Quantum transport in disordered graphene

1. for references to experiments: A.Geim and K.Novoselov Nature Mat. 6, 183 (2007) Quantum transport in disordered graphene K Kechedzhi, E McCann J Robinson, H Schomerus T Ando, B Altshuler V Falko

2. Material science versus seductive beauty of Dirac fermions: effect of different types of disorder (intra- and intervalley scattering) on quantum transport characteristics in graphene. Weak localisation or weak anti-localisation? Universal conductance fluctuations. UCF correlation function thermometry of graphene. __________________________________________________________ Metallic (high-density) regime pFl >>1

3. Interference correction: weak localisation effect… WL = enhanced backscattering in time-reversal-symmetric systems

4. Random (path-dependent) phase factor due to a magnetic field Interference correction: weak localisation effect… Broken time-reversal symmetry, e.g., due to a magnetic field B suppresses / kills the weak localisation effect WL magnetoresistance

5. Berry phaseπ conduction band valence band Chiral electrons: isospin direction of a plane wave is linked to the electron momentum

6. Electron ‘chirality’ has been seen directly in ARPES of graphene Mucha-Kruczynski, Tsyplyatyev Grishin, McCann, VF, Boswick Rotenberg - arXiv:0711.1129 ARPES of heavily doped graphene synthesized on silicon carbide Bostwick et al - Nature Physics, 3, 36 (2007)

7. WAL = suppressed backscattering for Berry phaseπelectrons WAL = suppressed backscattering for Berry phaseπelectrons • chiral electrons • chiral electrons ... but … WL = enhanced backscattering for non-chiral electrons in time-reversal-symmetric systems

8. WAL = suppressed backscattering for Berry phaseπelectrons • chiral electrons Suzuura, Ando - PRL 89, 266603 (2002) ... but … WL = enhanced backscattering for non-chiral electrons in time-reversal-symmetric systems

9. Some types of disorder lead to a similar effect. • chiral electrons McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006) for bilayers: Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007) ... however … weak trigonal warping leads to a random phase difference, δfor long paths.

10. time-inversion symmetry ... and, finally, … Inter-valley scattering restores the WL behaviour typical for electrons time-inversion symmetric systems McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006) for bilayers: Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)

11. … and, finally, some proper theory…. valley index sublattice index, ‘isospin’

12. atomic-range distortion of the latticebreakingA-B symmetry Intervalley-scattering disorder 4x4 matrix in the isospin-valley space Coulomb potential of remote charges in the substrate

13. Translation iso/pseudo-spin vectors realise a 4-dimensional representation of the symmetry group of the honeycomb lattice Generating elements:

14. Translation Rotation iso/pseudo-spin vectors realise a 4-dimensional representation of the symmetry group of the honeycomb lattice Generating elements:

15. Translation Rotation Mirror reflection iso/pseudo-spin vectors realise a 4-dimensional representation of the symmetry group of the honeycomb lattice Generating elements:

16. Symmetry operations and transformations of matrices 4-components wave-functions arrange a 4D irreducible representations of the lattice symmetry group. The 16D space of matrices can be separated into irreducible representations of the symmetry group G Generators of the groupG{T,C6v} :

17. Examples of convenient 4x4 matrices valley ‘pseudospin’ matrices: SU2 Lie algebra with: sublattice ‘isospin’ matrices: SU2 Lie algebra with:

18. Irreducible matrix representation of G{ T,C6v } four 1D-representations four 2D-representations one 4D-representation Σ(x,y)Λ(x,y)

19. Time inversion ofΣ, Λmatrixes: invariant time-inversion symmetry invert sign invariant Time-reversal

20. Full basis of symmetry-classified 4x4 matrices symmetric invert sign symmetric sublattice ‘isospin’ matrices: SU2 Lie algebra with: valley ‘pseudospin’ matrices: SU2 Lie algebra with: 16 generators of group U4

21. monolayer Hamiltonian in the ΣxΛ representation Dirac term warping term the most general form of time-reversal-symmetric disorder McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006)

22. Microscopic origin of various disorder terms Comes from potential of charged impurities in the substrate, deposits on its surface (water-ice) and doping molecules screened by electrons in graphene. It is believed to dominate in the momentum relaxation in the existing GraFETs.

23. Fictitious ‘magnetic field’: Foster, Ludwig - PRB 73, 155104 (2006) Morpurgo, Guinea - PRL97, 196804 (2006) Suppresses the interference of electrons in one valley, similarly to the warping effect in the band structure. Lattice deformation – bond disorder

24. Lattice deformation – bond disorder Foster, Ludwig - PRB 73, 155104 (2006) Morpurgo, Guinea - PRL97, 196804 (2006) The phase coherence of two electrons propagating in different valleys is not affected (real time-reversal symmetry is preserved).

25. intra-valley AB disorder A different energy on A and B sites opens a gap and thus suppresses chiralty of electrons. B Intra-valley disorder ΛzΣsus suppresses the interference of electrons in one valley, but has the opposite sing in the two valleys, K and K’, at the rate

26. Characterized by the intervalley scattering rate Inter-valley disorder Induced by deposits on the graphene sheet, points of mechanical contact with the substrate, atomic defects, and sample edges. valley-off-diagonal matrix:

27. Aleiner, Efetov - PRL 97, 236801 (2006) Renormalisation of effective disorder

28. Aleiner, Efetov - PRL 97, 236801 (2006) Ostrovsky, Gornyi, Mirlin, PRB 74, 235443 (2006) Foster, Aleiner, PRB 77, 195413 (2008)

29. Adsorbate-induced disorder in graphene (e.g. H on graphene) Robinson, Schomerus, Oroszlany, VF - arXiv:0808.2511

30. leading terms do not contain valley operators Λ, thus, they remain invariant with respect to valley transformationsSU2Λ. WL correction Particle-particle correlation function ‘Cooperon’ relaxation rate of the corresponding ‘Cooperon’

31. All types of symmetry breaking disorder inter-valley + intra-valley disorder Trigonal warping inter-valley disorder The only surviving mode Morpurgo and Guinea, PRL97, 196804 (2006) McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006) same valley inter-valley

32. Magnetoresistance of graphene for ‘fast’ inter-valley scattering: usual WL magnetoresistance cut at ‘slow’ inter-valley scattering: neither WL nor WAL

33. WL magnetoresistance is sample-dependent, due to a sample-dependent inter-valley scattering strength. Narrow ribbon of graphene has strong inter-valley scattering due to edges and is expected to show robust WL magnetoresistance H.B. Heersche et al, Nature 446, 56-59 (2007) S.V. Morozov et al, PRL 97, 016801 (2006) F. Tikhonenko et al PRL 100, 056802 (2008) McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006) Kechedzhi, McCann, VF, Altshuler, PRL 98, 176806 (2007)

34. Effect of chirality of electrons and different types of disorder (intra- and intervalley scattering) on quantum transport characteristics in graphene. Weak localisation versus weak anti-localisation. __________________________________________________________ Universal conductance fluctuations. UCF correlation function thermometry of graphene. Metallic (high-density) regime pFl >>1

35. Diffusion pole: same valley inter-valley Wide (2D) graphene sheet, 1<α<4 Narrow ribbon, α=1 2D: Kechedzhi, Kashuba, VF PRB 77, 193403 (2008) UCF in graphene Tikhonenko et.al. PRL (2008)

36. Heating by current T=? Bolotin, Sikes, Jiang, Fudenberg, Hone, Kim, and Stormer, cond-mat:08021367 Drude conductivity has a weak T-dependence S.V. Morozov et. al. (2007) Correlation thermometry function of graphene ribbons Poor thermal contact due to atomic mass difference Determining T from weak localisation or interaction correction to conductivity is obscured by ‘what symmetry class’ issue. Amplitude of UCF has complicated dependence on T, due to crossover between symmtry classes Alternative: correlation function spectroscopy of UCF.

37. Correlation thermometry function of graphene ribbons

38. Correlation thermometry function of graphene ribbons Width at half maximum of the correlation function of UCF Temperature from the correlation function cryostat temperature Correlation function of UCF can be used to measure temperature of electrons in graphene nanoribbon Kechedzhi, Horsell, Tikhonenko, Savchenko, Gorbachev, Lerner, VF - arXiv:0808.3211

39. Theory of graphene at Lancaster NP junction in graphene: focusing, caustics, Veselago lens for electrons. Cheianov, VF – PR B 74, 041403 (2006) Cheianov, VF, Altshuler - Science 315, 1252 (2007) Weak localisation and WL magneto-resistance in graphene, UCF. Friedel oscillations and RKKY interaction. Random resistor network model of minimal conductivity in graphene. McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler – PRL 97, 146805 (2006) Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007) Cheianov, VF – PRL 97, 226801 (2006) Cheianov, VF, Altshuler, Aleiner – PRL 99, 176801 (2007) Bilayer graphene: band structure, Berry phase 2π, Landau levels, QHE. McCann, VF - PRL 96, 086805 (2006); Abergel, VF - PR B 75, 155430 (2007) Novoselov, McCann, Morozov, VF, Katsnelson, Zeitler, Jiang, Schedin, Geim - Nature Physics 2, 177 (2006) Theory of graphene ARPES and optics (visibility of graphene flakes, magneto-phonon resonance). Mucha-Kruczynski, Tsyplyatyev, Grishin, McCann, VF, Boswick, Rotenberg - arXiv:0711.1129 Abergel, Russell, VF – APL 91, 063125 (2007) Goerbig, Fuchs, Kechedzhi, VF – PRL 99, 087402 (2007)