Dynamic vortex phases and pinning in superconductors with twin boundaries

C. Reichhardt and C.J. Olson
Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120
and Department of Physics, University of California, Davis, California 95616

Franco Nori
Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120
(Received 23 August 1999)

We investigate the pinning and driven dynamics of vortices interacting with twin boundaries using large scale molecular-dynamics simulations on samples with near one million pinning sites. For low applied driving forces, the vortex lattice orients itself parallel to the twin boundary and we observe the creation of a flux gradient and vortex-free region near the edges of the twin boundary. For increasing drive, we find evidence for several distinct dynamical flow phases which we characterize by the density of defects in the vortex lattice, the microscopic vortex flow patterns, and orientation of the vortex lattice. We show that these different dynamical phases can be directly related to microscopically measurable voltage-current V(I) curves and voltage noise. By conducting a series of simulations for various twin boundary parameters we derive several vortex dynamic phase diagrams.

I. INTRODUCTION

Twin boundaries are a very common defect found in YBa₂Cu₃O_7-x (YBCO) and their pinning properties have been extensively studied using Bitter decoration,² torque magnetometry,³ magnetization,^4-7 transport,⁸ magneto-optical imaging^9-14 and theoretical studies.^15-17 Many of the earlier experiments on twinned YBCO samples found conflicting evidence for the role of twin boundaries in vortex pinning. In particular, the magneto-optical measurements by Duran et al. ⁹ had shown that twin boundaries act as areas of reduced pinning that allow easy flux penetration, whereas studies by Vlasko-Vlasov et al. ¹⁰ found the twin boundaries to be barriers to flux motion. Further magneto-optical studies, ^11-14 systematic computer simulations, ¹⁷ and transport measurements ⁸ have shown that these conflicting results can be resolved when the direction of the Lorentz force with respect to the twin boundary is considered. The twin boundary (TB) acts as an easy-flow channel when the Lorentz force is parallel to the twin, but acts as a strong barrier for forces perpendicular to the TB.

A very systematic simulational study, using samples with of the order of a million pinning sites, by Groth et al.¹⁷ of the angular dependence of the Lorentz force with respect to the twin boundary showed that, when the angle between the Lorentz force and the twin is large, a portion of the vortices get trapped inside the twin. This produces a pileup effect leading to a higher density of vortices on one side of the twin in agreement with observations by several groups including, for example, Vlasko-Vlasov et al.,¹⁰ Welp et al.,¹² and Wijngaarden et al.¹⁴ At lower angles between the Lorentz force and the twin, simulations ¹⁷ show that the flux moves in channels along the twin boundary while some guided motion of vortices along the edge of the twin still occurs. At the lowest angles the flux flows most easily along the twin with a number of vortices escaping from the twin and forming a flame pattern flux profile in agreement with magneto-optical experiments.^9,10,14,18

Recently interest in vortex systems has strongly focused on driven phases and dynamic phase transitions of vortices interacting with random or periodic defects in superconductors. The anisotropic pinning properties of twin boundaries as well as the possibility of tuning the strength of the twin boundary pinning make these defects quite distinct from random pinning or periodic pinning arrays, so that new dynamical phases can be expected to appear.

In systems containing random pinning, experiments using transport measurements,^19-21 voltage noise measurements,^22,23 vibrating reed measurements,²⁴, neutron scattering,²⁵, and Bitter-decoration,²⁶ as well as simulational work,^27,28 and work based on perturbation and/or elasticity theory²⁹ indicate that, at the depinning transition, the vortex lattice may disorder and undergo plastic flow in which vortices change nearest neighbors as mobile portions of the vortex lattice tear past pinned portions. At higher drives the vortex lattice may reorder and exhibit elastic or ordered flow. An intriguing question is whether specific types of plastic flow exist, and how they could be distinguished. Simulations with randomly placed pinning indicate the possible existence of at least two kinds of plastic flow. The first type consists of well-defined channels of mobile vortices flowing through the rest of the pinned vortex lattice.^30-33 A second type consists of intermittent or avalanching motion in which only a few vortices are mobile at any given time, but over time all the vortices take part in the motion so that well defined channels are not observed.^31-33

Recent simulations using the time-dependent Ginzburg-Landau equations at T=0 of vortices interacting with twin boundaries have suggested the possibility of the existence of three distinct flow phases which include two plastic flow phases and an elastic flow phase.¹⁸ Due to the nature of these simulations it was only possible to consider three different driving currents for each pinning parameter; so that V(I) curves, voltage noise signals, and the evolution of the vortex order as a continuous function of increasing driving force could not be extracted, nor could the evolution of the flow phases with the system parameters be determined.

In order to examine the microscopic dynamics of vortices interacting with twin boundaries we have performed large scale molecular-dynamics simulations for a wide variety of twin parameters which allow us to carefully compare the different kinds of plastic flow as a driving force is continuously increased. Our results in this work complement our previous simulational work on twin-boundaries,¹⁷ where we considered only the case of very slow driving that occurs as a magnetic field is increased. In Ref. 17 we considered flux-gradient-driven vortices and we focused on the magnetic-flux front profiles and compared them to magneto-optical images. In this paper we focus on the microscopic aspects of current-driven, as opposed to flux-gradient-driven, vortex motion and structure as well as on transport measures.

II. SIMULATION

To model pinning in the bulk, we divide our system into a 1000 x 1000 grid where each grid element represents a pinning site. The pinning density n_p is 496/, which is within experimentally determined values. At each pinning site (l,m) the pinning force f_l,m^thr is chosen from a uniform distribution [0,f_p], where f_p is the maximum possible pinning force. If the magnitude of the force produced by the other vortices, driving force and twin boundaries acting on a vortex located on a pinning site (l,m) is less than the threshold pinning force f_l,m^thr, the vortex remains pinned at the pinning site. If the force on the vortex is greater than f_l,m^thr, then the effective pinning force f_i^vp drops to zero and the vortex moves continuously until it encounters a pinning site that has a threshold force greater than the net force on the vortex. The pinning therefore acts as a stick-slip friction force with the following properties:
f_i^vp=-f_i^net, f_i^net < f_l,m^thr (3)
and
f_i^vp=0, f_i^net > f_l,m^thr. (4)

For the twin boundary pinning, we have considered a large number of models, all giving similar results. The simplest model that is most consistent with experiments is that of an attractive well containing stick-slip pinning with a different maximum threshold force f_p^TB than that of the bulk pinning outside the TB, f_p. This model of pinning is very similar to the one inferred from the measurements in Ref. 5 where the TB channel has strong depth variations. The ratio f_p^TB/f_p is expected to vary as a function of temperature. In the case predicted for low T (Ref. 1) where f_p^TB/f_p < 1, the twin boundary acts as an easy flow channel for certain angles.¹⁷ On the other hand, at higher T, f_p^TB/f_p > 1, and the twin acts as a barrier to flux flow. This second case is the most similar to the simulations conducted in Ref. 18 where the twin boundary was modeled as a line of parabolic pinning. In our simulations we can mimic the effects of temperature by varying the ratio of f_p^TB/f_p.

The twin boundary itself is modeled as an attractive parabolic channel with a width denoted by 2. The force on the ith vortex due to the kth the twin boundary is
(5)
where d_ik^TB is the perpendicular distance between the ith vortex and the kth twin boundary.

The driving force representing the Lorentz force from an applied current is modeled as a uniform force on all the vortices. The driving force is applied in the x-direction and is slowly increased linearly with time. We examine the average force in the x-direction
(6)
as well as the average force in the y-direction
(7)
These quantities are related to macroscopically measured voltage-current V(I) curves.

We also measure the density of 6-fold coordinated vortices P₆. Strong plastic flow causes an increase in the number of defects and a corresponding drop in P₆, while elastic flow is associated with few or no defects. Another measure of order in the lattice is the average height of the first-oder peaks in the structure factor S(k)
(8)

The defect density can also be correlated with the voltage noise power spectra S()
(8)
A vortex lattice that is flowing plastically should produce a large amount of voltage noise. To measure the quantity of noise produced, we integrated the noise power over one frequency octave. ^22,23

III. DYNAMIC PHASES

At higher drives, as shown in Fig. 1(c), and 1(d) with f_d=0.35f₀, there is a transition to a more disordered flow and the vortices start to cross the twin boundary. The overall vortex structure [Fig. 1(c)] is more disordered than it was at lower drives [Fig. 1(a)]. Unlike the guided plastic motion phase, the vortices pinned along the twin boundary are only temporarily trapped, and occasionally escape from the twin and are replaced by new vortices intermittently. The vortex trajectories shown in Fig. 1(d) also indicate that some vortex guiding still occurs. We label this phase the plastic motion (PM) phase. At even higher driving currents we observe a transition from the plastic flow phase to an elastic motion (EM) phase where the effect of the twin boundary becomes minimal, as shown in Fig. 1(e), and 1(f) for f_d=1.25f₀. Here, the vortex lattice reorders [Fig. 1(e)], the vortices flow along the direction of the applied Lorentz force [Fig. 1(f)], and no build-up of the flux near the twin appears.

IV. CURRENT-VOLTAGE CHARACTERISTICS AND VORTEX STRUCTURE

The vortex lattice becomes slightly more disordered in this plastic flow phase as indicated by the drop in P₆ and the smaller drop in < S(k) >. As f_d is increased further, V_y gradually decreases, but remains finite as vortices cross the twin at an increasing rate. When V_y approaches zero, near f_d/f₀=0.85, the vortex lattice reorders as indicated by the increase in P₆ and < S(k) >. We note that the reordering transition in P₆ is considerably sharper than that typically observed in simulations with random pinning.

V. NOISE MEASUREMENTS

The noise power is relatively low in the GPM phase, increases to a large value in the PM phase, and then gradually decreases as the EM phase is approached. In the GPM regime, although tearing of the vortex lattice occurs at the boundaries between the pinned and flowing vortices, the vortex trajectories follow fixed channels and a large portion of the vortex lattice remains ordered. This very orderly vortex motion produces little noise. In the PM phase, the vortex lattice is highly disordered and the trajectories follow continuously changing paths so the corresponding voltage noise power is high. This difference in noise power between the static and changing channels for vortex flow agrees well with results obtained in systems with strong random pinning. In such systems, when the vortex flow follows fixed winding channels that do not change with time, low noise power is observed above the depinning threshold.^28,33 Similarly, when the pinning is weak and the vortices move in straight fixed lines, low voltage noise is observed. ^28,33 This latter case agrees well with the result seen here in the GPM and EM phases, when the vortices follow straight paths and produce little noise power.

VI. DYNAMIC PHASE DIAGRAMS

It might be expected that the transition out of the guided plastic motion phase would fall at f_d=f_p^TB, when the vortices are able to depin from the twin boundary. Since vortex interactions are important, however, in actuality the vortex density increases on one side of the twin while a lower vortex density appears on the other side. This localized flux gradient produces an additional force on the vortices at the twin boundary, depinning them at a driving force f_d < f_p^TB. The additional force from the flux-gradient is not spatially uniform, unlike the driving force, so some of the vortices will depin before others in a random manner. Once the applied driving force and the gradient force are large enough to start depinning vortices from the twin, the flux lines enter the plastic flow phase. The effect of the pinning on the vortices does not fully disappear until f_d > f_p^TB, however, which is seen in the existence of a finite V_y. We also note that there is a pinned phase where no vortex motion occurs when f_d < f_p.

By changing the vortex density we can examine the effects of changing the effective vortex-vortex interaction. In Fig. 5(b) we plot the phase diagram constructed from a series of simulations in which the vortex density is varied. As the vortex density decreases the GPM-PM and the PM-EM transition lines shift to higher drives. This is because lower values of B (or N_v) increase the effective pinning force and shift the boundary to higher values of f_d.

VII. CONCLUSION

Twin boundaries correspond to one type of correlated pinning. Another type involves periodic arrays of pinning sites. ³⁴ The dynamic phase diagrams of these structures with correlated pinning are also under current intense investigation.

We recently became aware of the experiments in Ref. 35 which measure both the longitudinal and transverse voltage signal for vortices driven in samples with unidirectional twin boundaries. When the vortices are driven at 52^o with respect to the twin boundaries, at low temperatures the vortex motion deviates strongly from the direction of drive with a component moving along the twin boundary. Using this experimental set-up it should be possible to observe both the transverse and longitudinal vortex velocity as a function of applied current.

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