Marek Jaworski Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ ow 32/46, 02-668 Warszawa, Poland Surface losses and self-pumping effects in a long Josephson junction — a semi-analytical approach arXiv:1001.0671v1 [cond-mat.supr-con] 5 Jan 2010 The flux-flow dynamics in a long Josephson junction is studied both analytically and numerically. A realistic model of the junction is considered by taking into account a nonuniform current distri- bution, surface losses and self-pumping effects. An approximate analytical solution of the modified sine-Gordon equation is derived in the form of a unidirectional dense fluxon train accompanied by two oppositely directed plasma waves. Next, some macroscopic time-averaged quantities are calculated making possible to evaluate the current-voltage characteristic of the junction. The re- sults obtained by the present method are compared with direct numerical simulations both for the current-voltage characteristics and for the loss factor modulated spatially due to the self-pumping. The comparison shows very good agreement for typical junction parameters but indicates also some limitations of the method. PACS numbers: 74.50.+r, 05.45.Yv, 85.25.Cp In recent years the flux-flow (FF) dynamics in a long Josephson junction has attracted considerable attention in view of possible applications in superconducting mm- wave electronics $^{1–4}$. The FF mode appears in a junction immersed in a sufficiently large external magnetic field and can be described briefly as a unidirectional viscous flow of a dense train of fluxons (magnetic flux quanta). Recently, the FF oscillators have found various appli- cations, e.g. in mm-wave integrated receivers $^{4–7}$, however some important parameters, such as the radiation line- width are still not satisfactory. Therefore, there is still a need for more adequate description of a real Joseph- son junction operating in the FF mode, by taking into account some additional factors which may affect the current-voltage ( I - V ) characteristic of the junction and change its working conditions. In the majority of papers dealing with long Josephson junctions (see e.g. Refs. 8–14) a simplified version of the the sine-Gordon (sG) equation is considered, usually ne- glecting surface losses and assuming uniform distribution of the bias current density. Only recently, a few papers have been published 15–18 taking into account a nonuni- form bias current distribution and its influence on the junction behavior. Moreover, in Refs. 16,17 a realistic model of the FF oscillator has been investigated both ex- perimentally and numerically, including general bound- ary conditions, nonuniform bias current profile, surface losses as well as self-pumping effects 19 related to addi- tional tunneling of quasiparticles due to the Josephson radiation. The aim of the present paper is to present an analytical approach to the modified sG equation, which takes into account:(i) nonuniform current distribution, (ii) surface losses, and (iii) spatial modulation of the loss factor re- sulting from the self-pumping effect. Contrary to Ref. 16 we assume standard open-circuit boundary conditions to make the influence of various effects more pronounced. The paper is organized as follows. In Sec. II we for- mulate the problem, i.e. we present a generalized sG equation subject to open-circuit boundary conditions at the junction ends. Approximate analytical solutions to the sG equation are discussed in Sec. III. We start with a linearized (small-amplitude) solution and apply appro- priate boundary conditions. Next, some large-amplitude corrections are introduced, we discuss also possible self- pumping effects and their influence on the I - V character- istic. In Sec. IV analytical results are compared with di- rect numerical simulations. In particular, we discuss the influence of surface losses both for junctions of moderate length and for more realistic structures, such as very long junctions with small damping and strongly nonuniform current distribution. Sec. V contains concluding remarks, we indicate also possible extensions of the method by tak- ing into account more general boundary conditions. In the particular case of uniform current distribution, the present method makes it possible to obtain fully an- alytical closed-form expressions describing both the su- perconducting phase within the junction and the I - V characteristic. Such a solution can be regarded as the first-order approximation which appears sufficiently ac- curate for some moderate junction parameters. How- ever, in the general case, particularly for very long and weakly damped junctions, such an analytical approxi- mation is only the first step in an iterative procedure, which has to be performed numerically. Thus, in spite of analytical expressions describing the superconducting phase within the junction, the present approach has been named “semi-analytical”. Nevertheless, the present analysis can be easily extended to include also more general boundary conditions. I. INTRODUCTION