1. 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

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

3. 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?

4. 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)

5. ? 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 ?

6. Optical Methods to Probe SWNTs
O'Connell, et al. Science 297, (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

7. 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

8. 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): (1999)
Yu, Z. and Brus, L.
J. Phys. Chem. B 105(6): (2001)

9. 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 ( nm)
log Intensity

10. 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

11. 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

12. 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.

13. 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 Brookhaven National Labs.

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

15. 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. (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.
Transition Energies (eV)
Substantial differences in the absolute energies from theory:
Semis: > 200 meV
Metals: > 150 meV

17. 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.

18. 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!

19. 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.

20. 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

21. 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

22. Connecting Different Data Sets
Kane CL, Mele EJ. PRL (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!!!

23. 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

24. 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!!!

25. 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

26. Extra Slides

27. 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)

28. 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

29. 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

30. 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.

31. 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)

32. 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

33. 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.

34. 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

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

36. 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.

37. Metallic Armchair Tubes - Lineshape

38. 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.”

39. Lineshape Analysis
Rayleigh Scattering
Exciton Model eV
Band Model

40. 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.

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

42. Lineshape Analysis
Rayleigh Scattering
Exciton Model eV
Band Model