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Quasiparticle scattering and local density of states in graphene Cristina Bena (SPhT, CEA-Saclay) with Steve Kivelson (Stanford) C. Bena et S. Kivelson, Phys. Rev. B 72, 125432 (2005), cond-mat/0408328. C. Bena, to appear.
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Outline Graphene band structure Local density of states (LDOS) and Fourier transform scanning tunneling spectroscopy (FTSTS) Intuitive arguments for FTSTS T-matrix calculation for the LDOS and FTSTS spectra
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Graphene band structure Tight binding Hamiltonian Band structure b1b1 b3b3 b2b2 cc cc
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Graphene band structure Hexagonal Brillouin zone Zero energy = corners of BZ Higher energies = lines (circles, triangles, hexagons) Fermi points → nodal quasiparticles with linear dispersion
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Scanning tunneling microscopy (STM) measurements Density of states as a function of energy and position: ρ(x,E) At each position: ρ(E) Fixed energy E, scan entire sample → ρ(x) Analyze ρ(x): take Fourier transform (FTSTS) → patterns
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Density of states in the absence of impurity scattering Uniform in space Free Green’s function: Spectral function Density of states {Tr[G(k,E)}
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Impurity scattering Intuitive picture: Intuitive picture: Impurity generates scattering between quasiparticles with same energy Corresponding Friedel oscillations in the LDOS with wavevectors given by change in momenta of quasiparticles FTSTS spectra → peaks at wavevectors corresponding to scattering
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Impurity scattering potential Local (delta-function) in space → uniform in momentum Single site scattering (sublattice basis) Uniform interband (diagonal sub-band basis) U 0 C (x) C (x) (x)→ U 0 C (k 1 ) C (k 2 )
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T-matrix approximation Green’s function in imaginary time T-matrix approximation G 0 (k 1 ) G 0 (k 2 ) T G(k 1,k 2 ) Tr{Im[ )]}
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T-matrix approximation TV V V G0G0G0G0 For V independent of k
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Results: FTSTS spectra Low energy → high intensity points (scattering between corners of BZ) Higher energy → h igh intensity lines Shape of lines depends on energy (circles, triangles, hexagons)
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Friedel oscillations (LDOS) Undoped graphene Oscillations in LDOS at a specific energy Strongly dependent on form of impurity scattering 1/r (C. Bena, S. Kivelson, PRB 2005), 1/r 2 (V. Cheianov, V. Falko PRL 2006) (linearized band structure) 1/r 2 (C. Bena to appear ) (full band structure) Friedel oscillations in total charge depend on doping and have extra factor of 1/r ω =0.5 eV
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Impurity resonances Average LDOS: Also LDOS at impurity site, or on a neighboring site Impurity → low energy resonance V=2.5eV V= ∞
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Conclusions and future directions Lines of high intensity in FTSTS spectra due to impurity scattering STM measurements on graphene could reveal physics at all energies Test Fermi liquid picture Other type of impurities (Coulomb) may yield different physics.
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