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).