P. Ostrovsky et al- Anomalous localization and quantum Hall effect in disordered graphene
Transcript of P. Ostrovsky et al- Anomalous localization and quantum Hall effect in disordered graphene
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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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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
Graphene samples
Suspended sample Hall bar
Micro-mechanical cleavage Epitaxial growth
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
I d i
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
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)
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
Introd ction
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
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!
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
Introduction
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
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!
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
Introduction
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
Introduction
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IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0Ballistic transport
Experimental factsModel
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]
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionOdd nti tion
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Odd quantizationOrdinary quantizationAbsence of quantization
Outline
1 IntroductionExperimental facts
Model
2 Anomalous Quantum Hall effect
Odd quantization
Ordinary quantizationAbsence of quantization
3 Absence of localization at B a 0
Unitary class
Symplectic class4 Ballistic transport
Clean system
Disordered system
Single parameter scaling
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionOdd quantization
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Odd quantizationOrdinary quantizationAbsence of quantization
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?
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IntroductionOdd quantization
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Odd quantizationOrdinary quantizationAbsence of quantization
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionA l Q H ll ff
Odd quantization
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Odd quantizationOrdinary quantizationAbsence of quantization
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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Anomalous Quantum Hall effectAbsence of localization at B = 0
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect Unitary class
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Unitary classSymplectic class
Outline
1
IntroductionExperimental facts
Model
2 Anomalous Quantum Hall effect
Odd quantization
Ordinary quantizationAbsence of quantization
3 Absence of localization at B a 0
Unitary class
Symplectic class4 Ballistic transport
Clean system
Disordered system
Single parameter scaling
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect Unitary class
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Unitary classSymplectic class
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect Unitary class
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Anomalous Quantum Hall effectAbsence of localization at B = 0
Ballistic transport
Unitary classSymplectic class
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect Unitary class
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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???
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Clean systemDisordered system
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Absence of localization at B = 0Ballistic transport
Disordered systemSingle parameter scaling
Outline
1
IntroductionExperimental facts
Model
2 Anomalous Quantum Hall effect
Odd quantization
Ordinary quantizationAbsence of quantization
3 Absence of localization at B a 0
Unitary class
Symplectic class
4 Ballistic transport
Clean system
Disordered system
Single parameter scaling
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Clean systemDisordered system
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Absence of localization at B = 0Ballistic transport
Disordered systemSingle parameter scaling
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)
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Ab f l li i B 0
Clean systemDisordered system
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Absence of localization at B = 0Ballistic transport
Disordered systemSingle parameter scaling
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]
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Ab f l li ti t B 0
Clean systemDisordered system
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Absence of localization at B = 0Ballistic transport
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B 0
Clean systemDisordered system
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Absence of localization at B = 0Ballistic transport
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
!
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B 0
Clean systemDisordered system
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Absence of localization at B = 0Ballistic transport
Single parameter scaling
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B 0
Clean systemDisordered systemSi l li
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Absence of localization at B = 0Ballistic transport
Single parameter scaling
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!
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
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]
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0
Clean systemDisordered systemSingle parameter scaling
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Ballistic transportSingle parameter scaling
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0
Clean systemDisordered systemSingle parameter scaling
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Ballistic transportSingle parameter scaling
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0B lli i
Clean systemDisordered systemSingle parameter scaling
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Ballistic transportSingle parameter scaling
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
IntroductionAnomalous Quantum Hall effect
Absence of localization at B = 0B lli ti t t
Clean systemDisordered systemSingle parameter scaling
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Ballistic transportS g e p e e g
Single parameter scaling
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
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene
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)
Ostrovsky, Schssler, Gornyi, Mirlin Localization and QHE in graphene