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ROLE OF DISORDER IN SUPERCONDUCTING TRANSITION

Sudhansu S. Mandal

IACS, Kolkata

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IANDERSON’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):

What happens in presence of electron-electron interaction in 2-d superconductor?

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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 ifthickness

Graybeal and Beasley, 1984

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By RG analysis

Finkelstein, 1987

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Fiory and Hebard, 1984

Anderson and independently by Abrikosov and Gorkov’s theory seems to be correct upto O

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Haviland, Liu and Goldman, 1989; Liu et. et. al. , 1993

Superconductor-Insulator Transition in thin film Superconductor

What is the nature of this transition?

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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.

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Kapitulnik et al, 1992; 1995

Near quantum phase transition,

Goldman et al, 1998

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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

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Fisher 1990

Phase only model for bosons:

In dirty boson model, insulator and superconductor are self dual. When thesystem switches from superconductor to insulator, both charges and vorticesmove. Flow of Cooper pairs results Vortices moving at rightAngles to the current, produce a Voltage

Self duality suggests at the transition,

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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.

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In absence of magnetic field, BdG equation

is numerically solved.

Ghosal, Randeria, Trivedi, 1998 and 2001

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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.

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Dubi, Meir, and Avishai, 2007

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Sacepe et al, 2008

LDOS remains gapped in theinsulating side of SIT.

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Dependence of Tc on disorder in the regime of not so strong disorder

Fiory and Hebard, 1984 Chand et al, 2009

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Chand et al, 2009

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Homes et al, 2004

Uemura relation for works only for underdoped cuprates.

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Ginzburg-Landau like Free Energy Functional

Gorkov’s Equations :

In the normal state,

Superconducting gap function :

(Disorder average is taken)

SSM & Ramakrishnan

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Free Energy Functional

(Perturbative expansion when is small)

(Independent of disorder !)

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By self-consistent harmonic approximation,

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Chand et al, 2009Fiory and Hebard, 1984

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THANK YOU!