Physical Review B Rapid Communications 62, R14657 (2000)

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

Alejandro B. Kolton,1 Daniel Domínguez,1 Cynthia J. Olson2 and Niels Grønbech-Jensen3,4

1 Centro Atómico Bariloche, 8400 S. C. de Bariloche, Rio Negro, Argentina
2 Department of Physics, University of California, Davis, California 95616
3 Department of Applied Science, University of California, Davis, California 95616
4 NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720

(Received 11 September 2000)

We study a three-dimensional model of driven vortices in weakly coupled layered superconductors with strong pinning. Above the critical force Fc, 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 an "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 Fs 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]. For a long time, it was believed that the high-current phase would have crystalline order. Recently, it has been found that different kinds of order are possible at high currents, depending on pinning strength and dimensionality [2,3,4,5,6,7]. This has led to numerous theoretical [2,3], experimental [4] and numerical studies [5,6,7]. 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 experimentally [8] 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 and the in-plane structural order take place in a sequence of dynamical phases 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 λ|| [10]. Previous simulations of driven vortices in 3D superconductors have been performed using Langevin dynamics of short-range interacting particles [11] or the driven isotropic 3D XY model [12].
The equation of motion for a pancake located in position Ri=(ri,zi)=(xi,yi,ni d), (z ≡ ∧c), is:
η dri
dt
 =
∑
j ≠ i 
 Fv(ρij,zij)+
∑
p 
 Fp(ρip) + F  ,
(1)
where ρij=|ri−rj| and zij=|zi−zj| are the in-plane and inter-plane distance between pancakes i,j, ρip = |ri−rp| is the in-plane distance between the vortex i and a pinning site at Rp=(rp,zi), η is the Bardeen-Stephen friction, and F=[(Φ0)/(c)]J×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 Fp=−2 Ap e−(ρ/ap)2 r/ap2, where ap is the pinning range. The magnetic interaction between pancakes Fv(ρ,z)=Fρ(ρ,z) ∧r is given by [10,13]:
Fρ(ρ,0)
=
Av
ρ
 ⎡
⎣ 		1− λ||
Λ
(1−e−ρ/ λ||)⎤
⎦
(2)
Fρ(ρ,zn)
=
− λ||
Λ
Av
ρ
 [e−|zn|/λ||−e−Rn/λ||]   .
(3)
Here, R=√{z2+ρ2} and Λ = 2 λ||2/d is the 2D thin-film screening length. An analogous model to Eqs. (2-3) was used in [14]. We normalize length scales by λ||, energy scales by Av=ϕ02 / 4 π2 Λ, and time is normalized by τ = ηλ||2/Av. We consider Nv pancake vortices and Np pinning centers per layer in Nl rectangular layers of size Lx×Ly, and the normalized vortex density is nv=B λ||2 /Φ0=(a0/λ||)2. We consider nv=0.29 with Ly=16λ|| and Lx=√3/2Ly, Nl=8 and Nv=64. We take a pinning range of ap=0.2, a large pinning strengh of Ap/Av=0.2, with a high density of pinning centers np=3.125 nv. The model of Eq.(2-3) is valid in the limit d << λ|| << Λ. We take d/λ||=0.01, which corresponds to BSCCO compounds [10]. Moving pancake vortices induce a total electric field E=[(B)/(c)]v×z, with v=[1/(Nv Nl)]∑ivi. 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=0.01τ and averages are evaluated in 16384 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.
fig1.png
Figure 1: Vortex trajectories in the first five layers: (a) F=0.6, (b) F=1.1, (c) F=2.0. (d) F=3.9. Surface intensity plot of the averaged in-plane structure factor S(k): (e) F=0.6, (f) F=1.1, (g) F=2.0. (h) F=3.9.
We start with a qualitative description of the different steady states that arise as a function of increasing force. In Figure 1(a-d) we show the vortex trajectories { Ri(t)} for typical values of F by plotting the positions of the pancakes in five of the layers for all t. In Fig.1(e-h) we show the average in-plane structure factor S(k) = 〈[1/(Nl)]∑n|[1/(Nv)]∑i exp[ik·rni(t)]|2〉, with k=(kx,ky). Above the depinning critical force Fc, we find the following dynamical phases. (i)Plastic flow (Fc < F < Fp): 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 (Fp < F < Ft): 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 k·F=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 (Ft < F < Fs): 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 (k·F ≠ 0) have appeared [Fig.1(g)]. The location of channels is correlated in the c-axis. (iv)Coherent solid (F > Fs): The channels become more straigth with small transverse wandering [Fig.1(d)]. The S(k) shows well defined peaks for all k in the reciprocal lattice [Fig.1(h)]. The dynamical phases (i)-(iii) are similar to the ones found previously in 2D thin films [7].
fig2.png
Figure 2: (a) Velocity-force curve, left scale, black points, dV/dF (differential resistance), right scale, white points. (b) Intensity of the Bragg peaks. For smectic ordering S(G1), Ky=0, (×) symbols. For longitudinal ordering S(G2,3), Ky ≠ 0, (+) symbols. (c) Diffusion coefficient for transverse motion Dx, (up triangles), left scale. Longitudinal displacements 〈[∆y(t)]2〉 for a given t as a function of F, (squares), right scale.
The characteristic forces Fc,Fp,Ft,Fs separating the different dynamical phases are obtained from the analysis of the in-plane and out of plane structural and dynamical correlations. We show in Fig.2 the in-plane structure factor and temporal fluctuations, obtained in the same way as for 2D [7]. In Fig.2(a) we plot the average velocity V=〈Vy(t)〉 = 〈[1/(NvNl)]∑n,i[(dyni)/(dt)]〉, in the direction of the force as a function of F and its corresponding derivative dV/dF (differential resistance). The force Fp corresponds to the peak in the differential resistance. We also see a small second maximum in dV/dF for a force between Ft and Fs [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=2π/a0∧x, corresponding to smectic ordering, and the average of the peaks corresponding to longitudinal ordering at G2=±2π/a0(1/2,√3/2) and G3=±2π/a0(−1/2,√3/2). We see that at Fp the smectic peak rises up from zero, then at Ft it reaches an almost constant value and later at Fs it has a small jump. The longitudinal peak has a small finite value for forces above Fp, and only at Fs shows a significant increment towards a large value. Comparing with the previous 2D results [7], we can make the reasonable assumption that for Fp < F < Ft there is only short-range smectic order (since there is activated transverse diffusion beteween elastic channels, see below), for Ft < F < Fs there is probably quasi-long range smectic order but short range longitudinal order, and above Fs there is both transversal and longitudinal order (quasilong-range or long-range). What is new, compared with the 2D thin film case [7], is that above a force Fs there is a significant amount of longitudinal order. This may correspond either to a moving crystalline phase (if there is long-range order) or to a moving Bragg glass (if there is quasilong-range order) [3]. We have verified that, for a given F > Fs, there is both longitudinal and transversal order for system sizes of Nl×Nv=5×36,5×64,8×64,8×100,10×100. However, a detailed finite size analysis is not possible with these few small samples. We complement our discussion of the in-plane physics with the study of the temporal fluctuations, which are shown in Fig.2(c). We calculate the transverse diffusion coefficient Dx from the average quadratic transverse displacements of vortices from their center of mass position (―Xn,―Yn) , [1/(NvNl)]∑i[xi(t)−―Xni(t)−xi(0)+―Xni(0)]2 ≈ Dxt. We find that Dx is maximum at Fp in coincidence with the peak in dV/dF. Below Fp diffusion is through the intrincate network of plastic channels, above Fp diffusion is through activated jumps between elastic channels. Dx sharply drops to zero at Ft, indicating that transverse displacements are localized in the transverse solid phase [7]. The drift from the center of mass of longitudinal displacements 〈[∆y(t)]2〉 = 〈[yi(t)−―Yni(t)−yi(0)+―Yni(0)]2〉 is superdiffusive for F < Fs, similar to the results observed in 2D films [7]. For F > Fs the longitudinal displacements become frozen in a constant value 〈[∆y(t)]2〉 < a0/Nl, as it is shown in Fig.2(c). Since in-plane displacements are localized and there are large transversal and longitudinal Bragg peaks, we call this phase a coherent solid.
Fig3.png
Figure 3: (a) Correlation parameter in c-direction of instantaneous configurations Cz(n) vs F for n=1,2,3,4 interplane distance. (b) Trajectories overlap correlation parameter in c-direction On vs F for n=1,2,3,4 interplane distance. (c) Voltage fluctuations in c-direction 〈δ2Vc 〉 vs F.
Let us now discuss how the ordering along the c-axis takes place. We analyze the pair distribution function: g(ρ,n)=[(LxLy)/(Nv)]〈∑i ≠ jδ(ρ−ρij) δn,nij 〉. From g(ρ,n) we define a correlation function along c-axis Cz(n)=limρ→ 0 g(ρ,n). Short-range ordering will be given by a finite Cz(n=1), meaning that pancakes in neighboring planes are coupled and a "vortex line" can therefore be defined. In principle, an exponential decay Cz(n) ∼ exp(−n/ξz) would define a correlation length for the vortex line [9]. On the other hand, long-range ordering will be given by Cz(n→∞)→ C∞z > 0. In Fig.3(a) we show Cz(n) as a function of F for n=1,2,3,4. We see that at Fp there is an onset of short-range order along the c-axis with a finite Cz(n=1). At higher forces between Fp and Ft the other Cz(n > 1) start to rise. The absence of correlations for F < Fp means that pancake motion is completely random and uncorrelated between different planes. Therefore, we propose that the plastic flow regime corresponds to a pancake gas. Above Fp, 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 phase as an entangled line liquid. Above Ft, Cz(n) is finite for all n considered and tends to saturate upon increasing n. This indicates that vortex lines become more stiff above Ft. We also analyzed the c-axis correlation between averaged vortex densities. We first define ρv(r,n,t)=[1/(Nv)]∑i δ(r−rni(t)) taking a coarse-graining scale ∆r=a0/2 (results do not vary much for ∆r=a0/4). The regions where the average density 〈ρv(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 On=Cρ(n)/Cρ(0), with Cρ(n)=[(Lx Ly)/(Nl)][∑m ∫dr 〈ρv(r,m)〉〈ρv(r,m+n) 〉] − 1. This is shown in Fig.3(b). We see that On also has an onset at Fp. For Fp < F < Ft, we have some overlap of the elastic channels that decreases with increasing n, consistent with the entangled line-liquid picture. More interestingly, at Ft the overlap function On 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 Ft. A striking result is that we find On ≈ 1 above Fs, i.e., a perfect c-axis coupling of elastic channels (within the scale  ∼ a0/4). 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 Vn,n+1(r,t) = [(Φ0)/(2πc)][(d)/(dt)]ϕn,n+1(r,t), with ϕn,n+1 the superconducting phase difference between planes n and n+1. A good approximation for pancakes at rn,i is to write ϕn,n+1(r,t)=∑i[f(r−rn,i)−f(r−rn+1,i)] with f(r) ≈ arctan(x/y). We can therefore estimate the c-axis voltage fluctuations as 〈δ2 Vc〉 = ∑n ∫[〈V2n,n+1(r,t) 〉−〈Vn,n+1(r,t) 〉2 ] dr ≈ A ∑n [〈Vn2〉−〈Vn〉2]−[〈Vn·Vn+1〉− 〈Vn〉·〈Vn+1〉]; with Vn(t) = [1/(Nv)]∑ivn,i(t), and the constant A ∼ logΛ if L > Λ or A ∼ log(L) otherwise. It is clear that 〈δ2 Vc〉 = 0 for pancakes moving with the same velocity in all planes. We see in Fig.3(c) that the voltage fluctuations have a maximum at Fp. For F > Fp, 〈δ2 Vc〉 decreases, and above Fs it reaches an almost F-independent value. The fact that 〈δ2 Vc〉 does not vanish above Fs is consistent with the result that Cz(n) < 1 for all values of F in Fig.3(a). In other words, while transverse displacements are strongly correlated along the c direction for large forces [Fig.3(b)], the longitudinal displacements in different planes are weakly correlated .
In conclusion, we have clearly distinguished different dynamical phases in 3D layered superconductors considering both in-plane and c-axis ordering [16]. 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 ρc resistivity and in-plane current-voltage response [18].
We acknowledge discussions with L.N. Bulaevskii, P.S. Cornaglia, F. de la Cruz, Y. Fasano, M. Menghini. This work has been supported by ANPCYT (Proy. 03-00000-01034), by Fundación Antorchas (Proy. A-13532/1-96), Conicet, CNEA and FOMEC (Argentina); by CLC and CULAR (Los Alamos), and by the Director, Office of Adv. Sci. Comp. Res., Division of Mathematical, Information, and Computational Sciences of the U.S.D.O.E. (contract number DE-AC03-76SF00098).

References

[1]
R. Thorel et al., J. Phys. (Paris) 34, 447 (1973)
[2]
A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett. 73, 3580 (1994).
[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, ibid. 57, 7705 (1998); S. Scheidl and V. M. Vinokur, ibid. 57, 13800 (1998).
[4]
S. Bhattacharya and M. J. Higgins, Phys. Rev. Lett. 70, 2617 (1993); M. C. Hellerqvist et al., ibid. 76, 4022 (1996); U. Yaron et al., Nature (London) 376, 743 (1995); F. Pardo et al, ibid. 396, 348 (1998).
[5]
H. J. Jensen et al., Phys. Rev. Lett. 60, 1676 (1988); A.-C. Shi and A. J. Berlinsky, ibid. 67, 1926 (1991). N. Grønbech-Jensen, A. R. Bishop and D. Domínguez, ibid. 76, 2985 (1996); C. J. Olson, C. Reichhardt and F. Nori, ibid. 80, 2197 (1998).
[6]
K. Moon, R. T. Scalettar and G. Zimányi, Phys. Rev. Lett. 77, 2778 (1996); S. Ryu et al., ibid. 77, 5114 (1996). S. Spencer and H. J. Jensen, Phys. Rev. B 55, 8473 (1997); C. J. Olson, C. Reichhardt and F. Nori, Phys. Rev. Lett. 81, 3757 (1998); D. Domínguez, ibid. 82, 181 (1999).
[7]
A. B. Kolton, D. Domínguez, N. Grønbech-Jensen, Phys. Rev. Lett. 83, 3061 (1999).
[8]
D. López et al., Phys. Rev. Lett. 76, 4034 (1996).
[9]
See for example P. Olsson and S. Teitel, Phys. Rev. Lett. 82, 2183 (1999) and references therein.
[10]
J. R. Clem, Phys. Rev. B. 43, 7837 (1991).
[11]
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, ibid. 81, 1497 (1998); C. J. Olson, R. T. Scalettar, G. T. Zimányi, cond-mat/9909454.
[12]
D. Domínguez, N. Grønbech-Jensen and A.R. Bishop, Phys. Rev. Lett. 78, 2644 (1997).
[13]
A. B. Kolton, D. Domínguez and N. Grønbech-Jensen, Physica C, to be published; C. J. Olson and N. Grønbech-Jensen, Physica C, to be published.
[14]
D. Reefman, H. B. Brom, Physica C 213, 229 (1993).
[15]
N. Grønbech-Jensen, Int. J. Mod. Phys. C 7, 873 (1996); Comp. Phys. Comm. 119, 115 (1999).
[16]
For weak pinning, the ordering transition in the c-axis may occur in a single step and also a pronounced second peak in dV/dF is observed, see C. J. Olson, N. Grønbech-Jensen, A. B. Kolton and G. T. Zimányi, preprint.
[17]
O. Tsui et al., Phys. Rev. Lett. 73, 724 (1994); L. Bulaevskii, M. Maley and M. Tachiki, ibid. 74, 801 (1994).
[18]
M. Menghini et al., unpublished.


File translated from TEX by TTHgold, version 4.00.

Back to Home