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

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 19 2005

NT Bands: Space quantization kcirc k COT Optics ARL May 19 2005

Symmetry and Electronic Structure Armchair - metallic NT Zigzag - semiconductor NT COT Optics ARL May 19 2005

Tight-Binding Hamiltonian of Graphene b=0.14 nm B COT Optics ARL May 19 2005

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 19 2005

SWNT Periodic Table Semi-metals Semiconductors Metals Two (primary) classes and two (secondary groups) Semi-metals Semiconductors Metals COT Optics ARL May 19 2005

SWNT Periodic Table Primary and secondary gap semiconductors White, Mintmire Rotkin, unpublished COT Optics ARL May 19 2005

Symmetry and Electronic Structure DoS - density of states COT Optics ARL May 19 2005

After R Smalley and B Weisman COT Optics ARL May 19 2005

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 COT Optics ARL May 19 2005

1D Exciton: Classical model Using effective 1D Coulomb potential we derive dimensionless Schrödinger equation Bulashevich, et.al., Intl.J. of Nanoscience, 2, 521 (2003) 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 19 2005

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 19 2005

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 19 2005

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 Bulashevich, et.al., Intl.J. of Nanoscience, 2, 521 (2003) COT Optics ARL May 19 2005

Exciton: Direct Interaction Selfconsistent calculation requires the frequency-dependent dielectric function e(w) – obtained in Random Phase Approximation Bulashevich, et.al., Intl.J. of Nanoscience, 2, 521 (2003) COT Optics ARL May 19 2005

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: COT Optics ARL May 19 2005

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 Bulashevich, et.al., Intl.J. of Nanoscience, 2, 521 (2003) COT Optics ARL May 19 2005

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 Bulashevich, et.al., Intl.J. of Nanoscience, 2, 521 (2003) COT Optics ARL May 19 2005

BSE: Diagrammatic approach v 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 19 2005

Exciton: Variational solution Approximate solution of BSE - analytical expression to evaluate importance of separate terms The functional to minimize COT Optics ARL May 19 2005

e-h Coulomb Interaction Compare terms of e-h interaction: - screening - depolarization - exchange - kinetic energy Instability region aB COT Optics ARL May 19 2005

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 COT Optics ARL May 19 2005

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 19 2005