Post on 22-Aug-2020
Electronic properties of graphene - II
Vladimir Falko
helped byE.McCann, V.Cheianov
K.Kechedzhi, D.Abergel, A.RussellT.Ando, B.Altshuler, I.Aleiner
Monolayer grapheneBilayer graphene
Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.
Effect of trigonal warping and the Lifshitz transition.
Landau levels and the quantum Hall effect in bilayer and monolayer graphene.
Interlayer asymmetry gap in bilayers.
Graphene optics.
Bilayer [Bernal (AB) stacking]
Bilayer [Bernal (AB) stacking]
In the vicinity of each of K points
Bilayer [Bernal (AB) stacking]
In the vicinity of each of K points
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 bilayergrapheneeV4.01 ≈γ
yx ipp +=π
McCann, VFPRL 96, 086805
(2006)
emm 05.0~
yx ipp +=π
)( pn rr
ψψψ πσπ 2222 3 i
i
ee =→×
prBerry phase 2π
(for a monolayer = π)
σππ
πϕ
ϕ rr⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
−
−−
+− n
eH m
pi
i
mp
m 22
2
22
2
21
2
22
00
0)(0ˆ
)2sin,2(cos)( ϕϕ=pn rr
Monolayer grapheneBilayer graphene
Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.
Effect of trigonal warping and the Lifshitz transition.
Landau levels and the quantum Hall effect in bilayer and monolayer graphene.
Interlayer asymmetry gap in bilayers.
Graphene optics.
1γ
Hops between A and via B~ BA~
Direct inter-layer hops between A and ,~B 1.0~3
vv
3γ
Inter-layer asymmetry (electric field across the structure, effect of a substrate/overlayer)
‘trigonal warping’
31 ~ mvpB >>−λ
strong magnetic field
31 ~ mvpB <−λ
weak magnetic field
212104~ −× cm
+K−K
( ) 2114
210~2 22
13 −= cmNvv
vL hπ
γ
21110~ −< cmNN L
*8NNNL <<
Lifshitz transition
Berry phase:2π = 3π - π
Summary of band structure:chiral electrons in monolayer and bilayer graphene
valley ‘trigonal warping’ terms
dominant at a high magnetic fieldand in high-density structures
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛=
+
+
0)(0
00
2
2
1 ππ
μπ
πςvH
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛= +
+
00
0)(0
21
32
2
2 ππ
ςπ
πv
mH
1
1
−=
+=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
ς
ς
AB
BA
1
1
~
~
−=
+=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
ς
ς
AB
B
A
Monolayer grapheneBilayer graphene
Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.
Effect of trigonal warping and the Lifshitz transition.
Landau levels and the quantum Hall effect in bilayer and monolayer graphene.
Interlayer asymmetry gap in bilayers.
Monolayer and bilayer graphene optics.
2D Landau levels
semiconductor QW / heterostructure
(GaAs/AlGaAs)
σxy ( ) )2
hge
-1
-2 -1-3
2 31
-3
-2
3
1
2
integer QHEin semiconductors
eBghn
a
cnmm
pH ωππππh
r
)(42 2
12
+⇒+
==++
yxyx
zce
ippipp
lBArotAip
−=+=
=−∇−=+ππ ;
,rrr
hr
nnnB ϕπϕ
πϕλ +++ =
=
11
0 0
energies / wave functions
+π
π
)(0 rrϕ
1ϕ
2ϕ
yxyx
zce
ippipp
lBArotAip
−=+=
=−∇−=+ππ ;
,rrr
hr
Landau levels and QHE
0... 10 === −JJJ ϕπϕπ
εψψ =
00
,...,0
10 =⇒⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ − εϕϕ J
2D Landau levels of chiral electrons
J=1 monolayerJ=2 bilayer
( )
( )( ) ⎟
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−++
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−− +
+
ABBA
J
J
J
J
~~
00
00
ππ
ππ
valleyindex
also, two-fold real spin degeneracy
4J-degenerate zero-energy Landau level
Monolayer, J=1 , Berry’s phase π
McClure, Phys. Rev. 104, 666 (1956)Haldane, Phys.Rev.Lett. 61, 2015 (1988)
Zheng, Ando Phys. Rev. B 65, 245420 (2002)
Bilayer, J=2, Berry’s phase 2π
8-fold degenerate ε=0 Landau level for electrons with degree of chirality J=2
McCann, VF - Phys. Rev. Lett. 96, 086805 (2006)
↑↓
εψψ =
emm 05.0~
B
vnλ
ε 2±=±
)1( −±=± nncωε hemm 05.0~
↑↓
1γ
Hops between A and via B~ BA~
Direct inter-layer hops between A and ,~B 1.0~3
vv
3γ
Effect of thetrigonalwarping
term
8-fold degeneratezero-energy Landau level
↑↓
↑↓
-2-4 40
n (1012 cm-2)2
2
4
6
0
ρ xx
(kΩ
)
-2-4 40
n (1012 cm-2)2
2
4
6
0
ρ xx
(kΩ
)
1L graphene 2L graphene
db
EE
pp
σ xy
(4e2
/h) 1
2
-1-2
-4
0
-3
4 c
3
σ xy
(4e2
/h) 1
2
-1-2
-4
0
-3
4 a3
Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer grapheneK.Novoselov, E.McCann, S.Morozov, VF, M.Katsnelson, U.Zeitler, D.Jiang, F.Schedin, A.Geim
Nature Physics 2, 177 (2006)
1γ
How robust is the degeneracy of Landau level in bilayer graphene?
010 == εε
3γ
Direct inter-layer A hops (warping term, Lifshitz trans.)
B~ 10 εε =
Distant intra-layer AA,BB hops )3(210~~ 2
0
41 −γγγδ
cωδεε h=− || 01
McCann, VF - PRL 96, 086805 (2006)
dEz=Δ
Inter-layer asymmetry(substrate, gate)
Δ=− || 01 εε
Monolayer grapheneBilayer graphene
Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.
Effect of trigonal warping and the Lifshitz transition.
Landau levels and the quantum Hall effect in bilayer and monolayer graphene.
Interlayer asymmetry gap in bilayers.
Graphene optics.
T. Ohta et al – Science 313, 951 (‘06)(Rotenberg’s group at Berkeley NL)
Interlayer asymmetry gap in bilayer grapheneMcCann, VF - PRL 96, 086805 (2006)
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ−
Δ+
ξξ0
0 inter-layer asymmetry gap
(controlled usingelectrostatic gate)
McCann, VF - PRL 96, 086805 (2006)McCann - cond-mat/0608221
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ−
Δ+
ξξ0
0 inter-layer asymmetry gap
(controlled usingelectrostatic gate)
Interlayer asymmetry gap in bilayer graphene
Band mini-gap in bilayer graphene can be controlled electrically, so that a bilayer graphene transistor can be driven into a
pinched-off (insulating) state.
Monolayer grapheneBilayer graphene
Band structure of bilayer graphene, ‘chiral’ electrons and Berry’s phase 2π.
Effect of trigonal warping and the Lifshitz transition.
Landau levels and the quantum Hall effect in bilayer and monolayer graphene.
Interlayer asymmetry gap in bilayers.
Graphene optics.
Why can one see graphene in an
optical microscope?Abergel, VF - PR B 75, 155430 (2007)
%52 2
<ceh
π
00 /)( RRRV −=
Abergel, Russell, VF - Appl. Phys. Lett. 91, 063125 (2007)
Graphene flakes are visible when the oxide layer in SiO2/Si wafer acts
as clearing optical film if
sN
s
21
2sin2 +
−=
αελ
visibility,
o
nms10
300=
=
δα
o
ms10
1=
=
δα
μ
Blake, Hill, Castro Neto, Novoselov, Jiang, Yang, Booth, Geim - Appl. Phys. Lett. 91, 063124 (2007)