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A Critical Look at Criticality AIO Colloquium, June 18, 2003 Van der Waals-Zeeman Institute Dennis de Lang The influence of macroscopic inhomogeneities on the critical behavior of quantum Hall transitions Slide 2 Leonid Ponomarenko Dr. Anne de Visser WZI, UvA Prof. Aad Pruisken ITF, UvA Co-workers/Supervision : Slide 3 Outline : Quantum Hall Effect: essentials quantum phase transitions (critical behavior) motivation Experiments and remaining puzzles PI vs. PP transitions Modelling macroscopic inhomogeneities Conclusions and Outlook Slide 4 Quantum Hall Effect : Basic Ingredients 2D Electron Gas (disorder!) Low Temperatures (0.1-10 K) High Magnetic Fields (20-30 T) Slide 5 InGaAs Spacer (InP) Si-doped InP Substrate (InP) E F (Fermi Energy) The making of a 2DEG MBE/MOCVD/CBE/LPE: Slide 6 InGaAs Spacer (InP) Si-doped InP Substrate (InP) E F (Fermi Energy) The making of a 2DEG - II Slide 7 Hall bar geometry: Etching & Contacts V xx V xy I I The making of a 2DEG - III 4-point resistance measurement: Slide 8 Drude (classical): Magnetotransport: (Ohms law) The Hall Effect: Classical Slide 9 Magnetotransport: xy =h/ie 2 i =1 i =2 i =4 The Hall Effect: Quantum (Integer) Slide 10 2D Density of States (DOS) B>0: DOS becomes series of functions: Landau Levels energy separation: B=0: 2D DOS is constant Slide 11 2D states (B=0,T=0) are localized, but extended states in center of Landau Levels B>0: DOS becomes series of functions: Landau Levels energy separation: B=0: 2D DOS is constant broadening due to disorder 2D Density of States (DOS) Slide 12 Scaling theory : (Pruisken, 1984) Localization length: c Phase coherence length: L p (effective sample size) ij ~ g ij (T - (B-B c )) = p/2 p relates L (sample size) and T relates localization length and B Localized to extended states transition Slide 13 Integer quantum Hall effect Slide 14 Transitions: Transitions: Extended states current travels through the bulk Universality? Plateaus: Plateaus: Quantum Hall states: bulk is localized. Current travels on the edges (edge states) T 0 behavior? Integer quantum Hall effect Slide 15 Motivation Universality? 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? Slide 16 Outline : Quantum Hall Effect: essentials quantum phase transitions (critical behavior) motivation Experiments and remaining puzzles PI vs. PP transitions Modelling macroscopic inhomogeneities Conclusions and Outlook Slide 17 Measuring T dependence in PP transitions Slide 18 Historical benchmark experiments on PP (Wei et al., 1988) =1.5 =2.5 =3.5 =1.5 =2.5 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 Slide 19 Our own benchmark experiment on PI de Lang et al., Physica E 12 (2002); to be submitted to PRB Slide 20 Our own benchmark experiment on PI Hall resistance is quantized (T 0) =0.57 (non-Fermi Liquid value !!) Inhomogeneities can be recognized, explained and disentangled Contact misalignment Macroscopic carrier density variations Pruisken et al., cond-mat/0109043 [h/e 2 ] Slide 21 Our own benchmark experiment on PP Something is not quite right =0.48 =0.35 Slide 22 L. Ponomarenko, AIO colloq. December 4, 2002 Leonids density gradient explanation Ponomarenko et al., cond-mat/0306063, submitted to PRB Slide 23 Leonids density gradient explanation L. Ponomarenko, AIO colloq. December 4, 2002 Slide 24 Leonids density gradient explanation L. Ponomarenko, AIO colloq. December 4, 2002 Slide 25 Outline : Quantum Hall Effect: essentials quantum phase transitions (critical behavior) motivation Experiments and remaining puzzles PI vs. PP transitions Modelling macroscopic inhomogeneities Conclusions and Outlook Slide 26 Modelling preliminaries: Transport results can be explained by means of density gradients. n 2D n 2D (x,y) Resistivity components: ij ij (x,y) Electrostatic boundary value problem Slide 27 Scheme I Calculate the homogeneous through Landau Level addition/substraction PI = exp(-X) ; PI =1 X= 0 (T) PI = ( PI ) -1 e.g. PI = PP ( PI ) 2 +( PI ) 2 PP(k) = PI(k) PP(k) = PI(k) + k PP(k) = ( PP(k) ) -1 k=0k=1k=2 Slide 28 Scheme II Expansion of j i, 0, H to 2 nd order in x,y 0 (x,y)= 0 (1+ x x+ y y+ xx x 2 + yy y 2 + xy xy) H (x,y)= H (1+ x x+ y y+ xx x 2 + yy y 2 + xy xy) j x (x,y)= j x (1+a x x+a y y+a xx x 2 +a yy y 2 +a xy xy) j y (x,y)= j y (1+b x x+b y y+b xx x 2 +b yy y 2 +b xy xy) 22 parameters Slide 29 Scheme III Appropriate boundary conditions & limitations: L/2 W/2 ? - L/2 - W/2 j y ( y= W/2 ) = 0 (b.c.) j = 0 (conservation of current) E = 0 (electrostatic condition) Slide 30 Scheme IV 1.j x, j y using b.c. 2.E i = ij j j 3.V x,y = dx,y E x,y 4.I x = dy j x 5.R =V / I Result ONLY in terms of ij, ij : xx = xx ( 0, H, ij, ij ) xy = xy ( 0, H, ij, ij ) use Taylor expansion in x,y to obtain ij, ij as function of n x and n y : n(x,y) =n 0 (1+n x /n 0 x + n y /n 0 y) Slide 31 Results: 1.5 % gradient along x Slide 32 Slide 33 Slide 34 Results: 3.0 % gradient along y Slide 35 Slide 36 Slide 37 Results: realistic gradient along x,y n x < n y < 5% Slide 38 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 (