Fluctuation conductivity of thin films and nanowires near a parallel- field-tuned superconducting quantum phase transition A. V. Lopatin Argonne National Laboratory Collaborators: Nayana Shah, Valerii Vinokur 1. Quantum critical point in thin superconducting films and wires in parallel magnetic field 2. Fluctuation conductivity: review of different contributions 3. Fluctuation conductivity of homogeneously disordered films and wires in the vicinity of the quantum critical point. 4. Magneto-resistance at low temperatures. 5. Comparison with experimental data 6. Related theoretical works on magneto-resistance

Quantum critical point in superconductors with magnetic impurities. Example: Superconductivity in the presence of the magnetic impurities is described in terms of the pairbreakin parameter α ~ magnetic impurity concentration Critical temperature: Abrikosov Gorkov Tc α QCP Classical thermal fluctuations Quantum fluctuations Classical region was studied by Ramazashvili and Coleman ( PRL 97 ) and Mineev Sigrist ( PRB 2001 ) based on TDGL: Disadvantage – concentration of impurities cannot be tuned.

QCP in thin superconducting wires and films in parallel magnetic filed. Thin wire or film in parallel magnetic field. Thickness (diameter) is less then the coherence length H H Critical temperature ? Usadel equation close to the critical temperature: D – diffusion coefficient f – Green- function Depairing parameter α d – diameter of the wire t – thickness of the film Critical temperature: Depairing parameter is a function of the magnetic filed !

Tc Fluctuation conductivity around the QCP. Depairing parameter can be tuned by magnetic filed Tc QCP Classical thermal fluctuations Fluctuation conductivity ? Quantum fluctuations Zeeman effect? Zeeman effect can be neglected as long as diameter ( thickness ) mean free time

Review of different contributions to fluctuation conductivity. Aslamazov-Larkin: I Contribution due to local superconducting regions that appear due to thermal fluctuations Gives a positive contribution since it represents an additional channel for conductivity. Superconducting regions that appear due to thermal fluctuations Diagram: Pair propagators In case of dirty superconducting films Current vertices In case of QPT, at zero temperature one expects less singular contribution.

a) b) - Dephasing parameter Maki – Thompson contribution. Physical meaning: Scattering of an electron by a Cooper pair Pair propagator Cooperon MT contribution does not have a prescribed sign a) b) In dirty 2D films: - Dephasing parameter a) small dephasing parameter - stronger than AL correction b) large dephasing parameter - weaker than AL correction

= Density of states contribution Physical meaning: Density of states on the Fermi level is reduced due to proximity to the superconducting state Negative sign impurity = In homogeneously disordered films at temperatures close to the critical DOS contribution is always small. Exception – granular metals where the DOS contribution may be the dominant one

AL MT DOS Fluctuation conductivity at zero temperature. Analogues diagrammatic approach Pair propagator: Low temperatures: All diagrams are of the same order ! AL MT DOS

Fluctuation conductivity at zero temperature. Final answer at T=0: D – diffusion coefficient Plot of dimensionless correction Negative magnetoresistance ! Time dependent Ginzburg-Landau approach would not give the correct result Critical exponent 1 Numerical coefficient b : b=0.386 for d=1 b=0.070 for d=2

Comparison with other corrections at T=0 1. Localization correction - sensitive to the magnetic filed: Results in: Magnetoresistance: due to WL correction: due to proximity to superconducting QCP : 2. Altshuler-Aronov correction Not sensitive to the magnetic filed !

Finite temperatures: Largest contribution comes from the AL correction Total correction: Zero temperature correction Finite temperature correction Classical regime: Intermediate regime: Quantum regime: In the quantum regime the fluctuation correction is essentially temperature independent

Resistivity behavior as a function of magnetic field ( temperature ). Tc α Classical thermal fluctuations a. b. Quantum regime a. Dependence on temperature T for fixed α b. Dependence on α for fixed temperature T T R R

Experiments. Amorphous InO films in parallel field: Gantmakher, Golubkov, Dolgopolov, Shashkin, Tsydynzhapov (2000) Expected to be microscopically granular Qualitative agreement Perpendicular filed Parallel filed

H Tc Experiment with hollow cylinder Depairing parameter: When Liu at. al. Science 294,2332 (2001) Lowest temperature 20mk Monotonic dependence on temperature ! – not consistent with the theory.

Other related theoretical studies 1. Fluctuation conductivity in granular metals: I. S. Beloborodov and K. B. Efetov, PRL 82, 3332 (1999)   At T=0 DOS diagram has the largest contribution. Negative magnetoresistance at T=0. 2. Fluctuation conductivity in thin films in perpendicular magnetic field. V. M. Galitski, A.I. Larkin, PRB 63, 174506 (2001) All three corrections were found to be of the same order at T=0 The total correction is negative. Al three examples give negative fluctuation correction to the conductivity

Conclusions 1. Fluctuation conductivity in the vicinity of parallel field induced superconducting QCP was found 2. Three regimes are identified 3. Zero temperature corrections negative 2d – agreement with theory 1d - ? 5. Effective functional approach ( time dependent Ginzburg- Landau) would fail at T=0. 6. Negative zero temperature fluctuation corrections in all other known cases.