Physical Review B Rapid Communications 78, 180507(R) (2008)

Transverse Commensurability Effect For Vortices in Periodic Pinning Arrays

C. Reichhardt and C. J. Olson Reichhardt

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 11 August 2008; revised manuscript received 21 October 2008; published 24 November 2008)

Using computer simulations, we demonstrate a new type of commensurability that occurs for vortices moving longitudinally through periodic pinning arrays in the presence of an additional transverse driving force. As a function of vortex density, there is a series of broad maxima in the transverse critical depinning force that do not fall at the matching fields where the number of vortices equals an integer multiple of the number of pinning sites. The commensurability effects are associated with dynamical states in which evenly spaced structures consisting of one or more moving rows of vortices form between rows of pinning sites. Remarkably, the critical transverse depinning force can be more than an order of magnitude larger than the longitudinal depinning force.


Matching effects for vortices in periodic pinning arrays have been studied extensively for different types of pinning lattice geometries [1,2,3,4,5,6]. As a function of magnetic field, the critical current passes through a series of peaks generated by commensurability effects that occur when the number of vortices equals an integer multiple of the number of pinning sites and the vortex ground state is an ordered crystalline structure [2,3,4]. When each pinning site can capture only one vortex, the excess vortices at fields above the first matching field are located in interstitial sites and depin first under an external drive [2,3,7,4,6]. Once the interstitial vortices are moving under a longitudinal drive, it is possible to apply an additional transverse drive in the direction perpendicular to the vortex motion. In this case, although the vortices are mobile in the longitudinal direction, they can remain pinned in the transverse direction and there can be a finite transverse critical depinning threshold.
The possibility of a transverse depinning threshold for moving vortices was initially predicted for systems with random pinning when the moving vortices form well defined channels [8], and transverse depinning thresholds in randomly pinned systems have been observed in numerical simulations [9] and experiments [10,11]. For vortices moving in the presence of a periodic pinning array, a finite transverse depinning threshold has been measured at high drives when all of the vortices are moving and has a value that depends on the angle between the longitudinal driving direction and a symmetry axis of the pinning lattice. Here, the most prominent transverse depinning thresholds and dynamical locking effects occur for driving along the principal axes of the pinning lattice [12,13]. This type of effect has been experimentally observed for colloidal particles moving over periodic substrates [14]. When only one vortex can be captured by each pinning site, motion of the vortices at low drives occurs as a flow of interstitial vortices between vortices that remain trapped at the pinning sites [2,3,6]. In this case, it is not known whether a transverse depinning threshold exists, and in general it is not known how the transverse depinning threshold varies with magnetic field.
It might be expected that the transverse depinning threshold would simply exhibit peaks at the same magnetic fields where peaks in the longitudinal depinning threshold appear. In this work, we demonstrate that although there are enhancements of the transverse depinning threshold at certain fields, these fields are not related to fields which produce peaks in the longitudinal depinning threshold, but are instead associated with dynamical matching conditions. The distinct dynamical matching effects appear because the moving vortices assume a different structure than the static vortex ground state. Dynamical commensurability effects occur when an integer number of moving interstitial vortex rows form between adjacent rows of pinning sites. The dynamical matching effects are much broader than the static matching effects and have maxima that encompass several static matching fields. An oscillatory critical current appears for the dynamical transverse commensurability effect. This is similar to the critical current oscillations seen for vortices depinning in artificial channels [15,16] or in layered or strip geometries [17,18], although in the channel, layer, or strip systems, the commensurations arise due to matching effects of the vortex ground state rather than the dynamical matching effects observed in the present work. Remarkably, we find that the transverse depinning threshold can be up to an order of magnitude larger than the longitudinal depinning threshold.
We numerically simulate a two-dimensional system with periodic boundary conditions in the x and y directions containing Nv vortices and Np pinning sites following a procedure similar to that used in previous simulations for vortices in periodic pinning arrays. The number of vortices is proportional to the applied magnetic field B=B∧z, which is normal to our simulation plane. The repulsive vortex-vortex interaction force is given by Fvvi = ∑Nvi ≠ jf0K1(Rij/λ)∧Rij, where K1 is a modified Bessel function, Rij=|Ri−Rj| is the distance between vortex i and j located at Ri and Rj, ∧Rij=(Ri−Rj)/Rij, f0 = ϕ02/(2πμ0λ3), ϕ = h/2e is the elementary flux quantum, and λ is the London penetration depth. The pinning sites are placed in a triangular lattice, and the field at which the number of pinning sites equals the number of vortices, Np=Nv, is defined as the matching field Bϕ. The individual pinning sites are modeled as parabolic traps of radius rp = 0.35λ and strength Fp=1.25, with Fpi=∑kNpFpf0(Rik/rp)Θ(rp−Rik)∧Rik, where Θ is the Heaviside step function, Rik=|Ri−Rpk|, ∧Rik=(Ri−Rpk)/Rij, and Rkp is the location of pin k. The overdamped equation of motion for a vortex i is
η d Ri
dt
 = Fvvi + Fpi + Fext,
(1)
where η = 1 is the viscous damping term. Fext represents the net force from an applied current and is given by Fext=FLDf0∧x + FTrDf0∧y, where the longitudinal drive FLD is applied in the x direction and the transverse drive FTrD is applied in the y direction. The initial vortex positions are obtained by simulated annealing. The drive is first applied in the longitudinal direction in increments of ∆FLD=0.0015, with 15000 simulation time steps spent at each current increment. Once the longitudinal drive reaches the desired value, it is held fixed while the transverse drive is increased from zero with the same current increment protocol. We find that our increment rate is sufficiently slow to avoid any transient effects. The longitudinal and transverse critical depinning thresholds, FLc and FTrc, are obtained by measuring the vortex velocity 〈Vα〉 = Nv−1∑iNv vi ·∧α, with α = x,y, and identifying the driving force at which 〈Vα〉 > 0.001.
Fig1.png
Figure 1: (a) The transverse critical depinning force FTrc vs B/Bϕ for a system with Bϕ=0.052ϕ0/λ2 and fixed longitudinal drive FLD = 0.6. The points a, c, and e refer to the fields illustrated in Fig. 2. The maxima are labeled according to the number of moving vortex rows between adjacent pinning rows: 1R, one row; 2R, two rows; 3R, three rows; and 4R, four rows. (b) A blow up of panel (a) for B/Bϕ > 2.0. The points a, c, and e refer to the fields illustrated in Fig. 3. (c) The transverse velocity 〈VTr〉 vs transverse force FTrD for B/Bϕ = 4.33 (open triangles), 6.0 (filled circles), 8.0 (filled squares), and 10 (open diamonds). (d) The scaled longitudinal velocity 〈VL〉(Nv/Np) (filled circles) versus longitudinal drive FDL and transverse velocity 〈VTr〉(Nv/Np) (open squares) vs transverse drive FDTr for B/Bϕ = 6.0 (left curves) and 2.0 (right curves).
We first study a system with a low pinning density to ensure that a portion of the vortices are located in the interstitial sites. The existence of a clearly defined depinning threshold which varies nonmonotonically with field is illustrated in Fig. 1(c), where we show 〈VTr〉 versus FDTr for B/Bϕ = 4.33, 6.0, 8.0, and 10.0 in a system with Bϕ = 0.052ϕ0/λ2 and fixed FLD=0.6. From a series of simulations, we obtain the variation in FcTr versus B/Bϕ plotted in Fig. 1(a). Four well defined maxima in FTrc appear that are centered near B/Bϕ = 2.0, 6.0, 12.0, and 17.0. Figure 1(b) shows a blowup of the region B/Bϕ > 2.0, where the oscillation in FTrc can be seen more clearly. This oscillation is distinct from the matching effects observed for longitudinal depinning [1,3,4,5], where well defined peaks occur at integer matching fields. The maxima in Fig. 1(a) are much broader than in the longitudinal depinning case and each encompass three or more matching fields. Similarly, the minima in FcTr also each spread over several values of B/Bϕ.
For all fields B/Bϕ > 1.0, we find that FcTr is significantly larger than the longitudinal critical force FLc, as shown in Fig. 1(d) where we plot 〈VL〉(Nv/Np) versus FLD and 〈VTr〉(Nv/Np) vs FTrD. Here the velocities have been scaled by Np rather than Nv for presentation purposes. For Bϕ = 2.0, the transverse depinning threshold FTrc is about six times higher than the longitudinal depinning threshold FLc. Both depinning thresholds are lower for B/Bϕ = 6.0; however, FTrc is again much higher than FLc. We note that the curves in Figs. 1(c,d) do not extrapolate back to the origin because for FD << Fp some vortices remain pinned and the flow is plastic.
Fig2.png
Figure 2: The vortex positions (black dots), pinning site locations (open circles), and vortex trajectories (black lines) for the system in Fig. 1(a). (a), (b) B/Bϕ = 2.67, marked a in Fig. 1(a). (c), (d) B/Bϕ = 4.0, marked c in Fig. 1(a). (e), (f) B/Bϕ = 6.0, marked e in Fig. 1(a).
In Fig. 2(a) we show the vortex and pinning site positions for point a in Fig. 1(a) at B/Bϕ = 2.67 and in Fig. 2(b) we illustrate the vortex trajectories for FDTr <~FcTr, just below the transverse depinning transition. There is a single row of moving interstitial vortices between neighboring rows of pinning sites and the vortex lattice is anisotropic, with higher vortex density in the interstitial rows than in the pinned rows. The same vortex structure appears for 1.0 < B/Bϕ < 2.9, corresponding to the maximum in FcTr marked 1R in Fig. 1(a). In Fig. 2(c), we plot the vortex positions for B/Bϕ=4.0 at a minimum of FTrc found at the point marked c in Fig. 1(a). The interstitial rows are no longer uniform and consist of an interlacing of double rows with single rows. Figure 2(d) shows that the vortex trajectories at this field are more disordered. The vortex positions and trajectories at point e in the region marked 2R in Fig. 1(a) for B/Bϕ = 6.0 appear in Fig. 2(e,f). Here there are two well defined rows of moving vortices between adjacent pinning site rows.
Fig3.png
Figure 3: The vortex positions (black dots), pinning site locations (open circles), and vortex trajectories (black lines) for the system in Fig. 1(a). (a), (b) B/Bϕ = 8.0, marked a in Fig. 1(b). (c), (d) B/Bϕ = 10.67, marked c in Fig. 1(b). (e), (f) B/Bϕ = 17, marked e in Fig. 1(b).
We find that maxima in FTrc occur whenever there is an integer number of moving rows of interstitial vortices between neighboring pinning rows. Since the number of vortices in each interstitial row can vary over a considerable range without destroying the row structure, the maxima in FTrc are much broader than the peaks in FLc associated with longitudinal commensuration effects. The row structures become increasingly anisotropic with increasing field until a buckling transition occurs which marks the end of the maximum in FTrc. In Fig. 3(a) we illustrate the vortex positions for B/Bϕ = 8.0 at a minimum in FTrc found at the point marked a in Fig. 1(b). The interstitial vortices form a mixture of two and three interstitial rows between pinning site rows, producing the nonuniform trajectories shown in Fig. 3(b). At the maximum in FTrc marked c in Fig. 1(b), corresponding to B/Bϕ = 10.67, Fig. 3(c,d) shows that there are three well defined rows of moving vortices between adjacent pinning site rows. Similarly, Fig. 3(e,f) indicates that there are four interstitial vortex rows at B/Bϕ = 17, which falls on the maximum in FTrc at the point marked e in Fig. 1(b). Near B/Bϕ = 14, where FTrc passes through a minimum, the interstitial vortices form a mixture of three and four rows, while for B/Bϕ >~19 there is a mixture of four and five interstitial rows (not shown).
Commensurability effects generated by the presence of an integer number of vortex rows between line-like barriers have been observed for longitudinal vortex motion through channel geometries [15,16] as well as critical currents in layered materials [17], superconducting strips [18], and anisotropic pinning arrays [19]. In all these cases the commensurability occurs in the static vortex configurations. This is distinct from the transverse depinning maxima that we observe here, which arises due to commensurations in the dynamical interstitial vortex configuration. Ref. [16] predicts a dependence B ∝ (n−1)2 of the matching field on the number n of static rows for n ≥ 2; we expect a similar dependence for our dynamical vortex rows.
Fig4.png
Figure 4: (a) The longitudinal critical depinning force FLc vs Bϕ for B/Bϕ = 2.0 (filled squares) and B/Bϕ = 2.5 (open squares) and the transverse critical depinning force FTrc vs Bϕ for B/Bϕ = 2.0 (filled circles) and B/Bϕ = 2.5 (open circles) (b) FTrc vs the applied longitudinal force FLD/Fp for B/Bϕ = 3.0 (filled circles) and B/Bϕ = 3.67 (filled squares).
As shown in Fig. 1(c), the transverse depinning threshold is much higher than the longitudinal depinning threshold. In Fig. 4(a) we quantify this effect by plotting FcL and FcTr as a function of pinning density Bϕ for a commensurate field B/Bϕ = 2.0 where FcL passes through a peak and for an incommensurate field B/Bϕ=2.5. At the incommensurate field, FcL and FcTr are both reduced. At B/Bϕ = 2.0, FcL increases monotonically with Bϕ while FTrc shows a smaller increase; however, FTrc is significantly larger than FLc over the entire range of pinning densities studied. At the incommensurate field B/Bϕ=2.5, we find a similar trend; however, FLc increases much more slowly than FTrc with increasing Bϕ and at Bϕ=0.6, FTrc is nearly an order of magnitude larger than FLc. In addition to increasing with increasing Bϕ, the ratio FTrc/FLc increases with decreasing Bϕ as Bϕ approaches zero due to the different rates at which the two thresholds approach zero.
The higher value of FTrc compared to FLc can be understood by considering that longitudinal depinning occurs from the ground state configurations of the interstitial vortices [2,3] and is determined by the repulsive interactions between the vortices at the pinning sites and the interstitial vortices. In the ground state, the interstitial vortices occupy positions that lower the repulsion from the pinned vortices, and the initial longitudinal depinning occurs when the interstitial vortices begin to move between the pinning sites, such as in Fig. 2(a). For the transverse depinning, when the interstitial vortices are moving at a sufficiently high velocity in the longitudinal direction they do not have time to slip between the pinned vortices in the transverse direction, but instead come into close proximity with the pinned vortices and interact strongly with them, resulting in a high repulsive barrier for depinning. If the longitudinal drive is set to a lower value before the transverse drive is applied, the interstitial vortices have more time to pass between the pinned vortices and FTrc decreases. In Fig. 4(b) we illustrate this effect by plotting FTrc versus FLD/Fp for B/Bϕ = 3.0 and B/Bϕ = 3.67. In both cases FTrc increases from a low value with increasing FLD/Fp until reaching a maximum value at FLD/Fp=1.07 for B/Bϕ=3.0 and at FLD/Fp=1.12 for B/Bϕ=3.67. Above this drive, FTrc decreases with increasing FLD/Fp as FLD/Fp approaches 1 since the vortices at the pinning sites begin to depin for the higher longitudinal drives, reducing the magnitude of the transverse critical current.
In summary, we have shown that a new type of dynamical commensurability effect can occur for vortices in periodic pinning arrays. When interstitial vortices are moving between pinned vortices and an additional transverse force is applied, there is a finite transverse critical depinning force which oscillates with field. The oscillation is not simply related to the matching of the vortices with the number of pinning sites as in the case for the longitudinal depinning, but is associated with the dynamical structure of the vortices which allows for integer or non-integer numbers of rows of moving interstitial vortices between adjacent rows of pinning sites. The transverse commensurability effects are much broader than those seen for the longitudinal depinning and each maximum in the transverse depinning force can span several matching fields. Remarkably, the transverse depinning force can be more than an order of magnitude larger than the longitudinal depinning force due to the fact that the moving interstitial vortices are unable to move between the pinned vortices without coming close to the pinned vortices, which creates a strong repulsive barrier for transverse depinning.
This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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