Elastic scattering in graphene and consequences on transport

1
Elastic scattering in graphene and consequences
on transport

• Tero Heikkilä
• Theory group
• Low Temperature Laboratory
• Helsinki University of Technology

2
Motivation
A. Geim K. Novoselov, Nature Materials 6, 183
(2007)
K.S. Novoselov, et al., Nature 438, 197 (2005)
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Contents

• Basics sublattice (pseudospin) and reciprocal
  lattice (valley) indices, Dirac equation
• Conductivity in the clean limit (exists!)
• Different types of scattering
• Predictions away from half-filling
• Predictions at half-filling (Dirac point)

4
Dirac equation rough derivation
Graphene lattice in real space
... And in reciprocal space
Tight-binding Hamiltonian connecting nearest
neighbours
(courtesy of Matti Laakso)
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Dirac equation rough derivation
Define
Matrix form (in sublattice space)
(courtesy of Matti Laakso)
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Dirac equation rough derivation
Conduction band
K
K
Valence band
Dirac point
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Conductivity in conventional two-dimensional
conductors
electron density
scattering time
mobility
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Ballistic graphene
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Pseudodiffusion
Twordzydlo, et al., PRL (06)

• Ballistic graphene strip between reservoirs
• Solve by wave-function matching
• Reservoirs
• Transmission per channel at the Dirac point
  remains finite!
• Elsewhere the transmission tends towards unity

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Pseudodiffusion
Twordzydlo, et al., PRL (06)
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Pseudodiffusion
Twordzydlo, et al., PRL (06)
\Rightarrow similar diffusive-like behavior
also in supercurrent, Titov Beenakker (06)
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Ballistic limit

• Ballistic limit predictions
• Universal conductivity _at_ Dirac point
• away from the Dirac point
• What about disorder?

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Types of disorder
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Types of disorder
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Types of disorder
P. M. Ostrovsky, I. V. Gornyi A. D. Mirlin,
Phys. Rev. B 74, 235443 (2006)
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Types of disorder examples
P. M. Ostrovsky, I. V. Gornyi A. D. Mirlin,
Phys. Rev. B 74, 235443 (2006)
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Types of disorder examples
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Disorder and transport

• First away from half-filling
• Self-consistent Born (weak s-wave scattering)
• Unitary scatterers(strong scatterers)

P. M. Ostrovsky, I. V. Gornyi A. D. Mirlin,
Phys. Rev. B 74, 235443 (2006)
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Disorder and transport
Away from half-filling, the type of disorder is
not very important ( prefactors)
(short-range scattering x 2)
P. M. Ostrovsky, I. V. Gornyi A. D. Mirlin,
Phys. Rev. B 74, 235443 (2006)
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Weak (anti)localization?

• Weak localization arises due to enhanced
  backscattering
• Magnetic field suppresses the effect, resulting
  typically to a negative magnetoresistance
• (Pseudo)spin-active scattering may also lead to
  weak antilocalization, i.e., positive
  magnetoresistance

R. Tarkiainen, et al., (04)
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Weak (anti)localization?
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Weak (anti)localization?

• Experiments
• S. V. Morozov, et al., PRL 97 016801 (2006) no
  localization
• X. Wu, et al., PRL 98, 136801 (2007)
  antilocalization
• H. B. Heersche, et al., Nature 446, 56 (2007)
  sample-dep. small weak localization

B.
A.
single layer _at_Dirac point
single layer, finite gate
C.
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Disorder and transport

• Now _at_ Dirac point?
• Localization effects need to be considered!
• However, it seems that the conductivity _at_Dirac
  point does not suffer from localization

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Disorder and transport
_at_ the Dirac point, the value of conductivity
depends on the type of scattering
P. M. Ostrovsky, I. V. Gornyi A. D. Mirlin,
Phys. Rev. B 74, 235443 (2006)
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Disorder and transport
_at_ the Dirac point, the value of conductivity
depends on the type of scattering
Long-range scattering from charged impurities
and/or ripples
P. M. Ostrovsky, I. V. Gornyi A. D. Mirlin, PRL
98, 256801 (2007)
But according to S. Adam, E. H. Hwang, V. M.
Galitski, S. Das Sarma arXiv0705.1540
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Conclusions

• Peculiar electronic dynamics in graphene with new
  degrees of freedom allows in principle for many
  types of elastic scattering
• Recent experiments long-range scattering from
  Coulomb impurities is the dominant scattering
  mechanism

S. Adam, E. H. Hwang, V. M. Galitski, S. Das
Sarma arXiv0705.1540
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Our work (starting)
TTH, Matti Laakso and Pauli Virtanen

• Transport calculations in small structures aim
  to find generic features pseudospin-dependent
  interface scattering and its effect on ballistic
  behavior and resonant tunnelling
• Noninteracting model, able to calculate
  I(V)-curves, noise, full counting statistics,
  supercurrent...
• Supported by tight-binding numerics

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Our work (starting)
TTH, Matti Laakso and Pauli Virtanen
Fano factor
(Picture adapted from J. P. Robinson H.
Schomerus, arXiv0706.1506)