1. 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

2. Leonid Ponomarenko Dr. Anne de Visser WZI, UvA Prof. Aad Pruisken ITF, UvA Co-workers/Supervision :

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

4. Quantum Hall Effect : Basic Ingredients 2D Electron Gas (disorder!) Low Temperatures (0.1-10 K) High Magnetic Fields (20-30 T)

5. InGaAs Spacer (InP) Si-doped InP Substrate (InP) E F (Fermi Energy) The making of a 2DEG MBE/MOCVD/CBE/LPE:

6. InGaAs Spacer (InP) Si-doped InP Substrate (InP) E F (Fermi Energy) The making of a 2DEG - II

7. Hall bar geometry: Etching & Contacts V xx V xy I I The making of a 2DEG - III 4-point resistance measurement:

8. Drude (classical): Magnetotransport: (Ohm’s law) The Hall Effect: Classical

9. Magnetotransport:  xy =h/ie 2 i =1 i =2 i =4 The Hall Effect: Quantum (Integer)

10. 2D Density of States (DOS) B>0: DOS becomes series of  functions: Landau Levels energy separation: B=0: 2D DOS is constant

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)

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

13. Integer quantum Hall effect

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

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?

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

17. Measuring T –dependence in PP transitions

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

19. Our own ‘benchmark’ experiment on PI de Lang et al., Physica E 12 (2002); to be submitted to PRB

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 ]

21. Our own ‘benchmark’ experiment on PP Something is not quite right…  =0.48  =0.35

22. L. Ponomarenko, AIO colloq. December 4, 2002 Leonid’s density gradient explanation… Ponomarenko et al., cond-mat/0306063, submitted to PRB

23. Leonid’s density gradient explanation… L. Ponomarenko, AIO colloq. December 4, 2002

24. Leonid’s density gradient explanation… L. Ponomarenko, AIO colloq. December 4, 2002

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

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

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 = PP (   PI ) 2 +(   PI ) 2   PP(k) =   PI(k)   PP(k) =   PI(k) + k  PP(k) = (  PP(k) ) -1 k=0k=1k=2

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…

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)

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)

31. Results: 1.5 % gradient along x

34. Results: 3.0 % gradient along y

37. Results: ‘realistic’ gradient along x,y n x < n y < 5%

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 (<5%) can vary the value of the critical exponents of PP transitions by about 10-15%. Experimentally obtained values of the maximum of  xx 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.