Nonlinear motion of vortices in HTSC after interacting with external RF field

1. QFS’ 2012 Lancaster, UK Aug. 2012 Nonlinear motion of vortices in HTSCafter interacting with external RF field D.Apushkinskaya1, E.Apushkinsky2, and M.Astrov3 • Department of Mathematics, Saarland University, Saarbrücken, D-66041, Germany;e-mail: darya@math.uni-sb.de • 2. Experimental Physics Department, St.Petersburg State Polytechnical University, St.Petersburg, 195251 Russia;e-mail: apushkinsky@hotmail.com • Superconducting Magnet System Department, D.V.Efremov Scientific Research Institute of Electrophysical Apparatus, St.Petersburg,196641 Russia; e-mail: astrov@sintez.niiefa.spb.su Summary:A type II superconductor (SC) is placed in a dc magnetic field with the value above the lower critical one. This field penetrates inside SC in the form of individual vortices, which may be “pinned” to the SC crystal lattice in cites of its defect. Also an ac radio-frequency (RF) field (orthogonal to a dc field) is applied on the boundary of SC. The ac field simulates vortex oscillations through the pinning centers and induce a propagating sound wave which causes the vortex oscillations in the whole volume. If an external influence is sufficiently strong, we can neglect the vortex mass in the equation of the vortex motion. As a result, we get a parabolic free boundary problem, where a-priori unknown free boundary describes the set of points at which transition from SC state to the normal state or vice versa is realized. From the general theory of free boundary problems we get some information about the interface between SC and normal state, as well as about qualitative properties of the vortex tension and decay of the vortex oscillations. PARABOLIC OBSTACLE PROBLEM The model equation Eq. (2) can be treated as the following one dimensional parabolic obstacle problem QUALITATIVE DESCRIPTION A type II superconductor material is placed in a dc magnetic field B0. It is assumed that Bc1< |B0|<Bc2, where Bc1 and Bc2 denote lower and upper critical fields, respectively. In this case, there are many non-superconducting regions inside the sample formed by the cores of vortices [1]. The vortices have different lengths provided an intricate configuration of the superconducting grains. In addition, the vortices may be “pinned” to the superconductor crystal lattice in cites of its defect. Since the crystal defects are distributed inside a grain in a complicated manner, the configuration of every magnetic filament (vortex) is also quite complicated [2]. Besides dc field B0 an ac magnetic field B~ B0 is applied to SC sample. It is known that acoustic vibrations of superconductor crystal lattice, which registered as echo signals [3], can be excited by exposing to vortex lattice the electromagnetic oscillations. According to [4] and [5] this phenomenon has the following mechanism. The ac magnetic field B~ simulates vortex oscillations [6] on the SC surface (see Fig.1), they are transformed into lattice oscillations through the pinning centers and induce a propagating sound wave. Spreading into SC such a wave causes the vortex oscillations in the whole volume. (3) (4) For the coefficients a, b, c and the function f, we assume that (i) a and f are non-degenerate in QR, i.e., there exists 0 >0 such thata(x,t)> 0 and f(x,t)> 0 for any (x,t) QR; (ii) a, b, c and f belong to Cα,α/2(QR) for some α(0,1). Observe that {v>0} is a priori an unknown subset of QR. We denote by  the intersection of QR with the boundary of the set{v=0}. We will call  thefree boundary. Qualitative Properties of v (S1) Under assumptions (i)-(ii), the problem (3)-(4) has a unique solution v for suitable initial and boundary data [11]. (S2) From standard regularity theory for parabolic equations [12,13] it follows that v and v/x are continuous and the set {v=0}is closed. (S3) Up to a reduction of the size of the parabolic box, any solution v has a bounded first derivative in time and bounded second derivatives in space [14]. (S4) 2v/x2is discontinuous across . (S5) v/tis continuous for almost every t. More precisely, v/t  0 is continuous in the interior of the set {v=0}. Otherwise, if v(x0,t0) > 0, by standard parabolic estimates it follows that v/tis continuousin a neighborhood of (x0,t0). Therefore, the only difficulty is the regularity of v/t on the free boundary  . We emphasize that the continuity of v/tcannot be obtained everywhere in t. The relevant counterexample can be found in [14]. (S6) If, in addition, v/t≥ 0, then v/tis continuous for all t and satisfies v/t = 0 on  [14]. The assumption that v/t is nonnegative can be established in some special cases (special initial conditions, special boundary conditions and time independent coefficients). (S7)vis non-degenerate near the free boundary  The latter means that there exists an absolute constant c > 0 such that where Inequality (5) holds true for all points (x0,t0) belonging to the closure of the set {v>0}and for all ρsufficiently small [15]. It is evident that estimate (5) prevents the free boundary from being flat with the region {v=0}below (such impossible case is shown in Fig.2) MATHEMATICAL MODEL Similarly to [6] we write the equation of motion for a vortex displacement v for volume element dV. Fig.1. Vortex oscillation caused by ac field action (1) where, • is the viscous decay force. It takes place only at the vortex ends, due to radiation losses, and at the pinning centers, due to the energy transfer to the crystal lattice. This force increases if field and temperature increase [7] • Frestis the retrieving force. It summarizes the influence of other vortices as well as the voltage induced by randomly distributed pinning centers. This force is zero in the equilibrium state; for the nonequilibrium state it can be written as Here αL is the Labusch parameter, while the value of f(x,t) is determined by the distribution of pinning centers, the entanglement of the vortices, the anisotropic properties of SC and the transport current, if the latter exists [8]. The function χ{v>0} denotes the characteristic function of the set {v>0},i.e., Fig. 2. Such a case is not possible! Fig.3. Such a case is allowed • INTERPRETATION • Properties (S2) and (S3) guarantees that the propagation of superconductivity along the sample proceeds with the speed bounded for almost all t. • The latter means that only for isolated values of t the function v/t is unbounded or is not defined. Formation or destruction of superconducting carriers occur precisely in such moments. • From the nondegeneracy property (S7) it follows that the formation of superconducting carriers can not occur simultaneously on the spatial interval. The opposite case, i.e., simultaneous destruction of carriers on the spatial interval, is allowed (see Fig.3). • Due to property (S2) a vortex remains elastic at all values of x and t. • A free boundary  describes the set of points where transition from SC state to the normal state or vice versa is realized. In view of (FB2), for almost every t the set can be defined locally by the equation x=g(t) with bounded continuous function g satisfying • where C is an absolute constant. Thus, in the local coordinate system we have for almost all t the growth estimate for function describing . . (5) • is the Ampere force [9]. Here is the vortex lattice modulus of elasticity [10] • FT=FT(t)denotes the random force of thermal deviation, which we consider to be zero due to time averaging. The vortex mass m can be neglected under strong external action and Eq. (1) reduces to the form: The approximate solutions of Eq. (2) agree well with experimental data obtained for BiPbSrCaCuO samples [9].  In the present work we use the model equation to get a parabolic free boundary problem, where a-priori unknown free boundary describes the set of points at which transition from SC state to the normal state or vice versa is realized. Since the vortex deviations from the equilibrium state in positive and negative directions are absolutely symmetric, we restrict our analysis to the case v≥0 (2) Properties of the Free Boundary (FB1) Estimate (5) guarantees that can not appear or disappear suddenly, or is not “blurred”. (FB2) There exist a set *  such that for almost every time there is no point of  \ *. The set  * is locally contained in a C1/2-graph as a function of t [15]. The latter can be understand as follows. We say that the set * is defined locally by the equationx=g(t), t(t1,t2), for some bounded continuous function g and for points t,s(t1,t2), ts we have References Parabolic free boundary problem [11] A. Friedman, Variational principles and free-boundary problems (Robert E. Krieger Publishing Co. Inc., Malabar, FL, 1993). [12] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type (AMS, Providence, RI, 1967). [13] G.M. Lieberman, Second order parabolic differential equations (World Scientific Publishing Co. Inc., River Edge, NJ, 1996). [14] A. Blanchet, J. Dolbeault and R. Monneau, J. Math. Pures Appl. (9) 85, 371, (2006). [15] A. Blanchet, Nonlinear Anal. 65, 1362, (2006). Vortex lattice dynamics [1] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994) [2] T. Onogi, T. Ichiguchi and M. Ban, Hitachi Review. 39, 41, (1990) [3] M.P. Petrov, I.V. Pleshakov, A.P. Paugurt and et. al., Solid State Commun. 78, 893 (1991) [4] D. Dominguez, L. Bulaevskii, B. Ivlev, M. Maley and A. R. Bishop, Phys. Rev. B 51, 15649, (1995) [5] H. Haneda, T. Ishigura and M. Miriam, Appl. Phys. Lett. 68, 3335, (1996) [6] C. J. van der Beek, V. B. Geshkenbein and V. M. Vinokur, Phys. Rev. B 48, 3393, (1993). [7] J. Pankert, G. Marbach, A. Comberg, P. Lemmens, P. Froning and S. Ewert, Phys. Rev. Lett. 65, 3052, (1990). [8] A.M. Campbell and J.E. Evetts, Critical Currents in Superconductors (Taylor and Francis, London, 1972). [9] E.G. Apushkinsky, M.S. Astrov and V.K. Sobolevskii, Tech. Phys. 56, 788, (2011). [10] R. Labusch, Phys. Status Solidi A 32, 439, (1969). Acknowledgements. The first author was supported by the Russian Foundation of Basic Research (RFBR) through the grant number 11-01-00825 and by the Russian Federal Target Program 2010-1.1-111-128-033.