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

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

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

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

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

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)

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

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, 266603 (2002)

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, 146805 (2006) for bilayers: Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)

... 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, 146805 (2006) for bilayers: Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)

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

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

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

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

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

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

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

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

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

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

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)

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.

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, 155104 (2006) Morpurgo, Guinea - PRL 97, 196804 (2006)

Lattice deformation – bond disorder Foster, Ludwig - PRB 73, 155104 (2006) Morpurgo, Guinea - PRL 97, 196804 (2006) The phase coherence of two electrons propagating in different valleys is not affected (real time-reversal symmetry is preserved).

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

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

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

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

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

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.

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

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

H.B. Heersche et al, Nature 446, 56-59 (2007) S.V. Morozov et al, PRL 97, 016801 (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, 056802 (2008) McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler, PRL 97, 146805 (2006) Kechedzhi, McCann, VF, Altshuler, PRL 98, 176806 (2007)

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

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, 193403 (2008) Narrow ribbon, α=1 2D: Wide (2D) graphene sheet, 1<α<4

Correlation thermometry function of graphene ribbons Bolotin, Sikes, Jiang, Fudenberg, Hone, Kim, and Stormer, cond-mat:08021367 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.

Correlation thermometry function of graphene ribbons

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

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)