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 (arXiv:0704.3598), who report the first reliable calculation of Coulomb drag in graphene double layers.

Espoo, June 6, 2007

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.

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:

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.

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.

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: Ui=1/q; U12=exp(-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.

By evaluating the Matsubara sums by standard techniques one finds

Translationally invariant systems (a periodic system requires G’s):

Feynman diagrams (here for electron-impurity scattering)

Standard approximations are available for the charge and current vertices (ladder approxmation, or WL approximation).

Boltzmann limit:

where

(See, e.g., Mahan’s book.)

F: ”transport polarization”

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 V+M, where M is an effective phonon-mediated interaction between the layers. After this, good agreement is found.

(Consequences – cont’d)

• 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)

(Consequences – cont’d)

• Drag in quantizing magnetic fields. Good quality data exists, mainly from von Klitzing’s 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.

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.

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 1±cos factor comes from the two inequivalent K-points in the Brillouin zone (Berry phase factor)

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 T=0, and can be used to extract numerical results, and approximate analytical results.

Conclusions (cont’d)

• 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.....)

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.

These results are, in a sense, very disappointing. Coulomb drag in double-layer graphene seems to behave in an extremely conservative way.

More calculations, now for varying density at fixed separation, and for fixed density, for varying separation.

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.