---
abstract: 'We have investigated the vortex dynamics in superconducting thin film devices with non-uniform patterns of artificial pinning centers (APCs). The magneto-transport properties of a conformal crystal and a randomly diluted APC pattern are compared with that of a triangular reference lattice. We have found that in both cases the magneto-resistance below the first matching field of the triangular reference lattice is significantly reduced. For the conformal crystal, the magneto-resistance is below the noise floor indicating highly effective vortex pinning over a wide magnetic field range. Further, we have discovered that for asymmetric patterns the R vs. H curves are mostly symmetric. This implies that the enhanced vortex pinning is due to the commensurability with a stripe in the non-uniform APC pattern and not due to a rearrangement and compression of the whole vortex lattice. (submitted on 04/29/13 to APL)'
author:
- 'S. Guénon'
- 'Y. J. Rosen'
- 'Ali C. Basaran'
- 'Ivan K. Schuller'
title: Highly Effective Superconducting Vortex Pinning in Conformal Crystals
---

Vortex dynamics in superconducting thin films with artificial pinning centers (APCs) have been extensively studied in recent years. A central problem in this field is how to improve the critical current density of a superconducting device by choosing a suitable APC distribution. If the APCs are arranged in a hexagonal or a rectangular lattice, the critical current is increased for certain magnetic matching fields due to the commensurability with the Abrikosov vortex lattice [@Baert1995; @Harada1996; @Martin1997; @Castellanos1997; @Moshchalkov1998; @Morgan1998; @Villegas2003]. In order to increase the field range for vortex pinning, different APC distributions like quasiperiodic [@Kemmler2006; @Misko2010] or random [@Rosen2010; @Kemmler2009] lattices have been investigated. Recent theoretical papers are focusing on non-uniform APC distributions like hyperbolic-tessellation arrays [@Misko2012] and conformal crystals [@Ray2012]. A conformal crystal can be obtained by conformally mapping a semiannular section of a regular hexagonal lattice on a rectangle (see [@Ray2012; @Rothen1993; @Rothen1996]). In this transformation the angles are preserved, but a vertex density gradient along one side of the rectangle is introduced. Hence, a conformal crystal is a non-uniform pattern, in which on a small scale the vertices are arranged triangularly. On a large scale however, the vertex density changes continuously.\
A recent theoretical study [@Ray2012] has shown that the pinning in conformal crystals is significant stronger over a much wider field range than that found for other APCs with an equivalent number of pinning sites. These promising results have not yet been confirmed experimentally.\
In our experimental study we have investigated the vortex dynamics in superconducting thin film devices with non-uniform APC patterns. We compare the magneto-transport properties of conformal crystals and corresponding randomly diluted APC patterns with the properties of a triangular reference lattice. Additionally, we investigate whether an electric current applied to a superconducting micro bridge with a non-uniform APC pattern produces enhanced pinning due to vortex distribution rearrangement and compression. Because the current pushes the vortices against the edge of the sample, an asymmetric APC pattern with respect to the center line of the bridge could produce an asymmetric R vs. H curve when compared to the triangular array. In agreement with theoretical calculations, we find enhanced pinning by non-uniform APC patterns. However, contrary to the expectations the R vs. H curves are essentially symmetric in all cases indicating that the enhanced vortex pinning is not the result of a large scale redistribution of vortices. It appears rather that the vortices are effectively pinned locally in a stripe, in which the vortex lattice and the non-uniform APC pattern is commensurate.\

![(color online) Artificial pinning center patterns in the center of the micro bridge devices. A: symmetric randomly diluted, B and E: triangular, C: symmetric conformal crystal, D: asymmetric randomly diluted, F: asymmetric conformal crystal. Inset: Geometry of the sample under investigation. The micro bridge devices A-C and D-F are in series.[]{data-label="fig:pattern"}](fig_pattern)

Standard e-beam writing was used to prepare magnetic pinning sites [@Velez2008]. First, the APC dot patterns were written into PMMA resist on a SiO$_2$ coated Si substrate. Then, a 40 nm thick Cobalt film was deposited via e-beam evaporation and approximately 120 nm diameter Cobalt dots were obtained in a lift-off step. Micro bridges of a 100 nm thick, RF-sputtered Nb film were prepared using optical lithography and lift-off. A superconducting transition temperature of 8.6 K was measured.\
Fig. \[fig:pattern\] shows the centers of the APC patterns and the geometry of the sample. To avoid misalignment of the APC pattern with respect to the superconducting micro bridges, they were all written in a single scan with an approximate write field of $400 \times 400\,\mu\textnormal{m}^2$. The APC patterns are $100\,\mu$m long and $40\,\mu$m wide, to assure that the whole area between the voltage taps is filled with APCs. Device A and C are symmetric (the pinning site density increases to both edges of the micro bridge), while D and F are asymmetric (the pinning site density increases to the lower edge of the micro bridge). The triangular lattices B and E have a lattice constant of approximately $0.6\,\mu$m. The conformal crystals (C and F) were obtained by mapping the triangular lattice according to [@Ray2012], but, instead of choosing a semiannular region, a partial annular section was chosen with an opening angle of $\pi/2$. Hence, in this study the conformal crystal is deformed less than in [@Ray2012]. For the randomly diluted lattices (A and D) the vertices of the triangular lattice were randomly removed, so that the pinning site density is the same as that of the corresponding conformal crystals. All patterns were compressed by 18% in the direction perpendicular to the micro bridge edges.\
Magneto-resistance was measured in a liquid Helium cryostat with a superconducting magnet and a variable temperature insert. The sample was cooled with evaporated Helium gas and a temperature stability better than $1\,$mK was achieved using a heater in the sample mount. There was a temperature drift of a few mK on a time scale of a few minutes due to a change in the evaporation rate and therefore the time slot for acquiring an R vs. H was about one minute.\
In this study, it was very important to eliminate the different thermo-electric off-sets in the wiring. Therefore, we used for the symmetric patterns the DC polarity reversed mode of the Keithley transport electronics [@Daire2005]. In this mode the current source changes polarity in a low frequency square wave pattern and the voltage is read out synchronously. The signal is averaged by subtracting the opposite polarity voltages. For the asymmetric pattern we used an alternative technique to differentiate between the two current directions. We acquired four R vs. H (DC) curves by changing the current source polarity for consecutive curves. In this technique the average resistivity is given by $$\overline{R}(H)=\left[R_1(H)-R_2(-H)+R_3(H)-R_4(-H)\right]/4$$ The minus sign before the magnetic field in the second and fourth R(H) is necessary, because a change in the current direction changes the direction of the Lorentz-force.\

![(color online) Resistance vs. magnetic field of the micro bridge devices with symmetric APC patterns at different set temperatures (logarithmic scale). The ratio $T/T_c$ was estimated for the triangular reference lattice. Insets: Resistance vs. magnetic fields for small fields (linear scale).[]{data-label="fig:symmetric"}](fig_symmetric)

In order to investigate the flux flow resistance of non-uniform APC patterns we have measured the magneto-resistance at different temperatures near the superconducting transition. In all electrical transport measurements presented, we applied a relatively small $10\,\mu$A current to avoid Joule heating. With the series arrangement, we were able to consecutively measure the R vs. H curves of the three devices at each temperature by changing the voltage contacts. We used the micro bridge devices with a triangular APC pattern as a reference. The ratio $T/T_c$ was determined from the magneto-resistance vs. temperature dependence of the triangular reference pattern.\
Fig. \[fig:symmetric\] shows R vs. H curves of the devices with symmetric patterns at three different temperatures. The triangular reference lattices shows two matching minima at approximately 86 Oe and 172 Oe due to the relatively small diameter to separation ratio of the magnetic pinning dots [@Hoffmann2000]. The resistance of the device with a randomly diluted APC is considerably smaller than that of the triangular lattice below the first matching field. Above the first matching field, R vs. H is almost the same except in vicinity of the second matching field where the triangular lattice is smaller. The flux flow resistance of the device with the conformal crystal is smaller than that of the randomly diluted or the triangular APC pattern for all temperatures and magnetic fields except in vicinity of the matching fields (fig. \[fig:symmetric\] a for $T/T_c=0.994$). For field values smaller than the first matching field, the flux flow resistance is below the noise floor of the measurement. This indicates strong pinning which immobilizes the vortices. We emphasize that this behavior is reproducible and robust for all temperatures in the superconducting state (see fig. \[fig:symmetric\]).\

![(color online) Averaged resistance vs. magnetic field of the micro bridge devices with asymmetric APC patterns at different set temperatures (logarithmic scale). The ratio $T/T_c$ was estimated for the triangular reference lattice. Insets: Averaged resistance vs. magnetic fields for small fields (linear scale).[]{data-label="fig:asymmetric"}](fig_asymmetric)

The R vs. H of the asymmetric devices are very similar to the symmetric ones as shown in fig. \[fig:asymmetric\]. For resistances above approximately $30\,\textnormal{m}\Omega$, the R vs. H of the asymmetric conformal crystal and the randomly diluted pattern are symmetric. For small resistance values the R vs. H curves are slightly asymmetric. For instance, consider the asymmetric conformal crystal at $\textnormal{T}/\textnormal{T}_\textnormal{c}=0.994$ (see fig \[fig:asymmetric\] a). The difference between the magneto-resistance at field values of -30Oe and 30Oe is approximately 6.7m$\Omega$. But in this set of graphs the R vs. H of the triangular reference lattice is asymmetric, too. Between the magneto-resistance at the first matching field the difference is approximately 15m$\Omega$. Hence, the R vs. H asymmetry of the symmetric reference pattern is more pronounced. This indicates that the R vs. H asymmetry of the asymmetric pattern is smaller than the systematic error.\
In conclusion, we discovered that below the first matching field of the triangular reference lattice, the R vs. H of the randomly diluted and the conformal crystal is reduced. For the conformal crystal, the pinning effectively reduces the resistance below the noise floor over a wide magnetic field range.\
In order to investigate whether the enhanced pinning is due to a compression and rearrangement of the vortex lattice, we have measured the magneto-transport of an asymmetric conformal crystal. For a fixed current direction, the vortices are pushed towards the side with either a high or low APC density when the field direction is changed. If the vortex distribution is rearranged and compressed by the Lorentz force when it is pushed against the Bean-Livingston barrier at the micro bridge edge then the commensurability of the vortices with the APC distribution would depend on the field direction. This would result in an asymmetric R vs. H curve. However, our samples do not show a pronounced asymmetry when compared to the triangular lattice. We therefore suggest an alternative explanation:\
For every magnetic field smaller than the first matching field there exists a stripe in the non-uniform APC distribution, in which the APC density is more or less equal to the vortex density. The commensurability of the vortex lattice with the pinning sites in this stripe, which is parallel to the micro bridge edge, increases the vortex pinning and hinders or suppresses the movement of the whole vortex lattice. Although the conformal crystal and the randomly diluted lattice have the same APC density, the commensurability with the vortex lattice is considerably higher in the first one. Therefore, the pinning in the conformal crystal is more effective. This study might be helpful to enhance the critical current density in tape conductors, to reduce the 1/f-noise in SQUIDs [@Woerdenweber2000] or to improve the performance of microwave resonators [@Bothner2011] in magnetic fields.

We thank D. Ray, C.J. Olson Reichhardt, B. Janko and C. Reichhardt for sharing their preprint and useful conversations. This work was supported by the Office of Basic Energy Science, U.S. Department of Energy, under Grant No. DE FG03-87ER-45332.

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