Physical Review Letters 85, 5416 (2000)

Static and dynamic coupling transitions of vortex lattices in disordered anisotropic superconductors

C. J. Olson1, G. T. Zimányi1, A. B. Kolton2 and N. Grønbech-Jensen3,4

1.  Department of Physics, University of California, Davis, California 95616
2.  Centro Atomico Bariloche, 8400 S. C. de Bariloche, Rio Negro, Argentina
3.  Department of Applied Science, University of California, Davis, California 95616
4.  NERSC, 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 remniscent 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.
In highly anisotropic superconductors such as 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 [4,6,7,8,9,10], 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 [13], dislocation proliferations[12], dynamic effects [14], or a 3D to 2D transition in the vortex pancake lattice [4,7,15,16,18]. 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 [19], to a reordered state at high current [20,21,22,23]. 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 [20], but it is unclear where this transition could be located in relation to the 2D reordering transition, and how it is affected by the externally applied 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 transition.
The magnetic interactions between pancakes in a layered material have the same logarithmic form present in thin films, but are highly anisotropic [1,16,24]. We have performed simulations in which pancakes in all layers interact magnetically with long range interactions [25,26], in contrast to other simulations, which treated only nearest layer interactions [27,28]. Our approach compliments calculations based on Lawrence-Doniach models [29]. The pancake interactions are long range both in and between planes, and are treated according to Ref. [30]. Josephson coupling is neglected as a reasonable approximation for materials with large anisotropy γ [16].
The overdamped equation of motion, at T=0, for vortex i is given by fi = −∑j=1Nv∇i U(ρi,j,zi,j)+ fivp + fd = vi, where Nv is the number of vortices, and ρi,j and zi,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 [17]
U(ρi,j,0)=2dϵ0 ⎛
⎝ 		(1− d
2λ
 )ln R
ρ
 + d
2λ
 E1⎞
⎠

U(ρi,j,z)=−sm d2ϵ0
λ
 ⎛
⎝ 		exp(−z/λ)ln R
ρ
 − E2⎞
⎠
where ϵ0 = Φ02/(4πλ)2, λ is the London penetration depth, R = 22.6 λ, the maximum radial distance, E1 = ∫∞ρ dρ′ exp(−ρ′/λ)/ρ′, E2 = ∫∞ρ dρ′ exp(−√{z2 + ρ′2}/λ)/ρ′, d=0.005λ is the interlayer spacing, and ξ is the coherence length. We vary the relative strength of the interlayer coupling using the prefactor sm. The uncorrelated pins are modeled by parabolic traps that are randomly distributed in each layer. The vortex-pin interaction is given by fivp = ∑k=1Np,L (fp/ξp)(ri − rk(p)) Θ((ξp − |ri − rk(p) |)/λ), where the pin radius ξp=0.2λ, the pinning force fp=0.02f0*, and f0*=ϵ0/λ. The case of stronger pins is considered in [25]. The driving current must be increased slowly enough for the system to equilibrate at each drive [31]. Here fd is increased by 0.00025f0* every 35000 time steps. We have simulated a 16λ×16λ system with a vortex density of nv = 0.35/λ2 and a pin density of np = 1.0/λ2 in each of L=8 layers. This corresponds to Nv=89 vortices and Np=256 pins per layer, with a total of 712 pancake vortices. We have checked for finite size effects on systems containing up to 42 layers and 3738 pancakes.
Fig1.png

Fig1.png

Figure 1: (a) Critical current fc (filled circles) and interlayer correlation Cz (open squares) versus interlayer coupling strength sm. A transition from 2D to 3D behavior occurs at smc=2. (b) fc for samples with L= 4, 8, 12, and 16 layers. Filled diamonds: fc for sm=0.5, in the 2D regime. Open triangles: fc for sm=5.0, in the 3D regime. (c) Location of the coupling transition smc for different pinning strengths fp. (d) fc versus vortex density nv at sm=1.0 shows a sharp transition, with 3D lines at nv < 0.24/λ2 and 2D pancakes at nv > 0.24/λ2.
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 sm 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, Cz = 1 − 〈(|(ri,L−ri,L+1)|/(a0/2))Θ( a0/2 − |(ri,L−ri,L+1)|)〉, where a0 is the vortex lattice constant. A clear sharp drop in Cz with decreasing sm appears in Fig. 1(a) at sm=2.0 for a sample with L=8 layers. The value smc at which the transition occurs depends upon the pinning strength fp, as indicated in Fig. 1(c). The 3D-2D transition is accompanied by a large increase in the critical current fc, as seen in Fig. 1(a). To determine fc, we find the average vortex velocity at each drive, Vx = (1/Nv)∑1Nv vx, and define fc as the driving force at which Vx > 0.0005. For weak interlayer coupling, sm ≤ 2.0, the different layers of the sample depin independently and fc is close to the value it would have in a 2D sample. Here, fc 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 sm > 2.0 average the random pinning over their length and become much less effectively pinned. Therefore fc decreases with increasing number of layers as seen in Fig. 1(b).
When the magnetic field H increases, the distance ρ 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 sm ∝ 1/H. To check this argument, we have performed simulations in which we fix sm=1 and vary H directly by changing the vortex density nv. As shown in Fig. 1(d), this method also produces a sharp decoupling transition and an associated jump in jc. 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 [32]. The behavior in fc 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.
Fig2.png
Figure 2: (a) Diamonds: Vx; heavy line: dV/dI, for a sample with sm=1.5 that undergoes dynamic recoupling. (b) P6, the fraction of sixfold coordinated vortices, for sm=1.5. (c) Cz, the interlayer correlation, for sm=1.5. (d) Diamonds: Vx; heavy line: dV/dI, for a sample with sm=4.0, which remains coupled at all drives. (e) P6, the fraction of sixfold coordinated vortices, for sm=4.0. (f) Cz, the interlayer correlation, for sm=4.0.
Fig3.png
Figure 3: Phase diagram for varying interlayer coupling sm and driving force fd. Filled circles: depinning line. X's: smectic transition line (in 3D phase only). Triangles: in-plane reordering line. Diamonds and heavy dashed line: recoupling line. For samples with sm ≤ 2.0, a dynamic transition into the 3D reordered state occurs at high drives. Inset: Vortices (filled circles) and vortex trajectories (lines) at two different currents. (a) and (b): Vortex trajectories in top and bottom layers, respectively, for a system with sm=1.5 at fd/f0*=0.0125, in the 2D plastic flow regime. The vortex positions and flow are different in each layer. (c): Vortex trajectories for a system with sm=1.5 at fd/f0*=0.02, in the 3D reordered flow regime.
We next consider the question of a possible dynamic recoupling transition by applying a driving force. When the vortex lattice begins to move for fd > fc, it undergoes plastic tearing due to the strong pinning in our sample. For decoupled samples with sm ≤ 2.0 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 Fig. 3(a,b). When high drives are applied, however, the pancakes recouple into lines and all planes begin to move in unison [Fig. 3(c)].
The reordering transition is shown in more detail in Fig. 2(a-c). We prepare an ordered lattice and then increase fd slowly enough to allow the lattice to relax into its pinned ground state for fd < fc, which is disordered for sm ≤ 2.0. As the drive is increased above fc, for a sample with sm=1.5, the recoupling transition in Cz [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 Fig. 2(d-f), a sample with sm=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 [22].
We summarize the behavior of the system in the phase diagram of Fig. 3. The coupled vortices with sm > 2.0 undergo plastic flow of stiff lines, pass through a smectic state, and finally reorder into a crystalline-like 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 sm=5 we observe elastic depinning directly into the 3D ordered state, with no plastic flow. Decoupled vortices with sm ≤ 2.0 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.
Fig4.png
Figure 4: (a) Vx(I) curves for samples with different values of sm. From right to left, sm=0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, and 7.0. (b) dV/dI curves for the same samples, from right to left, as listed above. (c) Filled squares: The value of dV/dI0, the maximum in the dV/dI curve, for samples with different sm. dV/dI0 increases significantly as the transition value of sm is approached from below. Open circles and dashed line: fc versus sm. (d) dV/dI for a single sample with sm=1.0 illustrating the double peak feature. The first broad peak (left arrow) appears just above the 2D depinning current of fd/f0*=0.010 and is associated with 2D decoupled plastic flow. The second sharp peak (right arrow) is associated with the dynamic recoupling transition.
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 sm 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 fc. To understand the reordering peak, note that when the driven 2D decoupled lattice recouples, it must cross from the low Vx response in the 2D plastic limit up to the higher 3D elastic Vx response curve at the recoupling transition. The location of the depinning transition is not affected by the value of sm since it is purely a 2D effect; however, the recoupling transition is shifted to lower driving currents as sm 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 sm=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 sm=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 DE-AC03-76SF00098.

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