1

Effect of density gradients on Effect of density gradients on magnetotransport magnetotransport

of quantum Hall systemsof quantum Hall systems

L. Ponomarenko

2

Participants:Participants:

Anne de Visser and Dennis de Lang

A. Pruisken (ITF, UvA)

G. Galeev (IRE, Moscow)

H. Kunzel (HHIN, Berlin)

3

Outline:Outline:

Introduction– Quantum Hall Effect– Quantum Critical Point– Motivation

Results– Experiment– Explanation

Outlook

4

IntroductionIntroduction

Two-dimensional electron gas (2DEG)– applications:

• high frequency devices

– fundamental research:• Quantum Hall Effect• Weak localization problem

5

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

i = 6

i = 5

i = 4

T = 60 mK

xx (

k)

B (T)

0

2

4

6

xy (k

)

Quantum Hall EffectQuantum Hall Effect

Vxx

Vxy2ie

hxy

22xyxx

ijij

hiexy /2

0xx

0xx 0xx

0xx

0xx

0xx

6

Band StructureBand Structure

E

DOS

EF

EE

DOSDOS

EF

EF

B 0

*m

eBE

B = 0

7

Effect of Disorder (T = 0)Effect of Disorder (T = 0)

Order

Disorder

E

E

DOS

DOS

E

E

X

X

Filling factor:

eB

nh

8

Quantum Phase TransitionQuantum Phase Transition

Takes place at T = 0Critical behavior

– Localization length

For non-interacting electrons

cBB~

3

7

9

ExperimentExperiment

1 2 3 4 50

2

4

6

1.80 K 1.05 K 0.45 K 0.189 K

xx (

k)

B (T)

0

10

20

30

xy (

k)

10

CriticalityCriticality

InGaAs/InP (H.P.Wei et al., PRL, 1988)

– half width of xx scales as

– maximum slope of xy scales as

= 0.42

Localization length exponent and critical exponent :

TB ~

TB

MAX

xy ~

2,3

7

,2

p

p

11

However…However…

AlGaAs/GaAs– S.Koch et al. (PRB, 1991):

ranges from 0.36 to 0.81

– H.P.Wei et al. (PRB, 1992):

scaling only below 0.2 K

Semicircle relation

0.0

0.5

n+1n

xx

(e2 /h

)

xy (e

2/h)

12

Introduction (Summary):Introduction (Summary):

Transition between two plateaus is a quantum phase transition

Critical exponent is not universal

Experimental data do not obey

“semicircle” law

13

-8 -6 -4 -2 0 2 4 6 80.0

0.5

1.0

1.5

2.0

2.5 Vxx

Vxx

T = 400 mK

GaAs/AlGaAs quantum well

Field (T)

xx (

k)

Different contacts and field polarityDifferent contacts and field polarity

Antisymmetry:

1 2

3 4

5 6

)(BRxx )( BRxx

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0 2 4 6 80

5

10

15

B/B

(%

)

filling factor

Field (T)

xy (

k)

2 3 4 5 60

1

2

Hall ResistanceHall Resistance

Same for both field polarities, but PP transitions on different contacts take place at different fields

1 2

3 4

5 6

Vxy Vxy

)()( BRBR xyxy

)()( BRBR xyxy

)(BRxy )(BRxy

eB

nh

15

Antisymmetry results from Antisymmetry results from density gradient !density gradient !

)()(

)()(

DBCA

DCBA

VVVV

VVVV

A

B D

C

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Antisymmetry results from Antisymmetry results from density gradient !density gradient !

A

B D

C

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How to obtain correct data?How to obtain correct data?

Illumination

Averaging data from different contacts and for both field polarities

4.5 4.6 4.7 4.8 4.90.0

0.2

0.4

0.6

B-

B+

after illuminationT = 189 mK

Rxx

(k

)

Field (T)4.5 4.6 4.7 4.8 4.9

0.0

0.2

0.4

0.6

B-

B+

after illuminationT = 189 mK

Rxx

(k

)

Field (T)4.5 4.6 4.7 4.8 4.9

0.0

0.2

0.4

0.6after illumination

T = 189 mK

Rxy (k

)

Rxx

(k

)

Field (T)

4.0

4.4

4.8

5.2

4.5 4.6 4.7 4.8 4.90.0

0.2

0.4

0.6after illumination

T = 189 mK

Rxy (k

)

Rxx

(k

)

Field (T)

4.0

4.4

4.8

5.2

Gradient is 0.25 %

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Recovered semicircleRecovered semicircle

By proper averaging a “semicircle” behavior can be obtained from slightly inhomogeneous sample

5 60.0

0.5

xx (

e2 /h)

xy

(e2/h)5 6

0.0

0.5

xx (

e2 /h)

xy

(e2/h)5 6

0.0

0.5

xx (

e2 /h)

xy

(e2/h)5 6

0.0

0.5

xx (

e2 /h)

xy

(e2/h)5 6

0.0

0.5

xx (

e2 /h)

xy

(e2/h)5 6

0.0

0.5

xx (

e2 /h)

xy

(e2/h)5 6

0.0

0.5

xx (

e2 /h)

xy

(e2/h)

19

SummarySummary

Antisymmetry effectExplained by density gradientCriterion for sample selection and

reliable temperature rangeSemicircle recovered

20

OutlookOutlook

Reliable scaling experiment on plateau-plateau transition

Plateau-insulator transition

21

OutlookOutlook

0 5 10 15 200

50

100

150

200

250 T = 370 mK T = 615 mK T = 1.2 K T = 1.9 K T = 2.9 K T = 4.2 K

17.23

R

xx (

k)

B (T)

0

10

20

30

40

B-

B+

Rxy (k

)