SLIDE 1 Igor Gornyi
Karlsruhe Institute of Technology
Collaboration:
Mikhail Titov (Nijmegen) Boris Narozhny (Karlsruhe) Pavel Ostrovsky (Stuttgart) Michael Schütt (Karlsruhe) Alexander Mirlin (Karlsruhe)
- Phys. Rev. B 85, 195421 (2012)
+ arXiv:1205.5018
Coulomb drag in graphene
Chernogolovka, 14 September 2012
SLIDE 2
SLIDE 3
What is Coulomb drag?
Coulomb drag = response of the passive layer to a current in the active layer mediated by Coulomb interaction
SLIDE 4 Coulomb drag measurements
Gramila, Eisenstein, MacDonald et.al ., Phys. Rev. Lett. 66, 1216 (1991)
SLIDE 5
Theory: 2D
Pogrebinskii (1977) – introduced Coulomb drag Zheng, MacDonald (1993) – memory function Jauho, Smith (1993) – kinetic equation Kamenev, Oreg (1995) – diagrammatics Flensberg et al. (1995) – diagrammatics ...
SLIDE 6 Coulomb drag: why interesting?
- no drag without interaction:
probe of inter-electron correlations
- provides information about inelastic processes,
phase-coherent phenomena
- drag is related to particle-hole asymmetry
Drag in graphene near the Dirac point ?
SLIDE 7
Particle-hole asymmetry
Example: strong magnetic field (i) Curvature normal positive drag (ii) Landau levels DoS anomalous oscillatory drag IG, Mirlin, von Oppen (2004)
SLIDE 8
Drag in 2D: standard theory
SLIDE 9
Drag in graphene
SLIDE 10 Dirac spectrum at low energies electron-hole symmetry at the Dirac point linear spectrum – no Galilean invariance
–
non-trivial single-layer conductivity
small interlayer distance d
News in graphene
SLIDE 11 Drag in graphene: experiment
single-gate device
Kim, Jo, Nah, Yao, Banerjee, and Tutuc, Phys. Rev. B 83, 161401 (2011)
SLIDE 12 Drag in graphene: experiment
single-gate device
Kim, Jo, Nah, Yao, Banerjee, and Tutuc, Phys. Rev. B 83, 161401 (2011)
SLIDE 13 Drag in graphene: experiment
double-gate device
Tutuc and Kim, Solid State Comm. (2012)
SLIDE 14 Drag in graphene: experiment
double-gate device
Tutuc and Kim, Solid State Comm. (2012)
SLIDE 15 –
“clean” substrate and spacer – BN
–
smaller inter-layer spacing d = 1-10 nm
Drag in graphene: experiment
Gorbachev, Geim, Novoselov, Ponomarenko et al. ArXiv:1206.6626
SLIDE 16 double-gate device
Gorbachev, Geim, Novoselov, Ponomarenko et al. ArXiv:1206.6626
Drag in graphene: experiment
SLIDE 17 double-gate device
Drag in graphene: experiment
Gorbachev, Geim, Novoselov, Ponomarenko et al. ArXiv:1206.6626
SLIDE 18 double-gate device
Drag in graphene: experiment
Gorbachev, Geim, Novoselov, Ponomarenko et al. ArXiv:1206.6626
SLIDE 19
disordered graphene
SLIDE 20 Second-order perturbation theory
non-linear susceptibility (rectification function) screened interlayer interaction
Narozhny, Titov, IG, and Ostrovsky, PRB (2012)
SLIDE 21 arbitrary values of chemical potential controlled perturbation theory
–
weak interaction static screening is sufficient
–
dominant scattering mechanism is due to disorder
– qualitatively not important:
- plasmons
- spectrum renormalization
- energy dependence of disorder
scattering time
non-degenerate relativistic gas Fermi liquid
experimental condition
Drag in disordered graphene
SLIDE 22
SLIDE 23 comparison with experiment
courtesy of L. Ponomarenko
SLIDE 24
ultra-clean graphene
SLIDE 25
Graphene: no Galilean invariance, relativistic dynamics
SLIDE 26
Clean vs. disordered graphene
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Kinetic theory of the drag
Linearized kinetic equation:
SLIDE 28 Inelastic scattering in graphene
Linear spectrum:
Velocity is not equivalent to momentum: momentum conservation does not prevent current relaxation
- Finite transport rate due to inelastic e-e scattering
Collinear scattering singularity: momentum conservation = energy conservation
- Fast thermalization within a given direction
Kashuba '08, Fritz, Müller, Schmalian, Sachdev '08
SLIDE 29
Collinear scattering singularity
SLIDE 30
Double-layer graphene
Only two modes (velocity and momentum) in each layer: Fast unidirectional thermalization between layers: Kinetic (integral) equation reduces to a 3x3 matrix equation! Hydrodynamics: total momentum + particle currents
SLIDE 31
Hydrodynamic equations
SLIDE 32 Scattering rates: Golden Rule
close to the Dirac point away from the Dirac point
SLIDE 33
Drag resistivity
Equal layers: Non-equal layers near the Dirac point
Finite drag at the Dirac point in the clean case!
SLIDE 34
neutrality point
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Drag rate: beyond Golden Rule
Exactly at the Dirac point: finite third-order drag
SLIDE 36
Diffusive regime
Conventional (second-order) drag vanishes at the Dirac point Third-order (Levchenko & Kamenev 2008) drag dominates
SLIDE 37 Correlated disorder
Ballistic regime: Diffusive regime: interlayer Cooper mode (IG, Yashenkin, Khveshchenko '99)
Correlated elastic scattering in the two layers with common impurities
SLIDE 38 Correlated disorder
Correlated elastic scattering in the two layers with common impurities Moderately correlated disorder:
SLIDE 39
2nd vs. 3rd order drag
SLIDE 40
Drag at the Dirac point
SLIDE 41
SLIDE 42 Magnetodrag
Gorbachev, Geim, Novoselov, Ponomarenko et al. ArXiv:1206.6626
SLIDE 43 Summary
Coulomb drag in graphene:
- Perturbation theory
- Kinetic theory (3 mode hydrodynamics)
- Clean graphene (equilibrated drag)
- Peak at the Dirac point
3rd order drag, drag with correlated disorder
