1. Quantum transport in disordered graphene
for references to experiments:
A.Geim and K.Novoselov
Nature Mat. 6, 183 (2007)
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. Interference correction: weak localisation effect…
Random (path-dependent) phase factor
due to a magnetic field
Broken time-reversal symmetry, e.g., due to a magnetic field B suppresses / kills the weak localisation effect
WL magnetoresistance

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

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

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

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

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

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

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

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

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

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

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

16. Symmetry operations and transformations of matrices
Generators of the group G{T,C6v} :
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

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

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-reversal invariant invert sign Time inversion of Σ, Λ matrixes:
symmetry
invert sign
invariant
Time inversion of Σ, Λ matrixes:

20. Full basis of symmetry-classified 4x4 matrices
sublattice ‘isospin’ matrices:
SU2 Lie algebra with:
valley ‘pseudospin’ matrices:
SU2 Lie algebra with:
invert sign
symmetric
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, (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. Lattice deformation – bond disorder
Fictitious ‘magnetic field’:
Suppresses the interference of electrons in one valley, similarly to the warping effect in the band structure.
Foster, Ludwig - PRB 73, (2006)
Morpurgo, Guinea - PRL 97, (2006)

24. Lattice deformation – bond disorder
Foster, Ludwig - PRB 73, (2006)
Morpurgo, Guinea - PRL 97, (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
different energy on A and B sites opens a gap and thus suppresses chiralty of electrons. B
Intra-valley disorder Λ zΣsus suppresses the interference of electrons in one valley, but has the opposite sing in the two valleys, K and K’, at the rate

26. 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:
Characterized by
the intervalley
scattering rate

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

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

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

30. WL correction relaxation rate of the corresponding ‘Cooperon’
Particle-particle correlation function ‘Cooperon’
relaxation rate of the corresponding ‘Cooperon’
leading terms do not contain valley operators Λ , thus, they remain invariant with respect to valley transformations SU2Λ.

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

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

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

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

36. Correlation thermometry function of graphene ribbons
Bolotin, Sikes, Jiang, Fudenberg, Hone, Kim, and Stormer, cond-mat:
Poor thermal contact due to atomic mass difference
Heating by current
T=?
Drude conductivity
has a weak T-dependence
S.V. Morozov et. al. (2007)
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:

39. Theory of graphene at Lancaster
NP junction in graphene: focusing, caustics, Veselago lens for electrons.
Cheianov, VF – PR B 74, (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, (2006)
Kechedzhi, McCann, VF, Altshuler – PRL 98, (2007)
Cheianov, VF – PRL 97, (2006)
Cheianov, VF, Altshuler, Aleiner – PRL 99, (2007)
Bilayer graphene: band structure, Berry phase 2π, Landau levels, QHE.
McCann, VF - PRL 96, (2006); Abergel, VF - PR B 75, (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:
Abergel, Russell, VF – APL 91, (2007)
Goerbig, Fuchs, Kechedzhi, VF – PRL 99, (2007)