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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
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Leonid PonomarenkoDr. Anne de VisserWZI, UvAProf. Aad
PruiskenITF, UvACo-workers/Supervision:
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Outline:Quantum Hall Effect:essentialsquantum phase transitions
(critical behavior)motivation
Experiments and remaining puzzlesPI vs. PP transitions
Modelling macroscopic inhomogeneitiesConclusions and Outlook
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Quantum Hall Effect: Basic Ingredients2D Electron Gas
(disorder!)Low Temperatures (0.1-10 K)High Magnetic Fields (20-30
T)
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InGaAsSpacer (InP)Si-doped InPSubstrate (InP)EF (Fermi Energy)
The making of a 2DEGMBE/MOCVD/CBE/LPE:
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InGaAsSpacer (InP)Si-doped InPSubstrate (InP)EF (Fermi Energy)
The making of a 2DEG - II
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Hall bar geometry: Etching & ContactsVxxVxyIIThe making of a
2DEG - III4-point resistance measurement:
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Drude (classical):Magnetotransport:(Ohms law)The Hall Effect:
Classical
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Magnetotransport:rxy=h/ie2i =1i =2i =4The Hall Effect: Quantum
(Integer)
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2D Density of States (DOS)B>0:DOS becomes series of
d-functions:Landau Levels energy separation:
B=0:2D DOS is constant
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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)
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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
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Integer quantum Hall effect
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Universality? T 0 behavior? Integer quantum Hall effect
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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?
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Outline:Quantum Hall Effect:essentialsquantum phase transitions
(critical behavior)motivation
Experiments and remaining puzzlesPI vs. PP transitions
Modelling macroscopic inhomogeneitiesConclusions and Outlook
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Measuring T dependence in PP transitions
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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
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Our own benchmark experiment on PIde Lang et al., Physica E 12
(2002); to be submitted to PRB
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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]
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Our own benchmark experiment on PPSomething is not quite
rightK=0.48K=0.35
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L. Ponomarenko, AIO colloq. December 4, 2002Leonids density
gradient explanationPonomarenko et al., cond-mat/0306063, submitted
to PRB
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Different contacts and field polarity
Antisymmetry:
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Leonids density gradient explanationL. Ponomarenko, AIO colloq.
December 4, 2002
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Hall Resistance
Same for both field polarities, but PP transitions on different
contacts take place at different fields
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Leonids density gradient explanationL. Ponomarenko, AIO colloq.
December 4, 2002
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How to obtain correct data?
Illumination
Averaging data from different contacts and for both field
polarities
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Outline:Quantum Hall Effect:essentialsquantum phase transitions
(critical behavior)motivation
Experiments and remaining puzzlesPI vs. PP transitions
Modelling macroscopic inhomogeneitiesConclusions and Outlook
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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
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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
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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
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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)
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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)
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Results: 1.5 % gradient along x
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Results: 1.5 % gradient along x
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Results: 1.5 % gradient along x
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Results: 3.0 % gradient along y
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Results: 3.0 % gradient along y
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Results: 3.0 % gradient along y
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Results: realistic gradient along x,ynx< ny < 5%
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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 (