Physical Review B Rapid Communications 61, R3811 (2000)

Transverse Depinning in Strongly Driven Vortex Lattices with Disorder

C.J. Olson and C. Reichhardt

Department of Physics, University of California, Davis, CA 95616

(Received 4 October 1999)

Using numerical simulations we investigate the transverse depinning of moving vortex lattices interacting with random disorder. We observe a finite transverse depinning barrier for vortex lattices that are driven with high longitudinal drives, when the vortex lattice is defect free and moving in correlated 1D channels. The transverse barrier is reduced as the longitudinal drive is decreased and defects appear in the vortex lattice, and the barrier disappears in the plastic flow regime. At the transverse depinning transition, the vortex lattice moves in a staircase pattern with a clear transverse narrow-band voltage noise signature.
The dynamics of driven vortex lattices interacting with disorder exhibit a wide variety of interesting nonequilibrium behavior and dynamic phase transitions. Experiments [1,2,3,4,5,6,7,8], simulations [9,10,11,12,13,14], and theory [9,15,16,17,18,19] suggest that at low drives the vortex lattice is disordered and exhibits plastic or random flow while at higher drives the lattice can undergo a reordering transition and flow elastically. In this highly driven state it was suggested by Koshelev and Vinokur [9] that the flux lattice forms a moving crystal. In subsequent theoretical work, Giamarchi and Le Doussal [15,16] proposed that the reordered state is actually an ordered moving glass phase and that the vortices travel in highly correlated static channels. In other work [17,18,19], it has been proposed that these channels may be decoupled, producing a smectic structure. Simulations [11,12,13,14] and experiments [7] have found evidence for both smectic as well as more ordered moving vortex lattice structures.
A particularly intriguing prediction of the theory of Giamarchi and Le Doussal is that, in the highly driven phase, the moving lattice has a barrier against a transverse driving force, resulting in the existence of a finite transverse critical current. This transverse critical current has been observed in simulations by Moon et al. [11] and Ryu et al. [12] in the highly driven phase. Large transverse barriers have also been seen in systems containing periodic pinning [20]. Also, recent experiments [8] involving STM images of moving vortices reveal that the vortex lattice moves along one of its principle axes, which may not be in the direction of the drive (as predicted by [21]), suggesting that the moving lattice could be stable to a small transverse force component.
Although the existence of a transverse critical current has been confirmed in simulations, there has been no numerical study of the properties of the critical current, such as the dependence of the barrier size on the strength of the longitudinal drive or on the defectiveness of the vortex lattice. It would also be very interesting to understand the dynamics of the vortices at the transverse depinning transition, and relate this to experimental measures such as voltage noise spectra.
In this work we report a simulation study of the transverse depinning transition in driven vortex lattices interacting with random disorder. We find a finite transverse depinning barrier at high longitudinal drives, when the vortex lattice is defect free and moves in correlated 1D channels along a principle axis. For lower drives the transverse barrier is reduced but still present, even in the decoupled channel limit when adjacent channels slip past each other and some defects in the vortex lattice appear. For the lowest drives the flux lattice becomes highly defected as it enters the plastic flow phase, and the transverse barrier is lost. In the high driving limit, at the transverse depinning transition the vortex lattice spends most of its time moving along the longitudinal direction, but periodically jumps in the transverse direction by one lattice constant. The vortex lattice thus moves in a staircase fashion, keeping its principle axis aligned in the original longitudinal direction. As the transverse force increases the frequency of the jumps in the transverse direction increases, producing a clear washboard signal in the transverse velocity detectable for transverse drives up to ten times the transverse depinning threshold.
We consider a 2D slice of a system of superconducting vortices interacting with a random pinning background. The applied magnetic field H=H∧z is perpendicular to our sample, and we use periodic boundary conditions in x and y. The T=0 overdamped equation of motion for a vortex is:
fi = ηvi = fivv + fivp + fd + fiT  ,
(1)
where fi is the total force acting on vortex i, vi is the velocity of vortex i, and η = 1 is the damping coefficient. The repulsive vortex-vortex interaction is fivv = ∑j=1NvAvf0K1(|ri −rj|/λ)∧rij where ri is the position of vortex i, λ is the penetration depth, f0=Φ02/8π2λ3, the prefactor Av is set to 3 [13], and K1(r/λ) is a modified Bessel function which falls off exponentially for r > λ, allowing the interactions to be cut off at r = 6λ for computational efficiency. The vortex density is nv = 0.75/λ2 giving the number of vortices Nv=864 for a sample of size 36λ×36λ. The pinning is modeled as randomly placed attractive parabolic traps of radius rp=0.3λ with fivp = (fp/rp)(|ri − rk(p)|)Θ(rp − |ri − rk(p)|)∧r(p)ik, where r(p)k is the location of pin k, Θ is the Heaviside step function, ∧rij = (ri − rj)/|ri − rj| and ∧r(p)ik = (ri − r(p)k)/|ri − r(p)k|. The pin density is np = 1.0/λ2 and the pinning force is fp = 1.5f0. The Lorentz force from an applied current J=J∧y is modeled as a uniform driving force fd on the vortices in the x-direction. We initialize the vortex positions by performing simulated annealing with fd/f0 = 0.0. We then gradually increase fd to its final value by repeatedly increasing fd by 0.004f0 and remaining at each drive for 104 time steps, where dt = 0.02. If we increase the drive more rapidly than this, the reordered vortex lattice that forms at higher drives may fail to align its principle axis in the direction of the driving. Slow increases in fd always produce an aligned lattice. Once the final fd value is reached we equilibrate the system for an additional 2×104 steps and then begin applying a force in the transverse direction fdy which we increase by 0.0001f0 every 104 time steps. We monitor the transverse velocities Vy = (1/Nv)∑Nvi=1vi·∧y to identify the transverse critical current.
fig1.png
Figure 1: Transverse voltage Vy versus transverse driving force fdy. Diamonds: A sample with no pinning, fp/f0=0, shows no transverse barrier at any drive (here, fd/f0=3.0). Squares and circles: a sample with pinning strength fp/f0=1.5 and longitudinal critical current fcx/f0=0.5. Squares: For a longitudinal driving force fd/f0 = 1.0 the vortices flow plastically (left inset) and there is no transverse barrier. Circles: For a longitudinal driving force fd/f0 = 3.0 the vortices flow in well-defined channels (right inset) and a transverse barrier of fcy/fcx = 0.01 appears. Insets: Filled circles represent vortices and lines show the paths traveled by the vortices while moving over randomly spaced pinning sites (not shown). Left inset: plastic flow; right inset: channel flow.
In Fig. 1 we show Vy versus the transverse drive fdy at longitudinal drives of fd/f0 = 1.0 and fd/f0=3.0 for a system with a longitudinal depinning threshold of fcx/f0 ≈ 0.5. For fd/f0 = 3.0 the vortex lattice is free of defects and the vortices move in well defined 1D channels as seen from the vortex trajectories in the right inset, in agreement with previous simulations [13]. In this case there is clear evidence for a transverse barrier with fcy/fcx ≈ 0.01, approximately 100 times smaller than the longitudinal depinning threshold, in agreement with earlier simulations [11,12]. Thus, the vortex lattice resists changing its direction of motion. For fd/f0 = 1.0, the vortex lattice is highly defected and the 1D channel structure is lost (as seen in the left inset of Fig. 1). In this case the transverse barrier is absent since the lattice has no particular alignment and can readily change the direction of its motion. In the absence of pinning, fcy=0 for all drives fd, as indicated by the top curve in Fig. 1.
fig2.png
Figure 2: Circles: Transverse critical force fcy/fcx versus longitudinal driving force fd. Diamonds: Corresponding fraction of six-fold coordinated vortices, P6. The transverse critical force saturates for a reordered vortex lattice, fd/f0 > 1.8, but drops as the lattice becomes defected, reaching zero at fd/f0 ≈ 1.0. Inset: Transverse critical force for different system sizes L=24λ, 36λ, 48λ, and 60λ. Circles: fd/f0 = 1.5; Squares: fd/f0 = 2.0; Diamonds: fd/f0 = 3.0.
We ran a series of simulations in which the final longitudinal drive fd was varied in order to determine the dependence of the magnitude of the transverse barrier on the magnitude of the longitudinal drive, as well as on the density of defects in the vortex lattice. In Fig. 2 we plot the resulting transverse depinning thresholds fcy and the fraction of six-fold coordinated vortices P6 (calculated from the Voronoi or Wigner-Seitz cell construction) as a function of fd. For longitudinal drives fd/f0 > 1.5, there are no defects in the vortex lattice (indicated by the fact that P6 ≈ 1.0) and fcy is roughly constant, fcy/fcx ≈ 0.01. Below fd/f0 <~1.5 defects begin to appear in the vortex lattice as adjacent moving channels decouple, and the overall vortex lattice develops a moving smectic structure [13]. The transverse critical current fcy, which is still finite in this phase, becomes progressively reduced as more defects are generated. At the lowest drives, fd/f0 >~1.0, the transverse critical force is lost when the 1D channels are completely destroyed and the vortex lattice enters the amorphous plastic flow phase shown in the left inset of Fig. 1. The dislocations in the lattice, which were aligned perpendicular to the vortex motion at fd/f0 > 1.0, become randomly aligned at the transition to plastic flow. The loss of fcy thus coincides with the loss of alignment of the defects and the destruction of the 1D channels.
We checked the effect of finite system size on the magnitude of the transverse critical force for fd/f0 = 1.5, 2.0 and 3.0 for different system sizes (L = 24λ, 36λ, 48λ and 60λ). In the inset of Fig. 2, we show that fcy/fcx is not affected by the system size. These results support the idea that it is the presence of defects in the lattice that reduce or destroy the transverse barrier, rather than any matching effect with the system size, and that as long as some form of channeling occurs the barrier is still present.
fig3.png
Figure 3: Consecutive simulation images of vortex motion just above the transverse critical force, fdy/fcy = 0.011. Circles represent vortices and lines indicate paths followed by the vortices. A particular row of vortices has been highlighted. In (a) the vortices are moving in the direction of the applied longitudinal drive. In (b) the vortex lattice changes its direction of motion and follows a different lattice vector until it has translated by one lattice constant. In (c) the lattice again switches direction and continues to flow in the longitudinal channels. The frequency at which the vortex lattice hops from channel to channel increases with increasing transverse drive. For transverse drives further above the transverse depinning transition (not shown), the step-like motion seen above is superimposed on a slow continuous drift in the transverse direction.
In order to view the dynamics of the vortex lattice at the transverse depinning threshold we plot in Fig. 3 the vortex positions and trajectories for the same system shown in Fig. 1 with fd/f0 = 3.0 and fdy/fcx = 0.011. For clarity we have highlighted a particular row of vortices. In Fig. 3(a) the principle axis of the vortex lattice is aligned with the direction of the drive and the ordered lattice is moving along this axis. In Fig. 3(b) the entire vortex lattice has translated by one lattice constant in the transverse direction. During the transition the lattice moves at an angle to the longitudinal drive, following a different axis of the lattice. Once the vortices have moved one lattice constant transverse to the drive, they begin moving along the same channels that formed before the transverse translation. After this, the vortices move along the longitudinal direction once again, as in Fig. 3(c), before jumping by another lattice constant in the transverse direction. At the transverse depinning transition, the vortex lattice thus moves in a staircase like manner, always keeping its principle axis aligned in the direction of the original longitudinal drive and always translating along one of the axes of the lattice. As fdy is increased the frequency of jumps in the transverse direction increases. At larger fdy values, the lattice begins to drift continuously in the transverse direction without realigning, but still has a distinct step-like motion superimposed on this drift. If fdy is increased to a high enough value, we find that the vortex lattice reorients itself with the net driving force via the creation of a grain boundary. This will be discussed in more detail elsewhere. The transverse depinning transition is unlike the longitudinal depinning transition in that the latter may occur through plastic deformations of the lattice and the generation of a large number of defects when the pinning is strong. In contrast, the transverse depinning transition is elastic, and never plastic, regardless of the pinning strength.
The fact that the vortices move periodically by a lattice constant in the transverse direction is a result of the fact that the longitudinal channels followed by the vortices are uniquely determined by the underlying disorder [15]. The vortices jump from one of these stable channels to another, giving the same effect as a washboard potential. This periodic effect occurs only for a moving lattice in which 1D channels have formed; to a stationary lattice, the disorder would appear random.
fig4.png
Figure 4: (a) Transverse voltage signal Vy as a function of time for a sample with longitudinal drive fd/f0=3.0 at four different transverse driving currents fdy. Bottom to top: fdy/fcx=0.011, fdy/fcx=0.014, fdy/fcx=0.016, and fdy/fcx=0.20. Each pulse corresponds to the vortex lattice moving one lattice constant in the transverse direction. (b) Power spectrum S(ν) of the transverse voltage noise signal Vy for a sample with fd/f0=3.0 and fdy/fcx=0.016. A clear narrow band signature appears at ν = 6.0×10−5. (c) Location of the narrow band peak ν for different transverse driving forces fdy.
A consequence of the staircase-like vortex motion just above the transverse depinning threshold is that the net transverse vortex velocity at a fixed fdy should show a clear washboard frequency which should increase for increasing fdy. In Fig. 4(a) we plot Vy for samples with fdy held fixed at several different values just above the transverse depinning threshold for a system with fd/f0=3.0. For fdy/fcy = 0.011 the Vy shows periodic pulses which correspond to the correlated transverse jumps of the vortex lattice seen in Fig. 3. The flat portions of the voltage signal correspond to time periods when the lattice is moving only in the longitudinal direction, between hops. For increasing transverse drive the frequency of these pulses also increases. The additional structure in the Vy voltage pulses at lower values of fdy is characteristic of the underlying pinning, and varies for different disorder realizations. It occurs when the vortex lattice moves slightly unevenly, with a small wobble, but no defects or tearing occur in the lattice. The main feature of large periodic pulses is always observed, and the wobble dies away at larger transverse drives. In Fig. 4(b) we show that the Fourier transform of the velocity signal Vy for a driving force of fdy/fcy = 0.016 exhibits a resonance frequency at ν = 6.0×10−5 inverse MD steps. In Fig. 4(c), the resonant frequency increases linearly with fdy. We find that this resonance persists for fdy up to ten times larger then the transverse depinning threshold. It should be possible to detect this washboard frequency with Hall-noise measurements.
In recent experiments employing an STM to directly image a slowly moving vortex lattice [22], evidence for staircase-like motion of the flux lattice has been observed. In these experiments the direction of the driving force could not be directly controlled, but was assumed to be at a slight angle with respect to the principle vortex lattice vector, so that a transverse component of the driving force was present. Further experiments in which the magnitude and direction of the drive can be directly controlled are needed; however, experimental imaging techniques such as STM or Lorentz microscopy [23] seem highly promising.
In summary we have investigated the transverse depinning of moving vortex lattices interacting with random disorder. We find that for high longitudinal drives where the vortex lattice is defect free a finite transverse barrier forms. For lower drives where defects in the vortex lattice form and the vortex lattice has a smectic structure the transverse barrier is reduced but still finite. In the highly disordered plastic flow phase the transverse barrier is absent. The transverse depinning transition is elastic, unlike the plastic longitudinal depinning transition, and near this transition the vortex lattice moves in a staircase-like fashion. We observe a washboard frequency in the transverse voltage signal which can be detected for transverse drives up to ten times the depinning drive.
We thank T. Giamarchi, P. Kes, P. Le Doussal, F. Nori, R. Scalettar, and G. Zimányi for helpful discussions. We acknowledge support from CLC and CULAR, administered by the University of California.

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