# Vortex matching at 6 T in YBa2Cu3O7-δ thin films by imprinting a 20 nm-
periodic pinning array with a focused helium ion beam

Max Karrer  Physikalisches Institut, Center for Quantum Science (CQ) and
LISA+, University of Tübingen, Auf der Morgenstelle 14, 72076, Tübingen,
Germany  Bernd Aichner  Faculty of Physics, University of Vienna,
Boltzmanngasse 5, 1090 Vienna, Austria  Katja Wurster  Physikalisches
Institut, Center for Quantum Science (CQ) and LISA+, University of Tübingen,
Auf der Morgenstelle 14, 72076, Tübingen, Germany  César Magén  Instituto de
Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, 50009
Zaragoza, Spain  Christoph Schmid  Physikalisches Institut, Center for Quantum
Science (CQ) and LISA+, University of Tübingen, Auf der Morgenstelle 14,
72076, Tübingen, Germany  Robin Hutt  Physikalisches Institut, Center for
Quantum Science (CQ) and LISA+, University of Tübingen, Auf der Morgenstelle
14, 72076, Tübingen, Germany  Barbora Budinská  Faculty of Physics, University
of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria  Vienna Doctoral School in
Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
Oleksandr V. Dobrovolskiy  Faculty of Physics, University of Vienna,
Boltzmanngasse 5, 1090 Vienna, Austria  Reinhold Kleiner  Physikalisches
Institut, Center for Quantum Science (CQ) and LISA+, University of Tübingen,
Auf der Morgenstelle 14, 72076, Tübingen, Germany  Wolfgang Lang
<EMAIL_ADDRESS>Faculty of Physics, University of Vienna,
Boltzmanngasse 5, 1090 Vienna, Austria  Edward Goldobin  Dieter Koelle
<EMAIL_ADDRESS>Physikalisches Institut, Center for Quantum Science
(CQ) and LISA+, University of Tübingen, Auf der Morgenstelle 14, 72076,
Tübingen, Germany

###### Abstract

Controlled engineering of vortex pinning sites in copper-oxide superconductors
is a critical issue in manufacturing devices based on magnetic flux quanta. To
address this, we employed a focused He-ion beam (He-FIB) to irradiate thin
YBa2Cu3O7-δ films and create ultradense hexagonal arrays of defects with
lattice spacings as small as 20 nm. Critical current and magnetoresistance
measurements demonstrate efficient pinning by an unprecedentedly high matching
field of 6 T visible in a huge temperature range from the critical temperature
$T_{c}$ down to 2 K. These results show that He-FIB irradiation provides
excellent opportunities for the development and application of superconducting
fluxonic devices based on Abrikosov vortices. In particular, our findings
suggest that such devices can operate at temperatures far below $T_{c}$, where
superconductivity is robust.

##  I Introduction

Superconductivity is a macroscopic quantum phenomenon characterized by the
long-range coherence of charge carriers that make up the superconducting
condensate. Most technically viable superconductors, including the copper-
oxide superconductors (HTS) with high critical temperature $T_{c}$, belong to
the class of type-II superconductors. These materials allow the magnetic field
to penetrate in the form of Abrikosov vortices of supercurrent, which carry
the quantized magnetic flux $\Phi_{0}=h/(2e)$, where $h$ is Planck’s constant
and $e$ is the elementary charge.

The properties and the interaction of Abrikosov vortices are controlled by two
characteristic lengths: the Ginzburg-Landau coherence length $\xi$ and the
London penetration depth $\lambda$ [1]. In the vortex center, the
superconducting order parameter is suppressed to zero on the length scale
$\xi$, thus creating a normal-conducting vortex core of approximately $2\xi$
diameter. As the formation of this normal-conducting core, i.e., the
suppression of superconductivity, requires energy, the vortex will be
preferentially pinned to a non-superconducting defect.

The magnetic field and the supercurrents circulating around the vortex core
decay exponentially on a characteristic scale given by the London penetration
depth $\lambda>\xi$, which determines the magnetic vortex-vortex interaction
range. For thin superconducting films with thickness $t_{z}\lesssim\lambda$,
the so-called Pearl length [2] $\Lambda=2\lambda^{2}/t_{z}$ replaces
$\lambda$.

Importantly, Ginzburg-Landau theory predicts the temperature dependence
$\xi(T)=\xi(0)(1-T/T_{c})^{-1/2}$, which implies that $\xi(T)\to\infty$ at
temperature $T\to T_{c}$. The same temperature dependence is predicted for
$\lambda(T)$ [1]. Consequently, at temperatures near $T_{c}$, the size of the
vortex cores and their interaction range rises significantly. For the
anisotropic compound YBa2Cu3O7-δ (YBCO) used here as a prototypical HTS, the
in-plane parameters are $\xi_{ab}(0)\approx 1.2\,\mathrm{nm}$ and
$\lambda_{ab}(0)\approx 250\,\mathrm{nm}$ (for epitaxial thin films) [3, 4,
5].

A single Abrikosov vortex can store a classical data bit [6] and can be
manipulated by several methods [7]. Vortices and interacting vortex ensembles
have been suggested to build superconducting devices like vortex ratchets [8],
cellular automata [9, 10], superconducting diodes, vortex-based memory cells
[11] and artificial vortex ice [12]. Although some of these concepts have been
realized in experiments [13, 14, 15], the fundamentally possible degree of
complexity and miniaturization could not yet be exploited. Hence, previous
experiments with artificial vortex pinning sites were restricted by the
employed etching techniques and had to be operated at temperatures $T\to
T_{c}$ where the interaction length $\Lambda(T)$ is longer than the spacing
$a$ between the pins [16, 17, 18]. However, one aims to create devices that
work at temperatures below $\sim T_{c}/2$ where superconductivity is robust
and thermodynamic fluctuations do not reduce the collective vortex effects.
Accordingly, the goal of a novel technique is to design and fabricate
arrangements of many pinning sites in YBCO with diameters $d\gtrsim
2\xi_{ab}(T_{c}/2)$ and spacings $a$ such that $d<a\ll\Lambda(T_{c}/2)$.

Thin films of YBCO are usually patterned by etching away material, which leads
to a landscape with varying thickness. This process is extremely delicate and
becomes even more challenging for nanoscale patterning due to the potential
loss of oxygen through the open side faces of the remaining material. However,
the loss of oxygen ultimately results in the degradation of the
superconducting properties.

To overcome this issue, a masked light-ion beam from an implanter has been
employed to fabricate regions with suppressed superconductivity [19, 20, 21,
22, 23, 24, 25, 26, 27]. The impacting ions create mainly Frenkel defects of
the oxygen atoms in the YBCO crystal [28], while maintaining the
crystallographic framework and the surface of the film intact. The binding
energies are typically in the range of 1 to $3\,\,\mathrm{eV}$ [29, 30] for O
atoms in the CuO chains and about $8\,\mathrm{eV}$ [30] for the O atoms in the
CuO2 planes of YBCO. Point defects caused by ion impact break the
superconducting carrier pairs and suppress $T_{c}$. These defects also scatter
charge carriers and increase the normal-state resistivity. He+ ions with
moderate energy are ideal [31] for creating enough point defects to completely
suppress superconductivity at a practically available fluence of
$\phi<10^{16}\,\,\mathrm{ions/cm}^{2}$ while still penetrating a YBCO film
with a thickness of $\lesssim 100\,\,\mathrm{nm}$ [32]. However, columns of
non-superconducting material of diameters [15] $d$ and spacings [33] $a$
achieved by masked ion irradiation of YBCO are presently limited to $d\sim
70\,\mathrm{nm}$ and $a\sim 100\,\mathrm{nm}$.

In this work, we utilize the focused He-ion beam (He-FIB) of a helium ion
microscope (HIM) to achieve a spatial resolution well below 70 nm. We
exemplify the advantages of the HIM’s unique properties [34] by showcasing
vortex-matching effects in a set of hexagonal pinning arrays. The matching
effects show up in measurements of the critical current $I_{c}$ and resistance
$R$ vs magnetic field $B$, and they are remarkably robust, extending to $B$ as
high as 6 T, and persisting at $T$ from $T_{c}$ down to 2 K. Our results
represent a significant advancement in the fabrication of vortex pinning
landscapes in thin YBCO films.

##  II Experimental methods

###  II.1 Sample fabrication

Figure 1: Fabrication of nanopatterns by He-FIB: (a) XRD $\theta$–$2\theta$
scan of the YBCO film before patterning (to create bridge H20); inset shows an
$\omega$ scan of the (005) peak. (b) Optical micrograph of the bridge H20 with
electrical connections after He-FIB irradiation within the area depicted by
the yellow rectangle. (c) Sketch of the He-FIB patterning process. (d)
Secondary electron HIM image of a reference sample with a hexagonal
$a=20\,\mathrm{nm}$ array after irradiation with an excessive dose
$D=60\,\mathrm{ki/dot}$. The white spots are blister-forming amorphous
regions.

For the studies presented here, thin YBCO films were grown epitaxially on
(100)-oriented single crystal (LaAlO3)0.3(Sr2AlTaO6)0.7 (LSAT) substrates by
pulsed laser deposition (PLD). A 20-nm-thick Au film was deposited in situ by
electron beam evaporation right after YBCO film growth, which serves for
electrical contacting and protecting the superconducting film from
environmental influences. The film thicknesses, see Table 1, are small enough
to permit complete penetration of the He-FIB. An excellent $c$-axis
orientation of the films is confirmed by X-ray diffraction (XRD). Figure 1(a)
shows a $\theta$–$2\theta$ scan of the 37-nm-thick YBCO film, and the inset
shows the rocking curve ($\omega$ scan) of the YBCO (005) peak with a full
width at half maximum (FWHM) of $0.08\,^{\circ}$.

Table 1: Parameters of YBCO microbridges with pinning arrays reported in this paper. The length indicates the distance between the voltage probes. The critical temperature $T_{c}$ is determined as the resistive transition’s inflection point. The first magnetic matching field $B_{1}$ is calculated from Eq. (1) and confirmed by $I_{c}(B)$ and $R(B)$ measurements. name | width | length | thickness | $a$ | $D$ | $T_{c}$ | $B_{1}$
---|---|---|---|---|---|---|---
| ($\mu$m) | ($\mu$m) | (nm) | (nm) | (ki/dot) | (K) | (T)
H30 | 8.0 | 20.0 | $26\pm 1$ | $30\pm 0.6$ | 10 | 75 | 2.7
H25 | 6.5 | 17.0 | $26\pm 1$ | $25\pm 0.5$ | 10 | 63 | 3.8
H20 | 5.0 | 13.5 | $37\pm 2$ | $20\pm 0.5$ | 10 | 52 | 6.0
H15 | 4.0 | 10.0 | $26\pm 1$ | $15\pm 0.4$ | 10 | n/a | (10.6)

To fabricate microbridges on each chip, we used a combination of
photolithography and Ar ion milling to pattern the Au/YBCO bilayer.
Subsequently, the Au layer was removed with Lugol’s iodine from the YBCO
bridge structures to be exposed to He-FIB irradiation. An optical image of a
microbridge with an illustration of the electrical connections is shown in
Fig. 1(b). The dimensions of different bridges are chosen to maintain the same
number of irradiated dots for various pinning site spacings and to minimize
the irradiation time for the denser lattices (see Tab. 1).

###  II.2 He-FIB irradiation

The He-FIB irradiation was performed in a HIM (Zeiss Orion NanoFab) at the
University of Tübingen. The sample chips were attached to aluminum sample
stubs and grounded with silver paste. After mounting in the HIM, they were
thermally acclimated for roughly 60 min before the irradiation procedure
began. For simplicity, we will refer to the number of ions hitting a certain
spot on the surface of the sample as the “dose” $D$, which will be given in
units of ki/dot, i.e., 1000 ions per dot. In Fig. 1(c), the fabrication of a
hexagonal array of columnar defects (CDs) by He-FIB is illustrated.

Notably, the dose $D=10\,$ki/dot was carefully selected to induce point
defects and locally suppress $T_{c}$ in the YBCO film, while avoiding
amorphization within the CDs. To achieve this, the dose must not exceed a
critical amorphization dose $D_{c}\approx 13\,$ki/dot (see section IV.1.1).
However, if $D<D_{c}$, the areas subjected to irradiation do not exhibit any
visible surface degradation and cannot be seen by HIM imaging. To monitor the
CD array fabrication process and visualize it, an auxiliary sample (shown in
Fig. 1(d)) was irradiated with a dose $D\gg D_{c}$. The scanning transmission
electron microscopy (STEM) study presented in section IV.1.1 reveals that, for
He-FIB doses above the amorphization threshold, the amorphous region causes
local swelling, resulting in a bulge of the surface, as shown in Fig. 1(d).

Apart from the number of impinging ions, the lateral intensity profile of the
He ion beam is also crucial. For our typical HIM parameters (SpotControl = 5
and an aperture of $10\,\mathrm{\mu m}$), we assume a Gaussian intensity
profile with an estimated FWHM beam diameter $\varnothing\sim 9\,$nm. The
“SpotControl” (SC) value for a given aperture diameter determines the width of
the beam diameter. A higher spot control value produces a narrower beam
profile at the expense of lower ion fluence and therefore longer irradiation
time. For example, a narrow beam is required to draw lines across a YBCO
microbridge to create Josephson junctions [35, 36]. On the other hand, a lower
SpotControl value widens the beam and can lead to unwanted defects in the
inter-CD areas, as discussed in section IV.1.2.

Throughout this work, we used a He-FIB with an energy of $30\,\mathrm{keV}$.
The irradiated area is limited by the field of view (FoV) of the HIM, and for
optimum resolution of the HIM’s beam deflection circuitry, the FoV should be
$30\times 30\,\mathrm{\mu m^{2}}$ or smaller. This provides a beam positioning
accuracy of approximately $0.5\,\mathrm{nm}$.

Several hexagonal arrays have been produced with spacings
$a=30,\,25,\,20,\,15\,$nm and doses ranging from $D=1$ to $10\,$ki/dot. The
different values of $a$ were chosen to test the ultimate resolution that can
be achieved. Here, we present the results for $D=10\,\mathrm{ki/dot}$, which
demonstrates the strongest pinning.

###  II.3 Transport measurements

The electrical measurements were carried out in a Physical Properties
Measurement System (PPMS) equipped with a 9 T superconducting solenoid and a
variable temperature insert with a Cernox resistor for in-field temperature
control (Quantum Design). For all experiments, the magnetic field was oriented
perpendicular to the sample surface ($x,y$-plane), i.e., the field is along
the $z$-direction. To rule out thermoelectric signals, resistance measurements
were taken with constant excitation currents of 1.2 to $2.4\,\mu$A in both
polarities. The magnetic field was swept in both polarities at fixed
temperatures, and perfect mirror symmetry was observed. Furthermore, the
results are independent of the magnetic field sweep direction and no
hysteresis [25] was observed.

The critical current $I_{c}$ is determined from isothermal voltage $V$ vs
current $I$ measurements using a voltage criterion of $135\,\mathrm{nV}$,
which is above the noise floor of $40\,\mathrm{nV}$ and corresponds to an
electric field $E=100\,\mathrm{\mu V/cm}$. Supplementary resistance
measurements in magnetic fields up to 12.5 T were performed in an Oxford
Instruments helium bath cryostat.

##  III Simulations

Figure 2: Monte Carlo simulations of $T_{c}$ suppression induced by He-FIB
irradiation (with $\varnothing=9\,$nm) of YBCO; only the central parts of
larger hexagonal patterns are shown. The $x$-$y$-plots show cuts at $z=15$ nm
depth below the film surface, while the $x$-$z$-plots show cuts at $y=0$ nm.
(a) CD lattices with various lattice spacings; $D=10\,$ki/dot. (b) CD lattices
with $a=20$ nm and various $D$.

Our experiments were preceded by simulations of the expected $T_{c}(x,y,z)$
suppression patterns, which are summarized in Fig. 2. Briefly, atom
displacement cascades due to He-FIB irradiation were calculated using the
SRIM/TRIM program [37]. Individual He+ atom’s impact points were shifted to
simulate the Gaussian beam intensity distribution and the hexagonal
irradiation pattern. The resulting three-dimensional landscapes of defects can
be converted to local $T_{c}$ values by using experimental data from uniformly
large-area irradiated thin YBCO films as calibration [38].

In Fig. 2(a), the red dots indicate hexagonal arrays of well-formed
nanocolumns with suppressed $T_{c}$. They are well separated down to $a=20$
nm, leaving superconducting percolation paths in between. Because of the
straggling of the ion trajectories, these inter-CD regions have a finite but
smaller defect density and hence also slightly reduced $T_{c}$ values. In
contrast, the simulation of an array with $a=15$ nm reveals little $T_{c}$
contrast between defect columns and inter-CD regions, making the observation
of matching effects unlikely.

Furthermore, proper dose selection is crucial, as demonstrated in Fig. 2(b)
for a lattice with spacings $a=20$ nm. Doses of $D\ll 10\,\mathrm{ki/dot}$ do
not entirely suppress superconductivity. They solely decrease $T_{c}$ locally,
resulting in a weaker pinning potential, as shown in Fig. 4(a). Conversely,
doses $D>10\,\mathrm{ki/dot}$ provide no additional benefit since the
suppression of $T_{c}$ is already maximal. On the contrary, an excessive
number of defects causes crystal lattice instabilities, which spread into
neighboring regions and widen the defect columns.

##  IV Results and discussion

###  IV.1 Optimization of irradiation parameters

####  IV.1.1 Irradiation dose

Figure 3: Cross-sectional HAADF-STEM image of the lamella extracted from a
hexagonal array of CDs with the lattice constant $a=20\,\mathrm{nm}$, length
$2000\,\mathrm{nm}$ in $x$-direction. The array has a dose gradient along the
$x$ direction given by
$D(x)=689\,\mathrm{i/dot}\cdot(N+1)=689\,\mathrm{i/dot}\cdot(1+x/a^{\prime})$,
where $N=0\ldots 115$ is the CD’s column number along the gradient direction
and $a^{\prime}=\sqrt{3}/2a\approx 17.32\,\mathrm{nm}$ is the distance between
the CD’s columns in $x$ ($D$ gradient) direction. (a) and (b) show zoomed
images in the region with preserved crystal structure and in the region with
onset of amorphization ($x_{c}\approx 320\,\mathrm{nm}$, $D_{c}\approx
13\,\mathrm{ki/dot}$) correspondingly. (c) shows the overall view of the
entire array. The scales below the image show the coordinate $x$ within the
array and the corresponding dose $D(x)$. Each small tick on the $D$ axis
corresponds to a position of a defect column (i.e., the distance between the
ticks is $a^{\prime}$). (d) The intensity profile (white) overlayed onto a
selected part of the lamella’s STEM image show a slight contrast modulation
with the periodicity $a^{\prime}$. The red arrows indicate the locations of
the irradiated spots.

To maintain the crystalline framework of YBCO and still suppress its
superconductivity, it is essential to determine the optimal number of 30 keV
He+ ions (referred to as dose $D$) to irradiate one dot of the pinning
lattice. If the dose exceeds the threshold value $D_{c}$, it causes the
breakdown of the crystalline structure, making the YBCO film amorphous. To
identify $D_{c}$, a hexagonal array of dots was used with a distance of $a=20$
nm. The dots were irradiated with increasing doses in steps of 689 i/dot from
row to row, linearly from 0.689 ki/dot to 80 ki/dot, leaving the other
irradiation parameters the same as for sample H20.

This array was characterized by aberration-corrected scanning transmission
electron microscopy (STEM) in a probe-corrected FEI Titan 60-300 operated at
300 kV, equipped with a CEOS corrector for the condenser system, which
provides a probe size below 1 Å. High-angle annular dark field (HAADF) in STEM
was used to provide Z-contrast images. The cross-sectional specimen was
prepared by Ga+ FIB in an FEI Helios 650 Dual Beam FIB system.

The cross-sectional STEM image of the entire irradiated region is shown in
Fig. 3(c). At low doses, the crystalline structure of YBCO is preserved, as
can be seen in the magnified detail depicted in Fig. 3(a). At a critical dose,
$D_{c}\approx 13\,\mathrm{ki/dot}$, the crystalline structure of YBCO
disappears in the STEM image, indicating the amorphization of the film, as
displayed in Fig. 3(b). Note that at $D\lesssim D_{c}$, the YBCO layer already
begins to swell, but the surface of the substrate remains flat. Only at
substantially higher $D$ the substrate is damaged too, and individual beam
tracks become clearly visible on the right side of Fig. 3(c). This observation
proves that the reported vortex-matching effects are not caused by local
strain from the substrate, but rather originate from subtle modifications of
the YBCO structure.

The intensity profile for $D\lesssim D_{c}$ shown as an overlay to the STEM
image in Fig. 3(d) reveals a reduction with increasing $x$, i.e., increasing
$D$, that begins when the YBCO lattice expands and reaches its lowest value in
the amorphized region. Intriguingly, it can be clearly seen that for
$D<D_{c}$, the layered structure of YBCO is conserved, yet the intensity
profile shows little dips with a periodicity of $a^{\prime}=\sqrt{3}/2a$. Such
an intensity depression is probably caused by the enhanced disorder in the
columnar defect channels. The width of the dips is $\leq 10$ nm, supporting
our previous estimate for the diameter of the CDs.

For lower dose values that do not cause visible damage, no signatures of the
irradiation are visible in STEM images. Notwithstanding, the CDs created with
lower doses strongly influence the behavior of magnetic flux quanta, as seen
in electric transport measurements. The observation of magnetic-field
commensurability effects with the flux-line lattice is a direct probe of
efficient vortex pinning in periodic pinning landscapes. Minima of the
resistivity and maxima of the critical current appear at so-called matching
fields

$B_{k}=k\Phi_{0}/A,$ (1)

where $k$ is an integer or rational number of vortices contained within the
unit cell of the hexagonal pinning array with area
$A=\frac{\sqrt{3}}{2}a^{2}$. The field $B_{1}$ corresponds to the flux density
of one vortex per pinning site. When the magnetic field $B>B_{1}$, one of two
scenarios may occur: either some or all CDs are filled with multiquanta
vortices when the CD’s radius is larger than the (in-plane) coherence length
or interstitial vortices enter between the CDs [39].

The question of how lower doses $D\ll D_{c}$ influence the efficiency of the
pinning lattices was investigated in several microbridges prepared on the same
substrate. The bridges were patterned with a hexagonal $a=30$ nm CD lattice
using different doses but otherwise identical conditions. Figure 4(a) shows
the resistivity $\rho$ vs. applied magnetic field $B$ of these bridges.
Because $T_{c}$ decreases with increasing $D$, the data were taken at the same
normalized $T/T_{c}\sim 0.96$. Although commensurability effects are barely
visible at $D=1$ ki/dot, they appear at higher doses. Still, the $\rho(B)$
curves of the bridges with $D\leq 3.5$ ki/dot are similar and do not show
matching dips at $B_{2}$. Only for $D=10$ ki/dot, there is a clear matching
effect at $B_{2}$, and, moreover, the minimum at $B_{1}$ is remarkably
pronounced. These experimental results confirm the simulation results
presented in Fig. 2(b) and point to an optimal dose $D=10$ ki/dot for our
experiments.

Figure 4: Vortex matching effects in the resistivity vs field curves of
irradiated YBCO bridges. (a) Magnetoresistivity of four samples at
$T/T_{c}\sim 0.96$ after irradiation with an $a=30$ nm hexagonal pattern using
SC = 4, and doses from 1 to 10 ki/dot. (b) Comparison of different spot
control (SC) settings of the HIM. Two hexagonal pinning arrays with $a=40$ nm
were prepared with a dose $D=2.0$ ki/dot on the same substrate. The
magnetoresistivity at 86.5 K ($T/T_{c}\sim 0.96$) is plotted for SC = 4 and 5.

####  IV.1.2 Spot Control

The “Spot Control” (SC) value specifies the strength of the HIM’s condenser
lens, thus setting the position of the crossover point in the ion optical
column. For a given aperture diameter, a higher spot control value thus
produces a narrower beam profile at the expense of lower ion fluence and
longer irradiation time. For SC = 5 used in this study we estimate a beam
diameter of full width at half maximum (FWHM) $\varnothing\approx 9\,$ nm on
the sample surface. At a dose of 10 ki/dot the dwell time of the beam for a
single spot is $\approx 3$ ms, and the total irradiation time of the bridge is
approximately 40 minutes for the lattice spacing $a=30$ nm.

The processing time can be reduced by a factor of about two using SC = 4. To
test the consequences, the resistivities at 86.5 K are compared as a function
of the applied magnetic field in Fig. 4(b). The two bridges with a hexagonal
CD pattern of $a=40$ nm were manufactured from the same YBCO film and treated
identically, except for the choice of SC. Despite the similarity of the vortex
matching characteristics, the resistivity of the SC = 4 bridge is higher due
to a wider tail of the ion beam profile, resulting in a higher defect density
between the irradiated spots.

###  IV.2 Resistivity and critical current

The microbridges described in Tab. 1 were further studied by electrical
transport measurements. Figure 5(a) depicts the resistivity $\rho(T)$ of all
four microbridges with arrays written using $D=10\,\mathrm{ki/dot}$. The
samples with $a$ = 30, 25, and 20 nm exhibit a metallic temperature dependence
in the normal state and a critical temperature that decreases with smaller
spacing. The slope’s invariance indicates that no oxygen is lost during
irradiation [32], while the rising offset of a straight line extrapolated from
the normal state indicates an increase in the number of defects in the
percolation paths between the CDs. This observation is consistent with the
decline of $T_{c}$ and the simulation results presented in Fig. 2(a).

In contrast, the array with $a=15\,\mathrm{nm}$ displays a semiconducting
$\rho(T)$ dependence and does not become superconducting even at our lowest
measurement temperature of $2\,\mathrm{K}$. When the CDs are placed at such a
close spacing, they begin to overlap and impede the superconducting
percolation paths between the CDs. This observation from the resistivity
measurements for variable $a$ supports our estimate of the He-FIB’s fluence
profile $\varnothing=9\,\mathrm{nm}$ in the simulations.

Figure 5: (a) Resistivity vs temperature (at $B=0$) of YBCO microbridges
patterned with a hexagonal array of CDs of various spacings $a$ by He-FIB
irradiation with dose $D=10\,\mathrm{ki/dot}$. Extrapolations of the normal-
state resistivities are shown by dashed lines. (b) Color-coded contour plot of
the resistance of sample H20 as a function of temperature and magnetic field.
The dotted line represents the onset of superconductivity, $T_{c,onset}$,
taken as 90 percent of the normal-state resistance value at 100 K,
$R(100\,\text{K})$. The dashed-dotted line estimates the irreversibility line
$T_{irr}$, using a criterion of 0.1 percent of $R(100\,\text{K})$.

In order to further investigate the superconducting transition of sample H20,
we measured $R(T)$ for different magnetic fields, as shown in Fig. 5(b). The
dotted line represents the onset of superconducting percolation paths between
the CDs, $T_{\mathrm{c,onset}}(B)$, which is defined as
$R(T_{\mathrm{c,onset}},B)=0.9R(100\,\mathrm{K},B)$. The steep slope
$-dB/dT_{\mathrm{c,onset}}$ implies that the upper critical field
$B_{c2}(0\,\mathrm{K})$ is very high. Remarkably, no features in
$T_{\mathrm{c,onset}}(B)$ are observed at the first matching field
$B_{1}=6.0\,$T (horizontal dashed line in Fig. 5(b)). This suggests that in
the strong fluctuation and vortex liquid regime near
$T_{\mathrm{c,onset}}(B)$, the current paths between the CDs are basically
unaffected by the pinning array and the slope of $T_{c,onset}(B)$ is
comparable to that of unirradiated films. In contrast, we observe a distinct
peak of the irreversibility temperature $T_{\mathrm{irr}}(B)$, defined as
$R(T_{\mathrm{irr}},B)=10^{-3}R(100\,\mathrm{K},B)$, right at $B_{1}$. This
indicates that the onset of dissipation caused by moving vortices is
significantly shifted to higher temperatures when the vortices can arrange
themselves commensurable with the CD array.

Figure 6: Effects of vortex commensurability in sample H20. (a) Critical
current vs. applied magnetic field at various temperatures. The inset shows
the critical current normalized to its value at the matching field $B_{1}$.
(b) Resistance vs. applied magnetic field at temperatures from 17.5 K (dark
blue symbols) to 60 K (dark red symbols) in 2.5 K steps. The inset shows
details at higher temperatures.

The vortex matching effects can be investigated more directly by critical
current $I_{c}(B)$ (static probe) or resistance $R(B)$ (dynamic property of
moving vortices) measurements. The $I_{c}(B)$ curves in Fig. 6(a) – taken at
six different temperatures from 2 to 25 K – show peaks at $B_{1}=6.0\,$T,
which remain visible even down to $T=2\,$K. The inset of the figure highlights
these peaks by normalizing $I_{c}$ to $I_{c}(B_{1})$. As the temperature
increases, the pinning force of intrinsic defects decreases faster than that
of CDs [40], which makes the peaks even more pronounced. It is worth noting
that previous works [16, 41, 17, 18, 22, 42, 24, 43, 44, 25, 45, 26, 46, 47,
33] did not identify commensurability effects at such high magnetic fields.
Furthermore, the vortex matching observed in our samples continues to occur
even at very low temperatures. These observations are a natural consequence of
ultradense and strong pinning sites created by He-FIB irradiation.

From Fig. 6(a) it is evident that the pinning effect remains strong not only
at $B_{1}$ but also throughout the field range $0\leq B<B_{1}$, where
commensurability effects are less significant. The minimum value of $I_{c}$,
which roughly corresponds to the weakest pinning sites, is reached at about
$4\,\mathrm{T}$ at low temperatures. At the lowest temperature $T=2\,$K,
$I_{c}(B=4\,\,\mathrm{T})$ is only twice as low as $I_{c}(B=0)$. However, for
$B>B_{1}$ the pinning becomes less effective and $I_{c}$ decays more rapidly.
While at $B<B_{1}$ every vortex can, in principle, be solely pinned in a CD,
this is not the case for $B>B_{1}$. Then, an arrangement of multiquanta
vortices in the CDs or interstitial vortices occurs, which is energetically
less favorable. Thus, the value of $B_{1}$ sets an upper limit to the typical
range of strong pinning that is relevant for practical applications. This
implies that shifting $B_{1}$ to high magnetic fields is essential, which is
possible with the method presented here.

On the high-temperature side, up to about 60 K, the vortex matching effects
are probed via reduced vortex mobility, resulting in minima of the resistance
as shown in Fig. 6(b). This suggests that even at 80 % of the normal-state
resistance value $R(100\,\text{K})$, correlations between vortices do exist.
Hence, the vortex ensemble is not entirely melted but instead moves through a
plastic vortex flow [48]. Voltage-current isotherms in sample H30 indicate
that the interaction of artificial and intrinsic defects results in an ordered
Bose glass state [49, 50].

Figure 7: Resistance vs. applied magnetic field at various temperatures of the
YBCO microbridge H20 patterned with a hexagonal array of CDs (spacing $a=20$
nm and dose $D=10\,\mathrm{ki/dot}$). Some data were collected in both
polarities of the magnetic field and confirm the symmetry $R(B)=R(-B)$.

The magnetoresistance of sample H20 was investigated up to 12.5 T in a
separate series of measurements. Figure 7 shows the $R(B)$ curves at
temperatures from 17 to 55 K. Some measurements were taken at both magnetic
field polarities to confirm the perfect mirror symmetry of the data. We do not
find matching features (dips in $R(B)$) at $B_{2}=12.0$ T, though minima in
$R(B)$ at $B_{2}$ are visible for the bridge H30 with a wider hexagonal $a=30$
nm CD array (see Fig. 4(a)). In addition, $I_{c}(B)$, measured on the H20
bridge at lower temperatures, did not reveal any commensurability signatures
at $B_{2}$. One possible explanation is that the dynamics of interstitial
vortices changes when the channels between the CDs become narrow. Such details
of vortex-matching properties in ultradense pinning landscapes deserve further
investigation.

##  V Conclusions

To summarize, irradiating thin YBCO films with a 30 keV focused He ion beam of
undercritical dose $D<D_{c}$ is an excellent approach for introducing user-
defined patterns of suppressed critical temperature. After evaluating the
appropriate irradiation conditions, we demonstrated the ability to create
ultra-dense hexagonal pinning arrays with a spacing as low as
$a=20\,\mathrm{nm}$. These arrays exhibit strong pinning across a wide range
of magnetic fields up to the first matching field of $6\,\mathrm{T}$. The
overlap of neighboring CDs sets the present experimental minimum separation of
the pins at $15\,\mathrm{nm}<a\leq 20\,\mathrm{nm}$. However, it is possible
that further adjustment of irradiation parameters could slightly reduce this
constraint.

Our experiments have shown that vortex-matching effects can be observed even
at low temperatures of 2 K, where superconductivity is stable and not affected
by fluctuations. This paves the road for designing more effective pinning
landscapes that can operate at several tesla and at temperatures much lower
than the critical temperature, avoiding the undesirable vortex liquid and
fluctuation regimes.

Our findings have expanded the range of temperatures and magnetic fields that
can be studied with _regular_ artificial pinning landscapes. In order to bring
the theoretical concepts in the field of fluxonics [8, 6, 11, 10] to practical
use and to facilitate high operating frequencies, it is essential to confine
vortices to designed features with the smallest possible size [9]. With our
findings, it is now possible to create complex pinning patterns with narrow
spacing that can be utilized in applications such as vortex-based cellular
automata for fast logic with low power consumption [9, 10]. Future
developments in He-FIB instrumentation, like a multibeam technique [51], would
further augment the versatility of the presented method.

###### Acknowledgements.

We are grateful for the insightful discussions with John Notte (ZEISS
Semiconductor Manufacturing Technology) and Gregor Hlawacek (Helmholtz-Zentrum
Dresden-Rossendorf). B.B. acknowledges financial support from the Vienna
Doctoral School in Physics (VDSP). O.D. acknowledges financial support from
the Austrian Science Fund (FWF) under grant I6079-N (FluMag). C. M.
acknowledges financial support from Regional Gobierno de Aragón through
Project No. E13_23R, including FEDER funding and from the EU Horizon 2020
programme under the Grant Agreement No. 823717-ESTEEM3. The authors
acknowledge the use of instrumentation as well as the technical advice
provided by the National Facility ELECMI ICTS, node “Laboratorio de
Microscopías Avanzadas (LMA)” at “Universidad de Zaragoza”. The research was
funded in whole, or in part, by the Austrian Science Fund (FWF) grant I4865-N
and the German Research Foundation (DFG), grant KO 1303/16-1. For the purpose
of open access, the authors have applied a CC-BY public copyright license to
any Author Accepted Manuscript version arising from this submission. The
research is based upon work from COST Actions CA21144 (SuperQuMap), CA19140
(FIT4NANO), and CA19108 (Hi-SCALE) supported by COST (European Cooperation in
Science and Technology).

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