Transverse Transport / the Hall and Nernst Effects

1. Fluctuation conductivity in disordered superconducting films: Transverse Transport / the Hall and Nernst Effects Usadel equation for fluctuation corrections Alexander Finkel‘stein

2. Fluctuation Conductivity in Disordered Superconducting Films: Konstantin Tikhonov (KT) TA&MU Karen Michaeli (KM) Pappalardo Fellow at MIT and Georg Schwiete (GS) FU Berlin “Fluctuation Hall conductivity in Superconducting Films” N. P. Breznay, KM, KT, AF, andAharonKapitulniksubmitted ” The Hall Effect in Superconducting Films” KM, KT, and AF PRB accepted, arXiv 12036121 “Fluctuation Conductivity in Disordered Superconducting Films” KT,GS, and AF PRB 85, 1745272012

3. Outlook for two parts of the talk ( I, II ): • I: Effect of fluctuations is more pronounced for the • transverse components of the transport (e.g., the Hall and Nernst effects) • as compared to the longitudinal components: • II: • We developed an approach to the calculation of fluctuation conductivity in the framework of the Usadel equation. The approach has clear technical advantages compared to the standard diagrammatic techniques. • We generalized results for fluctuation corrections to arbitrary (B,T) and compared various asymptotic regions with previous studies. • The approach has also been applied to the calculation of Hall conductivity (and also checked by comparison with the diagrammatic calculation). • We hope that the formalism proves useful for studies of fluctuations out-of-equilibrium and in superconductor-normal metal hybrid systems.

4. The Nernst Coefficient The Nernst signal Y. Wang, et al 2005 twice off-diagonal effect / usually “twice“ small / this appeared not true for the superconducting fluctuations

5. Under the approximation of the constant density of states: For a non-constant density of states example of “twice” smallness This fact makes the Nernst effect very favorable for studying fluctuations a-la para-conductivity (e.g., Aslamazov-Larkin). There is no Drude terms to compete with ! The only source for the Nernst signal are superconducting fluctuations

6. Nernst Effect – Conventional Superconductors The Nernst signal The strong Nernst signal above Tc cannot be explained by the vortex-like fluctuations. A. Pourret, et al 2007 the fluctuations of the order parameter cause the effect.

7. Why the Nernst Signal Created by the Superconducting Fluctuations is so strong, even stronger than in the Hall effect? no need for “particle-hole” asymmetry in the fluctuation propagator to get the transverse thermo-electric coefficient axy (unlike axx orsxy, which are only “once” transverse ) “Particle-Hole” asymmetry: twice off-diagonal effect / usually “twice“ small / not true for the discussed problem

8. αxy Agreement with the experiment (no fitting parameters; TC and diffusion coefficient were taken from independent measurements) Karen Michaeli & AF Experimental data from A. Pourret, et al 2007 film of thickness “Fluctuations of the superconducting order parameter as an origin of the Nernst Effect” EPL, 86 (2009); Phys Rev B 80 (2009) “Quantum kinetic approach for studying thermal transport in the presence of electron-electron interactions and disorder” Phys Rev B 80 (2009) Serbin et al. Phys. Rev. Lett. 2009

9. the Hall Signal Created by the Superconducting Fluctuations N. P. Breznay et al. submitted

10. Fluctuation corrections to conductivity due to SC fluctuations: phenomenology Advantage: physical transperancy Shortcomings

11. The Hall effect very close to Tc; result that can be obtained by the phenomenological approach A. Aronov, S. Hikami, and A.Larkin (1995)

12. KM, KT, and AF submitted , arXiv 12036121

13. the Hall Signal Created by the Superconducting Fluctuations KM, KT, and AF PRB accepted, arXiv 12036121 Two types of the contributions depending on the mechanism of deflection in the transverse direction: quasiparticles or superconducting modes The standard set of the diagrams (but in the case of Hall, lot of cancellations!) plus the overlooked one, which is a reminiscent of the DOS correction to the Hall conductivity. flux technique (M. Khodas and A.F. 2003)

14. B-T Phase Diagram

15. T ordered r QCP B-T Phase Diagram B-induced QCP

16. Transverse transport in the vicinity of the critical points; there are regions where Hall correction does not depend on Hall effect

17. According to the third law of thermodynamics when The Nernst Coefficient αxxcontributes negligible in comparison toαxy The Peltier coefficient is related to the flow of entropy

18. The Peltier Coefficient near the quantum critical point Since the transverse signal is non-dissipative the sign of the effect is not fixed. Transverse transport in the vicinity of the critical point is very peculiar

19. Fit of the data obtained by the Kapitulnik group N. P. Breznay, KM, KT, AF, and Aharon Kapitulnik, submitted

20. Usadel equation: the bridge between phenomenology and diagrammatics (Eilenberger 1968; Usadel 1970) Start with action with electron-electron interaction in the Cooper channel decoupled via D (Hubbard-Stratonovich transformation): Single particle Hamiltonian: where There is a separation of scales: Low energy physics in the diffusive limit is contained in the reduced function

21. Usadelequation: cont. One can write closed (nonlinear) equation for the reduced g: Current density can also be expressed in terms of g: Averaging with respect to: with Closed scheme Gaussian approximation

22. Usadelequation: solution In the regime of Gaussian fluctuations, the solution of the Usadel equation can be found by a perturbative expansion around the metallic solution: with Fermi distribution scalar potential GL action can be written as follows

23. For B=0 a similar formalism was developed by Volkov et al (1998) and more recently by Kamenev and Levchenko (2007) Three mechanisms of the corrections • dn is the correction to the quasiparticle density of states as would be measured by a tunneling probe • dD is the renormalization of the diffusion coefficient due to coherent Andreev scattering • js is the supercurrent density f, f* etc. parametrize deviations of g from the metallic solution, f~CD

24. Fluctuationcorrections to conductivitydue to superconductingfluctuations • Kubo formula • Disorder dressing • Both fermionic and • bosonic degrees present

25. B-T Phase Diagram for the longitudinal transport III Asymptotic results for fluctuation conductivity- contact with known limiting cases I II IV

26. kOm Resistance curves for different temperatures II “criticality” zoomed image Magnetotransport starting in the region of the QCP and for large magnetic fields

27. Thequantumcriticalregime There are two distinct regimes: Low temperature: Sign change! Classical regime: We recover the result obtained by Galitski, Larkin (2001) [In contrast to more recent study by Glatz, Varlamov, Vinokur (2011)]

28. Fluctuation conductivity in superconducting films Effect of fluctuations is more pronounced for the transverse components of the transport as compared to the longitudinal components: Here we demonstrate a theoretical fit of the recent data obtained by the A. Kapitulnik group (Stanford) for the Hall conductivity in superconducting Tantalum Nitride (TaNx) films.* A large contribution to the Hall conductivity near the superconducting transition arising due to the fluctuations has been tracked to temperatures well above Tc=2.75K and magnetic fields well above the upper critical field, Hc2. Quantitative agreement has been found between the data and the calculations based on the microscopic analysis of the superconducting fluctuations in the disordered films. *Studying fluctuation effects in the Hall conductivity is an experimental challenge in systems with high carrier concentration and large longitudinal resistance. N. P. Breznay et. al submitted Phys. Rev. B

29. Conclusion • We developed an approach to the calculation of fluctuation conductivity in the framework of the Usadel equation. The approach has clear technical advantages compared to diagrammatic techniques. • Calculation can be performed in the scalar gauge rather than with the tme-dependent vector potential (no analytical cntinuation is needed). • We generalized results for fluctuation corrections to arbitrary (B,T) and compared various asymptotic regions with previous studies • (where asymptotics are calculated separately). • The approach has also been applied to the calculation of the Hall conductivity. • The approach provides a more transparent physical structure. • We hope that the formalism proves useful for studies of fluctuations out-of-equilibrium and in superconductor-normal metal hybrid systems.

30. Magnetoresistance Almost vertical  Intersection point 0.35 K 0.76 K TiN-film, Tc~0.6 K Line of maxima in magnetoresistance Baturina et al. (2003)

31. Our fit of the data obtained by the Kapitulnik group N. P. Breznay et al. 2012

34. Thequantumcriticalregime There are two distinct regimes: Low temperature: Sign change! Classical regime: We recover the result obtained by Galitski, Larkin (2001) [In contrast to more recent study by Glatz, Varlamov, Vinokur (2011)]