2D Fluctuations: Pictures From Exhibition Andrei Varlamov Varlamov - - PowerPoint PPT Presentation

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In collaboration with In collaboration with Anatoly Larkin Anatoly Larkin and Yuri and Yuri Ovchinnikov Ovchinnikov

2D Fluctuations: Pictures From Exhibition

“Dynamics and Relaxation in Complex Quantum and Classical Systems and Nanostructures” July, 26, 2006, Dresden, Germany

Andrei Andrei Varlamov Varlamov CNR CNR -

• INFM, COHERENTIA,

INFM, COHERENTIA, Rome Rome, , Italy Italy

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Outline Outline

• Specifics of fluctuations in

Specifics of fluctuations in 2D superconductor 2D superconductor

• The width of critical region in

The width of critical region in magnetic field magnetic field

• Relation between T

Relation between TBKT

BKT and T

and T BCS

BCS

• Fluctuation renormalization of the

Fluctuation renormalization of the Josephson Josephson current current

Strong vortex-antivortex fluctuations in

type II superconducting films

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Modulus and Phase fluctuations Modulus and Phase fluctuations

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Phase fluctuations Modulus fluctuations

GL picture BKT picture

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The width of critical region in The width of critical region in magnetic field magnetic field

• a. H=0

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• b. H≠0, T-Tc<< Tc

LLL approximation

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• c. H - Hc2(0) << Hc2(0)

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Critical region width Critical region width

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Superconducting transition in 2D film Superconducting transition in 2D film

• a. Mean field theory

and

• b. Fluctuation GL theory

Tc0

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• c. BKT theory

Interpolation formula:

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Fluctuation renormalization of the Fluctuation renormalization of the Josephson Josephson current current

Mean field in the vicinity of Tc gives:

2D 3D

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GL fluctuation theory gives:

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Exponential tail in Exponential tail in Josephson Josephson junction close to junction close to T Tc

c

+

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Ic(T) Tc0 Tc TBKT

GL perturbative result Non-perturbative resul AB mean field result:

• a. with Tc0
• b. with Tc

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• In the immediate vicinity of transition the thermodynamic

and transport properties of the film are determined by the large size (R>>ξ(T)) vortex-antivortex pairs

• Beyond the critical region Gi<τ<<1 usually the

thermodynamic and transport properties are determined by the long wave-length fluctuation of the order parameter.

• The cornerstone of the presented theory is the fact that the

energy of the vortex-antivortex pair tends zero when R<ξ(T)) and proliferation of such cheap pairs determines the fluctuation thermodynamics in the region of temperatures below transition.

Strong vortex Strong vortex-

• antivortex

antivortex pair pair fluctuations in type II SC film fluctuations in type II SC film

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Partition function

Gas approximation: where is the contribution of the isolated pair of size δ ξ(T), with all 0<δ<1

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The order parameter ∆δ(r) has two zeros

• f the opposite vorticity at the distance 2 δ ξ(T).

Corresponding free energy functional: Fp is the difference between the state with one v-a pair and the ground state with the homogeneous order parameter 2 δ ξ(T).

Order parameter

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Solution of the GL equation

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Contribution to the free energy of the pairs of size 2δ ξ(T) is Functional inegration in our model is equivalent to the account for contributions of all discs of sizes 0<δ<1 Steepest descent approximation results in

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The corresponding free energy The steepest descent approximation is valid when i.e. beyond the critical region, where is valid the concept of small vortex-intivortex pairs itself

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Fluctuation heat capacity

Differentiation of the second term results in the well known positive contribution which occurs due to the long-wavelength order parameter fluctuations:

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Differentiation of the first term gives the contribution

• f the small vortex-antivortex fluctuations:

it is much larger than the former contribution and has the opposite sign with respect to it, smearing the heat capacity jump. In the region of temperatures

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GL results GL +V-A

TBKT T C(T)