PHYSICAL REVIEW B VOLUME 44, NUMBER 2 1 JULY 1991-II Dynamics of a superconductive filament in the constant-voltage regime Boris A. Malomed P.P. Shirkou Institute for Oceanology, 28 Krasikou Street, Moscour, 117259, U S.S.R. Andreas Weber Physikalisches Institut der Universitat Bayreuth, 8580 Bayreuth, Postfach 101951, Federal Republic of Germany (Received 28 February 1991) The system of time-dependent Ginzburg-Landau (GL) equations governing the dynamics of a dirty superconductive filament near T, is solved numerically with the boundary conditions corresponding to a constant voltage between the edges of the filament. It is demonstrated that, while at small values of the length I of the filament only a time-periodic regime exists, a period doubling occurs at larger I. With the subsequent increase of I. , several independent frequencies arise, each corresponding to a periodically appearing phase-slip center. For I sufficiently large, the quasiperiodic regime seems to become chaotic. These results are similar to those obtained previously for a model GL equation with a frequency inhomogeneity, and are in contrast with the situation for the same system in the fixed-current regime, where only time-periodic states are known. I. INTRODUCTION In this work, we present numerical simulations of the dynamics of a superconducting filament. The transverse dimensions are taken small compared to the coherence length and the penetration depth and so one can con- sider a one-dimensional system and neglect the eAects due to the magnetic field of the current. The correspond- ing system of dynamical Ginzburg-Landau equations for dirty superconductors is well known: This state has been found numerically, and it proved to be stable. ~ The aim of the present paper is to investigate numer- ically the dynamics of the filament under fixed-voltage conditions. One may think of this filament as connect- ing two bulk superconductors maintained at two dift'erent potentials. The boundary conditions to Eqs. (1) and (2) are then e(z = 0) = O, e(z = 0) — = 1, 4(z = I) = y, 4(z = L) = e (4) (2) where @(z, t) is the order parameter in the filament, 4(z, t) is the electric potential, j(t) is current Sowing through the filament, and c is a parameter stemming from the microscopic theory. It equals 5.79 for nonmagnetic impurities and 12 for magnetic impurities (strong depair- ing). The current j may vary in time, but cannot depend on z. The usual dimensionless units are used. From the experimental point of view two dif- ferent situations are of interest. Either one has a fixed current j or a fixed de'erence of poten- tials (voltage) between the ends of the filament. The former situation has been studied in detail previously. ' It is well known that the stationary solu- tion corresponding to the purely superconductive state 4 = 0, 4 = F exp( — iQz), Qz = 1 — F, j = Fz Q ex- ists linearly stably below the critical value j, = ~. In ~27 a narrow range between jul;„and j~a„(( j,) one also has a stable time-periodic state with (spatial) periodi- cally placed phase-slip centers (PSC's), i.e. , points where the order parameter vanishes periodically in time. Above j the purely superconducting state is globally unsta- ble, i.e. , localized fluctuations nucleate the normal state. where p is the difference of the potentials and L is the length of the filament. The current j may be an arbitrary function of time, which is to be found together with the unknown quantities @(z, t) and 4(z, t). We set c = 5.79 and solve the boundary value problem based on Eqs. (1)— (4) regarding L and p as control parameters. II. NUMERICAL SIMULATIONS Our numerical technique was based on solving Eqs. (1) and (2) separately. From Eq. (2) one can determine 4 and j for a given function 4'(z, t) Inserting th. e results into Eq. (1) we determine 4 for the next time step. Equa- tion (1) is solved by means of a Newton-Kantarowich scheme (linearization with the Taylor expansion), while Eq. (2) is integrated directly. Due to the separation of the equations the time step of the numerical scheme has to be chosen rather small and usually was taken as 0.025. The mesh of the spatial grid employed was typically 0.02. We proceed with the description of the results. From the boundary conditions [Eqs. (3) and (4)], one expects that the quantity y plays the role of a driving frequency for the nonlinear system under consideration. Indeed, the numerical solution for I suKciently small demonstrates a sharp peak in the power spectrum of the current j(t) just 875 1991 The American Physical Society