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

“Dynamics and Relaxation in Complex Quantum and Classical Systems and Nanostructures” July, 26, 2006, Dresden, Germany 2D Fluctuations: Pictures From Exhibition Andrei Varlamov Varlamov Andrei CNR - - INFM, COHERENTIA, INFM, COHERENTIA, CNR Rome, , Italy Italy Rome In collaboration with Anatoly Larkin Anatoly Larkin In collaboration with and Yuri Ovchinnikov Ovchinnikov and Yuri 11/08/2006 1 11/08/2006 1

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 T BKT and T BCS BKT and T � BCS � Fluctuation renormalization of the Fluctuation renormalization of the � Josephson current current Josephson � Strong vortex-antivortex fluctuations in type II superconducting films 11/08/2006 2 11/08/2006 2

Modulus and Phase fluctuations Modulus and Phase fluctuations 11/08/2006 3 11/08/2006 3

Modulus fluctuations Phase fluctuations GL picture 11/08/2006 4 11/08/2006 4 BKT picture

The width of critical region in The width of critical region in magnetic field magnetic field a. H=0 11/08/2006 5 11/08/2006 5

b. H ≠ 0, T-T c << T c LLL approximation 11/08/2006 6 11/08/2006 6

c. H - Hc 2 (0) << Hc 2 (0) 11/08/2006 7 11/08/2006 7

Critical region width Critical region width 11/08/2006 8 11/08/2006 8

Superconducting transition in 2D film Superconducting transition in 2D film a. Mean field theory T c0 and b. Fluctuation GL theory 11/08/2006 9 11/08/2006 9

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c. BKT theory Interpolation formula: 11/08/2006 11 11/08/2006 11

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Fluctuation renormalization of the Fluctuation renormalization of the Josephson current current Josephson 2D 3D Mean field in the vicinity of T c gives: 11/08/2006 13 11/08/2006 13

GL fluctuation theory gives: 11/08/2006 14 11/08/2006 14

Exponential tail in Josephson Josephson Exponential tail in junction close to T T c junction close to c + 11/08/2006 15 11/08/2006 15

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I c (T) GL perturbative result AB mean field result: b. with T c a. with T c0 Non-perturbative resul T c0 11/08/2006 T BKT T c 17 11/08/2006 17

Strong vortex- -antivortex antivortex pair pair Strong vortex fluctuations in type II SC film fluctuations in type II SC film • 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 11/08/2006 18 11/08/2006 below transition. 18

Partition function Gas approximation: where is the contribution of the isolated pair of size δ ξ (T), with all 0< δ <1 11/08/2006 19 11/08/2006 19

Order parameter The order parameter ∆ δ ( r ) has two zeros of the opposite vorticity at the distance 2 δ ξ (T). Corresponding free energy functional: 2 δ ξ (T). F p is the difference between the state with one v-a pair and the ground state with the homogeneous order parameter 11/08/2006 20 11/08/2006 20

Solution of the GL equation 11/08/2006 21 11/08/2006 21

Contribution to the free energy of the pairs of size 2 δ 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 11/08/2006 22 11/08/2006 22

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 11/08/2006 23 11/08/2006 23

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: 11/08/2006 24 11/08/2006 24

Differentiation of the first term gives the contribution of the small vortex-antivortex fluctuations: In the region of temperatures it is much larger than the former contribution and has the opposite sign with respect to it, smearing the heat capacity jump. 11/08/2006 25 11/08/2006 25

C(T) GL results GL +V-A T BKT T 11/08/2006 26 11/08/2006 26