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Transport properties in graphene and graphene bilayer
Mikito Koshino (Tohoku Univerisity)
Tsuneya Ando (Tokyo Institute of Techonology)Kentaro Nomura (RIKEN)Shinsei Ryu (UC Berkley)
Acknowledgments
Graphenes
Graphene: single-layer graphite
Manchester’s web page
10 micron 20 micron
Manchester’s web page
Experiments:Novoselov et al.,Science 306, 666 (2004)Novoselov et al., Nature 438, 197 (2005)Zhang et al., Nature 438, 201 (2005)
EF
V(x)
n np
Electrons on graphene
v
v
v
-- Constant velocity -- Klein Tunneling
Relativistic 2D Dirac electronsK’K
K
K’
K’
px
py
E
px
py
E
K
Outlines
Tsuneya Ando (Tokyo Institute of Techonology)Kentaro Nomura (RIKEN)Shinsei Ryu (UC Berkley)
Acknowledgments
Electronic transport and localization effectof graphene / bilayer graphene
Graphene: “massless Dirac electron”
Bilayer graphene: “massive and gapless”
Band structure of graphene
K’K
K
K
K’
K’
A B
Velocity:
Effective Hamiltonian: “massless Dirac Fermion”
Atomic structure
0 = 3 eV
a = 0.246nm
McClure, Phys. Rev. 104, 666 (1956).
A, B: “pseudospin degree of freedom”
Band structure: gapless & linear
Dirac point
Zero-gap semiconductor or metal?
Conductivity of usual metal
D : Density of statesvF : Fermi velocity : Scattering time
Graphene
EF
EF
… Is conductivity zero or finite?
Dirac point:
--- Finite velocity
--- Zero density of states
Self-consistent Born approximation Shon and Ando, JPSJ, 67, 2421 (1998)
Self energy
Conductivity
Hamiltonian:
Disorder potential:
Self-consistent Born approximation:
1/ WConductivity of graphene(Short-range scatterers)
Shon and Ando, JPSJ, 67, 2421 (1998)
Fermi energy
Clean
Dirty
--- At Dirac point
independent of disorder
--- Off Dirac point
Cf. Long-range scatterersNoro, Koshino, Ando JPSJ, 79, 094713 (2011)
Self-consistent Born approximation
Bilayer graphene
Hamiltonian
Effective mass:
McCann and Fal’ko, PRL 96, 086805 (2006)
A1 B1 A2 B2 0 ~ 3 eV
Interlayer1 ~ 0.4 eV
0.334nm
Massive chiral particle(near E=0)
A1 B2
“AB-stacking”
“Massive, but still zero gap”
Conductivity of bilayer graphene
Conductivity 0.03
0.08
0.15
1
MK and Ando, PRB 73, 245403 (2006)MK, New J. Phys., 11, 095010 (2009)
( ~ 10.1k)
Experiment (suspended bilayer):Feldman et al, Nature Physics 5, 889 (2009)
Dirac point of bilayer:
--- Finite density of states
Self-consistent Born approximation
Conductivity at Dirac point:
--- Zero velocity
Absence of back-scattering
Direction of pseudo spin
Hamiltonian: V(r) : Disorder potential (scalar)
V(r) cannot flippseudo-spin
Cf. Absence of localization in metallic carbon nanotubeAndo and Nakanishi, JPSJ. 67, 1704 (1998)Ando, Nakanishi, and Saito, ibid. 67, 2857 (1998)
Localization in graphene
Conductivity (Thouless number)
SCBA
g (in
log
scal
e)
Conductivity increasesas L increases
L =6,8,10,12,14
Hamiltonian:
Fermi Energy
“Symplectic”
EF
Disorder potential
Beta function Nomura, Koshino, Ryu, PRL 99 (2007)
Hikami, Larkin, Nagaoka (1980)
2D (conventional)
Beta function(Kubo-formula conductivity)
symplectic
orthogonal
(g)
0
0
g
Scaling theory
Graphene(anti-localizaton)
System size
Conductance
Beta function
L L+dL
2DEG withspin-orbit coupling
(anti-localizaton)(localizaton)
How to tell localized / extended?
Extended state Localized state
Ene
rgy
Ene
rgy
0
0
E
Thouless number:
Conductivity:
energy sensitive toboundary condition
insensitive
How to tell a localized orextended?
Absense of localization
Graphene 2DEGwith SO-couplingKramers doublets
(time reversal symmetry)
localized phase
Localization prohibitted Localization allowed
Cf. Topological Insulator : Fu and Kane, PRB 76, 045302 (2007)
# of levels crossing at single energy= 1, 3, 5, ….
# of levels crossing at single energy= 0, 2, 4,….
Nomura, Koshino, Ryu, PRL 99 (2007)
Localization in bilayer graphene
SCBA
Monolayer
Koshino, PRB 78, 155411 (2008)
Bilayer
Conductivity increases as L increases Conductivity decays as L increases
SCBA
(localziation)(anti-localziation)
g (in
log
scal
e)
g (in
log
scal
e)
Cf. Weak localization of bilayer: Gorbachev et al, PRL 98, 176805 (2007)Kechedzhi et al, PRL 98, 176806 (2007)
Fermi EnergyFermi Energy
Pseudo-spin and back-scattering
absence ofback-scattering
presence ofback-scattering
anti-localization localization
Monolayer Bilayer
Gap opening in bilayer graphene
Theories:McCann, Phys. Rev. B 74, 161403(R) (2006).Castro et al., Phys. Rev. Lett. 99, 216802 (2007).Nilsson, et al., Phys. Rev. Lett. 98,126801 (2007).
Perpendicular E-fieldopens a band gap
Experiments:Oostinga et al., Nature Mater. 7, 151 (2008).Ohta, et al., Science 313, 951 (2006).
Bilayer graphene:
E
Back gateBilayer
graphene
E-field
Top gate
Single-valley Hall conductivity
--- Non-zero Hall conductivity induced in each single valleys(opposite between K and K’)
Koshino, PRB 78, 155411 (2008)
EFEF
Insulator “Quantum Hall state”
QuantumHall transition
Gapped bilayer graphene:
(related to Berry phase 2)
Quantum Hall transition
Extended Localized
Energy
1 2 3xy = 0
Usual quantum Hall (QH) systems:
Extended states exist between different QH phases
xy
Energy
Conductivity (EF= 0)
Analog of quantum Hall transition
Divergence oflocalization length
Koshino, PRB 78, 155411 (2008)
Gap width Localization lengthg
(in lo
g sc
ale)
Gap width
Single-vally Hall conductivity (EF= 0)
quantum Hall transition
xyK
/(e2 /h
)
2
L: small
large
Phase diagram
EF /
Gap
wid
th
Hall plateau diagramHall conductivity
EF /
A
B
C
D
A
BC
D
States extended only at boundary
Koshino, PRB 78, 155411 (2008)
--- Analog to QH transition controlled by E-field (no B-field)--- Observed as divergence of the localization length
Summary
Graphene (massless Dirac)
--- Metallic at Dirac point--- Absence of localization
(c.f., topological insulator)
Bilayer graphene(massive & gapless)
--- Presence of localization--- Gap opening induced delocalized states
(analog of QH transition)
Electronic transport and localization effectof graphene / bilayer graphene
Conductivity in smooth impurities
Exp: Conductivity measurementNovoselov et al., Nature 438, 197 (2005)
Conductivity at Dirac point
“Missing ”
Theory:(short-range)
Noro, Koshino, Ando JPSJ, 79, 094713 (2011)
Strength of disorder
Minimum conductivity is NOT universal Min
imum
con
duct
ivity long-range
short-range
d (potential range)
electron density
Localization and valley mixing
Suzuura, et al. PRL 89, 26660 (2002)
Disorder potential
Long-range⇒ K,K’ decoupled (Symplectic)
Short-range⇒ K,K’ mixed (Orthogonal)
K K’
short-range
long-range
time-reveral
ky
kx
E
Time reversal symmetry
ky
kx
E
Band structure of cabon nanotube (CNT)
a1
a2
LL
L
armchair
zigzag chiral
L = n1a1 + n2a2Chiral vector:
…. metallic…. semiconducting
metallic semiconducting
Localization of metallic carbon nanotube
EF
ConductanceNear EF = 0
Absense of back scattering
right-goingleft-going
Conductance never decays,constant atAndo and Nakanishi, JPSJ. 67, 1704 (1998);
Ando, Nakanishi, and Saito, ibid. 67, 2857 (1998).