1. R OLE OF D ISORDER IN S UPERCONDUCTING T RANSITION Sudhansu S. Mandal IACS, Kolkata HRI 1

2. A NDERSON ’ S T HEOREM 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): 2 HRI What happens in presence of electron-electron interaction in 2-d superconductor?

3. HRI 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 Graybeal and Beasley, 1984

4. HRI 4 By RG analysis Finkelstein, 1987

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

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

7. 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. 8 Kapitulnik et al, 1992; 1995 Near quantum phase transition, Goldman et al, 1998

9. HRI 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: 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 Ramakrishnan, 1989

10. HRI 10 Fisher 1990 Phase only model for bosons: 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. HRI 11 Fermionic microscopic lattice model Partition function after Hubbard-Stratonovic transformation: Considering only thermal phase fluctuation, resultant partition function Phase of the order parameter is ignored in mean field BdG approximation.

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

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

15. HRI 15 Dubi, Meir, and Avishai, 2007

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

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

18. HRI 18 Chand et al, 2009

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

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

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

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23. HRI 23 By self-consistent harmonic approximation,

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

25. HRI 25