Electron Interactions and Nanotube Fluorescence Spectroscopy C.L. Kane & E.J. Mele

Electron Interactions and Nanotube Fluorescence Spectroscopy C.L. Kane & E.J. Mele Large radius theory of optical transitions in semiconducting nanotubes derived from low energy theory of graphene Phys. Rev. Lett. in press cond-mat/ 0403153 Brief Introduction to nanotubes Independent electron model for optical spectra D interactions: nonlinear scaling with 1/R 1D interactions: excitons Short Range Interactions: exciton fine structure

Carbon Nanotubes as Electronic Materials Gate Source Drain A Molecular Quantum Wire ~ 1 nm ~ 1 µm Tans et al. (Nature 1998) Ballistic Conductor Field Effect Transistor Logic Gates

Carbon Nanotubes as Optical Materials Photoluminescence Electroluminescence & Photoconductivity Nanotubes in surfactant micelles Bachillo et al. (00). Photoluminescence from individual suspended nanotubes Lefebvre et al. (003). Infrared Emission, photoconductivity in individual nanotube field effect devices Freitag et al., (IBM) 004

Carbon Nanotube : Wrapped Graphene C = n 1 a 1 + n a Tubes characterized by [n 1,n ] or Radius : R = C /π Chiral Angle : 0 < θ < 30 ο Chiral Index : ν = n 1 -n mod 3 = 0,1,-1

Electronic Structure Metal Finite Density of States (DOS) at Fermi Energy Semiconductor Gap at Fermi Energy Graphene Zero Gap Semiconductor Zero DOS metal

Low Energy Theory of Graphene Brillouin Zone Effective Mass Model: Massless Dirac Hamiltonian H = v eff F Eq ( ) =± v q F ψ σ i v 0.53 ev nm F = ψ Tight Binding model: v = 3 γ a / ; γ =.5 ev F 0 0

Wrap it up... Flat Graphene: A zero gap semiconductor Periodic boundary conditions on cylinder: n 1 -n = 0 mod 3 1D Metal n 1 -n = +/-1 mod 3 Semiconductor

Near-infrared Photoluminescence from Single-wall Carbon Nanotubes O Connel et al. (Science 0) Bachillo et al. (Science 0) Excitation (661 nm) Emission (> 850 nm)

Nanotube Fluorescence Spectroscopy O Connel et al. (Science 0) Bachillo et al. (Science 0) c c 1 v 1 v Each peak in the correlation plot corresponds to a particular species [n 1,n ] of semiconducting nanotube GOAL: Understand observed transition energies in terms of low energy properties of an ideal dimensional graphene sheet.

Free Electron Theory of Nanotube Bandgaps Systematic expansion for large radius, R Zeroth order: 0 vf n En = (n = 1,, 4, 5, ) 3 R

Free Electron Theory of Nanotube Bandgaps Systematic expansion for large radius, R Zeroth order: 0 vf n En = (n = 1,, 4, 5, ) 3 R Trigonal Warping Correction E TW.. n ( 1) n ν n sin3θ R Curvature and Trigonal Warping: Vary as 1/R Alternate with band index n Alternate with chiral index ν Vanish for armchair tubes, θ=0 Curvature Correction C n sin3θ E n ( 1) ν Different dependence on n R The large R limit is most accurate for nearly armchair tubes: θ ~ 0 E n0 describes tight binding gaps accurately for R >.5 nm

Nanotube Assignments from Pattern of sin 3θ/R Deviations Experimental Ratio Plot Theory (includes sin3θ/r deviations) By comparing the experimental and theoretical ratio plots the [n 1,n ] values (and hence R and θ) for each peak can be identified. Corroborated by Raman spectroscopy of the radial breathing mode.

The Ratio Problem Free electron theory predicts E / E11 for R Consequence of linear dispersion of graphene E E 11 1.75 Increasing diameter

Scaling of Optical Transition Energies (Kane,Mele 04) Free electrons for θ=0 0 E ( ) v /3 nn R = Fn R ν sin 3θ / R deviations are clear Separatrix between ν=+1 and ν= 1describes nearly armchair tubes with θ=0, where sin 3θ/R deviations vanish.

Scaling of Optical Transition Energies (Kane,Mele 04) Free electrons for θ=0 0 E ( ) v /3 nn R = Fn R ν sin 3θ / R deviations are clear Nearly armchair [p+1,p] tubes Separatrix between ν=+1 and ν= 1describes nearly armchair tubes with θ=0, where sin 3θ/R deviations vanish.

Scaling of Optical Transition Energies (Kane,Mele 04) Nearly armchair [p+1,p] tubes Free electrons for θ=0 0 E ( ) v /3 nn R = Fn R Ratio Problem: E / E 11 < Blue Shift Problem: E R E R 0 nn( ) > nn( ) Worse for large R Nonlinear scaling E nn (R)=E(q n =n/3r) accounts for both effects.

Electron Interactions in large radius tubes For πr>>a electron interactions can be classified into three regimes, which lead to distinct physical effects. Long Range Interaction : (r > πr) One Dimensional in character Strongly bound excitons Intermediate Range Interaction : (a < r < πr) Two Dimensional in character Nonlinear Scaling with n/r Short Range Interaction : (r~a) Atomic in character Exciton Fine Structure

Long Range Interaction : (r > πr) V( z) = e ε z Renormalize Single Particle Gap Increase observed energy gap Leads to exciton binding Decrease observed energy gap Single Particle and Particle hole gaps both scale linearly with 1/R : E 0 G ~ v / R F / m*~ v F R E R R f e ε ( ) ~ ( v F / ) ( / v F) Gap renormalization and exciton binding largely cancel each other.

Cancellation of gap renormalization and exciton binding: Single Particle excitation: Self energy ~ e /εr Depends on dielectric environment Particle-hole excitation: Bound exciton is unaffected by the long range part of the interaction. The cancellation is exact for an infinite range interaction Coulomb Blockade Model : Bare gap: Interaction energy: U N / Single particle gap + U Particle-hole gap

Intermediate Range Interaction: (a < r < πr) Leads to nonlinear q log q dispersion of graphene. Responsible for nonlinear scaling of E 11 (n/r). Short Range Interaction: (r ~ a) Leads to fine structure in the exciton spectrum: S=0,1, etc. Splittings ~ e a / R

Interactions in D Graphene Gonzalez, Guinea, Vozmediano, PRB 99 σ nrnr ()(') H = v F drψ ψ + e drdr' i r r ' Renormalized Quasiparticle Dispersion: g Eq ( ) = v F q 1+ log 4 Λ q = e / v F Singularity due to long range Coulomb interaction V(q) = πe /q. g

Interactions in D Graphene Gonzalez, Guinea, Vozmediano, PRB 99 σ nrnr ()(') H = v F drψ ψ + e drdr' i r r ' Renormalized Quasiparticle Dispersion: g Eq ( ) = v F q 1+ log 4 Λ q = e / v F Singularity due to long range Coulomb interaction V(q) = πe /q. Dielectric Screening in Dimensions g g screened = g/ε Π static (q) = q/4v F ε static = 1+gπ/

Interactions in D Graphene Gonzalez, Guinea, Vozmediano, PRB 99 σ nrnr ()(') H = v F drψ ψ + e drdr' i r r ' Renormalized Quasiparticle Dispersion: g Eq ( ) = v F q 1+ log 4 Λ q Singularity due to long range Coulomb interaction V(q) = πe /q. Dielectric Screening in Dimensions g = e / v F g screened = g/ε Π static (q) = q/4v F ε static = 1+gπ/ Scaling Theory g = g( Λ) ; v =v ( Λ) F F dg 1 = g d ln Λ 4 Marginally Irrelevant q ln q correction is exact for q 0 Marginal Fermi Liquid

Compare D Theory with Experiment E = E( q = n/3 R) nn n g Eq ( ) = vfq 1+ log 4 v F Free electron Theory D Interacting Theory =.47 ev nm F Λ q e g = = 1. ε.5 ε v

Compare D Theory with Experiment E = E( q = n/3 R) nn n g Eq ( ) = vfq 1+ log 4 v F Free electron Theory D Interacting Theory =.47 ev nm F Λ q e g = = 1. ε.5 ε v

Compare D Theory with Experiment Free electron Theory D Interacting Theory Nearly armchair [p+1,p] tubes E = E( q = n/3 R) nn n g Eq ( ) = vfq 1+ log 4 v F =.47 ev nm F Λ q e g = = 1. ε.5 ε v The optical spectra reflects the finite size scaling of the D Marginal Fermi Liquid

Exciton effects: Compute particle-hole binding due to statically screened interaction (similar to Ando 97). Related Work: Spaturu et al (Berkeley) PRL 03 Perebeinos et al (IBM) PRL 04 E/E 11 0 Lowest exciton dominates oscillator strength for each subband. Lineshape for absorption is not that of van Hove singularity. Large bandgap renormalization mostly cancelled by exciton binding.

? Scaling behavior: E n (R) = E( q n = n/3r ) 3R E v n F nn ( R ) nn vfn 1 e 3RΛ Eexciton( R) = cn + log 3R 4ε vf n c n ~ independent of n Log(3RΛ/n)

Exciton Fine Structure K e h K Degenerate exciton states: e : k = K or K' ; s= or h : k = K or K' ; s = or 16 states Degeneracy lifted by short range (q~1/a) interactions: K K V(q=K) K K K K+G V(q=G) K+G K Effective D Contact Interaction: HC = dru αβγδ abcdψaα( r) ψbβ( r) ψc γ( r) ψdδ( r) U ~ H C e a ea ~ 4 π Rξ ξ ~ exciton size ~ πr

Exciton Eigenstates: Classify by momentum, spin, parity under C rotation q=± K ; S = 0 ; ~30 mev (R~.5nm) q= 0 ; S = 0 ; odd Optically Allowed q= 0 ; S = 0 ; even q= 0 ; S = 1 ; odd q=± K ; S = 1 ; Dark States q = 0 ; S = 1 ; even See also Zhao, Mazumdar PRL 04

Conclusion Fluorescence spectroscopy data for nearly armchair tubes is well described by a systematic large radius theory. D interactions: - q log q renormalization of graphene dispersion. - Non linear scaling with 1/R. - Explains ratio problem and blue shift problem. 1D interactions - Lead to large gap enhancement AND large exciton binding - Largely cancels in optical experiments revealing D effects. Short Range interactions -Lead to fine structure in exciton levels -Dark Ground State Experiments: measure single particle energy gap - Tunneling (complicated by screening) - Photoconductivity - Activated transport