1. 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.

2. 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

3. Graphene band structure  Tight binding Hamiltonian  Band structure b1b1 b3b3 b2b2 cc cc

4. Graphene band structure  Hexagonal Brillouin zone  Zero energy = corners of BZ  Higher energies = lines (circles, triangles, hexagons)  Fermi points → nodal quasiparticles with linear dispersion

5. 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

6. 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)}

7. 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

8. 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 )

9. 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[ )]}

10. T-matrix approximation TV V V G0G0G0G0 For V independent of k

11. 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)

12. 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

13. Impurity resonances  Average LDOS:  Also LDOS at impurity site, or on a neighboring site  Impurity → low energy resonance V=2.5eV V= ∞

14. 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.