Static and Dynamic Coupling Transitions of Vortex Lattices in Disordered Anisotropic Superconductors

C. J. Olson,¹ G. T. Zimanyi,¹ A. B. Kolton,² and N. Gronbech-Jensen^3,4
1Department of Physics, University of California, Davis, California 95616
²Centro Atomico Bariloche, 8400 S. C. de Bariloche, Rio Negro, Argentina
³Department of Applied Science, University of California, Davis, California 95616
⁴NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720
(Received 20 March 2000)

We use three-dimensional molecular dynamics simulations of magnetically interacting pancake vortices to study vortex matter in disordered, highly anisotropic materials such as BSCCO. We observe a sharp 3D-2D transition from vortex lines to decoupled pancakes as a function of relative interlayer coupling strength, with an accompanying large increase in the critical current reminiscent of a second peak effect. We find that decoupled pancakes, when driven, simultaneously recouple and order into a crystalline-like state at high drives. We construct a dynamic phase diagram and show that the dynamic recoupling transition is associated with a double peak in dV/dI.

PACS numbers: 74.60.Ge, 74.60.Jg

In highly anisotropic superconductors such as Ba₂Sr₂CaCu₂O₈ (BSCCO), the vortex lattice is composed of individual pancake vortices [1]. These pancakes, which interact both magnetically and through Josephson coupling, align under certain conditions into elastic lines resembling those found in isotropic superconductors. Three-dimensional (3D) line-like behavior has been observed in transformer geometry measurements [2], muon-spin-rotation [3], and neutron scattering [4]. Under different conditions, however, the pancake vortices in each layer move independently of the other layers, and the system acts like a stack of independent thin film superconductors. Such two-dimensional (2D) behavior has also been seen in transformer experiments [5]. Thus in layered superconductors two different effective dimensionalities of the vortex pancake lattice may appear, each with different characteristic properties.

Layered superconductors exhibit a striking second peak in magnetization measurements [4,6-10], corresponding to an abrupt increase in the critical current of the material as the applied field is increased. This second peak is especially sharp in BSCCO, as shown in local Hall probe measurements [7,9,10] and recent Josephson plasma frequency measurements [11]. The lattice appears ordered at fields below the transition and disordered above. There seems to be no widely accepted agreement on the mechanism behind this transition, although numerous scenarios have been suggested, including vortex entanglement [12], dislocation proliferations [13], dynamic effects [14], or a 3D to 2D transition in the vortex pancake lattice [4,7,15-17]. The effect of strong disorder on a possible 3D-2D transition is unclear, and also it is not known how the transport properties would be affected by a 3D-2D transition.

In 2D systems with uncorrelated pinning, the vortex lattice can be dynamically reordered by an applied driving current, passing from a glassy state at zero drive, through plastic flow [18], to a reordered state at high current [19-22]. In layered systems, when the inter-plane interactions of pancakes is weak enough that the pancakes are decoupled, the pancakes on each plane should reorder when a high enough driving current is applied. This reordering may also be accompanied by a dynamically driven recoupling transition [19], but it is unclear where this transition could be located in relation to the 2D reordering transition, and how it is affected by the external magnetic field.

To address the issue of possible 3D-2D transitions in disordered anisotropic materials, we have developed a simulation which allows for decoupling by incorporating the correct magnetic interactions between vortex pancakes. We show that a sharp 3D-2D transition occurs when the relative strength of interlayer and intralayer pancake interactions is varied, and that this transition is associated with a sharp increase in the critical current. Furthermore, the system exhibits a rich array of dynamic 3D phases when driven by a current. We show that 2D decoupled pancakes can be dynamically recoupled in a transition that occurs simultaneously with the dynamic ordering in each plane. We construct a phase diagram as a function of interlayer coupling and applied driving force, and show that a sharp, experimentally observable second peak in dV/dI is associated with the dynamic recoupling.

The magnetic interactions between pancakes in a layered material have the same logarithmic form present in thin films, but are highly anisotropic [1,16,23]. We have performed simulations in which pancakes in all layers interact magnetically with long range interactions [24,25], in contrast to other simulations, which treated only nearest layer interactions [26,27]. Our approach compliments calculations based on Lawrence-Doniach models [28]. The pancake interactions are long range both in and between planes, and are treated according to Ref. [29]. Josephson coupling is neglected as a reasonable approximation for materials with large anisotropy [16].

The overdamped equation of motion, at T=0, for vortex i is given by , where N_v is the number of vortices, and and z_i,j are the distance between pancakes i and j in cylindrical coordinates. The system has periodic boundaries in-plane and open boundaries in the z direction. The magnetic energy between pancakes is [30]
,
where =/(4), is the London penetration depth, R = 22.6, the maximum radial distance, , , and d=0.005 is the interlayer spacing. We vary the relative strength of the interlayer coupling using the prefactor s_m. The uncorrelated pins are modeled by parabolic traps that are randomly distributed in each layer. The vortex-pin interaction is given by , where the pin radius =0.2, the pinning force f_p=0.02f₀^*, and f₀^*=/. The case of stronger pins is considered in [24]. The driving current must be increased slowly enough for the system to equilibrate at each drive. Here f_d is increased by 0.00025f₀^* every 35 000 time steps. We have simulated a 16 x 16 system with a vortex density of n_v = 0.35/ and a pin density of n_p = 1.0/ in each of L=8 layers. This corresponds to N_v=80 vortices and N_p=256 pins per layer, with a total of 640 pancake vortices. We have checked for finite size effects on systems containing up to 42 layers and 3360 pancakes.

In the equilibrium state of the vortex lattice at zero drive, we find a sharp 3D-2D transition from vortex pancakes to vortex lines when the strength s_m of the interlayer coupling is decreased, as shown in Fig. 1(a). We quantify the transition by measuring the spatial correlation of pancakes in neighboring planes, C_z=1-< (|(r_i,L-r_i,L+1)|/(a₀/2))(a₀/2-|(r_i,L-r_i,L+1)|)>, where a₀ is the vortex lattice constant. A clear sharp drop in C_z with decreasing s_m appears in Fig. 1(a) at s_m=2.0 for a sample with L=8 layers. The value s_m^c at which the transition occurs depends upon the pinning strength f_p, as indicated in Fig. 1(c). The 3D-2D transition is accompanied by a large increase in the critical current f_c, as seen in Fig. 1(a). To determine f_c, we find the average vortex velocity at each drive, , and define f_c as the driving force at which V_x > 0.0005. For weak interlayer coupling, s_m 2.0, the different layers of the sample depin independently and f_c is close to the value it would have in a 2D sample. Here, f_c is insensitive to the number of layers in the system, as can be seen by comparing the data from samples with L=4 to 16 in Fig. 1(b). In contrast, coupled lines of pancakes at s_m > 2.0 average the random pinning over their length and become much less effectively pinned. Therefore f_c decreases with increasing number of layers as seen in Fig. 1(b).

When the magnetic field H increases, the distance between pancakes in the same plane decreases, but the distance d between planes is unchanged. Thus the in-plane interaction between pancakes increases with H but the inter-plane interaction remains constant. We can therefore model a change in H by rescaling the strength of the in-plane and inter-plane interactions, such that s_m1/H. To check this argument, we have performed simulations in which we fix s_m=1 and vary H directly by changing the vortex density n_v. As shown in Fig. 1(d), this method also produces a sharp decoupling transition and an associated jump in j_c. Our results support the suggestion that the sharp second peak observed in magnetization measurements results from a dimensional change in the vortex lattice from weakly pinned 3D vortex lines to strongly pinned 2D pancakes as the magnetic field is increased [31]. The behavior in f_c that we observe indicates that a sharp change in transport properties can occur in a disordered system as a result of a change in the effective dimensionality. Furthermore, the sharpness of the transition found both here and in experiments, as well as hysteretic and superheating/supercooling effects, suggests that the 3D-2D transition is first order.

We next consider the question of a possible dynamic recoupling transition by applying a driving force. When the vortex lattice begins to move for f_d > f_c, it undergoes plastic tearing due to the strong pinning in our sample. For decoupled samples with s_m2.0, each plane performs independent 2D plastic flow, as seen by the different vortex positions and trajectories in the top and bottom layer of the sample shown in Figs. 3(a) and 3(b) below. When high drives are applied, however, the pancakes recouple into lines and all planes begin to move in unison [Fig. 3(c) below].

The reordering transition is shown in more detail in Figs. 2(a)-2(c). We prepare an ordered lattice and then increase f_d slowly enough to allow the lattice to relax into its pinned ground state for f_d< f_c, which is disordered for s_m2.0. As the drive is increased above f_c, for a sample with s_m=1.5, the recoupling transition in C_z [Fig. 2(c)] is sharp and occurs simultaneously with the in-plane reordering transition indicated by P₆ [Fig. 2(b)]. Furthermore, a sharp peak in dV/dI appears at the transition, which will be discussed in more detail below. In contrast, as seen in Figs. 2(d)-2(f), a sample with s_m=4.0 that is above the static 2D-3D transition contains vortices that move as stiff 3D lines, and shows the same reordering transitions seen in previous work on effectively two-dimensional systems [21].

We summarize the behavior of the system in the phase diagram of Fig. 3. The coupled vortices with s_m > 2.0 undergo plastic flow of stiff lines, pass through a smectic state, and finally reorder into a crystallinelike state at high drives. The plastic flow and smectic regions shrink as the number of layers is increased, and finally disappear, so that for L=16 and s_m=5 we observe elastic depinning directly into the 3D ordered state, with no plastic flow. Decoupled vortices with s_m2.0 depin into a 2D plastic flow phase in which each layer moves independently. The vortices switch abruptly from 2D to 3D behavior at the recoupling transition line, so they directly enter the 3D ordered state. The dynamic recoupling transition line and in-plane reordering transition line fall on top of each other in the phase diagram.

As shown in Fig. 4(b), the dynamic recoupling and simultaneous reordering are associated with a sharp peak in the dV/dI curve. This peak is distinct from the broader peak in dV/dI associated with plastic flow of the vortex lattice, as indicated in Fig. 4(d). The sharp peak disappears into the background value of dV/dI and is not observed when the interlayer coupling becomes too weak. The height of the reordering peak increases rapidly as the static 2D-3D transition value of s_m is approached from below, as indicated in Fig. 4(c), and simultaneously the current at which the reordering peak appears shifts downwards towards the depinning current f_c. To understand the reordering peak, note that when the driven 2D decoupled lattice recouples, it must cross from the low V_x response in the 2D plastic limit up to the higher 3D elastic V_x response curve at the recoupling transition. The location of the depinning transition is not affected by the value of s_m since it is purely a 2D effect; however, the recoupling transition is shifted to lower driving currents as s_m increases. As a result, the rising V(I) curve becomes steeper as the static transition point is approached from below, as can be seen for s_m=1.0, 1.5, and 2.0 in Fig. 4(a). The maximum value of dV/dI, denoted dV/dI₀, correspondingly increases, as shown in Fig. 4(c). When the static 2D-3D transition is crossed at s_m=2.0, the second sharp peak disappears. The behavior of this sharp peak in dV/dI, associated with the dynamic recoupling transition, should be experimentally observable in transport measurements performed at fields approaching the second peak from above, when the vortex pancakes are expected to be decoupled.

In summary, we have used a 3D molecular dynamics simulation employing the magnetic interactions of pancake vortices to study the dynamic phases of vortex matter in disordered highly anisotropic materials such as BSCCO. As a function of the relative interlayer coupling strength, we observe a sharp 3D-2D transition from vortex lines to decoupled pancakes. We find an abrupt large increase in the critical current as the 3D-2D line is crossed in a direction corresponding to increasing H, with decoupled pancakes being much more strongly pinned. At driving currents well above depinning, we find that the decoupled pancakes simultaneously recouple and reorder into a crystalline-like state at high drives. We construct a phase diagram as a function of interlayer coupling and show that the recoupling transition coincides with the single-layer recrystallization transition. We show that the recrystallization is associated with an experimentally observable double peak in dV/dI and that the peak height grows rapidly as the static recoupling transition point is approached from below.

We acknowledge helpful discussions with L.N. Bulaevskii, D. Dominguez, C. Reichhardt, and R.T. Scalettar. This work was supported by CLC and CULAR (LANL/UC), by the NSF-DMR-9985978, and by the Director, Office of Adv. Scientific Comp. Res., Div. of Math., Information, and Comp. Sciences, U.S. DoE Contract No. DE-AC03-76SF00098.

1. J.R. Clem, Phys. Rev. B 43, 7837 (1991).
2. D. Lopez et al., Phys. Rev. Lett. 76, 4034 (1996).
3. S.L. Lee et al., Phys. Rev. Lett. 71, 3862 (1993).
4. R. Cubitt et al., Nature (London) 365, 407 (1993).
5. R. Busch et al., Phys. Rev. Lett. 69, 522 (1992).
6. M. Daeumling, J.M. Seuntjens, and D.C. Larbalestier, Nature 346, 332 (1990); H. Safar et al., Phys. Rev. Lett. 68, 2672 (1992).
7. T. Tamegai et al., Physica (Amsterdam) 213C, 33 (1993).
8. G. Yang et al., Phys. Rev. B 48, 4054 (1993).
9. E. Zeldov et al., Nature (London) 375, 373 (1995)
10. B. Khaykovich et al., Phys. Rev. Lett. 76, 2555 (1996); Phys. Rev. B 56, R517 (1997).
11. M.B. Gaifullin et al., Phys. Rev. Lett. 83, 3928 (1999).
12. D. Ertas et al., Physica (Amsterdam) 272C, 79 (1996).
13. T. Giamarchi and P. Le Doussal, Phys. Rev. B 55, 6577 (1997).
14. L. Krusin-Elbaum et al., Phys. Rev. Lett. 69, 2280 (1992).
15. G. Yang et al., Phys. Rev. B 48, 4054 (1993).
16. L.I. Glazman and A.E. Koshelev, Phys. Rev. B 43, 2835 (1991); L.L. Daemen et al., Phys. Rev. Lett. 70, 1167 (1993); Phys. Rev. B 47, 11 291 (1993).
17. M.V. Feigel'man, V.B. Geshkenbein, and A.I. Larkin, Physica (Amsterdam) 167C, 177 (1990); V.M. Vinokur, P.H. Kes, and A.E. Koshelev, ibid. 168C, 29 (1990); G. Blatter et al., Phys. Rev. B 54, 72 (1996); A.E. Koshelev and V.M. Vinokur, ibid. 57, 8026 (1998).
18. S. Bhattacharya and M.J. Higgins, Phys. Rev. Lett. 70, 2617 (1993).
19. A.E. Koshelev and V.M. Vinokur, Phys. Rev. Lett. 73, 3580 (1994); P. Le Doussal and T. Giamarchi, Phys. Rev. B 57, 11 356 (1998); L. Balents, M.C. Marchetti, and L. Radzihovsky, ibid. 57, 7705 (1998).
20. K. Moon et al., Phys. Rev. Lett. 77, 2778 (1996).
21. C.J. Olson, C. Reichhardt, and F. Nori, Phys. Rev. Lett. 81, 3757 (1998).
22. A.B. Kolton, D. Dominguez, and N. Gronbech-Jensen, Phys. Rev. Lett. 83, 3061 (1999).
23. A.I. Buzdin and D. Feinberg, J. Phys (Paris) 51, 1971 (1990).
24. A.B. Kolton et al., Physica (Amsterdam) 341C-348C, 1007 (2000); A.B. Kolton et al., Phys. Rev. B (to be published).
25. C.J. Olson et al., Physica (Amsterdam) 341C-348C, 1083 (2000).
26. N.K. Wilkin and H.J. Jensen, Phys. Rev. Lett. 79, 4254 (1997).
27. A. van Otterlo, R.T. Scalettar, and G.T. Zimanyi, Phys. Rev. Lett. 81, 1497 (1998).
28. S. Ryu et al., Phys. Rev. Lett. 77, 2300 (1996).
29. N. Gronbech-Jensen, Comput. Phys. Comm. 119, 115 (1999).
30. E.H. Brandt, Rep. Prog. Phys. 58, 1465 (1995).
31. G. Blatter et al., Rev. Mod. Phys. 66, 1125 (1994).

Back to Home CJOR
Last Modified: 1/1/02