electronic properties and the quantum hall effect in bilayer graphene vladimir falko
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Electronic properties and the quantum Hall effect in
bilayer graphene
Vladimir Falko
Geim’s group at ManchesterNovoselov et al - Nature 438, 197 (2005)Novoselov et al - Nature Physics 2, 177 (2006)
Kim-Stormer group at Columbia University NYZhang et al - PRL 94, 176803 (2005)Zhang et al - Nature 438, 201 (2005)
Morpurgo’s group at TU-Delft S-Graphene-S Josephson effect transistor Conference ‘Graphene Week’, MPI-PKS Dresden (2006)
Ultra-thin graphitic films: from flakes to micro-devices
Novoselov et al - Science 306, 666 (2004)
Monolayer and bilayer graphene
Berry phase, degeneracy of the zero-energy Landau level, and the QHEMcCann, 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)
Relevance of Fermi surface warping and symmetry-breaking defects for weak localisation and WL magnetoresistance in graphene:
monolayer McCann, Kechedzhi, VF, Suzuura, Ando, Altshuler - PRL 97, 146805 (2006)
bilayer Kechedzhi, McCann, VF, Altshuler – PRL 98, 176806 (2007)
NP junctions: focusing, caustics and Veselago lens for electronsCheianov, VF - PR B 74, 041403 (2006)
Cheianov, VF, Altshuler - Science 315, 1252 (2007)
Random resistor network model for the minimal conductivity of graphene with inhomogeneous charge density
Cheianov, VF, Altshuler, Aleiner (2007)
Specifics of Friedel oscillations in monolayer graphene Cheianov, VF – PRL 97, 226801 (2006)
Berry phase, degeneracy of the zero-energy Landau level, and the QHE in bilayer graphene
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)
• Tight-binding-model analysis leading to ‘chiral’ electrons characterised by the Berry phase Jπ.
• Landau levels and quantum Hall effect in bilayer graphene.
• Trigonal warping in bilayer graphene.
• FIR magneto-optical properties of bilayer graphene.
- bonds
hybridisation forms strong directed bonds which determine a honeycomb lattice structure.
2sp
C
Carbon has 4 electrons in the outer s-p shell
)(zp orbitals determine conduction properties of graphite
0
Wallace, Phys. Rev. 71, 622 (1947)Slonczewski, Weiss, Phys. Rev. 109, 272 (1958)
0ie 3/2ie
3/2ie
11 p
ARPES: heavily doped graphene synthesized on silicon carbideA. Bostwick et al – Nature Physics, 3, 36 (2007)
Bilayer [Bernal (AB) stacking]
Bilayer [Bernal (AB) stacking]
Closest neighbour intra-layer hops
yx ipp
Bilayer [Bernal (AB) stacking]
Closest neighbour approximation (questions about the effect of the next-neighbour hops are welcome!)
ARPES: heavily doped bilayer graphene synthesized on silicon carbideT. Ohta et al – Science 313, 951 (2006)(Rotenberg’s group at Berkeley NL)
McCann, VFPRL 96, 086805
(2006)
Fermi level in undoped bilayer graphene
eV4.01
yx ipp
McCann, VFPRL 96, 086805
(2006)
emm 05.0~
yx ipp
)( pn
Berry phase Jπ(for a monolayer π
for a bilayer 2π )
iJi
Jee 32
2
Degree of chirality J
Monolayer grapheneBilayer graphene
Tight-binding-model analysis:
‘chiral’ electrons and the Berry phase Jπ.
Landau levels and quantum Hall effect in bilayer and monolayer graphene.
Effect of trigonal warping
Infra-red and FIR magneto-optics in graphene.
2D Landau levels
semiconductor QW / heterostructure
(GaAs/AlGaAs)
xy ( ) )2
h
ge
-1
-2 -1-3
2 31
-3
-2
3
1
2
integer QHEin semiconductors
eB
ghn
a
cnmm
pH
)(42 2
12
yxyx
zce
ippipp
lBArotAip
;
,
nnnB
11
0 0
energies / wave functions
)(0 r
1
2
yxyx
zce
ippipp
lBArotAip
;
,
Landau levels and QHE
0... 10 JJJ
00
,...,0
10
J
2D Landau levels of chiral electrons
J=1 monolayerJ=2 bilayer
A
B
B
A
J
J
J
J
~
~
0
0
0
0
valleyindex
also, two-fold real spin degeneracy
4J-degenerate zero-energy Landau level
McClure, Phys. Rev. 104, 666 (1956)
Haldane, Phys.Rev.Lett. 61, 2015 (1988)
Zheng and Ando Phys. Rev. B 65, 245420 (2002)
McCann and VFPhys. Rev. Lett. 96, 086805 (2006)
4J-degenerate zero-energy Landau level for electrons with degree of chirality J
emm 05.0~
-2-4 40
n (1012 cm-2)
2
2
4
6
0
xx
(k
)
-2-4 40
n (1012 cm-2)
2
2
4
6
0
xx
(k
)
1L graphene 2L graphene
db
EE
pp
xy
(4e2
/h)
1
2
-1
-2
-4
0
-3
4 c
3 x
y (4e2
/h)
1
2
-1
-2
-4
0
-3
4 a3
Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene
K.Novoselov, E.McCann, S.Morozov, V.Fal’ko, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim Nature Physics 2, 177 (2006)
1
How robust is the degeneracy of Landau level in bilayer graphene?
010
3
Direct inter-layer A hops (warping term, Lifshitz trans.)
B~
10 McCann, VF - PRL 96, 086805 (2006)
Monolayer grapheneBilayer graphene
Tight-binding-model analysis:
‘chiral’ electrons and the Berry phase Jπ.
Landau levels and quantum Hall effect in bilayer and monolayer graphene.
Effect of trigonal warping
Infra-red and FIR magneto-optics in graphene.
1
Hops between A and via B~
BA~
Direct inter-layer hops between A and ,~B 1.0~3
v
v
3
Inter-layer asymmetry (electric field across the structure, effect of a substrate/overlayer)
‘trigonal warping’ term
31 ~ mvpB
strong magnetic field
31 ~ mvpB
weak magnetic field
212104~ cm
KK
211
4
210~2 22
13 cmNvv
vL
21110~ cmNN L
*8NNNL
Lifshitz transition
iyx peipp
8-fold degeneratezero-energy Landau level
1
How robust is the degeneracy of Landau level in bilayer graphene?
010
3
Direct inter-layer A hops (warping term, Lifshitz trans.)
B~
10
Distant intra-layer AA,BB hops )3(210~~ 2
0
41
c || 01
McCann, VF - PRL 96, 086805 (2006)
dEz
Inter-layer asymmetry(substrate, gate)
|| 01
Spontaneous symmetry breaking due to e-e interactions
T. Ohta et al – Science 313, 951 (‘06)(Rotenberg’s group at Berkeley NL)
Interlayer asymmetry gap in bilayer graphene
McCann, VF - PRL 96, 086805 (2006)
inter-layer asymmetry gap
(controlled usingelectrostatic gate)
Lifting degeneracy of Landau levels in bilayer graphene
McCann, VF - PRL 96, 086805 (2006)McCann - cond-mat/0608221
0
0
0
0
02
2
p
p
m
longer than next neighbour in-plane AA and
BB hops (weak)
inter-layer asymmetry
(controlled usinggate voltage)
m2
1
310~~ 20
41
Monolayer grapheneBilayer graphene
Tight-binding-model analysis:
‘chiral’ electrons and the Berry phase Jπ.
Landau levels and quantum Hall effect in bilayer and monolayer graphene.
Effect of trigonal warping
Infra-red and FIR magneto-optics in graphene.
Abergel, VF - PR. B 75, 155430 (2007)
E
E
E
E
E
E
Infrared absorptions due to inter-LL transitions
σ + , Mz=+1 σ - , Mz=-1
Electronic Properties of Graphene-Based Nanostructures ICTP Trieste Italy, 25-29 August 2008
ESF Conference Graphene Week ‘08Obergurgl, Austria, XX March or YY April 2008
(if we and ESF agree on dates)
KK
pp
tt
0)(
0
0
02
2
1
vH
0
0
0
)(0
2
132
2
2
v
mH
Berry phase π
‘trigonal warping’valley symmetry of wave vector K is lower
than the hexagonal crystalline symmetry
Berry phase 2π
1
1
A
B
B
A
1
1
~
~
A
B
B
A
KK
pp
yx
yx
ipp
ipp
''''1 KKKKantisymmKKsymmKK CCCCg
''''2 KKKKantisymmKKsymmKK CCCCg
Weak localisation correction
may be suppressed
by the intervalley scattering
due to atomically
sharp scatterers
or edges
i
can be suppressed
only by decoherence
Berry phase π
killed bytrigonal warping
reflectingthe asymmetry
in each valley
Berry phase 2π
)()( pEpE
KK
pp
Berry phase π
‘slow’ inter-valley scattering:neither WL nor WAL
magnetoresistance
‘fast’ inter-valley scattering: usual WL magnetoresistance cut at
''''1 KKKKantisymmKKsymmKK CCCCg
''''2 KKKKantisymmKKsymmKK CCCCg
Weak localisation magnetoresistance
i
)0()( RBR
i
iB
B
D
B 0~
ii D
B0~
E. McCann, K.Kechedzhi, V.Fal'ko, B.Altshuler,
in preparation
E. McCann, K.Kechedzhi, V.Fal'ko, H.Suzuura,
T.Ando, B.Altshuler, cond-mat/0604015
S.V. Morozov et al, cond-mat/0603826(Manchester group)
Weak localisation magnetoresistance