Geometrically controlled ratchet effect with collective vortex motion

Rectified flux motion arising from the collective effect of many interacting vortices is obtained in a specially designed superconducting device. Ratchet structures with different asymmetric pinning potentials are generated by tuning the size, depth, and distribution of triangular blind-antidots in a high-temperature superconducting film. We experimentally and theoretically demonstrate that the amplitude and sign of the rectified vortex motion can be finely tuned with the pattern geometry. Two different dynamical regimes depending on the nature of vortices initiating the dissipation are identified, which can control the rectified vortex motion.

Single particles in an asymmetric potential show a ratchet effect, biasing or rectifying their motion, when subjected to non-equilibrium fluctuations [1][2][3]. The fundamental properties of ratchets have been studied in very different types of particles such as colloidal suspensions [4,5], molecules [6], electrons [7] or quantized magnetic flux in superconductors [8][9][10][11]. Controlled motion of particles is important to provide deeper understanding of several microscopic ratchet systems and they are potentially useful for many novel electronic and molecular devices such as rectifiers, pumps, switches or transistors [12,13].
In the case of controlled transport of magnetic flux quanta, the density and size of particles (vortices) as well as the strength of their interaction can be tuned with two external controllable parameters, i.e. temperature, T, and magnetic field, H. Ratchet effects in these systems are intimately related to the assymetry of vortex pinning arrays. Several studies have shown that besides vortex pinning effects, inter-particle interactions strongly affect vortex dynamics, giving rise to a rich phenomenology of the rectified response. Drift reversals are observed when multiple interacting particles are moving in the ratchet potential, which can depend on the magnetic field, temperature, and AC frequency [9,10,14,15] or even transverse rectification effects [15,16]. In general, in these systems the vortex dynamics is determined by the motion of a few interacting particles, since the ratchet effect disappears at high fields, whereas the study of systems containing many particles and their collective interaction is much less explored and may significantly change the transport properties from the few-particles case.
High-temperature superconductors (HTS) show a very rich H-T vortex phase diagram, making them ideally suited for the study of collective vortex interactions and dynamical flow phases in complex ratchet systems with numerous vortices. Collective effects in guided vortex motion were observed in HTS systems with triangular blind-antidots [17]. However, tailoring the ratchet vortex pinning potential to control the polarity and amplitude of rectified vortex motion in systems with many interacting particles has not been achieved until now. In this paper, we construct a system in which collective interactions on ratchets with many vortices can be manipulated through geometry controlled asymmetric pinning potentials that may provide a route to explore new implementations of ratchets in solid state devices. The robust features of net flux of a large number of particles (vortices) are observed in YBCO films with ratchet pinning potentials created by triangular blindantidots, where both the polarity and amplitude of the rectified effect can be entirely controlled with the geometry of the patterns, instead of by external parameters such as magnetic field, temperature or frequency, as in most of the ratchet systems studied. Our ratchet device presents rectified motion up to high fields and different vortex dynamic regimes are identified depending on the rectification polarity.
High-quality epitaxial YBa 2 Cu 3 O x 7− (YBCO) 250 nm thick films were grown on single-crystalline LaAlO 3 substrates by using a metalorganic decomposition methodology based on the trifluoroacetate precursor route [18]. Transport bridges of 100 μm in length and 20-40 μm in width were patterned in the standard four-probe configuration by optical lithography. Asymmetric potentials were generated on the bridges by patterning arrays with different size, depth, and distribution of triangular blind-antidots via focus ion beam (FIB) and electron beam lithography (EBL). In the former, a direct local etching of antidot array was obtained with the FIB Ga + ions whereas for the EBL we performed a wet etching of a patterned PMMA resist layer with orthophosforic acid. For all samples the pattern was designed keeping one side of the triangles along the current bridge and the other two forming an angle θ with the driving force, thus producing an asymmetry along the vortex motion direction (see figure 1(a)). The patterning conditions were optimized to avoid any damaging of the YBCO layer and good values of critical temperature, T c = 89-92 K, and critical current density, J (77K) c = 2-4 MAcm −2 were obtained in all the samples. Figure 1(b) shows an atomic force microscopy (AFM) micrograph of one of the patterned YBCO bridges with a blind-antidot lattice of triangles with a lateral size of 1 μm and 70 nm in depth. In-field electrical transport measurements were carried out in a quantum design PPMS. Current-voltage (I-V) curves were measured by using the standard four-probe method with the magnetic field applied perpendicular to the film plane.
Rectified vortex motion in our ratchet superconducting systems has been evaluated by inverting the direction of the driving force (J B ⃗ × ⃗ ) along the asymmetric pinning potential, created by the triangular array of antidots, and determining the critical current density difference at which the vortices start to move (dissipate) in each configuration. The potential is asymmetric across the transport bridge (i.e. along the y ⃗ direction shown in figure 1(a)) thus we apply positive and negative dc current along x⃗ and determine the values of J c , which we refer as J c + and J c − , respectively. , changes when comparing arrays with different geometry and also their sign can be reversed (as observed in sample A3). This hints that the steepness and sign of the potential barrier that determines the hard moving direction can be modulated with the specific geometry of the pattern. It is convenient to define a dimensionless parameter as: red antidot total θ = where s antidot is the maximum area of blind-antidots perpendicular to the current flow (red area in figure 1(c)), s total the cross-section area of the unpatterned bridge (dashed squared area in figure 1(c)) and θ the angle between the tilted edges of the triangle and the driving force direction. This parameter, which considers the maximum area reduction of the current cross-section due to the presence of blind-antidots, and the asymmetry introduced by the triangular shape will help us to understand the rectified flux flow observed in our systems.
Considering equation (1) and for the particular geometry of samples A1, A2 and A3 we obtain S red = 15, 12.7 and 8.2 respectively. So, according to this parameter, the samples with higher S red values present a positive rectification (given that positive corresponds to J J c c > + − for positive applied magnetic fields), whereas the sample with negative rectification (A3) presents a lower value of S red . Figure 2(b) displays the strength of the ratchet effect, determined by the normalized maximum J c rectification at positive magnetic fields, as a function of the S red , for several systems containing different arrays of triangles, including measurements performed at various temperatures (77, 65 and 50 K). A direct correlation between the strength of the rectified vortex motion and the S red value of the patterned bridges can be observed. This correlation is essentially temperature independent within the range studied, thus showing that the amplitude and direction of the ratchet effect, determined by the steepness and sign of the asymmetric potential barrier, is mainly controlled by the geometry of patterned antidots. It is important to note that there is a crossover value of S red ∼ 9 which separates the systems with positive and negative current rectification, producing therefore a geometrically controlled ratchet reversal. These experimental results indicate that the final current cross-section of the patterned bridge, characterized by S red , must be the key parameter determining the sign and the amplitude of the ratchet rectification.
Although a complete modeling of the system would require full 3D simulations, a simplified 2D model can explain the effect of the geometry on the J c rectification effect. Consider an infinitely long superconducting bar that contains several U-shaped notches (also infinitely long, simulating the blind-antidots). The calculation is based on assuming that the superconductor is in the critical state with a critical-current density depending on the local magnetic field through a Kim-type descending function [19,20] 1T. We will assume that dissipation is first produced where the current density reaches its maximum value J c0 . For this purpose, we calculate if these dissipative regions start either below the blind-antidots or in the space between them. To quantify it, we discretize the superconductor in N M × rectangular elements (see figure 3). Starting with a constant critical current density J k at each k-element and using an iterative procedure assuming that Biot-Savart law and the above J H ( ) c | | relation are satisfied, we find the critical-current density distribution all through the section of the superconductor. As a criterion, we consider that the dissipative regions are those in which the current density is above J 0.95 c0 . In figure 3 we show how, depending on the size of the notches, maximum dissipative areas are located either in some notched regions, driven by a vortex within the antidots (internal vortices) or outside them, driven by vortices located between the antidots (external vortex). The asymmetry observed in the dissipation areas (figures 3(b) and (c)) is due to the asymmetry of the z-component of the field created by the SC. This self-field compensates for the applied field in the left part of the SC shifting the largest values of J c also to the left [19,20]. Although this model, being symmetric with respect to the direction of the current flow, does not account for the ratchet effect, it is consistent with the fact that the two regimes of dissipation appear. This result points out that by changing the geometry (i.e. cross-section) of the ratchet system in a proper way one can tune the nature (internal or external) and amount of vortices initiating the dissipation. Once the dissipation is set, the sign and strength of the ratchet effect will present two different regimes. Positive rectification occurs when flux motion is induced by internal vortices. In this case, for J c + and positive magnetic field, internal vortices are flowing against the base of the triangles ( figure 1(d)), requiring a higher driving force than necessary to go against to the tilted edges, when the current polarity is reversed (for J c − ), and thus J c Δ > 0. The other type of system with negative rectification, happens when motion is initiated by external vortices, which feel the opposite asymmetry of the pinning potential, rectifying the motion along the opposite direction ( figure 1(e)). More complex numerical simulations considering the presence of these two types of vortices initiating the dissipation in asymmetric pinning sites [21][22][23] would be required in order to quantitatively describe the observed ratchet effect.
We can now go a step further in this analysis and study the flux dynamics in our collective ratchet systems when dissipation is produced either by internal or external vortices. Vortex velocities have been evaluated by measuring the voltage drop across the bridge, V = lBv, with B the magnetic flux density, v the average vortex velocity and l the distance between voltage contacts. The rectification of vortex motion is characterized by the difference in measured voltage for a fixed value of dc current flowing in the positive and negative x direction, We used the maximum value of J c (J c − or J c + depending on the ratchet system) to determine the difference in measured voltage.  ⩾ for regions outside and below notches, respectively. The overall dimensions for the strips are 30 μm long × 250 nm thick. In the strips with five and six notches, these have widths of 1 and 3 μm, respectively. All notches have a depth of 75 nm (vertical scale is enlarged for clarity). rectification, respectively. It is worth noting that the maximum rectified vortex velocity (proportional to V Δ ) increases as the value of S red in the system is enhanced/reduced for samples presenting positive/negative rectification, in agreement with an enhancement of the ratchet strength, as shown in figure 2(b) figure 4). The average number of vortices per triangle, n, at the values of the magnetic field where we find maximum rectification, are n ∼ 4-35 and n ∼ 10-100 for systems with positive and negative rectification, depending on the triangle size. These results strongly suggest that different interaction mechanisms are responsible for rectified flux flow in systems governed by internal and external vortex motion. Figure 4(b) shows the temperature dependence H v 0 max μ for systems with positive (red open symbols) and negative (blue closed symbols) rectification. The grey region in the graph corresponds to a low-field region of single-vortex pinning where J c is nearly independent of field and vortices mainly interact individually with the different defects present in the sample (either intrinsic or artificial). At higher fields (white region), collective effects start to be important and vortex matter enters in a regime in which vortex-vortex and vortex-defect interactions start to compete [24][25][26]. It is worth noting that for systems with positive rectification (driven by internal vortex motion), the maximum rectified effect occurs at lower magnetic fields than for those with negative rectification (governed by external vortex motion). In the latter case, maximum effects are obtained deep inside the region where vortex-defect interactions strongly compete with collective effects. Based on these observations one can figure out that the nature of the vortex initiating the dissipation has a significant role in the dynamics of the ratchet effect. Systems with negative rectification require strong vortex-vortex interactions to work. This kind of ratchet may be described if one assumes that external vortices do not directly interact with the asymmetric blind-antidots but they feel the asymmetric pinning potential through collective interactions with internal pinned vortices within the triangles [17,27]. Conversely, for systems with positive rectification, the field range where maximum effect is obtained appears in a region with weak vortex-vortex interactions and thus in this case it is reasonable to assume that collective interactions are not essencial to generate the ratchet potential. The asymmetry in this case would come from the nano-wall pinning created at the edges of the blind triangles [28].
In conclusion, we have demonstrated geometrically controlled rectified vortex motion effects in YBCO films patterned with asymmetric triangular blind-antidots. In these systems, both the steepness and sign of the ratchet potential can be tailored with the geometry (size, depth, and distribution) of the patterned triangles. Geometrically inverse rectification effects can be well described by numerical simulations considering the final cross-section of patterned bridges. Two different vortex dynamic regimes have been identified in positive and negative rectification systems. In the former, ratchet effects are determined via direct interaction between vortices and asymmetric patterns. In the latter, the ratchet dynamics is mainly governed by strong collective vortex interactions. Using a system based on a high-temperature superconductor we are able to explore the physics of ratchet systems with many interacting vortices, providing a useful toolbox for controlling and manipulating transport of multiple particles at the nanoscale. In these systems, the sign of rectified effects does not show any temperature or field dependence but is fixed by the structure geometry.