Driven vortices in three-dimensional layered superconductors: Dynamical ordering along the c-axis

Alejandro B. Kolton,¹ Daniel Dominguez,¹ Cynthia J. Olson,² and Niels Gronbech-Jensen^3,4
1Centro Atomico Bariloche, 8400 San Carlos de Bariloche, Rio Negro, Argentina
²Department of Physics, University of California, Davis, California 95616
³Department of Applied Science, University of California, Davis, California 95616
⁴NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720
(Received 11 September 2000)

We study a 3D model of driven vortices in weakly coupled layered superconductors with strong pinning. Above the critical force F_c, we find a plastic flow regime in which pancakes in different layers are uncoupled, corresponding to a pancake gas. At a higher F, there is a "smectic flow" regime with short-range interlayer order, corresponding to an entangled line liquid. Later, the transverse displacements freeze and vortices become correlated along the c-axis, resulting in a transverse solid. Finally, at a force F_s the longitudinal displacements freeze and we find a coherent solid of rigid lines.

It is well-known that an external current can induce an ordering of the vortex structure in superconductors with pinning.¹ Recently, it has been found that different kinds of order are possible at high currents, depending on pinning strength and dimensionality.^2-7 This has led to numerous theoretical,^2,3 experimental,⁴ and numerical studies.^5-7 A crystal-like structure, which could be either a perfect crystal ² or a Bragg glass, ³ is only possible in d=3 at large drives. In d=2, or in d=3 for intermediate currents, a transverse glass is expected, with order only in the direction perpendicular to the driving force. ^3,6,7 In the equilibrium vortex phase diagram, the behavior of vortex line correlations along the direction of the magnetic field (c-axis) has been intensively discussed both experimentally⁸ and theoretically.⁹ In the case of driven vortices, little is known on how the c-axis line correlations would behave in the different dynamical regimes. Here we will address this issue starting from the less favorable case: weakly coupled superconducting planes with strong pinning. We will show how the order along the c-axis take place in a sequence of dynamical regimes upon increasing current.

We study pancake vortices in a layered superconductor, considering the long-range magnetic interactions between all the pancakes and neglecting Josephson coupling.¹⁰ This model is adequate when the interlayer periodicity d is much smaller than the in-plane penetration length .¹⁰ Previous simulations of driven vortices in 3D superconductors have been performed using Langevin dynamics of short-range interacting particles¹¹ or the driven isotropic 3D XY model.¹²

The equation of motion for pancakes located in positions R_i=(r_i,z_i)=(x_i,y_i,n_id), , with pinning sites at R_p=(r_p,z_p), is:
(1)
where =|r_i-r_j|, z_ij=|z_i-z_j|, =|r_i-r_p|, is the Bardeen-Stephen friction, and and F=(/c)J x z is the driving force due to an in-plane current J. We consider a random uniform distribution of attractive pinning centers in each layer with , where a_p is the pinning range. The magnetic interaction between pancakes is given by^10,13
(2,3)
Here, and =2/d is the 2D thin-film screening length. An analogous model was used in Ref. 14. We normalize length scales by , energy scales by A_v=/4, and time is normalized by =/A_v. We consider N_v pancake vortices and N_p pinning centers per layer in N_l rectangular layers of size L_x x L_y, and the normalized vortex density is n_v=B/=(a₀/)². We consider n_v=0.29 with L_y=16 and L_x=/2L_y, N_l=8, and N_v=64. We take a pinning range of a_p=0.2, a large pinning strengh of A_p/A_v=0.2, with a high density of pinning centers n_p=3.125n_v. The model of Eqs. (2) and (3) is valid in the limit d. We take d/=0.01, which corresponds to Bi-Sr-Ca-Cu-O compounds.¹⁰ Moving pancake vortices induce a total electric field E=(B/c)v x z, with . We study the dynamical regimes in the velocity-force curve at T=0, solving Eq. (1) for increasing values of F=Fy.¹³ We use periodic boundary conditions both in the planes and in the z direction and interactions between all pancakes in all layers are considered.¹³ The periodic long-range in-plane and inter-plane interaction is evaluated using Ref. 15. The equations are integrated with a time step of t=0.01 and averages are evaluated in 16 384 integration steps after 2000 iterations for equilibration. Each simulation is started at F=0 with a triangular vortex lattice and slowly increasing the force in steps of F=0.1 up to values as high as F=8.

We start with a qualitative description of the different steady states that arise as a function of increasing force. In Figs. 1(a)-1(d) we show the vortex trajectories R_i(t) for typical values of F by plotting the positions of the pancakes in five of the layers for all t. In Figs. 1(e)-1(h) we show the average in-plane structure factor , with k=(k_x,k_y). Above the depinning critical force F_c, we find the following dynamical regimes. (i)Plastic flow (F_cp): Pancakes flow in an intrincate network of "plastic" channels similar to the behavior found in 2D.^5,7 The motion in different planes is completely uncorrelated, [Fig.1(a)] and there is no signature of order in the structure factor [Fig.1(e)]. (ii)Smectic flow (F_pt): The motion organizes in "elastic" channels that are almost parallel and separated by a distance ~a₀, see Fig. 1(b). Small and broad "smectic" peaks appear in S(k) for =0 [Fig.1(f)]. There are "activated" jumps of pancakes between channels. Along the c-direction the channels tend to align sitting on top of each other between neighboring planes. (iii)Transverse solid (F_ts): There are well defined channels in all the planes and the pancakes do not jump between channels [Fig.1(c)]. The structure factor has sharp smectic peaks and small "longitudinal" peaks (0) have appeared [Fig.1(g)]. The location of channels is correlated in the c-axis. (iv)Coherent solid (F > F_s): The channels become more straight with small transverse wandering [Fig.1(d)]. The S(k) shows well defined peaks for all k in the reciprocal lattice [Fig.1(h)]. This is a dynamical regime with respect to the 2D thin-film results.⁷

Let us now characterize in detail these dynamical regimes. First, we analyze the in-plane structure factor and temporal fluctuations. In Fig. 2(a) we plot the average velocity , in the direction of the force and the differential resistance dV/dF as a function of F. The force F_p corresponds to the peak in the differential resistance. We also see a small second maximum in dV/dF for a force between F_t and F_s. ¹⁶ In Fig. 2(b) we plot the magnitude of the peaks in the in-plane structure factor. We show the peak height at G₁=2/a₀, corresponding to smectic order, and the average of the peaks corresponding to longitudinal order at G₂=2/a₀(1/2,/2) and G₃=2/a₀(-1/2,/2). We see that at F_p the smectic peak rises up from zero, then at F_t it reaches an almost constant value and later at F_s it has a small jump. The longitudinal peak has a small finite value for forces above F_p, and only at F_s shows a significant increment towards a large value. We have studied the S(k) for sizes of N_v x N_l=36 x 5, 64 x 8, 100 x 8, 100 x 10. The trend is as follows: for F_p < F < F_t, S(G₁) is finite but strongly size dependent and S(G_2,3)0; for F_t < F < F_s, S(G₁) is weakly size dependent and S(G_2,3) is finite but strongly size dependent; for F > F_s, S(G₁) and S(G_2,3) do not show significant changes with size. The characteristic forces F_p, F_t, F_s do not vary much with system size [see the inset in Fig. 2(b)]. Comparing this with our previous 2D studies,⁷ we can make the assumption that for F_p < F < F_t there is only short-range smectic order, for F_t < F < F_s there is probably quasi-long range smectic order but short range longitudinal order, and above F_s there is both transversal and longitudinal order (quasi-long-range or long-range). Compared with 2D, there is a force F_s above which there is a significant amount of crystalline order. This may correspond either to a moving crystal (if there is long-range order) or to a moving Bragg glass (if there is quasi-long-range order). ³ We complement our discussion of the in-plane physics with the study of the temporal fluctuations, which are shown in Fig. 2(c) for two system sizes N_v x N_l= 64 x 8, 100 x 10. We obtain the transverse diffusion coefficient D_x from the transverse displacements from the center of mass , as . We find that D_x is maximum at F_p in coincidence with the peak in dV/dF. Below F_p diffusion is through the intrincate network of plastic channels, above F_p diffusion is through activated jumps between elastic channels. D_x has a drop to zero at F_t, indicating that transverse displacements are localized in the transverse solid regime.⁷ The drift from the center of mass of longitudinal displacements is superdiffusive for F < F_s, as found for 2D films.⁷ The interesting result is that for F > F_s the longitudinal displacements become frozen in an almost time-independent value < [y(t)]² >a₀², as it is shown in the inset of Fig.2(c). We use as a criterion for defining F_s the point where < [y(t)]² >=a₀², evaluated at the longest time t of the simulation for each force. We find that this criterion for F_s does not depend on system size (comparing 64 x 8 with 100 x 10). The behavior of < y² > vs F is smoother in a larger system (100 x 10). We see that the value of F_s obtained in this way is in agreement with the small range of forces where the longitudinal order peaks S(G_2,3) have a significant increase [Fig. 2(b)].

Let us now study how the ordering along the c-axis takes place. We first look at the overlap of the vortex trajectories in the the c axis. We start by defining the averaged vortex density taking a coarse-graining scale r=a₀/2 (results do not vary much for r=a₀/4). The regions where < (r,n) > is large define the paths of steady-state vortex motion. We can thereby calculate the overlap function of vortex trajectories between different planes as , with . This is shown in Fig.3(a). We see that O_n has an onset at F_p. For F_p < F < F_t, we have a finite overlap of the elastic channels that decreases with increasing n. More interestingly, at F_t the overlap function O_n becomes independent of n. This means that there is long-range c-axis coupling of the path of the elastic channels. When transverse displacements become localized in the x-direction, they also become locked in the c-direction. Thus, the freezing of in-plane transverse displacements occurs simultaneously with a transverse disentanglement of flux lines at F_t. A striking result is that we find O_n1 above F_s, i.e., a perfect c-axis coupling of trajectories (within the scale ~a₀/4). Therefore, the behavior of O_n is correlated with the regimes found for the in-plane magnitudes. We now analyze the pair distribution function: . From g(,n) we define the correlation function along the c-axis . Short-range ordering will be given by a finite C_z(n=1), meaning that pancakes in neighboring planes are coupled and a "vortex line" can therefore be defined. In principle, an exponential decay C_z(n)~exp(-n/) would define a correlation length for the vortex line. ⁹ Long-range ordering will be given by . In Fig. 3(b) we show C_z(n) as a function of F for n=1,2,3,4. We see that at F_p there is an onset of short-range order along the axis with a finite C_z(1). The onset of correlations for more planes, n > 1, occurs in the range of forces between F_p and F_t. For higher forces there is an overall increase of C_z(n) with increasing F, but we do not see clear features at F_t and F_s, as we found for O_n and the in-plane magnitudes. Also, since there are few planes, the behavior of C_z(n) with n can not be easily extrapolated. However, comparing O_n with C_z(n) we can extract the following conclusions. The absence of correlations for Fp means that pancake motion is completely random and uncorrelated between different planes. Therefore, the plastic flow regime corresponds to a pancake gas. Above F_p, in the smectic flow regime, it is possible to define a vortex line with short range correlations along the c-axis. Since there are in-plane jumps between elastic channels (i.e., cutting and reconnection of flux lines) we may consider this regime as an entangled line liquid. Above F_t, C_z(n) is finite for all n considered, suggesting that vortex lines become stiffer for increasing forces. The behavior of O_n above F_t suggest that the lines disentangle along the transverse direction for F > F_t. Another interesting point to consider is the correlation of vortex velocities. If vortices in different planes move at different velocities, they will induce a Josephson voltage difference along the c-axis given by (r,t)=/2c)(d/dt)(r,t), with the superconducting phase difference between planes n and n+1. A good approximation for pancakes at r_n,i is to write with f(r)arctan(x/y). We can therefore estimate the c-axis voltage fluctuations as ; with , and the constant A~log if L > or A~log(L) otherwise. We see in Fig.3(c) that the voltage fluctuations have a maximum at F_p. For F > F_p, < V_c > decreases, and above F_s it reaches an almost F-independent value. The fact that < V_c > does not vanish above F_s is consistent with the result that C_z(n) < 1 for all values of F in Fig.3(b).

In conclusion, we have analyzed different dynamical regimes in 3D layered superconductors considering both in-plane and c-axis ordering. The onset of short-range c-axis correlations could be studied experimentally with plasma resonance measurements.¹⁷ The long-range ordering along the c-axis could be studied through simultaneous measurements of resistivity and in-plane current-voltage response.¹⁸

We acknowledge discussions with L.N. Bulaevskii, P.S. Cornaglia, F. de la Cruz, Y. Fasano, and M. Menghini. This work was supported by ANPCYT (PICT-03-00000-01034), by Fundacion Antorchas (A-13532/1-96), Conicet, CNEA and FOMEC (Argentina); by NSF-DMR-028535, CLC and CULAR (LANL/UC), and by the Office of Adv. Sci. Comp. Res., Div. of Math. Inf. and Comp. Sci. of the U.S. D.O.E. (Contract No. DE-AC03-76SF00098).

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