# Dynamical ordering in the c-axis in 3D driven vortex lattices

###### Abstract

We present molecular dynamics simulations of driven vortices in layered superconductors in the presence of an external homogeneous force and point disorder. We use a model introduced by J.R.Clem for describing 3D vortex lines as stacks of 2D pancake vortices where only magnetic interactions are considered and the Josephson interlayer coupling is neglected. We numerically evaluate the long-range magnetic interaction between pancake vortices exactly. We analyze the vortex correlation along the field direction on (c-axis). We find that above the critical current, in the “plastic flow” regime, pancakes are completely uncorrelated in the c-direction. When increasing the current, there is an onset of correlation along the c-axis at the transition from plastic flow to a moving smectic phase. This transition coincides with the peak in the differential resistance.

## 1 Introduction

The prediction [1] of a dynamical phase transition upon increasing drive, from a fluidlike plastic flow regime [2] to a coherently moving solid [1] in a moving vortex lattice has motivated many recent theoretical [3], experimental [4, 5, 6], and simulation work [7, 8, 9, 10, 11, 12, 13]. In a previous work we have studied the dynamical regimes in the velocity-force curve (voltage-current) in 2D thin films [10] and found two dynamical phase transitions above the critical force. The first transition, from a plastic flow regime to a smectic flow regime, is characterized by the simultaneous occurrence of a peak in differential resistance, isotropic low frequency voltage noise and maximum transverse diffusion. The second transition, from a smectic flow regime to a frozen transverse solid, is a freezing transition in the transverse direction where transverse diffusion vanishes abruptly and the Hall noise drops many orders of magnitude. In other 2D simulations the peak in differential resistance was found to coincide with the onset of orientational order [8] and a maximum number of defects [9]. Experimentally, the position of this peak was taken by Hellerqvist et al. [5] as an indication of a dynamical phase transition. In this paper we show that in driven 3D layered superconductors, the peak in differential resistance also coincides with the onset of correlation along the c-axis.

## 2 Model

Simulations of vortices in 3D layered superconductors have been done previously using time–dependent Ginzburg–Landau–Lawrence–Doniach equations (two layers), [12], the 3D XY model [13] and Langevin dynamics of interacting particles [14, 15]. Here we study the motion of pancakes vortices in a layered superconductor with disorder, with an applied magnetic field in the c-direction and with an external homogeneous current in the layers (ab-planes). We use a model introduced by J.R. Clem for a layered superconductor with vortices in the limit of zero Josephson-coupling between layers [16]. We simulate a stack of equally spaced superconducting layers with interlayer periodicity , each layer containing the same number of pancake vortices. The equation of motion for a pancake located in position (with z-axis in c-direction) is:

(1) |

where and are the in-plane and inter-plane distance between pancakes , is the in-plane distance between the vortex and a pinning site at (pancakes interact only with pinning centers within the same layer), is the Bardeen-Stephen friction, and is the driving force due to an applied in-plane current density . We consider a random uniform distribution of attractive pinning centers in each layer with , where is the pinning range. The magnetic interaction between pancakes is given by:

(2) |

(3) |

Here, , is the penetration length parallel to the layers, and is the 2D thin-film screening length. A analogous model to Eqs. (2-3) was used in [14]. We normalize length scales by , energy scales by , and time is normalized by . We consider pancake vortices and pinning centers per layer in rectangular layers of size , and the normalized vortex density is . Moving pancake vortices induce a total electric field , with . We study the dynamical regimes in the velocity-force curve at , solving Eq. (1) for increasing values of . We consider a constant vortex density in layers with , , and pancake vortices per layer. We take a pinning range of , pinning strengh of , with a density of pinning centers on each layer. We use periodic boundary conditions in all directions and the periodic long-range in-plane interaction is dealt with exactly using an exact and fast converging sum [17]. The equations are integrated with a time step of and averages are evaluated in integration steps after iterations for equilibration (when the total energy reaches a stationary value). Each simulation is started at with an ordered triangular vortex lattice (perfectly correlated in c-direction) and slowly increasing the force in steps of up to values as high as .

## 3 Results

We start by looking at the vortex trajectories in the steady state phases. In Figure 1(a-b) we show a top view snapshot of the instantaneous pancake configuration for two typical values of . In Figure 2(a-b) we show the vortex trajectories for the same two typical values of by plotting all the positions of the pancakes in all the layers for all the time iteration steps. In Fig. 3(a) we plot the average vortex velocity and its corresponding derivative . We also study the pair distribution function: , in the direction of the force as a function of

(4) |

In Fig. 3(b) we plot a correlation parameter along c-axis defined as:

(5) |

as a function of for . Below a critical force, , all the pancakes are pinned and there is no motion. At the characteristic force, , we observe a peak in the differential resistance. At pancake vortices start to move in a few channels , as was also seen in 2D vortex simulations [2]. A typical situation is shown in Fig. 2(a). In this plastic flow regime we observe that the motion is completely uncorrelated along c-direction, with for as shown in Fig. 3(b). In Fig. 1(a) and 2(a) we see that this situation corresponds to a disordered configuration of pancakes and to an uncorrelated structure of plastic channels along c-axis. At there is an onset of correlation along the c-axis and pancakes vortices start to align forming well defined stacks or vortex lines. This onset of c-axis correlation corresponds to the transition from plastic flow to a moving smectic phase (a complete discussion of the translational as well as temporal order will be discussed elsewhere [18]). For we observe that the structure of smectic channels is very correlated in the c-direction.

## 4 Discussion

In a system with layers we have found that there is a clear onset of c-axis correlation with increasing driving force, where the pancake vortices start to align forming well defined stacks moving in smectic flow channels. Below this transition, in the plastic flow regime, these stacks of pancakes are unstable. Also, in [12] an enhancement of c-axis correlations with increasing drive was observed in a bilayered system.

We have further found [18] that the in-plane properties are in well correspondence with the ones obtained in 2D thin films simulations [10]. A better understanding of the effects of c-axis correlations in pancake motion on each layer can be obtained by studying translational and temporal order in larger systems, through the analysis of the structure factor, voltage noise, and in-plane [11] as well as inter-plane velocity-velocity correlation functions [18].

In conclusion we have analyzed the vortex correlation along the field
direction (c-axis) in the velocity-force characteristics at and found that
above the critical current there is an onset of c-axis correlation in the
transition between a plastic flow regime to a smectic flow regime. This transition
coincides with the peak in the differential resistance. Experimentally, this effect
could be studied with measurements of c-axis resistivity as a function of an applied
current parallel to the layers [19].

We acknowledge discussions with L.N. Bulaevskii, P.S. Cornaglia, F. de la Cruz, Y. Fasano, M. Menghini and C.J. Olson. This work has been supported by a grant from ANPCYT (Argentina), Proy. 03-00000-01034. D.D. and A.B.K. acknowledge support from Fundación Antorchas (Proy. A-13532/1-96), Conicet, CNEA and FOMEC (Argentina). This work was also supported by the Director, Office of Advanced Scientific Computing Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy under contract number DE-AC03-76SF00098.

## References

- [1] A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett. 73, 3580 (1994).
- [2] H. J. Jensen et al., Phys. Rev. Lett. 60, 1676 (1988); A.-C. Shi and A. J. Berlinsky, Phys. Rev. Lett. 67, 1926 (1991).
- [3] T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. 76, 3408 (1996); P. Le Doussal and T. Giamarchi, Phys. Rev. B 57, 11356 (1998); L. Balents, M. C. Marchetti and L. Radzihovsky, Phys. Rev. B 57, 7705 (1998); S. Scheidl and V. M. Vinokur, Phys. Rev. E 57, 2574 (1998); Phys. Rev. B 57, 13800 (1998).
- [4] M. J. Higgins and S. Bhattacharya, Physica C 257, 232 (1996).
- [5] M. C. Hellerqvist et al., Phys. Rev. Lett. 76, 4022 (1996).
- [6] F. Pardo et al, Nature (London) 396, 348 (1998).
- [7] K. Moon, R. T. Scalettar and G. Zimányi, Phys. Rev. Lett. 77, 2778 (1996); S. Spencer and H. J. Jensen, Phys. Rev. B 55, 8473 (1997).
- [8] S. Ryu et al., Phys. Rev. Lett. 77, 5114 (1996).
- [9] C. J. Olson, C. Reichhardt and F. Nori, Phys. Rev. Lett. 81, 3757 (1998).
- [10] A. B. Kolton, D. Domínguez, N. Grønbech-Jensen, Phys. Rev. Lett. 83, 3061 (1999).
- [11] D. Domínguez, Phys. Rev. Lett. 82, 181 (1999).
- [12] I. Aranson, A. Koshelev and V. Vinokur, Phys. Rev. B 56, 5136 (1997).
- [13] D. Domínguez, N. Grønbech-Jensen and A.R. Bishop, Phys. Rev. Lett. 78, 2644 (1997).
- [14] D. Reefman, H. B. Brom, Physica C 213, 229 (1993).
- [15] S. Ryu, D. Stroud, Phys. Rev. B, 54, 1320 (1996); N. K. Wilkin, H. J. Jensen, Phys. Rev. Lett. 21, 4254 (1997); A. van Otterlo, R. T. Scalettar, G. T. Zimámyi, Phys. Rev. Lett. 81, 1497 (1998); C. J. Olson, R. T. Scalettar, G. T. Zimányi, cond-mat/9909454.
- [16] J. R. Clem, Phys. Rev. B. 43, 7837 (1990).
- [17] N. Grønbech-Jensen, Int. J. Mod. Phys. C 7, 873 (1996); Comp. Phys. Comm. 119, 115 (1999).
- [18] A. B. Kolton, D. Domínguez, N. Grønbech-Jensen, to be published.
- [19] M. Menghini (private communication).