Quasiparticle scattering and local density of states in graphene

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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

ca
b3
b2
b1
cb
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

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Scanning tunneling microscopy (STM) measurements

• Density of states as a function of energy and
  position ?(x,E)
• At each position ?(E)
• Fixed energy E, scan entire sample ? ?(x)
• Analyze ?(x) take Fourier transform (FTSTS) ?
  patterns

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Density of states in the absence of impurity
scattering

• Uniform in space
• Free Greens function
• Spectral function
• Density of states

TrG(k,E)
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Impurity scattering

• 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)

U0Ca (x) Ca (x) d(x)?U0Ca (k1) Ca (k2)
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T-matrix approximation
Greens function in imaginary time
T
G0(k2)
G0(k1)
T-matrix approximation
G(k1,k2)
TrIm
)
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T-matrix approximation
T
V
V
V
G0
For V independent of k
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Results FTSTS 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)

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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/r2 (V. Cheianov, V. Falko
• PRL 2006)
• (linearized band structure)
• 1/r2 (C. Bena to appear)
• (full band structure)
• Friedel oscillations in
• total charge depend
• on doping and have
• extra factor of 1/r

?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

V2.5eV
V8
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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.