Quasiparticle scattering and local Quasiparticle scattering and local density of states in graphenedensity of states in graphene

Cristina Bena (SPhT, CEA-Saclay)Cristina Bena (SPhT, CEA-Saclay)

with Steve Kivelson (Stanford)with Steve Kivelson (Stanford)

C. Bena et S. Kivelson, Phys. Rev. B C. Bena et S. Kivelson, Phys. Rev. B 7272, 125432 (2005), , 125432 (2005), cond-mat/0408328.cond-mat/0408328.C. Bena, to appear.C. Bena, to appear.

OutlineOutline

Graphene band structureGraphene band structure

Local density of states (Local density of states (LDOSLDOS) and ) and Fourier transform scanning Fourier transform scanning tunneling spectroscopy (tunneling spectroscopy (FTSTSFTSTS))

Intuitive arguments for Intuitive arguments for FTSTSFTSTS

T-matrixT-matrix calculation for the calculation for the LDOSLDOS and and FTSTSFTSTS spectra spectra

Graphene band structureGraphene band structure Tight binding HamiltonianTight binding Hamiltonian

Band structureBand structure

b1

b3 b2

c

c

Graphene band structureGraphene band structure

Hexagonal Brillouin Hexagonal Brillouin zonezone

Zero energy = corners Zero energy = corners of of BZBZ

Higher energies = lines Higher energies = lines (circles, triangles, (circles, triangles, hexagons)hexagons)

Fermi points → Fermi points → nodal nodal quasiparticles quasiparticles withwith linear dispersionlinear dispersion

-202

-2 0 2

0

1

2

3

-202

0

1

2

3

Scanning tunneling microscopy Scanning tunneling microscopy (STM) measurements(STM) measurements

Density of statesDensity of states as as a function of energy a function of energy and position: and position: ρρ(x,E)(x,E)

At each position: At each position: ρρ(E)(E)

Fixed energy E, Fixed energy E, scan entire sample scan entire sample →→ ρρ(x)(x)

Analyze Analyze ρρ(x):(x): take take Fourier transform Fourier transform (FTSTS)(FTSTS) →→ patternspatterns

Density of states in the absence of Density of states in the absence of impurity scatteringimpurity scattering

Uniform in spaceUniform in space Free Green’s function:Free Green’s function:

Spectral functionSpectral function

Density of statesDensity of states

{Tr[G(k,E)}

Impurity scatteringImpurity scattering

Intuitive picture:Intuitive picture:

Impurity generates Impurity generates scattering between scattering between quasiparticles quasiparticles with same with same energyenergy

Corresponding Corresponding Friedel Friedel oscillations in the LDOSoscillations in the LDOS with with wavevectors given by wavevectors given by change in momenta of change in momenta of quasiparticlesquasiparticles

FTSTS spectra FTSTS spectra → → peakspeaks at at wavevectors corresponding wavevectors corresponding to scattering to scattering

Impurity scattering potentialImpurity scattering potential

Local (delta-function) in space Local (delta-function) in space → → uniform in uniform in momentummomentum

Single site scattering (Single site scattering (sublatticesublattice basisbasis))

Uniform interband (diagonal Uniform interband (diagonal sub-band basissub-band basis))

U0C(x) C(x) (x)→U0C(k1) C(k2)

T-matrix approximationT-matrix approximation

Green’s function Green’s function in imaginary timein imaginary time

T-matrix approximation GG00(k(k11)) GG00(k(k22))

TT

G(kG(k11,k,k22))

Tr{Im[ )]}

T-matrix approximationT-matrix approximation

TT VV VV VV

GG00

For For VV independent of independent of kk

Results: Results: FTSTS spectraFTSTS spectra

Low energy → high intensity points (scattering between corners of BZ)

Higher energy → high intensity lines

Shape of lines depends on energy (circles, triangles, hexagons)

Friedel oscillations (LDOS)Friedel oscillations (LDOS) Undoped grapheneUndoped graphene Oscillations in LDOS at a specific energyOscillations in LDOS at a specific energy Strongly dependentStrongly dependent on form of impurity scattering on form of impurity scattering 1/r1/r (C. Bena, S. Kivelson, (C. Bena, S. Kivelson,

PRB 2005)PRB 2005), ,

1/r1/r2 2 (V. Cheianov, V. Falko(V. Cheianov, V. FalkoPRL 2006) PRL 2006) (linearized band structure)(linearized band structure)1/r1/r2 2 (C. Bena to appear(C. Bena to appear) )

(full band structure)(full band structure)

Friedel oscillations in Friedel oscillations in total charge depend total charge depend on doping and have on doping and have extra factorextra factor of of 1/r1/r

15 20 25 30 35 40

0.2

0.4

0.6

0.8

1

4.6 4.8 5.2 5.4 5.6 5.8 6

12

13

14

15 ω=0.5 eV

Impurity resonancesImpurity resonances

Average LDOS:Average LDOS:

Also LDOS at Also LDOS at impurity site, or impurity site, or on a neighboring on a neighboring sitesite

ImpurityImpurity →→ low low energy resonanceenergy resonance

V=2.5eV

V=∞

Conclusions and future directionsConclusions and future directions

Lines of high intensityLines of high intensity in FTSTS in FTSTS spectra due to impurity scatteringspectra due to impurity scattering

STM measurementsSTM measurements on graphene could on graphene could reveal physics at reveal physics at all energiesall energies

Test Fermi liquid pictureTest Fermi liquid picture Other type of impurities (Coulomb) Other type of impurities (Coulomb)

may yield may yield different physics.different physics.