Role of Disorder in Superconducting Transition

1. Role of Disorder in Superconducting Transition HRI Sudhansu S. Mandal IACS, Kolkata

2. Anderson’s Theorem • Statement:Superconducting order parameter and hence superconducting transition temperature is independent of (weak) non-magnetic disorder. Cooper pairs are formed with time reversed partners. In a time reversal invariant system, the superconducting order parameter remains invariant. • Validity:. For , there are several states localized within the energy . Ma and Lee, 1985 • Solution of Gorkov’s equation (near critical temperature): HRI What happens in presence of electron-electron interaction in 2-d superconductor?

3. In the dirty limit, the Coulomb interaction is renormalized and the process of repulsion of electrons with opposite momenta and spins becomes stronger. Maekawa and Fukuyama, 1984 The system is considered as 2-d if thickness HRI Graybeal and Beasley, 1984

4. By RG analysis HRI Finkelstein, 1987

5. Fiory and Hebard, 1984 HRI Anderson and independently by Abrikosov and Gorkov’s theory seems to be correct upto O

6. Superconductor-Insulator Transition in thin film Superconductor HRI What is the nature of this transition? Haviland, Liu and Goldman, 1989; Liu et. et. al. , 1993

7. Quantum Phase Transition: The transition can be initiated by changing certain control parameter such as thickness of film, electron concentration, disorder, magnetic field. Depending on whether Anderson localisation in the normal state or vanishing of superconducting transition temperature occurs earlier, one of these types of transitions occurs. HRI

8. Goldman et al, 1998 Kapitulnik et al, 1992; 1995 HRI Near quantum phase transition,

9. Theoretical Models: Josephson lattice model: Tendency of global phase coherence is offset by charge fluctuations. This model predicts discontinuous S-I transition. Experimentally, the transition is continuous. Phase only model of 2-d SC: Ramakrishnan, 1989 HRI Josephson junction lattice model overestimates the energy cost of time dependent phase fluctuation. Quantum fluctuations are stronger. At T=0, all states are localised in 2-d disordered systems. The stiffness becomes very small. The critical sheet resistance is estimated to be

10. Fisher 1990 Phase only model for bosons: HRI In dirty boson model, insulator and superconductor are self dual. When the system switches from superconductor to insulator, both charges and vortices move. Flow of Cooper pairs results Vortices moving at right Angles to the current, produce a Voltage Self duality suggests at the transition,

11. Fermionic microscopic lattice model Partition function after Hubbard-Stratonovic transformation: Considering only thermal phase fluctuation, resultant partition function HRI Phase of the order parameter is ignored in mean field BdG approximation.

12. Ghosal, Randeria, Trivedi, 1998 and 2001 In absence of magnetic field, BdG equation is numerically solved. HRI

13. HRI

14. Fermionic microscopic lattice model Partition function after Hubbard-Stratonovic transformation: Considering only thermal phase fluctuation, resultant partition function HRI Phase of the order parameter is ignored in mean field BdG approximation.

15. HRI Dubi, Meir, and Avishai, 2007

16. Sacepe et al, 2008 LDOS remains gapped in the insulating side of SIT. HRI

17. Dependence of Tc on disorder in the regime of not so strong disorder HRI Fiory and Hebard, 1984 Chand et al, 2009

18. Chand et al, 2009 HRI

19. Homes et al, 2004 HRI Uemura relation for works only for underdopedcuprates.

20. Ginzburg-Landau like Free Energy Functional Gorkov’s Equations : SSM & Ramakrishnan In the normal state, HRI Superconducting gap function : (Disorder average is taken)

21. Free Energy Functional (Independent of disorder !) (Perturbative expansion when is small) HRI

22. HRI

23. By self-consistent harmonic approximation, HRI

24. HRI Fiory and Hebard, 1984 Chand et al, 2009

25. THANK YOU! HRI