimental Results It is tempting to attribute many of the strain-induced changes observed experimentally, as reported in Chapter 3, to the theoretically predicted changes in the electronic band structure of nanotubes described in this chapter. Among these are, the large drop in Raman intensity observed with applied strain [see Figure 3.10(b) and (d)], which could be attributed to large changes in the van Hove singularities and hence, resonant transition energies. However, this data was found to correlate better with the number of strain steps rather than with the absolute strain, indicating irreversible changes in nanotube-nanotube coupling within a bundle. The irreversible changes are found in metallic nanotubes, which have different Raman spectra than semiconducting nanotubes. Because of the strong electron-phonon coupling in nanotubes, the G band Raman spectra of metallic and semiconducting nanotubes are qualitatively very different. 9 In metallic nanotubes, 57 the lower frequency component of the G band (G - ) exhibits a broad lineshape and is significantly downshifted in frequency with respect to its counterpart in semiconducting nanotubes. 10 While the G – Raman feature in metallic nanotubes is generally attributed to phonon coupling to the continuum of electronic states, the precise nature of this feature is not understood. There have been several conflicting reports in the literature, including a Peierls-like mechanism 11 12 and a nanotube bundling effect. 13 The dramatic narrowing of the G band of metallic nanotubes under strain, as shown in Figure 3.13, can be explained through a strain-induced bandgap, as predicted by equation (4-4). However, these observed strain-induced changes, again, correlate more strongly with the chronological strain step than with absolute strain. Instead, these irreversible changes with strain are thought to be caused by decoupling of nanotubes within the bundle. This decoupling has been found to diminish the broad, low-energy mode of the G band of carbon nanotubes. 13 In addition to this analytical tight binding model, much work has been done simulating nanotubes under strain using both ab initio and molecular dynamics calculations. Ab initio calculations can be used to predict, from first principles, the electronic structure of carbon nanotubes under various perturbations, such as strain. Ogata, et al., performed both density functional theory (DFT) calculations and tight binding calculations of nanotubes under strain. 14 They found the tight binding predictions to be in good agreement with the DFT results. Ito, et al., also performed ab initio calculations of the electronic band structure of carbon nanotubes under axial strain, and found results that were consistent with those of Ogata, et al., at moderate 58 strains. 15 It should be noted that these numerical simulations are generally limited to high symmetry nanotubes (i.e. armchair and zigzag), which have fewer atoms in the unit cell than chiral nanotubes. Molecular dynamics calculations can be used to predict atomistically the mechanical properties, such as Young’s modulus and defect formation, of nanotubes under strain. Zhang, et al., used tight binding molecular dynamics calculations to investigate the plastic deformation of carbon nanotubes under strain. These simulations predict that the strain required to form 5-7 defects is strongly dependent on chiral angle. 16 Prylutskyy, et al., used a modified tight binding model to predict the Young’s moduli for (5,5) and (10,0) nanotubes to be 1.10 TPa and 1.20 TPa, respectively, in good agreement with experimental results. 17 Zhang, et al., predict a Young’s modulus for SWNTs using a nanoscale continuum theory to be 705 GPa, which is also in agreement with experiments and other simulations. 18 Breaking strains between 10% and 15% have been predicted near 0 K using molecular dynamics simulations for various zigzag nanotubes such as (9,0) and (20,0). 19 The breaking strains in this simulation correspond to fracture stresses in the range of 65 to 93 GPa. This study found that 5-7 defects have little effect on the strength of nanotubes, only lowering their failure strain by a few tenths of a percent. Calculation of the Vibrational Structure of Carbon Nanotubes Since the atomic structure of carbon nanotubes is closely related to that of graphene, their vibrational properties are also related. Graphene’s phonon dispersion relation consists of six basic vibrational modes corresponding to six phonon branches. 59 Six modes exist because there are two atoms in the unit cell and three degrees of freedom: radial, tangential-in-plane, and tangential-out-of-plane. The three acoustic modes correspond to the low energy modes with zero energy at the Γ point of the Brillouin zone as shown in Figure 4.3. 20 . The two acoustic branches that are linear correspond to the in-plane modes: radial (bond stretching) and tangential (bond- bending). The out-of-plane (transverse) mode has a k 2 dependence near the origin, for both acoustic transverse modes and optical transverse modes (~865 cm -1 ). Figure 4.3: (a) Phonon dispersion curves, plotted along high symmetry directions, for a two-dimensional graphene sheet. (b) The corresponding density of states vs. phonon energy for phonon modes in units of states/1C-atom/cm -1 . 20 The graphene phonon dispersion relation can be derived from a fourth nearest-neighbors force constant model. Figure 4.4 shows the fourth-nearest- neighbors for the A and B carbon atoms, the two non-identical atoms of the graphene 60 unit cell. The force constant parameters, shown in Table 4.1, were obtained from fitting the two-dimensional phonon dispersion relations as determined experimentally from neutron scattering or electron energy loss spectroscopy measurements. 20 These parameters are used in a force constant tensor, K (ij) , which couples the vibrations of the i th and j th atoms of the fourth nearest-neighbors model. The dynamical matrix elements are given by the product of K (ij) and the phase difference factor, ) exp( ij R k Δ ⋅ i . The eigenvalues of the dynamical matrix give the phonon dispersions, () K m D 2 ω . Figure 4.4: Fourth-nearest-neighbors for an (a) A carbon atom and (b) B carbon atom. First nearest neighbors are shown with open circles, second nearest neighbors are shown with solid squares, third nearest neighbors are shown with open squares, and fourth nearest neighbors are shown with open hexagons. The circles connecting atoms are guides for the eye. 20 61 Table 4.1: Force constant parameters for graphite in units of 10 4 dyn/cm. Here, the subscripts r, ti, and to, correspond to radial, transverse in-plane, and transverse out-of-plane, respectively. 20 The nanotube phonon dispersion relation can be obtained from that of graphene through zone-folding. This is similar to the approach used to determine the electronic dispersion relations, described in Chapter 1. Carbon nanotubes have 2N atoms per unit cell, where N is the number of hexagons in the unit cell. The 6N phonon dispersion relations for the x, y, and z vibrations for each atom are folded into the one-dimensional Brillouin zone of a carbon nanotube along the K 2 direction. The phonon dispersion relations depend on the chirality, (m,n), of the nanotube due to the discretization of the K 1 vector imposed by the periodic boundary conditions. The one-dimensional phonon energy dispersion relation is related to the two- dimensional energy relations for a graphene sheet through the following equation, () T k T N m k k m D m D π π μ μ ω ω μ ≤ < − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = - ; 1 ..., 0, ; 6 ..., 1, , 1 2 2 2 1 K K K , (4-7) where () K m D 2 ω denotes the two-dimensional energy dispersion relations for a graphene sheet, k is a one-dimensional wave vector, and T is the magnitude of the one-dimensional translation vector T. 20 This equation implies that the nanotube phonon dispersion is derived from projections of the N cutting lines onto the 62 graphene phonon dispersion relation. The nanotube phonon dispersion relations can be approximated by taking these projections onto the graphene phonon dispersions, derived from the fourth nearest-neighbors force constant model, even if the graphene dynamical matrix is still used, as shown in Figure 4.5. This dispersion relation agrees with that in Saito, et al., who calculated their dispersion relation using a nanotube dynamical matrix. 20 For (10,10) nanotubes (N = 20), there are 66 distinct phonon branches, despite having 120 degrees of vibrational freedom. This is because 12 modes are non-degenerate and 54 modes are doubly-degenerate. 20 Figure 4.5: Comparison of phonon dispersion relations using a graphene dynamical matrix for a (10,10) nanotube. The phonon dispersion relation was calculated using our MATLAB code as shown in Appendix 4. The y-axis shows the phonon frequency in cm -1 and the x-axis shows the k vector magnitude from 0 to T π . 63 Vibrational Structure of Carbon Nanotubes under Axial Strain The application of axial strain to a nanotube can be modeled using the fourth- nearest-neighbors force constant model. In the zone-folding model mentioned previously, we lengthen the axial component of the carbon-carbon bonds by a factor of ε, causing the distances and angles from the A and B atoms to the first through fourth nearest neighbors to change, as was shown in equations (4-1) through (4-3). Using the graphene dynamical matrix closely approximates models using the nanotube dynamical matrix, which differs in that it takes into account the nanotube curvature. We have coded a MATLAB program which simulates the effect of strain on the nanotube phonon dispersion relation using a graphene dynamical matrix (see Appendix 4). We have calculated the effect of strain on a (10,10) nanotube, as can be seen in Figure 4.6. As more strain is applied, the degenerate bands split, and some bands increase in frequency while others decrease. Most notable of these bands are the highest energy bands occurring for μ = 0 in Equation (4-7), which have frequencies near 1600 cm -1 at k = 0. These Raman active bands correspond to the nanotube G band, which corresponds to the tangential motion of atoms on the nanotube surface. As strain is applied, the higher energy band corresponding to the longitudinal optical mode of the nanotube (G + band) upshifts, and the lower energy band corresponding to the transverse optical mode of the nanotube (G - band) downshifts. 64 Figure 4.6: Phonon dispersion relations for a (10,10) nanotube with (a) 0%, (b) 1%, (c) 2%, and (d) 3% axial strain for Raman resonant phonon frequencies near. One can see the longitudinal optical (upper curve) and transverse optical (lower curve) modes separate and shift in opposite directions. While our simulations predict a strain-induced upshift of G + and downshift of G - , our experiments have shown consistent downshifts of both the G + and G - bands. The reason for this discrepancy lies in the force constants listed in Table 4.1, which will become inaccurate under high strains as the bonds become weaker. Furthermore, the rate at which these bonds weaken with strain (that is, the anharmonic term in the interatomic potential) is not known. 21 Reflecting the change in bond strength with strain in the force constant matrix would give a more accurate depiction of axial strain on a carbon nanotube and perhaps reflect the true shifts of the longitudinal optical and transverse optical modes with strain. Another simplification of the model, which is of less importance, is that the graphene dynamical matrix is used instead of 65 a nanotube dynamical matrix. A 6N × 6N nanotube dynamical matrix is much larger than a 6 × 6 graphene dynamical matrix, where N can exceed 1,000 for chiral nanotubes with large unit cells. Thus, simulation times increase by a factor of N 2 . These simulations, however, take into account nanotube curvature through axial rotations of the force constant tensor. 20 Conclusion In this chapter, we have discussed modeling of the electronic and vibrational structure of carbon nanotubes both in equilibrium and under strain. A tight-binding model of the electronic structure predicts strain-induced shifts in nanotube subband transition energies, changes in the density of states, as well as changes in the Raman intensity. Furthermore, a fourth-nearest neighbor force constant model predicts the phonon dispersion relations for carbon nanotubes under axial strain. This model shows downshifts in the Raman-active bands with increasing strain, as observed experimentally. 66 Chapter 4 Endnotes 1 L. Yang, M. P. Anantram, J. Han, and J. P. Lu, Phys. Rev. B 60, 13874 (1999). 2 M. Lucas and R. J. Young, Phys. Rev. B 69, 085405 (2004). 3 R. B. Capaz, C. D. Spataru, P. Tangney, M. L. Cohen, and S. G. Louie, Phys. Status Solidi (b) 241, 3352 (2004). 4 S. B. Cronin, Y. Yin, A. Walsh, R. B. Capaz, A. Stolyarov, P. Tangney, M. L. Cohen, S. G. Louie, A. K. Swan, M. S. Ünlü, B. B. Goldberg, and M. Tinkham, Phys. Rev. Lett. 96, 127403 (2006). 5 Y. N. Gartstein, A. A. Zakhidov, and R. H. Baughman, Phys. Rev. B 68, 115415 (2003). 6 S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. B 72, 035425 (2005). 7 A. Jorio, A. G. Souza Filho, G. Dresselhaus, M. S. Dresselhaus, R. Saito, J. H. Hafner, C. M. Lieber, F. M. Matinaga, M. S. S. Dantas, and M. A. Pimenta, Phys. Rev. B 63, 241404(R) (2001). 8 A. G. Souza Filho, N. Kobayashi, J. Jiang, A. Grüneis, R. Saito, S. B. Cronin, J. Mendes Filho, Ge. G. Samsonidze, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. Lett. 95, 217403 (2005). 9 M. S. Dresselhaus, G. Dresselhaus, A. Jorio, A. G. Souza Filho, and R. Saito, Carbon 40, 2043 (2002). 10 F. Wang, M. Y. Sfeir, L. Huang, X. M. Henry Huang, Y. Wu, J. Kim, J. Hone, S. O'Brien, L. E. Brus, and T. F. Heinz, Phys. Rev. Lett. 96, 167401 (2006). 11 O. Dubay, G. Kresse, and H. Kuzmany, Phys. Rev. Lett. 88, 235506 (2002). 12 S. Reich, C. Thomsen, and J. Maultszch, Carbon Nanotubes: Basic Concepts and Physical Properties, 1st edition (Wiley-VCH, Weinheim, Germany, 2004), p.168. 13 C. Jiang, K. Kempa, J. Zhao, U. Schlecht, U. Kolb, T. Basché, M. Burghard, and A. Mews, Phys. Rev. B 66, 161404(R) (2002). 14 S. Ogata and Y. Shibutani, Phys. Rev. B 68, 165409 (2003). 15 T. Ito, K. Nishidate, M. Baba, and M. Hasegawa, Surface Science 514, 222 (2002). 67 16 P. Zhang, P. E. Lammert, and V. H. Crespi, Phys. Rev. Lett. 81, 5346 (1998). 17 Yu. I. Prylutskyy, S. S. Durov, O. V. Ogloblya, E. V. Buzaneva, P. Scharff, Comp. Mat. Sci. 17, 352 (2000). 18 P. Zhang, Y. Huang, P. H. Geubelle, P. A. Klein, K. C. Hwang, Intl. J. Sol. Struct. 39, 3893 (2002). 19 T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. Ruoff, Phys. Rev. B 65, 235430 (2002). 20 R. Saito, M. S. Dresselhaus, and G. Dresselhaus. Physical Properties of Carbon Nanotubes. Imperial College Press, UK (1998). 21 N. R. Raravikar, P. Keblinski, A. M. Rao, M. S. Dresselhaus, L. S. Schadler, and P. M. Ajayan, Phys. Rev. B 66, 235424 (2002). 68 Chapter 5 Surface Enhanced Raman Spectroscopy of Individual Carbon Nanotubes Raman spectroscopy is a powerful tool that gives the precise vibrational energies of molecules, which provide the fingerprint for unique chemical identification. As such, Raman spectroscopy is extremely useful for a vast number of applications. However, the Raman scattering cross-section of most molecules is extremely small, which generally limits its potential uses. Surface-enhanced Raman spectroscopy (SERS) can be used to improve the small Raman intensities, thus, making Raman spectroscopy related applications more practical. Reports of SERS enhancement factors span a wide range from two orders of magnitude 1 to fourteen orders of magnitude. 2 Since this previous work primarily involved nanoparticles in solution and roughened metal surfaces, it is not possible to image the exact geometry of the nanoparticle/molecule complex. Consequently, several unexplored factors remain, including the separation between the nanoparticle and Raman active molecule, the number of molecules on each nanoparticle, the number of nanoparticles within the focal volume, and the extent to which nanoparticles couple plasmonically to each other. A more complete understanding of SERS will enable reliable single molecule spectroscopy and bring forth applications and analytical techniques achievable with handheld spectrometers. Surface Enhanced Raman Spectroscopy Metal nanoparticles, whose size is small compared to the wavelength of light, exhibit a surface plasmon resonance phenomenon. The surface plasmons generate 69 extremely high electric fields near their surface. When a Raman-active molecule sits close to the nanoparticle surface, great enhancements of Raman signals occur. This effect is known as surface enhanced Raman spectroscopy or SERS. The SERS effect was first produced in 1974 by Martin Fleischman, who noticed large enhancements of the Raman signal from pyridine adsorbed on roughened silver. 3 Three years later, two groups discovered the enhanced signal could not be accounted for by the chemical concentration alone. 4 5 However, both electromagnetic and chemical effects contribute to the SERS enhancement. 6 In normal Raman scattering, the Stokes Raman signal intensity is proportional to the Raman cross section, the excitation laser intensity, and the number of molecules that are within the probe volume, as shown in the left image of Figure 5.1. 6 In these situations, the Raman cross sections are so small that at least ~10 8 molecules are necessary to generate a measurable Raman scattering signal. 6 However, in SERS, where molecules are attached to metallic nanostructures, the surface-enhanced Stokes Raman signal intensity is proportional to the Raman cross section of the adsorbed molecule, the excitation laser intensity, and the number of molecules that are involved in the SERS process, as seen in the right of Figure 5.1. 6 The number of molecules involved in the SERS process can be smaller than the number involved in the normal Raman scattering probed volume. 6 The quantification of the enhancement over normal Raman scattering is known as the SERS enhancement factor. 70 Figure 5.1: Comparison of normal Raman Scattering (left) and surface-enhanced Raman Scattering (right). In normal scattering, the conversion of laser light I L into Stokes scattered light I NRS is proportional to the Raman cross section σ R free, the excitation laser intensity I( ν L ), and the number of molecules N that are within the probe volume. The right depicts a SERS experiment where σ R ads describes the increased Raman cross section of the adsorbed molecule, also known as “chemical” enhancement. A( ν L ) and A( ν S ) are the field enhancement factors at the laser and Stokes frequency, respectively. N’ is the number of molecules involved in the SERS process. 6 In the bottom of Figure 5.1, A( ν L ) and A( ν S ) represent field enhancement factors of the laser and Stokes fields. 6 These enhancement factors arise from enhanced local optical fields at the position of the molecule near the metal surface due to excitation of electromagnetic resonances, which arise from the collective oscillation of conduction electrons in the small metallic structures. These oscillations are referred to as surface plasmons. 6 Because both the excitation field, as well as the Raman scattered field, contribute to the overall Raman enhancement, the SERS signal is proportional to the fourth power of the electric field enhancement factor. 6 Maximum values for electromagnetic enhancement are on the order of 10 6 to 10 7 for isolated particles of isolated single colloidal silver and gold spheroids. 6 71 Closely spaced interacting nanoparticles to provide additional field enhancement, particularly near the gap, resulting in electromagnetic SERS enhancement factors up to 10 8 according to theoretical calculations. 6 Theory predicts strong enhancement of electromagnetic fields near sharp features and regions with large curvature through a so-called lightning rod or antenna effect, which may exist near silver and gold nanostructures. 6 7 Silver nanoparticles, grown on glass substrates, have shown average SERS enhancements of 10 7 . 7 The plasmon resonance frequencies involved generally depend on the size, shape, and material of the metallic nanoparticles and their environment. 8 The individual dipole oscillators of small isolated particles within colloidal cluster structures were found to couple experimentally. 6 The excitation over the entire cluster is non-uniform and tends to be spatially localized in SERS “hot spots.” 6 The non-uniform distribution of excitation results from the fractal geometry of the clusters where the excitations are considered neither as surface plasmon polaritons nor as independent localized surface plasmons. 6 Electromagnetic field variations within these hot spots can exceed 10 5 , implying local electromagnetic SERS enhancement factors of more than 10 10 . 8 Chemical enhancement is due to specific interactions involving electronic coupling between the molecule and metal that enhances the Raman signal. 6 The chemical enhancement can be clearly seen in experiments involving the adsorption of CO and N 2 on metals. These molecules differ by a factor of 200 in their SERS intensities in the same experimental conditions despite their nearly identical polarizabilities. 8 Possible mechanisms for chemical enhancement include a 72 resonance Raman effect due to a new metal-molecule charge transfer electronic transition or a dynamic charge transfer between the metal and molecule. The roughness of the metallic surface plays an important role by facilitating the transfer of photon-excited electrons to the molecule from the metal. 6 The magnitude of these chemical enhancements has been estimated to reach not more than factors of 10 to 100, and is therefore considered a small effect relative to the fourth-power dependence on the electromagnetic field. 6 Kneipp et al. obtained SERS enhancement factors of 10 14 using colloidal silver nanoparticles by correlating the SERS intensity of a single molecule to an equivalent intensity derived from 10 14 non-enhanced molecules. 2 In another experiment, a 10 μm-diameter bundle of nanotubes, in contact with a silver nanoparticle cluster, exhibited an enhanced anti-Stokes G band due to vibrational pumping from the extremely high electric fields of the nanoparticles. 9 Measurements of DNA with controlled interparticle spacing of bound gold nanoparticles have yielded 10 3 to 10 6 -fold enhancements in the Raman intensity. 10 Three-dimensional nanocrescents, nanoshells, and nanorods have also demonstrated Raman enhancement capabilities. 11 12 13 Attempts to explain this phenomenon with classical electrodynamics indicate that particle proximity, surface roughness, and nanoparticle size all contribute to the SERS effect. 11 14 15 Atomic force microscopy (AFM) and scanning electron microscopy (SEM) studies of bulk samples, where roughness and nanoparticle size are varied, have been performed to further understand the SERS effect. 16 Though offering interesting results, these experiments 73 reveal general characteristics of large ensembles but offer little information about the individual nanoparticles and molecules involved in the SERS process. Surface Enhanced Raman Spectroscopy of Individual Carbon Nanotubes We have conducted a systematic study of SERS on individual carbon nanotubes. A focused laser spot is scanned over the sample to identify SERS “hot spots”. The geometry of both the metal nanoparticles and carbon nanotubes at a given “hot spot” are imaged using SEM. Raman spectra taken before and after depositing nanoparticles are used to determine the enhancement factor of the SERS “hot spot” region. In this scheme, the carbon nanotubes serve as Raman active molecules to study the SERS phenomenon. Unlike most single molecules, the uniquely large aspect ratio and conducting properties of carbon nanotubes allows them to be imaged by SEM. Burnout of these robust molecules allows quantification of the heat generated by the SERS plasmonic phenomenon. Chemical vapor deposition (CVD) is used to grow nanotubes from iron (III) nitrate catalyst nanoparticles in methane gas at 900ºC for 10 minutes on a Si/SiO 2 substrate. 17 A numbered grid is patterned on the substrate using electron-beam lithography. This grid allows the laser spot to be aligned optically with a single isolated carbon nanotube. A 6nm thick film of Ag is deposited on the entire sample area in an electron-beam metal evaporator at a pressure of 5x10 -7 Torr. Since a 6nm deposition of Ag is not enough to form a complete continuous film, nanometer-sized islands are formed which serve as the SERS enhancing nanoparticles. Typical nanoparticle sheet densities are ~2500/ μm 2 for a 6nm thick Ag film. 18 Though the 74 plasmon resonance of silver lies in the ultraviolet wavelength range, it is shifted into the visible spectrum due to the nanoparticle geometry. 19 Raman spectra are taken with a Renishaw inVia Reflex micro-Raman spectrometer using a 100X objective lens with a 0.5 μm spot size. Employing a high precision automated microscope stage (Prior, Inc.), local spectra were taken over several hundred square micrometers on the numbered fiducial grid. Spectra were collected at the locations of approximately 100 nanotubes before and after depositing silver nanoparticles. Only those nanotubes exhibiting a finite Raman signal before and after are reported here. Imaging the SERS Nanoparticle Geometry After silver nanoparticle deposition, spatial mappings of Raman spectra and SEM images were taken of the nanotubes deposited on the fiducial grid. The long gold strips that can be seen in these SEM images serve to electrically ground the nanotubes and give them better contrast in the SEM images. We were able to obtain high resolution images of the nanoparticles that make up the SERS hot spots, as seen in Figure 5.2. We magnified three regions that exhibited SERS hot spots, labeled (a), (b), and (c). 75 Figure 5.2: SEM image of a numbered fiducial grid, together with gold strips, carbon nanotubes and deposited silver nanoparticles (upper left). Magnified images of regions exhibiting SERS enhancement are also shown. The white circles represent the size and location of the laser spot used to measure SERS enhanced Raman spectra. The high magnification images shown in Figure 5.3, Figure 5.4, and Figure 5.5 show the SERS hot spots near the carbon nanotubes, shown in Figure 5.2(a), (b), and (c), respectively. In these regions, silver nanoparticles coat the surface with a density of ~2500 nanoparticles/ μm 2 . This corresponds to 490 nanoparticles in the 0.5 μm diameter laser spot. Only nanoparticles directly connected to the long lithographically patterned gold strips appear bright. 20 Though many nanotubes and nanoparticles can be observed in these images, the SEM resolution is not high 76 enough to resolve neither the precise shape and configuration of the nanoparticles nor the ~2 nm gaps between nanoparticles. Figure 5.3: Magnified image of the region indicated in Figure 5.2(a). Note the silver nanoparticles covering the nanotubes near the center of the circle, representing the laser spot size. 77 Figure 5.4: Magnified image of the region indicated in Figure 5.2(b). Note the silver nanoparticles covering the nanotubes near the center of the circle, representing the laser spot size. 78 Figure 5.5: Magnified image of the region indicated in Figure 5.2(c). Note the silver nanoparticles covering the nanotubes near the center of the circle, representing the laser spot size. The nanotubes and gold strip appear bright in the SEM images due to the electron beam induced charge contrast. 20 As a consequence of this contrast, only the Ag nanoparticles touching carbon nanotubes appear bright, even though the entire sample is coated uniformly with nanoparticles. Similarly, only those nanotubes that are in contact with the lithographically patterned gold strip appear bright. One of our initial aims in conducting this experiment was to image the exact geometry of the SERS hot spot. The size and shape of these hot spots depends on the nanoparticles’ shape, relative position, and orientation with fractal small-particle composites generating the most intense hot spots. 8 However, we discovered that there were almost 490 nanoparticles in the Raman focal volume and that the 79 scanning electron microscope’s resolution was too low to resolve the 2 nm gaps required. Measuring the SERS Enhancement Factor Figure 5.6 shows an SEM image of CVD grown nanotubes coated with Ag nanoparticles and lithographically defined features that allow correlation between microscopy and spectroscopy. The two Raman spectra in Figure 5.6(b) were collected from the “laser spot” region indicated in the SEM image, before and after depositing Ag nanoparticles. Both spectra were collected at the same location with a 120 second integration time using a 633 nm laser at 2 mW. G band Raman spectra were fit using a mixed Lorenzian/Gaussian lineshape after subtracting a linear baseline. Before depositing the Ag nanoparticles, the G band Raman mode, appearing at 1594.6 cm -1 , shows an intensity of 58 photon counts. After depositing the Ag nanoparticles, the G band intensity reaches 19,450 photon counts. This factor of 335 enhancement is due to the surface plasmon resonance of the Ag nanoparticles, which couples the incident light into the nanotube very effectively. Several major silicon peaks from the underlying substrate can be seen at 303, 520, and 1000 cm -1 and have not undergone an increase in intensity. Also, the spectrum taken with Ag nanoparticles exhibits a broad background due to luminescence from the Ag nanoparticles. 80 Figure 5.6: (a) SEM image of nanotubes covered with a film of Ag nanoparticles. The white circle indicates the size and location of the laser spot. (b) Raman spectra of the nanotube from (a) before and after Ag deposition. The SERS spectra in Figure 5.7 and Figure 5.8 also show a broad background. The nanotube in Figure 5.7 has a G band frequency of 1597.4 cm -1 and an intensity of about 200 counts before silver nanoparticle deposition. After deposition, the G band frequency upshifts to 1603.1 cm -1 with an intensity of about 20,000 counts with a SERS enhancement factor of 78,000. The nanotube in Figure 5.8 has a G band frequency of 1590.0 cm -1 and an intensity of about 1000 counts before silver nanoparticle deposition. After deposition, the G band frequency upshifts to 1590.1 cm -1 with an intensity of about 15,000 counts with a SERS enhancement factor of 15,200. 81 Figure 5.7: Raman spectra taken before (top) and after (bottom) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 2 in Table 5.1. 82 Figure 5.8: Raman spectra taken before (top) and after (bottom) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 3 in Table 5.1. The SERS enhancement factors of five nanotubes were measured in this fashion and are listed in Table 5.1. Three of the five nanotubes measured in this study show a significant upshift in the G band frequency after the deposition of Ag nanoparticles. We also observed upshifts in the radial breathing mode (RBM), D band, and G’ band Raman peaks. In most cases, the RBM, D band, and G’ band only appear in the SERS spectra due to the weak Raman signals of the pristine nanotubes. Possible reasons for these upshifts are charge transfer from the metal nanoparticles to the nanotubes and surface pressure exerted on the nanotubes by the Ag nanoparticles. 21 22 Nanotubes’ high surface-to-volume ratios make their Raman spectra extremely sensitive to slight modifications of their surface. The Raman 83 spectra of nanotubes have been shown to upshift as free charge is increased. 23 24 Mechanical pressure is also known to increase the vibrational frequency of carbon nanotubes. 21 25 26 According to the work of Venkateswaran, et al., 21 a 3 cm -1 upshift of the G band corresponds to a pressure of 0.5 GPa. 21 While these extremely high pressures may seem unlikely, surface forces on the nanometer scale have been shown to exert pressures of this magnitude. 27 Nanotube G band frequency upshift (Δω G ) Raman intensity ratio (I SERS /I 0 ) SERS enhancement factor 1 0.2 cm -1 335 134,000 2 5.8 cm -1 195 78,000 3 0.2 cm -1 38 15,200 4 5.1 cm -1 8.6 3,440 5 4.3 cm -1 27 10,800 Table 5.1: SERS related effects of five carbon nanotubes. The Raman data for the first entry is shown in Figure 5.6(b) and the rest are shown in Appendix 5. In our measurements, we observe enhancements in the Raman intensity ratio (I SERS /I 0 ) up to 335 measured from nanotubes with a 0.5 μm-diameter laser spot. Although the number of Ag nanoparticles in the 0.5 μm laser spot is approximately 490, it is likely that the SERS enhanced signal originates from the location of only one nanoparticle or between two nanoparticles and hence a 25 nm or shorter segment of the nanotube. We infer this is true because of the sheer rarity of finding a SERS “hot spot” and the large strength of the Raman signal when they were found. In this case, the SERS-active region is at most 25 nm in size, which gives an effective area that is 400 times smaller than the 500 nm diameter laser spot. Thus, the true 84 enhancement factor is 400 times larger than the intensity ratio from the measured spectra, giving our intensity ratio of 335 a total SERS enhancement factor of 134,000, as indicated in Table 5.1. The reason for taking the ratio of the areas [(25 nm) 2 /(500 nm) 2 ] instead of the lengths (25 nm/500 nm) is that these length scales are much smaller than the wavelength of light. Therefore a photon passing 20 nm away from a nanotube can still be absorbed (or scattered). It is, therefore, important to include the area of the SERS enhanced volume, rather than just the length, in determining the SERS enhancement factor. Plasmonic Heating The plasmonic charge oscillating on the surface of the Ag nanoparticles causes Joule-like heating in the nanoparticles, producing elevated temperatures. Figure 5.9 shows SEM images of a nanotube before and after being burned out by these high temperatures. Since it is known that nanotubes burn out in air above 600°C, 28 29 30 the results shown in Figure 5.9 demonstrate that Ag nanoparticles can reach temperatures in excess of 600°C when irradiated with laser powers of 20mW. We observed this SERS-induced burnout in a total of three nanotubes. It should also be noted that all of the nanotubes shown in these SEM images were irradiated with 20 mW of laser light. However, only the SERS-active region exhibited burnout, further demonstrating the spatial rarity of such an event. The three nanotubes exhibiting burnout were from a different set of nanotubes than those shown in Figure 5.6 and Table 5.1, which were not exposed to laser powers above 2 mW. Furthermore, Raman spectra taken from these burnt out nanotubes were not taken 85 before adding the Ag nanoparticles. Therefore, no correlation between burnout threshold and SERS enhancement factor could be made. In the future, the configuration of nanoparticles on nanotubes can be found using similar experiments. With the use of TEM, the nanoparticles adjacent to nanotubes can be imaged with a magnification high enough to observe their shape, size, and spacing. Additionally, using AFM pushing software or other techniques, nanoparticles can be added one- by-one to more clearly understand their effect on the Raman spectra of carbon nanotubes. Figure 5.9: Images of a nanotube before and after Ag nanoparticle deposition and subsequent laser irradiation of a SERS “hot spot.” Plasmonic heating of gold nanoparticles has been demonstrated to heat biological tissues to between 40 and 50°C for noninvasive cancer treatment therapy. 31 Temperatures exceeding 215°C have been reached for gold nanoparticles illuminated with low power laser (1.2 × 10 10 W/m 2 ) in order to perform plasmon assisted chemical vapor deposition of PbO nanowires. 32 Higher temperatures are expected for the plasmon resonance of a nanoparticle dimer due to the strong plasmon resonance. 86 Theoretical calculations of the SERS phenomenon have been carried out by Oubre, et al., who performed finite-difference time-domain (FDTD) simulations of two nearly touching metallic nanoshells irradiated with laser light. 33 Though our experiments use solid non-spherical nanoparticles, they are resonant with 633nm wavelength laser light and have been found to be qualitatively similar to metallic nanoshells in similar experiments. 34 These FDTD simulations show that nanoshells spaced 1.5 nm apart will have a SERS enhancement three orders of magnitude higher than that of individual nanoshells. 33 The maximum enhancement occurs in a small region in between the nanoshells and decays quickly away from this region. 33 For the randomly distributed nanoparticles in our study, the probability is extremely low that two very closely spaced nanoparticles will lay on either side a carbon nanotube. Our measurements show that the occurrence rate of SERS hot spots is very low, which reflects the small effective volume of SERS enhancement. In the future, more effective techniques for generating SERS that make better use of the nanoparticle geometry will improve the probability of this rare event and enable related applications. Conclusion The SERS enhancement of individual carbon nanotubes is measured through the use of correlated micro-Raman spectroscopy and scanning electron microscopy (SEM). A fiducial grid enables the ability to obtain Raman spectra and SEM images of carbon nanotubes before and after the deposition of silver nanoparticles. The Raman intensity of carbon nanotubes was observed to increase 134,000-fold through 87 the SERS phenomenon. The localized burnout of nanotubes in SERS hot spot regions is attributed to the heat dissipated by the immense plasmonic charge oscillating in the Ag nanoparticles. 88 Chapter 5 Endnotes 1 K. Kneipp, G. Hinzmann and D. Fassler, J. 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Doll, and L. P. Lee, Adv. Mat. 17, 2683 (2005). 12 Y. Yang, L. Xiong, J. Shi, and M. Nogami, Nanotechnology 17, 2670 (2006). 13 M. Suzuki, Y. Nidome, N. Terasaki, K. Inoue, Y. Kuwahara, and S. Yamada, Jpn. J. Appl. Phys. 43, L554 (2004). 14 H. Xu, J. Aizpurua, M. Käll, and P. Apell, Phys. Rev. E 62, 4318 (2000). 15 F. J. García-Vidal and J. B. Pendry, Phys. Rev. Lett. 77, 1163 (1996). 16 R. M. Stöckle, V. Deckert, C. Fokas, and R. Zenobi, Appl. Spectroscopy 54, 1577 (2000). 17 J. Kong, H. T. Soh, A. M. Cassell, C. F. Quate, H. J. Dai, Nature 395, 878 (1998). 89 18 S. E. Roark, D. J. Semin, and K. L. Rowen, Anal. Chem. 68, 473 (1996). 19 J. Okumu, C. Dahmen, A. N. Sprafke, M. Luysberg, G. von Plessen, M. Wuttig, J. Appl. Phys. 97, 094305 (2005). 20 T. Brintlinger, Y.-F. Chen, T. Dürkop, E. Cobas, M. S. Fuhrer, J. D. Barry, and J. Melngailis, Appl. Phys. Lett. 81, 2454 (2002). 21 U. D. Venkateswaran, E. A. Brandsen, U. Schlecht, A. M. Rao, E. Richter, I. Loa, K. Syassen, and P. C. Eklund, Phys. Stat. Sol. 223, 225 (2001). 22 S. Lefrant, J. P. Buisson, J. Schreiber, O. Chauvet, M. Baibarac, and I. Baltog, Synthetic Metals 139, 783 (2003). 23 S. B. Cronin, R. Barnett, M. Tinkham, S. G. Chou, O. Rabin, M. S. Dresselhaus, A. K. Swan, M. S. Ünlü, and B. B. Goldberg, App. Phys. Lett. 84, 2052 (2004). 24 P. Corio, P. S. Santos, V. W. Brar, Ge. G. Samsonidze, S. G. Chou, and M. S. Dresselhaus, Chem. Phys. Lett. 370, 675 (2003). 25 S. Reich, C. Thomsen, J. Maultzsch, Carbon Nanotubes: Basic Concepts and Physical Properties, Wiley-VCH, Weinheim, Germany, Vol. 1, 107-111 (2004). 26 R. Kumar and S. B. Cronin, J. Nanosci. Nanotechnol. 8, 122 (2008). 27 S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 94, 167401 (2004). 28 F. Cataldo, Fullerenes, Nanotubes, and Carbon Nanostructures 10, 293 (2002). 29 I. W. Chiang, B. E. Brinson, A. Y. Huang, P. A. Willis, M. J. Bronikowski, J. L. Margrave, R. E. Smalley, and R. H. Hauge, J. Phys. Chem. B 105, 8297 (2001). 30 Kenji Hata, Don N. Futaba, Kohei Mizuno, Tatsunori Namai, Motoo Yumura, Sumio Iijima, Science 306, 1362 (2004). 31 D. Pissuwan, S. M. Valenzuela, and M. B. Cortie, Trends in Biotechnology 24, 62 (2006). 32 D. A. Boyd, L. Greengard, M. Brongersma, M. Y. El-Naggar, and D. G. Goodwin, Nano Lett. 6, 2592 (2006). 33 C. Oubre and P. Nordlander, J. Phys. Chem. B 109, 10042 (2005). 90 34 P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, Nano Lett. 4, 899 (2004). 91 Chapter 6 Carbon Nanotubes Strained by a Top-Down Microprocessing Method The method of applying strain to carbon nanotubes pinned to PDMS substrates, as described in Chapter 3 of this thesis, is appropriate for scientific investigation. However, for practical applications, a top-down method using standard microprocessing steps is desired. The ultimate goal is to create strain- induced band gaps in carbon nanotube devices, thereby converting the metallic nanotubes to the more desirable semiconducting state. This chapter investigates the feasibility of this idea. Because of their high growth temperatures, it is currently infeasible to grow nanotubes on PDMS. Therefore, methods such as solution deposition and stamping are required to transfer them to PDMS. The former causes nanotubes to form bundles, making it difficult to measure individual nanotubes repeatedly. As found in Chapter 3, carbon nanotube bundles have a low rate of strain transfer, making it difficult to induce metal-to-semiconductor transitions. Furthermore, nanotubes slip on PDMS when strained, severely limiting the amount of strain that can be applied to nanotubes. Methods for achieving higher strains are desired for studying defect formation and nanotube breakdown. In this chapter, we investigate alternative techniques for straining nanotubes, which involve partially suspending nanotubes across a gap that is progressively made deeper through chemical etching. This method allows individual nanotubes under axial strain to be measured by Raman spectroscopy without slipping on the surface. It also enables greater strains to be achieved by straining individual 92 nanotubes (grown by CVD) rather than in nanotube bundles (grown by laser ablation). Experimental Setup The etch setup consists of several steps as illustrated in Figure 6.1. Aligned nanotubes are grown on a silicon wafer with a 500 nm oxide using a method developed by Prof. Chongwu Zhou’s research group at USC. 1 Catalyst nanoparticles were deposited by nanosphere lithography where closely packed polystyrene beads are used as shadow-masks for metal deposition. Aligned nanotubes are then grown from these catalyst islands with CH 4 and C 2 H 4 gas feedstock. 1 An SEM image of nanotubes grown using this method are shown in Figure 6.2. The location of these nanotubes were noted during SEM imaging, enabling them to be subsequently found optically with the optical microscope integrated in our Raman spectrometer. Using a high power 532nm laser, we burn holes in the underlying Si/SiO 2 substrate in the form of a cross to designate the location of the nanotubes. Next, a 3 μm diameter tungsten wire, supplied by the California Fine Wire Company (Grover Beach, CA), is carefully placed perpendicularly across the aligned nanotubes to serve as a shadow-mask. The sample is then covered with a circular TEM grid consisting of 50 μm wide stripes separated by 50 μm as shown in Figure 6.3(a). Unlike standard TEM grids that consist of a grid of intersecting horizontal and vertical bars, this TEM grid consists only of horizontal bars. This TEM grid serves as a shadowmask on the sample, which is then coated with 5 nm of chromium and 25 nm of gold in a metal 93 evaporator, creating a pattern of stripes with 3 μm gaps, such as the one shown in Figure 6.3(b). Raman spectroscopy is performed to identify resonant nanotubes in this gap. Atomic force microscopy is also performed to determine the depth of the gap. The gap is then etched by dipping it in a bath of hydrofluoric acid (Buffer-HF Improved, Transene Company, Inc., Danvers, MA) for a set amount of time and then subsequently dipping it in baths of water and blow-drying it with an airgun. The HF has been empirically found to etch the SiO 2 layer of the substrate at a rate of 2.5 nm/s. Figure 6.1: Processing steps for HF experiment. (1) First, a piece of silicon with a 500nm oxide is obtained. (2) Aligned nanotubes are transferred to the surface. (3) Using shadowmasking and metal evaporation, 3 μm gaps are created between gold contacts. (4) Finally, the oxide is etched using HF. 94 Figure 6.2: SEM image of aligned nanotubes grown using the method of Ryu, et al. 1 Figure 6.3: (a) TEM grid used as a shadow-mask for top-down microprocessing shown at 40x optical magnification. (b) SEM image of aligned nanotubes between two gold contacts. The horizontal gap was created by placing a 3 μm tungsten wire over the sample during metal evaporation. 95 As the acid etches the silicon oxide, the nanotube becomes detached and adheres to the substrate in almost the same configuration. AFM images from a previous experiment of nanotubes in a progressively etched gap can be seen in Figure 6.4. After subsequent etches, the nanotubes are observed to maintain the same position and accumulate residue. The nanotubes were located optically in the same region and were resonant after every etch. The strain is generated in the nanotubes after the sample is dried and the nanotube is sucked to the bottom of the trench by the large surface tension as shown in Figure 6.6. Since the ends of the nanotube are pinned underneath the metal contacts, the adhesion of the nanotubes to the bottom of the gap results in an elongation of the nanotube from ℓ to ℓ + Δ ℓ. If the nanotube assumes a configuration as shown in the bottom right image of Figure 6.1, then the elongation is given by, () () θ θ tan 2 sin 2 h h − = Δ l (6-1) where h is the etched depth and θ is the contact angle. For an ℓ of 3 μm, an h of 0.5 μm and a contact angle of 45°, l l Δ is about 80%. 96 Figure 6.4: AFM images (left column and top), optical images (middle column), and Raman spectra (right column) from an etched gap containing a resonant carbon nanotube from a prior experiment. The top row of images show the AFM measurements of the gap before etching and after the first etch. Note how the nanotubes are barely visible. The second row of images corresponds to a 34 nm deep gap, the third row corresponds to a 61 nm deep gap, and the bottom corresponds to a 112 nm deep gap. The black oval on the AFM images corresponds to the resonant nanotube. This nanotube’s location is also indicated by the green laser dot in the middle column of optical images for each etch. The G band Raman spectra are taken for each etch as shown in the right column. This data shows that the nanotubes remain unperturbed from their positions as the underlying substrate is etched and dried. Furthermore, the nanotubes appear thicker due to the accumulation of residue. 97 As trenches are progressively etched, the amount of residue on nanotubes increases. During the etching process, the nanotubes are submerged in a solution of hydrofluoric acid which contains particles that adhere to the nanotube surface. Performing section analyses of the nanotubes’ AFM images, we obtain the height of the nanotubes depicted in Figure 6.4 after each progressive etch, as shown in Figure 6.5. By calculating the difference between the maximum and minimum nanotube heights, we obtain error bars for each data point. The height of the nanotubes increases rapidly for the first few etches and then saturates thereafter. Furthermore, the magnitude of the error bars increases for the first few etches and then diminishes greatly after the last etch. This data indicates that the nanotubes are accumulating residue. The residue at first accumulates along the nanotube length somewhat heterogeneously and then finally accumulates homogeneously as indicated by the small error bars on the final data point. 98 Figure 6.5: Plot of nanotube heights versus etch depth. The heights were obtained through section analysis of four nanotubes. The heights were averaged for each depth. Error bars reflect the range of nanotube heights for each etch. After removal of the etchant through submersion in water and air-drying, the nanotubes are bound to the substrate through van der Waals forces and, therefore, strained, just as the nanotube shown in the right image of Figure 6.6. If critical point drying was used in place of our process, the nanotube would become completely suspended with no strain, just as in the left image of Figure 6.6. This is accomplished by taking the solution from subcritical to supercritical fluid, avoiding the gas-liquid interface due to the equivalent gas and liquid densities. Using supercritical fluids avoids the surface tension that occurs between the gas-liquid interface. In our experiment, this surface tension is the cause of the strain in the nanotubes. 99 Figure 6.6: SEM images of suspended nanotubes dried (a) with and (b) without critical point drying. Note how the nanotube dried in the critical point dryer in (a) is suspended yet the nanotube dried without a critical point dryer in (b) adheres to the substrate due to the high surface tension of the water. The relatively large surface forces pin the nanotube under strain. In our experiment, three etches were performed on five nanotubes that were resonant with a 633 nm wavelength laser were measured. The first etch of 20 seconds in duration resulted in a 50 nm etch of the underlying SiO 2 . The second etch of 180 seconds and resulted in an additional 450 nm increase in depth for a total of 500 nm. A third etch, lasting 28 seconds resulted in an additional 82 nm increase in total depth to 582 nm. At this point, severe disruption of all nanotubes was observed to the point that only one nanotube (NT5) continued to span the gap. This disruption can be seen in the SEM images of Figure 6.7. From these images, it is apparent that NT5 has broken free from the contact and is no longer strained. 100 Figure 6.7: SEM images of Gaps 2 (upper left), 3 (upper right) and 5 (lower middle) after being etched 582 nm. Note the severely displace nanotubes in all three images and also how NT5 in the lower middle image (upper nanotube) has broken away from the contact. Strain-Induced Changes in the Raman Spectra Although only one nanotube out of the five nanotubes measured experienced a significant and consistent downshift of the G band, all nanotubes experienced significant changes in their D band intensity. As mentioned in Chapter 3, the 101 intensity of the D band, relative to the G band, gives a measure of the amount of defects in a nanotube. 2 A “defect” can be considered anything that breaks the sp 2 symmetry of the nanotube. This so-called D/G ratio was found to increase significantly in three out of five nanotubes measured in this fashion, as shown in Figure 6.8. The D/G ratio of one nanotube (NT1) increased on the first etch and then decreased on the second etch. Another nanotube (NT2) did not consistently have a D band. NT5 was found to be resonant even after the third etch, where it appears the nanotube became unstrained, possibly due to etching underneath the gold electrode. Figure 6.8: D band intensity divided by the G + band intensity versus trench depth for four out of five nanotubes measured. Note how three nanotubes show consistent upshifts in this ratio for the first three points, indicating a relative increase of the D band intensity with respect to the G + band intensity. Further note how NT5 decreases after the third etch, but not to the initial value. 102 These changes in the D/G ratio indicate changes are occurring in these nanotubes that are dependent on the etching process. Since nanotubes are relatively unaffected by HF etchant and not all nanotubes that experienced an increase in their D/G ratios experienced a downshift in the G band indicative of strain, the cause of this increase is difficult to determine. Possible causes are 1.) secondary products of etching interfering with nanotubes, 2.) minor defects acting as nucleation sites for larger defects, exacerbated by some aspect of the etching process, 3.) formation of surface complexes during etching, or 5.) the presence of chemical species on the nanotube surface. The last possibility is the most likely. The fact that the D/G ratio of NT5 relaxed to a value higher than the pre-strain value after becoming unstrained indicates that irreversible defects have occurred. The D/G ratios are similar in magnitude and independent of the G band downshift for most nanotubes, so it is likely that the reactants are irreversibly damaging the nanotube. Strain-Induced Changes in the G Band Four out of five measured nanotubes experienced upshifts in their G band with an increase in strain. Two nanotubes, NT2 and NT4, experienced two upshifts while NT1 experienced an upshift and then a downshift and NT3 experienced a downshift and then an upshift as shown in Figure 6.9. These upshifts range from 1 cm -1 to 12 cm -1 . After the first strain, the G + band frequencies range from 1589 cm -1 to 1604 cm -1 . However, after the second strain, the G + band frequencies range from only 1594 cm -1 to 1599 cm -1 . 103 Figure 6.9: Graphs of changes in G + band frequency with etch depth. Note how NT1-NT4 experience upshifts in frequency while NT5 experiences radical downshifts, for the first two etches. After the third etch (not shown), only NT5 spans the gap and downshifts to its pre-strain frequency. In our prior strain experiments conducted on PDMS polymers, upshifts of ~30 cm -1 in the G band frequency with strain as well as a characteristic change from metal to semiconductor were observed. 3 The nanotubes in these trench etched experiments do not have the same upshifts nor do they have the same changes in Raman lineshape indicative of a metal-semiconductor transition. In the PDMS strain experiments on semiconducting nanotubes, the nanotubes achieved G band frequencies higher after slipping on the PDMS surface and compressing than their original pre-strain frequency, as shown in Figure 3.3(c). Since a slip-and-compress mechanism can not explain this upshift, the precise nature of this upshift is difficult to determine, especially since the configuration of individual nanotubes is unknown. One possibility could be that the nanotubes reconfigure into bundles as they become detached from the metal contacts and aggregate, as shown in the SEM images of Figure 6.7, causing upshifts in their G band Raman spectra. 104 Strain-Induced Downshifts of the Raman Modes Only one nanotube out of five experienced a large downshift in the G + band frequency. The raw spectral data for this nanotube is shown in Figure 6.10. The G + band frequency, as shown in Figure 6.9, decreases linearly from 1579 cm -1 to 1548 cm -1 , for a total 31 cm -1 downshift. According to Cronin, et al., the G + band downshifts in frequency of 28 cm -1 /% axial strain, imply that 1.1% axial strain was applied to the nanotube. 2 Using equation (6-1) and assuming symmetric contact angles for both sides of the nanotube, we calculate a contact angle for 500 nm of etching to be 13.8°, given that the nanotube here spans an exceptionally wide gap 11 μm wide due to irregularly formed contacts. This implies that a 6.9 μm length of nanotube is in direct contact with the substrate. For comparison, after the first etch of 50 nm, the contact angle was 15.1°, with only a 10.6 μm length of nanotube in direct contact with the substrate. Between the first and second etches, almost 35% of the nanotube lost contact with the substrate. These calculations give a rough idea of the geometry of the nanotube after etching and non-critical drying. 105 Figure 6.10: Raman spectra measured from nanotube sample NT5 before and after successive strains induced by the trench etch process. Note how the G band shape changes from two peaks to one broad peak and then to a peak with two shoulders. Also note the change in intensity and downshift ofthe D band. Additionally, the G’ band greatly downshifts with trench etch depth for NT5 as shown in Figure 6.11. Downshifts in the G’ band are indicative of axial strain being applied to the nanotube. 2 After 50 nm of etching, the G’ band frequency downshifts from 2636 cm -1 to 2625 cm -1 . When the trench depth reaches 500 nm, the G’ band apparently splits into two peaks: a broad peak centered at 2594 cm -1 and a narrow peak centered at 2579 cm -1 . After the final etch where the nanotube relaxes, the peaks cease splitting and upshift to the initial frequency. This two-peak structure in the G’ band has been found in unstrained nanotubes where the S E 44 peak is within the resonant window for the laser energy and the S E 33 peak is within the resonant window for the scattered photon. 4 This phenomenon, however, has only been observed in semiconducting nanotubes. However, since the nanotube undergoes strain, a change in the density of states could bring the E 44 and E 33 peaks into the resonant windows for the laser and for the scattered photon, respectively. 106 Out of the four possible chiralities for this nanotube, as discussed in the next section, the only one which has two closely spaced peaks near the E 33 resonance is the (9,9) nanotube. It is possible that strain displaces these peaks enough to give the G’ peak a two-peak shape. Figure 6.11: G’ band of NT5 after subsequent etches. Note the downshift by almost 57 cm -1 after the second etch as well as the asymmetry of the peak. Here, the G’ band splits into two peaks. After the final etch, the G’ band upshifts to its original frequency. Radial Breathing Mode A radial breathing mode was observed in only one nanotube out of the five that were measured using this scheme. The RBM and anti-Stokes RBM appeared at 185.5 cm -1 and -185.5 cm -1 , respectively, with a Stokes/anti-Stokes intensity ratio of about 5.3 after 500nm of etching, as shown in Figure 6.12. This RBM corresponds to a nanotube diameter of 1.3 nm. 5 According to a Kataura plot calculated using a tight binding model, a nanotube with a diameter of 1.3 nm resonant with a 633 nm laser will be metallic with a chirality of either (16,1), (9,9), (15,3), or (13,4), as shown in Figure 6.13. The appearance of the RBM for only the third etch indicates 107 a strain-induced ΔE ii , as calculated in Chapter 4, shifting onto resonance with strain. Using equation (4-6), we calculate the ΔE ii to be -0.0124 eV using Г r = 8 meV and E l = E ii = 1.95 eV and E ph = 230 meV which corresponds to a 185.5 cm -1 RBM. Additionally, a silicon peak can be seen near 330 cm -1 which has roughly the same intensity in all spectra. Figure 6.12: Radial breathing mode and anti-Stokes radial breathing mode Raman spectra of a nanotube before etching, after 50 nm of etching, after 500 nm of etching, and after 582 nm of etching. The radial breathing mode in the 500 nm case appears at about 185.5 cm -1 . 108 Figure 6.13: Section of a Kataura plot calculated using a tight binding model for a region consisting 1.2-1.4 nm diameter (vertical blue bar) nanotubes resonant with 633 nm laser with a 0.1 eV FWHM (horizontal red bar). The nanotubes falling within this region are all metallic with chiralities of (16,1), (9,9), (15,3), and (13,4). Electrical Measurements Since the nanotubes are in contact with metal electrodes, it is possible to perform electrical measurements of nanotubes spanning the gap for each etch. Since many nanotubes span each gap, the electrical measurements reflect an ensemble measurement of all nanotubes, including non-resonant and weakly-Raman resonant nanotubes. However, for the case of NT5, we can observe a change in the electrical behavior of the nanotubes through I-V measurements as shown in Figure 6.14. After the first etch, the resistance decreased from the pre-etch value of 90 k Ω 109 to 88 k Ω. However, after the second etch, the resistance greatly increased to 4.0 M Ω. This large increase in resistance can be explained by nanotubes becoming detached from the electrodes and decreasing the total area through which current flows between the electrodes. However, the 2 k Ω decrease observed after the first etch may be due to the removal of residues by HF that were previously diminishing the conductivity of the nanotubes. Since the precise locations of nanotubes were unknown between etches, it is not possible to distinguish the effects of strain, nanotube detachment, and etching of underlying oxide from each other on the electrical behavior. Figure 6.14: I-V curve of NT5 after subsequent etches. The calculated resistance decreases by 2 k Ω after the first etch and then increases to 4.0 M Ω after the second etch. No signal was detected after the third etch. 110 Conclusion In this experiment, we observed changes in carbon nanotubes as strain was applied through top-down microprocessing. Though we only observed strain- induced Raman G band downshifts in one nanotube, we were able to calculate the change in E ii . Furthermore, we were able to perform electrical measurements of nanotubes as they were strained. It was found that HF etching can induce defects in nanotubes through the accumulation of residue, even if they are not strained. The metal electrodes were found to collapse under large strain. Additionally, the strain experienced by NT5 was not as high as we expected due to the low contact angles. Nonetheless, this experiment provides a proof-of-principle demonstration for this method of strain as well as an identification of its shortcomings. 111 Chapter 6 Endnotes 1 K. Ryu, A. Badmaev, L. Gomez, F. Ishikawa, B. Lei, and C. Zhou, J. Am. Chem. Soc. 129, 10104 (2007). 2 S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 93, 167401 (2004). 3 R. Kumar and S. Cronin, Phys. Rev. B 75, 155421 (2007). 4 M. S. Dresselhaus, G. Dresselhaus, A. Jorio, A. G. Souza Filho, and R. Saito, Carbon 40, 2043 (2002). 5 S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, and R. B. Weisman, Science 298, 2361 (2002). 112 Chapter 7 Conclusion Summary We have demonstrated that straining carbon nanotubes can cause metal-to- semiconductor transitions, debundling-induced changes in Raman spectra, and changes to the nanotube density of states. These results have been modeled using tight-binding calculations and zone-folding. Furthermore, we demonstrated large nanotube enhancements through the SERS effect as well as the burnout of carbon nanotubes. To determine the effects of strain on nanotube bundles and to create metal-to- semiconductor transitions, we strained nanotube bundles on PDMS, as discussed in Chapter 3. We showed that reversible effects happen to semiconducting nanotubes while metallic nanotubes irreversibly upshift in their G band spectra. In Chapter 4, we theoretically modeled the electronic and vibrational structure of carbon nanotubes. We calculated the phonon dispersion of carbon nanotubes under axial strain. In Chapter 5, we showed that by measuring the same nanotubes repeatedly, we can determine their SERS enhancements precisely. We also demonstrated that the SERS effect can be used to burn out carbon nanotubes. Finally, in Chapter 6, we demonstrated that by straining nanotubes to the bottom of a gap by top-down microprocessing, we can induce large downshifts in their G band spectra and calculate the change in E ii . We also showed the presence of a two-peak G’ band, which indicates a change in the E 33 and E 44 energies. 113 Future Outlook The experiments on nanotube strain and the SERS effect described in this thesis present new possibilities for studying the Raman spectra in carbon nanotubes. The metal-to-semiconductor transitions and debundling phenomenon we observed present insight into their future use as transistors consisting entirely out of carbon. Further developments of our dispersion relations under axial strain will give insight into the electron-phonon coupling within carbon nanotubes. The strain-induced changes to carbon nanotubes’ optical properties can be utilized in interesting and novel applications. One such application is microcracking detection. For example, high pressure steam pipes used in oil refineries and other industrial plants develop microcracks over time, which can lead to catastrophic explosions as the cracks grow in size. 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Length of unit vector gamma0 = 2.9; % eV. Gamma_0, the overlap integral. %%%%%%%%%%%%%%%%% % PROGRAM START % %%%%%%%%%%%%%%%%% disp('n and m must be nonnegative integers with n>=m') n = input('n = '); m = input('m = '); if mod((n-m),3) == 0, % condition for metallic nanotubes disp('Nanotube is metallic.') else disp('Nanotube is semiconducting.') end % Diameter Calculation % -------------------- % The diameter can be calculated by taking the absolute value of the chiral vector % (the vector that stretches from one carbon atom back to itself on a graphene sheet % corresponding to an unrolled nanotube) and dividing by pi. The chiral vector (n*a_1 + m*a_2) % essentially stretches around circumference of the nanotube. % % The following formula is from M.S. Dresselhaus, et al. "Raman spectroscopy on % isolated single wall carbon nanotubes." Carbon 40, p. 2043-2061 (2002) diameter = (a.*sqrt(n.^2 + m.^2 + m.*n))./pi; disp(['Diameter = ', num2str(diameter),' nm']) 124 % Chiral Angle Calculation % ------------------------ % The chiral angle is the angle between the a_1 vector and the chiral vector % % The following formula is from M.S. Dresselhaus, et al. "Raman spectroscopy on % isolated single wall carbon nanotubes." Carbon 40, p. 2043-2061 (2002) thetar = atan((m.*sqrt(3))./(2*n+m)); %in radians theta = thetar*180/pi; % in degrees disp(['Chiral angle = ', num2str(theta),' degrees (pi/', num2str(pi/thetar),' radians)']) % Dispersion Relation % ------------------- % The relation between energy and wavenumber % % Equations taken from % Saito, et al. "Physical Properties of Carbon Nanotubes," Imperial College Press, London, 2003. Rtest = (n-m)./(3*gcd(n,m)); if round(Rtest) == Rtest, %i.e., Rtest is an integer R=3; else R=1; end G = gcd(n,m); Ch = sqrt(n^2+m^2); % Length of chiral vector q = 2*(n^2+n*m+m^2)/(R*G); %number of graphene hexagons in the unit cell T = sqrt(3)*pi*diameter/gcd(2*n+m,2*m+n);% length of translational vector dR = gcd((2*n+m),(2*m+n)); if n == m, % Armchair k = -pi/T:pi/(256*T):0; % k is the wavenumber in the axial direction of the % nanotube (a.k.a. k_z) elseif m == 0, % Zigzag k = -pi/(3*a):pi/(256*a):0; else % Chiral k = -a*pi/T^2:a*pi/(256*T^2):a*pi/T^2; end kc = (-q/2 + 1):1:q/2; % allowed wavenumbers in radial direction K1 = kc*(2*pi/(sqrt(n^2+n*m+m^2)*a))^2; % reciprocal lattice vector corresponding to chiral vector. % Coefficient of kc comes from solving for K1 in armchair configuration. K2 = 2*pi*k/a; % reciprocal lattice vector corresponding to translational vector. 125 % Coefficient of k comes from solving for K2 in armchair configuration. fig1 = figure; hold on; for J = 1:length(kc), % Uncomment the following 3 lines to show how close to completing calculation. if (mod(J,10) == 0 | J == length(kc)), disp(['(iteration ',num2str(J),'/',num2str(length(kc)),')']) % displays how close to completing calculation. end for I = 1:length(k), kx = (a*sqrt(3)/(4*pi))*( (m+n)*K1(J) + (m-n)*K2(I)/dR ); %kx in terms of K1 and K2. ky = (a/(4*pi))*( -(m-n)*K1(J) + 3*(m+n)*K2(I)/dR ); % ky in terms of K1 and K2 % K1 and K2 were in terms of b1 and b2, which were converted to in terms of kx and ky and solved for kx and ky. E1(J,I) = sqrt(1+4*cos(kx*a*sqrt(3)/2)*cos(ky*a/2)+4*(cos(ky*a/2)).^2); % Dispersion for graphene E2(J,I) = - sqrt(1+4*cos(kx*a*sqrt(3)/2)*cos(ky*a/2)+4*(cos(ky*a/2)).^2); % Dispersion for graphene end plot(k,E1(J,:)) plot(k,E2(J,:)) end hold off; title(['Dispersion relation for (',num2str(n),',',num2str(m),') nanotube']); if (n == m | m == 0), % Armchair and Zigzag xlabel('{\itk}'); set(gca,'XTick',[min(k),max(k)]); set(gca,'XTickLabel',{'X','Gamma'}); %Change numerical labels on axis axis tight; axis square; set(gcf,'position',[281 338 435 306]) else % Chiral xlabel('{\itk}T/\pi'); axis tight; axis([min(k) max(k) -1.5 1.5]); set(gcf,'position',[406 142 232 527]) end ylabel('E({\itk})/\gamma_0'); set(gcf,'name','Dispersion Relation') % Density of States % ----------------- % % From Reich, et al., "Carbon Nanotubes," WILEY-VCH Verlag, Darmstadt, Germany pp. 44-47 (2004). 126 mg = -500:500; %-200:200 is good. Em = abs(3*mg-n-m)*a*gamma0/(diameter*sqrt(3)); %in eV E = gamma0*linspace(-4,4,length(mg)); %in eV for I = 1:length(mg), for J = 1:length(mg), if (abs(E(I)) > abs(Em(J))), g(I,J) = abs(E(I))./sqrt(E(I).^2-Em(J).^2); else g(I,J) = 0; end end n(I) = 4*a*sum(g(I,:))/(pi^2*diameter*gamma0); end figure; plot(E,n,'g') title(['Density of states']); xlabel('E (eV)'); ylabel('n(E) (states/eV)'); set(gcf,'position',[5 40 435 306]) set(gcf,'name','Density of States') % Eii calculation % --------------- % Energy separation of peaks in DOS calculation. for I = 1:length(E), % Find the index of where the energy is zero for the DOS calculation if E(I) == 0, nIndex = I; end end figure; hold on; J = 1; for I = (nIndex+1):length(n), if ((n(I)>n(I-1)) & (2*E(I)>1)), % find when a value is larger than previous value -> peak % (van Hove singularity) & ignore gap around 0 Eii(J) = 2*E(I); % Eii is twice this value since it goes from peak to peak text(J+0.2,Eii(J),[num2str(Eii(J)),'eV']); J = J+1; end end plot(1:length(Eii),Eii,'r+') title('E_{ii}'); xlabel('i') ylabel('E_{ii} (eV)') set(gcf,'position',[586 40 435 306]) set(gcf,'name','Eii') hold off; figure(fig1) %make figure1 come out front 127 Appendix 2 Review of Related Literature The discovery of carbon nanotubes in 1991 has given birth to a pan-global interest in their electronic, mechanical, and optical properties. Due to the nature of their bond strength and geometric arrangement, they present an interesting system to study both from an applied and a fundamental point of view. Specifically, applying axial strain to a carbon nanotube not only affects its physical structure, but also its electronic and vibrational structure, and hence its optical properties. We present here a survey of techniques and experiments that explore the optical properties of carbon nanotubes under axial strain. Previous measurements of nanotubes under strain The optical properties of carbon nanotubes are very sensitive to strain, and provide a multitude of information about their physical and electronic structure. The vast majority of optical measurements of carbon nanotubes under strain utilize resonance Raman spectroscopy, which provides a precise measure of the vibrational and electronic energies of carbon nanotubes. 1 From these spectra, information regarding the electronic resonance energies, nanotube diameter and chirality, abundance of defects, 1 temperature, 2 3 4 5 6 and relative amount of strain 17 7 8 9 10 11 12 13 14 15 16 17 18 can be ascertained with relatively low intensity laser light. One of the Raman features that is most sensitive to strain is the G band, which corresponds to the optical phonon mode in graphite and is observed at ~1590 cm -1 , as shown in Figure A2.1. This phonon mode downshifts due to a softening/weakening of the C-C bonds. 17 7 8 9 10 11 12 13 14 15 16 17 Conversely, the G band shifts upward when nanotubes 128 are compressed. 18 The G’ band (~2700 cm -1 ) is also observed to downshift with strain. While this Raman peak typically has a broader linewidth than the G band, its high frequency allows small changes in the relative percent of the vibrational energy to be easily observed, making this mode very sensitive to strain. The D band Raman mode (~1350 cm -1 ) provides additional information about nanotubes under strain. The D band is only observed when the sp 2 symmetry of the nanotube is broken. Therefore, strain-induced distortion of the graphitic hexagons, as well as strain- induced defects, will break the sp 2 symmetry of the nanotube and lead to a rise in the D band Raman intensity. Also, while under strain, nanotube ropes can de-bundle, leading to shifts in their electronic energies. 19 20 The radial breathing mode (RBM), which corresponds to the radial motion of atoms, has not been observed to shift under strain, however, the intensity of this peak is sensitive to strain. 8 11 The change in RBM intensity reflects a shift in the electronic transition energies through the resonant Raman process. Photoluminescence (PL) measurements of nanotubes under strain have also been performed, providing very precise measurements of the electronic transition energies and bandgaps of semiconducting nanotubes. 21 129 Figure A2.1: Typical Raman spectrum from a semiconducting carbon nanotube. Measurements of Nanotube Composite Materials Nanotube-epoxy and nanotube-polymer composites have been studied extensively under strain, where nanotubes have been added to increase the strength of the composite material. In these composites, nanotubes are usually mixed with an epoxy resin or other polymer material by sonication for 6-24 hours and then cured, leaving the nanotubes suspended within the elastic polymer. 10 11 13 16 17 18 22 23 In these composite measurements, strains ranging from -1% to 4% are achieved by clamping the ends of the composite and then bending, straining, or compressing the middle while Raman spectra are taken. Figure A2.2 shows a schematic diagram of one approach used to induce strain in a nanotube composite material. In other experiments, nanotubes were deposited on the surface of an elastic polymer and strained as the polymer was stretched. In these composite measurements, thousands of nanotubes are measured simultaneously, resulting in a Raman signal that is an ensemble average of many different nanotube chiralities, all with very different 130 electronic and vibrational properties. Also, these ensembles consist of nanotubes positioned at various angles with respect to the direction of applied strain, and therefore experience different amounts of strain. In some of these experiments, the polymer-nanotube mixture was extruded to align the nanotubes in a highly oriented fashion. 16 Figure A2.2 shows the strain induced downshift of the G’ band Raman mode of SWNTs dispersed in an epoxy resin. A downshift of 15 cm -1 /% tensile strain using the technique is depicted in the figure. Figure A2.2: Left: Schematic diagram of a four-point bending method for inducing tensile strain in a composite material. Right: Strain-induced downshift of the G’ band Raman mode of nanotubes dispersed in epoxy resin. 22 Reprinted with permission from [22], C. A. Cooper, R. J. Young, M. Halsall, Composites: Part A 32, 401 (2001). © 2001, Elsevier. A wide range of values have been reported for the rate of G band downshift with strain, ε ω ∂ ∂ / , as measured by various research groups. Values ranging 131 from -0.17 cm -1 /% strain up to -36 cm -1 /% strain have been observed for nanotube composites. 17 18 Table A2.1 summarizes the strain-induced changes in the Raman spectra reported in the literature. For nanotube bundles in composite materials, the bundles couple directly to the surrounding matrix, transferring most of the tensile load axially through the nanotubes on the bundle circumference, resulting in shifts of the G band by ~2.5 cm -1 /%. 18 The nanotubes on the interior of the bundle do not couple strongly to the surrounding matrix and therefore experience relatively little strain or compression. The G band frequency of the Raman spectra of nanotubes in composites can downshift in a two stage process, first remaining unchanged for up to ~2% strain, indicating sticking and slipping of nanotubes, 12 and then downshifting by 0.75-40 cm -1 /% strain, indicating axial stretching of carbon nanotubes. 17 9 12 The Raman spectra of nanotubes under strain and compression exhibit similar mechanical responses to the surrounding composite, implying that the Young’s moduli in tension and compression are similar, at least for moderate strains. 14 MWNTs in composites are found to carry more strain in compression than in tension, with a compression Young’s modulus 20% higher than that for tension. These MWNT composites demonstrate a difference in the magnitude of the G’ band frequency shift for compression (7 cm -1 /% compression) and tension (1 cm -1 /% strain). 23 The relatively small shifts with tension are attributed to nanotubes oriented at an angle with respect to the strain direction that are more easily loaded in compression. 23 Furthermore, MWNTs are found to carry tensile loads in their outermost nanotubes, while all nanotubes in the MWNT carry the compressive load. 23 Flow-aligned nanotubes in polymer composites experience a G’ band frequency downshift of 2-3 cm -1 /% 132 strain. 15 Nanotubes strained within polymers exhibit strain saturation where, beyond a certain point, additional strain does not shift the Raman bands. 13 This has been reported to occur beyond ~1% strain for SWNTs. 13 16 Table A2.1 summarizes the strain-induced changes in the Raman spectra of nanotubes observed by several research groups. Raman Shifts of Nanotubes Under Axial Strain Experiment Method Max. Strain G ω Δ G' ω Δ Hadjiev, et al. 18 Bending nanotube-epoxy composite -0.18% (compression) 2.5 cm -1 n/a Jiang, et al. 17 Strained membrane containing nanotubes 0.05%-0.3% -36 cm -1 /% n/a Li, et al. 9 Strained CNT film on rubber 0.030%-0.094% -31 cm -1 /% n/a Lucas, et al. 10 11 Strained epoxy/SWNT composite 0.7% n/a -10.7 cm -1 /% Ruan, et al. 12 Tensile loading of MWNTs in polyethylene fibers 0% – 8% -0.17 to -0.75 cm -1 / % n/a Schadler, et al. 23 Compression and strain of CNT/epoxy composite -1% to 1% n/a 6 cm -1 /% (compression); +1 cm -1 /% (tension) Wood, et al. 14 15 Diamond anvil cell; mini- tensile machine 0-1.5% n/a -9 cm -1 /% Zhao, et al. 13 Straining of nanotube- polymer composite 0-4% n/a -18 cm -1 /% Zhao, et al. 16 17 Straining of nanotube- polymer composite 0-1% n/a -9 cm -1 /% Cooper, et al. 22 Tensile deformation from 4-point bending rig 0-1.2% n/a -15 cm -1 /% Cronin, et al. 7 8 AFM manipulation of single nanotube 0.06%-1.65% -32 cm -1 /% -27.7 cm -1 /% Table A2.1: Strain-induced changes in the Raman spectra using various straining techniques. G ω Δ and G' ω Δ are the shifts observed in the G and G’ bands, respectively. Ajayan, et al., performed both SEM and Raman spectroscopy on nanotube- polymer composites under strain. 24 Their SEM images reveal micro-cracking, which causes inhomogeneity in the strain of the measured nanotubes. Nanotubes 133 suspended across a crack experience a very large amount of strain, while the remainder of the nanotubes experiences relatively little strain. This micro-cracking causes uncertainties in the measurement due to the inhomogeneous distribution of strain and the ambiguity in the magnitude of the strain. In addition to strain-induced shifts in the G and G’ band Raman modes, Lucas and Young observed changes in the intensity of the radial breathing mode (RBM) of nanotubes, indicating that the resonant electron transition energies shift on and off of resonance with applied strain. 10 11 The photoluminescence (PL) measurements of Arnold, et al., show shifts in the interband energies due to axial compression at low temperatures that are in good agreement with the theoretical work of Yang, et al. 25 Individual Nanotube Measurements In another approach, resonance Raman spectroscopy is measured from individual nanotubes, before and after applying strain to the specific nanotube being measured. In this approach, only one nanotube with a specific chirality and orientation is measured at a time, thus providing results that are easier to interpret than ensemble measurements on nanotube composites. 7 8 In this method, strain is induced using an AFM tip to push nanotubes on a silicon substrate that are pinned between two metal pads. Figure A2.3 shows a nanotube that was strained in this fashion. The strong van der Waals forces hold the nanotube fixed after displacement with the AFM tip. Because the ends of the nanotubes are held fixed by metal pads, this transverse displacement results in an elongation of the nanotube, and hence an axial strain. The magnitude of the strain is determined from the nanotube geometries 134 in the AFM images taken before and after AFM manipulation. The pre-strain length is subtracted from the post-strain length, giving the elongation of the nanotube. Spectra taken with 0.5 μm spatial resolution demonstrate that the strain extends throughout the length of the nanotube and is not localized to the displaced region. 8 This implies that the surface forces hold the nanotube very strongly in the transverse direction but are weak in the axial direction. Therefore, the axial strain propagates along the entire length of the nanotubes and is not confined to the displaced region as was previously thought. 26 Figure A2.3: AFM image showing a carbon nanotube after inducing strain with an AFM tip. 7 Reprinted with permission from [7], S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 94, 167401 (2004). © 2004, American Physical Society. In these individual nanotube measurements, the Raman spectra of nanotubes were generally found to be more sensitive to strain than in nanotube composites. Axial strains as low as 0.06% produce easily resolvable shifts (2 cm -1 ) in the Raman frequencies. 7 The G band was observed to downshift by 16 cm -1 with 0.5% strain, or 32 cm -1 /% strain and revert back to its initial frequency upon breaking, as shown in Figure A2.4. 7 The G’ band was found to downshift by ~27.7 cm -1 /% strain, roughly two times larger than those observed in nanotube composites. 7 Under the applied 135 strain, the D band also downshifted by 15 cm -1 . However, no change in the D band intensity was observed, corroborating that these deformations are elastic. Figure A2.4 shows the G band Raman spectra of an individual carbon nanotube before and after inducing 1.65% strain. Upon initial strain, both G + and G - peaks downshift significantly. After the nanotube was strained beyond its breaking point, the Raman peaks resumed their original pre-strain values, indicating the elasticity of these strain deformations. Figure A2.4: G band Raman spectra for an individual SWNT before and after inducing 1.65% strain, and after breaking the nanotube with an AFM tip. Data taken after the SWNT was broken shows relaxation of the Raman peaks back to their original positions. 7 Reprinted with permission from [7], S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 94, 167401 (2004). © 2004, American Physical Society. The behavior of metallic nanotubes under strain was found to be quite different from that of semiconducting nanotubes. In metallic nanotubes, the intensity of the RBM Raman peak was observed to change dramatically with applied strain. 8 This indicates that, as strain is applied, the electronic transition energies of the metallic nanotubes (E ii ) shift, making them either more resonant or less resonant with 136 the incident laser energy. From these changes observed in the RBM spectra, the expected strain-induced changes in E ii can be predicted using the model described in the following section. Carbon Nanotubes under Axial Strain When axial strain is applied to SWCNTs, several changes occur both to the nanotube physics as well as to the Raman spectra. For moderate strains, the axial component of the bonds of the nanotube increases in length. This, in turn, affects the Raman spectra by downshifting the frequencies of the G + band. 27 Also, the gaps between the van Hove singularities increase, decrease, or stay the same with strain, depending on the nanotube chirality. 28 However, the circumference of the nanotube changes little for most nanotubes with strain due to the small Poisson’s ratio and small range of non-deforming strains. 29 These changes in the van Hove singularities can cause the nanotube to shift off of resonance as strain is applied. In addition, the narrow crossing point between the conduction and valence bands in metallic nanotubes can open up with strain and bring about semiconducting behavior. 29 With high strains, nanotubes can undertake defects and even break. With a Young’s Modulus of ~1 GPa, nanotubes have higher tensile strengths than steel and Kevlar. 30 Nanotubes have been calculated to have breaking stresses of 30 GPa occurring at strains of 5.3%. 30 When strains on the nanotube approach this number, permanent defects form in the nanotube in the form of 5-7-7-5 defects known as Stone-Wales defects. 31 These defects, form from the rotation of a carbon bond to induce a heptagon-pentagon pair, which weaken the nanotube and accelerate its 137 failure. 31 The D band intensity has been linked to an increase in defects, 32 but due to the complex double resonance process involved in the D band, this relationship has not been quantified. Despite this knowledge of the behavior of carbon nanotubes under axial strain, many issues remain unresolved. In particular, there is little data regarding the precise changes to Raman spectra as nanotubes transition to the point of irreversible strain and accumulate Stone-Wales defects. Furthermore, incremental measurements have not been conducted. These incremental measurements would give a “play-by- play” picture of what is happening to the vibrational modes of the system. This thesis explores these avenues of research as well as another effect known as surface enhanced Raman spectroscopy, which involves the enhancement of Raman signals from nanotubes in the presence of metallic nanoparticles. Conclusion In this background, the optical response of nanotubes to applied strain is discussed. Current research regarding the strain dependence of nanotube-polymer composite materials, as well as individual nanotubes, is reviewed. Raman spectroscopy serves as a sensitive tool for studying axial strains as low as 0.06%, which produce easily resolvable shifts (2 cm -1 ) in the Raman frequencies. The elongation of the C-C bond results in a downshift of the D, G and G’ bands of the Raman spectra. In nanotube-polymer composite materials, the large variability in the magnitude of these strain-induced downshifts is attributed to the transfer of strain, which differs vastly depending on how the nanotube composite is prepared. On 138 average, the strain-induced downshifts are smaller for composite materials than for individual nanotubes, indicating that there is a low rate of strain transfer from the bulk composite to the nanotubes. In order to optimize this strain transfer, the nanotubes should be isolated as individual nanotubes, rather than in bundles, to maximize the surface contact between the host polymer matrix and the nanotubes. Results of measurements on individual nanotubes are also reviewed. A particular emphasis is given to the important new physics revealed by the strain- induced changes in the Raman spectra of individual nanotubes. These observations demonstrate the elasticity of nanotubes after exceeding their breaking strain. The large anisotropy in the surface forces that hold the nanotube fixed on the substrate was also revealed in these measurements. In addition to changes in the vibrational energies, strain-induced changes in the electronic energies are also observed by resonant Raman spectroscopy. These changes are consistent with tight binding calculations, which predict strain-induced modulations of the electronic energy levels of nanotubes. These tight biding calculations also predict an (n-m) mod 3 family dependence in the strain-induced shifts. 139 Appendix 2 Endnotes 1 M. S. Dresselhaus, G. Dresselhaus, A. Jorio, A. G. Souza Filho, R. Saito, Carbon 40, 2043 (2002). 2 F. Huang, K. T. Yue, P. Tan, and S.-L. Zhang, Appl. Phys. Lett. 84, 4022 (1998). 3 H. D. Li, K. T. Yue, Z. L. Lian, Y. Zhan, L. X. Zhou and S. L. Zhang, Appl. Phys. Lett. 76, 2053 (2000). 4 N. R. Raravikar, P. Kelinski, A. M. Rao, M. S. Dresselhaus, L. S. Schadler, and P. M. Ajayan, Phys. Rev. B 66, 235424 (2002). 5 M. Z. Atashbar, and S. Singamaneni, Appl. Phys. Lett. 86, 123112 (2005). 6 S. Chiashi, Y. Murakami, Y. Murakami, Y. Miyauchi, E. Einarsson and S. Maruyama, Chem. Phys. Lett. 386, 89-94 (2004). 7 S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 94, 167401 (2004). 8 S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. B 72, 035425 (2005). 9 Z. L. Li, P. Dharap, S. Nagarajaiah, E. V. Barrera, J. D. Kim, Adv. Mat. 16, 640 (2004). 10 M. Lucas and R. J. Young, Phys. Rev. B 69, 085405 (2004). 11 M. Lucas and R. J. Young, Comp. Sci. and Tech. 64, 2297 (2004). 12 S. L. Ruan, P. Gao, T. X. Yu, Polymer 47, 1604 (2006). 13 Q. Zhao, M. D. Frogley, H. D. Wagner, Composites Sci. and Tech. 62, 147 (2002). 14 J. R. Wood, Q. Zhao, M. D. Frogley, E. R. Meurs, A. D. Prins, T. Peijs, D. J. Dunstan, H. D. Wagner, Phys. Rev. B 62, 7571 (2000). 15 J. R. Wood, Q. Zhao, and H. D. Wagner, Composites A – App. Sci. and Mfctg. 32, 391 (2001). 16 Q. Zhao and H. D. Wagner, Composites A – App. Sci. and Mfctg. 34, 1219 (2003). 140 17 Q. Zhao and H. D. Wagner, Phil. Trans. Royal Soc. London Series A-Math. Phys. Engg. Sci. 362, 2407 (2004). 18 V. G. Hadjiev, M. N. Iliev, S. Arepalli, P. Nikolaev, and B. S. Files, App. Phys. Lett. 78, 3193 (2001). 19 A. M. Rao, J. Chen, E. Richter, U. Schlecht, P. C. Eklund, R. C. Haddon, U. D. Venkateswaran, Y.-K. Kwon, and D. Tománek, Phys. Rev. Lett. 86, 3895 (2001). 20 R. Kumar and S. B. Cronin, Phys. Rev. B. 75, 155421 (2007). 21 K. Arnold, S. Lebedkin, O. Kiowski, F. Hennrich, and M. M. Kappes, Nano Lett. 4, 2349 (2004). 22 C. A. Cooper, R. J. Young, M. Halsall, Composites: Part A 32, 401 (2001). 23 L. S. Schadler, S. C. Giannaris, and P. M. Ajayan, Appl. Phys. Lett. 73, 3842 (1998). 24 P. M. Ajayan, L. S. Schadler, C. Giannaris, and A. Rubio, Adv. Mat. 12, 751 (2000). 25 K. Arnold, S. Lebedkin, O. Kiowski, F. Hennrich, and M. M. Kappes, Nano Lett. 4, 2349 (2004). 26 D. Bozovic, M. Bockrath, J. H. Hafner, C. M. Lieber, H. Park, and M. Tinkham, Phys. Rev. B 67, 033407 (2003). 27 S. B. Cronin, A. K. Swan, M. S. Ünlü, B. B. Goldberg, M. S. Dresselhaus, and M. Tinkham, Phys. Rev. Lett. 93, 167401 (2004). 28 L. Yang and J. Han, Phys. Rev. Lett. 85, 154 (2000). 29 E. D. Minot, Y. Yaish, V. Sazonova, J.-Y. Park, M. Brink, and P. L. McEuen, Phys. Rev. Lett. 90, 156401 (2003). 30 R. Kumar and S. B. Cronin, J. Nanosci. Nanotechnol. 8, 122 (2008). 31 B. G. Demczyk, Y. M. Yang, J. Cumings, M. Hetman, W. Han, A. Zettl, and R. O. Ritchie, Mat. Sci. and Eng. A 334, 173 (2002). 32 A. C. Dillon, P. A. Parilla, J. L. Alleman, T. Gennett, K. M. Jones, and M. J. Heben, Chem. Phys. Lett. 401, 522 (2005). 141 Appendix 3 Supplemental PDMS Strain Data Figure A3.1: Supplemental strain data of nanotubes on PDMS from a nanotube known as “Slab 2.” Note how the G + shows an irreversible change in nanotube behavior from low frequencies to high frequencies. 142 Figure A3.2: Additional strain data of a nanotube on PDMS known as “Slab3- Grid1-Tube5” which was already shown in Figure 3.11. Note the linear reversible behavior of the G + band and D band compared to the G - band. 143 Figure A3.3: Strain data for a nanotube on PDMS known as “Slab3-Grid1- Tube6.” Note the negative curvature trend in the beginning that could not be repeated for the G band. 144 Figure A3.4: Strain data for a nanotube on PDMS known as “Slab3-Grid1- Tube7.” Note the irreversible behavior in the G - band. 145 Figure A3.5: Strain data for a nanotube on PDMS known as “Slab3-Grid1- Tube8.” Note the irreversible trend in the G’ spectrum. 146 Appendix 4 MATLAB Dispersion Code for the calculation of Phonon Dispersion Relations in Carbon Nanotubes under Axial Strain % CNT_phonons_Strain_2.m by Adam Bushmaker and Rajay Kumar % % CNT_phonons_strain_2.m by Adam bushmaker <bushmake@usc.edu> % Built off CNT_phonons_strain.m by Rajay Kumar recieved Feb 21, 2008 % % CNT_phonons_strain.m by Rajay Kumar <rkumar@alum.mit.edu> % Built off of legacy code provided by Adam Bushmaker for graphene phonons % 2008-Feb-20 % % Program to graph the phonon dispersion relations for a nanotube under % axial strain. This program uses a graphene dynamical matrix, not a % nanotube dynamical matrix, so the plots are somewhat inaccurate. % 3x3 K matrix for each atom we're considering %put together into dynamical matrix, with appropriate phase delay terms %find k vectors of interest (subbands) - apply zone-folding to Graphene %model %find characteristic polynomial and roots for each point %plot roots (frequencies) vs. k to get bands close all; clear all; delta = 255; % gradation of points. The higher delta is, the more points there will be in the plots hbar = 6.58211814e-16; %eVs cmInvPEReV = 8065.541154; % 1/cm*eV a = 0.249; %nm rt3 = sqrt(3); Mc = 1.99430717e-26; %kg %%%%%%%%%%%%%%%%%%%% % NANOTUBE SECTION % %%%%%%%%%%%%%%%%%%%% % Obtain (n,m) for nanotube disp('n and m must be nonnegative integers with n>=m') n = input('n = '); m = input('m = '); disp('Strain (decimal, i.e. 0.1 means 10% strain)') epsilon = input('strain = '); 147 theta = atan((m.*sqrt(3))./(2*n+m)); %Chiral angle in radians d_R = gcd((2*n+m),(2*m+n)); dt = (a.*sqrt(n.^2 + m.^2 + m.*n))./pi; %NT diameter Ch = a*sqrt(n^2+m^2+n*m); % Length of chiral vector T = sqrt(3)*pi*dt/d_R;% length of translational vector, also direction of strain N = 2*(n^2+n*m+m^2)/d_R; %number of graphene hexagons in the unit cell kNT = 0:pi/(delta*T):pi/T; % k is the wavenumber in the axial direction of the nanotube (a.k.a. k_z) kc = 0:N;% allowed wavenumbers in radial direction (cutting line index) K1 = kc*(2*pi/(sqrt(n^2+n*m+m^2)*a))^2; % reciprocal lattice vector corresponding to chiral vector. % Coefficient of kc comes from solving for K1 in armchair configuration. K2 = kNT; % reciprocal lattice vector corresponding to translational vector. %%%%%%%%%%%%%%%%%%%% % GRAPHENE SECTION % %%%%%%%%%%%%%%%%%%%% %------------------------------------------------------------------- ------ % Force constants (N/m) phr = x translation (PHi Radial), phti = y translation (PHi Transverse Inplane), % phto = z translation (PHi Transverse Out of plane) phr(1) = 365.0; phti(1) = 245.0; phto(1) = 98.2; phr(2) = 88.0; phti(2) = -32.3; phto(2) = -4.0; phr(3) = 30.0; phti(3) = -52.5; phto(3) = 1.5; phr(4) = -19.2; phti(4) = 22.9; phto(4) = -5.8; %------------------------------------------------------------------- ------- %jb = band index %jt = neighbor tier index %jr = jr'th atom in tier, from the positive x axis rotating CCW for jt = 1:4 K0(jt,:,:) = [phr(jt) 0 0;0 phti(jt) 0;0 0 phto(jt)]; end %3x3 unrotated force tensors for nearest neighbor tiers 1-4 dR(1) = a/rt3; dR(2) = a; dR(3) = 2*dR(1); dR(4) = a*sqrt(7/3); %radius of nearest neighbor tiers 1-4 % If the chiral angle were zero, then Th(1,1) would be off by -30 deg = % -pi/6=5pi/6. The chiral angle is further subtracted from this. Now the % x axis of fig. 9.1 of Saito is aligned with the chiral vector. % So, to convert these angles such that the chiral vector is at 0 deg, % ThNT(i,j)=Th(i,j)+pi/6-theta 148 Th(1,1)=0; Th(1,2)=2*pi/3; Th(1,3)=4*pi/3; %angles for first tier of atoms x3 Th(2,1)=1*pi/6; Th(2,2)=3*pi/6; Th(2,3)=5*pi/6; Th(2,4)=7*pi/6; Th(2,5)=9*pi/6; Th(2,6)=11*pi/6; %angles for second tier x6 Th(3,1)=1*pi/3; Th(3,2)=3*pi/3; Th(3,3)=5*pi/3; %angles for third tier x3 {first tier angles + pi/3 (60 deg)} x41 = 2.5; y41 = sqrt(3)/2; Th(4,1) = atan(y41/x41); %angles for fourth tier x6 Th(4,6) = 2*pi-Th(4,1); Th(4,2) = -Th(4,1)+2*pi/3; Th(4,3) = Th(4,1)+2*pi/3; Th(4,4) = Th(4,2)+2*pi/3; Th(4,5) = Th(4,3)+2*pi/3; ThNT=Th; % Calculate ThNT from Th. Also, % make vectors with POLAR COORDINATES from center (A atom) to each atom % AtomVector(tier,atom_in_tier,:)=[radius;angle_from_Ch] for jt = 1:4 for jr = 1:6 ThNT(jt,jr)=Th(jt,jr)+(pi/6)-theta; % rotate so T aligns with y and Ch aligns with x if jt == 1 | jt == 3 if jr == 4 | jr == 5 | jr == 6 ThNT(jt,jr)=0; end end if (ThNT(jt,jr) >= 2*pi) ThNT(jt,jr)= ThNT(jt,jr)-(2*pi); end AtomVector(jt,jr,:)=[dR(jt);ThNT(jt,jr)]; %POLAR COORDINATES [radius;angle] AtomVectorRect(jt,jr,:)=[AtomVector(jt,jr,1)*cos(AtomVector(jt,jr,2) );AtomVector(jt,jr,1)*sin(AtomVector(jt,jr,2))]; %AtomVector in Cartesian Coordinates AtomStrainRect(jt,jr,:)=[AtomVectorRect(jt,jr,1);(1+epsilon)*AtomVec torRect(jt,jr,2)]; %multiplied by strain+1 % Convert to Polar, rotate back to normal coordinates. AtomStrain(jt,jr,:)=[sqrt(AtomStrainRect(jt,jr,1).^2+AtomStrainRect( jt,jr,2).^2);atan2(AtomStrainRect(jt,jr,2),AtomStrainRect(jt,jr,1))- (pi/6)+theta]; %convert back to polar if jt == 1 | jt == 3 if jr == 4 | jr == 5 | jr == 6 AtomStrain(jt,jr,:)=0; end 149 end if AtomStrain(jt,jr,2) < 0, AtomStrain(jt,jr,2)=AtomStrain(jt,jr,2)+2*pi; end if AtomStrain(jt,jr,2) >= 2*pi, AtomStrain(jt,jr,2)=AtomStrain(jt,jr,2)-2*pi; end end end %Need vectors with strain %Convert AtomVector to Rectangular Coordinates, multiply y/T- component by %strain+1, convert back to Polar % Calculate the rotated force constant matrices for jt = 1:4 for jr = 1:6 Um = [cos(AtomStrain(jt,jr,2)) -sin(AtomStrain(jt,jr,2)) 0;sin(AtomStrain(jt,jr,2)) cos(AtomStrain(jt,jr,2)) 0;0 0 1]; %The unitary rotation matrix for each atom Is = (dR(jt)^2)/(AtomStrain(jt,jr,1)^2); invSqrM = [Is 0 0;0 Is 0;0 0 Is]; Ktemp(:,:) = K0(jt,:,:); % Ktemp = Ktemp.*invSqrM; K(:,:,jt,jr) = Um\Ktemp*Um; %Um\ is equivalent to inv(Um), this line rotates the force constant matrix if jt == 1 | jt == 3 if jr == 4 | jr == 5 | jr == 6 K(:,:,jt,jr) = zeros(3,3); %make sure 4th, 5th and 6th elements of tier 1 and 3 matrices are =0 (no atoms there) end end end end %Convert Graphene code from Gamma to M, M to K, K to Gamma to new "swaths" %that are basically K2 vectors spaced apart by K1. Same approach except %now we don't have to manually type in three swaths - just go from one to %the other using K1 as the translational vector. Then convert these into %kx and ky for each swath. % Since we only look at the case where mu=0 (i.e., kc = 0), many for loops % will be commented out as we only care about the case for K1=0. %for I=1:length(K1), %swath number I = 1; % just calculate the first sub-band, the higher order sub- bands are not Raman active (well, 2 and 3 are, but they are weak) 150 for J=1:length(K2), % kx(I,J) = -K1(I)*sin(theta) + K2(J)*cos(theta); % That one minus sign was very annoying!!! kx(I,J) = 0*sin(theta) + K2(J)*cos(theta); ky(I,J) = 0*cos(theta) + K2(J)*sin(theta); end %end %build the 3x3 sub-matrices of the 6x6 dynamical matrix %for swath = 1:length(K1) %which swath? swath = 1; %only look at the first sub band for jb = 1:length(K2) %which band? Dab = zeros(3,3,length(K2)); Daa = zeros(3,3,length(K2)); D = zeros(6,6,length(K2)); % initialize the sub-matricies and the 6x6 dynamical matrix for jt = 1:4 for jr = 1:6 Ph = exp(i*AtomStrain(jt,jr,1)*(kx(swath,jb)*cos(AtomStrain(jt,jr,2))+ky( swath,jb)*sin(AtomStrain(jt,jr,2)))); % Here, call the new distances to atoms and the new angles Kk(:,:,jt,jr) = K(:,:,jt,jr)*Ph; %Kk is the phase shifted K force constant matrix if jt ~= 2 %& (jt~=1 & jr~=1) Dab(:,:,jb) = Dab(:,:,jb)-Kk(:,:,jt,jr);%*exp(- i*a*(rt3/2*kx1(jb)+ky1/2)); end if jt == 2 Daa(:,:,jb) = Daa(:,:,jb)- Kk(:,:,jt,jr);%+K(:,:,jt,jr); %stuff I commented out because I'm not sure of the correct formulation <AWB> end Daa(:,:,jb) = Daa(:,:,jb)+K(:,:,jt,jr); end end Dba = Dab(:,:,jb)'; Dbb = Daa(:,:,jb); %assuming the matrix is symmetric D(:,:,jb) = [Daa(:,:,jb) Dab(:,:,jb);Dba Dbb]; %build the 6x6 Dynamical matrix from the sub- matricies w(:,jb,swath) = sqrt(1/Mc*roots(poly(D(:,:,jb)))); %calculate the eigen values of the dynamical matrix end %end %Plotting section hold on; for jp = 1:6 band = 1; % for band = 1:length(K1) plot(kNT,cmInvPEReV*hbar*w(jp,:,band)) % 151 % plot(sqrt(kx(band,:).^2+ky(band,:).^2),cmInvPEReV*hbar*w(jp,:,band)) % % end end title(['Phonon dispersion relation for (',num2str(n),',',num2str(m),') nanotube with ',num2str(100*epsilon),'% axial strain']); xlabel('{\itk}'); ylabel('\omega [cm^{-1}]'); set(gca,'XTick',[0,pi/T]); set(gca,'XTickLabel',{'0','pi/T'}); %Change numerical labels on axis %axis tight; %axis square; xlim([0 pi/T]); ylim([1550 1600]); 152 Appendix 5 Supplemental SERS Data Figure A5.1: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 2 in Table 5.1. Figure A5.2: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 3 in Table 5.1. 153 Figure A5.3: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 4 in Table 5.1. Figure A5.4: Raman spectra taken before (black) and after (red) silver nanoparticle deposition demonstrating the SERS effect. This nanotube corresponds to Nanotube 5 in Table 5.1. 154 Figure A5.5: Zoomed-out SEM image of nanotubes in Figure 5.2(a). Figure A5.6: Zoomed-out SEM image of nanotubes in Figure 5.2(b) and (c).

Abstract (if available)

Abstract In this thesis, I present resonant Raman spectroscopy of individual carbon nanotube bundles under axial strains up to 17%. The main effect of this strain is to cause nanotube debundling. The G band Raman spectra of metallic and semiconducting nanotubes are found to respond differently to strain and debundling, giving insight into the nature of the broad metallic G- band lineshape. For metallic nanotubes, the G- band upshifts and becomes narrower with strain, making it appear more semiconductor-like. Surprisingly, this metal to semiconductor transition is irreversible with strain, indicating that nanotube-nanotube coupling plays a significant role in the observed G- band of metallic nanotubes. The vibrational and electronic properties of these nanotubes under strain are modeled using tight-binding calculations.

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Asset Metadata

Creator Kumar, Rajay (author)

Core Title Raman spectroscopy of carbon nanotubes under axial strain and surface-enhanced Raman spectroscopy of individual carbon nanotubes

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 04/10/2008

Defense Date 03/07/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag axial strain,carbon nanotubes,nanotechnology,nanotube bundle,OAI-PMH Harvest,Raman spectroscopy,surface-enhanced Raman spectroscopy

Language English

Advisor Zhou, Chongwu (committee chair), Cronin, Stephen B. (committee member), Thompson, Mark E. (committee member)

Creator Email rajaykum@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1096

Unique identifier UC1141466

Identifier etd-Kumar-20080410 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-55312 (legacy record id),usctheses-m1096 (legacy record id)

Legacy Identifier etd-Kumar-20080410.pdf

Dmrecord 55312

Document Type Dissertation

Rights Kumar, Rajay

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu