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.