Transverse Transport / the Hall and Nernst Effects Usadel equation for fluctuation corrections Alexander Finkel‘stein 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, and Aharon Kapitulnik submitted ” The Hall Effect in Superconducting Films” KM, KT, and AF PRB accepted, arXiv “Fluctuation Conductivity in Disordered Superconducting Films” KT,GS, and AF PRB 85, Fluctuation Conductivity in Disordered Superconducting Films:

Outlook for two parts of the talk ( ): Outlook for two parts of the talk ( I, II ): 3 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.

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

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 ! are superconducting fluctuations

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

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  xy (unlike  xx or  xy, which are only “once” transverse ) “Particle-Hole” asymmetry: twice off-diagonal effect / usually “twice“ small / not true for the discussed problem

Experimental data from A. Pourret, et al 2007 film of thickness α xy Agreement with the experiment (no fitting parameters; T C and diffusion coefficient were taken from independent measurements) “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 Karen Michaeli & AF 8

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

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

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

KM, KT, and AF submitted, arXiv

KM, KT, and AF PRB accepted, arXiv 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. the Hall Signal Created by the Superconducting Fluctuations Two types of the contributions depending on the mechanism of deflection in the transverse direction: quasiparticles or superconducting modes flux technique (M. Khodas and A.F. 2003) 13

B-T Phase Diagram 14

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

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

The Nernst Coefficient α xx contributes negligible in comparison to α xy The Peltier coefficient is related to the flow of entropy According to the third law of thermodynamics when 17

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 18

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

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

21 Usadel equation: 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 Usadel equation: 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 GL action can be written as follows Fermi distribution scalar potential

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

Fluctuation corrections to conductivity due to superconducting fluctuations  Kubo formula  Disorder dressing  Both fermionic and bosonic degrees present

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

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

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

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

Conclusion 29  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.

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

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

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