**7. References**


Clarke, J. & Braginsky, A. I. (2004). *The SQUID Handbook,* Vol. I, Wiley-VCH, ISBN 3-527- 40229-2, Weinheim, Germany

260 Superconductors – Materials, Properties and Applications

φ

ω

= 1.0 can be achieved.

Sons, ISBN 0471014699, New York, USA

(January 1999), pp. 43-45, ISSN 0028-0836

β

device. Further work in extending the present analysis to finite values of

*Dipartimento di Fisica "E. R. Caianiello", Università degli Studi di Salerno, Italy* 

. Averaging of the rapidly varying quantities in the

β

is necessary.

gives the effective dynamics of the two junctions in the system. In

and B can play the role of additional control parameters in the

, the critical current of the device is seen to depend on *A*, on the

and the amplitude *B* of the a. c. component of the applied magnetic flux in a

closed analytic form. From the analysis of the voltage vs. applied flux curves it can be

Experimental work confirming the predictions of the present analysis needs to be performed. As far as non-normalized quantities are concerned, for direct experimental confirmation of the present results, we finally notice that the junction dynamics evolves with characteristic frequencies the order of 1 THz. Therefore one needs to run the experiment with very rapidly oscillating signals (10 THz or more) in such a way that normalized

Ambegaokar, V. & Halperin, B. I. (1969). Voltage Due to Thermal Noise in the dc Josephson Effect. *Phys. Rev. Lett.,* Vol.22, No.25, (June 1969), pp. 1364-1366, ISSN 0031-

Barone, A. & Paternò, G. (1982). *Physics and Applications of the Josephson Effect,* John Wiley &

Baselmans, J. J.; Morpurgo, A. F.; Van Wees, B. J. & Klapwijk, M. (1999). Vortices with Half Magnetic Flux Quanta in "Heavy Fermion" Superconductors . *Nature,* Vol.397, No.6714,

Bishop, A. R. & Trullinger, S. E. (1978). Josephson Junction Threshold Viewed as a Critical Point. *Phys. Rev. B,* Vol.17, No.5, (March 1978), pp. 2175-2182, ISSN 1098-0121 Bocko, M. F.; Herr, A. M. & Feldman, M. J. (1997). Prospect for Quantum Coherence Computation Using Superconducting Electronics. *IEEE Transactions on Applied Superconductivity,* Vol.7, No.2, (June 1997), pp. 3638-3641, ISSN 1051-

Bulaevkii, L. N.; Kuzii, V. V. & Sobyanin, A. A. (1977). Superconducting System with Weak Coupling to the Current in the Ground State. *JETP Lett.,* Vol.25, No.7, (April 1997), pp.

Chesca, B. (1999). Magnetic field dependencies of the critical current and of the resonant modes of dc SQUIDs fabricated from superconductors with s+idx2-y2 order parameter symmetries. *Ann. Phys.(Leipzig),* Vol.8, No.6, (September 1999), pp. 511-522, ISSN 1521-

analysis to first order in the parameter

β = 0

differential equation for

ω

argued that the quantities

ω

particular, for

frequencies of

R. De Luca

**Author details** 

**7. References** 

9007

8223

3889

290-294, ISSN 0021-3640

frequency

	- Wollman, D. A.; Van Harlingen, D. J.; Lee, W. C.; Ginsberg, D. M. & Leggett, A. J. (1993). Experimental Determination of the Superconducting Pairing State in YBCO from the phase coherence of YBCO-Pb dc SQUIDs. *Phys. Rev. Lett.,* Vol.71, No.13, (September 1993), pp. 2134-2137, ISSN 0031-9007

**1. Introduction**

http://dx.doi.org/10.5772/48358

**1.1. The essential physical background**

**Pinning Potential** 

the effective magnetic penetration depth.

sample is strongly nonlinear.

work is properly cited.

It is well-known that a type-II superconductor, while exposed to a magnetic field **B** whose magnitude is between the lower and upper critical field, is penetrated by a flux-line array of Abrikosov vortices, or *fluxons* [1–3]. Each vortex contains one magnetic flux quantum, <sup>Φ</sup><sup>0</sup> <sup>=</sup> 2.07 <sup>×</sup> <sup>10</sup>−<sup>15</sup> Wb, and the repulsive interaction between vortices makes them to arrange in a triangular lattice, with the vortex lattice parameter *aL* � <sup>√</sup>Φ0/*<sup>B</sup>* where *<sup>B</sup>* <sup>=</sup> <sup>|</sup>**B**|. A vortex is often simplified by the hard-core model [4], where the core is a cylinder of normal material with a diameter of the order of the coherence length. In this model, the magnetic field is constant in the core but decays exponentially outside the core over a distance of the order of

**Microwave Absorption by Vortices in** 

**Chapter 11**

**Superconductors with a Washboard** 

Valerij A. Shklovskij and Oleksandr V. Dobrovolskiy

Additional information is available at the end of the chapter

In an ideal material, the vortex array would move with average velocity **v** under the action of the Lorentz force **F***<sup>L</sup>* essentially perpendicular to the transport current. Due to the nonzero viscosity experienced by the vortices when moving through a superconductor, a faster vortex motion corresponds to a larger dissipation. In experiments, inhomogeneities are usually present or can intentionally be introduced in a sample [5] which may give rise to local variations of the superconducting order parameter. This may cause the vortices to be pinned. By this way, the resistive properties of a type-II superconductor are determined by the vortex dynamics, which due to the presence of pinning centers can be described as the motion of vortices in some *pinning potential* (PP) [6]. In particular, randomly arranged and chaotically distributed point-like pinning sites give rise to an ubiquitous, *isotropic* (*i*) pinning contribution, as said of the "background nature". Depending on the relative strength between the Lorentz and pinning forces, the vortex lattice can be either pinned or on move, with a nonlinear transition between these regimes. Thus, the current-voltage characteristics (CVC) of such a

> ©2012 Shklovskij and Dobrovolskiy, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

©2012 Shklovskij and Dobrovolskiy, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.
