# Perspective: Challenges and Transformative Opportunities in Superconductor
Vortex Physics

Serena Eley<EMAIL_ADDRESS>Department of Physics, Colorado School of
Mines, Golden, Colorado, USA  Andreas Glatz  Materials Science Division,
Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois 60639, USA
Department of Physics, Northern Illinois University, DeKalb, Illinois 60115,
USA  Roland Willa  Theory of Condensed Matter Physics, Karlsruhe Institute of
Technology, Karlsruhe, Germany  Heidelberger Akademie der Wissenschaften,
Heidelberg, Germany

###### Abstract

In superconductors, the motion of vortices introduces unwanted dissipation
that is disruptive to applications. Fortunately, material defects can
immobilize vortices, acting as vortex pinning centers, which engenders
dramatic improvements in superconductor material properties and device
operation. This has motivated decades of research into developing methods of
tailoring the disorder landscape in superconductors to increase the strength
of vortex pinning. Yet efficacious materials engineering still alludes us. The
electromagnetic properties of real (disordered) superconducting materials
cannot yet be reliably predicted, such that designing superconductors for
applications remains a largely inefficient process of trial and error. This is
ultimately due to large gaps in our knowledge of vortex dynamics: the field is
challenged by the extremely complex interplay between vortex elasticity,
vortex-vortex interactions, and material disorder.

In this Perspective, we review obstacles and recent successes in understanding
and controlling vortex dynamics in superconducting materials and devices. We
further identify major open questions and discuss opportunities for
transformative research in the field. This includes improving our
understanding of vortex creep, determining and reaching the ceiling for the
critical current, advanced microscopy to garner accurate structure-property
relationships, frontiers in predictive simulations and the benefits of
artificial intelligence, as well as controlling and exploiting vortices in
quantum information applications.

††preprint: AIP/123-QED

##  I Introduction

Distinguished for their ability to carry high dissipation-less currents below
a critical temperature $T_{c}$, superconductors are used in motors,
generators, fault-current limiters, and particle accelerator magnets. Their
impact spans beyond these examples of large-scale applications, also affecting
nanoscale devices. Perhaps most renown for their key role in the quantum
revolution, superconductors constitute building blocks in current and next-
generation devices for computing and sensing. For example, superconducting
photons detectors feature high-resolutions due to high kinetic inductance and
a sharp superconductor-to-normal phase transition. Moreover, superconductors
can be configured to form anharmonic oscillators that can be exploited in
quantum computing.

Figure 1: Frontiers in vortex matter research. Black line represents the
vortex core. Yellow region shows how the density of superconducting electron
pairs decays towards the center of the core (of size $\sim\xi$, coherence
length). Blue plane (with arrows) represents amplitude of supercurrent,
circulating around the core of radius up to the penetration depth $\lambda$.
Figure 2:  Examples of vortex structures (curved blue lines) that are
predicted to form in different defect landscapes under the influence of an
applied current. Imaging these structures and defects, would allow us to
establish the crucial connection between vortex excitations, vortex-defect and
vortex-vortex interactions, Lorentz forces, and resulting vortex phases that
is needed for efficacious defect engineering.

Notwithstanding these successes, the performance of superconducting devices is
often impaired by the motion of vortices—lines threading a quantum
$\Phi_{0}=h/2e$ of magnetic flux through the material (see Fig. 1). Propelled
by electrical currents and thermal/quantum fluctuations, vortex motion is
dissipative such that it limits the current-carrying capacity in wires, causes
losses in microwave circuits, contributes to decoherence in qubits, and can
also induce phase transitions. Understanding vortex dynamics is a formidable
challenge because of the complex interplay between moving vortices, material
disorder that can counteract (pin) vortex motion, and thermal energy that
causes vortices to escape from these pinning sites. Furthermore, as depicted
in Fig. 2, in three-dimensional samples (bulk crystals or thick films),
vortices are elastic objects that form complicated shapes as they wind through
the disorder landscape, reshaping and moving under the influence of current-
induced Lorentz forces.

These complexities encumber predictability: we can neither predict
technologically important parameters in superconductors nor prescribe an ideal
defect landscape that optimizes these parameters for specific applications.
Though modifying the disorder landscape, e.g. using particle-irradiation or by
incorporating non-superconducting inclusions into growth precursors, can
engender dramatic enhancements in the current carrying capacity, these
processes are often designed through a trial-and-error approach. Furthermore,
the optimal defect landscape is highly-material dependent. This is because the
efficacy of pinning centers depends on the relationship between their geometry
and the vortex structure, the latter being determined by parameters of the
superconductor such as the coherence length $\xi$, penetration depth
$\lambda$, and the anisotropy $\gamma$, see Fig. 1. For example, though
particle irradiation has successfully doubled the critical current in cuprates
and certain iron-based superconductors, the same ions and energies do not even
produce universal effects in materials belonging to the same class of
superconductorsTamegai  _et al._ (2012).

Though we can indeed tune the disorder landscape, we certainly do not have
full control of it. Defects such as stacking faults, twin boundaries, and
dislocations are often intrinsic to materials and their densities are
challenging to tune. As a further complication to understanding vortex-defect
interactions, superconductors often have mixed pinning landscapes, i.e.,
containing multiple types of defects. Though these landscapes immobilize
vortices over a broader range of conditions (temperatures and fields) than
landscapes containing only one type of defect, it is challenging to infer the
vortex structures that form within these materials and no techniques currently
exist to fully image these structures and vortex-defect interactions on a
microscopic level.

Generally speaking, achieving a materials-by-design approach first entails
garnering a sufficient microscopic understanding of vortex-defect and vortex-
vortex interactions, then incorporating these details into simulations.
Significant headway has been made along these lines with the implementation of
large-scale time-dependent Ginzburg-Landau (TDGL) simulations to study vortex
motion through disordered media. Spearheaded by the Argonne National
Laboratory, this effort has accurately modeled critical currents $J_{c}$ in
thin films (2D), layered and anisotropic 3D materials, as well as isotropic
superconductors Sadovskyy  _et al._ (2015a, 2016a); Glatz  _et al._ (2016);
Sadovskyy  _et al._ (2017); Kimmel  _et al._ (2019). Additionally, it has
determined the optimal shape, size, and dimensionality of defects necessary to
maximize $J_{c}$, depending on the magnitude and orientation of the magnetic
field Koshelev  _et al._ (2016); Sadovskyy  _et al._ (2016b); Kimmel,
Sadovskyy, and Glatz (2017); Sadovskyy  _et al._ (2019). Backed by good
agreement with experimental and analytic results for simple geometries Willa
_et al._ (2015); Willa, Geshkenbein, and Blatter (2015, 2016); Willa, Marziali
Bermúdez, and Pasquini (2018), the utility of the numerical routine has
successfully been extended to previously unknown territories, optimizing
pinning geometries outside the scope of analytic methods Glatz  _et al._
(2016); Kimmel, Sadovskyy, and Glatz (2017); Koshelev  _et al._ (2016); Kwok
_et al._ (2016a); Papari  _et al._ (2016); Sadovskyy  _et al._ (2015a, 2016a,
2016b, 2019); Willa  _et al._ (2018a); Glatz  _et al._ (2020). In fact, these
TDGL simulations have unveiled new phenomena—such as a small peak in
$J_{c}(B)$ at high fields that is caused by double vortex occupancy of
individual pinning sites.Willa  _et al._ (2018b) The Argonne team has even
deployed mature optimization processes based on targeted evolution using
genetic algorithms. Sadovskyy  _et al._ (2019) This is a remarkable step
towards the goal of critical-current-by-design.

A critical-current-by-design must consider thermal fluctuations, which
dramatically impact the critical current due to the effects of rapid
thermally-induced vortex motion (thermal creep). Creep, which manifests as a
decay in the persistent current over time, is rarely problematic in
low-$T_{c}$ superconductors as it is typically quite slow. Consequently, Nb–Ti
solenoids in magnetic resonance imaging systems can operate in persistent
mode, retaining a fairly constant magnetic field for essentially indefinite
time periods. However, creep is fast in high-$T_{c}$ superconductors,
restricting applications and reducing the effective $J_{c}$.

For the sake of power and magnet applications, the goals are clear—maximize
the critical current and minimize creep. Regarding the former, there is much
room for improvement: no superconductor containing vortices has ever achieved
a $J_{c}$ higher than 25% of its theoretical maximum, which is thought to be
the depairing current $J_{d}=\Phi_{0}/(3\sqrt{3}\pi\mu_{0}\xi\lambda^{2}$).
Regarding creep, we are fighting a theoretical lower bound.Eley  _et al._
(2017a) This lower bound positively correlates with a material’s Ginzburg
number $Gi=(\gamma^{2}/2)(k_{\mathrm{B}}T_{c}/\varepsilon_{sc})^{2}$, which is
the ratio of the thermal energy to the superconducting condensation energy
$\varepsilon_{sc}=(\Phi_{0}^{2}/2\pi\mu_{0}\xi^{2}\lambda^{2})\xi^{3}$. The
implications are grim: creep is expected to be so fast in potential, yet-to-
be-discovered room-temperature superconductors rendering them unsuitable for
applications. The caveat is that this lower bound is limited to low
temperatures and fields (single vortex dynamics), and collective vortex
dynamics could be key to achieving slow creep rates.

Though superconducting sensing and computing applications do not require high
currents, vortices still pose a nuisance by limiting the lifetime of the
quantum state in qubits Oliver and Welander (2013), inducing microwave energy
loss in resonators Song  _et al._ (2009a), and generally introducing noise. It
is known that dissipation from vortex motion reduces the quality factor in
superconducting microwave resonators, which are integral components in certain
platforms for quantum sensors and the leading solid-state architecture for
quantum computing (circuit-QED)Wallraff  _et al._ (2004); Blais  _et al._
(2004); Krantz  _et al._ (2019); Müller, Cole, and Lisenfeld (2019). They are
used to address and readout qubits as well as mediate coupling between
multiple qubits. Consequently, resonator stability can be essential for qubit
circuit stability. Moreover, thermally activated vortex motion can contribute
to $1/f$ noise and critical current fluctuations Trabaldo  _et al._ (2019);
Van Harlingen  _et al._ (2004a) in quantum circuits and is a suspected source
of the dark count rate in superconducting nanowire single-photon detectors
Bulaevskii  _et al._ (2011); Yamashita (2013).

In these quantum circuits, vortices appear due to pulsed control fields,
ambient magnetic fieldsSong  _et al._ (2009b), and the self-field generated by
bias currents Yamashita (2013). Mitigating the effects of vortices requires
heavy shielding to block external fields and careful circuit design to control
their motion, the latter of which is quite tricky. The circuit should include
structures to trap vortices away from operational currents and readout as well
as narrow conductor linewidthsStan, Field, and Martinis (2004) to make vortex
formation less favorable. However, these etched structures may exacerbate
another major source of decoherence—parasitic two-level fluctuators—defects in
which ions tunnel between two almost energetically equivalent sites, which act
as dipoles and thus interact with oscillating electric fields during device
operation.Müller, Cole, and Lisenfeld (2019) Hence, designing quantum circuits
that are robust to environment noise is not trivial and has become a topic of
intense interest.Müller, Cole, and Lisenfeld (2019); Oliver and Welander
(2013)

Despite all of the aforementioned application-limiting problems caused by
vortices, they are not pervasively detrimental to device performance. For
example, vortices can trap quasiparticles—unpaired electrons that are a third
source of decoherence in superconducting quantum circuits—boosting the quality
factor of resonatorsNsanzineza and Plourde (2014) and the relaxation time of
qubitsWang  _et al._ (2014). Furthermore, vortices can host elusive, exotic
modes that are in fact useful for topological qubits, which are predicted to
be robust to environmental noise that plagues other quantum device
architectures. To exploit these modes in computing, we must control the
dynamics of their vortex hosts. Hence, in general, these disparate goals of
eliminating or utilizing vortices for applications both require an improved
understanding of vortex formation, dynamics, and, ultimately, control.

The goal of this Perspective is to present opportunities for transformative
advances in vortex physics. In particular, we start by addressing vortex creep
in Sec. III.1, which notes limited knowledge of non-thermal creep processes
and how recent increases in computational power will enable full consideration
of creep in simulations. Second, in Sec. III.2 we explore the true maximum
achievable critical current, and the need to simultaneously exploit multiple
pinning mechanisms to surpass current records for $J_{c}$. Next, Sec. III.3
discusses vortex-induced losses in response to AC magnetic fields and
currents, with a focus on the impact on superconducting RF cavities used in
accelerators and quantum circuits. We examine how the quantum revolution has
handled the vortex problem for computing, while sensing applications
necessitate further studies. As solving the aforementioned problems requires
advanced computational algorithms, we then proceed to discuss future uses of
artificial intelligence to understand the vortex matter genome in Sec. III.4.
Finally, in Sec. III.5, we recognize that most experimental studies use
magnetometry and electrical transport studies to _infer_ vortex-defect
interactions, and discuss the frontiers of microscopy that could lead to
observing these interactions as well as accurately determining defect
densities.

##  II Background

Superconductors have the remarkable ability to expel external magnetic fields
up to a critical value $H_{c1}$, a phenomena that is known as the Meissner
Effect. Though surpassing $H_{c1}$ quenches superconductivity in some
materials, the state persists up to a higher field
$H_{c2}=\Phi_{0}/2\pi\mu\xi^{2}$ in type-II superconductors. In this class of
materials, $H_{c1}$ can be quite small (several $\text{\,}\mathrm{mT}$)
whereas $H_{c2}$ can be extremely large (from a few tesla up to as high as
$120\text{\,}\mathrm{T}$)Miura  _et al._ (2002), such that the interposing
state between the lower and upper critical fields consumes much of the phase
diagram and defines the technologically relevant regime. This mixed state
hosts a lattice of vortices, whose density scales with the magnetic field
$n_{v}\propto B$. We should also note that, in addition to globally applied
fields, self-fields from currents propagating within a superconductor can also
locally induce vortices.

Each vortex carries a single flux quantum $\Phi_{0}$ and the core defines a
nanoscale region through which the magnetic field penetrates the material. As
such, the vortex core is non-superconducting, of diameter $2\xi(T)$, and
surrounded by circulating supercurrents of radii up to $\lambda(T)$, as
depicted in Fig. 1. Given the dependence of the vortex size on these material-
dependent parameters, vortices effectively look different in different
materials—for example, they are significantly smaller in the high-temperature
superconductor $\mathrm{Y}\mathrm{Ba}_{2}\mathrm{Cu}_{3}\mathrm{O}_{y}$
(YBCO), where $\xi(0)=$1.6\text{\,}\mathrm{nm}$$, than in Nb, in which
$\xi(0)=$38\text{\,}\mathrm{nm}$$.Wimbush (2015)

Vortex motion constitutes a major source of dissipation in superconductors.
Propelled by currents of density $\bm{J}$ and thermal/quantum energy
fluctuations, vortices experience a Lorentz force density
$\bm{F}_{L}=(\bm{J}\times\bm{B})/c$ accompanied by Joule heating that weakens
superconducting properties. It is this cascading process that is responsible
for the undesirable impacts on applications, for which examples were provided
in Sec. I.

###  II.1 Fundamentals of vortex pinning

Figure 3: Enhancement in $J_{c}$ or $M\propto J_{c}$ in (a) an oxygen-ion-
irradiated Dy2O3-doped commercial YBCO film grown by American Superconductor
CorporationLeroux  _et al._ (2015); Eley  _et al._ (2017a) in a field of
$5\text{\,}\mathrm{T}$, (b) a BaZrO3-doped (Y0.77Gd0.23)Ba2Cu3Oy film grown by
Miura et al. Miura  _et al._ (2013a), (c) a BaZrO3-doped BaFe2(As1-xPx)2 film
grown by Miura et al.Eley  _et al._ (2017b), and (d) a heavy-ion-irradiated
NbSe2 crystalEley  _et al._ (2018). Measurements by S. Eley. Insets show
transmission electron micrographs of defect landscape, from Refs. [Eley  _et
al._ , 2017b; Miura  _et al._ , 2013a; Eley  _et al._ , 2017a, 2018].

.

Immobilizing vortices constitutes a major research area, in which the most
prominent benchmark to assess the strength of pinning is the critical current
$J_{c}$. Bean and Livingston (1964); Zeldov  _et al._ (1994a, b); Brandt
(1999); Willa, Geshkenbein, and Blatter (2014); Gurevich (2014, 2017, 2018);
Kubo and Gurevich (2019); Dhakal, Ciovati, and Gurevich (2020) Once vortices
are present in the bulk, crystallographic defects such as point defects,
precipitates, twin boundaries, stacking faults, and dislocations provide an
energy landscape to trap vortices. Depending on the defect type and density,
one of two mechanisms are typically responsible for vortex pinning: weak
collective effects from groups of small defects or strong forces exerted by
larger defects.

Originally formulated by Larkin and Ovchinnikov Larkin and Ovchinnikov (1979),
the theory of weak collective pinning describes how atomically small defects
alone cannot apply a sufficient force on a vortex line to immobilize it.
However, the collective action of many can indeed pin a vortex. In the case of
a random arrangement of small, weak, and uncorrelated pinning centers, the
average force on a straight flux line vanishes. Then, only fluctuations in the
pinning energy (higher order correlations) are capable of producing a net
pinning force. Considering this, weak collective pinning theory phenomenology
finds that the resulting critical current should scale _quadratically_ with
the pin density $n_{p}$, i.e., $J_{c}\propto n_{p}^{2}$, see Ref. [Blatter
_et al._ , 1994].

On the other hand, strong pinning results when larger defects each plastically
deform a vortex line and a low density of these defects is sufficient to pin
it. Competition between the _bare_ pinning force $f_{p}$ and the vortex
elasticity $\bar{C}$ generates multi-valued solutions. Because of this, a
proper averaging of the effective force $\langle f_{\mathrm{pin}}\rangle$ from
individual pins is non-zero and results in a critical current
$J_{c}=n_{p}\langle f_{\mathrm{pin}}\rangle$. Here, the critical current
reached by strong pins depends _linearly_ on the defect density. While
conceptually simpler than weak collective pinning, it has taken significantly
longer to develop a strong pinning formalism. With its completion in the early
2000s, the formalism enabled computing numerous physical observables,
including the critical current Blatter, Geshkenbein, and Koopmann (2004);
Koopmann, Geshkenbein, and Blatter (2004), the excess-current characteristic
Thomann, Geshkenbein, and Blatter (2012, 2017); Buchacek  _et al._ (2018,
2019a, 2019b); Buchacek, Geshkenbein, and Blatter (2020), and the $ac$
Campbell response Willa  _et al._ (2015); Willa, Geshkenbein, and Blatter
(2015, 2016); Willa, Marziali Bermúdez, and Pasquini (2018).

Defects merely trap the vortex state into a metastable minimum. Thermal and
quantum fluctuations release vortices from pinning sites, and this activated
motion of vortices from a pinned state to a more stable state is called vortex
creep. In the presence of creep, the critical current $J_{c}$ is no longer a
distinct boundary separating a dissipation-free regime from a dissipative one.
Experimentally, this manifests as a power-law transition $V=J^{n}$ between the
superconducting state in which $V=0$ and Ohmic behavior. The creep rate
$S\equiv-d\ln(J)/d\ln(t)$ then becomes $\propto 1/n$ and can be assessed by
fitting the transitional regime in the current-voltage characteristic or
measuring the temporal decay of an induced persistent current. Measurements of
the vortex creep rate also provide access to microscopic details such as the
effective energy barriers $U^{*}=T/S$ surmounted and whether single vortices
or bundles are creeping.

Various methods of tailoring the disorder landscape in superconductors have
proven successful in remarkably enhancing the critical current. Figure 3 shows
examples of cuprates, iron-based superconductors, and low-$T_{c}$ materials
that have all benefited from incorporating inclusions. Defects can be added
post-growth, using techniques such as particle irradiation,Leroux  _et al._
(2015); Eley  _et al._ (2017a); Tamegai  _et al._ (2012); Averback and de la
Rubia (1997); Fang  _et al._ (2011); Gapud  _et al._ (2015); Goeckner  _et
al._ (2003); Haberkorn  _et al._ (2015a, b, 2012); Iwase  _et al._ (1988); Jia
_et al._ (2013); Kihlstrom  _et al._ (2013); Kirk and Yan (1999); Konczykowski
_et al._ (1991); Matsui  _et al._ (2012); Nakajima  _et al._ (2009); Roas  _et
al._ (1989); Salovich  _et al._ (2013); Sun  _et al._ (2015); Swiecicki  _et
al._ (2012); Taen  _et al._ (2012, 2015); Thompson  _et al._ (1992); Vlcek
_et al._ (1993); Zhu  _et al._ (1993); Leonard  _et al._ (2014) or during the
growth process by incorporating impurities to the source material.Miura  _et
al._ (2013a); Haugan  _et al._ (2004); Horide  _et al._ (2013); Miura  _et
al._ (2011); Palau  _et al._ (2011); Wimbush  _et al._ (2010) Though these
processes induce markedly different disorder landscapes, both can effectuate
remarkable increases in $J_{c}$. However, the conditions necessary to improve
electromagnetic properties are highly material-dependent—this lack of
universality renders defect landscape engineering a process of trial-and-
error.

Particle irradiation can induce point defects (vacancies, interstitial atoms,
and substitutional atoms), larger random defects, or correlated disorder
(e.g., amorphous tracks known as columnar defects). Notably, the critical
current in commercial YBa2Cu3O7-δ coated conductors was nearly doubled through
irradiation with protonsJia  _et al._ (2013), oxygen-ionsLeroux  _et al._
(2015); Eley  _et al._ (2017b), gold-ionsRupich  _et al._ (2016), and silver-
ions. Furthermore, iron-based superconductors have also been shown to benefit
from particle irradiationTamegai  _et al._ (2012). To incorporate larger
defects, such as nanoparticle inclusions, numerous groupsMacManus-Driscoll
_et al._ (2004); Feighan, Kursumovic, and MacManus-Driscoll (2017); Miura  _et
al._ (2013a, 2011, 2016, 2017) have introduced excess Ba and $M$ (where $M$=
Zr, Nb, Sn, or Hf) into growth precursors. This results in the formation of
randomly distributed 5-20 nm sized non-superconducting Ba$M$O3 nanoparticles
or nanorods. This method has produced critical currents that are up to seven
times higher than that in films without inclusions,Miura  _et al._ (2013b)
therefore, has become one of the leading schemes for enhancing $J_{c}$.

The enhancement achieved by inclusions and irradiation is often restricted to
a narrow temperature and field range, partially because $\xi$ and $\lambda$
are temperature dependent, whereas the defect sizes and densities are fixed.
Another reason for the limited range of the enhancement is that, under the
right conditions, certain fast moving vortex excitations may form. For
example, in materials containing parallel columnar defects, double-kink
excitations form at low fields and moderate temperatures that result in fast
vortex creep concomitant with reduced $J_{c}$. Maiorov  _et al._ (2009) Mixed
pinning landscapes, composed of different types of defects, can indeed enhance
$J_{c}$ over a broader temperature and field range than inclusions of only one
type and one size. More work is indeed necessary to optimize this.

###  II.2 Thermally activated vortex motion

Vortex creep is a very complex phenomenon due to the interplay between vortex-
vortex interactions, vortex-defect interactions, vortex elasticity, and
anisotropy. Blatter  _et al._ (1994); Feigel’man  _et al._ (1989) Willa  _et
al._ (2020)These interactions determine $U_{act}(T,H,J)$, a generally unknown
regime-dependent function. The simplest creep model, proposed Anderson and
Kim, neglects the microscopic details of pinning centers and considers
vortices as non-interacting rigid objects hopping out of potential wells of
depth $U_{act}\propto U_{p}|1-J/J_{c}|$. However, as elastic objects, the
length of vortices can increase over time under force from a current and
vortex-vortex interactions are non-negligible at high fields. As such, the
Anderson-Kim model’s relevance is limited to low temperatures and fields.

At high temperatures and fields, collective creep theories, which consider
vortex elasticity, predict an inverse power law form for the current-dependent
energy barrier $U_{act}(J)=U_{p}[(J_{c}/J)^{\mu}-1]$, where $\mu$ is the so-
called glassy exponent that is related to the size and dimensionality of the
vortex bundle that hops during the creep processBlatter, Geshkenbein, and
Vinokur (1991). To capture behavior across regimes, the interpolation formula
$U_{act}(J)=(U_{p}/\mu)\left[(J_{c}/J)^{\mu}-1\right]$ is commonly used, where
$\mu\rightarrow-1$ recovers the Anderson-Kim prediction. Combining this
interpolation formula with the creep time
$t=t_{0}e^{U_{act}(J)/k_{\mathrm{B}}T}$, we find the persistent current should
decay over time as $J(t)=J_{c0}[1+(\mu
k_{\mathrm{B}}T/U_{p})\ln(t/t_{0})]^{-1/\mu}$ and that the thermal vortex
creep rate is

$\displaystyle S\equiv\Big{|}\frac{d\ln J}{d\ln
t}\Big{|}=\frac{k_{\mathrm{B}}T}{U_{p}+\mu k_{\mathrm{B}}T\ln(t/t_{0})},$ (1)

where $\ln(t/t_{0})\sim 25\text{-}30$. Because the magnetization $M\propto J$,
creep can easily be measured by capturing the decay in the magnetization over
time using a magnetometer. Moreover, as seen from Eq. (1), knowledge of
$S(T,H)$ provides access to both $U_{p}$ and $\mu$. Hence, creep measurements
are a vital tool for revealing the size of the energy barrier, its dependence
on current, field, and temperature, and whether the dynamics are glassy or
plastic. It is important to note that Eq. (1) is typically used to analyze
creep data piecewise—it can rarely be fit to the entire temperature range.
Creep rates are not predictable and no analytic expression exists that broadly
captures the temperature and field dependence of creep. Both $U_{p}$ and $\mu$
have unknown temperature dependencies, which is a major gap in our ability to
predict vortex creep rates.

###  II.3 Predictive vortex matter simulations

Simulating the the behavior of vortex matter Blatter  _et al._ (1994); Brandt
(1995); Crabtree and Nelson (1997); Nattermann and Scheidl (2000); Blatter and
Geshkenbein (2003); Kwok  _et al._ (2016b) has a long history. Though the
value of such simulations was realized long ago Brandt (1983a, b), the
efficacy to produce accurate results in materials containing complex defect
landscapes is considered a recent success, tied to improvements in
computational power. Specifically, we can now numerically solve more realistic
models, ranging from Langevin dynamics to time-dependent Ginzburg-Landau
(TDGL) equations to fully microscopic descriptions, including Usadel and
Eilenberger, Bogoliubov-de Gennes, and non-equilibrium Keldysh-Eilenberger
quantum transport equations. While the phenomenological TDGL equations
describe vortex matter realistically on lengths scales above the
superconducting coherence lengths, full microscopic equations are needed to
describe, e.g., the vortex core accurately. This, however, means that the
system sizes, which can be simulated down to the nanoscale using microscopic
models, are quite limited, while TDGL can simulate macroscopic behavior
including most dynamical features of vortex matter.

Figure 4: (a) 3D STEM tomogram of a 0.5 Dy-doped YBCO sample. Image processing
is discussed in Ref. [Leroux  _et al._ , 2015]. (b) Critical current $J_{c}$
as a function of the magnetic field $B$ applied along the c-axis of YBCO. The
simulated field dependence (circles, red curve) with only the nanoparticles
observed by STEM tomography in the sample with 0.5 Dy doping exhibits almost
the same exponent $\alpha$, for $J_{c}\propto B^{-\alpha}$, as the experiment
(triangles, green curve). Adding $2\xi$ diameter inclusions to the simulation
makes the dependence less steep (squares, blue curve), which yields an
exponent very similar to the experimental one in the sample with 0.75 Dy
doping (stars, yellow curve). (c) Snapshot of the TDGL vortex configuration
with applied magnetic field and external current for the same defect structure
as in the experiment (a). Isosurfaces of the order parameter close to the
normal state are shown in red and follow both vortex and defect positions. The
amplitude of the order parameter is represented on the backplane of the volume
where blue corresponds to maximum order parameter amplitude. Arrows indicate
the experimental and simulated $J_{c}$ dependencies.

The Langevin approach only considers vortex degrees of freedom, while mostly
neglecting elasticity and vortex-vortex interactions, which are nonlocal
effects. Hence, its accuracy is limited to when inter-vortex separations are
significantly larger than $\xi$, vortex pinning sites are dilute, or the
superconducting host is sufficiently thin that vortices can be considered 2D
particles. Nevertheless, this simple picture reveals remarkably rich,
dynamical behavior – notably realizing a dependence of $J_{c}$ on the strength
and density of pinning centers Brandt (1983a, b), thermal activation of
vortices from pinning sites Koshelev (1992), a crossover between plastic and
elastic behavior Cao, Jiao, and Ying (2000); Scala  _et al._ (2012), and
dynamic ordering of vortex lattices at large velocities Koshelev and Vinokur
(1994); Moon, Scalettar, and Zimányi (1996).

However, vortex elasticity is indeed an influential parameter in bulk systems.
It results in vortex phases that are characterized by complex vortex
structures, glassy phases that do not exist in 2D systems, as well as other
interesting characteristics Ertaş and Kardar (1996); van Otterlo  _et al._
(2000); Bustingorry, Cugliandolo, and Domínguez (2007); Luo and Hu (2007,
2010); Koshelev and Kolton (2011); Dobramysl  _et al._ (2013). Herein lies the
strength of the TDGL approach, which is a good compromise between complexity
and fidelity. It describes the full behavior of the superconducting order
parameter Schmid (1966) and therefore represents a ‘mesoscopic bridge’ between
microscopic and macroscopic scales. Notably, it surpasses the Langevin
approach by (i) describing all essential properties of vortex dynamics,
including inter-vortex interactions with crossing and reconnection events,
(ii) possessing a rigorous connection to the microscopic Bardeen-Cooper-
Schrieffer theory in the vicinity of the critical temperature Gor’kov (1959),
and (iii) considering realistic pinning mechanisms. Regarding pinning, it can
specifically account for pinning due to modulation of critical temperature
($\delta T_{c}$-pinning) or mean-free path ($\delta\ell$-pinning), strain,
magnetic impurities Doria  _et al._ (2007), geometric pinning through
appropriate boundary conditions, and, generally, weak to strong pinning
regimes—all beyond the reach of the Langevin approach. Consequently, the TDGL
formulation is arguably one of the most successful physical models, describing
the behavior of many different physical systems, even beyond superconductors
Aranson and Kramer (2002).

In its early days, the TDGL approach was used to study depinning, plastic, and
elastic steady-state vortex motion in systems containing twin and grain
boundaries as well as both regular and irregular arrays of point or columnar
defects. Gropp  _et al._ (1995); Crabtree  _et al._ (2000) Those simulations
were predominately used to illustrate the complex dynamics of individual
vortices because computational limitations prohibited the study of large-scale
systems with collective vortex dynamics. Only later did simulation of about a
hundred vortices in two-dimensional systems become possible, resulting in
predictions for, e.g., the field dependence of $J_{c}$ in thin films with
columnar defects Palonen, Jäykkä, and Paturi (2012).

A 2002 article by Winiecki and Adams Winiecki and Adams (2002) deserves credit
as one of the first simulation-based studies of vortex matter in three-
dimensional superconductors that produced a realistic electromagnetic
response. Later, in 2015, Koshelev et al. Koshelev  _et al._ (2016) achieved a
major technical breakthrough by investigating optimal pinning by monodispersed
spherical inclusions. The simulated system size of
$100\xi\\!\times\\!100\xi\\!\times\\!50\xi$ was much larger than any
previously studied system, enabling even more realistic simulations of the
collective vortex dynamics than previous works. Their computational approach
is based on an optimized parallel solver for the TDGL equation Sadovskyy  _et
al._ (2015b), which allows for simulating vortex motion and determining the
resulting electrical transport properties in application-relevant systems. The
efficacy of this technique is best demonstrated in a study Sadovskyy  _et al._
(2016a) that applied the same approach to a ‘real’ pinning landscape by
incorporating scanning transmission electron microscopy tomography data of Dy-
doped YBa2Cu3O7-δ films Ortalan  _et al._ (2009); Herrera  _et al._ (2008),
and the results showed almost quantitative agreement of the field and angular
dependent critical current with experimental transport measurements, see Fig.
4.

Finally, we discuss applying TDGL calculations to commercial high-temperature
superconducting tapes, which typically consist of rare earth (RE) or yttrium
barium copper oxide (REBCO) matrices. Specifically, Ref. [Sadovskyy  _et al._
, 2016b] simulated vortex dynamics in REBCO coated conductors containing self-
assembled BaZrO3 nanorods, and reported good quantitative match to
experimental measurements of $J_{c}$ versus the applied magnetic field-angle
$\theta$. Most notably, the simulations demonstrated the non-additive effect
of defects: adding irradiated columnar defects at a 45∘ angle with the nanorod
(c-) axis removes the $J_{c}(\theta=0^{\circ})$ peak of the nanorods and
generates a peak at $\theta=45^{\circ}$ instead. This study then went beyond
simply reproducing experimental behavior, and predicted the optimal
concentrations of BaZrO3 nanorods that are necessary to maximize $J_{c}$,
which it found to be 12-14% of $J_{d}$ (at specific $\theta$)—far higher than
had been experimentally achieved in similar systems. This approach is
certainly more efficient than the standard trial-and-error approach, growing
and measuring samples with a large variety of defect landscape.

These recent successes in accurately predicting $J_{c}$ in superconductors
based on the microstructure highlight how close we are to the ultimate goal of
tailoring pinning landscapes for specific applications with well-defined
critical current requirements. Constituting the new critical-current-by-design
paradigm, Kwok  _et al._ (2016b); Glatz  _et al._ (2020) the routine use of
TDGL simulations for efficient defect landscape optimization is a
transformative opportunity in vortex physics, as is expanding these
computational successes to include the use of artificial intelligence
algorithms. Furthermore, microscopic and far-from-equilibrium simulations of
vortex matter beyond the TDGL approach require significant computational
resources and are only now becoming feasible. We will discuss related
developments in Sec. III.4.

##  III Transformative Opportunities

###  III.1 Vortex Creep

In this section, we identify major opportunities to accelerate our
understanding of thermally-activated vortex hopping (thermal creep) and non-
thermal tunneling (quantum creep) between pinning sites. Only limited
situations are amenable to an analytic treatment of vortex creep: these
include thermal depinning of single vortices and vortex bundles in the regime
of weak collective pinning. In the strong pinning regime, e.g., for columnar
defects, we must consider complicated excitations that form during the
depinning process. Activation occurs via half-loop formationSonin and Horovitz
(1995), which is depicted in Fig. 2. During this process, the vortex nucleates
outside of its pinned position, and the curved unpinned segment grows over
time as a current acts on it, until the entire vortex eventually leaves the
pinning site. Because half-loop formation likely occurs in a range of high-
current-carrying materials, which may contain amorphous tracks, nanorods, or
twin boundaries, numerical treatment of vortex creep within the strong pinning
framework is of significant interest.

The first task involves studying creep of isolated vortices, pinned by a
single strong inclusion or columnar defect. In accordance with analytic
predictions, an increase in temperature shifts the characteristic depinning
current below $J_{c}$, rounds the $IV$ curves and affects the excess-current
characteristic far beyond $J_{c}$Buchacek  _et al._ (2019a, b); Buchacek,
Geshkenbein, and Blatter (2020). The next steps will involve studying multiple
vortices, more defects, and mixed defect landscapes, which will indeed
increase the complexity of the problem, warranting computational assistance.

Recent advances in computational power and high-performance codes will enable
tackling these challenges, which involve long simulation times at
exponentially low dynamics. Instead of simulating the thermal relaxation of a
metastable configuration in a single ’linear’ simulation, the same
configuration can be simulated in parallel, i.e., experiencing fluctuations
along different ’world lines’. This accelerates the search for a rare
depinning event, after which parallel computations are interrupted and
restarted from new depinned configurations.

Figure 5: Creep at reduced temperature $T/T_{c}$ = 1/4 and a field of
$\mu_{0}H=1\textnormal{ T}$ for different superconductors plotted versus
$Gi^{1/2}$. The open symbols indicate materials for which the microstructure
has been modified either by irradiation or incorporation of inclusions. The
solid grey line represents the limit set by $Gi^{1/2}(T/T_{c})$. The result
predicts that the creep problem even in yet-to-be-discovered high-$T_{c}$
superconductors may counteract the benefits of high operating temperatures.
Material from S. Eley, et al. Nat. Mater. 16, 409–413 (2017). Copyright 2017,
_Nature Publishing Group._

In a 2017 paperEley  _et al._ (2017a), we found that the minimum achievable
thermal creep rate in a material depends on its Ginzburg number $Gi$ as $S\sim
Gi^{1/2}(T/T_{c})$, shown in Fig. 5. Our result is limited to the Anderson-Kim
regime and considered pinning scenarios with analytically determined pinning
energies $U_{P}$. It also somewhat gravely predicts that there is a limit to
how much creep problem in high-$T_{c}$ superconductors, which tend to have
high $Gi$, can be ameliorated, such that we may expect the performance of yet-
to-be discovered room temperature superconductors to be irremediably hindered
by creep. However, YBCO films containing nanoparticles demonstrate non-
monotonic temperature-dependent creep rates $S(T)$, such that $S$ dips to
unexpectedly low values at intermediate temperatures outside of the Anderson-
Kim regimeEley  _et al._ (2017a). This dip, thought to be induced by strong
pinning from nanoparticles, suggests that collective pinning regimes may hold
the key to inducing slower creep rates that dip below our proposed lower limit
in the Anderson-Kim regime. A numerical tacking of the vortex creep problem
would improve our theoretical understanding of creep and answer a major open
question in vortex physics – what is indeed the slowest achievable creep rate
in different superconductors?

Our finding of the lower limit to the creep rate reduces the guesswork in
trial-and-error approaches to optimizing the disorder landscape and improves
our ability to select a material for applications requiring slow creep. Yet,
ultimately, a material’s quantum creep rate actually sets its minimum
achievable creep rate. This is a regime that is has received relatively little
attention—there have been few theoretical and experimental studies of quantum
creep. Theoretical models are limited to tunneling barriers induced by weak
collective pinningBlatter, Geshkenbein, and Vinokur (1991); Blatter and
Geshkenbein (1993) and columnar defects,Chudnovsky, Ferrera, and Vilenkin
(1995) though most materials have very complex, mixed pinning landscapes. Most
experimental work has focused on cuprates, determining a crossover temperature
of $\sim$ 8.5-11 K in YBCO films,Åkerman  _et al._ (2001); Landau and Ott
(2000); Luo  _et al._ (2002), 1.5-2 K in YBCO crystals,Hoekstra  _et al._
(1999); Landau and Ott (2000) 5-6 K in Tl2Ba2CaCu2O8 films,Hoekstra  _et al._
(1999); Tejada, Chudnovsky, and García (1993), 17 K in TlBa2CaCu2O7-δ,Åkerman
_et al._ (2001) 30 K in HgBa2CaCu$2$O6+δ.Åkerman  _et al._ (2001) Klein et
al.Klein  _et al._ (2014) studied an iron-based superconductor, finding
crossover around 1 K in Fe(Te,Se). No studies have been conducted in materials
containing inclusions nor using any systematic tuning of the energy barrier.

Furthermore, the crossover between the thermal and quantum creep is unclear.
As previously mentioned, the Anderson-Kim model’s relevancy is limited to low
temperatures $k_{\mathrm{B}}T\ll U_{p}$ in which $S$ is expected to increase
approximately linearly with temperature. A linear fit to this regime often
extrapolates to non-zero $S$ at $T=0$, suggestive of non-thermal creep. In
fact, it is common to perfunctorily attribute this extrapolation to quantum
creep without conducting measurements in the quantum creep regime. However,
there are compelling discrepancies between typical experimental results in
this context and theory. For example, theory predicts that the tunneling
probability should decrease with bundle size, whereas experiments often
observe the opposite trend (positive correlation between low temperature $S$
and field)Lykov (2013). Theory also predicts a quadratic, rather than linear,
temperature-dependent $S(T\rightarrow 0)$Lykov (2013); Hoekstra  _et al._
(1999). That is, quantum creep may be thermally assistedBlatter, Geshkenbein,
and Vinokur (1991), and not simply present itself as a temperature-independent
creep rate at low temperatures. An even more confounding result is that
Nicodemi et al.Nicodemi and Jeldtoft Jensen (2001) predicted non-zero creep
rates at $T=0$ using Monte Carlo simulations based on a purely classical
vortex model and reconciled it with non-equilibrium dynamics.

It has also been suggested that the overall measured creep rate is simply the
sum of the thermal and quantum components.Åkerman  _et al._ (2001) However, in
some iron-based superconductors,Haberkorn  _et al._ (2012, 2014) $S$ is fairy
insensitive to $T$ or even decreases with increasing $T$ up to fairly high
fractions of $T_{c}$. Hence, either quantum creep is a significant component
at surprisingly high temperatures or the creep rate dramatically decreases at
temperatures below the measurement base temperature, motivating the need for
lower temperature creep measurements. Superconductors with high normal-state
resistivity $\rho_{n}$ and low $\xi$, such as high-$T_{c}$ cuprates, are the
best candidates for having measurable quantum creep rates. This is because the
effective quantum creep rate is predicted to be

$\displaystyle\\!\\!\\!S_{q}=\begin{cases}-(e^{2}\rho_{n}/\hbar\xi)\sqrt{J_{c}/J_{d}},&\\!\\!\\!\\!\text{if
}L_{c}<a_{0}\\\
-(e^{2}\rho_{n}/\hbar\lambda)(a_{0}/\lambda)^{4}(a_{0}/\xi)^{9}(J_{c}/J_{d})^{9/2},&\\!\\!\\!\\!\text{if
}L_{c}>a_{0}\end{cases}\\!\\!$ (2)

where $L_{c}$ is the length of the vortex segment (or bundle) that tunnels
Blatter  _et al._ (1994). Determining the dependence of the quantum creep rate
on material parameters in superconductors would fill a major gap in our
understanding of vortex physics. This would significantly contribute towards a
comprehensive model of vortex dynamics, and reveal whether creep may induce
measurable effects in quantum circuits, which typically operate at millikelvin
temperatures.

###  III.2 Pinning at the extreme: Can the critical current reach the
depairing current?

Cooper pairs constituting the dissipationless current in superconductors will
dissociate when their kinetic energy surpasses their binding energy.
Theoretically, this could be achieved by a sufficiently high current, termed
the depairing current, $J_{d}$. Consequently, $J_{d}$ is recognized as the
theoretical maximum achievable $J_{c}$, such that $J_{c}/J_{d}$ is often
equated with the efficiency $\eta$ of the vortex pinning landscape, which may
be confusing as a perfect defect would not produce $J_{c}=J_{d}$.Wimbush
(2015)

The most successful efforts to carefully tune the defect landscape obtain
$J_{c}/J_{d}$ of only 20-30%.Civale (2019); Selvamanickam  _et al._ (2015) As
exemplified by a series of samples we have measured, see Fig. 6, most samples
produce $J_{c}/J_{d}<5\%$, whereas $J_{c}/J_{d}$ is routinely higher for
coated conductors (REBCO films). Though this at first appears to be a far cry
from the ultimate goal, some surmise that this is indeed near the maximum that
can be achieved by immobilizing vortices by means of core pinning, which
merely refers to a vortex preferentially sitting in potential wells defined by
a defect to minimize the energy of its core. Wimbush et al. Wimbush (2015)
present a compelling argument that core pinning can obtain a maximum
$J_{c}/J_{d}$ of only $30\%$—A current equivalent to $J_{d}$ would produce a
Lorentz force $f_{d}=J_{d}\Phi_{0}=4B_{c}\Phi_{0}/3\sqrt{6}\mu_{0}\lambda$,
with $B_{c}=\Phi_{0}/2\sqrt{2}\pi\lambda\xi$ the thermodynamic critical field.
At the same time, the condensation energy $\varepsilon_{sc}$ produces a
characteristic pinning force
$f_{p}^{core}\sim\varepsilon_{sc}/\xi\approx\pi\xi^{2}B_{c}^{2}/2\mu_{0}$,
such that the ratio of the maximal core pinning force to the depairing Lorentz
force is

$\displaystyle f_{p}^{core}/f_{d}$ $\displaystyle=3\sqrt{3}/16\approx 32\%.$
(3)

Similarly, Matsushita Matsushita (2007) performed a more precise calculation
by considering the effects of the geometry of the flux line, and found that
$f_{p}^{core}/f_{d}\approx 28\%$. Hence, decades of work in designing the
defect landscape to pin vortex cores may have indeed nearly accomplished the
maximum efficiency achievable by means of core pinning.

Figure 6:  Critical current $J_{c}$ normalized to the depairing current for
various superconductors at $T=$5\text{\,}\mathrm{K}$$ and
$\mu_{0}H=$0.3\text{\,}\mathrm{T}$$. The data includes Dy2O3-doped YBa2Cu3O7-δ
commercial coated conductors and B$M$O3-doped Y0.77Gd0.33Ba2Cu3O7-δ films
(where $M=$ Sn, Zr, or Hf), all grown via metal organic deposition.

If the ultimate goal of $J_{c}=J_{d}$ cannot be obtained by core pinning
alone, are there other mechanisms to immobilize vortices that could indeed
produce $J_{c}/J_{d}>30\%$? Magnetic interactions between vortices themselves
or vortices and magnetic inclusions can also restrict the motion of a
vortex—referred to as magnetic pinning. Herein lies a transformative
opportunity to make large strides towards approaching $J_{c}=J_{d}$. Magnetic
pinning alone or in combination with core pinning may produce unprecedentedly
high values for $J_{c}$, though this mechanism has received considerably less
attention than core pinning because it is quite complicated to actualize.

A high density of vortices tend to arrange themselves into a hexagonal
lattice, and one pinned to a defect via core pinning may restrict the motion
of its neighbors, subsequently affecting its neighbors’ neighbors due to
magnetic vortex-vortex interactions, which occur over a length scale of
$\lambda$. A magnetic inclusion provides another opportunity for inflicting
magnetic and core pinning on a vortex. Again, following the arguments of
WimbushWimbush (2015), we can compare the pinning force induced by core
pinning to that of magnetic pinning. The magnetic Zeeman energy
$\varepsilon_{mag}=\frac{1}{2}\int_{A}M\cdot B\,dA$ produced by a strong
ferromagnet is much greater than the condensation energy and may be several
orders of magnitude greater than the core pinning energy. However, it is
unclear whether the resulting pinning force is greater because it occurs over
the longer length scale of $\lambda$ versus $\xi$, i.e.,
$f_{p}^{mag}\sim\varepsilon_{mag}/\lambda$, such that

$f_{p}^{mag}/f_{p}^{core}\approx 2\sqrt{2}(\mu_{0}M/B_{c}).$ (4)

Hence, the advantage depends on the ratio of the magnetization of the pinning
site to the thermodynamic critical field. Independent of whether $f_{p}^{mag}$
surpasses $f_{p}^{core}$, concomitant mechanisms would produce an additive
effect that may surpass current record values of $J_{c}$. Yet, ferromagnets in
proximity to superconductors can locally degrade superconductivity by inducing
pair breaking, such that it is challenging to incorporate ferromagnetic vortex
pinning centers without compromising the superconducting state. This
complication combined with the typical materials science considerations of
incorporating inclusions that do not induce too much strain on the surrounding
superconducting matrix will make this a challenge all but insurmountable.

In addition to magnetic pinning, exploiting geometric pinning provides another
potentially transformative opportunity to dramatically boost $J_{c}$ in
superconductors. In clean, narrow (sub-micron) superconducting strips,
geometric restrictions can induce self-arrest of vortices recovering the
dissipation-free state at high fields and temperatures due to surface/edge
(Bean-Livingston barrier) Bean and Livingston (1964) or geometric Zeldov  _et
al._ (1994a, b); Brandt (1999); Willa, Geshkenbein, and Blatter (2014)
pinning. Figure 7 depicts an example of geometric vortex pinning around two
superconducting strips.

Moreover, at a fixed applied current, the magnetoresistance (MR) shows
oscillations with increasing magnetic field, indicating the penetration of
complete vortex rows into the system Papari  _et al._ (2016). Therefore, these
MR oscillations are a way to determine the vortex structure in nanoscale
superconductors. At very high fields, the vortex lattice in these strips
starts to melt. Combining magnetoresistance measurements and numerical
simulations can then relate those MR oscillations to the penetration of vortex
rows with intermediate geometrical pinning, where the vortex lattice remains
unchanged, and uncover the details of geometrical melting. This opens the
possibility to control vortices in geometrically restricted nanodevices and
represents a novel technique of ‘geometrical spectroscopy’. Combined use of MR
measurements and large-scale simulations would reveal detailed information of
the structure of the vortex system. A similar re-entrant behavior was observed
in superconducting strips in a parallel field configuration: Here, high fields
lead to ‘vortex crowding’, in which a higher density of vortex lines starts to
straighten, therefore reducing the Lorenz force on the vortices. The result is
an intermediate dissipation-less state Wang  _et al._ (2017).

The situation becomes more complex when one considers nanosized
superconducting strips and bridges, in which vortex pinning is dictated by an
intricate interplay of surface and bulk pinning. As described above, in the
case of a very narrow bridge, $J_{c}$ is mostly defined by its surface
barrier, whereas in the opposite case of very wide strips, it is dominated by
its bulk pinning properties. However, understanding the intermediate regime,
where the critical current is determined both by bulk pinning and by the Bean-
Livingston barrier at the edge of a strip is of great interest in small
superconducting structures and wires.

Figure 7:  Field profile in a double-strip geometry before penetration of
vortices Willa, Geshkenbein, and Blatter (2014). The arrangement and geometry
(e.g. width $w$ and thickness $d$) of the specimen significantly influence the
relative importance of Bean-Livingston, geometric and bulk pinning. Material
from R. Willa, _ETH Zurich Research Collection_ , see Ref. [Willa, 2016].

Recent studies Kimmel  _et al._ (2019) revealed that while bulk defects arrest
vortex motion away from the edges, defects in their close vicinity promote
vortex penetration, thus suppressing the critical current. This phenomenon is
also quite important in the study of superconducting radio-frequency cavities.
Furthermore, the role of defects near the penetrating edge is asymmetric
compared to the exit edge of a superconducting strip. This complex interplay
of bulk and edge pinning open new opportunities for tailored pinning structure
for a given application. In the simple case of the straight strip with
similar-type spherical defects, an optimized defect distribution can have a
more than 30% higher critical current density than a homogeneously disorder
superconducting film.

The need for high-current, low-cost superconductors continues to grow with new
applications in lightweight motors and generators, as well as strong magnets
for high-energy accelerators, NMR machines, or even Tokamak fusion reactors.
Many of these applications require large magnetic fields and therefore large
critical currents, which is both a fundamental research and engineering
challenge as it requires reliably fabrication of uniform, km-long high-
performance superconducting cables having an optimal pinning microstructure.

Consequently, there are two main aspects that must be addressed for large-
field applications: (i) determining the best possible pinning landscape and
geometry for a targeted application and (ii) controlling fabrication of long
superconducting cables to incorporate an optimized pinning landscape with
highest possible uniformity. Both of these aspects are part of the critical-
current-by-design paradigm Kwok  _et al._ (2016b). We will describe these in a
more general context in section III.4. Taking advantage of the modern
computational approaches described there in combination with experiments opens
novel pathways to new materials for large-field applications, in particular
the use of high-$T_{c}$ superconducting material instead of more traditional
choice of elemental Nb or Nb-based compounds.

###  III.3 Superconducting RF cavities and Quantum Circuits

Figure 8:  Surface disorder and multilayers in SRF cavities. (a) Sketch
showing how vortices (red) parallel to the surface of a cavity wall penetrate
the wall (outside the superconductor, the red lines illustrate field lines),
(b) intercalating insulating layers (SI[SI]S) will cause vortex pancakes to
form and might limit the penetration depth of vortices Gurevich (2006, 2017).
(c) Simulation snapshot of surface vortex penetration into a type-II
superconductor having spherical defects (yellow) near the surface in an AC
magnetic field parallel to the surface. Vortex lines are shown in red. The
planar projection shows the superconducting order parameter amplitude.

####  III.3.1 Superconducting RF cavities

Most studies of vortex dynamics in superconductors are conducted using DC
currents and static magnetic fields. Yet, the need to understand vortex
dynamics under AC magnetic fields or AC currents is rapidly increasing, as
these are the operating conditions e.g. for superconducting radio frequency
(SRF) cavities and quantum circuits for sensing and computing. Superconductors
are desirable for RF devices because the minimal resistance enables very high
quality factors $Q$, a metric that indicates high energy storage with low
losses and narrow bandwidths. SRF cavities are used in current and next-
generation designs for particle accelerators used in, e.g., high energy
physics. In addition to $Q$, the maximum accelerating field, $E_{a}$, is
another important metric for SRF cavity performance. The goal is often to
maximize $Q$ at low drive powers, which is essential for large accelerating
fields and reduced demands on cryogenic systems that are responsible for
cooling the cavities. Similarly, higher $E_{a}$ is desirable, as it indicates
larger reachable particle energies.

Elemental niobium (Nb) and Nb-based compounds are the material of choice for
all current accelerator applications. Advances in the fabrication of Nb-
cavities have pushed their performance to extraordinary levels Padamsee (2009,
2017), with $Q$-values approaching $2\times 10^{11}$ and $E_{a}$ in excess of
$45$ MV/m in Nb Grassellino  _et al._ (2017), and $E_{a}\sim$ 17 MV/m for
Nb3Sn resonators. In both cases, the magnetic field reached is above the lower
critical field of the material, but below the theoretically predicted
superheating field, at which vortices would spontaneously penetrate even a
perfect cavity wall, shown in Fig. 8a.

Further increases in $E_{a}$ require a conceptual breakthrough in our
understanding of Nb-cavity performance limits or new constituent materials.
New material candidates being considered include Nb3Sn, NbTiN, MgB2, Fe-based
superconductors, and engineered multilayer or stratified structures (see Fig.
8b). SRF cavities operate at temperatures well below $T_{c}$ and high enough
frequencies to drive the superconductor into a metastable state, near
breakdown. The resulting period approaches intrinsic time scales, such as
vortex nucleation and quasi-particle relaxation times.

While the experimental progress in improving the performance and quality
factor of SRF cavities has been impressive Padamsee (2017), e.g., the counter-
intuitive increase of the quality factor with nitrogen doping, it is mostly
based on trail-and-error approaches. A deep fundamental understanding is
important to make more systematic progress, requiring new theoretical and
computational studies. Because the cavities operate out-of-equilibrium, a
phenomenological description based on TDGL theory can only serve as a rough,
qualitative guide. Developing a fundamental theory describing the nonlinear
and non-equilibrium current response of SRF cavities requires a microscopic
description based on quantum transport equations for non-equilibrium
superconductors. A microscopic description, however, is challenging because
the RF currents under strong drive conditions (i.e., high field frequencies
and amplitudes near the breakdown/superheating field) affect both the
superconducting order parameter and the kinetics of quasiparticles, all of
which have to be treated self-consistently. This endeavor requires development
of numerical approaches to solve the quantum transport equations, based on the
Keldysh and Eilenberger formulations of non-equilibrium superconductivity in
the strong-coupling limit. The Keldysh-Eilenberger quantum transport equations
are, in general, non-local in space-time, non-linear, and in many physical
situations involve multiple length and time scales. Solving these equations
requires considerable computational resources, which are now becoming
available with exa-scale computing facilities.

Herein lies a transformative opportunity to dramatically boost the performance
of SRF cavities. Namely, researchers are now equipped to develop microscopic
theoretical models, and incorporate them into computational codes, to reveal
the origin and mechanisms that limit the accelerating field of SRF cavities.
The acquired knowledge will then guide materials optimization to maximize the
critical currents, superheating fields, and quench fields. Reaching the
theoretical limits for these parameters necessitates suppressing vortex
nucleation.

At high RF magnetic field amplitudes, screening currents near the vacuum-
superconductor interface can nucleate Abrikosov vortices that can quench the
cavity, see Fig. 8c. This _vortex breakdown_ depends on (i) the amplitude and
frequency of the surface field, (ii) the cavity’s surface roughness, and (iii)
the type, distribution, and size of defects near the interface. The impact of
near-surface defects on vortices is diametric: they may reduce the potential
barrier for vortex nucleation Kimmel  _et al._ (2019), but also pin nascent
vortices generated by nucleation at the surface, preventing a vortex avalanche
and substantial dissipation. Given this, there are various possible optimal
microstructures for the SRF constituent materials: (i) a “clean”
superconductor with a maximum surface barrier, (ii) a superconductor with a
thin (few $\xi$) defect-free surface and nanoscale defects in its bulk, or
(iii) some special spatial gradient in the size and/or density of defects.

Large-scale TDGL simulation can be applied to study the conditions under which
vortex avalanches form, devise mechanisms that are effective at mitigating
these avalanches and, more generally, gain insight into the flux penetration
under RF field conditions. Furthermore, by coupling the TDGL and heat
transport equations, this method can study _hot spots_ , providing insight on
avoiding the formation of these hot spots in SRF cavity walls. As previously
mentioned, though TDGL cannot produce quantitative results, it can serve as a
useful guide to experiments and also provide insight into simulations based on
microscopic transport equations. Lastly, though discussed in the context of
SRF cavities, these new computational methods can be applied to
superconducting cables for AC power applications.

####  III.3.2 Vortices in Quantum Circuits

Figure 9:  (a) An optical image showing multiple $\lambda/4$ resonators
multiplexed to a common feed line and surrounded by a ground plane containing
holes to pin vortices and suppress vortex formation, (b, c) Scanning electron
micrographs of superconducting CPW resonator without (b) and with (c) holes.
(d) Quality factor $Q_{i}$ versus $B_{\perp}$ for varying hole density
$\rho_{h}$. The field at which the vortex density matches the hole density
(each hole is filled with one vortex) is plotted with a color-matched vertical
line. Above this threshold field, additional vortices are not pinned by the
holes but instead only weakly pinned by film defects and interstitial pinning
effects. (e) $\Delta f_{r}/f_{r}$ versus $B_{\perp}$ for varying $\rho_{h}$.
Reprinted with permission from Ref. [Kroll  _et al._ , 2019]. Copyright 2019,
_American Physical Society_.

##### Energy loss due to vortices.

Similar to SRF cavities, superconducting circuits for quantum information also
operate at RF/microwave frequencies and are affected by vortices.
Specifically, along with parasitic two-level fluctuators and quasiparticles,
vortices are a considerable source of energy loss in superconducting quantum
circuitsMüller, Cole, and Lisenfeld (2019); Oliver and Welander (2013);
Martinis, Ansmann, and Aumentado (2009). These energy loss mechanisms create a
noisy environment in which the qubit irreversibly interacts stochastically,
rather than deterministically. Consequently, the evolution of the quantum
state is unpredictable, increasingly deviating over time from predictions
until the qubit state is eventually lost. This is called decoherence, which
limits the amount of time $T_{1}$ over which information is retained in qubits
to the microsecond range and there is typically a large spread in the $T_{1}$
times for each qubit in multi-qubit systemsFinke (2019).

Vortices appear in superconducting quantum circuits due to stray magnetic
fields, self-fields generated by bias currents, and pulsed control fields. In
addition to limiting $T_{1}$ in qubits, thermally-activated vortex motion can
cause significant noise in superconducting circuits and reduce the quality
factor $Q$ of superconducting microwave resonatorsSong  _et al._ (2009b, a);
Kroll  _et al._ (2019). To mitigate this, techniques have been developed to
either prevent vortex formation or trap vortices in regions outside of the
path of operating currents. Shielding circuits from ambient magnetic fields
and narrowing linewidths constituting the deviceVan Harlingen  _et al._
(2004b); Samkharadze  _et al._ (2016) significantly reduce the vortex
population. For example, for a line of width $w$, flux will not enter until
the field surpasses $\Phi_{0}/w^{2}$. Because of this realization, flux qubits
typically contain linewidths of $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
therefore exclude vortex formation up to a threshold magnetic field of roughly
$2\text{\,}\mathrm{mT}$, which is 20 times larger than the Earth’s magnetic
fieldVan Harlingen  _et al._ (2004b).

Though shielding has enabled remarkable headway in improving the stability of
superconducting qubits for computing applications, it is not a complete
solution. A reasonable amount of shielding can only suppress the field by a
small amount, which may be insufficient if devices must operate in high-field
environments. Moreover, shielding may render devices useless in quantum
sensing applications in which the purpose of the device is to sense the
external environment. This has sparked research on further modifications to
the device design and on understanding the effects of a magnetic field on
different architectures of quantum circuits, including transmon
qubitsSchneider  _et al._ (2019) and superconducting resonatorsBothner  _et
al._ (2017); Kroll  _et al._ (2019), which are integral components to readout.

In addition to shielding, another common remedy for the vortex problem in
superconducting circuits involves micropatterning arrays of holes in the
ground plane to serve as vortex traps and reduce the prevalence of vortex
formation within that perforated areaSong  _et al._ (2009a); Bothner  _et al._
(2011, 2012); Chiaro  _et al._ (2016), see Fig. 9. For example, Bothner et al.
found that $Q(B=$4.5\text{\,}\mathrm{mT}$)$ is a factor of 2.5 higher in Nb
resonators containing flux trapping holes in the ground plane Bothner  _et
al._ (2011) compared to without the holes. However, Chiaro et al.Chiaro  _et
al._ (2016) showed that, without careful design, these features can increase
the density of and subsequently losses from parasitic two-level fluctuators,
thought to primarily form at surfaces and interfaces. Moreover, coplanar
waveguide resonators were recently found to be more robust to external
magnetic fields when the superconducting ground plane area is reduced, which
lowers the effective magnetic field inside the cavity, and by coupling the
resonator inductively instead of capacitively to the microwave feedline,
shielding the feedlineBothner  _et al._ (2017).

The methods we have discussed here engendered tremendous advances in
suppressing the vortex problem in superconducting quantum circuits, however,
the details are material-dependent. Likewise optimal mitigation strategies may
be material-dependent. For example, Song et al. compared the microwave
response of vortices in superconducting Re (rhenium) and Al coplanar waveguide
resonators with a lattice of flux-trapping holes in the ground plane.
Generally, in both systems, vortices shift the resonance frequency $f_{0}$,
broaden the resonance dip $|S_{21}|(f)$, and reduce the quality factor $Q$.
However, vortices in the Al resonators induce greater loss and are more
sensitive to flux creep effects than in the Re resonators. The Al resonator
experienced a far more substantial fractional frequency shift $df/f_{0}$ with
increasing frequency than the Re resonator. Furthermore, while the loss $1/Q$
due to vortices increased with frequency for Re, it decreased for Al.

Most research on the microwave response of vortices in quantum circuits is
limited to AlSong  _et al._ (2009b, a); Chiaro  _et al._ (2016); Nsanzineza
and Plourde (2014); Wang  _et al._ (2014), NbBothner  _et al._ (2012, 2011);
Stan, Field, and Martinis (2004); Kwon  _et al._ (2018); Golosovsky,
Snortland, and Beasley (1995), NbTiNSamkharadze  _et al._ (2016); Kroll  _et
al._ (2019), and ReSong  _et al._ (2009a). Whereas Al and Nb are used in
commercial quantum computers, superconducting nitrides (TiN, NbN, NiTiN)Sage
_et al._ (2011); Ohya  _et al._ (2014); Vissers  _et al._ (2012); Leduc  _et
al._ (2013); Sandberg  _et al._ (2012); Chang  _et al._ (2013); Kerman  _et
al._ (2006); Barends  _et al._ (2010a, b); Bruno  _et al._ (2015) and Re have
garnered substantial attention because they may suffer less from parasitic
two-level fluctuators, which are particularly problematic in oxides and at
interfacesMüller, Cole, and Lisenfeld (2019). Nitrides and Re are known to
develop thinner oxide layers than Al and Nb, and can be grown epitixally on
common substratesDumur  _et al._ (2016); Wang  _et al._ (2009); Vissers  _et
al._ (2010). To develop a generic understanding of how to design quantum
circuits that are resilient to ambient magnetic fields and control vortices in
circuits made of next-generation materials, we must study circuits consisting
of broader ranges of materials, perform further studies on nitride-based
circuits, investigate different designs for flux trapping, and conduct imaging
studies that can observe rather than infer the efficacy of vortex pinning
sites. There have been a few studies that imaged vortices in superconducting
strips, which provided guidance on appropriate line widths to preclude vortex
formationStan, Field, and Martinis (2004); Kuit  _et al._ (2008). To build
upon this, imaging studies (using e.g. a scanning SQUID or magnetic force
microscope) of devices would inform on the efficacies of flux trapping sites,
reveal locations in which vortices form, and track vortex motion.

##### Vortices in topological quantum computing schemes.

Up until now, we have discussed vortices exclusively as a nuisance, which is
indeed the case for a broad range of applications. A notable exception lies in
the burgeoning field of topological quantum computing, in which vortices serve
as hosts for Majorana modesLiu  _et al._ (2019). Qubits encoded using Majorana
modes are predicted to be relatively robust to noise, thus have long coherence
times. One way to realize this is to couple a superconductor to a topological
insulator, induce vortices in the superconductor, and Majorana states are
predicted to nucleate in the vortex core. (Also note that Majorana modes have
been theorized to exist in other systems) Grosfeld and Stern (2011); Lee  _et
al._ (2019); You  _et al._ (2014); Cheng, Lutchyn, and Das Sarma (2012);
Budich, Walter, and Trauzettel (2012); Vijay and Fu (2016); Alicea (2012).
Initially elusive, signatures of Majorana vortex modes have been recently
observed in a variety of systems, including the iron-based superconductor
$\mathrm{Fe}\mathrm{Te}_{x}\mathrm{Se}_{1-x}$ Chiu  _et al._ (2020); Ghazaryan
_et al._ (2020), EuS islands overlying gold nanowires Manna  _et al._ (2020),
superconducting Al layer encasing an InAs nanowireVaitiekenas  _et al._
(2020), and Bi2Te3/NbSe2 heterostructuresSun  _et al._ (2016). To exploit
these modes for computing, we must be able to control their vortex hosts.
Consequently, vortex pinning research will be beneficial to vortex-based
topological quantum computing applications.

###  III.4 Vortex matter genome using artificial intelligence: Critical-
current-by-design

Figure 10:  Critical-current-by-design using: (a) Genetic algorithms to
optimize critical currents. Starting with a superconductor having intrinsic
defects, genetic algorithms can be used to optimize the defect structure by
mutation of defects and targeted selection of landscapes with larger critical
currents. A mutation of a defect (or several defects at once) can be done by,
e.g., translation, resizing, deletion, or splitting as sketched in the defect
sequence on the right. Overall, this procedure creates “generations” of
mutated defect configurations and only the best is selected and chosen to be
the seed for the next generation shown in the partial tree on the left
(circles/dots represent configurations, where the large numbered one is the
best). Using neural networks and machine learning (ML) to predict the best
mutations could further improve the targeted selection approach Sadovskyy  _et
al._ (2019). (b) ML/Artificial intelligence (AI) to improve and tailor defect
landscapes in superconductors.  1 illustrates how AI models can be used to
predict pinning landscapes from synthesis parameters and, vice versa, to
predict synthesis parameters like precursor concentrations, pressures, and
temperatures in, e.g., vapor deposition methods, for a targeted pinning
landscape. The models need to be trained by experimental or simulation data
sets.  2 similarly shows how to directly predict critical current
dependencies, like field orientation dependencies, from pinning topologies and
vice versa. Again, the underlying model is trained by experimental and
simulation data.

Over the years, research in superconductivity and vortex pinning has produced
large amounts of experimental and simulation data on microstructures,
synthesis, and critical current behavior. More recently, artificial
intelligence (AI) and machine learning (ML) approaches have enabled
revolutionary advances in the fields of image and speech recognition as well
as automatic translation, and are now finding an increasing number of
applications in scientific research areas that deal with massive data sets,
like particles physics, structural biology, astronomy, and spectroscopy.
Combining these will enable novel approaches to predict pinning landscapes in
superconductors for the future design of materials with tailored properties by
using sophisticated ML algorithms and AI models. This has become a promising
approach within the critical-current-by-design paradigm, which refers to
designing superconductors with desired properties using sophisticated
numerical methods replacing traditional trial-and-error approaches. These
properties include maximizing critical currents, achieving robust critical
currents with respected to variations of the pinning landscape (which is
important for large-scale commercial applications), or attaining uniform
critical currents with respect to the magnetic field orientation.

The next step towards advancing the use of AI/ML approaches for critical-
current-by-design may be to build upon the genetic algorithms implemented in
Ref. [Sadovskyy  _et al._ , 2019] to optimize pinning landscapes for maximum
$J_{c}$. This approach utilizes the idea of targeted selection inspired by
biological natural selection. In contrast with conventional optimization
techniques, such as coordinate descent, in which one varies only a few
parameters characterizing the entire sample, targeted evolution allows
variations in each defect individually without any a-priori assumptions about
the defects’ configuration. This essentially means that one solves an
optimization problem with, theoretically, infinite degrees of freedom. Ref.
[Sadovskyy  _et al._ , 2019] demonstrated the feasibility of this approach for
clean samples as well as ones with preexisting defects, e.g. as found in
commercial coated conductors. The latter, therefore, provides a post-synthesis
optimization step for existing state-of-the-art wires and a promising path
toward the design of tailored functional materials.

However, the mutations of the defects required for the genetic algorithm [see
Fig. 10(a)] were chosen randomly. Those mutations generate “generations” of
pinning landscapes, of which the best is chosen by targeted selection and then
used as a seed for the next generation. Using a simple machine learning
approach could further enhance the convergence of this method, by performing
only mutations which have higher probabilities of enhancing the critical
current. Besides superconductors, this methodology can be used to improve the
intrinsic properties of other materials where defects or the topological
structure plays an important role, such as magnets or nanostructured
thermoelectrics.

Going beyond these ML-improved simulations, one can build quantitative data-
driven AI approaches for superconductors that will enable, e.g. predicting the
critical current phase diagram and extracting the defect morphology
responsible for its performance directly from the existing accumulated
experimental and simulation data without actual dynamics simulations. Here, we
will discuss two potentially transformative opportunities, summarized in Fig.
10(b). The first application is motivated by the need for reliably producing
uniform superconductors on macroscopic commercial scales. This requires a deep
understanding of material synthesis processes e.g., for self-assembled pinning
structures in a superconducting matrix (see Fig. 10(b),  1). Materials at the
forefront of this quest are REBCO films with self-assembled oxide inclusions
in the form of nanorods and nanopartices Obradors and Puig (2014); Kwok  _et
al._ (2016b). For example, BaZrO3 (BZO) nanorods that nucleate and grow during
metal organic chemical vapor deposition (MOCVD) have proven particularly
effective for pinning vortices Majkic  _et al._ (2017).

The major difficulties in achieving consistent and uniform critical currents
in REBCO tapes containing BZO nanorods are the interplay of many parameters
controlling the deposition process (temperature of the substrate and of the
precursor gases, deposition rate, precursor composition, etc.) and strong
sensitivity of the microstructure to small variations in these parameters.
Even for the same nominal level of Zr additives, significant variations in
nanorod diameter, size distribution, spacing distribution, and angular splay
have been observed. Physical factors controlling these variations remain
poorly understood. For example, the nanorods’ diameter may be a mostly
equilibrium property resulting from the interplay of strain and surface
energies or caused by kinetic effects controlled by surface diffusion of
adatoms and deposition rate.

These complexities have precluded the development of predictive models.
However, making use of the accumulated experimental data sets and possibly
synthesis/kinetic growth simulation data (Monte Carlo or molecular dynamic
simulations, which are also still in an exploratory phase), allows building
ML/AI models to predict pinning landscapes for given synthesis parameters as
described above or, more relevant for commercial application, the prediction
of synthesis parameters for a desired, uniform pinning landscape.

To constitute a complete vortex-pinning genome, a second notable milestone is
using AI to predict $J_{c}$ for a given pinning landscape based solely on data
recognition (disregarding TDGL simulations) and, conversely, predicting the
necessary pinning landscape to produce a desired $J_{c}$. In fact, the latter
cannot be achieved by direct simulations. Typical data sets, both experimental
and simulation-based, contain information on defect structures, critical
currents, and other transport characteristics for a wide range of magnetic
fields and temperatures. Creating an organized database of this information
would enable (i) quickly accessible critical current values for a wide range
of conditions, (ii) an effective mapping of simulation parameters onto
experimental measurements, and (iii) using the data as training sets for AI-
driven predictions of defect structures for desired applications.

Experimentally, microstructures are routinely probed by transmission electron
microscopy (TEM) and, less directly, by x-ray diffraction (XRD). In contrast
to simulation data, which contains all information about pinning landscapes,
the extracted information is usually rather limited, since TEM only allows
imaging of thin slices of the material and only detects relatively large
defects. A full 3D tomography of defect landscapes [cf. Fig. 4(a)] is very
time consuming and expensive, and therefore currently typically infeasible.
The resulting AI/ML models will also allow for a cross-validation of the
simulation-based data with available experimental data on materials properties
in superconductors with different defect microstructures.

Overall, this AI/ML approach will directly reduce the cost and development
time of commercial superconductors and, in particular, accelerate their design
for targeted applications. To estimate the benefit of such an approach, one
can consider, for example, a pinning landscape defined by 9 parameters. Using
traditional interpolation in this 9-dimensional parameter space one would need
to have a certain number of data points per parameter. For the modest case of
15 data points per direction one would need to simulate (measurements are
infeasible) $15^{9}\approx 4\cdot 10^{10}$ pinning landscapes, which –
assuming 15 minutes per simulation on a single GPU – results in a total
simulation time of a million GPU years. This simulation time is beyond current
capabilities, even on the largest supercomputers. However, surrogate ML models
can reduce this to approximately $10^{4}$ simulations, while maintaining the
same resulting accuracy (see seed studies in, e.g., Ref. [Crombecq  _et al._ ,
2011]).

In this section, we mentioned the complications associated with 3D tomographic
imaging of a superconductor’s microstructure to supply complete information
for simulations. In the next section, we detail the limitations of tomographic
imaging and other advanced microscopy techniques, many of which will be
revolutionized by improvements in computational power and the application of
advanced neural networks. This in turn will have a transformative impact on
vortex physics.

###  III.5 Advanced microscopy to better understand vortex-defect interactions

####  III.5.1 Quantitative point-defect spectroscopy

We have discussed the role of point defects in suppressing vortex motion via
weak collective pinning. Notably, the theory of weak collective pinningLarkin
and Ovchinnikov (1979) has attracted significant attention as it can explain
the origin of novel vortex phases, e.g. vortex glass Fisher (1989); Fisher,
Fisher, and Huse (1991) and vortex liquid Nelson (1988) phases, as well as the
associated vortex melting phase transition Brandt (1989); Houghton, Pelcovits,
and Sudbø (1989). It cannot, however, be used to predict $J_{c}$ in single
crystals, whose defect landscape is dominated by point defects. This
limitation is not necessarily reflective of gaps in weak collective pinning
theory itself, but rather the fact that point defect densities are typically
unknown because they are extremely challenging to measure over a broad spatial
range.

Consequently, point defects are the dark matter of materials. Herein lies yet
another transformative opportunity in vortex physics. Developing a technique
to accurately measure point defect profiles and subsequent systematic studies
correlating point defects, $J_{c}(B,T)$, and $S(B,T)$ may lead to recipes for
predictably tuning the properties of superconductors, most directly impacting
crystals and epitaxial materials that lack a significant contributions from
strong pinning centers. The most promising routes for quantitative point
defect microscopy include scanning transmission electron microscopy (STEM),
atom probe tomography (APT), atom electron tomography (AET), and positron
annihilation lifetime spectroscopy (PALS). Here, we primarily focus on STEM
combined with AET, then will introduce APT and PALS as other techniques with
atomic-scale resolution that are relatively untapped opportunities to reveal
structure-property relationships in superconductors.

In scanning transmission electron microscopy (STEM), an imaging electron beam
is transmitted through a thin specimen, such that detectors can construct a
real-space image of the microstructure and collect other diffraction data. In
superconductors, STEM studies have revealed a panoply of defects, including
columnar tracks, defect clusters, dislocations, twin boundaries, grain
boundaries, and stacking faults. These studies can also provide information on
other pertinent microstructural properties, including strained regions that
induce variations in the superconducting order parameter, therefore,
preferential regions for vortices to sit in an otherwise defect-free
landscape. To identify dopants (e.g. BaHfO3 nanoparticles), STEM is also often
performed in conjunction with analytical techniques, such as energy dispersive
x-ray spectroscopy.

To understand the ability of STEM to determine point defect densities in
superconductors, we must first understand what limits the spatial resolution
and throughput. Older STEMs cannot resolve point defects due to imperfections
(aberrations) in the objective lenses and other factors that set the
resolution higher than the wavelength of the imaging beam. Atomic resolution
was finally achieved upon the advent of transformational aberration correction
schemes, which were first successfully demonstrated in the late 1990s and have
been increasingly widely adopted over the past decadeBatson, Dellby, and
Krivanek (2002); Rose (2005); Dahmen  _et al._ (2009); Ophus (2017). In fact,
the spatial resolution of an aberration-corrected STEM has now fallen below
the Bohr radius of $0.53\text{\,}\mathrm{pm}$ Kisielowski  _et al._ (2008);
Erni  _et al._ (2009); Alem  _et al._ (2009); Shibata (2019).

Though point defects can now be imaged in superconductors, it is not
straightforward to determine point defect densities. A single scan captures a
small fraction of the sample, which may not be representative of defect
distributions throughout the entire specimen. Accordingly, low throughput
prevents collecting a sufficiently large dataset to provide a reasonably
quantitative picture of defect concentrations. One of the limiting factors for
throughput is the detector speed, which has recently improved significantly
owing to the development of direct electron detectors such as active pixel
sensors (APS) and hybrid pixel array detectors (PAD). These detectors have
higher quantum efficiency, operate at faster readout speeds, and have a
broader dynamic range than conventional detectors—charge-coupled devices
(CCDs) coupled with scintillators Ophus (2017).

Enabled by fast detectors, the advent of 4D-STEMOphus (2019) is another
recent, major milestone that is a significant step towards determining point
defect densities. Note that 4D-STEM involves collecting a 2D raster scan in
which 2D diffraction data is collected at each scanned pixel, generating a 4D
dataset containing vast microstructural information. In addition to high-speed
direct electron detectors, computational power was prerequisite for 4D-STEM
implementation, in which massive datasets can be produced: see Ref. [Ophus,
2019] for an example in which a single 4D-STEM image recorded in
$164\text{\,}\mathrm{s}$ consumes $420\text{\,}\mathrm{GB}$. Hence, over the
past few years, this has warranted efforts to develop fast image simulation
algorithms Ophus (2017) and schemes to apply deep neural networks to extract
information, such as defect species and locationZiatdinov  _et al._ (2017).
Furthermore, STEMs can be used for electron tomography, in which images that
are collected as the sample is incrementally rotated are combined to create a
3D image of the microstructure.Miao, Ercius, and Billinge (2016)

Aberration correction, high-speed detectors, and the data revolution are
transformative advances that will certainly accelerate progress in
understanding structure-property relationships in superconductors.
Nevertheless, there are more salient impediments to an atomic-scale
understanding of the true sample under study, including artifacts from sample
preparation techniquesSchaffer, Schaffer, and Ramasse (2012) and beam
scattering within thick samples. To remedy the latter, materials are often
deposited onto membranes, though this may not present a representative picture
of the defect landscape when the sample is in a different form (e.g. thicker
and on a different substrate).

Atom probe tomography (APT) is another microscopy technique with atomic-scale
resolution, and it also provides 3D compositional imaging of surface and
buried features. Over the past decade, it has become increasingly popular due
to the development of a commercial local-electrode atom probe (LEAP). For APT,
the specimen must be shaped as a needle and an applied electric field
successively ejects layers of atoms from the surface of the specimen towards a
detector. By means of time-of-flight mass spectroscopy, the detector
progressively collects information on the position and species of each atom,
reconstructing a 3D tomographic image of the specimen that can span
$0.2\text{\times}0.2\text{\times}0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$
with resolution of $0.10.5\text{\,}\mathrm{nm}$ Petersen  _et al._ (2011). As
each atom is individually identifiable, it can provide remarkably revealing
information on the microstructure. Similar to STEM, sample preparation and
data processing are bottlenecks; APT also currently suffers from limited
detection efficiencyKelly and Miller (2007). Furthermore, the analyzed volume
(field of view) is currently too small to be sufficiently representative of
the sample to provide accurate quantitative details on point defect
concentrations. The biggest complication, however, may be that the defect
landscape of the APT specimen, shaped as a needle, may dramatically differ
from the material in the form in which we typically study its electromagnetic
properties.

Lastly, positron annihilation spectroscopy is a hitherto untapped opportunity
to correlate vacancy concentrations with electrical transport properties in
superconductors. This non-destructive technique can determine information
about vacancies and larger voids in a material by bombarding it with positrons
at $50\text{\,}\mathrm{eV}$ to $30\text{\,}\mathrm{keV}$ acceleration
energies,Gidley, Peng, and Vallery (2006); Or  _et al._ (2019) then measuring
the time lapse between the implantation of positrons and emission of
annihilation radiation. Upon implantation, positrons thermalize in $\sim
10\text{\,}\mathrm{ps}$ then either interact with an electron and annihilate
or form a positronium atom (electron-positron pair)Stráský  _et al._ (2018).
Positronium atoms will then ricochet off the walls of voids and eventually
annihilate, releasing a $\gamma$-ray that can be detected with integrated
$\gamma$-ray detectors. The lifetime of the positron can provide information
on void sizes and concentration of vacancies: longer lifetimes correspond to
larger voids and higher vacancy densities.

PALS has been used for decades to sensitively detect vacancies and vacancy
clusters in metals and semiconductorsSchultz and Lynn (1988a); Gidley, Peng,
and Vallery (2006) as well as probe subnanometer, intermolecular voids in
polymersPethrick (1997); Gidley, Peng, and Vallery (2006). Depth profiling is
possible on the nm to the $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
scaleSchultz and Lynn (1988b); Wagner  _et al._ (2018); Peng  _et al._ (2005);
Gidley, Peng, and Vallery (2006) by tuning the positron implantation energy
and, though some systems have beam scanning capabilities enabling lateral
resolution, spatial resolution is generally quite poor due to large beam spot
sizes and positron diffusion. In most systems, the spot size is typically
$\sim$1\text{\,}\mathrm{mm}$$. However, PALS instruments containing
_microprobes_ are capable of spot sizes that are smaller than
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ David  _et al._ (2001); Gigl  _et
al._ (2017). For example, in 2017, Gigl et al.Gigl  _et al._ (2017) developed
a state-of-the-art system with a minimum lateral resolution of
$33\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and maximum scanning range of
$19\text{\times}19\text{\,}\mathrm{mm}$. Regarding speed, the system can scan
an area of $1\text{\times}1\text{\,}\mathrm{mm}$ with a resolution of
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in less than 2 minutes, which is
considered to be an exceptionally short time frameGigl  _et al._ (2017).
Moreover, David et al.David  _et al._ (2001) reported a remarkably small spot
diameter of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in a setup with a short
scanning range of $0.2\text{\times}0.6\text{\,}\mathrm{mm}$. Unfortunately,
further improvements to beam focus may be ineffectual and spatial resolution
comparable to electron microscopy is unreachable. The spatial resolution is
ultimately limited by lateral straggle: the positron diffusion length is
roughly several hundreds of nanometers in a perfect crystal, which limits the
spot size even if the beam focus is improved Schultz and Lynn (1988b). Ongoing
efforts to advance PALS include improving theoretical methods for
interpretation of experimental results, advancing theoretical descriptions of
positron physics (states, thermalization, and trapping), incorporating sample
stages that allow tuning sample environmental conditions (e.g. temperature,
biasing), and improving the efficiency of beam moderators (which convert
polychromatic positron beams to monochromatic beams)Schultz and Lynn (1988b).

####  III.5.2 Cryogenic microstructural analysis for accurate determination of
structure-property relationships

Accurately correlating the formation of different vortex structures and
intricacies of vortex-defect interactions with electromagnetic response is not
trivial. Typically, conventional microscopy is performed under conditions that
differ from a material’s actual working environment: structural
characterization of superconductors is routinely conducted at room temperature
whereas accessing the superconducting regime requires cryogenic temperatures
and is probed using electromagnetic stimuli. Yet we know that temperature
changes significantly impact the microstructure, causing strain-induced phase
separation and altering defects such as dislocations. Electromagnetic stimuli
may similarly impact the defect landscape. Hence, another transformative
opportunity in vortex physics is cryogenic structural characterization of
superconductors under the influence of electromagnetic stimuli, which requires
advances in microscopy.

Scanning transmission electron microscopy combined with spectroscopic analysis
is one of the most informative methods of gathering structural and chemical
analysis at the atomic-scale. Accurate determination of structure-property
relationships require in-situ property measurements conducted concommitently
with microscopy. Recent, rapid advances in in-situ transmission electron
microscopy have been fueled by the introduction of a variety of commercial in-
situ sample holders that allow for electrical biasing, heating, magnetic
response, and mechanical deformation of nanomaterialsMcDowell, Jungjohann, and
Celano (2018). These new capabilities have accelerated progress in a variety
of fields, including battery electrochemistry, liquid-phase materials growth,
bias-induced solid-state transformations in e.g. resistive switching devices
for memory and neuromorphic applications, gas-phase reactions and catalysis,
solid-state chemical transformations e.g at interfaces between semiconductors
and metallic contacts, and mechanical behavior McDowell, Jungjohann, and
Celano (2018). For in-situ TEM to be beneficial to superconductors, samples
must be cooled to cryogenic temperatures and studied under the influence of
magnetic fields.

Developed in the 1960s, liquid helium cooled stages have been used to study
superconductors, solidified gases, and magnetic domains Goodge  _et al._
(2020), initially without the benefit of aberration-corrected systems with
atomic-scale resolution. More recently, cryogenic STEM with atomic scale
resolution has been used to study quantum materials, including low-temperature
spin states. However, these studies have been limited to a single temperature
that was above the boiling point of the choice cryogen (liquid helium or
liquid nitrogen)Goodge  _et al._ (2020), due to thermal load, whereas variable
temperature capabilities are requisite for probing phase transitions and the
effects of thermal energy. To this end, there has recently been a push to
develop advanced sample holders with stable temperature controlGoodge  _et
al._ (2020); Hum (2019). One of the most promising efforts is led by
Hummingbird Precision Machine, a company that is developing a double-tilt,
cryo-electrical biasing holder for TEMs that allows samples to be concurrently
cooled to liquid helium temperatures and electrically biased, while undergoing
atomic-scale structural imaging Hum (2019). Because of such industry
involvement in the development and commercialization of cryogenic sample
holders and, more generally, the rapid pace of in-situ TEM (e.g. the number of
in-situ TEM papers doubled between 2010 and 2012Taheri  _et al._ (2016)) we
expect to see large advancements in this identified challenge over the next
several years.

####  III.5.3 Cross-sectional imaging of vortex structures

In Sec. I, we discussed how competition between pinning forces, vortex
elasticity, and current-induced forces results in complicated vortex
structures, such as double-kinks, half-loops, and staircases. Typically, we
may conjecture which structures have formed based on the microstructure and
applied field orientation. Subsequent correlations are made between the
presumed structures and the magnetization or transport results, which may be
suggestive of a specific vortex phase. However, without direct proof of the
structures, we cannot unequivocally correlate distinct excitations with
specific vortex phases. For example, in a study of a NbSe2 crystal containing
columnar defects tilted 30∘ from the c-axis, magnetization results evinced
glassy behavior when the field was aligned with the c-axisEley  _et al._
(2018). As these conditions are likely to produce vortex staircases, the
question arose whether (and why) vortex staircases would create a vortex glass
phase.

Direct imaging of vortex-defect interactions, in a way that captures the
vortex structure overlaid on the atomic-scale structure, would enable
unambiguous determination of the phases produced by specific vortex
excitations. Accordingly, development of advanced microscopy techniques that
can produce cross sectional images is another transformative opportunity in
vortex physics. In this section, we summarize common techniques for imaging
superconducting vortices, detail their limitations, and describe the features
of an advanced instrument that could accelerate progress in understanding and
designing materials with predetermined vortex phases.

Figure 11: Lorentz Transmission Electron Microscopy (LTEM) image of vortices
in Bi2Sr2CaCu2O8+δ, irradiated to induce columnar defects. Trapped vortices
can be distinguished from free ones, based on their shape and contrast (lower
contrast for vortices trapped in columnar defects). Plan-view LTEM images have
provided useful information on vortex dynamics, though the 3D vortex structure
within the bulk is hidden from view. Reprinted with permission from Ref.
[Kamimura  _et al._ , 2002]. Copyright 2002, _The Physical Society of Japan_.
TEM images of (b) a heavy-ion-irradiated NbSe2 crystalEley  _et al._ (2018)
and (c) a BaZrO3-doped (Y0.77Gd0.23)Ba2Cu3Oy film grown by Miura et al. Miura
_et al._ (2013a). Permission to use TEM image in (c) granted by M. Miura. Red
line is a cartoon of how a vortex might wind through the disorder landscape.
Advanced microscopy techniques designed to capture this structure is a
transformative opportunity in vortex physics.

Lorentz TEM (LTEM), which exploits the Aharonov-Bohm effect to capture
magnetic contrast, was first used by Tonomura to image superconducting
vortices Tonomura (2006) and has played a major role in identifying new
materials that host exotic magnetic phases. However, in LTEM, the objective
lens serves the dual purpose of applying a field and observing the response of
the specimen, and is therefore limited to plan-view imaging. That is, for out-
of-plane magnetic fields applied to a thin film, the technique can only image
magnetic contrast across the film’s surface—such that the vortex structure
itself and interactions with defects within the bulk are out-of-view.

Building upon Tonomura’s initial work, HitachiHarada  _et al._ (2008);
Kawasaki  _et al._ (2000) developed a unique, specialized system containing a
multipole magnet that can apply fields up to $50\text{\,}\mathrm{mT}$ at
various orientations with respect to the sample. Though this system still
produces plan-view images, rather than cross-sectional images (revealing the
full vortex structure), variations in the contrast of the imaged vortex
section has provided remarkable evidence of vortex pinning and useful
information on vortex-defect interactions.Kamimura  _et al._ (2002); Harada
(2013); Tonomura  _et al._ (2002, 2001) For example, Fig. 11(a) shows a LTEM
image in which a vortex trapped within a columnar defect can be identified by
its shape and contrast, compared to untrapped vortices.

The most promising technique for directly imaging vortex-defect interactions
may be differential phase contrast microscopy (DPC)Lubk and Zweck (2015).
Conducted in a TEM, DPC is one of the best tools for quantitatively imaging
nanoscale magnetic structures. In a TEM, an illuminating electron beam is
deflected by electromagnetic fields within a material. DPC microscopy
leverages these deflections to directly image electric and magnetic fields
within materials at atomic resolutionDekkers and de Lang (1974); Chapman  _et
al._ (1978). Consequently, scanning the beam (STEM) produces spatial maps of
nanoscale magnetic field contrast to complement the atomic-scale structural
information resolved by a transmitting beam. Accordingly, STEM-DPC is an
invaluable tool in nanomagnetism research, used to image magnetic domainsLee
_et al._ (2017); Chen  _et al._ (2018) and canted structures such as
skyrmionsMcVitie  _et al._ (2018); Matsumoto  _et al._ (2016); Schneider  _et
al._ (2018). Most notably, it is one of few techniques that can unequivocally
identify new magnetic phases and exotic magnetic quasiparticles in real-space.
More generally, it can also image nanoscale electric fieldsMüller  _et al._
(2014); Hachtel, Idrobo, and Chi (2018); Shibata  _et al._ (2012, 2017);
Yücelen, Lazić, and Bosch (2018) in materials and devices. To image vortex-
defect structures in an STEM capable of DPC, the sample stage would need to be
cryogenically cooled and the chamber should contain a magnet. Complications
will include designing the system in a way in which the magnetic field does
not significantly distort the beam.

##  IV Summary and Outlook

In this Perspective, we have highlighted the pivotal role that vortices play
in superconductors and how improving our ability to control vortex dynamics
will have an immediate impact on a range of applications. Herein we discussed
major open questions in vortex physics, which include the following:

  * •

How do thermal and quantum vortex creep depend on material parameters and how
can we efficiently consider creep in predictive simulations?

  * •

What is the highest attainable critical current $J_{c}$?

  * •

How do we optimize vortex pinning in quantum circuits and controllably exploit
vortices in certain schemes for topological computing?

  * •

Given the multitude of variables that govern $J_{c}$, what computational
methods can improve the efficacy of the critical-current-by-design approach?

  * •

What is the relationship between $J_{c}$ and point defect densities as well as
vortex structures and vortex phases?

To answer these and other identified questions, we delineated five major
categories of near-term transformative opportunities: The first involves
applying recent advances in analytical and computational methods to model
vortex creep, and performing more extensive experimental investigations into
quantum creep. Second, we discussed how critical currents higher than the
current record of 30% $J_{d}$ may be obtained by implementing a combination of
core pinning and magnetic pinning. This is a promising route for dramatic
advancements in large-scale applications—achieving higher currents densities
enables smaller motors and generators as well as higher field magnets. Third,
we noted that vortices do not only hamper large-scale applications, but also
induce losses in nanoscale quantum circuits. Though shielding circuits has
proven effective in minimizing vortex formation, quantum senors may require
exposure to the environment, necessitating a better understanding of vortex
dynamics in circuits. Furthermore, vortices are desirable for use in quantum
information applications, in which case we must study how to manipulate single
flux lines to implement braiding and entanglement of Majorana Bound States.
Fourth, the recent advent of high-performance computational tools to study
vortex matter numerically has pushed us to the verge of predicting a
superconductor’s electrical transport properties based on the material and
microstructure. However, the quest to automatically tailor a defect landscape
for specific applications requires considering a fairly high-dimensional
parameter space. To enable an effective mapping between simulations and
experiments and manage the multitude of variables, we propose to apply self-
adjusting machine learning algorithms that use neural networks. Fifth and
finally, to accurately determine structure-property relationships, we need to
experimentally measure and routinely consider point defect densities, which
are challenging to determine. We therefore highlighted the prevailing
microscopy techniques for point defect measurements, which include 4D-STEM and
positron annihilation lifetime spectroscopy.

###### Acknowledgements.

S.E. acknowledges support from the National Science Foundation DMR-1905909.
A.G. was supported by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, Materials Sciences and Engineering Division. R.W.
acknowledges funding support from the Heidelberger Akademie der Wissenschaften
through its WIN initiative (8. Teilprogramm).

## data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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