A Critical Look at Criticality

A Critical Look at CriticalityAIO Colloquium, June 18, 2003Van der Waals-Zeeman InstituteDennis de LangThe influence of macroscopic inhomogeneities on the critical behavior of quantum Hall transitions

Leonid PonomarenkoDr. Anne de VisserWZI, UvAProf. Aad PruiskenITF, UvACo-workers/Supervision:

Outline:Quantum Hall Effect:essentialsquantum phase transitions (critical behavior)motivation

Experiments and remaining puzzlesPI vs. PP transitions

Modelling macroscopic inhomogeneitiesConclusions and Outlook

Quantum Hall Effect: Basic Ingredients2D Electron Gas (disorder!)Low Temperatures (0.1-10 K)High Magnetic Fields (20-30 T)

InGaAsSpacer (InP)Si-doped InPSubstrate (InP)EF (Fermi Energy) The making of a 2DEGMBE/MOCVD/CBE/LPE:

InGaAsSpacer (InP)Si-doped InPSubstrate (InP)EF (Fermi Energy) The making of a 2DEG - II

Hall bar geometry: Etching & ContactsVxxVxyIIThe making of a 2DEG - III4-point resistance measurement:

Drude (classical):Magnetotransport:(Ohms law)The Hall Effect: Classical

Magnetotransport:rxy=h/ie2i =1i =2i =4The Hall Effect: Quantum (Integer)

2D Density of States (DOS)B>0:DOS becomes series of d-functions:Landau Levels energy separation:

B=0:2D DOS is constant

B>0:DOS becomes series of d-functions:Landau Levels energy separation:

B=0:2D DOS is constantbroadening due to disorder2D Density of States (DOS)

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/2c p relates L (sample size) and T c relates localization length x and BLocalized to extended states transition

Integer quantum Hall effect

Universality? T 0 behavior? Integer quantum Hall effect

MotivationUniversality? T 0 behavior? QHE transitions are second order (quantum) phase transitions there should be an associated critical exponent

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?

Measuring T dependence in PP transitions

Historical benchmark experiments on PP(Wei et al., 1988)n=1.5n=2.5n=3.5n=1.5n=2.5InGaAs/InPH.P.Wei et al. (PRL,1988): PP=0.42 (left)AlGaAs/GaAsS.Koch et al. (PRB, 1991): ranges from 0.36 to 0.81H.P.Wei et al. (PRB, 1992): scaling (PP=0.42 ) only below 0.2 K

Our own benchmark experiment on PIde Lang et al., Physica E 12 (2002); to be submitted to PRB

Our own benchmark experiment on PIHall resistance is quantized (T 0)

k=0.57 (non-Fermi Liquid value !!)Inhomogeneities can be recognized, explained and disentangledContact misalignmentMacroscopic carrier density variations

Pruisken et al., cond-mat/0109043 [h/e2][h/e2]

Our own benchmark experiment on PPSomething is not quite rightK=0.48K=0.35

L. Ponomarenko, AIO colloq. December 4, 2002Leonids density gradient explanationPonomarenko et al., cond-mat/0306063, submitted to PRB

Different contacts and field polarity

Antisymmetry:

Leonids density gradient explanationL. Ponomarenko, AIO colloq. December 4, 2002

Hall Resistance

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

Leonids density gradient explanationL. Ponomarenko, AIO colloq. December 4, 2002

How to obtain correct data?

Illumination

Averaging data from different contacts and for both field polarities

Modelling preliminaries:Transport results can be explained by means of density gradients. n2D n2D(x,y)Resistivity components: rij rij (x,y) Electrostatic boundary value problem

Scheme ICalculate the homogeneous r0, rH through Landau Level addition/substractionr0PI = exp(-X) ; rHPI =1 X=Dn/n0(T)sPI = (rPI)-1 e.g. s0PI =r0P(r0PI)2+(rHPI)2s0PP(k) = s0PI(k) sHPP(k) = sHPI(k) + k rPP(k) = (sPP(k))-1k=0k=1k=2

Scheme IIExpansion of ji, r0 , rH to 2nd order in x,yr0(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

Scheme IIIAppropriate boundary conditions & limitations:L/2W/2?- L/2- W/2jy(y=W/2) = 0 (b.c.)j = 0 (conservation of current)E = 0 (electrostatic condition)

Scheme IVjx, jy using b.c.Ei = rij jjVx,y= dx,y Ex,yIx=dy jxR =V / IResult ONLY in terms of aij, bij :rxx = rxx(r0, rH, aij, bij ) rxy =rxy (r0, rH, aij, bij ) use Taylor expansion in x,y to obtain aij, bij as function of nx and ny : n(x,y) =n0 (1+nx/n0 x + ny/n0 y)

Results: 1.5 % gradient along x

Results: 3.0 % gradient along y

Results: realistic gradient along x,ynx< ny < 5%

Conclusions 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 (

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A Critical Look at Criticality