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A Critical Look at Criticality

The influence of macroscopic inhomogeneities on the

critical behavior of quantum Hall transitions

Dennis de Lang

AIO Colloquium, June 18, 2003

Van der Waals-Zeeman Institute

Quantum Hall Effect:

essentials

quantum phase transitions (critical behavior)

motivation

Experiments and remaining puzzles

PI vs. PP transitions

Modelling macroscopic inhomogeneities

Conclusions and Outlook

Quantum Hall Effect: Basic Ingredients

2D Electron Gas (disorder!)

Low Temperatures (0.1-10 K)

High Magnetic Fields (20-30 T)

4-point resistance measurement:

I

I

Vxy

Vxx

Hall bar geometry: Etching & Contacts

B=0:

2D DOS is constant

B>0:

DOS becomes series of d-functions:

Landau Levels

energy separation:

2D states (B=0,T=0) are localized, but

extended states in center of

Landau Levels

2D Density of States (DOS)

B=0:

2D DOS is constant

broadening due to disorder

B>0:

DOS becomes series of d-functions:

Landau Levels

energy separation:

Localized to extended states transition

Scaling theory : (Pruisken, 1984)

Localization length: x~| B-Bc| -c

Phase coherence length: Lf ~ T-p/2

(effective sample size)

rij ~ gij(T -k(B-Bc)) ; k = p/2cp relates L (sample size) and Tc relates localization length x and B

Plateaus: “Quantum Hall states”: bulk is localized. Current travels on the edges (edge states)

Transitions: “Extended states”current travels through the bulk

Integer quantum Hall effect

T 0 behavior?

Universality?

T 0 behavior?

QHE transitions are second order (quantum) phase transitions…

… there should be an associated critical exponent

Universality?

… since all LLs are in principle identical, the critical exponent of each transition should be in the same universality class.

How does macro-disorder

result in chaos?

Quantum Hall Effect:

essentials

quantum phase transitions (critical behavior)

motivation

Experiments and remaining puzzles

PI vs. PP transitions

Modelling macroscopic inhomogeneities

Conclusions and Outlook

Historical ‘benchmark’ experiments on PP

- InGaAs/InP
- H.P.Wei et al. (PRL,1988): PP=0.42 (left)
- AlGaAs/GaAs
- S.Koch et al. (PRB, 1991):
- ranges from 0.36 to 0.81
- H.P.Wei et al. (PRB, 1992): ’scaling’ (PP=0.42 ) only below 0.2 K

n=1.5

n=2.5

n=2.5

n=1.5

n=3.5

(Wei et al., 1988)

Our own ‘benchmark’ experiment on PI

de Lang et al., Physica E 12 (2002); to be submitted to PRB

Our own ‘benchmark’ experiment on PI

Hall resistance is quantized (T 0)

k=0.57 (non-Fermi Liquid value !!)

Inhomogeneities can be recognized, explained and disentangled

[h/e2]

[h/e2]

Contact misalignment

Macroscopic carrier density variations

Pruisken et al., cond-mat/0109043

Leonid’s density gradient explanation…

L. Ponomarenko, AIO colloq. December 4, 2002

Ponomarenko et al., cond-mat/0306063, submitted to PRB

Leonid’s density gradient explanation…

L. Ponomarenko, AIO colloq. December 4, 2002

Leonid’s density gradient explanation…

L. Ponomarenko, AIO colloq. December 4, 2002

Quantum Hall Effect:

essentials

quantum phase transitions (critical behavior)

motivation

Experiments and remaining puzzles

PI vs. PP transitions

Modelling macroscopic inhomogeneities

Conclusions and Outlook

Transport results can be explained by means of density gradients.

n2Dn2D(x,y)

Resistivity components:

rij rij (x,y)

Electrostatic boundary value problem

Calculate the ‘homogeneous’ r0, rH through Landau Level addition/substraction

r0PI = exp(-X) ; rHPI =1 X=Dn/n0(T)

r0P

sPI= (rPI)-1 e.g.s0PI =

(r0PI)2+(rHPI)2

s0PP(k) = s0PI(k)sHPP(k) = sHPI(k) + k

rPP(k)= (sPP(k))-1

k=0

k=1

k=2

Expansion of ji, r0 , rH to 2nd order in x,y…

r0(x,y)= r0(1+axx+ayy+axxx2+ayyy2+axyxy)

rH(x,y)= rH(1+bxx+byy+bxxx2+byyy2+bxyxy)

jx(x,y)= jx (1+axx+ayy+axxx2+ayyy2+axyxy)

jy(x,y)= jy (1+bxx+byy+bxxx2+byyy2+bxyxy)

22 parameters…

Appropriate boundary conditions & limitations:

W/2

?

- W/2

- L/2

L/2

jy(y=W/2) = 0 (b.c.)

j = 0 (conservation of current)

E = 0 (electrostatic condition)

Result ONLY in terms of aij, bij : rxx= rxx(r0, rH, aij, bij ) rxy =rxy (r0, rH, aij, bij )

Scheme – IV

- jx, jy using b.c.
- Ei = rijjj
- Vx,y= dx,y Ex,y
- Ix=dy jx
- R =V / I

… use Taylor expansion in x,y to obtain aij, bij as function of nx andny :n(x,y) =n0 (1+nx/n0 x + ny/n0 y)

Results: ‘realistic’ gradient along x,y

nx< ny < 5%

Realistic QH samples show different critical exponents for different transitions within the same sample.

Inhomogeneity effects on the critical exponent can only be disentangled at the PI transition.

Density gradients of a few percent (<5%) can vary the value of the critical exponents of PP transitions by about 10-15%.

Experimentally obtained values of the maximum of sxx often show a noticable T-dependence. This can be explained by a carrier density ‘gradient’ along the width of the Hall bar. It is also an indication that the obtained critical exponent is underestimated.

Reported ‘universal’ values of PP transition exponents should be viewed with great care and scrutiny.

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