# Microscopic Examination of SRF-quality Nb Films through Local Nonlinear
Microwave Response

Chung-Yang Wang  Quantum Materials Center, Department of Physics, University
of Maryland, College Park, Maryland 20742, USA  Carlota Pereira  CERN,
European Organization for Nuclear Research, 1211 Geneva 23, Switzerland
Stewart Leith  CERN, European Organization for Nuclear Research, 1211 Geneva
23, Switzerland  Guillaume Rosaz  CERN, European Organization for Nuclear
Research, 1211 Geneva 23, Switzerland  Steven M. Anlage  Quantum Materials
Center, Department of Physics, University of Maryland, College Park, Maryland
20742, USA

###### Abstract

The performance of superconducting radio-frequency (SRF) cavities is sometimes
limited by local defects. To investigate the RF properties of these local
defects, a near-field magnetic microwave microscope is employed. Local third
harmonic response ($P_{3f}$) and its temperature-dependence and RF power-
dependence are measured for one Nb/Cu film grown by Direct Current Magnetron
Sputtering (DCMS) and six Nb/Cu films grown by High Power Impulse Magnetron
Sputtering (HiPIMS) with systematic variation of deposition conditions. Five
out of the six HiPIMS Nb/Cu films show a strong third harmonic response that
is likely coming from a low-$T_{c}$ surface defect with a transition
temperature between 6.3 K and 6.8 K, suggesting that this defect is a generic
feature of air-exposed HiPIMS Nb/Cu films. One possible origin of such a
defect is grain boundaries hosting a low-$T_{c}$ impurity such as oxidized Nb.
Time-Dependent Ginzburg-Landau (TDGL) simulations are performed to better
understand the measured third harmonic response. The simulation results show
that the third harmonic response of RF vortex nucleation caused by surface
defects can qualitatively explain the experimental data. Moreover, the density
of surface defects that nucleate RF vortices, and how deep an RF vortex
travels through these surface defects, can be extracted qualitatively from
third harmonic response measurements. From the point of view of these two
properties, the best Nb/Cu film for SRF applications can be identified.

##  I Introduction

In high-energy physics, there is continued interest in building next-
generation particle accelerators (for example, the International Linear
Collider, ILC) using bulk Nb superconducting radio-frequency (SRF) cavities
Padamsee (2017); Bambade  _et al._ (2019). For the ILC, around 10000 SRF
cavities will be built.

The quality of an SRF cavity is typically quantified by its quality factor
(Q-factor) as a function of the accelerating gradient for the particle beam.
Real-world materials are not perfect. The Q-factors of SRF cavities are
usually below their theoretical predictions. In particular, as the
accelerating gradient, and hence the RF magnetic field on the Nb surfaces,
becomes strong, the Q-factor drops significantly (this is called the Q-slope)
Ciovati (2006); Gurevich (2006a, 2012, 2017). Such a Q-slope phenomenon limits
the RF field supported by the SRF cavities, which then limits the performance
of the particle accelerator. Besides the Q-slope phenomenon, quenches are also
frequently observed in many SRF cavities Champion  _et al._ (2009); Bao and
Guo (2019); Posen  _et al._ (2015a). One reason for a quench is that a
superconductor is locally heated up to exceed its critical temperature and
loses superconductivity. Both the Q-slope and defect-nucleated quenches
indicate that the performance of SRF cavities is limited by breakdown events
below the theoretically predicted intrinsic critical field of the
superconductor Gurevich (2006a, 2012); Padamsee (2017). These breakdowns are
sometimes caused by uncontrolled local defects Kneisel  _et al._ (2015);
Antoine (2019); Weingarten (2023). Candidates of defects in SRF cavities
include oxides Yoon  _et al._ (2008); Proslier  _et al._ (2008); Romanenko and
Schuster (2017); Semione  _et al._ (2019, 2021), impurities Russo (2007);
Kharitonov  _et al._ (2012), grain boundaries Carlson  _et al._ (2021); Lee
_et al._ (2007); Polyanskii  _et al._ (2011); Köszegi  _et al._ (2017); Wang
_et al._ (2018, 2022a); Lee  _et al._ (2020), dislocations Wang  _et al._
(2022a); Romanenko and Padamsee (2010); Bieler  _et al._ (2010), surface
roughness Wang  _et al._ (2022b); Ries  _et al._ (2020), etc. To make high
Q-factor SRF cavities that operate to high accelerating gradients, it is
necessary to understand these defects, in particular their influence on the RF
properties of SRF cavities. Therefore, there is a need to understand in detail
the RF properties of these local defects.

In SRF material science, various kinds of techniques have been developed to
characterize SRF cavities and SRF materials. For example, researchers
routinely measure the Q-factor Dhakal  _et al._ (2014) and residual resistance
Gonnella  _et al._ (2014) of SRF cavities. However, it is costly and time-
consuming to fabricate and measure an entire cavity. As a result, many
measurements are performed on coupon samples of SRF materials, including
measurements of RF quench field Posen  _et al._ (2015a); Kleindienst  _et al._
(2015); Keckert  _et al._ (2021a) and surface resistance Kleindienst  _et al._
(2015); Arzeo  _et al._ (2022); Keckert  _et al._ (2021b, a).

Another quantity of interest is the vortex penetration field because SRF
cavities are expected to operate best in the Meissner state (vortex-free) to
avoid dissipation due to vortex motion James  _et al._ (2012); Posen  _et al._
(2015a); Ries  _et al._ (2020); Carlson  _et al._ (2021); Antoine  _et al._
(2019). Superconductors show strong nonlinearity in the presence of vortices
and show relatively weak nonlinearity in the vortex-free Meissner state. The
nonlinear electrodynamic response arises when properties of the superconductor
(such as the superfluid density) become time-dependent during the RF cycle.
One manifestation of nonlinearity is that the superconductor creates response
currents to the stimulation at frequencies other than the driving frequency.
Utilizing the connection between vortices and nonlinearity, the vortex
penetration field can be determined by measuring the third harmonic response
of a superconductor subjected to a time-harmonic magnetic field Lamura  _et
al._ (2009). In particular, the vortex penetration field of thin films and
multilayer structures have been studied with such alternating current (AC)
(kHz regime) third harmonic response magnetometry Antoine  _et al._ (2010,
2011, 2013); Katyan and Antoine (2015); Aburas  _et al._ (2017); Antoine  _et
al._ (2019); Ito  _et al._ (2019, 2020).

The techniques described above (Q-factor, residual resistance, RF quench
field, surface resistance, vortex penetration field, etc.) help physicists to
characterize the global properties of SRF materials. However, none of them can
directly study the local RF properties of SRF materials.

Motivated by the need to study RF properties of local defects, we successfully
built and operated a near-field magnetic microwave microscope using a scanned
loop (the original version) Lee  _et al._ (2000); Lee and Anlage (2003); Lee
_et al._ (2005a, b); Mircea  _et al._ (2009) as well as a magnetic writer from
a magnetic recording hard-disk drive (the microwave microscope adopted in this
work) Tai  _et al._ (2011, 2012, 2014a, 2014b, 2015); Oripov  _et al._ (2019).
The spatial resolution of the local probe of the microwave microscope adopted
in this work is in the sub-micron scale, and the frequency is in the range of
several GHz. Since the presence of a vortex is closely related to the third
harmonic response, we measure the third harmonic response and its dependence
on temperature and RF field amplitude. With this microwave microscope, the
local RF properties of SRF materials are explored by measuring locally-
generated third harmonic response.

Bulk Nb is the standard choice for fabricating SRF cavities. The main reason
is that Nb has the highest critical temperature ($T_{c}=9.3$ K) and the
highest first critical field ($B_{c1}=180$ mT) of all the pure metals at
ambient pressure. Besides bulk Nb, there are some candidate alternative
materials for SRF applications Valente-Feliciano (2016), including Nb film on
Cu Arbet-Engels  _et al._ (2001); Russo (2007); James  _et al._ (2012); Ries
_et al._ (2020); Sublet  _et al._ (2015); Arzeo  _et al._ (2022); Roach  _et
al._ (2012); Rosaz  _et al._ (2022), Nb3Sn on bulk Nb substrate Lee  _et al._
(2020); Posen  _et al._ (2015b, a); Posen and Hall (2017); Trenikhina  _et
al._ (2017); Ilyina  _et al._ (2019); Carlson  _et al._ (2021), multilayer
structure (superconductor-insulator-superconductor structures, for instance)
Gurevich (2006b); Kubo  _et al._ (2014); Kubo (2016); Wang  _et al._ (2022b);
Gurevich (2017); Antoine  _et al._ (2010, 2019), etc. The potential benefits
of using materials other than bulk Nb would be a higher $T_{c}$ and a
potentially higher critical field $B_{c}$. Here we focus on Nb films on Cu.

The development of the deposition of Nb films onto Cu cavities has a long
history Calatroni (2006). In particular, the first Nb/Cu cavities were
produced at CERN in the early 1980s Benvenuti  _et al._ (1984). Motivations
for Nb thin film technology for SRF applications include better thermal
stability (Nb/Cu cavities allow operation at 4 K, rather than 2K, because of
the superior thermal conductivity of Cu) and reducing material cost (high
purity Nb costs around 40 times more than Cu). The performance of bulk Nb
cavities is approaching the intrinsic limit of the material. On the contrary,
Nb/Cu cavities typically suffer from serious Q-slope problems Aull  _et al._
(2015); Palmieri and Vaglio (2015), which limits their use in high
accelerating fields. Solving the Q-slope problem in Nb/Cu cavities is
essential for making them competitive for use in high-field accelerators.

In this work, we use our near-field magnetic microwave microscope to study
local RF properties (on sub-micron scales, at several GHz) of SRF-quality
Nb/Cu films produced at CERN. In particular, surface defects of these Nb/Cu
films are the main focus of this work.

The outline of this paper is as follows: In Sec. II, we describe the
experimental setup. In Sec. III, we show the experimental results of these
Nb/Cu films and focus on surface defect signals. In Sec. IV, we perform Time-
Dependent Ginzburg-Landau (TDGL) simulations to better understand the
experimental results, and compare these Nb/Cu films. In Sec. V, we summarize
the experimental results of these Nb/Cu films and identify the best one for
SRF application.

##  II Experimental setup

Figure 1: Schematic of experiment setup. The microwave source (MW source)
sends a signal of $P_{RF}sin^{2}(\omega t)$ to the magnetic writer (Probe),
which then generates $B_{RF}sin(\omega t)$ acting on the sample surface
locally. The sample response is collected by the same magnetic writer, and the
third Fourier component $P_{3f}sin^{2}(3\omega t)$ is studied. The dashed line
box represents the cryostat.

The setup of our near-field magnetic microwave microscope (identical to that
described in Ref. Oripov  _et al._ (2019)) is described in this section. A
schematic of the setup is shown in Fig. 1.

The heart of our microwave microscope is the magnetic writer head (provided by
Seagate Technology) that is used in conventional hard-disk drives. The central
part of the magnetic writer head is basically a solenoid that generates a
localized RF magnetic field. The solenoid is in the sub-micron scale, which
sets the spatial resolution of our microscope. In the setup, a Seagate
magnetic writer head is attached to a cryogenic XYZ positioner (sub-micron
spatial resolution) and used in a scanning probe microscope fashion. The probe
is in contact with the sample during the third harmonic measurement. However,
the surfaces of the probe and sample are not perfectly flat, resulting in a
finite probe-sample separation $h$, estimated to be less than 1 micron.

The microwave source signal $P_{RF}sin^{2}(\omega t)$ is sent to the probe
(magnetic writer head) by its built-in and highly-engineered transmission
line. The probe then produces a local (sub-micron scale) RF magnetic field
$B_{RF}sin(\omega t)$ acting on the sample surface. The superconducting sample
then generates a screening current on the surface in an effort to maintain the
Meissner state. This screening current generates a response magnetic field
that is coupled back to the same probe, creating a propagating signal whose
third harmonic component $P_{3f}sin^{2}(3\omega t)$ is measured by a spectrum
analyzer at room temperature. $P_{3f}$ is measured because it arises from both
the nonlinear Meissner effect Groll  _et al._ (2010); Makita  _et al._ (2022),
and when a vortex penetrates the sample surface and forms a vortex semi-loop
Gurevich and Ciovati (2008) (as discussed in Sec. I), either due to an
intrinsic mechanism or local defects (weak spot for a vortex to penetrate).

$P_{3f}$ measurements of superconductor nonlinear response show tremendous
dynamic range, often more than 50 dB Tai  _et al._ (2011, 2012, 2014b, 2015);
Oripov  _et al._ (2019). The excellent instrumental nonlinear background of
our measurements ($\sim$ -155 dBm) allows for very sensitive measurements of
superconductor nonlinearity and its variation with temperature, driving RF
power, location, and probe-sample separation. Note that measurements are
recorded in dBm and later converted to linear power for further study.

To improve the signal-to-noise ratio, microwave filters are installed as
follows. Low pass filters are installed between the microwave source and the
probe to block the unwanted harmonic signals generated by the microwave
source. High pass filters are installed between the probe and the spectrum
analyzer to block the fundamental input frequency signal from reaching the
spectrum analyzer and producing unwanted nonlinear signals.

Measurements are performed with a variety of fixed input frequencies between
1.1 GHz and 2.2 GHz, while varying temperature and applied RF field amplitude.
No external DC magnetic field is applied. The measured residual DC field near
the sample at low temperatures is around 35 $\mu$T, as measured by a cryogenic
3-axis magnetometer.

The base temperature for a sample in the cryostat is around 3.5 K. The sample
and the thermometer are both directly mounted on the cold plate to ensure good
thermalization.

##  III Experimental results

Figure 2: Photo of the seven Nb films on one common Cu substrate. The samples
are prepared at CERN.

In this work, we study seven Nb films deposited on one common Cu substrate, as
shown in Fig. 2. One of the samples is prepared by Direct Current Magnetron
Sputtering (DCMS) with zero bias, and the sample thickness is around 3.5
$\mu$m. The other six samples are prepared by High Power Impulse Magnetron
Sputtering (HiPIMS), with bias from 0 V to 125 V, and the sample thickness
ranges from 1.15 $\mu$m to 1.5 $\mu$m. The preparation of the seven Nb/Cu
films is discussed in Appendix A. In the following, the HiPIMS 25 V bias Nb/Cu
sample is discussed in detail, and the results of all the seven Nb/Cu samples
are summarized and compared in Table 2 and in Table 3.

Figure 3: Representative data for $P_{3f}$ as a function of temperature for
the HiPIMS 25 V bias Nb/Cu sample. The input frequency is 1.86 GHz and the
input power is 2 dBm. The blue dots are the raw data, and the red curve is the
$P_{3f}$ averaged over 0.05 K range bins. Inset: enlargement of the figure
above 7 K.

Fig. 3 shows the representative data for the third harmonic response power
$P_{3f}$ as a function of temperature at a fixed location on the HiPIMS 25 V
bias Nb/Cu sample. A representative measurement protocol is as follows. The
sample is warmed up to 10 K (above $T_{c}$), and then the microwave source is
turned on with fixed input frequency ($\omega/{2\pi}$=1.86 GHz) and input
power ($P_{RF}$=+2 dBm), and then $P_{3f}$ is measured as the sample is
gradually cooled down to 3.5 K. In other words, the surface of the sample
experiences a fixed RF field $B_{RF}sin(\omega t)$ in a sub-micron scale area
during the cooldown process from 10 K to 3.5 K.

The measured $P_{3f}(T)$ in Fig. 3 can be decomposed into three segments: the
region above 9.1 K, the region below 6.7 K, and the region in between. The
magnetic writer head (the probe of our microwave microscope) itself has
temperature-independent nonlinearity. $P_{3f}$ above 9.1 K comes from this
probe background and is indeed temperature-independent. A transition around
9.1 K can be seen in the inset of Fig. 3. This transition around 9.1 K comes
from the intrinsic nonlinear response of the Nb film. The strongest $P_{3f}$
signal here shows up below 6.7 K. In the following, such a $P_{3f}$ onset
temperature is called the $P_{3f}$ transition temperature and is denoted as
$T_{c}^{P_{3f}}$. Compared to the signal around 9.1 K, the onset around 6.7 K
is dramatic. Such a signal suggests that some mechanism shows up at and below
6.7 K that produces strong nonlinearity. In summary, $P_{3f}(T)$ in Fig. 3 can
be understood as the combination of three different sources of nonlinearity:
temperature-independent probe background, that related to intrinsic Nb
response, and mechanisms showing up at and below 6.7 K. The mechanisms leading
to strong $P_{3f}$ below 6.7 K are extrinsic and are likely due to surface
defects. Note that $P_{3f}$ below 6.7 K is much stronger than the intrinsic Nb
signal around 9.1 K, suggesting that our local $P_{3f}$ measurement is
sensitive to surface defects.

Figure 4: Measured linear power scale $P_{3f}(T)$ from strong (red) to weak
(purple) input power for the HiPIMS 25 V bias Nb/Cu sample. The input
frequency is 1.86 GHz. Compared to the raw data shown in Fig. 3, here the
probe background is subtracted. Note that input power (in dBm scale) is
proportional to $B_{RF}^{2}$, and hence $B_{RF}^{red}=1.78B_{RF}^{purple}$.

Since the main objective of this work is investigating RF properties of
surface defects, the strong $P_{3f}$ below 6.7 K is the main focus in the
following.

To further study the nature of $P_{3f}$ below 6.7 K, $P_{3f}(T)$ for various
input powers $P_{RF}$ (and hence various applied RF field amplitudes $B_{RF}$)
are measured. For each measurement, the sample is warmed up to 10 K and then
cooled down to 3.5 K while experiencing an applied RF field with fixed input
frequency and input power. Since the measured $P_{3f}$ is the combination of
probe background and sample contribution, the probe background (probe
background is obtained by averaging the magnitude of $P_{3f}(T)$ between 9.5 K
and 10 K) is subtracted from the total signal to isolate the sample signal.
The process is repeated six times, each time with a different input power. The
results of the six $P_{3f}(T)$ of different input powers/RF field amplitudes
are shown in the linear format in Fig. 4.

In Fig. 4, all the six $P_{3f}(T)$ exhibit a two-peak feature, and their
$T_{c}^{P_{3f}}$ are consistently around 6.7 K. The two-peak feature suggests
that there might be two distinct mechanisms, or features, of nonlinearity. For
both peaks, as the RF field amplitude increases (purple to red in Fig. 4), the
$P_{3f}(T)$ maximum increases; in addition, the $P_{3f}(T)$ maximum and the
$P_{3f}(T)$ low-temperature end both show up at a lower temperature. These
features of $P_{3f}(T)$ are listed in Table 1. We will see that the four
features of $P_{3f}(T)$ listed in Table 1 are the key features of all the
nonlinear data in the sense that they show up in all the Nb/Cu films
measurement results (see Fig. 4 and Fig. 6) and also in numerical simulations
of superconductor nonlinear response (see Fig. 9, Fig. 13, and Fig. 14).

Table 1: The four key features of $P_{3f}(T)$ in our measurements of Nb/Cu
films.

As discussed in Sec. I, $P_{3f}$ is strong in the presence of magnetic
vortices. In Sec. IV, we explore this idea further and show that the four key
features of $P_{3f}(T)$ (Table 1) can be understood by vortex nucleation
(either due to intrinsic mechanisms or extrinsic surface defects) and the
vortex penetration field $B_{vortex}^{RF}(T)$.

Figure 5: Statistics of one-dimensional scan for
$P_{3f}(T=5.7\quad\textrm{K},\quad P=-5\quad\textrm{dBm},\quad
f=1.86\quad\textrm{GHz})$ for the HiPIMS 25 V bias Nb/Cu sample. The
coefficient of variation (the ratio of the standard deviation to the mean) is
0.08. Inset: spatial distribution of this one-dimensional scan.

So far the measurements are taken at one single location on the surface of the
sample. How about the situation at other locations? In particular, does the
defect signal (the strong $P_{3f}$ below 6.7 K) show up at other locations as
well? To answer this question, a one-dimensional (1D) scan is performed for a
40 $\mu$m range with a step size of 0.2 $\mu$m. Temperature and input power
are chosen to be T = 5.7 K and P = -5 dBm, and our goal is to determine the
homogeneity of the signal from the lower temperature peak of the blue curve in
Fig. 4. The 1D scan result is shown in Fig. 5. According to this scanning
result, the $P_{3f}$ signal from the defect is quite consistent and uniform at
the $\mu$m-scale (coefficient of variation = 0.08). One possible explanation
of such uniformity is that the size of one single defect is at the nm-scale,
and hence the $P_{3f}$ signal of our sub-micron scale measurements comes from
the contributions of multiple defects. A measurement technique with nm-scale
spatial resolution is required to explore the RF properties of one single
defect (scan across one single grain boundary, for example).

Figure 6:  $P_{3f}(T)$ for two input powers for the HiPIMS 75 V bias Nb/Cu
sample with an input frequency of 1.98 GHz (a) and for the HiPIMS 125 V bias
Nb/Cu sample with an input frequency of 1.66 GHz (b). Table 2: Summary of the
defect signal (red) and the Nb signal (blue) of the seven Nb/Cu samples.

So far only the HiPIMS 25 V bias Nb/Cu sample has been discussed. $P_{3f}$ and
its temperature dependence and input power dependence are studied for the
other six Nb/Cu samples in the same manner as the HiPIMS 25 V bias Nb/Cu
sample. The HiPIMS 25 V bias Nb/Cu sample $P_{3f}(T)$ shows the two-peak
feature, suggesting that the measured $P_{3f}(T)$ is the superposition of two
defect signals. On the other hand, for some of the Nb/Cu samples (75 V bias,
100 V bias, 125 V bias), the $P_{3f}(T)$ defect signal shows a single-peak
structure, suggesting that the measured $P_{3f}(T)$ captures only one defect
signal. Fig. 6 shows the representative $P_{3f}(T)$ for the 75 V bias sample
(Fig. 6 (a)) and the 125 V bias sample (Fig. 6 (b)). For both Fig. 6 (a) and
(b), $P_{3f}(T)$ displays Features 1, 2, and 3 (Table 1), suggesting that
these features are quite universal. Although we have no access to the
temperature regime below 3.5 K, Feature 4 (Table 1) is likely to be present
based on $P_{3f}(T)$ above 3.5 K.

Besides these common features, Fig. 6 (a) and Fig. 6 (b) do have a qualitative
difference. In Fig. 6 (a), a stronger RF field amplitude leads to a stronger
$P_{3f}$ (the red curve is above the blue curve) for all temperatures. In Fig.
6 (b), on the contrary, for $T>4.4$ K, a stronger RF field amplitude leads to
a weaker $P_{3f}$ (the red curve is below the blue curve). In Sec. IV.4, we
will show that such a difference could be related to how deep an RF vortex
semi-loop penetrates into a sample through a surface defect.

The results of the defect signal and the Nb signal of all seven Nb/Cu samples
are summarized in Table 2. For the six HiPIMS Nb/Cu samples, it is quite
universal that $P_{3f}$ measurements reveal the intrinsic Nb signal around 9 K
and the extrinsic defect signal at low temperatures. Specifically, the defect
signal with $T_{c}^{P_{3f}}$ between 6.3 K and 6.8 K is observed for five out
of the six HiPIMS Nb/Cu samples, suggesting that such a defect is a generic
feature for these HiPIMS Nb/Cu samples. Moreover, the defect signals are
always much stronger than the intrinsic Nb signal around 9 K, like the
situation shown in Fig. 3. Scanning measurements are performed for three
HiPIMS samples (25 V bias sample, 50 V bias sample and 125 V bias sample), and
the results show that the defect signals are quite uniform at the $\mu$m-scale
for all three samples. Data for the HiPIMS 50 V bias sample and the 100 V bias
sample are shown in Appendix B.

One may wonder if hysteresis shows up in $P_{3f}(T)$ measurements. To check
hysteresis, we compare $P_{3f}(T)$ in a warm-up process to $P_{3f}(T)$ in a
cool-down process for a sample showing a defect signal (Appendix C). As shown
in Appendix C, $P_{3f}(T)$ does not exhibit a clear hysteresis.

##  IV Discussion

###  IV.1 Introduction to numerical simulations

In our measurements, the applied field is a localized RF magnetic field
instead of a uniform DC magnetic field. In addition, the configuration of the
RF magnetic field produced by the probe is non-uniform but is similar to the
field produced by a point dipole. In other words, the situation is different
from the case of “a constant DC magnetic field parallel to the sample
surface”. As a result, it is required to study the third harmonic response
more carefully.

The Time-Dependent Ginzburg-Landau (TDGL) model is widely used for studying
vortex behavior in superconductors Oripov and Anlage (2020); Wang  _et al._
(2022b); Carlson  _et al._ (2021); Pack  _et al._ (2020); Kato (1999); Lara
_et al._ (2015); Dobrovolskiy  _et al._ (2020); Hernández and Domínguez
(2008). In particular, vortices (see Sec. IV.2) and the proximity effect (see
Sec. IV.3) are incorporated naturally in TDGL simulations and hence TDGL is a
good tool for SRF material science. To better understand our data, numerical
simulations of the TDGL equations are performed. The full TDGL equations must
be solved in this case because the superconductor is subjected to a time-
dependent and inhomogeneous RF magnetic field. We do not assume or impose any
spatial symmetries in the model, and solve Maxwell’s equations for the dipole
in free space above the superconductor, as well as inside the superconductor
Oripov and Anlage (2020). The TDGL equations solved are the same as those
discussed in Ref. Oripov and Anlage (2020).

In the simulations, the smooth and flat superconducting sample occupies the
$z<0$ region, and the magnetic writer probe is approximated to be a pointlike
magnetic dipole with a sinusoidal time-dependent magnetic moment
$(M_{dp}sin(\omega t),0,0)$ (namely an RF magnetic dipole pointing in the x
direction) whose frequency is 1.7 GHz ($\omega/{2\pi}$=1.7 GHz). The RF dipole
locates at $(0,0,h_{dp})$ with $h_{dp}$=400 nm. That is, the RF magnetic
dipole is above the superconducting sample and parallel to the surface of the
sample. Since the magnetic field produced by the RF dipole is non-uniform, the
peak RF magnetic field amplitude experienced by the superconductor is
specified and is denoted as $B_{pk}$. Here $(x,y,z)=(0,0,0)$ is the location
in the sample that experiences the strongest field, and hence $B_{pk}$ is the
RF field amplitude at $(x,y,z)=(0,0,0)$. Because the RF field is localized,
nontrivial dynamics of the sample (vortex nucleation, for example) would show
up only in the region that is underneath the RF magnetic dipole (namely near
$(x,y,z)=(0,0,0)$), while the region that is far away from the RF magnetic
dipole would be in the vortex-free Meissner state.

In the simulations, TDGL is used to calculate the time evolution of the order
parameter and the vector potential as the superconducting sample is stimulated
by the time-dependent RF field produced by the horizontal point dipole above
it. (There is no DC magnetic field in the simulations.) With the order
parameter and the vector potential, the screening current and hence the
magnetic field associated with that screening current (response of the sample)
can be calculated. The response magnetic field is calculated at the location
of the point dipole and it is assumed that this time-varying magnetic field
induces a voltage wave that propagates up to the spectrum analyzer at room
temperature. $\sqrt{P_{3f}}$ is proportional to the third Fourier component of
the magnetic field (generated by the screening current) at the dipole
location.

Most TDGL treatments assume a 2D sample and fields that are uniform in the
third dimension. This oversimplifies the problem. It creates “artificial
features that extend uniformly” in the third dimension and thus creates
infinitely long vortices in the superconductor. Our approach (multi-domain 3D
simulation) does not make such unrealistic assumptions. In addition, our
experiment and model examine the properties of magnetic vortex semi-loops,
which are thought to be the generic types of RF vortex excitations created at
the surface of SRF cavities Gurevich and Ciovati (2008).

In the following, we first consider the case of a defect-free bulk Nb
($T_{c}=9.3$ K) (Sec. IV.2) for developing central concepts relevant to
nonlinear response and then propose a surface defect model (Sec. IV.3) whose
$P_{3f}$ shares common features with the experimental results (the four key
features in Table 1). After that, we consider the surface defect model with
various defect heights (Sec. IV.4) and show that $P_{3f}(T)$ can reveal how
deep an RF vortex semi-loop penetrates into a sample through a surface defect.
We then study another surface defect model (Sec. IV.5) and compare its
$P_{3f}(T)$ with the $P_{3f}(T)$ of the first surface defect model and show
that $P_{3f}(T)$ can reveal how many RF vortex semi-loops are nucleated by
surface defects in each half of the RF cycle. With physical insights gained by
these simulations, the Nb/Cu films can be compared further (Sec. IV.6), and
the best Nb/Cu film for SRF applications can be identified.

Parameters for all the TDGL simulations are given in Appendix D.

###  IV.2 Calculated bulk Nb nonlinear response

Figure 7:  Time-domain analysis for the dynamics of RF vortex semi-loops in
bulk Nb. A time-dependent magnetic moment (RF dipole) is 400 nm above the
superconductor surface, pointing in the x-direction, and producing
$B_{pk}=61.6$ mT at T=8.23 K. (a)-(f) show the square of the normalized order
parameter ($|\psi/\psi_{\infty}|^{2}$) on the XZ plane cross-section at
different times during the first half of the RF period. The black region is
where $|\psi/\psi_{\infty}|^{2}<0.03$. (g) shows $|\psi/\psi_{\infty}|^{2}$ on
the YZ plane cross-section at the same moment as (d). (h) RF field at
$(x,y,z)=(0,0,0)$ versus time during the first half of the RF period. (i)
Phase change for a closed contour (on the YZ plane) that is large enough to
enclose the entire nontrivial region. Red crosses in (h) and (i) correspond to
snapshots (a)-(g). Figure 8:  (a) TDGL simulation result of $P_{3f}(T)$ for a
fixed RF field amplitude ($B_{pk}=61.6$ mT) imposed by a point dipole source
for bulk Nb. From left to right, $P_{3f}$ is weak at low temperatures (below
8.1 K), then increases with temperature (between 8.1 K and 8.8 K), and drops
with the temperature at high temperatures (above 8.8 K). (b)-(e) show
$|\psi/\psi_{\infty}|^{2}$ on the YZ plane cross-section at 8.04 K, 8.14 K,
8.42 K, and 8.60 K, respectively. The black region is where
$|\psi/\psi_{\infty}|^{2}<0.03$. These snapshots are taken at $\omega
t=0.6\pi$. (f) shows $\Delta\theta/2\pi$ versus time for the four
temperatures. Figure 9: (a) TDGL simulation result of $P_{3f}(T)$ for three
different RF field amplitudes for bulk Nb. (b) Schematic of the vortex
penetration field $B_{vortex}^{RF}(T)$ (the black curve) and the temperature
range of $P_{3f}(T)$’s bell-shaped structure for the three RF field amplitudes
(the three colorful horizontal lines). The y-axis is the RF field amplitude.
Note that (a) and (b) share the same horizontal axis and the same color-coded
RF field amplitudes.

Unlike a DC vortex whose behavior shows no time dependence, an RF vortex shows
nontrivial dynamics, and should be examined in a time-domain manner. Here we
demonstrate the time-domain analysis (focusing on the dynamics of RF vortices)
for a specific RF field amplitude ($B_{pk}=61.6$ mT) and a specific
temperature (8.23 K). (Material parameters of Nb are used in the defect-free
bulk Nb simulations. See Appendix D.)

The dynamics of RF vortex semi-loops for bulk Nb during the first half of an
RF cycle (frequency=1.7 GHz, period=$5.88*10^{-10}$s) is shown in Fig. 7, for
a fixed RF field amplitude ($B_{pk}=61.6$ mT) and a fixed temperature (8.23
K). Fig. 7 (a)-(g) show the space and time dependence of the square of the
normalized order parameter ($|\psi/\psi_{\infty}|^{2}$) (here 1 means full
superconductivity and 0 means no superconductivity); the black region is where
$|\psi/\psi_{\infty}|^{2}<0.03$. Since the order parameter is suppressed
significantly at the center of a vortex core, a vortex can be visualized by
tracking the black region. Here $\psi_{\infty}=\psi_{\infty}(T)$ is the value
of the order parameter deep inside bulk Nb at temperature T.

In the early stage of the RF cycle, there is no RF vortex (Fig. 7 (a) and
(b)), and then an RF vortex semi-loop that is parallel to the direction of the
RF dipole (which points in the x direction) shows up (Fig. 7 (c), (d), (e) and
(g)). The RF vortex semi-loop disappears later in the RF cycle (Fig. 7 (f)).

Besides examining the spatial distribution of the order parameter, another
signature of vortices is the phase of the order parameter. Because Ginzburg-
Landau theory is based on the existence of a single-valued complex
superconducting order parameter ($\psi=|\psi|e^{i\theta}$), the phase $\theta$
must change by integral multiples of $2\pi$ in making a closed contour (see
equation (4.45) in Tinkham (2004)), namely

$\oint ds\cdot\nabla\theta=2\pi N,$

where N is a positive or negative integer, or zero. The integral
$\frac{1}{2\pi}\oint ds\cdot\nabla\theta$ is quantized, and corresponds to the
number of vortices enclosed by the closed contour.

Fig. 7 (i) shows the value of the integral $\frac{1}{2\pi}\oint
ds\cdot\nabla\theta$ (namely $\Delta\theta/2\pi$) as a function of time. The
contour is on the YZ plane and is large enough to enclose the entire
nontrivial region. Note that Fig. 7 (h) and Fig. 7 (i) share a common
horizontal axis. Based on Fig. 7 (i), there are no vortices at the moments of
(a), (b) and (f), and there is one vortex at the moments of (c), (d) and (e),
which agrees with the order parameter analysis (Fig. 7 (a)-(g)).

The time-domain analysis described here (space and time dependence of the
order parameter (Fig. 7 (a)-(g)) and $\Delta\theta/2\pi$ (Fig. 7 (i))) is
applied to all TDGL simulations whenever we check whether or not there are RF
vortex semi-loops.

Equipped with the picture of RF vortex nucleation, now let’s move on to the
resulting $P_{3f}$. The simulation result of $P_{3f}(T)$ for a fixed RF field
amplitude ($B_{pk}=61.6$ mT) for bulk Nb is shown in Fig. 8 (a). The bell-
shaped structure $P_{3f}(T)$ in Fig. 8 (a) can be decomposed into three
segments (separated by the two dashed vertical black lines) and can be
understood with the vortex penetration field $B_{vortex}^{RF}(T)$ (see Fig. 9)
and the strength of superconductivity. An RF vortex semi-loop shows up when
$B_{pk}>B_{vortex}^{RF}(T)$. Below 8.1 K, $B_{vortex}^{RF}(T)>B_{pk}$ and
hence the entire bulk Nb is in the vortex-free Meissner state (see Fig. 8 (b)
and the purple curve in (f)), whose nonlinear response is weak. As temperature
increases, $B_{vortex}^{RF}(T)$ decreases and hence $B_{pk}$ would be greater
than $B_{vortex}^{RF}(T)$ at a certain temperature depending on the strength
of the RF stimulus. In this simulation ($B_{pk}=61.6$ mT), from the full time-
domain simulation (see the discussion for Fig. 7) one finds that there is one
RF vortex semi-loop (underneath the RF magnetic dipole) that penetrates the
surface of the bulk Nb when the temperature is around 8.14 K (see Fig. 8 (c)
and the blue curve in (f)). Roughly speaking, this implies that
$B_{vortex}^{RF}(T=8.14\textrm{K})\approx 61.6$ mT. $B_{vortex}^{RF}(T)$
drops/vortex nucleation is favorable as temperature increases, and indeed the
second RF vortex semi-loop shows up around 8.6 K (see Fig. 8 (e) and the red
curve in (f)) and thus $P_{3f}$ increases with temperature between 8.1 K and
8.8 K. Besides examining the order parameter (Fig. 8 (b)-(e)), Fig. 8 (f) also
shows how the vortex number changes with temperature.

The nonlinear response of the superconductor is determined not only by the
number of vortices (as described above in the language of
$B_{vortex}^{RF}(T)$) but also by the strength of superconductivity. As the
temperature approaches the transition temperature of a superconductor, its
superconductivity and hence nonlinear response becomes weak. Such a
temperature dependence leads to the decreasing tail of $P_{3f}(T)$ above 8.8 K
in Fig. 8 (a).

The simulation indicates that $P_{3f}$ and the presence of RF vortex semi-
loops are indeed closely related. Specifically, $P_{3f}$ is weak as the bulk
Nb is in the vortex-free Meissner state (below 8.1 K) and is strong in the
presence of RF vortex semi-loops (above 8.1 K).

Fig. 9(a) summarizes the simulation results of $P_{3f}(T)$ for three different
RF field amplitudes for bulk Nb. For all three RF field amplitudes, $P_{3f}$
is weak at low temperatures ($B_{pk}<B_{vortex}^{RF}(T)$, vortex-free Meissner
state), arises at high temperatures ($B_{pk}>B_{vortex}^{RF}(T)$, RF vortex
semi-loops), and then drops with temperature as the temperature is near the
critical temperature. For the red curve, the first vortex semi-loop shows up
around 8.14 K, which implies $B_{vortex}^{RF}(T=8.14\textrm{K})\approx
61.6\textrm{mT}$; for the blue curve, the first vortex semi-loop shows up
around 8.84 K, which implies $B_{vortex}^{RF}(T=8.84\textrm{K})\approx
39.2\textrm{mT}$. Fig. 9(b) illustrates $B_{vortex}^{RF}(T)$ and the RF field
amplitude dependence of the temperature range of $P_{3f}(T)$’s bell-shaped
structure. In this RF field amplitude-temperature phase diagram, the vortex-
free Meissner state occupies the region below $B_{vortex}^{RF}(T)$ and RF
vortex semi-loops show up in the region above $B_{vortex}^{RF}(T)$. It is
clear that the $P_{3f}(T)$’s bell-shaped structure extends to lower
temperatures as the RF field amplitude becomes stronger (from the blue to the
green to the red in Fig. 9(a) and (b)) because of the temperature dependence
of $B_{vortex}^{RF}(T)$, and this explains Feature 4 in Table 1. Note that the
four key features of $P_{3f}(T)$ (Table 1) are clearly observed in Fig. 9(a)
(except that the onset temperature of $P_{3f}(T)$ is equal to but not below
9.3 K since this simulation is for a defect-free bulk Nb), suggesting that
these key features are signatures of RF vortex nucleation.

Equipped with the intuition of $P_{3f}$ and RF vortex semi-loops and their
temperature dependence and RF field amplitude dependence for the defect-free
bulk Nb, now let’s move on to the case with surface defects, which is the main
focus of this paper.

###  IV.3 Simulation of vortex nucleation in grain boundaries

Figure 10: Sketch of the side view (XZ plane) of the grain boundary model. The
origin is marked by a blue dot. The RF dipole is represented by the blue
arrow. The schematic is not to scale ($h_{dp}$=400 nm and $h_{GB}$=200 nm).
Since the physics around the origin plays a dominant role, the region far away
from the origin is set to be defect-free bulk Nb to reduce computational time.
As a surface defect model, the grain boundary has a finite height $h_{GB}$.
The top view of the region indicated by the red dashed line is shown in Fig.
11 (b), and the top view of the region indicated by the green dashed line is
shown in Fig. 18. Figure 11: A top view schematic of the grain boundary model.
(a) Illustration of Nb grains and grain boundaries, with red being Nb grains
and blue being grain boundaries filled with impurities. The realization of
this illustration used in the TDGL simulations is shown in (b). (b) The
distribution of critical temperature on the XY plane around the origin
(corresponds to the region indicated by the red dashed line in Fig. 10), with
red being Nb and blue being low-$T_{c}$ impurity with
$T_{c}^{\textrm{impurity}}=4$ K. $(x,y,z)=(0,0,0)$ is at the center. The RF
dipole locates at $(x,y,z)=(0,0,400\textrm{nm})$ and points in the x
direction. The white dashed curve indicates the grain boundary that is roughly
parallel to the x direction. Because the model works well only for the region
around the center, only the screening current inside the white dashed
rectangle is collected when calculating $P_{3f}$. (c) A snapshot of the
distribution of normalized order parameter $|\psi/\psi_{\infty}|$ obtained by
a TDGL simulation with the temperature being 5.4 K (higher than
$T_{c}^{\textrm{impurity}}$) and $B_{pk}=56.6$ mT. This snapshot is taken at
the end of an RF cycle, namely when the RF field drops to zero ($\omega
t=2\pi$ and hence $B_{RF}sin(\omega t)=0$). The normalized order parameter of
the dark blue region is around 0.15. Figure 12: TDGL simulation result of
$P_{3f}(T)$ for the grain boundary model shown in Fig. 10 and Fig. 11. Here
$h_{GB}=200$ nm and $B_{pk}=56.6$ mT. Inset: enlargement of the figure above
8.1 K. Figure 13: TDGL simulation result of $P_{3f}(T)$ for two different RF
field amplitudes for the grain boundary model shown in Fig. 10 and Fig. 11.
Here $h_{GB}=200$ nm.

It is known that surface defects can serve as weak spots for RF vortex
nucleation. The dynamics of RF vortices penetrating a sample surface through
surface defects can be analyzed from two aspects: how many RF vortices are
nucleated by surface defects, and how deep do RF vortices travel into a sample
through surface defects in half an RF cycle? These two ideas could apply to RF
vortex nucleation by various surface defects (grain boundaries, dislocations,
etc.). In the following, we consider RF vortex nucleation in the case of grain
boundaries, but the lessons of the features of $P_{3f}$ and RF vortex
nucleation should be generic.

As discussed in Sec. I, various kinds of surface defects could exist in SRF
materials. One possible scenario of surface defects is that the grain
boundaries of Nb are filled with the oxide of Nb Sung  _et al._ (2013). The
oxidation of Nb when it is exposed to air is a well-known phenomenon Yoon  _et
al._ (2008); Proslier  _et al._ (2008); Romanenko and Schuster (2017); Semione
_et al._ (2019, 2021); Halbritter (1987). In such oxidation, oxygen forms a
solid solution in Nb and produces materials with critical temperatures below
the bulk $T_{c}$ of pure Nb (9.3 K) DeSorbo (1963); Koch  _et al._ (1974);
Weingarten (2023). Nb samples with higher oxygen content tend to have a lower
critical temperature. For instance, $T_{c}$ drops to around 7.33 K for 2%
oxygen content, and drops to around 6.13 K for 3.5% oxygen content Koch  _et
al._ (1974).

Motivated by the oxidation of Nb in grain boundaries, here we model the oxide
of Nb as a low-$T_{c}$ material (impurity phase) and consider a surface defect
model that the grain boundaries of Nb are filled with the low-$T_{c}$
material. Such grain boundaries might serve as weak spots for vortex
nucleation. As shown later in this section, the proximity effect is active in
the grain boundaries. Here we consider one possible toy model realization of
“Nb grain boundaries filled with low-$T_{c}$ material”. Of course, the toy
model (a grain boundary model) considered here is just one possible scenario
of surface defects that might be able to qualitatively explain the
experimental results.

The RF dipole locates at $(0,0,h_{dp})$ and hence vortex semi-loops first show
up near $(x,y,z)=(0,0,0)$. Therefore, the physics around the origin plays a
dominant role. As an approximation, surface defects (Nb grain boundaries
filled with low-$T_{c}$ material) are introduced near the origin (illustrated
in Fig. 10), while the region far away from the origin is set to be defect-
free bulk Nb to reduce computational time. As a result, only the region around
the origin characterizes the grain boundary scenario accurately, and thus the
screening current is collected only from such an accurate region when
calculating $P_{3f}$.

Fig. 10 shows the side view of the grain boundary model. As a surface defect
model, the grain boundary extends (in the z direction) from the sample’s
surface to a finite depth $h_{GB}$ (grain boundary height), and the region
below is set to be bulk Nb. ($z=0$: sample surface; $0>z>-h_{GB}$: defect
whose XY cross-section is shown in Fig. 11 (b); $z<-h_{GB}$: bulk Nb). Here
the grain boundary height $h_{GB}$ is set to be 200 nm.

The side view of the grain boundary model is shown in Fig. 10, and the top
view ($(x,y,z=0)$) is shown in Fig. 11. Fig. 11 (a) is an illustration of an
Nb surface containing Nb grains (red) and grain boundaries filled with a
low-$T_{c}$ impurity (blue). The realization used in the TDGL simulations is
shown in Fig. 11 (b). Fig. 11 (b) is a top view of the sample’s critical
temperature distribution around the origin of the grain boundary model: the
red region means Nb with $T_{c}=9.3$ K, and the blue region means low-$T_{c}$
impurity with $T_{c}^{\textrm{impurity}}=4$ K. (Parameters of the grain
boundary model are given in Appendix D.) The origin is at the center of Fig.
11 (b). The geometry is intentionally asymmetric in both the x direction and
the y direction to prevent symmetry-induced artifacts. A top view of the
sample’s critical temperature distribution with a broader scope (containing
the defect region together with the bulk Nb region) is shown in Appendix E.

An RF vortex semi-loop that nucleates in the sample tends to be parallel to
the direction of the RF dipole, which points in the x direction. Therefore, as
$B_{pk}>B_{vortex}^{RF}(T)$, RF vortex semi-loops show up in the grain
boundaries that are roughly parallel to the x direction. There is one grain
boundary in Fig. 11 (b) that is roughly parallel to the x direction and it is
marked by a white dashed curve. Grain boundaries are well-characterized inside
the white dashed rectangle (the accurate region), and hence the screening
current contribution to the signal recovered at the location of the dipole is
collected only from inside the white dashed rectangle when calculating
$P_{3f}$.

It is worth mentioning that the proximity effect shows up naturally in TDGL
simulations. Fig. 11 (c) shows a snapshot of the distribution of normalized
order parameter $|\psi/\psi_{\infty}|$ (here 1 means full superconductivity
and 0 means no superconductivity) obtained by a TDGL simulation with the
temperature being 5.4 K and $B_{pk}=56.6$ mT. This snapshot is taken at the
end of an RF cycle, namely when the RF field drops to zero ($\omega t=2\pi$
and hence $B_{RF}sin(\omega t)=0$). Due to the proximity effect, the
normalized order parameter of the dark blue region is around 0.15 but not
zero, even though the temperature (5.4 K) is higher than
$T_{c}^{\textrm{impurity}}$ (4 K). As a result, a grain boundary filled with a
low-$T_{c}$ impurity can host RF vortices even for
$T>T_{c}^{\textrm{impurity}}$.

Fig. 12 shows the simulation result of $P_{3f}(T)$ for the grain boundary
model shown in Fig. 10 (side view of the model) and Fig. 11 (top view of the
model). Compared to the $P_{3f}$ around 9 K, the $P_{3f}$ between 4.5 K and 6
K is much stronger. In other words, in the presence of surface defects,
$P_{3f}$ generated by surface defects is much stronger than the intrinsic
$P_{3f}$ of Nb. In the following, we focus on the $P_{3f}$ generated by
surface defects.

Fig. 13 shows the simulation result of $P_{3f}(T)$ for two different RF field
amplitudes for the grain boundary model. At low temperatures, the sample is in
the Meissner state and $P_{3f}$ is weak. As temperature increases, RF vortex
semi-loops nucleate in the grain boundary marked by the white dashed curve in
Fig. 11 (b) and result in strong $P_{3f}$. (The existence of RF vortex semi-
loops is verified by examining the order parameter in a time-domain manner as
described in Fig. 7.) This can be interpreted as $B_{pk}>B_{vortex}^{RF}(T)$,
where $B_{vortex}^{RF}(T)$ is the vortex penetration field of the region
around that specific grain boundary. Note that RF vortex semi-loops show up in
the grain boundary, indicating that the grain boundary serves as the weak spot
for RF vortex nucleation.

Here the upper onset temperature of $P_{3f}$ is around 5.9 K. The grain
boundary model is a mixture of the $T_{c}^{\textrm{Nb}}=9.3$ K Nb and the
$T_{c}^{\textrm{impurity}}=4$ K impurity, and hence $T_{c}^{P_{3f}}$ is
between 4 K and 9.3 K. Of course, the numerical value of $T_{c}^{P_{3f}}$
depends on how the Nb and the impurity are distributed. Our objective with
this model is not to propose a specific microstructure of the sample, but to
illustrate the generic nonlinear properties of a proximity-coupled defective
region of the sample.

Similar to the case of bulk Nb (Fig. 9(a)), $P_{3f}(T)$ in Fig. 13 also
exhibits the four key features (Table 1). Therefore, the ideas illustrated in
Fig. 9(b) could describe the physics of the grain boundary model, with
$T_{c}^{P_{3f}}$ being 5.9 K but not 9.3 K and $B_{vortex}^{RF}(T)$ being the
vortex penetration field of the region around the specific grain boundary, but
not a bulk property. The fact that experimental data (Fig. 4 and Fig. 6) and
TDGL simulation of the grain boundary model (Fig. 13) both exhibit the four
key features (Table 1) and a $T_{c}^{P_{3f}}$ below 9.3 K suggests that “RF
vortex semi-loops nucleate in grain boundaries full of low-$T_{c}$ impurity”
is indeed one of the possible mechanisms of the observed $P_{3f}$ signals for
the HiPIMS Nb/Cu samples.

###  IV.4 Effect of height of grain boundaries on $P_{3f}(T)$

Figure 14: TDGL simulation result of $P_{3f}(T)$ for two different RF field
amplitudes for the grain boundary model shown in Fig. 10 and Fig. 11 for
$h_{GB}=150$ nm (a) and for $h_{GB}=200$ nm (b) (identical to Fig. 13). (c)
Crossing temperature $T^{*}$ as a function of grain boundary height $h_{GB}$.

If the RF field amplitude is not very strong, RF vortex semi-loops would stay
in grain boundaries ($0>z>-h_{GB}$) instead of entering the bulk Nb regions.
Therefore, $h_{GB}$ determines how deep an RF vortex semi-loop penetrates into
a sample through a surface defect. In Sec. IV.3, the grain boundary height
$h_{GB}$ is set to be 200 nm. Here we consider the effect of varying $h_{GB}$,
with everything else being the same as described in Sec. IV.3.

Fig. 14 (a) and (b) show the results for $h_{GB}=150$ nm and $h_{GB}=200$ nm,
respectively. Fig. 14 (a) is similar to Fig. 6 (a), in the sense that a
stronger RF field amplitude leads to a stronger $P_{3f}$ (the red curve is
above the blue curve) for all temperatures; Fig. 14 (b) is similar to Fig. 6
(b), in the sense that $P_{3f}(T)$ shows a “crossing” effect: a stronger RF
field amplitude leads to a weaker $P_{3f}$ (the red curve is below the blue
curve) for temperatures close to $T_{c}^{P_{3f}}$. The temperature where the
red curve ($P_{3f}(T)$ with a strong RF field amplitude) and the blue curve
($P_{3f}(T)$ with a weak RF field amplitude) cross is denoted as $T^{*}$. For
Fig. 14 (a), there is no crossing and hence $T^{*}=T_{c}^{P_{3f}}=5.9$ K. For
Fig. 14 (b), $T^{*}=5.27\textrm{K}<T_{c}^{P_{3f}}=5.9\textrm{K}$. Fig. 14 (c)
shows how crossing temperature $T^{*}$ changes with grain boundary height
$h_{GB}$. For a shallow grain boundary (small $h_{GB}$), there is no crossing
and hence $T^{*}=T_{c}^{P_{3f}}$. The crossing shows up when the grain
boundary is beyond a critical depth. As grain boundary height becomes larger,
the crossing effect becomes more significant ($T^{*}$ becomes smaller, which
means that the temperature window that a stronger RF field amplitude leads to
a weaker $P_{3f}$ becomes larger) and eventually tends to saturate.

The crossing effect can be understood as follows. In experiments/simulations,
$P_{3f}$ is collected by the magnetic writer/at the RF dipole location, which
is at $z=h_{dp}$ (above the sample surface). For a weak RF field amplitude,
the RF vortex semi-loop in the grain boundary stays close to the sample’s
surface ($z=0$). As the RF field amplitude increases, the RF vortex semi-loop
in the grain boundary is pushed toward the bottom of the grain boundary
($z=-h_{GB}$)(see Appendix F), which means that the RF vortex semi-loop is
farther away from the magnetic writer/the RF dipole location ($z=h_{dp}$), and
hence the measured $P_{3f}$ becomes weaker. Such a phenomenon shows up only
when the RF vortex semi-loop in the grain boundary can be pushed far away from
the sample surface. For a shallow grain boundary, the RF vortex semi-loop
always stays just below the sample surface instead of penetrating deep into
the sample, and hence $P_{3f}$ doesn’t decrease as the RF field amplitude
increases (no crossing effect).

The crossing effect is quantified by $T^{*}$, which could be related to how
deep an RF vortex semi-loop penetrates into a sample through a surface defect
(denoted as $h_{\textrm{penetration}}^{\textrm{defect}}$).

###  IV.5 Simulation of a defect model with two grain boundaries

Figure 15:  A grain boundary model containing two grain boundaries that are
roughly parallel to the x direction. (a) The distribution of critical
temperature on the XY plane around the origin, with red being Nb and blue
being low-$T_{c}$ impurity with $T_{c}^{\textrm{impurity}}=4$ K.
$(x,y,z)=(0,0,0)$ is at the center. The RF dipole locates at
$(x,y,z)=(0,0,400\textrm{nm})$, which is above the center of the image, and
points in the x direction. The two white dashed curves indicate the grain
boundaries (the top grain boundary and the bottom grain boundary) that are
roughly parallel to the x direction. Because the model works well only for the
region around the center, only the screening current inside the white dashed
circle is collected when calculating $P_{3f}$. (b) TDGL simulation result of
$P_{3f}(T)$ for the grain boundary model shown in (a). Here $h_{GB}=240$ nm,
and $B_{pk}=56.6$ mT.

In the grain boundary model discussed in Sec. IV.3 and Sec. IV.4, there is
only one single grain boundary underneath and roughly parallel to the RF
dipole, and thus RF vortex semi-loops nucleate in one single grain boundary
and $P_{3f}(T)$ shows a single-peak feature. Such a scenario corresponds to
the case that the sample’s grain boundary density is low. For a sample whose
grain boundary density is high, it can be modeled as a grain boundary model
that contains two grain boundaries underneath and roughly parallel to the RF
dipole.

Here we consider a grain boundary model that contains two grain boundaries
that are near the origin and roughly parallel to the x direction. The basic
setting of the model is the same as described in Sec. IV.3. The only
difference is how the Nb and the impurity are distributed horizontally, as
shown in Fig. 15 (a). Fig. 15 (a) shows the top view of the critical
temperature distribution and Fig. 15 (b) shows the TDGL simulation result of
$P_{3f}(T)$ for this model.

$P_{3f}(T)$ in Fig. 15 (b) exhibits a two-peak feature, which can be
understood as follows. At low temperatures, the sample is in the Meissner
state and $P_{3f}$ is weak. As temperature increases, around 4.93 K an RF
vortex semi-loop nucleates in the top grain boundary and results in the lower
temperature peak, and then around 5.67 K another RF vortex semi-loop nucleates
in the bottom grain boundary and results in the higher temperature peak. The
nucleation of the two RF vortex semi-loops is verified by monitoring
$\Delta\theta/2\pi$ (the same analysis as shown in Fig. 8). Note that the two-
peak feature of the $P_{3f}(T)$ in Fig. 15 (b) (simulation) is also observed
in Fig. 4 (measurement).

In this model, RF vortex semi-loops nucleate in both grain boundaries and thus
result in the two-peak feature of $P_{3f}(T)$. On the contrary, in Sec. IV.3
and Sec. IV.4, there is only one RF vortex semi-loop nucleate in one grain
boundary and thus results in the single-peak feature of $P_{3f}(T)$. In other
words, the number of $P_{3f}(T)$ peaks is an indication of the density of
surface defects that nucleate RF vortex semi-loops.

###  IV.6 Nb/Cu samples comparison

Table 3: Comparison of the seven Nb/Cu samples. The first column shows
qualitative assignments of the density of surface defects that nucleate RF
vortex semi-loops ($\rho^{\textrm{defect}}$) and the second column shows
qualitative assignments of how deep an RF vortex semi-loop penetrates into a
sample through a surface defect
($h_{\textrm{penetration}}^{\textrm{defect}}$).

Larger values of $P_{3f}$ are closely related to the presence of RF vortex
semi-loops. By studying $P_{3f}$, we investigate surface defects that nucleate
RF vortex semi-loops, which are the kinds of surface defects that are closely
related to the RF performance of SRF cavities. Surface defects that don’t
nucleate RF vortex semi-loops are beyond consideration here. For example, if a
grain boundary is too narrow (much smaller than the coherence length of a
sample) to nucleate an RF vortex semi-loop, it would be less likely to produce
large amounts of $P_{3f}$. Such a grain boundary is likely to be harmless to
SRF applications. In the following, surface defects refer to the kind of
surface defects that nucleate RF vortex semi-loops.

Equipped with the simulation results, $P_{3f}(T)$ of surface defects can be
analyzed further. First, the number of $P_{3f}(T)$ peaks can be related to the
density of surface defects ($\rho^{\textrm{defect}}$). Single-peak $P_{3f}(T)$
corresponds to a low density of surface defects, and two-peak $P_{3f}(T)$
corresponds to a high density of surface defects. Second, the crossing effect
of $P_{3f}(T)$ can be related to how deep an RF vortex semi-loop penetrates
into a sample through a surface defect
($h_{\textrm{penetration}}^{\textrm{defect}}$). $T^{*}=T_{c}^{P_{3f}}$ (no
crossing) corresponds to a small $h_{\textrm{penetration}}^{\textrm{defect}}$
(shallow), and $T^{*}<T_{c}^{P_{3f}}$ (crossing) corresponds to a large
$h_{\textrm{penetration}}^{\textrm{defect}}$ (deep).

The assignment of qualitative surface defect properties of all seven Nb/Cu
samples are summarized in Table 3. The first column shows an estimate for the
density of surface defects ($\rho^{\textrm{defect}}$) and the second column
shows how deep an RF vortex semi-loop penetrates into a sample through a
surface defect ($h_{\textrm{penetration}}^{\textrm{defect}}$). RF vortex
penetration through surface defects is one of the main enemies of SRF
applications. To achieve good SRF performance, the surface defect density
should be low and the defect should be shallow. From the point of view of
these two properties, the HiPIMS 75 V bias Nb/Cu sample is the best sample
among the five HiPIMS Nb/Cu samples that have non-zero voltage bias.

In the numerical simulations of this work, surface defects that nucleate RF
vortex semi-loops are modeled as grain boundaries. Since $P_{3f}$ is
associated with the behavior of RF vortex semi-loops, it is likely that the
ideas of $\rho^{\textrm{defect}}$ (how many RF vortex semi-loops nucleated by
surface defects) and $h_{\textrm{penetration}}^{\textrm{defect}}$ (how deep RF
vortex semi-loops travel) could apply to surface defects other than grain
boundaries (dislocation tangles, for example).

##  V Conclusion

In this work, we study seven Nb/Cu films that are candidates for SRF
applications. Local $P_{3f}$ measurements reveal surface defects with $P_{3f}$
onset temperatures between 6.3 K and 6.8 K for five out of the six HiPIMS
Nb/Cu samples, indicating that such defects are a generic feature of these
air-exposed HiPIMS Nb/Cu films. $P_{3f}$ coming from the low-$T_{c}$ surface
defect is much stronger than the intrinsic Nb response around 9 K, suggesting
that our local $P_{3f}$ measurement is sensitive to surface defects. With the
capability of $\mu$m-scale scanning, it is found that such a defect is quite
uniform in space on the $\mu$m-scale.

TDGL simulations are performed to analyze the experimental results further. In
particular, the simulations suggest that the density of surface defects that
nucleate RF vortices and how deep RF vortices travel through these surface
defects can be extracted qualitatively from our local $P_{3f}$ measurements.
From the point of view of these two properties, the HiPIMS 75 V bias Nb/Cu
sample is the best sample for SRF applications.

##  VI Acknowledgement

The authors would like to thank Javier Guzman from Seagate Technology for
providing magnetic write heads. C.Y.W. would like to thank Bakhrom Oripov and
Jingnan Cai for helpful discussions. This work is funded by the U.S.
Department of Energy/High Energy Physics through grant No. DESC0017931 and the
Maryland Quantum Materials Center.

##  Appendix A Sample preparation

Table 4: Sample preparation parameters of the seven Nb/Cu films studied in
this paper.

Here we discuss how the seven Nb/Cu films studied in this paper are prepared.

The substrate used for the Nb coatings is a 2 mm thick, oxygen-free electronic
(OFE) copper disk measuring 75 mm in diameter. Prior to coating, the substrate
disk is degreased using commercial detergent. The sample is then chemically
polished using a mixture of sulfamic acid
($\textrm{H}_{\textrm{3}}\textrm{NSO}_{\textrm{3}}$, 5 g/L), hydrogen peroxide
($\textrm{H}_{\textrm{2}}\textrm{O}_{\textrm{2}}$, 5% vol.), n-butanol (5%
vol.) and ammonium citrate (1 g/L) heated up at 72${}^{0}\textrm{C}$ for 20
minutes. After polishing, the disk is rinsed with sulfamic acid to remove the
build-up of native oxide and cleaned with de-ionized water and ultra-pure
ethanol.

The Cu substrate is mounted on an ultra-high vacuum (UHV) stainless steel
chamber equipped with a rotatable shutter to expose in turn the areas to be
coated, and the chamber is then connected to a sputtering system. Both
assemblies are performed inside an ISO5 cleanroom, and the sputtering
apparatus is described in detail in Rosaz  _et al._ (2022). The entire system
is transported to the coating bench where it is coupled to the pumping group
and gas injection lines, and pumped down to about $1*10^{-7}$ mbar. The
pumping group and the sputtering system undergo a 48-hour bakeout at
200${}^{0}\textrm{C}$, during which a 4-hour activation of the Non-Evaporable
Getter (NEG) pump is performed. The temperature of the UHV chamber is
maintained at 150${}^{0}\textrm{C}$ until the start of the coating. After
cooling down, the system reaches a base pressure around $9.3*10^{-10}$ mbar.
Ultra-pure krypton (99.998%) is injected into the system until a process
pressure of $2.3*10^{-3}$ mbar is reached. The seven coatings are then
performed according to the deposition parameters outlined in Table 4. One of
the samples is prepared by Direct Current Magnetron Sputtering (DCMS) with
zero bias, and the coating thickness is around 3.5 $\mu$m. The other six
samples are prepared by High Power Impulse Magnetron Sputtering (HiPIMS), with
bias voltages ranging from 0 V to -125 V, with coating thicknesses ranging
from 1.15 $\mu$m to 1.5 $\mu$m.

During the coating process, the cavity temperature was monitored with an
infrared thermal sensor (OMEGA OS100-SOFT) and kept constant at
150${}^{0}\textrm{C}$. The HiPIMS plasma discharge was maintained using a
pulsed power supply (Huettinger TruPlasma HighPulse 4006) and the negative
bias voltage was applied to the samples using a DC power supply (TruPlasma
Bias 3018). The DCMS discharge was maintained using a Huettinger Truplasma
3005 power supply. The discharge and bias voltages and currents were monitored
throughout the entire coating process using voltage (Tektronix P6015A) and
current (Pearson current monitor 301×) probes whose signals are recorded by a
digital oscilloscope (Picoscope 2000). After the coating, the samples were
cooled down to room temperature, after which the chamber was vented with dry
air. The Nb layer thickness is measured by X-ray fluorescence via the
attenuation method.

##  Appendix B Data for the HiPIMS 50 V bias and 100 V bias Nb/Cu sample

Figure 16:  (a) and (b) show $P_{3f}(T)$ for the HiPIMS 50 V bias Nb/Cu sample
with an input frequency of 2.16 GHz in a strong input power regime and in a
weak input power regime, respectively. (c) shows $P_{3f}(T)$ for the HiPIMS
100 V bias Nb/Cu sample with an input frequency of 1.18 GHz.

Here we show the representative data for $P_{3f}(T)$ generated by surface
defects for the HiPIMS 50 V bias and 100 V bias Nb/Cu sample. For the HiPIMS
50 V bias sample, $P_{3f}(T)$ with $T_{c}^{P_{3f}}$ around 6.4 K is observed
in a strong input power regime (Fig. 16 (a)), and $P_{3f}(T)$ with
$T_{c}^{P_{3f}}$ around 6.8 K is observed in a weak input power regime (Fig.
16 (b)). For the HiPIMS 100 V bias sample, $P_{3f}(T)$ with $T_{c}^{P_{3f}}$
around 6.5 K is observed (Fig. 16 (c)).

##  Appendix C Check for hysteresis in $P_{3f}$ measurements

Figure 17: Check for hysteresis in $P_{3f}(T)$ measurement for the HiPIMS 125
V bias Nb/Cu sample. The input frequency is 1.88 GHz and the input power is
-25 dBm.

Here we check whether or not $P_{3f}(T)$ measurements exhibit hysteresis. The
HiPIMS 125 V bias Nb/Cu sample first undergoes a cool down from 10 K to 3.6 K
with zero RF field. The microwave signal is turned on after the temperature
stabilizes at 3.6 K. The sample then gradually warms up from 3.6 K to 10 K
(warm up $P_{3f}(T)$), and then cools down from 10 K to 3.6 K (cool down
$P_{3f}(T)$), with the microwave signal being turned on in this process. As
shown in Fig. 17, such $P_{3f}(T)$ measurement does not exhibit a clear
hysteresis.

The measurement of Fig. 17 is performed one year and a half after the
measurement of Fig. 6 (b). It is quite possible different regions of the
sample surface are studied in these two measurements. This might explain why
Fig. 17 ($T_{c}^{P_{3f}}$=4.8 K) and Fig. 6 (b) ($T_{c}^{P_{3f}}$=6.8 K) show
different $P_{3f}$ signals.

##  Appendix D Parameters for TDGL simulations

Table 5: Values of parameters used in TDGL simulations for bulk Nb (Sec. IV.2)
and for the two surface defect models (Sec. IV.3, Sec. IV.4 and Sec. IV.5)

Values of parameters used in TDGL simulations are summarized in Table 5. The
“impurity” sector specifies the material parameters of the low-$T_{c}$
impurity in the two surface defect models (Sec. IV.3, Sec. IV.4 and Sec.
IV.5).

Material parameters (penetration depth $\lambda$, Ginzburg-Landau parameter
$\kappa$, etc.) of Nb films vary from one sample to sample. For the Nb part in
the TDGL simulations, we adopt the material parameters of bulk Nb instead of
Nb films for simplicity.

The choice of the material parameters of the low-$T_{c}$ impurity in the two
surface defect models is based on the following considerations. For
simplicity, the Ginzburg-Landau parameter of the low-$T_{c}$ impurity is taken
to be the same as that of bulk Nb ($\kappa^{impurity}=\kappa^{Nb}=1.1$). In
experiments, typically the measured $P_{3f}(T)$ is strong at low temperatures
(surface defects) and is weak around 9 K. To simulate such a feature, a
mechanism that suppresses $P_{3f}(T)$ at high temperatures is required.

One possible mechanism is that superconductivity near the sample surface,
including both the impurity region and the Nb region, is suppressed
significantly when the low-$T_{c}$ impurity is above its transition
temperature ($T_{c}^{impurity}=4K$), namely when $T>4$ K. As the temperature
increases significantly above $T_{c}^{impurity}$ (around 9 K, for example),
superconductivity is weak (and thus all of the responses related to the
strength of superconductivity are weak, including $P_{3f}$) even in the Nb
region, due to the proximity effect. The question is, what kind of material
parameters should be assigned to the low-$T_{c}$ impurity to achieve this
feature at high temperatures?

The Ginzburg-Landau free energy is given by
$\alpha\psi^{2}+\frac{1}{2}\beta\psi^{4}$. One can naively extend Ginzburg-
Landau theory to the regime above the transition temperature of a
superconductor, where $\alpha$ becomes positive
($\alpha\propto\alpha_{0}(T-T_{c})$, with $\alpha_{0}/\beta\propto
1/\lambda^{2}$). In that case, a large and positive $\alpha$ (of the impurity)
means that superconductivity is strongly suppressed, which means that the
superconductivity in nearby superconductors (the Nb region in our case) is
weak due to the proximity effect. A large and positive $\alpha$ above $T_{c}$
implies a large $|\alpha|$ below $T_{c}$, which, in turn, implies a small
penetration depth $\lambda$ ($\alpha_{0}/\beta\propto 1/\lambda^{2}$). As a
result, we choose a small $\lambda^{impurity}$ (16.3 nm).

##  Appendix E A top view for the grain boundary model in Sec. IV.3

Figure 18: A broader view of the distribution of critical temperature on the
XY plane for the grain boundary model discussed in Sec. IV.3. The region shown
here corresponds to the region indicated by the green dashed line in Fig. 10.

Fig. 11 (b) shows the region around the origin (which plays the dominant role
in RF vortex nucleation and $P_{3f}$) for the grain boundary model discussed
in Sec. IV.3. Compared to Fig. 11 (b), Fig. 18 shows the setup over a broader
range as indicated by the green dashed line in Fig. 10. Fig. 18 contains the
entire defect region (the five Nb grains and the blue region) and part of the
bulk Nb region as shown in Fig. 10.

##  Appendix F Snapshots of a vortex in a grain boundary for the grain
boundary model in Sec. IV.3

Figure 19: A snapshot of the vortex core obtained by the TDGL simulation for
the grain boundary model in Sec. IV.3 for $B_{pk}=56.6$ mT (a) and for
$B_{pk}=67.9$ mT (b), with the temperature being 5.4 K and $h_{GB}=200$ nm.
This snapshot is taken when the RF field reaches its maximum in an RF cycle
($\omega t=\pi/2$ and hence $B_{RF}sin(\omega t)=B_{RF}$). The snapshot shows
the distribution of the square of the normalized order parameter on the YZ
plane immediately below the dipole around the grain boundary that nucleates an
RF vortex semi-loop. The black region is where
$|\psi/\psi_{\infty}|^{2}<0.001$.

An RF vortex semi-loop is roughly parallel to the direction of the RF dipole,
which points in the x direction, and hence the cross-section of the RF vortex
semi-loop is on the YZ plane. Fig. 19 visualizes an RF vortex semi-loop in the
grain boundary marked by the white dashed curve in Fig. 11 (b), with the
vortex core corresponding to the black region, where
$|\psi/\psi_{\infty}|^{2}<0.001$. The RF vortex semi-loop penetrates the
sample surface and the vortex core is around 93.5 nm deep for $B_{pk}=56.6$ mT
(Fig. 19 (a)) and is around 110.6 nm deep for $B_{pk}=67.9$ mT (Fig. 19 (b)).

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