8www.intechopen.com 2 Will-be-set-by-IN-TECH BSCCO consists of a stack of SIS-type IJJs, so it is expected that the BSCCO should be regarded as a good sample to study the fluxon dynamics in the SIS Josephson junctions. Actually, there are many excellent works for the effect of magnetic field on the BSCCO IJJs. [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] It is sure that a pioneering work done by Sakai, Bodin and Pedersen(SBP) [20] is the starting point to understand the fluxon dynamics in BSCCO IJJs with a stacking structure.We adopt here a unified theory proposed by Machida and Sakai (MS) [29], which includes both the electric and magnetic field couplings between neighboring IJJs in multistacked structure. The MS-unified theory is a further extension of the model proposed by Sakai, Bodin and Pedersen (SBP).[20] Present paper is mainly based on my previous jobs. [35,36]
TheoryLet us consider a stacked system consisting of N-identical SIS-Josephson junctions such as IJJs in BSCCO high-T c superconductors, and set spacial coodinates x, y and z parallel to a, b and c axes of a BSCCO single crystal. If the external magnetic induction B ext is selected parallel to y-axis, i.e., B ext =(0, B ext ,0), and the spacial variation of the gauge-invariant phase difference ϕ ℓ for the ℓ-th SIS Josephson junction is assumed as a function of only x, then the ϕ ℓ (x, t) as a function of x and real time t satisfies a following coupled sine-Gordon(CSG) equation with a matrix form.[29]where λ J is the Josephson penetration depth given by Φ 0 /2πµ 0 d ′ L J c using a flux quantum Φ 0 , the vacuum permeability µ 0 and the critical current density J c . The Σ c and Σ L are matrices describing the electric and magnetic interactions between neighboring Josephson junctions and are written aswhere the coupling constants Σ c and Σ L are given by [29]The d ′ c and d ′ L are the effective electric and magnetic thickness given byusing electrode thickness w and barrier thickness d. The λ e and λ L are the Debye screening length and the London penetration depth, respectively. The J ℓ (x, t) in Eq.(2) is the current defined by j