Coulomb Drag in Graphene

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Coulomb Drag in Graphene
Antti-Pekka Jauho1,2 1MIC-Department of Micro and
Nanotechnology, DTU 2Laboratory of Physics, TKK
Two electronic systems in close proximity, yet
coupled via Coulomb interaction can exhibit the
phenomenon of Coulomb drag momentum transfer
from an active, current-carrying system to the
passive, open-circuit system, where a voltage is
induced. This is a remarkable transport
experiment where the entire signal is determined
by the effective Coulomb interaction between the
subsystems. In this talk I review the common
knowledge of Coulomb drag, and introduce a recent
preprint by Tse, Hu, and Das Sarma
(arXiv0704.3598), who report the first reliable
calculation of Coulomb drag in graphene double
layers.
Espoo, June 6, 2007
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First experiments on Coulomb drag a double
quantum well system Gramila et al. Phys. Rev.
Lett 66, 1216 (1991)

• Technological challenge how to couple
  independently two quantum wells which are only
  100 Å apart? Solution use multiple depletion
  gates, and sample thinning.
• Important result of first measurements the drag
  rate does not behave as 1/? ? T2, as one would
  expect from a phase-space argument. This implies
  that a detailed microscopic theory needs to be
  developed.

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Theory of Coulomb drag linear response
formulation
Following exactly the same steps as when deriving
the Kubo formula for electrical conductivity, one
can derive an expression for the
transconductivity, which gives the
transresisitivity (via matrix inversion), and
thereby the momentum transfer rate (??m/(ne2?) in
Drude spirit)
where the the indices (i,j) refer to subsystems,
and (??) to Cartesian coordinates. The
subsystems couple via the Coulomb interaction
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Matsubara formalism
where all operators are in interaction picture,
and
As usual, the denominator can be canceled, and
one only retains the topologically distinct
connected diagrams, each with unit weight.
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As an example, consider the 2nd order expansion
Triangle function
In the hatched region anything can happen.
This defines a very general starting point for
microscopic calculations of drag. Higher order
expansion leads to a screened U12.
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Effective interlayer interaction in the
RPA. Diagrammatic language thick dashed line
screened interlayer interaction, thick straight
line screened intralayer interaction, wavy
lines unscreened intra/interlayer interactions
Ui1/q U12exp(-qd)/q
The algebraic equations can be solved, to yield
the RPA dielectric function for the double-layer
system.
The zeroes of Re ?(q,?) yield the collective
excitations of the double layer system.
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By evaluating the Matsubara sums by standard
techniques one finds
Translationally invariant systems (a periodic
system requires Gs)
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Feynman diagrams (here for electron-impurity
scattering)
Standard approximations are available for the
charge and current vertices (ladder approxmation,
or WL approximation).
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Boltzmann limit
where
(See, e.g., Mahans book.)
F transport polarization
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Finally, putting everything together, the
standard result for Coulomb drag emerges (APJ
H. Smith PRB 47, 4420 (1993) Zheng and MacDonald
PRB 48, 8203(1993) these early papers used
either BE or MF)
Often one can take

• Thus, in essence, the calculation of drag is
  reduced to the determination of the screening, or
  polarization, or Lindhard function, of the
  individual subsystems. Hundreds of papers have
  used the above expression as a starting point,
  and its predictions for experiments have been
  tested thoroughly, usually with good results.
  Some of its main consequences are
• drag rate goes as d-4 for large layer
  separations, and as T2 for low temperatures.
  However, the original experiment of Gramila did
  not obey these rules! Resolution one needs to
  include phonon mediated drag. Modification V ?
  VM, where M is an effective phonon-mediated
  interaction between the layers. After this, good
  agreement is found.

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• (Consequences contd)
• This equation predicts a plasmon enhancement of
  drag at elevated temperatures (Flensberg and Hu,
  PRL 73, 3572 (1994), experiment Hill et al. PRL
  78, 2204 (1997)) Why do plasmons enhance the
  drag? If dielectric function vanishes, the
  effective interaction becomes large. To
  contribute, the enhancement must happen in the
  accessible phase-space therefore need elevated
  temperatures

Original prediction
Experiment with detailed modeling (beyond RPA, no
phonons)
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• (Consequences contd)
• Drag in quantizing magnetic fields. Good
  quality data exists, mainly from von Klitzings
  group. Data displayed some surprising features,
  including negative drag (which is very hard to
  understand phenomenologically). After ten years
  work the theory is now in decent shape, but it
  required some hard work by skillful theorists
  (Mirlin, Gornyi, von Oppen, Stern....). Main
  difficulty when is the transport polarization
  expressible in terms of a Lindhard function?

• Mesoscopic effects in drag (Price et al, Science
  316, 8203 (2007), see also introduction by Lerner
  in the same issue). The fluctuations are huge,
  i.e. they are larger than the mean value. These
  authors also present a qualitative theoretical
  model the effect. Simple theories predict
  qualitatively wrong results. A heroic
  experiment requires very low temperatures, and
  good noise resolution.

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Why study drag in graphene?

• Transport properties of graphene show unusal
  features, notably non-zero conductivity at zero
  bias voltage (which may not be universal, as
  early data seemed to suggest.
• Experiments made thus far have focused on
  single-particle properties therefore drag is
  highly relevant because of its direct dependence
  on many-body interactions
• Maybe the special dispersion law in graphene
  gives rise to some special effects?
• Graphene is an extremely clean system, it is
  very accurately strictly two-dimensional (no
  messy modeling of quantum wells is needed), and
  one can study drag at much shorter distances than
  is possible for 2D electron gas. Individual
  layer contacting does not seem to be a big
  problem.
• The drag can still be calculated from the
  standard expression, provided that (i) one uses a
  dielectric function appropriate for graphene, and
  (ii) the standard formula is generalized to
  account for the Berry phase structure of the
  graphene Hamiltonian.

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The drag is now calculated from
where the Gammas generalize the F-functions found
above, and are given by

• The differences are
• the impurity dressed current vertices are no
  longer just proportional to the momentum
• both intra and interband processes must be
  accounted for (lambda labels the conduction and
  valence bands)
• the 1cos? factor comes from the two
  inequivalent K-points in the Brillouin zone
  (Berry phase factor)

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• Some consequences
• If at least one of the graphene layers is
  undoped (or, equivalently, intrinsic, ???), the
  drag vanishes. Is this trivial? No because
  graphene has finite conductivity at zero doping,
  and therefore there is no a priori reason for the
  drag to vanish. However, both physical and
  mathematical arguments can be given to show that
  this is indeed the case (essentially, these are
  based on the symmetry around the Dirac point)
• The authors give explicit expressions for both
  intra and interband Gammas

These expressions result from pretty tough
integrations. They are valid at T0, and can be
used to extract numerical results, and
approximate analytical results.
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• Conclusions (contd)
• The results are quite insensitive to the nature
  of the impurity scattering (this could be
  intrinsic within the graphene sheet, or charged
  impurities in the substrate) as long as momentum
  dependence of the impurity transport relaxation
  time is weak
• In the limit of low T and large d, an
  approximate analytic formula can be derived

which is the same as found for the
two-dimensional electron gas!!
(disappointing.....)
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Scaled transresisitivity as a function of
temperature. Numerical results are in black,
analytic results in red. As distance is
increased, results agree better, as expected.
More calculations, now for varying density at
fixed separation, and for fixed density, for
varying separation.
These results are, in a sense, very
disappointing. Coulomb drag in double-layer
graphene seems to behave in an extremely
conservative way.
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How about plasmon enhancement? The authors
present some qualitative arguments here. Their
conclusion is that intraband excitations are
never relevant, because they stay out of the
relevant phase-space (here the situation differs
from 2DEG the difference can be traced to the
details of the Lindhard function). However, it
seems that interband excitation has some hope,
and perhaps can be probed experimentally. No
quantitative results are given. This requires a
tough calculation, because one needs finite
temperature Lindhard functions, and transport
polarization. An obvious project for a graduate
student/post-doc who does not shun hard
work.. How about phonons? This is still virgin
land. The formulas are there, but even in the
simpler case of two-dimensional electron gas
embedded in GaAs the required work was
formidable. A detailed understanding of the
phonon spectrum is necessary. Phonon damping
must be understood very well (this was a major
stumbling block in the early theories of
phonon-drag in GaAs). How about IQHE? To my
knowledge, no work reported yet. Could be
interesting.
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