1. Bardeen-Stephen flux flow law disobeyed in Bi 2 Sr 2 CaCu 2 O 8+δ G. Kriza, 1,2 A. Pallinger 1, B. Sas 1, I. Pethes 1, K. Vad 3, F. I. B.Williams 1,4 1 Research Institute for Solid State Physics and Optics, Budapest, Hungary 2 Institute of Physics, Budapest University of Technology and Economics, Budapest, Hungary 3 Institute of Nuclear Research, Debrecen, Hungary 4 Service de Physique de l’Etat Condensée, Direction Sciences de la Matière, Comissariat à l’Energie Atomique, Gif-sur-Yvette, France and

2. Aim: Measure Free Flux Flow (FFF) resitivity in the high-T c superconductor Bi 2 Sr 2 CaCu 2 O 8+δ (BSCCO)  B  ab Transport current J T Bardeen–Stephen law (BS) Abrikosov vortex

3. What is known of ρ FFF in high- T c superconductors? No clear experimental evidence for BS law in any high-T c SC (nor in any unconventional superconductor) No theory takes into account all the essential ingredients BS law This experiment

4. Complication: How to account for the pinning force? Pinning force  uncertainty in the total force F  uncertainty in velocity – force relation Solution Model pinning (e.g., to interpret surface impedance) Create conditions where pinning is irrelevant (our approach) Unpinned vortex liquid state near T c Apply high current so that the pinning force is negligible in front of the Lorentz force

5. Experiment BSCCO single crystal: B  ab Voltage response Current excitation: Voltage – current (V – I) characteristics a b c

6. Typical voltage – current characteristics V/I varies with I up to the highest current  pinning is not negligible R ab = dV/dI saturates (becomes current independent) at the high current. If R ab = dV/dI = const, then for I → , V/I → R ab The differential resistance R ab measures the high-current limit of the resistance V/I  We assume that R ab reflects the free flux flow limit

7. Temperature and field dependence of the high-current differential resistance R ab High temperature: sublinear B-depend- dence Low temperature: T and B independent resistance

8. Interpolating function: B c2 (T) = B c2 (0)[1– (T/T c ) 2 ] with B c2 (0) = 120 tesla to give as in Qiang Li et al., Phys. Rev. B 48, 9877 (1993). Empirical form for the high-current differential resistance R ab Empirical form for the B and T dependence of the resistance:

9. Dependence of the resistance R ab on the local resistivities  c and  ab t l V I Anisotropic quasi-2d sample: Strong anisotropy  c >>  ab  shallow current penetration  influence on the current density  R ab depends on both  c and  ab geometrical factor Valid for linear response and for asymptotically linear resistivities in the high-current limit Our samples are well in the thick sample limit  ab cc

10. How to disentangle  c and  ab from R ab ? V top I Multicontact method: V top  ab V bottom  c  ab cc V bottom This experiment was done by R. Busch, G. Ries, H. Werthner, G. Kreiselmeyer, and G. Saemann-Ischenko, Phys. Rev. Lett. 69, 522 (1992) Problem: Busch et al. measured in the I → 0 limit whereas we measured in the high-current limit

11. How to compare high-field and low-field resistances? Go to the unpinned liquid phase! With increasing current, the V-I curves are less and less nonlinear For T > T lin linear response  "unpinned liquid phase” (smooth crossover, no sharp change)

12. Magnetic field–temperature phase diagram B c2 upper critical field T FOT first order transition line T irr magnetic irreversibility line T 2nd second magnetization peak

13. Magnetic field–temperature phase diagram B c2 upper critical field T FOT first order transition line T irr magnetic irreversibility line T 2nd second magnetization peak Vortex liquid Glass?

14. Magnetic field–temperature phase diagram B c2 upper critical field T FOT first order transition line T irr magnetic irreversibility line T 2nd second magnetization peak Pinned liquid Unp. L. Unpinned liquid phase For T > T lin the V-I curves are linear Glass?

15. Analysis of the multicontact data of Busch et al. Phys. Rev. Lett. 69, 522 (1992) FIG. 3 Digitize isothermal sections Sort out data for which T > T lin (B) (unpinned liquid)

16. Single crystal resistance (same quantity as in our experiments) Busch et al., Phys. Rev. Lett. 69, 522 (1992) Reproduces B/B c2 scaling Reproduces logarithmic field dependence: The slope  = 0.23 is in good agreement with our results Unpinned liquid

17. In-plane (ab-plane) resistivity Busch et al., Phys. Rev. Lett. 69, 522 (1992)  ab also exhibits B/B c2 scaling Exponent of best power law fit: 0.75  0.01 (too good to be true?) Unpinned liquid

18. In-plane (ab-plane) resistivity Busch et al., Phys. Rev. Lett. 69, 522 (1992)  ab also exhibits B/B c2 scaling Exponent of best power law fit: 0.75  0.01 (too good to be true?) Unpinned liquid

19. Out-of-plane (c-axis) resistivity Busch et al., Phys. Rev. Lett. 69, 522 (1992)  c also exhibits B/B c2 scaling Given experimental forms for R ab and  ab, we can write an experimental form for  c using : Unpinned liquid Reproduces the maximum below B c2 seen earlier: G. Briceño, M. F. Crommie, and A. Zettl, Phys. Rev. Lett. 66, 2164 (1991); K. E. Gray and D. H. Kim, Phys. Rev. Lett. 70, 1693 (1993); N. Morozov et al., Phys. Rev. Lett. 84, 1784 (2000).

20. Comparison with thin film  ab data Z. L. Xiao, P. Voss-de Haan, G. Jakob, and H. Adrian, Phys. Rev. B 57, R736 (1998) Reasonable agreement but systematic deviation from power law (weaker than linear on log-log plot)

21. Comparison with thin film  ab data H. Raffy, S. Labdi, O. Laborde, and P. Monceau, Phys. Rev. Lett. 66, 2515 (1991) P. Wagner, F. Hillmer, U. Frey, and H. Adrian, Phys. Rev. B 49, 13184 (1994) M. Giura, S. Sarti, E. Silva,R. Fastampa, F. Murtas, R. Marcon, H. Adrian, and P. Wagner, Phys. Rev. B 50, 12920 (1994) Z. L. Xiao, P. Voss-de Haan, G. Jakob, and H. Adrian, Phys. Rev. B 57, R736 (1998) Different thin film results do not agree with each other Common feature: weaker than power law, stronger than logarithmic B-dependence  Thin film resistance is in-between single crystal R ab and  ab Macroscopic defects may force c-acis currents  Thin film resistance may be a sample-dependent mixture of  ab and  c (2d topology amplifies the effect of macroscopic defects)

22. Comparison with c-axis-configuration single crystal  c data N. Morozov, L. Krusin-Elbaum, T. Shibauchi, L. N. Bulaevskii, M. P. Maley, Yu. I. Latyshev, and T. Yamashita, Phys. Rev. Lett. 84, 1784 (2000) Excellent agreement with our empirical form (blue line):  = 0.20 from the fit is in excellent agreement with our result Discrepancy:  0 = 3.6 (k  cm) −1 from fit is different from  0 ≈ 8 (k  cm) −1 inferred by Morozov et al. (Resistance decreases more slowly above the maximum than the fitting function.) Empirical form

23. Quick summary of empirical forms

24. Pinned liquid Unp. L. Sharp crossover at T co from R ab = const (low T) to R ab  log B (high T) For T < T co : R ab = const For T > T co : R ab  log B Probable difference: intervortex correlations (interlayer?) Mismatch with thermodynamic vortex phases: No change in R ab when T irr and T lin crossed No anomaly in the low-current resistance at T co T co reflects transition in the dynamic vortex system? Unpinned liquid dynamically restored for T co < T < T lin ? Dynamic ordering of vortices below T co ? R ab = const R ab =  log B

25. Other high-T c materials: YBCO Y. Tsuchiya et al., PRB 63, 184517 (2001): Microwave surface impedance in YBCO ab- plane resistivity from microwave surface impedance (arb. units) Power law exponent agrees well with 3/4 in the high temperature limit. No agreement at low temperature, but this is not dynamic vortex system!

26.  cancellation of power law exponents In conventional superconductors:  ab  DOS(B)  c  DOS(B) In a superconductor with line nodes DOS(B)  B 1/2  sublinear B-dependence is not surprising, but the origin of exponent 3/4 is not clear. The response of the extended line nodes is not taken into account in existing theories. The cancellation of power law exponents may indicate a common origin of ab plane and c axis dissipation. Simultaneous in-plane and interplane phase slips? Theoretical calculations of this mechanism are in disagreement with our results. Cancellation of power law exponents in  ab and  c   ab  c = const

27. Conclusion Empirical forms for the magnetic field dependence of resistivities of BSCCO in the high-current limit: Some evidence that:  FFF  B 3/4 holds in YBCO as well We speculated about a dynamic transition in the vortex system See also: Á. Pallinger, B. Sas, I. Pethes, K. Vad, F. I. B.Williams, and G. Kriza, Phys. Rev. B (in press). Validity in other high-T c ? Theoretical underpinnings?