P. Ostrovsky et al- Anomalous localization and quantum Hall effect in disordered graphene

8/3/2019 P. Ostrovsky et al- Anomalous localization and quantum Hall effect in disordered graphene

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IntroductionAnomalous Quantum Hall effect

Absence of localization at B = 0Ballistic transport

Anomalous localization and quantum Hall effect

in disordered graphene

P. Ostrovsky1; 2 A. Schssler2 I. Gornyi2 ; 3 A. Mirlin2; 4 ; 5

1Landau ITP, Chernogolovka 2Forschungszentrum Karlsruhe

3Ioffe Institute, St.Petersburg 4Universitt Karlsruhe 5PNPI, St.Petersburg

Landau-100, Chernogolovka, 26 June 2008

Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene


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Outline

1 IntroductionExperimental facts

Model

2 Anomalous Quantum Hall effect

Odd quantization

Ordinary quantization

Absence of quantization

3 Absence of localization at B a 0

Unitary class

Symplectic class4 Ballistic transport

Clean system

Disordered system

Single parameter scaling



3/31



Experimental factsModel

Outline


Model


Odd quantization

Ordinary quantization

Absence of quantization


Unitary class


Clean system

Disordered system




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Graphene samples

Suspended sample Hall bar

Micro-mechanical cleavage Epitaxial growth


I d i


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Experiments on conductivityDensity dependence

Novoselov, Geim et al. 08 Zhang, Tan, Stormer, Kim 07

Conductivity is linear in density:

long-range Coulomb impurities

corrugations (ripples)


Introd ction


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Experiments on conductivityMinimal conductivity

Novoselov, Geim et al. 05 Zhang, Tan, Stormer, Kim 07

Minimal conductivity

of order e2=

h

temperature independenta

A no localization!


Introduction


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Experiments on QHE

Novoselov, Geim et al. 05 Novoselov, Geim, Stormer, Kim 07

Anomalous quantum Hall effect

only odd plateaus: xy a @ 2n C 1A 2e

2=

h

QHE transition at zero concentration

visible up to room temperature!


Introduction


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Clean graphene model

(a) (b)

2.46 A

m

k0K

K

K

K

K K

Tight-binding approximation

two sublattices: A, B

two valleys: K, KH

linear dispersion:" a v0 j pj

massless Dirac Hamiltonian:

K: Ha

v0 p KH : H

a v0

Tp

af

x ; yg

velocity: v0 % 108 cm/s

band width:

$ 1 eV


Introduction


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Disorder model

valleys decouple for long-range disorder

Dirac equation with disorder:

iv0 r C V@ x; yA a

two-component wave function a

f A ; Bg

Va

V

random field (with structure in sublattices)

Types of disorder

0 a1: random potential (charged impurities)

x, y: random vector potential (ripples)

z: random mass[Ludwig et al. 94; Nersesyan, Tsvelik, Wenger 94]


IntroductionOdd nti tion


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Anomalous Quantum Hall effectAbsence of localization at B = 0

Ballistic transport

Odd quantizationOrdinary quantizationAbsence of quantization

Outline


Model


Odd quantization

Ordinary quantizationAbsence of quantization


Unitary class


Clean system

Disordered system



IntroductionOdd quantization


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Ballistic transport


Decoupled valleys: paradox?

Conventional field theory [Pruisken 84, Khmelnitskii 84]

0 U

0.6

xx

n

n1

2

n1

xy

2 valleys 2 spina

A

1 0 1

3

2

1

0

1

2

3

2e

2

h

Experiment

1 0 1

3

2

1

0

1

2

3

2e

2

h

Why odd plateaus?

What is the RG flow?

When may this happen?

What are other options?


IntroductionOdd quantization


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Ballistic transport


Single valley conductivities

xx a 1

2Tr

jx @ GR

GA A jx @ GR

GA A

(bulk)

Ixy a

1

2Tr

jx @ GR

GAAjy@ G

RC

GAA

(bulk)

II

xya

ie

2Tr

@ xjy

yjx

A @ GR GA A

(edge)Boundary conditions important!

Single valleya

A infinite mass boundary condition

Ha

v0 p C m z ; m 3 I at the edge

Hall conductivity: 2 xy a

xy C1

2

| { z }

valley Kappears in -model

C

xy 1

2

| { z }

valley K 0


IntroductionA l Q H ll ff

Odd quantization


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Ballistic transport


Effective field theory: -model

Single valley (unitary -model with topological term a 2 xy C ):

S Q a1

4Str

xx

2@

r QA 2 C

xy C1

2

Qr xQr yQ

!

Weakly mixed valleys:

SQK ; QK0 a S QK C S QK0 C

mix

StrQKQK0

0

gU

2gU

xx

2e

2

h

2k1 2k 2k1

xy2e2h

1 0 1

n

3

2

1

0

1

2

3

xy

2e

2

h

c 0 c

TmixTmix

n2 0 n2

0

gU

2gU

1

0

1

n

xx

2e

2

h

xy

2e

2

h

Even plateau width $ @ = mix A

0 : 45, visible at T


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Ballistic transport

qOrdinary quantizationAbsence of quantization

Chiral disorder: Classical quantum Hall effect

Ripples D Abelian random vector potentialDislocations D non-Abelian random vector potential

AtiyahSinger theorem: Zero Landau level remains degenerate

aA no localization Aharonov, Casher 79

1 0 1

n

3

2

1

0

1

2

3

xy

2e

2

h

c 0 c

Ripples: odd plateaus

RipplesC

Dislocations: all non-zero plateaus


IntroductionAnomalous Quantum Hall effect Unitary class


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Ballistic transport

Unitary classSymplectic class

Outline

1

IntroductionExperimental facts

Model


Odd quantization



Unitary class


Clean system

Disordered system





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Ballistic transport


Unitary class

Generic single-valley disorder (e.g. charged impurities + ripples), B a 0a

A effective time-reversal symmetry broken

Unitary sigma model with xy a 0: anomalous -term with a

SQ

a

1

8

Str

xx @ r QA

2C

Qr xQr yQ

ln

0

d

ln

d

ln

L U

0

no localization, QHE criticality instead!

Minimal conductivity: a

4

U % @ 2: 0 2: 4A e2

=h




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Ballistic transport


Symplectic class

Random potential (e.g. charged impurities)a

A effective time-reversal symmetry preserved

Symplectic sigma model: anomalous

-term with a

!

SQ

a

xx

16Str

@r Q

A

2C

i

NQ

N

Q

a0

;1

ln

0

d

ln

d

ln

L

Sp

Sp

0

no localization! criticality?

Minimal conductivity: a 4 Sp $ e

2=

h, or

Absolute antilocalization:

3 I




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QAbsence of localization at B = 0

Ballistic transport

ySymplectic class

Scaling of conductance: numerical results

Bardarson, Tworzydo, Nomura, Koshino, Ryu 07

Brower, Beenakker 07

Absence of localization confirmed

Absolute antilocalization scenario

From ballistics to diffusion: single parameter scaling???



Clean systemDisordered system


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Disordered systemSingle parameter scaling

Outline

1

IntroductionExperimental facts

Model


Odd quantization



Unitary class

Symplectic class

4 Ballistic transport

Clean system

Disordered system






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Ballistic setup

W

L

rectangular sample with dimensions L W

large aspect ratio: W ) L

aA

boundary conditions (edge modes) irrelevantballistic regime: L ( l

aA treat disorder perturbatively

ideal contacts

perfect metallic leads (highly doped regions of graphene)



Ab f l li i B 0



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Transfer matrix technique

a

b

c

d

Scattering matrix vs. Transfer matrixc

b

a

a

d

a

t rH

r tH

a

d

c

d

a

a

b

a

2

t+ 1

rH tH 1

tH 1

r tH 1

3

a

b

Transport properties

determined by transmission eigenvalues Tn of t+ t

e.g. conductance G and Fano factor F

Ga

4e2

hTr@ t+ tA F a 1 Tr

@

t+

tA

2

Tr@ t+ tA

Clean limit: Tpy @ xA a

1C

p2y

p2y 2sinh2

q

p2y 2x

!

1

[Tworzydo et al. 06; Titov 07]



Ab f l li ti t B 0



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ySingle parameter scaling

Clean graphene: transmission distribution

Measure in channel space

P@

TA

dTa

2Wdpy

2

aA P

@T

A a

W

dpy

dT

Low energies: L 1

Expansion in small energy

P@ TA aW

2 L

1

Tp

1 T

4

1 C @ LA 2

2

p

1 T

arcosh3 1pT

1C

T

2 arcosh2 1pT

3 5

High energies: L 1

T@

pyA is a rapidly oscillating function

After averaging over oscillations

P@ TA aWj

j

2

K@

p

TA

E@

p

TA

Tp

1 T



Absence of localization at B 0



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ySingle parameter scaling

Conductance and Fano factor

0 2 4 6 8 100

2

4

6

8

0.0

0.1

0.2

0.3

0.4

L

G

4e2W

hL

F

Limit Conductance Fano factor

L ( 14e2

h

W

L

1 C 0: 101 @ LA 2 1

3

1 0: 05 @ LA 2

L ) 1e2

hWj j

1 Csin@ 2 L

4A

2p

@ L

A

3= 2

!

1

8

1 C9sin@ 2 L

4A

2p

@ L

A

3 = 2

!






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Ballistic transport experimentDanneau et al. 07

Setup

Rectangular sample

Temperature 4: 2 30 K

Large aspect ratio W=

La

24Ballistic limit L $ 200 nm

Observations

Conductance

G@ a 0A % 4e2

h

WL

Fano factor F@ a 0A % 1= 3

Conductance grows with

Fano factor decreases with




Clean systemDisordered systemSi l li


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Lowest-order disorder correction

Transfer matrix evolution

@ xA a 0 @ xA i

x

0

dxH 0 @ x xH

A zV @ xH A @ xH A

Gaussian white-noise disorderV

@x

;y

A a

V

@x

;y

A

h V

@x

;y

A

ia

0 h V2

@x

;y

A

ia

2

a 0 C x y z

Lowest order perturbative correctionLow energy

L ( 1:P

@T

AU3

@1

C AP

@T

A

The functional dependence is not changed!



Absence of localization at B = 0

Clean systemDisordered systemSi l t li


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0

Ballistic transportSingle parameter scaling

Higher order corrections

Second-order correction logarithmically diverges!

Example: zero energy, random potential 0

Conductance:G a

4e2

h

W

L

1 C 0 C 22

0

log @ L= aA C : : : | { z }

0 @ LA

Divergence is cut by the sample size L and lattice constant a

How to proceed?Include logarithmic terms into renormalized parameter

0 @ LA

aA Renormalization Group

[Dotsenko, Dotsenko 83, Ludwig et al. 94; Nersesyan et al. 94, Aleiner, Efetov 06]




Clean systemDisordered systemSingle parameter scaling


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Renormalization group

2D action for Dirac fermions in random potentialS a

d2xh

"

r

C i " C 0 @ " A2

i

Energy Disorder

1-loop

2-loop

d

dlog a

0 C 2

0 = 2 d

0

dlog a 2 20 C 2

3

0






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Solution to RG equations

0 @ A a 1

2log@

l0 = A @ A a

p

2 0 log @ l0 = A

l0 a ap

0e1=

2

0

0 a e 1= 2 0

UB

B

D

0 0

Ll0

L

log

L

RG stops when

$ L

aA ultra-ballistic [

@ AL ( 1]

@ A $ 1 a A ballistic [ @ A L ) 1]

0 @ A $ 1 a A diffusive

Crossover between regimes

UBD: L $ l0 A zero-energy mean free path

UBB: L $ F @ A a

p

2 0 log @ = 0 A = A Fermi wave length

BD: L $ l@ A a 2 0 log @ = 0 A 3= 2

= A

Ta 0 mean free path



Absence of localization at B = 0B lli i



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Results for conductance and noise

UB

B

D

0 0

Ll0

L

log

L

Regime Conductance Fano factor

UB4e2

h

W

L

1 C 0 C 0: 101@ LA

2

2 0 log @ l0 = LA

!

1

3

1 C0: 05@ LA 2

2 0 log @ l0 = LA

!

Be2

h

W

2 0 log @ l0 = LA

1

8

D8e2

h

log

0

1

3



Absence of localization at B = 0B lli ti t t



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Ballistic transportS g e p e e g


AssumeZero energy

Gaussian white-noise random potential

Transmission distribution is universal ! ! !

P@ TA aW

2 L

Tp

1 Twith

a

@

1 C 0 @ LA ; ultra-ballistics

G h= 4e2 ; diffusion[Diffusive limit: Dorokhov 83]

1 [4e2/h]

d log

d logL Unified scaling

dlog

dlog La

@

2@ 1A 2 = ; ballistic

1= ; diffusive


IntroductionAnomalous Quantum Hall effectAbsence of localization at B = 0

Ballistic transport


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Ballistic transport

Conclusions

Results1 Anomalous QHE

Decoupled valleys= )

odd quantum Hall effect

Mixed valleys = ) even plateaus appear

Chiral disorder (ripples)= )

classical Hall effect at the lowest LL

2 Absence of localization at Ba

0

Decoupled valleys = ) no localization

Charged impurities + ripples= )

quantum Hall critical state

3 Ballistic transport

Transmission distribution including disorderTwo-loop RG for random potential

Single parameter scaling at the Dirac point

PRL 98, 256801 (2007); PRB 77, 195430 (2008); in preparation (2008)


P. Ostrovsky et al- Anomalous localization and quantum Hall effect in disordered graphene

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Transcript of P. Ostrovsky et al- Anomalous localization and quantum Hall effect in disordered graphene