1. Many-Body Effects in the Optics of Single-Wall Nanotubes
Slava V. Rotkin
Physics Department
Lehigh University

2. Single-Wall Nanotubes (SWNT)
Vector components of the circumference Ch=n a1+m a2 become a NT index [n,m]
Rolling up a (10,10) nanotube Ch z
By Shigeo Maruyama, University of Tokyo, Japan
COT Optics ARL May

3. NT Bands: Space quantization
kcirc k
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4. Woman in an Armchair. A cubist painting by Pablo Picasso.
Symmetry and Electronic Structure
Armchair -
metallic NT
Woman in an Armchair. A cubist painting by Pablo Picasso.
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5. Symmetry and Electronic Structure
Armchair -
metallic NT
Zigzag -
semiconductor NT
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6. Tight-Binding Hamiltonian of Graphene
b=0.14 nm B
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7. Tight-Binding Hamiltonian of Graphene
Bare Hamiltonian has a symmetry group of the SWNT lattice:
which is included in TB approach via the vectors of the bonds
The energy is sensitive to the symmetry, because at certain points of the Brillouin zone (Fermi points) the cancellation of terms happens
[10,10] armchair
[17,0] zigzag
COT Optics ARL May

8. SWNT Periodic Table Semi-metals Semiconductors Metals
Two (primary) classes and two (secondary groups)
Semi-metals
Semiconductors
Metals
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9. SWNT Periodic Table Primary and secondary gap semiconductors
White, Mintmire
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10. Symmetry and Electronic Structure
DoS - density of states
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11. Symmetry and Electronic Structure
DoS - density of states
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12. After R Smalley and B Weisman
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13. After R Smalley and B Weisman
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14. 1D Exciton: Classical model
First calculations on the 1D exciton problems dated back to 50-s
The log divergence of EB has been noted
for two particles on a wire
Specifics of the NT is in
1. Coulomb interaction: quasi-1D
2. One-electron spectrum: Eg~1/R
We start with a classical
Coulomb term on a cylinder
general expression is too ugly for calculations
COT Optics ARL May

15. 1D Exciton: Classical model
Using effective 1D Coulomb potential we derive
dimensionless Schrödinger equation
certainly, solutions are the same as Loudon's ones and
binding energy diverges
However, the radius of the tube sets the cut-off: aB~R
Worse news: compare EB and Eg
EB > Eg therefore NT must be an excitonic insulator
The underlying physics: direct Coulomb term is large as compared to kinetic energy. e-h interaction is not screened. no repulsion term.
COT Optics ARL May

16. 1D Exciton: Model What is missing in the picture??
Direct terms and their screening
...
...
Valence electrons make a cloud around
the electron (and the hole) and move with
corresponding frequency.
All screening terms can be collected
in a dielectric function
...
Also there are
exchange terms
(smaller than direct)
with opposite sign
is the picture complete now?
COT Optics ARL May

17. Fourier of Coulomb Potential
0.5 1
1.5 2
2.5 3 qR
I|m|(|qR|) K|m|(|qR|)
Interaction on the cylinder: z, R, q
Different components of the Coulomb potential have different strength (and also different screening): m=0,1,2,3 from top to bottom
m=0 component is diverging
COT Optics ARL May

18. 1D Exciton: Full model Full solution must include
Direct term and its screening
Exchange term (no screening)
Selfconsistent equation for e-h
Bethe-Salpeter equation
Bethe-Salpeter equation (BSE) is still too complex to solve
BSE is still too complex
First approximation: no exchange interaction
COT Optics ARL May

19. Exciton: Direct Interaction
Selfconsistent calculation requires the frequency-dependent dielectric function e(w) – obtained in Random Phase Approximation
Substituting reduced mass and energy gap
we obtain dimensionless (no dependence on RNT) Schrödinger equation which is universal – up to higher order corrections
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20. Continuous vs Quantum Model
Dielectric function:
Continuous vs Quantum Model
Zero-order approximation: a uniform hollow dielectric cylinder with the tangential polarizability  :
Then the dielectric function reads
Match continuous dielectric model with Quantum Mechanics:
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21. Exciton Binding Energy
Graphics solution of the Universal Bethe-Salpeter equation for exciton:
dielectric function (solid line) and the exciton binding energy (dashed and dash-dotted lines) intersect at the self-consistent binding energy
ground state
first excited
state
ground state
first excited
state
COT Optics ARL May

22. Exciton Binding Energy
Graphics solution of the Universal Bethe-Salpeter equation for exciton:
dielectric function (solid line) and the exciton binding energy (dashed and dash-dotted lines) intersect at the self-consistent binding energy
ground state
first excited
state
-20
-15
-10 -5 5 10 15 20 -1
-0.5
z, R
Energy, Eg
COT Optics ARL May

23. BSE: Diagrammatic approachv c
Exchange term is ~ qR v c c
Direct term is ~ 1 v v
Therefore, screening (inf.diagram series) is needed for direct term only
exchange terms (smaller than direct)
COT Optics ARL May

24. Exciton: Variational solution
Approximate solution of BSE
- analytical expression to evaluate importance of separate terms
Gaussian terms can be integrated
Though derivation is lengthy
The functional to minimize
where the frequency dependent part is
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25. e-h Coulomb Interaction
Compare terms of e-h interaction:
- screening
- depolarization
- exchange
- kinetic energy
Instability
region aB
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26. Summary Exciton problem in SWNT requires
- careful many-body series expansion
- dynamic screening
- exchange interaction
Exciton binding energy is less than the gap because
- the dynamic screening prevents the excitonic instability
- exchange interaction term compensates the screening Coulomb potential
- the gap-to-binding-energy ratio is a universal constant
- the exciton-to-NT-radius ratio is also a universal constant
External screening may change the direct Coulomb term and thus the exciton energy
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27. Acknowledgements Students Yan Li (UIUC), Kirill Bulashevich (Ioffe)
Prof. Karl Hess (UIUC),
Prof. Umberto Ravaioli (UIUC),
Prof. Robert A. Suris (Ioffe),
Dr. Alexey Petrov (Ioffe),
Feigel Scholarship from Lehigh University
Partial support from DoE, NSF, ONR, ARO grants, and Arnold and Mabel Beckman Foundation
COT Optics ARL May

28. COT Optics ARL May

29. Dielectric function: Analytic limit
The states near the Fermi point give the main contribution to the polarizability. Expanding the matrix element and the energy denominator in powers of q we obtain an analytical expression for dielectric function
there 0 is resonance integral and A is frequency factor
The frequency factor A is singular then frequency approaches the energy gap and goes to unity in the static case
COT Optics ARL May

30. COT Optics ARL May