Probing the Electronic Structure of Carbon Nanotubes using Rayleigh Scattering Matthew Y. Sfeir Feng Wang Limin Huang X. M. H. Huang Mingyuan Huang Jim Hone Stephen O’Brien Tony Heinz Louis Brus Tobias Beetz Lijun Wu Yimei Zhu Jim Misewich

Imagining the Carbon Nanotube Structure Family of closely related molecules with hundreds of members and diameters ranging from 0.4 - 3.0 nm. Each nanotube is uniquely described by its diameter [dt] and chiral angle q. [Also can be labeled by (n,m)]

What is the real electronic structure of an arbitrary SWNT? Motivation and Open Questions Our understanding of SWNTs mainly comes from: Kataura Plot S22 S44 S33 S11 M11 M22 1 Single-particle theory. (LEFT) Assignments of luminescence data (Box 1) based on predictions from single-particle theory. Measurement and calcs. confirming the existence of many-body effects. All allowed nanotube transitions (eV) SWNT diameter What is the real electronic structure of an arbitrary SWNT?

Experimental Detemination of Excitons in SWNTs Two photon excitation spectra of individual fluorescence peaks Energy levels of transitions observed directly from 2-photon excitation and emission spectra Band edge 1s 2p The optical transitions in nanotubes are excitons, NOT interband transitions. F. Wang et al, Science 308, 838(2005)

? Theory Experiment Motivation and Open Questions 1. Assign the optical spectra of all SWNTs. As there is no accurate theory to guide optical assignments this requires independent measurement of electronic and physical structure. 2. From assignments, develop understanding of the influence of many-body and chirality effects in both metals and semiconductors. Theory Experiment ?

Optical Methods to Probe SWNTs O'Connell, et al. Science 297, 593-596 (2002). 1) Absorption Small cross section, has only previously been measured in ensemble samples. 2) Luminescence Only applicable to small diameter semiconducting tubes. 3) Resonance Raman Scattering Need to satisfy unknown resonance condition; very weak. Raman 4) Rayleigh Scattering Can probe all frequencies simultaneously with white light; can we observe resonant enhancement? cm-1

Linear Light Scattering Processes Resonance Rayleigh Scattering Elastic light scattering [white light scattering] wscattered = wincident Probe electronic structure – detectable at all w, but enhanced at electronic transition energies  similar to absorption spectra w Resonance Raman Scattering Inelastic light scattering – momentum transfer via fundamental excitations in material (monochromatic) Vibrational Raman – probe Raman active phonons wscattered = wincident  wphonon Strongly enhanced at energies in resonance with an electronic transition  Resonance Raman cm-1 shift from w

Rayleigh Scattering From Nanostructures Resonance Rayleigh scattering shown to closely resemble the absorption spectra. Silver Particles SWNT Bundles Michaels, Amy M.; Nirmal, M.; Brus, L. E. JACS 121(43): 9932-9939. (1999) Yu, Z. and Brus, L. J. Phys. Chem. B 105(6): 1123-1134. (2001)

Supercontinuum Radiation for Spectroscopy Smaller cross-section of carbon nanotubes demands a brighter light source compatible with confocal microscopy methods. Solution: white-light generation in an optical fiber: laser brightness with a large spectral bandwidth (450 - 1450 nm) log Intensity

Sample Fabrication and Characterization Growing Suspended Nanotubes Imaging Substrates with slits patterned by optical lithography and wet etching. Look at total integrated intensity on CCD to find tubes. Correlate to electron microscopy images. CVD directional growth with lengths > 100 microns: Isolated SWNT 10 m nanotube scattering slit edges

Resonance Rayleigh Scattering Spectra M. Sfeir et al, Science 306, 1540 (2004) Semiconducting Carbon Nanotube Two well separated S33 and S44 transitions for larger diameter tubes, S33 and S22 for smaller diameters. Metallic Carbon Nanotube Single M11 or M22 transition observed in the visible – sometimes split into two very close peaks by trigonal warping effect

Theoretical Rayleigh Scattering from a SWNT Treat SWNT as an infinite right cylinder with effective dielectric function. e = e1 + ie2 Band Model Exciton Model Energy Energy Peaks in the dielectric function give rise to peaks in the Rayleigh spectrum; resulting lineshape is similar for exciton or interband model.

Assigning the Optical Spectra For unambiguous assignment of optical transitions, we need a technique compatible with our sample geometry that provides an independent structural verification. TEM Image Diffraction Collaboration with the electron microscopy group @ Brookhaven National Labs.

Determining SWNT Structure by Electron Diffraction Analyze electron scattering signal from ~ 20 nm collimated electron beam. Gao, et. al., Appl. Phys. Let., 82(16) 2703. (2003) Equatorial Oscillation

Direct Correlation of the Electronic and Physical Structure M. Sfeir et al, Science accepted (2006) Energy (eV) Intensity a: experimental diffraction b: simulated diffraction (16, 11) Optical Transitions: S33 = 2.0 eV S44 = 2.3 eV Diameter: 1.83 nm Chiral Angle: 23.9 o

(16,11) Electronic Structure Comparisons to some commonly used theoretical treatments. S33 S44 Rayleigh Spectrum Resonance Energy Fit -Tight Binding Calcs Extended Tight Binding ETB + Many-body Corr. 1.99 2.27 2.00 2.30 1.79 2.14 1.63 1.93 1.88 2.15 Transition Energies (eV) Substantial differences in the absolute energies from theory: Semis: > 200 meV Metals: > 150 meV

Testing Fundamental Predictions of Electronic Structure A predicted chirality dependence leads to systematic deviations as a function of (n,m). Do many-body effects (which shift absolute energies) disrupt this pattern? Zoom of region of Kataura plot Ignoring chirality and many-body effects: S33 M11 However, the graphene energy dispersion is not a linear function of k.

Testing Fundamental Predictions of Electronic Structure A predicted chirality dependence leads to systematic deviations as a function of (n,m). Do many-body effects (which shift absolute energies) disrupt this pattern? Zoom of region of Kataura plot Spread within a transition series is not random and depends on chirality [(n,m)]. 2n+m=46 S33 Semiconducting: “family” behavior 2n+m=44 Within certain structural "families" (changing d and q), energies evolve in a predictable way within that group. M11 Metals: trigonal warping effect Splitting of transitions within a series with increasing chiral angle It is difficult to measure these effects experimentally because of little correlation between optical and physical data!

Semiconducting SWNTS 1. Constant Chiral Angle 2. Constant Diameter D dt = 0.12 nm D q = 5.3o We can use these three patterns to indirectly assign many of our spectra! 3. Families of Constant 2n+m Our data confirms some “family” behavior – the relationships between SWNTs with different diameters and chiral angles.

Experimental Verification of the Trigonal Warping Effect Metallic SWNTs Experimental Verification of the Trigonal Warping Effect Not detectable by luminescence of Raman scattering - shows unique capabilities of the Rayleigh scattering method. M11 M11 M22 q = 30o q = 25o q = 24o DE = 0 meV DE = 90 meV DE = 140 meV

2 3 1 Connecting Different Data Sets How do we compare nanotubes from different regions of the Kataura plot to develop a universal picture of excited states? We have optical data for: Small diameter semi SWNTs. [diameters < 1 nm] Strong many-body effects Large diameter semi SWNTs [diameters > 1.6 nm] Unknown many-body effects? Metallic nanotubes with diameters in between [1.3 nm] No many body-effects??? S22 S44 S33 S11 M11 M22 2 3 1

Connecting Different Data Sets Kane CL, Mele EJ. PRL 93 197402 (2004). Nanotube electronic structure dominated by 2D many-body effects (REAL graphene dispersion). 1D are negligible. S33 M11 S22 0.8 nm 1.3 nm 1.7nm 2.00 1.95 1.90 1.85 2.30 2.25 2.20 2.15 2.10 (10,10) (14,13) (7,6) We don’t know the real graphene energy dispersion: E(k) For SWNT transitions with energy E, determine k, and compare different nanotubes with similar k. If 1D effects are strong, this treatment will give large errors (metal vs. semi; diameter dependence). k This is the best theoretical fit to our data and implies that metals and semiconductors not very different!!!

Intertube Interactions Synthesis and Fabrication SWNT Project Synergy Many projects have contributed to and benefitted from the Rayleigh scattering project and furthered our understanding of SWNTs. Mechanical Electro-optics Transport Intertube Interactions Theory Raman Scattering Transfer Technology Heinz, Brus Hone Hone, Kim Heinz, Hone, BNL Heinz, Hone NSEC Synthesis and Fabrication O’Brien, Hone

Conclusions We have developed an optical method useful for identifying the optically allowed electronic transitions in individual carbon nanotubes. Rayleigh scattering spectra can be interpreted qualitatively using theory as a guide but direct structual characterization is necessary for assignments. We have begun building a set of assignments from correlated electron diffraction measurements and extending those using the expected evolution . An interesting picture of the excited states is emerging – we invite theoretical help with this problem!!!

Production and Growth: Jim Hone Limin Huang Henry Huang Mingyuan Huang Acknowledgements Optical Experiments: Feng Wang Yang Wu Tony Heinz Louis Brus Production and Growth: Jim Hone Limin Huang Henry Huang Mingyuan Huang Discussion: Mark Hybertsen Philip Kim Gordana Dukovic Jim Yardley Jim Misewich, BNL Electron Microscopy: Limin Huang Lijun Wu, BNL Yimei Zhu, BNL Tobias Beetz, BNL

Extra Slides

Examining “Family” Relations We have seen that our data progresses in the expected way for diameter and chirality changes. 47 41 37 44 40 43 38 46 1/dt Eii (eV)

Chirality Dependence in a Non-interacting Model Metal Semiconductor - I Semiconductor - II mod (n – m, 3) = 0 mod (n – m, 3) = 1 mod (n – m, 3) = 2 Metals: “trigonal warping effect” Semiconductors: “family” behavior

Trigonal Warping Effect in Metallic SWNT Constant energy contours of graphene dispersion  = 30o Saito et. al., PRB 61, 2981 (2000). Reich and Thomsen, PRB 62, 4273 (2000).  = 0o

Can we extend this technique to a single nanotube? The nanotube has an extremely small scattering cross-section. N2 SWNT (40 m long) Silver (50 nm) 10-27 cm2 10-14 cm2 10-10 cm2 Need a sufficiently bright broadband excitation source and a single nanotube in a controlled geometry and environment.

Structural Information from Raman Scattering Rayleigh Identifies MULTIPLE electronic transitions which can be used to satisfy the resonance condition needed for Raman. Resonance Raman: eV Resonantly excite at ONE energy (1.96 eV) dt = 1.89 nm (21,4) nanotube? Low chiral angle Phonon frequency (cm-1)

S33 S44 M11 M11(-) M11(+) M22(-) M22(+) (n,m) Eii (eV) Transition p-TB theory ETB theory ETB+MB correction (16,11) 2.00 S33 1.79 1.63 1.88   2.30 S44  2.14 1.93 2.15 (15,10) 1.92 1.77 2.01 2.44 2.29  2.06 2.26 (13,12) 2.09 1.9 1.73 1.97 2.52  2.36 2.35 (13,11) 2.19 1.99 1.82 2.56  2.42 2.38 (10,10) M11 (11,8) M11(-) 1.84 1.66 1.91 2.02 M11(+) 1.74 (20,14) 2.22 M22(-) 2.04 M22(+) 2.13

Using Rayleigh Information To Engineer Nanotube Devices Henry’s Transfer Method X. M. H. Huang, R. Caldwell, L. Huang, S. Jun, M. Huang, M.Y. Sfeir, L. Brus, S.P. O’Brien, J. Hone, Submitted, 2005. This gives us the ability to select a nanotube with specific properties and place it on a surface with spatial accuracy of several microns.

An Example with Electronic Transport Data 1. Optical Characterization: Semiconducting Nanotube dt ~ 1.9 nm (17,10) possible assignment from family plots 2. Transfer to Si substrate for transport measurements: Confirms semiconducting character

Scattering Spectra along the Nanotube: Single Tube to Small Bundle Semiconducting Metallic Transitions red-shift by 20 - 50 meV upon bundling.

Nanotube Bundling in a Y-Junction A+B A: Moderate sized bundle structure. B: Single semiconducting nanotube with dt ~ 1.9 nm. A+B: Merged structure. The resonances in tube B are red-shifted by 35 and 47 meV. The resonances in tube A are shifted by much less in the combined structure. This effect is consistent with dielectric screening of manybody effects.

Metallic Armchair Tubes - Lineshape

Polarization Dependence of Rayleigh Scattering X 5 Polarization along nanotube axis: Selection rules allow symmetric transitions between singularities in the DOS, J = 0. Perpendicular to nanotube axis: Selection rules allow J =1 transitions, but quenched due to “depolarization effect.”

Lineshape Analysis Rayleigh Scattering Exciton Model eV Band Model

Dielectric Function Optical response is dominated by the peaked joint density of states. We observe Rayleigh scattering that is resonantly enhanced near the absorption maxima.

Rayleigh Spectra Collected with a QTH Lamp Collection time > 4.5 hours per graph.

Lineshape Analysis Rayleigh Scattering Exciton Model eV Band Model