A Critical Look at Criticality

1. 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

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

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

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

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

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

9. The Hall Effect: Quantum (Integer) Magnetotransport: i =1 rxy=h/ie2 i =2 i =4

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

11. 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:

12. 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

13. Integer quantum Hall effect

14. 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?

15. Motivation… 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?

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 • 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)

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) 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

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

22. Leonid’s density gradient explanation… L. Ponomarenko, AIO colloq. December 4, 2002 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. n2Dn2D(x,y) Resistivity components: rij rij (x,y) Electrostatic boundary value problem

27. Scheme – I 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

28. Scheme – II 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…

29. Scheme – III 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)

30. 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)

31. Results: 1.5 % gradient along x

32. Results: 1.5 % gradient along x

33. Results: 1.5 % gradient along x

34. Results: 3.0 % gradient along y

35. Results: 3.0 % gradient along y

36. Results: 3.0 % gradient along y

37. Results: ‘realistic’ gradient along x,y nx< ny < 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 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.