Carbon nanotubes:

1
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. _________________________
_________________________________
3
Interference correction weak localisation effect
WL enhanced backscattering in
time-reversal-symmetric systems
4
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

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 - arXiv0711.1129
ARPES of heavily doped graphene synthesized on
silicon carbide Bostwick et al - Nature Physics,
3, 36 (2007)
7
... but
WL enhanced backscattering for non-chiral
electrons in time-reversal-symmetric systems
8
... but
WL enhanced backscattering for non-chiral
electrons in time-reversal-symmetric systems
9
... however
weak trigonal warping leads to a random phase
difference, d for long paths.
10
... 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
4x4 matrix in the isospin-valley space
Coulomb potential of remote charges in the
substrate
13
iso/pseudo-spin vectors realise a
4-dimensional representation of the
symmetry group of the honeycomb lattice
Generating elements
14
iso/pseudo-spin vectors realise a
4-dimensional representation of the
symmetry group of the honeycomb lattice
Generating elements
15
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
Generators of the group GT,C6v
17
Examples of convenient 4x4 matrices
sublattice isospin matrices
SU2 Lie algebra with
18
Irreducible matrix representation of G T,C6v
one 4D-representation S (x,y) ? (x,y)
19
Time-reversal
20
Full basis of symmetry-classified 4x4 matrices
sublattice isospin matrices
SU2 Lie algebra with
valley pseudospin matrices
SU2 Lie algebra with
16 generators of group U4
21
monolayer Hamiltonian in the Sx? 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
Lattice deformation bond disorder
24
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).
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 ? zSsus 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
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 -
arXiv0808.2511
30
(No Transcript)
31
All types of symmetry breaking disorder
inter-valley
same valley
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, 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.
35
UCF in graphene
Tikhonenko et.al. PRL (2008)
36
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 - arXiv0808.3211
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 2p, 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 -
arXiv0711.1129 Abergel, Russell, VF APL 91,
063125 (2007) Goerbig, Fuchs, Kechedzhi, VF
PRL 99, 087402 (2007)