Early stages of magnetization relaxation in superconductors
Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by in the narrow initial time interval and by at later times before the flux creep sets in. The characteristic times and are proportional to the viscous drag coefficient . Quantitative agreement with available experimental data is obtained for both conventional and high-temperature superconductors with exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicates a strong influence of pinning centers on the flux diffusion. Notwithstanding the complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient.
Magnetic flux penetrates a type-II superconductor in the form of discrete quantized vortices. Vortex structures in conventional and high-temperature superconductors display remarkable complexity both in equilibriumCrabtree and Nelson (1997); Brandt (1995) and dynamic regimes.Anderson and Kim (1964); Bardeen and Stephen (1965); Beasley et al. (1969); Tinkham (1996); Blatter et al. (1994); Yeshurun et al. (1996); Marchetti and Nelson (1990); Okada et al. (2012); Raes et al. (2012); Lin et al. (2012) Relaxation of the magnetic moment of superconductors is achieved through initial viscous flux flow Kunchur et al. (1993); Moshchalkov et al. (1989); *MoshchalkovPHYSICAB169-91; Pardo et al. (1998); Troyanovski et al. (1999); Deng et al. (2012); Monarkha et al. (2012) and slow, logarithmic in time, thermally activated creep.Feigel man et al. (1989); Vinokur et al. (1991); Abulafia et al. (1995); Burlachkov et al. (1998); Gurevich and Küpfer (1993) Thermally-assisted hopping of vortices and vortex bundles between local minima in the random pinning potential is characteristic of both the creep and the flux flow under a driving force. In the latter, the hopping gives rise to the viscous drag coefficient , where is the effective activation energy and is the temperature.Abulafia et al. (1995) A free flux flow regime can be realized at microwave frequencies (GHz) when the effect of the pinning is negligible. Measurements of surface impedance give viscous drag coefficients at low temperatures for all superconductors, e.g., conventional ,Hor et al. (2005) cuprates YBCO, BSCO,Pompeo and Silva (2008); Golosovsky et al. (1996) and pnictide LiFeAs.Okada et al. (2012) The order of magnitude is in accordance with the Bardeen-Stephen result for the viscous drag, , caused by dissipation in the vortex core ( is the flux quantum, is the normal-state resistivity, and is the upper critical field).Bardeen and Stephen (1965)
In this paper we study early stages of the flux dynamics after switching off the external magnetic field. We use a simple hydrodynamic approach: The local force the vortex experiences due to interaction with other vortices, the surface, and the local quenched disorder (pinning centers) is described by an effective viscosity .The same approach successfully describes the vortex creep, if supplemented by a phenomenological model of current-dependent or time-dependent activation energy, or , where is the critical current and is the characteristic time scale for flux creep.Feigel man et al. (1989); Vinokur et al. (1991); Abulafia et al. (1995); Burlachkov et al. (1998); Gurevich and Küpfer (1993)
We consider a model of massless vortex motion where the driving Lorentz force equals the viscous drag . Here, is the current density, is the vortex velocity, and is a viscous drag coefficient. For magnetic induction related to the vortex density , the force balance equation reads , with . Taking into account the continuity equation , the dynamics of the magnetic flux in a superconductor is described by the well-known nonlinear diffusion equationVinokur et al. (1991); Abulafia et al. (1995); Burlachkov et al. (1998)
We have solved Eq. (1) for three sample geometries: a slab, a square-shaped plate, and a disk (see Fig. 1). We assume the sample thickness along the field is sufficiently large and neglect stray fields on the top and bottom of the sample. Magnetic induction is directed along the sample symmetry axis and satisfies the following initial and boundary conditions: (i) is uniform within the sample at , , and (ii) vanishes at the sample edges for .
For a long superconducting slab of width , Eq. (1) reads
where at for . We can seek the solution in the form
with functions to be determined from Eq. (2) and . This gives the following set of differential equations:
with the initial conditions (). Here, the coefficients are given by
for and odd, and otherwise. The characteristic time constant is
Equations (4) are solved by truncating the system at sufficiently large ().
The induction for the slab is shown in Fig. 2 at various times , , , …, . We observe that the flux flow near the sample edges in the initial time interval is very fast, reaching the center of the slab () at time after switching off the field. This regime is followed by a slower flux flow taking place in the bulk of the sample.
The spatial dependence of the magnetic induction is in accordance with the previous results for the flux flow regime with constant activation energy.Burlachkov et al. (1998) In the presence of flux creep, which may take place in the center of the slab for , or at large times when remanent magnetization is small, a phenomenological model of current and field-dependent activation energy should be used.Feigel man et al. (1989); Vinokur et al. (1991); Abulafia et al. (1995); Burlachkov et al. (1998); Gurevich and Küpfer (1993) Note that the obtained shown in Fig. 2 is qualitatively different from the solution of Eq. (2) when the field is switched on at . In that case the magnetic field enters the sample in the form of a flux front propagating from the edges.Vinokur et al. (1991); Burlachkov et al. (1998); Landau and Lifshitz (1987); *bass_nonlinear_1998 Magnetic induction in the vicinity of the front is a linear function of the coordinate, , with and being the position and the velocity of the front. In our case, the field is switched off at and the flux escapes the sample with no front in formed even at . Indeed, at a sufficiently large distance from the edge, Eq. (2) can be linearized with respect to , which gives the exponential decay () characteristic of the linear diffusion.
In the following we study the dynamics of the average magnetic induction ( is the sample area) which is proportional to the magnetic moment that can be measured. There are two regimes of the flux dynamics in the system. At very short times after switching off the field, the flux flow is localized near the edges and is unaffected by the sample size. In this case, the solution for a half-infinite superconductor is a good approximation, .Bryksin and Dorogovtsev (1993) Here, is a dimensionless function to be determined from Eq. (2) for the half-infinite superconductor with the boundary conditions and . Using the above expression for and taking into account that it deviates significantly from in the vicinity of the edges, we find for the average induction , where is the perimeter of the sample. This gives the magnetic moment relaxation
with the time constant
where the numerical prefactor characterizes the spatial spread of away from the edges. Comparison with the numerical solution for is shown in the inset of Fig. 3 for different sample geometries. We find that Eq. (7) is a good approximation of the exact in the short initial time interval before the flux flow reaches the center of the sample. The flux flow in this time interval is very fast, leading to a reduction of the overall magnetic moment.
At times the flux flow extends through the whole sample, giving rise to the magnetization relaxation which depends on geometry. For the superconducting slab, the first-order approximation of Eqs. (4) for reads . Truncating Eqs. (4) at , a practically exact solution is obtained. This solution can be approximated by a simple formula, , which is very close to the exact one for . This suggests that the exact solution for the magnetic moment can be accurately approximated by
where is a number which depends on geometry. Fitting the exact numerical solution for to Eq. (9) we find , , and for the slab, square, and disk geometries, respectively (Fig. 3). The fitting ensures the smallest absolute error between exact and fitted for . As expected, the decay of is slower (that is, geometric factor is larger) for the slab than for the disk, other parameters being equal.
In what follows, we analyze available experimental data on and extract the characteristic time constant as well as the effective drag . Relaxation of the magnetic moment in BSCO single crystals is studied in Ref. Moshchalkov et al., 1989. Experimental data are shown in Fig. 4 (open circles) fitted to Eq. (9) (solid curve) with and . The fitting is performed for the initial time interval before logarithmic in time, thermally activated flux creep sets in. The linear time dependence of the inverse magnetic moment is shown in the inset of Fig. 4; the crossover between flux flow and flux creep regimes is seen as a dramatic change of the slope at .
Let us now extract . The dimensions of the sample used in the experiment are , which gives mT, where is the volume. Taking for the slab of the width , we obtain . This value for the effective vortex viscosity exceeds by six orders of magnitude the Bardeen-Stephen drag coefficient measured in BSCO.Golosovsky et al. (1996) Huge enhancement of the drag indicates a strong influence of the pinning on the vortex diffusion. Despite the complexity of the vortex motion in the presence of pinning and thermal agitation, the magnetization follows a simple algebraic time dependence, Eq. (9).
Vortex dynamics has been studied in using the decoration technique for visualization of flowing vortex lattices.Pardo et al. (1998) Magnetization measurements have been performed using the SQUID (superconducting quantum interference device) magnetometry. A crossover has been observed as a function of increasing flux density from a layered (smectic) flowing flux lattice in the disorder-dominated low-field limit to a more ordered (Bragg glass) lattice structure in the interaction-dominated high-field case. The observed time dependence of magnetization relaxation in the high-field limit (mT) is shown in Fig. 5. The regimes indicated in Fig. 5 correspond to the flux flow and to the quasistatic vortex motion. The solid curve in Fig. 5 is the fit of to Eq. (7) for the sample mm in size, which gives the relaxation time min and the viscous drag coefficient . We observe that the simple hydrodynamic model with an effective viscous drag force fits the data in the initial stages of magnetization relaxation where the vortex density is large and the flux flow takes place. The flux flow is localized near the edges, as corroborated experimentally by a small reduction of the magnetic moment of the sample over the measurement time and, more directly, by observing the static vortex structure in the center of the sample.Pardo et al. (1998) Large effective is clearly due to hopping caused by successive pinning and thermally-assisted depinning of vortices, as evidenced by studying single-vortex dynamics in pristine monocrystals by scanning tunneling microscopy.Troyanovski et al. (1999)
Magnetic moment relaxations in YBCO polycrystalDeng et al. (2012) and monocrystalMonarkha et al. (2012) are shown in Figs. 6 and 7. The relaxation in the polycrystalline YBCO (rectangular geometry, mm in size) is studied at , , , , and K. The initial stage of magnetization relaxation can be fitted by Eq. (7) describing the flux flow in the vicinity of the edges (Fig. 6, solid curves). This is in agreement with the observed small reduction of the overall magnetic moment during the measurement. The obtained effective viscosity strongly depends on temperature, ranging between and as the temperature is increased from K to K, see inset of Fig. 6. The extracted value is consistent with the value measured independently at the same temperature by studying the spatiotemporal change of the magnetization profile in a bulk YBCO sample in the flux-flow regime.Bondarenko et al. (2006) Taking and neglecting the temperature dependence of the effective activation energy as well as of the prefactor, we find in accordance with the previous results.Abulafia et al. (1995)
Magnetic moment relaxation in small YBCO monocrystal () is shown in Fig. 7.Monarkha et al. (2012) The data can be fitted with the effective viscous drag coefficients and at temperatures of K and K, respectively. The decrease of in such a narrow temperature range may be due to the proximity of the critical temperature (K) where fluctuations are pronounced. In addition, the sharp change in the relaxation rate observed at K and min suggests that the flux flow is inhomogeneous and made of large domains which, upon depinning, abruptly increase the magnetic moment relaxation rate.
In conclusion, we have studied vortex dynamics in type-II superconductors in the initial time interval before the flux creep sets in. We have used a simple phenomenological (hydrodynamic) model of nonlinear diffusion of massless vortices where pinning of the flux lines by material inhomogeneities, interaction with other vortices and the surface, and the Bardeen-Stephen dissipation in the vortex core are described by an effective viscous drag coefficient, . After switching off the external magnetic field, the vortex dynamics exhibits two distinct regimes before the creep sets in with logarithmic in time decay of remanent magnetization. In the beginning, the flux flow is localized near the edges and is independent of the sample size. At later times, this regime is followed by a slower flux flow involving the bulk of the sample. We find that magnetic moment relaxation in these regimes can be accurately approximated by for and for , where geometry-dependent and are proportional to .
We have analyzed available experimental data on early stages of magnetization relaxation after the magnetic field is instantaneously removed. We obtained quantitative agreement for both conventional and high-temperature superconductors, albeit with exceeding the Bardeen-Stephen value by many orders of magnitude. Huge enhancement of with respect to , as well as its exponential temperature dependence, indicates a strong influence of pinning and thermally assisted depinning of vortices on flux diffusion. We argue that early stages of magnetization relaxation can be modeled as a flux flow with an effective drag coefficient. This allows for a simple experimental determination of the bulk vortex viscosity, which cannot be accessed by the surface impedance measurements.
This research was supported by the Serbian Ministry of Science, Project No. 171027. Work by V.K. at the Ames Laboratory is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-07CH11358. M.V. acknowledges support by the DFG through SFB 767 and the hospitality of the Quantum Transport Group, Universität Konstanz, Germany, where part of this work was done.
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