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# Depth-resolved Characterization of Meissner Screening Breakdown in Surface
Treated Niobium

Edward Thoeng Email<EMAIL_ADDRESS>TRIUMF, 4004 Wesbrook Mall, Vancouver,
BC V6T 2A3, Canada  Department of Physics & Astronomy, University of British
Columbia,6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada  Md Asaduzzaman
TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada  Department of
Physics & Astronomy, University of Victoria,3800 Finnerty Road, Victoria, BC
V8P 5C2, Canada  Philipp Kolb  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T
2A3, Canada  Ryan M. L. McFadden  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC
V6T 2A3, Canada  Gerald D. Morris  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC
V6T 2A3, Canada  John O. Ticknor  Department of Chemistry, University of
British Columbia, 2036 Main Mall, Vancouver, BC V6T 1Z1, Canada  Stewart
Blusson Quantum Matter Institute, University of British Columbia,2355 East
Mall, Vancouver, BC V6T 1Z4, Canada  Sarah R. Dunsiger  TRIUMF, 4004 Wesbrook
Mall, Vancouver, BC V6T 2A3, Canada  Department of Physics, Simon Fraser
University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada  Victoria L.
Karner  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada  Derek
Fujimoto  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada  Tobias
Junginger  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada
Department of Physics & Astronomy, University of Victoria,3800 Finnerty Road,
Victoria, BC V8P 5C2, Canada  Robert F. Kiefl  TRIUMF, 4004 Wesbrook Mall,
Vancouver, BC V6T 2A3, Canada  Department of Physics & Astronomy, University
of British Columbia,6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
Stewart Blusson Quantum Matter Institute, University of British Columbia,2355
East Mall, Vancouver, BC V6T 1Z4, Canada  W. Andrew MacFarlane  TRIUMF, 4004
Wesbrook Mall, Vancouver, BC V6T 2A3, Canada  Department of Chemistry,
University of British Columbia, 2036 Main Mall, Vancouver, BC V6T 1Z1, Canada
Stewart Blusson Quantum Matter Institute, University of British Columbia,2355
East Mall, Vancouver, BC V6T 1Z4, Canada  Ruohong Li  TRIUMF, 4004 Wesbrook
Mall, Vancouver, BC V6T 2A3, Canada  Suresh Saminathan  TRIUMF, 4004 Wesbrook
Mall, Vancouver, BC V6T 2A3, Canada  Robert E. Laxdal Email<EMAIL_ADDRESS>TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada

###### Abstract

We report direct measurements of the magnetic field screening at the limits of
the Meissner phase for two superconducting Nb samples. The samples are
processed with two different surface treatments that have been developed for
superconducting radio-frequency cavity applications — a “baseline” treatment
and an oxygen-doping (“O-doping” ) treatment. The measurements show: 1) that
the screening length is significantly longer in the “O-doping” sample compared
to the “baseline” sample; 2) that the screening length near the limits of the
Meissner phase increases with applied field; 3) the evolution of the screening
profile as the material transitions from the Meissner phase to the mixed
phase; and 4) a demonstration of the absence of any screening profile for the
highest applied field, indicative of the full flux entering the sample.
Measurements are performed utilizing the $\beta$-detected nuclear magnetic
resonance ($\beta$-NMR) technique that allows depth resolved studies of the
local magnetic field within the first $100\text{\,}\mathrm{nm}$ of the
surface. The study takes advantage of the $\beta$-SRF beamline, a new facility
at TRIUMF, Canada, where field levels up to $200\text{\,}\mathrm{mT}$ are
available parallel to the sample surface to replicate radio frequency (RF)
fields near the Meissner breakdown limits of Nb.

## Introduction

Superconducting radio frequency (SRF) cavities are the enabling technology for
modern large-scale high-energy linear accelerator (linac) facilities. The SRF
cavities or resonators can efficiently produce high-amplitude RF
electromagnetic fields to accelerate charged particles [1, 2, 3]. For a fixed
final energy, the length of the linac is inversely proportional to the
accelerating gradient ($E_{\mathrm{acc}}$). The $E_{\mathrm{acc}}$ is
proportional to the peak RF electric and magnetic fields on the cavity
surface, with the surface magnetic field and associated induced screening
currents limited fundamentally by the bounds of the superconducting state. For
SRF cavity applications, the superconductor must operate in the Meissner
state. In a type-II superconductor, when the surface magnetic field exceeds
the limits of the Meissner state, the material enters the mixed phase where
quanta of magnetic flux circulated by vortices of supercurrents penetrate into
the bulk. The penetrating RF fields cause oscillation of vortices at RF
frequency that generate heat, leading to thermal instability and quench of
superconductivity. The quench field ultimately defines the maximum achievable
accelerating gradient in an SRF cavity. Achieving a higher $E_{\mathrm{acc}}$
ultimately requires a higher onset field for vortex penetration. Global
research is focused on new processes including surface doping [4], layered
structures [5, 6], or new materials [7] that will extend the Meissner state to
higher fields to support high gradient operation.

In the Meissner state, the RF magnetic fields are screened from the bulk
within a very thin layer ($\sim 100\text{\,}\mathrm{nm}$) of the inner cavity
wall of the superconducting surface. The screening profile is typically
exponential with a characteristic length given by the London penetration
depth, $\lambda$ [8]. For type-II superconductors, the lower critical field
($B_{c1}$) marks the field above which it is energetically favourable for the
superconductor to be in a mixed phase though the Meissner state can be
maintained in a meta-stable phase up to the superheating field
($B_{\mathrm{sh}}$) $\sim$ thermodynamic critical field ($B_{\text{c}}$), made
possible by the Bean-Livingston surface energy barrier [9, 6, 10]. Niobium
(Nb), a marginal type-II superconductor [11], is the most common material used
for SRF cavity fabrication. Niobium has the highest superconducting critical
temperature ($T_{c}$) among elemental (metallic) superconductors and the
highest value of $B_{c1}$ of known superconductors.

For clean niobium, $B_{c1}$ ($0\text{\,}\mathrm{K}$) is in the range of
[range-units=single]170180 [12] with $B_{\mathrm{sh}}$
($0\text{\,}\mathrm{K}$) estimated to be $\sim 240\text{\,}\mathrm{mT}$ [13].
The SRF community has managed to push cavity performance to peak surface RF
magnetic field values beyond $B_{c1}$ [14, 15]. Different heat treatments have
empirically been developed for processing SRF cavities —
$120\text{\,}\mathrm{\SIUnitSymbolCelsius}$ bake [16], “two-step bake” [15],
“N doping” [4], and “mid-T bake” [17, 18, 19] — that either reduce RF surface
resistance, increase the achievable gradient, or both. Material studies have
shown that these treatments involve modifications in the near surface
electronic mean free path as a function of distance into the material.
Researchers are also pursuing thin films and layered structures in order to
reduce the required quantity of Nb by using alternate subtrates or to increase
performance.

Despite decades of success in discovering empirical solutions to improve SRF
cavity performance, understanding the underlying mechanisms behind SRF
dissipation is still challenging. Conventional surface characterization
techniques have been adopted, rather than tailored, for SRF studies. Some
examples of the techniques employed are secondary-ion mass-spectrometry (SIMS)
[20, 21], magnetization measurements [22, 23], magneto-optical imaging [24],
positron annihilation spectroscopy [25], and neutron radiography [26]. These
studies have provided insight into how different recipes alter impurities at
the surface and affect the superconducting properties of SRF materials.

However, the key areas of performance enhancement, namely engineering the
surface layer with baking/doping or thin film coating of the Nb surface
motivate the development of a sample diagnostic, on the nanometer length
scale, to characterize the local field screening. A suitable characterization
technique for SRF studies ideally would be able to measure Meissner
screening/vortex penetration that fulfills three important criteria: 1) the
ability to measure the local magnetic field; 2) over a nanometer-scale depth
resolution; with 3) applied fields parallel to the sample surface up to $\sim
200\text{\,}\mathrm{mT}$, to push the material near the fundamental limits of
the vortex-free state. Non-destructive bulk measurements of the fields inside
superconductors have been performed using techniques utilizing spin-polarized
beams such as high-energy muons (so-called conventional muons with a nominal
energy of $4\text{\,}\mathrm{MeV}$) [27, 12] and neutrons [26]. Depth-resolved
and local measurements of the surface field in SRF samples have been performed
using the low-energy muon spin rotation (LE-$\mu$SR) technique[28, 29, 30].
However, the LE-$\mu$SR technique is limited to low applied fields of
$\leq$$30\text{\,}\mathrm{mT}$ [31, 32] — far less than the typical magnitude
of $B_{c1}$ or $B_{\mathrm{sh}}$ of Nb. Thus, the measurement of the detailed
screening profile of the near surface of an SRF material at parallel fields
near the limits of the Meissner state until now, has been unachievable.

All three criteria can now be simultaneously fulfilled with the recently
installed $\beta$-SRF beamline upgrade to the $\beta$-NMR facility at TRIUMF
(Vancouver, Canada), which allows nanometer depth-resolved surface
characterization of the local field with parallel-fields up to
$200\text{\,}\mathrm{mT}$ [33]. The $\beta$-NMR technique is complementary to
LE-$\mu$SR. Instead of low energy $\mu^{+}$ , the $\beta$-NMR technique
commonly uses nuclear-spin-polarized ^8Li^+ radioactive ions [34, 35, 36]. The
ions are significantly more massive than muons (and thus higher momentum for
the same energy) and much less deflected by the applied field that is parallel
to the sample surface, but transverse to beam momentum. The details of the
instrument design, implementation, and commissioning have been reported
elsewhere [33]. Briefly, the beamline adds electrostatic optics, sample
cryostat, and a Helmholtz coil to, respectively, 1) focus the radioactive ion
beam (RIB) onto the sample, 2) cool the sample in an ultra-high vacuum (UHV)
chamber, and 3) produce high magnetic fields up to $200\text{\,}\mathrm{mT}$
parallel to the sample surface. Depth control of the implanted ^8Li^+ is
accomplished by adjusting the sample potential bias to vary the energy of the
implanting ion beam, providing deeper penetration with higher beam energies.

We report here first experiments at $\beta$-SRF with strong parallel magnetic
fields on two Nb samples with preparations developed in the SRF community. Our
findings demonstrate: 1) a clear contrast of the magnetic penetration depth
between the two Nb samples; 2) that the screening length near the limits of
the Meissner phase increases with applied field; 3) the evolution of the
screening length in transitioning from Meissner to the mixed state; and 4)
sample dependent surface fields marking the transition from the Meissner state
to the mixed state. All findings demonstrate the potential of the $\beta$-SRF
beamline to inform SRF cavity research in terms of near-surface engineering.
This new facility and the methodology showcased in this publication open a new
path for future studies of novel baking/doping recipes as well as more complex
layered SRF systems.

## Results

Two samples with different surface SRF processing procedures (see ‘Methods’)
have been characterized at the $\beta$-SRF beamline with applied parallel
fields ranging from [range-units=single]100200. Sample 1 is prepared with a
baseline treatment (“baseline” ) and sample 2 is prepared with a heat
treatment at $400\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$3\text{\,}\mathrm{h}$ commonly known in the community as “mid-T bake” or
“O-doping” . The former produces ‘clean’ Nb with an electron mean-free path
($l$) of $\sim 800\text{\,}\mathrm{nm}$ while the heating associated with the
latter surface treatment dissolves the native oxide coating of the Nb which is
then diffused into the Nb subsurface, increasing the interstitial oxygen
concentration and lowering the near surface electronic mean free path [21,
37].

For these measurements, both samples are zero field cooled (ZFC) to the base
temperature ($\sim 4.5\text{\,}\mathrm{K}$) and, subsequently, magnetic fields
parallel to the sample surface are increased monotonically in steps, with
$\beta$-NMR measurements performed at each field increment. The ZFC procedure
and systematic ramping of the field steps are employed in order to remove
energy barrier and geometric barrier effects [38] from impacting the value of
initial flux penetration. At each applied field step, depth-resolved
measurements are taken at four to five different implantation energies.

The spin-lattice relaxation (SLR) technique [39, 40, 34, 35, 36] is used to
determine the magnetic field detected by the ^8Li probe at each depth and to
quantify the field screening profile $B(x)$. In these measurements, the
applied field is parallel to the initial spin-polarization direction of the
implanted ^8Li ions and polarization is lost through an energy exchange
between the ^8Li probes and their surrounding environment. The time-dependent
stochastic fluctuations of the local field that are transverse to the probe
spin direction induce transitions between the ^8Li nuclear spin levels that
result in “relaxation” of the initial spin state over time to thermal
equilibrium (approximately zero polarization). The SLR rate, commonly denoted
as $1/T_{1}\,$, characterizes the depolarization time-constant of the
$\ch{{}^{8}Li}\,$ nuclear spins after they are stopped inside the sample. A
static superimposed field tends to slow the process of depolarization and it
is the variation of $1/T_{1}\,$ for different probe depths that gives
information on the screening profile.

The measured SLR rates for the two samples are shown in fig. 1 as a function
of the applied field ($B_{\text{a}}$) and implantation energy ($E$). Dashed
lines in fig. 1 are simple linear fits to the data as a guide to the eye.
Given that $1/T_{1}\,\propto 1/B^{2}$, where $B$ is the local magnetic field
for a given implantation energy, there are several features of note. The
depolarization rate for lower applied fields is greater than at higher fields.
The rising slope of $1/T_{1}\,$ vs. $E$ for lower applied fields (i.e.,
$100\text{\,}\mathrm{mT}$) is a clear indication of Meissner screening, as the
magnetic field sampled by the ions is diminished for increasing depth. The
slope of $1/T_{1}\,$ decreases for higher applied fields indicating an
evolution from the Meissner state to the mixed state. Also note that the slope
in relaxation rate for $100\text{\,}\mathrm{mT}$ applied field is different
between the two samples indicating a surface-treatment dependence in the
screening profile. Finally, note that at $200\text{\,}\mathrm{mT}$ the
$1/T_{1}\,$ values are independent of implantation energy (i.e., depth),
indicating that the flux has substantially and uniformly entered into the
sample.

(a) “baseline” sample.

(b) “O-doping” sample.

Figure 1: SLR rate for two Nb samples at various applied fields,
$B_{\text{a}}$, as a function of implantation energy. The dashed lines are
simple linear fits to the measured relaxation rate to help guide the eye. The
diminishing slope as $B_{\text{a}}$ increases indicates the evolution from the
Meissner state to the mixed state.

A useful approximation to allow mapping $1/T_{1}\,$ to the average local
magnetic field is through a simple Lorentzian model (see e.g., [41]):

$\displaystyle\frac{1}{T_{1}(E)}\approx\frac{a}{b+\langle B(E)\rangle^{2}},$
(1)

where $E$ is the implantation energy, $\langle B(E)\rangle$ is the average
measured local magnetic field, and $a$ and $b$ are material dependent fit
parameters. The $1/T_{1}\,$ values can be explicitly mapped to $\langle
B(E)\rangle$ values by using a subset of the datapoints as a basis. The data
is taken at a constant temperature (at T$\sim 4.5\text{\,}\mathrm{K}$) to
remove any temperature dependence in the Lorentzian fit parameters. The basis
points include the $100\text{\,}\mathrm{mT}$ case where the material is
expected to be in the Meissner state and the $200\text{\,}\mathrm{mT}$ case
where the field is assumed to be fully in the normal state near $B_{c2}$ (for
$T\sim$4.5\text{\,}\mathrm{K}$)$. The analysis follows an ansatz approach
where a Meissner screening profile is assumed for $100\text{\,}\mathrm{mT}$
and the best fit to a Lorentzian is calculated by varying parameters $a$ and
$b$. A single exponential screening profile with some non-superconducting
“dead layer” is assumed for the Meissner state as:

$\displaystyle B(x)$ $\displaystyle=\begin{cases}B_{0}&\text{for $x\leq
d$,}\\\ B_{0}\exp{-(x-d)/\Lambda}&\text{for $x>d$}\end{cases}$ (2)

where $B_{0}$ is the field at the surface, $d$ is the “dead layer” in which
there is no Meissner screening [42, 43, 44], and $\Lambda$ is the length scale
parameter for the exponential decay.

Due to the finite volume of the sample, flux expulsion due to the
superconducting diamagnetism enhances the field at the surface of the sample,
$B_{0}$, with respect to the applied field ($B_{\text{a}}$) by a factor
$\mathcal{E}$ such that $B_{0}=\mathcal{E}B_{\text{a}}$. For our sample
geometries, the enhancement factor in the Meissner phase, $\mathcal{E}_{M}$,
is determined using the finite element software CST Studio Suite® [45] where
the sample is assumed to be perfectly diamagnetic. The average value of the
enhancement is determined by averaging over the Gaussian beam spot of the
^8Li^+ ions with typical size of $\sim 4\text{\,}\mathrm{mm}$ in diameter
[33]. The sample dimensions and enhancement factors for the Meissner state for
each sample are summarized in ’Methods: Sample Preparations’. For the
$100\text{\,}\mathrm{mT}$ case, $B_{0}=\mathcal{E}_{M}B_{\text{a}}$ is assumed
in the fit. For the applied field of $200\text{\,}\mathrm{mT}$, the
diamagnetic field enhancement is expected to be negligible as evidenced by the
flat $1/T_{1}\,$ dependence on implantation energy and so, $\mathcal{E}=1$ is
assumed in the fit.

The average field sampled by the ^8Li probes for a given implantation energy
is a function of the implantation depth distribution, $\rho_{E}(x)$, and the
depth profile $B(x)$. For any particular profile, we can calculate a single
value of $\langle B(E)\rangle$ as a function of energy via:

$\displaystyle\langle B(E)\rangle$ $\displaystyle=\int B(x)\rho_{E}(x)dx,$ (3)

where the Monte Carlo code SRIM [46] is used to calculate the depth
distributions at various beam energies (see ‘Methods: Data Analysis’).

The Meissner state screening profile, defined by parameters $\Lambda$ and $d$,
and the Lorentzian parameters $a$ and $b$ are varied to give the best fit of
the [list-units = single]100;200 $1/T_{1}\,$ values using the Lorentzian
function in eq. 1. Assuming $\langle B(E)\rangle$ (in ) and $1/T_{1}\,$ (in ),
the best fit ($a$, $b$) parameter values are ($63\,200\pm 1900.\text{\,}$
,$13\,300\pm 1100.\text{\,}$) and ($50\,600\pm 1300.\text{\,}$, $10\,000\pm
1000.\text{\,}$) for the “baseline” and “O-doping” sample, respectively. The
Lorentzian parameters are then used to map the remainder of the $1/T_{1}\,$
data to $\langle B(E)\rangle$. The simple mapping results in the energy
dependent screening profiles shown in fig. 2. Here, we note the evident
screening at lower applied fields that systematically evolves to no screening
as the applied fields increase to $200\text{\,}\mathrm{mT}$. Also note the
distinctly different screening profiles between the two samples.

(a) “baseline” sample

(b) “O-doping” sample

Figure 2: Shown are the average B-field as a function of implantation energy
for various applied parallel fields $B_{\text{a}}$ for the two samples. Dashed
lines are fits to the data assuming a single exponential function and “dead
layer”.

(a) “baseline” sample.

(b) “O-doping” sample.

Figure 3:  Extracted field profiles within the first $150\text{\,}\mathrm{nm}$
of the sample surface as a function of $B_{\text{a}}$. The shaded area
indicates the non-superconducting “dead layer”.

A more involved analysis is used to extract values of the screening length as
a function of applied field. The analysis assumes: 1) a more detailed
expression for the relaxation rate which takes into account the dipole-dipole
interaction between two different nuclear spins (i.e., the probe ^8Li and the
host ^93Nb nuclear spins) [40, 47] as described in ‘Methods: Data Analysis’;
and 2) that $1/T_{1}\,$ values are calculated not via the average field, but
via the actual field at each depth. The latter is a more accurate technique,
but it does not allow a direct mapping of $1/T_{1}\,$ data to an energy
dependent screening profile as afforded by the analysis used to produce fig.
2.

As with the previous technique, the 100 and 200 datasets (using the single
exponential screening model in eq. 2 and the expected surface field
enhancement factors for an applied field of 100 ) are used to establish the
Lorentzian-like parameters, now $B_{d}$ and $\tau_{c}$ from eq. 18. The
obtained $B_{d}$ and $\tau_{c}$ parameters are then used to extract the best
fit for all datasets at various applied fields, $B_{\text{a}}$. We further
introduce the normalized enhancement factor, $\mathcal{N}(B_{a})$, that
relates the enhancement of the surface field compared to the applied field
relative to the sample dependent maximum enhancement factor,
$\mathcal{E}_{M}$, in the Meissner state. The normalized enhancement factor is
defined through the relation:

$\displaystyle\mathcal{N}(B_{a})=\frac{\mathcal{E}(B_{\text{a}})-1}{\mathcal{E}_{M}-1}.$
(4)

For any superconducting volume regardless of its specific dimensions, the
normalized enhancement factor should decrease from 1 when the sample is
completely in the Meissner phase, to zero in the normal state. For values in
between the two limits, the volume exhibits some diamagnetic character. In our
analysis, $\mathcal{N}=1$ is assumed for an applied field of
$100\text{\,}\mathrm{mT}$ and $\mathcal{N}=0$ for an applied field of
$200\text{\,}\mathrm{mT}$. For other applied fields, the data is used to
extract $\mathcal{N}$.

Two approaches are used in the fits. In the first case, Case A, the single
exponential screening profile that decays to zero, eq. 2, is used for all
applied fields. This profile coincides with what we would expect from Meissner
screening. In the Meissner state, the value of the normalized enhancement
factor should be $\mathcal{N}$=1 and the extracted value of $\Lambda$
corresponds to the magnetic penetration depth $\lambda$. However, once flux
begins to break into the sample, the local field is a superposition of an
exponentially decreasing field due to the screening currents and an
exponentially rising field from the internal vortex distribution (see fig.
4(b)). The latter increases from zero at the surface towards the average bulk
equilibrium value of the internal vortices near the surface [48, 49]. The
resulting field profile remains a single exponential, but instead of decaying
to zero it reduces to the bulk equilibrium value. Therefore, once in the mixed
state the Case A analysis outputs an extracted $\Lambda$ that is larger than
$\lambda$, and the extracted enhancement factor underestimates the actual
enhancement factor.

A second analysis, Case B, is used to better characterize the screening
profile in the mixed state. Applied fields that yield a decrease of normalized
enhancement factor $\mathcal{N}$ from unity and an abrupt increase in
screening length $\Lambda$ in Case A are interpreted as indicating the
presence of a mixed state and the screening profile is modified to [49, 50]:

$\displaystyle B_{\text{Tot}}(x)$ $\displaystyle=\begin{cases}B_{0}&\text{for
$x\leq d$,}\\\ B_{\rm M}(x)+B_{\rm v}(x)&\text{for $x>d$,}\\\
\qquad=B_{0}\exp{-(x-d)/\lambda}+B_{\rm
in}\left[1-\exp{-(x-d)/\lambda}\right]\\\ \qquad=(B_{0}-B_{\rm
in})\exp{-(x-d)/\lambda}+B_{\rm in},\end{cases}$ (5)

where $B_{\rm M}$ ($B_{\rm v}$) is the Meissner (vortex) contribution to the
field profile, $B_{\rm in}$ is the equilibrium field inside the bulk due to
vortex entry, and the extracted characteristic length, $\lambda$, is the
magnetic penetration depth. In order to minimize the number of fit parameters
added to take into account vortex penetration, the applied field,
$B_{\text{a}}$, and the interior field, $B_{\rm in}$, are related through the
normalized enhancement factor as $B_{\rm in}/B_{\text{a}}=1-\mathcal{N}$. This
simple relation is derived from the boundary condition between vacuum and
superconductor layer assuming a constant effective magnetization in the volume
probed by the beam (see Supplementary Information: ‘Derivation of $B_{\rm
in}$’).

The reconstructed field profiles $B(x)$ at increasing $B_{\text{a}}$ are shown
in fig. 3 for Case A. The extracted fit outputs for Case A and Case B are
summarized in table 1 and table 2 for “baseline” and “O-doping” samples,
respectively. The fit outputs, as defined in eq. 2 and eq. 5, include the
screening profile length constant ($\Lambda$ in Case A, or $\lambda$ in Case
B), the surface field ($B_{0}$), and the normalized enhancement factor
($\mathcal{N}$), all as a function of applied field for each sample. We also
introduce the normalized screening depth, defined as the screening
characteristic depth normalized to the value at $100\text{\,}\mathrm{mT}$,
i.e., $\Lambda_{N}=\Lambda(B_{a})/\Lambda(100$). The plots of $\mathcal{N}$
vs. $B_{a}$ and $\Lambda_{N}$ vs. $B_{a}$ are shown in fig. 4(a) for both
samples and for each case. Note that the same values of $\mathcal{N}(B_{a})$
are obtained for both Case A and Case B for the “O-doping” sample. At
$B_{\text{a}}=$$150\text{\,}\mathrm{mT}$, despite the different $\Lambda$s
obtained for the two cases, the resulting field profiles are virtually
identical. For cases where the obtained best fit $\mathcal{N}$ values are not
at the lower (i.e., $\mathcal{N}=0$) or upper (i.e., $\mathcal{N}=1$) bound,
accurate error estimates of $\mathcal{N}$ can be obtained. They are shown in
tables 1 and 2 and in fig. 4(a).

(a)

(b)

Figure 4:  (a) Normalized surface field enhancement factor $\mathcal{N}$ and normalized screening length $\Lambda_{N}$ for the two samples and for each analysis case (Case A, without vortex, and Case B, with vortex, as described in the text) as a function of the applied field. (b) Expected screening profile from the Meissner ($\rm B_{M}$), vortex ($\rm B_{v}$), and total ($\rm B_{Tot}$) contributions for $B_{\text{a}}<B_{\text{entry}}$ (Meissner state) and for $B_{\text{a}}>B_{\text{entry}}$ with $B_{\rm in}<B_{\text{a}}$ in a mixed state with uniform vortex density.  $B_{\text{a}}$ [] | Case A | Case B
---|---|---
$B_{0}$ [] | $\mathcal{N}(B_{a})$ |  $\Lambda$ [] | $\Lambda_{N}$ |  $B_{0}$ [] | $\mathcal{N}(B_{a})$ |  $\lambda$ [] | $\Lambda_{N}$
100 | 108.1 | 1 | $42.8(13.2)\text{\,}$ | $1.0(0.4)\text{\,}$ | 108.1 | 1 | $42.8(13.2)\text{\,}$ | $1.0(0.4)\text{\,}$
110 | $111.3(3.8)\text{\,}$ | $0.14(0.42)\text{\,}$ | $65.6(5.6)\text{\,}$ | $1.5(0.5)\text{\,}$ | $116.6(0.7)\text{\,}$ | $0.74(0.08)\text{\,}$ | $35.1(7.4)\text{\,}$ | $0.8(0.3)\text{\,}$
125 | 125 | $0\text{\,}$ | $127.8(6.0)\text{\,}$ | $3.0(0.9)\text{\,}$ | $129.5(0.7)\text{\,}$ | $0.45(0.07)\text{\,}$ | $31.2(9.7)\text{\,}$ | $0.7(0.3)\text{\,}$
150 | 150 | $0\text{\,}$ | $187.3(10.7)\text{\,}$ | $4.4(1.4)\text{\,}$ | $155.0(1.2)\text{\,}$ | $0.41(0.10)\text{\,}$ | $47.9(19.0)\text{\,}$ | $1.1(0.6)\text{\,}$
200 | 200 | 0 | $\infty$ | $\infty$ | 200 | 0 | $\infty$ | $\infty$
Table 1: Extracted fit parameters from the $B(x)$ model for the “baseline” sample, measured at $\sim 4.5\text{\,}\mathrm{K}$. The thickness of the “dead layer” is estimated at $d=$$11.7(2.9)\text{\,}\mathrm{nm}$.  $B_{\text{a}}$ [] | $B_{0}$ [] | $\mathcal{N}(B_{a})$ | Case A | Case B
---|---|---|---|---
$\Lambda$ [] | $\Lambda_{N}$ |  $\lambda$ [] | $\Lambda_{N}$
100 | 104.3 | 1 | $146.9(22.6)\text{\,}$ | $1.0(0.2)\text{\,}$ | $146.9(22.6)\text{\,}$ | $1.0(0.2)\text{\,}$
110 | 114.7 | 1 | $219.8(14.3)\text{\,}$ | $1.5(0.3)\text{\,}$ | $219.8(14.3)\text{\,}$ | $1.5(0.3)\text{\,}$
125 | 130.4 | 1 | $272.5(19.4)\text{\,}$ | $1.9(0.3)\text{\,}$ | $272.5(19.4)\text{\,}$ | $1.9(0.3)\text{\,}$
150 | $153.8(2.8)\text{\,}$ | $0.58(0.42)\text{\,}$ | $624.7(229.0)\text{\,}$ | $4.3(1.7)\text{\,}$ | $355.2(161.4)\text{\,}$ | $2.4(1.2)\text{\,}$
200 | 200 | 0 | $\infty$ | $\infty$ | $\infty$ | $\infty$
Table 2: Extracted fit parameters from the $B(x)$ model for the “O-doping”
sample, measured at $\sim 4.5\text{\,}\mathrm{K}$. The thickness of the “dead
layer” is estimated at $d=$$24.4(11.2)\text{\,}\mathrm{nm}$.

## Discussion

The results reveal several notable features. They are: 1) the screening length
in the “O-doping” sample is $\sim 3$ times longer than in the “baseline”
sample; 2) the screening length in the “O-doping” sample increases
significantly as the applied field is increased; 3) flux-penetration into the
bulk occurs at a lower field in the “baseline” treatment than in the
“O-doping” sample; and 4) the “O-doping” sample possesses a larger non-
superconducting “dead layer” than the “baseline” sample.

First, the “O-doping” sample has a significantly larger penetration depth
compared to the “baseline” . This is consistent with the expectation that
mid-T baking ($400\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$3\text{\,}\mathrm{h}$) renders a “dirtier” (shorter mean free path)
subsurface due to native oxide (Nb2O5) dissolution and interstitial oxygen
diffusion [16, 21, 37]. In contrast, the “baseline” sample has a much shorter
penetration depth in the Meissner phase consistent with a cleaner Nb surface.

In type-II superconductors, the mixed state transitions to the normal state
and the superconducting screening collapses at $B_{c2}(t)$ with temperature
dependence [51, 52]:

$\displaystyle b_{c2}=\frac{1-t^{2}}{1+t^{2}},$ (6)

where $t=T/T_{c}(B=0)$ is the reduced temperature and
$b_{c2}=B_{c2}(T)/B_{c2}(T=0)$ is the reduced upper critical field at
temperature $T$. The relation in eq. 6 can be inverted to give the reduced
critical temperature at a reduced field $b=B_{0}/B_{c2}(T=0)$:

$\displaystyle t_{c2}$ $\displaystyle=\sqrt{\frac{1-b}{1+b}},$ (7)

where $t_{c2}=T_{c2}(B_{0})/T_{c2}(B=0)$.

The two-fluid model can be used to estimate the effective penetration depth,
$\lambda$(0,0), at $T=0$ K and $B\rightarrow 0$ via:

$\displaystyle\lambda(0,0)$
$\displaystyle=\lambda(t,b)\cdot{\sqrt{1-[t(b)]^{4}}},$ (8)

where the field-dependence of the penetration depth arises due to the field
dependence of $T_{c2}(B_{0})$ with the reduced temperature defined earlier as
$t(b)=T/T_{c2}(B_{0})$. In the local limit, the relation between
$\lambda(0,0,l)$ and the electron mean-free path ($l$)can be estimated from
the relation [8]:

$\displaystyle\lambda(0,0,l)=\lambda_{L}(T=0)\sqrt{1+\frac{\xi_{0}}{l}}.$ (9)

where $\lambda_{L}$ is defined as the London penetration depth in the limit of
pure ($l\rightarrow\infty$) and local ($\xi_{0}\rightarrow 0$) response and
$\xi_{0}$ is the Bardeen-Cooper-Schrieffer (BCS) coherence length [53].

Typical quoted values for clean niobium are $T_{c2}(B=0)$ $\sim
9.2\text{\,}\mathrm{K}$ [52, 23], $B_{c2}(T=0)$ $\sim 410\text{\,}\mathrm{mT}$
[52, 23], $\lambda_{L}$ $\sim 40\text{\,}\mathrm{n}\mathrm{m}$ [54, 55] and
$\xi_{0}$=$38\text{\,}\mathrm{n}\mathrm{m}$ [55]. Our “baseline” sample is
characterized by a residual resistivity ratio (RRR) =300, corresponding to a
mean free path of $l$ $\sim 810\text{\,}\mathrm{nm}$ from the empirical
relation: $l\text{[nm]}\approxeq 2.7\cdot\text{RRR}$ [1]. Applying eqs. 6, 7,
8 and 9 for our “baseline” sample with measured values of
$\lambda(T=$4.5\text{\,}\mathrm{K}$,B_{0}=$108.1\text{\,}\mathrm{mT}$,l=$810\text{\,}\mathrm{nm}$)=$
$42.8\text{\,}\mathrm{nm}$ gives $\lambda(0,0,l=810)=$
$39.0\text{\,}\mathrm{nm}$ or $\lambda_{L}$=$38.1\text{\,}\mathrm{nm}$ in good
agreement with expected values.

The same analysis can also be applied for the “O-doping” sample by using the
same input values of $T_{c}=$$9.2\text{\,}\mathrm{K}$,
$B_{c2}=$$410\text{\,}\mathrm{mT}$ and
$\lambda_{L}=$$38.1\text{\,}\mathrm{nm}$ The experimental value of
$\lambda(t,b)=$$146.9\text{\,}\mathrm{nm}$ gives a penetration depth of
$\lambda(0,0)=$$134.5\text{\,}\mathrm{nm}$ and a mean free path value of
$l=$$3.3\text{\,}\mathrm{nm}$ from eq. 9. Mean free path values of doped Nb
have been reported by other studies from fits to RF cavity measurements with
values in the range of [range-units=single]4200 [56, 57]. The extracted values
here are on the low side of this range, which might indicate additional
pollution in the surface layer and/or suppressed $T_{c}$ due to an elevated
concentration of oxygen impurities [58, 59]. This latter hypothesis can be
used to also look at a second key feature of the data from the “O-doping”
sample, the increase in screening length with applied field.

We consider first the break-in field implied by the data. Considering the
sample geometry and formulation by Brandt for a sample with rectangular cross-
section with side length $c$ in the direction of the applied field and side
$a$ as the slab thickness, flux will break into the center of a pin free
sample at an applied field of [38]:

$\displaystyle B_{\text{entry}}=B_{c1}\times\tanh(\sqrt{0.36c/a}).$ (10)

Using the experimentally measured value of $B_{c1}(0K)$$\approx
174\text{\,}\mathrm{mT}$ [12] and scaling this value to the measurement
temperature of $\sim 4.5\text{\,}\mathrm{K}$ using
$B_{c1}(T)=B_{c1}(0)[1-(T/T_{c})^{2}]$ [52] gives
$B_{c1}$($4.5\text{\,}\mathrm{K}$)$\approx 132\text{\,}\mathrm{mT}$. Noting
that the samples are slightly different in shape (see table 3) and applying
eq. 10, the entry field would be expected at an applied field of
$B_{\text{entry}}$$\sim 113\text{\,}\mathrm{mT}$ in the “baseline” and
$B_{\text{entry}}$$\sim 123\text{\,}\mathrm{mT}$ in the “O-doping” sample. In
the dirtier “O-doping” sample, we expect that the assumed $B_{c1}$ value is
reduced from the clean limit value and so this $B_{\text{entry}}$ value
represents an upper bound. There is indication from the normalized enhancement
factor fits that flux has broken into the “baseline” sample already at an
applied field of $110\text{\,}\mathrm{mT}$ ($\mathcal{N}<1$), but in the
“O-doping” case the extracted enhancement factor is consistent with the
Meissner state ($\mathcal{N}=1$) for applied fields from [range-
units=single]100125 (see fig. 4 (a)).

For the “O-doping” cases, the measured magnetic penetration depth $\lambda$
increases significantly as the surface field nears the boundary of the
Meissner state, as shown in fig. 4(a). A way to account for this sharp rise is
to consider a suppressed $T_{c}$. A decrease of $T_{c}$ for Nb has been
reported due to interstitial oxygen diffusion in Nb [58, 59]. A linear fit to
the tabulated values of $T_{c}$ vs O concentration from Koch et al. [58] is
approximately given by $T_{c}\approxeq T_{c0}-1.01\cdot x$, where
$T_{c0}=$$9.2\text{\,}\mathrm{K}$ is the $T_{c}$ of pure Nb and $x$ is the
atomic-% concentration of interstitial oxygen in Nb[58]. From eqs. 8 and 7,
and using $T_{c}$ as the free variable with fixed
$B_{c2}=$410\text{\,}\mathrm{mT}$$, a best fit to the three points in fig. 5
is for a suppressed $T_{c}$ $\sim 6.5\text{\,}\mathrm{K}$, with
$\lambda(0,0)$$\sim 103\text{\,}\mathrm{nm}$ and $l$ $\sim
6\text{\,}\mathrm{nm}$ obtained from eq. 9 by using the same values of
$\lambda_{L}$ and $\xi_{0}$ as inputs. This level of suppression would require
an oxygen concentration of a few atomic-%, much higher than typical values
quoted for “O-doping” treatments that are $\sim 1/10$-th of this concentration
($\lesssim 0.5$ [at%]) [21, 20]. Alternatively, the strong change in screening
profile, assuming $T_{c0}=$$9.2\text{\,}\mathrm{K}$, could also be fit by a
suppressed upper critical field with
$B_{c2}($0\text{\,}\mathrm{K}$)\sim$230\text{\,}\mathrm{mT}$$, or a
combination of suppressed $T_{c}$ and suppressed $B_{c2}$. The simultaneous
suppression of $T_{c}$ and $B_{c2}$ can be due to the presence of pair-
breaking magnetic impurities (see, e.g., [60, 61, 62]) as observed in
experimental measurements (see, e.g., [63]), and is consistent with
calculations from the microscopic Eilenberger’s equations [64].

Non-magnetic impurities in current carrying superconductors act as pair
breakers, in contrast to their pair conserving nature in zero current, thereby
reducing the pair potential $\Delta$ and superfluid density [65, 54, 66, 67].
The dependence of superfluid density on screening currents in the Meissner
state contributes to non-linear Meissner screening as
$\Delta^{2}(B_{0})\propto n_{s}(B_{0})\propto$1/$\lambda^{2}(B_{0})$ [54, 66].
The non-linear screening adds a field-dependent correction term to the
magnetic penetration depth on top of implicit $B_{0}$ dependence via
$T_{c2}(B_{0})$ in eq. 8. For s-wave superconductors it is predicted that the
penetration depth increases quadratically with field [68, 69]:

$\displaystyle\lambda_{\text{NLME}}(t,b)$
$\displaystyle=\lambda(t,b)\left[1+\beta(T)\left(\frac{B_{0}}{B_{c}}\right)^{2}\right].$
(11)

Here, $\lambda(t,b)$, $t$ and $b$ are defined in eq. 8 and $\beta(T)$ is the
quadratic prefactor. Ginzburg-Landau theory provides an analytical expression
for the prefactor $\beta(T\rightarrow T_{c})$, which depends only on the
Ginzburg-Landau parameter $\kappa$ (defined as $\lambda/\xi$ for $T\rightarrow
T_{c}$) via the following relation [70, 71]:

$\displaystyle\beta$
$\displaystyle=\frac{\kappa(\kappa+2^{3/2})}{8(\kappa+2^{1/2})^{2}}.$ (12)

Typical $\kappa$ values of $1\rightarrow 10$ (i.e., in the dirty limit
$\kappa\approx\lambda_{L}/l$) for various levels of doping in niobium give
$\beta\sim$0.1. For the range of values reported here, with
$B_{c}\sim$$200\text{\,}\mathrm{mT}$, the non-linear Meissner screening should
enhance the magnetic penetration depth at $125\text{\,}\mathrm{mT}$ only by
<2% above the value at $100\text{\,}\mathrm{mT}$. Our reported difference of
almost a factor of two is considerably larger than that.

More extreme variations of the screening profile near $B_{c1}$ could come from
additional pair-breaking from magnetic impurities or gap anisotropy, resulting
in a subgap state that is phenomenologically modeled by density of states with
Dynes broadening [67, 72, 66]. For example, magnetic impurities associated
with oxygen vacancies in the native surface oxide of Nb have been revealed by
tunneling measurements and have been attributed as an intrinsic component to
the residual RF resistance commonly encountered in SRF cavities [73, 74, 75,
76]. These subgap states could result in a stronger non-linear field
dependence of the Meissner screening similar to the “extreme non-linear
Meissner effect” in the gapless state as reported by Lee et al [77].

Figure 5: Strong field-dependence of the measured effective magnetic
penetration depth (data points) for the “O-doping” sample, compared to a
linear Meissner penetration depth model for various reduced critical
temperatures $T_{c}$. Note that the penetration depth values are plotted
against the surface fields, i.e., $B_{0}=\mathcal{E}B_{\text{a}}$.

Another possible mechanism to partially explain the strong dependence of
magnetic penetration depth with applied field near $B_{c1}$ is localized
nucleation of vortex loops at the surface [78, 79]. Given the doping used in
the “O-doping” sample, it would be expected to have an elevated $\kappa$ and a
reduced $B_{c1}$. However, given the observed surface field enhancement (i.e.,
$\mathcal{N}=1$) which indicates that the bulk of the material remains
diamagnetic (see fig. 4(a)), it is likely that the material is operating in a
regime above $B_{c1}$ in a meta-stable regime below $B_{sh}$. In this regime,
variations in the order parameter from local suppression of $T_{c}$,
impurities, inclusions, or field enhancement from surface roughness could
suppress locally the superheating barrier and be responsible for the surface
nucleation of vortex loops. Such vortices would not penetrate into the bulk,
but would increase in number as the applied field is increased, right up to
the applied field where flux penetration occurs. These vortex loops at the
surface could contribute to the large change in effective penetration depth
measured in the doped sample due to the superposition of the Meissner
screening profile and the additional surface field from nucleating surface
vortices.

Our results in Case B reveal interesting features of the mixed state. As noted
in fig. 4(b), after vortices have entered the bulk, the resulting screening
profile is a superposition of the exponentially decaying field due to the
Meissner screening and the exponentially rising contribution due to the vortex
entry that is zero at the surface and rises to the near surface vortex
density. While our simple model assumes a uniform flux distribution, the
details of the vortex density are defined by the sample geometry [38], the
average field in the bulk, the local pinning force [50, 79], as well as
additional field dependence of penetration depth $\lambda(B_{\text{a}})$ [65,
80, 67]. Figure 4(a) for Case B indicates that the extracted magnetic
penetration depth for the “baseline” sample has a reduced field dependence
compared to the “O-doping” sample, consistent with “clean” niobium and a
pairing potential that depends only weakly on the applied field. Another
interpretation is that some flux may have penetrated the “O-doping” sample
giving a non-zero $B_{\rm in}$ (eq. 5), but due to flux pinning the vortices
do not redistribute uniformly and remain near the surface after entry. Such an
interpretation could be consistent with an increasing screening profile as a
function of applied field, but with a surface field enhancement factor
$\mathcal{N}\sim 1$. Subsequent studies could be done to evaluate any
hysteresis in the screening profile. In general, future experiments could
explore near surface magnetic field in the mixed state with more detailed
measurements in this field regime.

A notable feature in the data is the shorter “dead layer” in the “baseline”
sample ($d$ = $11.7(2.9)\text{\,}\mathrm{nm}$) compared to the “O-doping”
sample ($d$ = $24.4(11.2)\text{\,}\mathrm{nm}$). The “dead layer” is a
universal feature of all superconductors and is a sample dependent property.
The origin of the “dead layer” has previously been attributed to surface
roughness (see e.g.[42, 43, 44]). As both samples are from the same batch with
the same buffered chemical polishing (BCP) etching step applied, it is
unlikely that surface roughness variation is the sole origin of the much
larger “dead layer” in the “O-doping” sample. Surface composition
characterization has been performed on duplicate witness samples with
equivalent treatment using time-of-flight SIMS (TOF-SIMS) and energy
dispersive X-ray spectroscopy (EDX), and has been reported by Kolb et al [81].
Both characterizations indicate an elevated level of carbon concentration at
the surface of the “O-doping” sample, possibly received during the
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$-bake. The presence of enhanced
carbon concentration at the surface is consistent with other studies, e.g.,
in-situ X-ray photoelectron spectroscopy (XPS), where furnace baking of Nb at
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $3\text{\,}\mathrm{h}$
dissolves the protective Nb2O5 layer below the critical thickness of
$1\text{\,}\mathrm{nm}$, thereby promoting surface carbon precipitation [82].
Strong impurity gradients over the first $20\text{\,}\mathrm{n}\mathrm{m}$
have been postulated to modify the magnetic screening profile to non-
exponential forms and push peak screening currents away from the surface [83,
84, 85, 37]. Such modified screening profiles with a dead-layer magnitude
similar to the “baseline” sample (i.e., $d$$\sim
12\text{\,}\mathrm{n}\mathrm{m}$) may be consistent with our data that we have
modeled simply as a single exponential with a larger “dead layer” and enhanced
penetration depth. It is also important to note that pair-breaking due to
screening currents will vary over the penetration depth $\lambda$. This pair-
breaking modifies the super-fluid density locally as a function of the
screening depth, i.e., $\Delta[B(x)]^{2}\propto
n_{s}(x)\propto$1/$\lambda(x)^{2}$, and therefore results in a non-exponential
magnetic screening profile at high field [80].

In summary, the performance of SRF materials near fundamental limits is of
critical importance to the accelerator community, as these limits set the
scale for the accelerators in terms of operating gradient. Relevant phenomena
in this regime have been explored in detail using two heat treated Nb samples,
both showing distinct behaviour as a result of their modified subsurface.
Theories which self-consistently take into account various pair breaking
effects (due to magnetic field, nonmagnetic and magnetic impurities, gap
anisotropy, proximity effect from normal metal overlayer) are being
extensively developed (see, e.g.,[67, 72, 86, 87, 84, 66, 88]). We anticipate
that the $\beta$-SRF facility will be an important tool to validate these
proposed theories, and future experiments will help refine our understanding
on how to advance and tailor SRF performance.

## Methods

### Sample Preparations

The sample size is constrained to roughly 10 x 10 x 2 by the mechanical
dimensions of the sample ladder in the $\beta$-SRF cryostat [33]. The Nb
samples are cutouts of RRR $\gtrsim 300$ Nb sheets sourced from ATI Wah Chang
(Albany, Oregon, USA). The samples are prepared simultaneously during
processing of bulk Nb RF cavities. All Nb samples first undergo a “baseline”
treatment as follows:

  1. 1.

BCP, a standard chemical etching solution for SRF cavities containing 2:1:1
volume ratio of H3PO4:HNO3:HF, to remove the first
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of damaged layers due to
cutting/machining. Studies of BCP-treated niobium report typical roughness
values of $\sim 1.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [89].

  2. 2.

Annealing in-house with a high-vacuum furnace at
$1400\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $\sim 4\text{\,}\mathrm{h}$
in order to remove pinning sources in the material [12]. During this annealing
step, all the samples are wrapped in a clean Nb foil.

  3. 3.

An additional $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ BCP is applied to
remove possible furnace contaminations.

  4. 4.

Ultrasound-cleaning with de-ionized water.

A second sample was additionally treated with a “mid-T bake” or “O-doping”
recipe where the cavity/sample receives a “bake” of
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $3\text{\,}\mathrm{h}$ under
ultra-high vacuum. The dimensions of both “baseline” and “O-doping” samples
are shown in table 3 with axis parameters, approximate beam spot size, and
applied field orientation defined in fig. 6. Also listed are
$\mathcal{E}_{M}$, the Meissner state surface field enhancement factor, and
$B_{\text{entry}}$, the expected applied field where flux would break into the
sample center, assuming $B_{c1}$=$174\text{\,}\mathrm{m}\mathrm{T}$ at
$0\text{\,}\mathrm{K}$.

Parameter | “baseline” | “O-doping”
---|---|---
a () | 1.8 | 1.7
b () | 9.8 | 12.4
c () | 8.1 | 12.6
$\mathcal{E}_{M}$ | 1.081 | 1.043
$B_{\text{entry}}$() @ $4.5\text{\,}\mathrm{K}$ | 113 | 123
Table 3: Sample dimensions and the calculated geometrical enhancement factor
$\mathcal{E}_{M}$ in the Meissner state from finite element simulations and
the applied field, $B_{\text{entry}}$, where entry into the bulk is expected.
Figure 6:  Sample dimensions and orientation with respect to the applied
magnetic field and beam implantation. The beamspot is roughly to scale.

###  $\beta$-NMR Experiment of SRF Samples

The $\beta$-NMR technique [34, 35] implants spin-polarized radioactive ions
and measures the local magnetic field at the stopping site of the probe via
detection of the emitted $\beta$-particles. The emission of the
$\beta$-particles is highly anisotropic and correlated with the direction of
the probe’s nuclear spin at the time of the decay. By monitoring the counts of
the $\beta$-particles, the local magnetic field can be measured via its
influence in reorienting the nuclear spin of the probe. Importantly,
$\beta$-NMR provides spatial resolution on the nanometer scale, achieved
through the control of the probe’s implantation energy.

The principle of $\beta$-NMR is similar to LE-$\mu$SR [31, 32], but differs in
the properties of the probe in use (e.g., radioactive lifetime $\tau$$\sim
1.2\text{\,}\mathrm{s}$ for ^8Li vs $\tau$$\sim
2.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ for $\mu^{+}$ ). This difference
makes the two techniques complementary, as well as allowing different
measurements of phenomena occuring over longer timescales. The radioactive
ions used in $\beta$-NMR need to be spin-polarized in-flight (via optical
pumping), whereas $\mu^{+}$ are intrinsically spin-polarized upon production.
A dedicated facility with RIB production, laser polarization, and beam
transport is needed before the beam can be delivered to the spectrometer for
depth-resolved measurements. The following discussion provides more details on
the RIB production, polarization, delivery, and measurements of the
$\beta$-NMR facility.

#### Spin-polarized Probe Production & Beam Transport

High intensity RIBs are routinely produced at the TRIUMF Isotope Separator and
Accelerator (ISAC) facility [90] using high-intensity protons from the
$500\text{\,}\mathrm{MeV}$ cyclotron as a driver to bombard a solid target.
The $\beta$-NMR radioisotope nuclei ^8Li (nuclear spin $I=2$; gyromagnetic
ratio $\gamma_{\text{\ch{^8 Li}}}\,/2\pi$ = 6.30198(8) MHz/T [91], electric
quadrupole moment $Q$ = $31.4\text{\,}\mathrm{\text{mb}}$ [92]; radioactive
lifetime $\tau_{\beta}$ = $\sim 1.21\text{\,}\mathrm{s}$ [93]; and mass
$A_{{}^{8}Li}$ = $8.02\text{\,}\mathrm{u}$) are produced with the isotope
separation on-line (ISOL) method where they are ionized, extracted, and
separated on-line to produce isotopically pure RIBs for various experiments.
At the TRIUMF $\beta$-NMR facility, low-energy ([range-units=single]2030)
^8Li^+ ions of $\sim{10}^{7}\text{\,}{\mathrm{s}}^{-1}$ are routinely
delivered for experiments [34, 35, 36]. During delivery to the experiment, the
ions are neutralized and spin-polarized in-flight at the polarizer facility
via collinear optical pumping with circularly polarized resonant laser light
[94, 95], yielding a high degree ($\sim 70\text{\,}\mathrm{\char
37\relax}$[96]) of nuclear spin-polarization. They are afterwards re-ionized
before delivery to the sample location.

#### High-parallel-field Spectrometer: “$\beta$-SRF beamline”

The spin-polarized ions can be sent for experiments on either one of two
existing beamlines: the $\beta$-NMR beamline with up to $9\text{\,}\mathrm{T}$
applied fields (perpendicular to sample surface & parallel to beam momentum),
and the $\beta$-NQR beamline with one spectrometer station capable of fields
up to $24\text{\,}\mathrm{mT}$ and a second new spectrometer, highlighted
here, capable of $200\text{\,}\mathrm{mT}$ applied fields (each with applied
field parallel to the sample surface & transverse to the beam momentum). Each
spectrometer is equipped with a magnet coil outside the UHV chamber, a pair of
scintillation $\beta$ detectors, an RF coil inside the UHV chamber, and a
cold-finger cryostat. The cryostat allows sample cooling down to $\sim
4\text{\,}\mathrm{K}$ and is mounted on an electrically isolated high voltage
(HV) platform to allow beam energy deceleration within a short distance from
the sample for depth-resolved measurements.

Studies of SRF materials require fields parallel to the sample surface to
emulate the RF magnetic field direction with respect to the SRF cavity wall
and to provide a uniform surface field over the beamspot. This is challenging
to achieve with a low-energy ion beam, due to the strong bending of the beam’s
trajectory in this field geometry. To overcome this, a novel beamline
extension has been designed and implemented [33], allowing for field strengths
of relevance for SRF materials to be employed.

### Data Analysis

The local magnetic field profile can be obtained via depth-resolved
measurements of the ^8Li SLR rate [41, 29]. At the TRIUMF $\beta$-NMR
facility, the SLR measurements are performed by monitoring the transient decay
of the probes’ spin polarization both during and following short pulses of
ions (a typical length is $\Delta=$$4\text{\,}\mathrm{s}$). The spin-
polarization can be inferred from the $\beta$-rates detected at the two
opposing (scintillator) detector pairs. The circular polarization of the
pumping laser is typically switched between pulses resulting to alternating
parallel (“+”) and anti-parallel (“-”) spin-polarizations to reduce systematic
errors. [35, 34] The combined $\beta$-rates (see, e.g., [97, 34]) result in
the experimental asymmetry which is proportional to the spin-polarization:

$\displaystyle A(t)$ $\displaystyle=\frac{1-r}{1+r}\equiv A_{0}P(t),\qquad
r\equiv\sqrt{\frac{(L_{+}/R_{+})}{(L_{-}/R_{-})}}$ (13)

where $L_{+/-}$ and $R_{+/-}$ are rates in the opposing counters for the $+/-$
polarization senses, and the initial asymmetry $A_{0}$ depends on the
properties of the geometry of the detectors (solid angle) and the intrinsic
asymmetry [96, 36].

Once implanted inside the sample, the ions individually interact with the
electromagnetic fields of the host lattice and their spins “relax”
(depolarize) towards a dynamic equilibrium value while the beam is on, and
then towards thermal equilibrium ($\approx 0$) when it is off, with a
characteristic SLR relaxation time ($1/T_{1}$). The magnitude of the magnetic
field at the implantation site acts to slow the relaxation. Implantation
depths can be varied by biasing the sample. By recording the relaxation rates
while varying the depth, the applied field, and/or the sample temperature,
material properties of interest can be extracted. In order to extract the SLR
rate in Nb, the asymmetry spectra are fit to an empirical depolarization
function $p(t,t^{\prime};1/T_{1})$, which is convolved with the rectangular
beam pulse of length $\Delta$, resulting to a bipartite asymmetry function:

$\displaystyle A(t)$
$\displaystyle=A_{0}\begin{cases}\cfrac{R_{0}\int_{0}^{t}\exp[-(t-t^{\prime})/\tau]p(t,t^{\prime};1/T_{1})dt^{\prime}}{N(t)}&\text{for
$t\leq\Delta$,}\\\
\cfrac{R_{0}\int_{0}^{\Delta}\exp[-(t-t^{\prime})/\tau]p(t,t^{\prime};1/T_{1})dt^{\prime}}{N(\Delta)}&\text{for
$t>\Delta$,}\end{cases}$ (14) $\displaystyle N(t)$
$\displaystyle=\int_{0}^{t}N(t,t^{\prime})dt^{\prime}=R_{0}\int_{0}^{t}\exp[-(t-t^{\prime})/\tau]dt^{\prime},$
(15)

where $N(t)$ is the total number of nuclei with mean lifetime $\tau$ that are
present at time $t$ after the beam has turned on and implanted at a constant
rate $R_{0}$.

The ^8Li^+ depolarization in thin film Nb using a stretched exponential
function has been shown to give a good description of ^8Li ’s spin-response at
low magnetic fields [47]. The SLR becomes mono-exponential in high applied
fields [98]. In our measurements, we note that a small fraction of the ^8Li^+
beam stops in the surrounding cryostat components, necessitating inclusion of
an additional “background” term during fitting. This component, however, is
small, slow-relaxing, and easily isolated from the signal from the Nb. The
total depolarization model (both from the Nb sample and the background signal)
can be written as:

$\displaystyle p_{z}(t,t^{\prime})\approx
f_{\ch{Nb}}\exp\left[-\left(\frac{t-t^{\prime}}{T_{1,\ch{Nb}}}\right)^{\beta}\right]+(1-f_{\ch{Nb}})\exp\left[-\left(\frac{t-t^{\prime}}{T_{1,\text{BG}}}\right)\right],$
(16)

where $f_{\ch{Nb}}$ is the amplitude fraction of the signal from the Nb sample
(with values of $f_{\ch{Nb}}$ ranging from 0.82 to 0.94 for “baseline” , and
0.83 to 0.93 for “O-doping” ), $1/T_{1,\ch{Nb}}$ is the relaxation rate of the
Nb sample, $1/T_{1,\text{BG}}$ is the relaxation rate of the background (with
a value that is shared for all runs in each sample, $1/T_{1,\text{BG}}=0.1$
for “baseline” and $1/T_{1,\text{BG}}=0.06$ for “O-doping” ), and $\beta$ is
the stretching exponent (shared for all runs in each sample, $\beta=0.81$ for
“baseline” , $\beta=0.89$ for “O-doping” ). The analysis of the asymmetry
spectra is performed using the Python application bfit [99] to yield the
$1/T_{1}\,$ value for each data set. Examples of the depolarization during
beam ON for three different ion energies are shown in fig. 7. The increase in
depolarization rate with increasing implantation energy is due to Meissner
screening, which reduces the field as a function of depth.

Figure 7:  Normalized asymmetry spectra and fits to the data of the Nb
components (i.e., with background contributions subtracted) at various
implantation energies for the “O-doping” sample at an applied field of
$100\text{\,}\mathrm{mT}$ and at $T$$\sim 4.5\text{\,}\mathrm{K}$. The
increase in depolarization rate with increasing energy (i.e., subsurface
stopping depth) is an indication of Meissner screening.

As summarized in the ‘Results’ section, the analysis employs a direct mapping
from the local magnetic field $B(x)$ with a corresponding Larmor frequency
$\omega_{i}(x)=\gamma_{i}\ B(x)$, to the local relaxation rate
$\frac{1}{T_{1}}(x)$, where $\gamma_{i}$ is the gyromagnetic ratio of the
$i$-th nuclear spin (see below). The largest contribution to relaxation at low
temperature in the superconducting state and at relatively low applied field
($<1\text{\,}\mathrm{T}$) is due to the dipolar interaction between the ^8Li
probe’s nuclear spin $I=2$ and the nuclear spin $S=9/2$ of the 100% abundant
host ^93Nb sample [98, 47]. The dipolar relaxation rate resulting from this
interaction can be expressed as a weighted sum of three Lorentzian functions,
$J_{n}(\omega_{i})$. Each of the Lorentzians is the $n$-quantum spectral
density function that models the stochastic fluctuations in the transverse
component of the local electromagnetic field. This Lorentzian spectral density
function is defined as:

$\displaystyle J_{n}(\omega_{i})$
$\displaystyle=\frac{\tau_{c}}{1+\omega_{i}^{2}\tau_{c}^{2}},$ (17)

where the fit parameter $\tau_{c}$ is an exponential correlation time
constant.

The local relaxation rate due to the dipolar interaction can be expressed as
[40, 47]:

$\displaystyle\frac{1}{T_{1}}(x)$
$\displaystyle=\frac{1}{T_{1}}\left[B(x)\right]$
$\displaystyle=\gamma_{\text{\ch{^8 Li}}}\,\gamma_{\text{\ch{^{93}
Nb}}}\,B_{d}^{2}\left\\{\frac{1}{3}J_{0}(\omega_{S}-\omega_{I})+J_{1}(\omega_{I})+2J_{2}(\omega_{S}+\omega_{I})\right\\},$
(18)

where $B_{d}$ is the magnitude of the fluctuating dipolar field at the ^8Li
site) and is another fit parameter, $\gamma_{\text{\ch{^8 Li}}}\,$ is the
gyromagnetic ratio of the ^8Li probe nuclei, and $\gamma_{\text{\ch{^{93}
Nb}}}\,$ ($=2\pi\times$ 10.439565(3388) /) [91] is the gyromagnetic ratio of
the ^93Nb host nuclei. To reconstruct the local field, modelled as eq. 2 or
eq. 5, we perform a chi-square minimization between the measured SLR rate and
an average of the local SLR rate from eq. 18 using the implantation
distribution for a given energy:

$\displaystyle\left\langle\frac{1}{T_{1}}\right\rangle_{E}=\int_{0}^{\infty}\rho_{E}(x)\frac{1}{T_{1}}(x)dx.$
(19)

Here $\rho_{E}(x)$ is the implantation distribution for ions with implantation
energy $E$. These distributions can be accurately simulated using Monte Carlo
codes such as SRIM [46] and are shown in fig. 8. The fit to the distribution
using a phenomenological distribution function (described in detail in [47])
is also shown. A $5\text{\,}\mathrm{nm}$ layer of Nb2O5 on a Nb substrate is
assumed in the simulation to represent a typical Nb system (see, e.g., [100,
81, 76]).

Figure 8:  Implantation distributions $\rho(x)$ vs. stopping depth $x$ from
SRIM Monte Carlo code [46] simulation, represented as coloured histograms
(with 60 bins), for various ^8 Li^+ energies. Solid lines are fits to a
phenomenological implantation distribution function (for details, see [47]).
The grey area indicates the commonly present Nb2O5 native-oxide layer with
typical thickness of $\sim 5\text{\,}\mathrm{nm}$.

## References

  * [1] Padamsee, H., Knobloch, J. & Hays, T.  _RF Superconductivity for Accelerators_.  Wiley Series in Beam Physics and Accelerator Technology (Wiley, 1998).
  * [2] Padamsee, H.  50 years of success for SRF accelerators—a review.  _Supercond. Sci. Technol._ 30, 053003, DOI: 10.1088/1361-6668/aa6376 (2017).
  * [3] Padamsee, H.  Future Prospects of Superconducting RF for Accelerator Applications.  _Rev. Accel Sci. Technol._ 10, 125–156, DOI: 10.1142/S1793626819300081 (2019).
  * [4] Grassellino, A. _et al._ Nitrogen and argon doping of niobium for superconducting radio frequency cavities: a pathway to highly efficient accelerating structures.  _Supercond. Sci. Technol._ 26, 102001, DOI: 10.1088/0953-2048/26/10/102001 (2013).
  * [5] Gurevich, A.  Maximum screening fields of superconducting multilayer structures.  _AIP Advances_ 5, 017112, DOI: 10.1063/1.4905711 (2015).
  * [6] Kubo, T.  Multilayer coating for higher accelerating fields in superconducting radio-frequency cavities: a review of theoretical aspects.  _Supercond. Sci. Technol._ 30, 023001, DOI: 10.1088/1361-6668/30/2/023001 (2016).
  * [7] Valente-Feliciano, A.-M.  Superconducting RF materials other than bulk niobium: a review.  _Superconductor Science and Technology_ 29, 113002, DOI: 10.1088/0953-2048/29/11/113002 (2016).
  * [8] Tinkham, M.  _Introduction to Superconductivity_.  Dover Books on Physics Series (Dover Publications, 2004).
  * [9] Bean, C. P. & Livingston, J. D.  Surface Barrier in Type-II superconductors.  _Phys. Rev. Lett._ 12, 14–16, DOI: 10.1103/PhysRevLett.12.14 (1964).
  * [10] Liarte, D. B. _et al._ Theoretical estimates of maximum fields in superconducting resonant radio frequency cavities: stability theory, disorder, and laminates.  _Superconductor Science and Technology_ 30, 033002, DOI: 10.1088/1361-6668/30/3/033002 (2017).
  * [11] Prozorov, R., Zarea, M. & Sauls, J. A.  Niobium in the clean limit: An intrinsic type-I superconductor.  _Phys. Rev. B_ 106, L180505, DOI: 10.1103/PhysRevB.106.L180505 (2022).
  * [12] Junginger, T. _et al._ Field of first magnetic flux entry and pinning strength of superconductors for rf application measured with muon spin rotation.  _Phys. Rev. Accel. Beams_ 21, 032002, DOI: 10.1103/PhysRevAccelBeams.21.032002 (2018).
  * [13] Posen, S., Valles, N. & Liepe, M.  Radio frequency magnetic field limits of Nb and Nb3Sn.  _Phys. Rev. Lett._ 115, 047001, DOI: 10.1103/PhysRevLett.115.047001 (2015).
  * [14] Watanabe, K., Noguchi, S., Kako, E., Umemori, K. & Shishido, T.  Development of the superconducting rf 2-cell cavity for cERL injector at KEK.  _Nucl. Instrum. Methods Phys. Res., Sect. A_ 714, 67–82, DOI: 10.1016/j.nima.2013.02.035 (2013).
  * [15] Grassellino, A. _et al._ Accelerating fields up to $49\text{\,}\mathrm{MV}\text{\,}{\mathrm{m}}^{-1}$ in TESLA-shape superconducting RF niobium cavities via $75\text{\,}\mathrm{\SIUnitSymbolCelsius}$ vacuum bake. Preprint at https://arxiv.org/abs/1806.09824 (2018).
  * [16] Ciovati, G.  Effect of low-temperature baking on the radio-frequency properties of niobium superconducting cavities for particle accelerators.  _J. Appl. Phys._ 96, 1591–1600, DOI: 10.1063/1.1767295 (2004).
  * [17] He, F. _et al._ Medium-temperature furnace baking of 1.3 GHz 9-cell superconducting cavities at IHEP.  _Supercond. Sci. Technol._ 34, 095005, DOI: 10.1088/1361-6668/ac1657 (2021).
  * [18] Posen, S., Transtrum, M. K., Catelani, G., Liepe, M. U. & Sethna, J. P.  Shielding Superconductors with Thin Films as Applied to rf Cavities for Particle Accelerators.  _Phys. Rev. Applied_ 4, 044019, DOI: 10.1103/PhysRevApplied.4.044019 (2015).
  * [19] Ito, H., Araki, H., Takahashi, K. & Umemori, K.  Influence of furnace baking on Q–E behavior of superconducting accelerating cavities.  _Prog. Theor. Exp. Phys._ 2021, DOI: 10.1093/ptep/ptab056 (2021).  071G01.
  * [20] Lechner, E. M. _et al._ SIMS Investigation of Furnace-Baked Nb.  In _Proc. SRF’21_ , no. 20 in International Conference on RF Superconductivity, 761–763, DOI: 10.18429/JACoW-SRF2021-THPFDV003 (2022).
  * [21] Lechner, E. M. _et al._ RF surface resistance tuning of superconducting niobium via thermal diffusion of native oxide.  _Appl. Phys. Lett._ 119, 082601, DOI: 10.1063/5.0059464 (2021).
  * [22] Turner, D. A., Burt, G. & Junginger, T.  No interface energy barrier and increased surface pinning in low temperature baked niobium.  _Sci. Rep._ 12, 5522, DOI: 10.1038/s41598-022-09023-0 (2022).
  * [23] Casalbuoni, S. _et al._ Surface superconductivity in niobium for superconducting RF cavities.  _Nucl. Instrum. Methods Phys. Res., Sect. A_ 538, 45–64, DOI: 10.1016/j.nima.2004.09.003 (2005).
  * [24] Balachandran, S. _et al._ Direct evidence of microstructure dependence of magnetic flux trapping in niobium.  _Sci. Rep._ 11, 5364, DOI: 10.1038/s41598-021-84498-x (2021).
  * [25] Wenskat, M. _et al._ Vacancy-Hydrogen Interaction in Niobium during Low-Temperature Baking.  _Sci. Rep._ 10, 8300, DOI: 10.1038/s41598-020-65083-0 (2020).
  * [26] Aull, S. _et al._ Suppressed Meissner-effect in Niobium: Visualized with polarized neutron radiography.  _Journal of Physics: Conference Series_ 340, 012001, DOI: 10.1088/1742-6596/340/1/012001 (2012).
  * [27] Grassellino, A. _et al._ Muon spin rotation studies of niobium for superconducting rf applications.  _Phys. Rev. Spec. Top. Accel Beams_ 16, 062002, DOI: 10.1103/PhysRevSTAB.16.062002 (2013).
  * [28] Romanenko, A. _et al._ Strong Meissner screening change in superconducting radio frequency cavities due to mild baking.  _Appl. Phys. Lett._ 104, 072601, DOI: 10.1063/1.4866013 (2014).
  * [29] McFadden, R. M. L. _et al._ Depth-resolved measurements of the Meissner screening profile in surface-treated Nb.  _Phys. Rev. Appl._ 19, 044018, DOI: 10.1103/PhysRevApplied.19.044018 (2023).
  * [30] Junginger, T., Laxdal, R., MacFarlane, W. A. & Suter, A.  SRF material research using muon spin rotation and beta-detected nuclear magnetic resonance.  _Frontiers in Electronic Materials_ 4, DOI: 10.3389/femat.2024.1346235 (2024).
  * [31] Prokscha, T. _et al._ The new $\mu$E4 beam at PSI: A hybrid-type large acceptance channel for the generation of a high intensity surface-muon beam.  _Nucl. Instrum. Methods Phys. Res., Sect. A_ 595, 317–331, DOI: 10.1016/j.nima.2008.07.081 (2008).
  * [32] Prokscha, T., Salman, Z. & Suter, A.  Low energy $\mu$SR.  In Blundell, S. J., De Renzi, R., Lancaster, T. & Pratt, F. L. (eds.) _Muon Spectroscopy: An Introduction_ , chap. 18, 274–282, DOI: 10.1093/oso/9780198858959.003.0018 (Oxford University Press, Oxford, 2021).
  * [33] Thoeng, E. _et al._ A new high parallel-field spectrometer at TRIUMF’s $\beta$-NMR facility.  _Rev. Sci. Instrum._ 94, 023305, DOI: 10.1063/5.0137368 (2023).
  * [34] MacFarlane, W. A.  Implanted-ion $\beta$NMR: A new probe for nanoscience.  _Solid State Nucl. Magn. Reson._ 68–69, 1–12, DOI: 10.1016/j.ssnmr.2015.02.004 (2015).
  * [35] MacFarlane, W. A.  Status and progress of ion-implanted $\beta$NMR at TRIUMF.  _Z. Phys. Chem._ 236, 757–798, DOI: doi:10.1515/zpch-2021-3154 (2022).
  * [36] Morris, G. D.  $\beta$-NMR.  _Hyperfine Interact._ 225, 173–182, DOI: 10.1007/s10751-013-0894-6 (2014).
  * [37] Lechner, E. M. _et al._ Oxide dissolution and oxygen diffusion scenarios in niobium and implications on the Bean–Livingston barrier in superconducting cavities.  _Journal of Applied Physics_ 135, 133902, DOI: 10.1063/5.0191234 (2024).  https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/5.0191234/19867662/133902_1_5.0191234.pdf.
  * [38] Brandt, E. H.  Superconductors in realistic geometries: geometric edge barrier versus pinning.  _Physica C_ 332, 99 – 107, DOI: 10.1016/S0921-4534(99)00651-6 (2000).
  * [39] Slichter, C. P.  _Spin-Lattice Relaxation and Motional Narrowing of Resonance Lines_ , 145–218 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1990).
  * [40] Mehring, M.  _Principles of High Resolution NMR in Solids_ (Springer, Berlin, 1983), 2 edn.
  * [41] Hossain, M. D. _et al._ Low-field cross spin relaxation of ${}^{8}\text{L}\text{i}$ in superconducting NbSe2.  _Phys. Rev. B_ 79, 144518, DOI: 10.1103/PhysRevB.79.144518 (2009).
  * [42] Lindstrom, M., Wetton, B. & Kiefl, R.  Modelling the effects of surface roughness on superconductors.  _Physics Procedia_ 30, 249–253, DOI: 10.1016/j.phpro.2012.04.084 (2012).  12th International Conference on Muon Spin Rotation, Relaxation and Resonance ($\mu$SR2011).
  * [43] Lindstrom, M., Wetton, B. & Kiefl, R.  Mathematical modelling of the effect of surface roughness on magnetic field profiles in type II superconductors.  _Journal of Engineering Mathematics_ 85, 149–177, DOI: 10.1007/s10665-013-9640-y (2014).
  * [44] Lindstrom, M., Y. Fang, A. C. & Kiefl, R. F.  Effect of surface roughness on the magnetic field profile in the meissner state of a superconductor.  _J. Supercond. Novel Magn._ 29, 1499–1507, DOI: 10.1007/s10948-016-3449-7 (2016).
  * [45] CST Studio Suite: Low-frequency Solver, Dassault Systémes.
  * [46] Ziegler, J. F., Ziegler, M. D. & Biersack, J. P.  SRIM - The stopping and range of ions in matter (2010).  _Nucl. Instrum. Methods Phys. Res., Sect. B_ 268, 1818–1823, DOI: 10.1016/j.nimb.2010.02.091 (2010).
  * [47] McFadden, R. M. L. _et al._ Depth-resolved measurement of the Meissner screening profile in a niobium thin film from spin-lattice relaxation of the implanted $\beta$-emitter ^8Li.  _Journal of Applied Physics_ 134, 163902, DOI: 10.1063/5.0175532 (2023).
  * [48] Brandt, E. H.  Properties of the distorted flux-line lattice near a planar surface.  _Journal of Low Temperature Physics_ 42, 557–584, DOI: 10.1007/BF00117431 (1981).
  * [49] Brandt, E. H.  Penetration of magnetic ac fields into type-II superconductors.  _Phys. Rev. Lett._ 67, 2219–2222, DOI: 10.1103/PhysRevLett.67.2219 (1991).
  * [50] Xie, W., Liu, Y.-H. & Wen, H.-H.  Generalized phenomenological model for the magnetic field penetration and magnetization hysteresis loops of a type-II superconductor.  _Phys. Rev. B_ 105, 014505, DOI: 10.1103/PhysRevB.105.014505 (2022).
  * [51] Tinkham, M.  Effect of fluxoid quantization on transitions of superconducting films.  _Phys. Rev._ 129, 2413–2422, DOI: 10.1103/PhysRev.129.2413 (1963).
  * [52] Finnemore, D. K., Stromberg, T. F. & Swenson, C. A.  Superconducting properties of high-purity niobium.  _Phys. Rev._ 149, 231–243, DOI: 10.1103/PhysRev.149.231 (1966).
  * [53] Bardeen, J., Cooper, L. N. & Schrieffer, J. R.  Theory of superconductivity.  _Phys. Rev._ 108, 1175–1204, DOI: 10.1103/PhysRev.108.1175 (1957).
  * [54] Gurevich, A.  Theory of RF superconductivity for resonant cavities.  _Superconductor Science and Technology_ 30, 034004, DOI: 10.1088/1361-6668/30/3/034004 (2017).
  * [55] Maxfield, B. W. & McLean, W. L.  Superconducting penetration depth of niobium.  _Phys. Rev._ 139, A1515–A1522, DOI: 10.1103/PhysRev.139.A1515 (1965).
  * [56] Maniscalco, J. T., Gonnella, D. & Liepe, M.  The importance of the electron mean free path for superconducting radio-frequency cavities.  _Journal of Applied Physics_ 121, 043910, DOI: 10.1063/1.4974909 (2017).
  * [57] Koufalis, P. N., Hall, D. L., Liepe, M. & Maniscalco, J. T.  Effects of Interstitial Oxygen and Carbon on Niobium Superconducting Cavities. Preprint at https://arxiv.org/abs/1612.08291 (2017).
  * [58] Koch, C. C., Scarbrough, J. O. & Kroeger, D. M.  Effects of interstitial oxygen on the superconductivity of niobium.  _Phys. Rev. B_ 9, 888–897, DOI: 10.1103/PhysRevB.9.888 (1974).
  * [59] DeSorbo, W.  Effect of dissolved gases on some superconducting properties of niobium.  _Phys. Rev._ 132, 107–121, DOI: 10.1103/PhysRev.132.107 (1963).
  * [60] Kogan, V. G. & Prozorov, R.  Orbital upper critical field of type-II superconductors with pair breaking.  _Phys. Rev. B_ 88, 024503, DOI: 10.1103/PhysRevB.88.024503 (2013).
  * [61] Kogan, V. G. & Prozorov, R.  Changing the type of superconductivity by magnetic and potential scattering.  _Phys. Rev. B_ 90, 180502, DOI: 10.1103/PhysRevB.90.180502 (2014).
  * [62] Kogan, V. G. & Prozorov, R.  Critical fields of superconductors with magnetic impurities.  _Phys. Rev. B_ 106, 054505, DOI: 10.1103/PhysRevB.106.054505 (2022).
  * [63] Bawa, A., Gupta, A., Singh, S., Awana, V. P. S. & Sahoo, S.  Ultrasensitive interplay between ferromagnetism and superconductivity in NbGd composite thin films.  _Sci. Rep._ 6, 18689, DOI: 10.1038/srep18689 (2016).
  * [64] Eilenberger, G.  Transformation of Gorkov’s equation for type II superconductors into transport-like equations.  _Z. Phys. A: At. Nucl._ 214, 195–213 (1968).
  * [65] Lin, F. P.-J. & Gurevich, A.  Effect of impurities on the superheating field of type-II superconductors.  _Phys. Rev. B_ 85, 054513, DOI: 10.1103/PhysRevB.85.054513 (2012).
  * [66] Gurevich, A.  Tuning microwave losses in superconducting resonators.  _Superconductor Science and Technology_ 36, 063002, DOI: 10.1088/1361-6668/acc214 (2023).
  * [67] Kubo, T.  Effects of nonmagnetic impurities and subgap states on the kinetic inductance, complex conductivity, quality factor, and depairing current density.  _Phys. Rev. Appl._ 17, 014018, DOI: 10.1103/PhysRevApplied.17.014018 (2022).
  * [68] Yip, S. K. & Sauls, J. A.  Nonlinear Meissner effect in CuO superconductors.  _Phys. Rev. Lett._ 69, 2264–2267, DOI: 10.1103/PhysRevLett.69.2264 (1992).
  * [69] Xu, D., Yip, S. K. & Sauls, J. A.  Nonlinear Meissner effect in unconventional superconductors.  _Phys. Rev. B_ 51, 16233–16253, DOI: 10.1103/PhysRevB.51.16233 (1995).
  * [70] Makita, J., Sundahl, C., Ciovati, G., Eom, C. B. & Gurevich, A.  Nonlinear Meissner effect in Nb3Sn coplanar resonators.  _Phys. Rev. Res._ 4, 013156, DOI: 10.1103/PhysRevResearch.4.013156 (2022).
  * [71] Ginzburg, V. L. & Landau, L. D.  On the Theory of superconductivity.  _Zh. Eksp. Teor. Fiz._ 20, 1064–1082, DOI: 10.1016/B978-0-08-010586-4.50035-3 (1950).
  * [72] Kubo, T.  Superheating fields of semi-infinite superconductors and layered superconductors in the diffusive limit: structural optimization based on the microscopic theory.  _Superconductor Science and Technology_ 34, 045006, DOI: 10.1088/1361-6668/abdedd (2021).
  * [73] Proslier, T., Kharitonov, M., Pellin, M., Zasadzinski, J. & Ciovati.  Evidence of surface paramagnetism in niobium and consequences for the superconducting cavity surface impedance.  _IEEE Trans. Appl. Supercond._ 21, 2619–2622, DOI: 10.1109/TASC.2011.2107491 (2011).
  * [74] Kharitonov, M., Proslier, T., Glatz, A. & Pellin, M. J.  Surface impedance of superconductors with magnetic impurities.  _Phys. Rev. B_ 86, 024514, DOI: 10.1103/PhysRevB.86.024514 (2012).
  * [75] Junginger, T., Wasserman, W. & Laxdal, R. E.  Superheating in coated niobium.  _Supercond. Sci. Technol._ 30, 125012, DOI: 10.1088/1361-6668/aa8e3a (2017).
  * [76] Wenskat, M. _et al._ Vacancy dynamics in niobium and its native oxides and their potential implications for quantum computing and superconducting accelerators.  _Phys. Rev. B_ 106, 094516, DOI: 10.1103/PhysRevB.106.094516 (2022).
  * [77] Lee, J. E. _et al._ Gapless superconductivity in Nb thin films probed by terahertz spectroscopy.  _Nat. Commun._ 14, 2737, DOI: 10.1038/s41467-023-38422-8 (2023).
  * [78] Genenko, Y. A., Snezhko, A. V., Troche, P., Hoffmann, J. & Freyhardt, H. C.  Magnetic self-field entry into a current-carrying type-II superconductor. III. General criterion of penetration for an external field of arbitrary direction.  _Phys. Rev. B_ 57, 1164–1172, DOI: 10.1103/PhysRevB.57.1164 (1998).
  * [79] Asaduzzaman, M., McFadden, R. M. L., Thoeng, E., Laxdal, R. E. & Junginger, T.  Measurements of the first-flux-penetration field in surface-treated and coated Nb: Distinguishing between surface pinning and an interface energy barrier. Preprint at https://arxiv.org/abs/2402.15500 (2024).
  * [80] Kubo, T.  Superfluid flow in disordered superconductors with dynes pair-breaking scattering: Depairing current, kinetic inductance, and superheating field.  _Phys. Rev. Res._ 2, 033203, DOI: 10.1103/PhysRevResearch.2.033203 (2020).
  * [81] Kolb, P. _et al._ Mid-T heat treatments on BCP’ed coaxial cavities at TRIUMF.  _Frontiers in Electronic Materials_ 3, DOI: 10.3389/femat.2023.1244126 (2023).
  * [82] Prudnikava, A., Tamashevich, Y., Makarova, A., Smirnov, D. & Knobloch, J.  In-situ Synchrotron X-Ray Photoelectron Spectroscopy Study of Medium-Temperature Baking of Niobium for SRF Application. Preprint at https://arxiv.org/abs/2312.06529 (2024).
  * [83] Checchin, M. & Grassellino, A.  High-field Q-slope mitigation due to impurity profile in superconducting radio-frequency cavities.  _Applied Physics Letters_ 117, 032601, DOI: 10.1063/5.0013698 (2020).
  * [84] Ngampruetikorn, V. & Sauls, J. A.  Effect of inhomogeneous surface disorder on the superheating field of superconducting RF cavities.  _Phys. Rev. Res._ 1, 012015, DOI: 10.1103/PhysRevResearch.1.012015 (2019).
  * [85] McFadden, R. M. L., Asaduzzaman, M. & Junginger, T.  Comment on “Strong Meissner screening change in superconducting radio frequency cavities due to mild baking” [Appl. Phys. Lett. 104, 072601 (2014)].  _Applied Physics Letters_ 124, 086101, DOI: 10.1063/5.0157000 (2024).
  * [86] Gurevich, A. & Kubo, T.  Surface impedance and optimum surface resistance of a superconductor with an imperfect surface.  _Phys. Rev. B_ 96, 184515, DOI: 10.1103/PhysRevB.96.184515 (2017).
  * [87] Kubo, T. & Gurevich, A.  Field-dependent nonlinear surface resistance and its optimization by surface nanostructuring in superconductors.  _Phys. Rev. B_ 100, 064522, DOI: 10.1103/PhysRevB.100.064522 (2019).
  * [88] Zarea, M., Ueki, H. & Sauls, J. A.  Effects of anisotropy and disorder on the superconducting properties of niobium.  _Frontiers in Physics_ 11, DOI: 10.3389/fphy.2023.1269872 (2023).
  * [89] Xu, C., Tian, H., Reece, C. E. & Kelley, M. J.  Enhanced characterization of niobium surface topography.  _Phys. Rev. ST Accel. Beams_ 14, 123501, DOI: 10.1103/PhysRevSTAB.14.123501 (2011).
  * [90] Dilling, J. & Krücken, R.  The experimental facilities at ISAC.  _Hyperfine Interact._ 225, 111–114, DOI: 10.1007/s10751-013-0886-6 (2014).
  * [91] Stone, N. J.  Table of recommended nuclear magnetic dipole moments: Part I — long-lived states.  INDC(NDS) 0794, International Atomic Energy Agency, Vienna (2019).
  * [92] Stone, N. J.  Table of nuclear electric quadrupole moments.  INDC(NDS) 0833, International Atomic Energy Agency, Vienna (2021).
  * [93] Fléchard, X. _et al._ Measurement of the ^8Li half-life.  _Phys. Rev. C_ 82, 027309, DOI: 10.1103/PhysRevC.82.027309 (2010).
  * [94] Levy, C. D. P. _et al._ Laser polarization facility.  _Hyperfine Interact._ 225, 165–172, DOI: 10.1007/s10751-013-0896-4 (2014).
  * [95] Li, R. _et al._ Recent upgrades and developments at TRIUMF’s laser nuclear-spin-polarization facility.  _Nucl. Instrum. Methods Phys. Res., Sect. B_ 541, 228–231, DOI: 10.1016/j.nimb.2023.05.041 (2023).
  * [96] MacFarlane, W. A. _et al._ The Initial State of Optically Polarized ^8Li^+ from the $\beta$-NMR in Bismuth.  _J. Phys. Conf. Ser._ 551, 012059, DOI: 10.1088/1742-6596/551/1/012059 (2014).
  * [97] Fujimoto, D.  Digging Into MUD with Python: mudpy, bdata, and bfit. Preprint at https://arxiv.org/abs/2004.10395 (2020).
  * [98] Parolin, T. J. _et al._ Nuclear magnetic resonance study of Li implanted in a thin film of niobium.  _Phys. Rev. B_ 80, 174109, DOI: 10.1103/PhysRevB.80.174109 (2009).
  * [99] Fujimoto, D.  bfit: A Python Application For Beta-Detected NMR.  _Journal of Open Source Software_ 6, 3598, DOI: 10.21105/joss.03598 (2021).
  * [100] Halbritter, J.  On the oxidation and on the superconductivity of niobium.  _Applied Physics A_ 43, 1–28, DOI: 10.1007/BF00615201 (1987).

## Acknowledgements

The authors of this work would like to thank: M. Cervantes, D. O. Rosales for
assistance with sample preparation, as well as TRIUMF CMMS, SRF, DAQ,
Operations and Beam Delivery groups for excellent technical support before and
during the experiment. Funding was provided through a Research Tools and
Infrastructure grant [SAPEQ-2015-00005] and Discovery grant [SAPIN-2019-00040]
and [SAPPJ-2020-00030] from the Natural Sciences and Engineering Research
Council of Canada (NSERC).

## Author contributions statement

E.T. and R.E.L. wrote the main manuscript with minor contributions from
R.M.L.M. , and all authors reviewed the manuscript. E.T. and P.K. prepared the
samples. E.T. , R.E.L. , M.A. , and R.M.L.M. designed the research. E.T. ,
R.E.L. , J.O.T. , M.A. , S.R.D. , G.D.M. , V.L.K. , D.F. , W.A.M. , R.L. , and
S.S. contributed to the $\beta$-NMR measurements. E.T. and R.E.L. performed
the analysis. Fitting routines were developed by E.T. and R.E.L. based on
original routines developed by R.M.L.M. R.E.L. , T.J. , R.F.K. , and W.A.M.
supervise the contributing participants of this experiment. R.E.L. , R.F.K. ,
and T.J. contribute to the funding acquisition.

## Data availability statement

The data and fitting routines supporting the findings of this study are
available from the corresponding authors upon reasonable request. Raw data
from the $\beta$-NMR experiments (experiment number M1963) are available for
download from: https://cmms.triumf.ca.

## Additional information

Competing interests: The authors declare no competing financial interests.