# Nonlinear Meissner effect in Nb3Sn coplanar resonators.

J. Makita1, C. Sundahl2, G. Ciovati1,3, C.B. Eom2, and A. Gurevich  Department
of Physics and Center for Accelerator Science, Old Dominion University,
Norfolk, Virginia 23528, USA
2Department of Materials Science and Engineering, University of Wisconsin-
Madison, Madison, Wisconsin 53706, USA
3 Thomas Jefferson National Accelerator Facility, Newport News, Virginia
23606, USA

###### Abstract

We investigated the nonlinear Meissner effect (NLME) in Nb3Sn thin film
coplanar resonators by measuring the resonance frequency as a function of a
parallel magnetic field at different temperatures. We used low rf power
probing in films thinner than the London penetration depth $\lambda(B)$ to
significantly increase the field onset of vortex penetration and measure the
NLME under equilibrium conditions. Contrary to the conventional quadratic
increase of $\lambda(B)$ with $B$ expected in s-wave superconductors, we
observed a nearly linear increase of the penetration depth with $B$. We
concluded that this behavior of $\lambda(B)$ is due to weak linked grain
boundaries in our polycrystalline Nb3Sn films, which can mimic the NLME
expected in a clean d-wave superconductor.

###### pacs:

74.25.-q, 74.25.Ha, 74.25.Op, 74.78.Na

The Meissner effect is one of the fundamental manifestations of the
macroscopic phase coherence of a superconducting state. Meissner screening
current density ${\bf J}=-en_{s}{\bf v}_{s}$ induced by a weak magnetic field
is proportional to the velocity $\bf{v}_{s}$ of the condensate. At higher
fields, the superfluid density $n_{s}$ becomes dependent on ${\bf v}_{s}$ due
to pairbreaking effects, resulting in the nonlinear Meissner effect (NLME) GL
; YS1 ; YS2 ; Dahm1 ; Dahm2 ; H1 ; H2 ; Prozorov . For a single band isotropic
s-wave superconductor, the NLME is described by:

${\bf J}=-\frac{\phi_{0}{\bf
Q}}{2\pi\mu_{0}\lambda^{2}}(1-\Upsilon\xi^{2}Q^{2}),\\\ $ (1)

where $\lambda$ is the London penetration depth, $\xi$ is the coherence
length, ${\bf Q}=m{\bf v}_{s}/\hbar=\nabla\chi+2\pi{\bf A}/\phi_{0}$, $m$ is
the quasiparticle mass, $\chi$ is the phase of the order parameter
$\Psi=\Delta e^{i\chi}$, ${\bf A}$ is the vector potential, $\phi_{0}$ is the
magnetic flux quantum, and the factor $\Upsilon(T,l_{i})$ depends on the
temperature $T$, the mean free path $l_{i}$ and details of pairing mechanisms.
Ginzburg and Landau (GL) were the first who obtained the field dependent-
correction to the penetration depth $\lambda(B)$ of the magnetic field ${\bf
B}$ parallel to a semi-infinite superconductor GL :

$\lambda(B)=\left[1+\frac{\kappa(\kappa+2^{3/2})B^{2}}{8(\kappa+2^{1/2})^{2}B_{c}^{2}}\right]\lambda,$
(2)

where $B_{c}=\phi_{0}/2^{3/2}\pi\lambda\xi$ is the thermodynamic critical
field and $\kappa=\lambda/\xi$ is the GL parameter.

In recent years, the NLME has attracted much attention as a probe of
unconventional pairing symmetries of moving condensates. Particularly, Yip and
Sauls YS1 ; YS2  showed that in a clean d-wave superconductor at
$k_{B}T<p_{F}v_{s}$ the supercurrent acquires a nonlinear singular term
$\propto|{\bf Q}|{\bf Q}$ strikingly different from that in Eq. (1), where
$p_{F}$ is the Fermi momentum. Yet Eq. (1) can describe a variety of nonlinear
electromagnetic responses, both in conventional and unconventional
superconductors. For instance Eq. (1) describes a clean d-wave superconductor
at high temperatures $T>p_{F}v_{s}/k_{B}$ or a d-wave superconductor with
impurities YS1 ; YS2 ; Dahm1 ; Dahm2 ; H1 ; H2 . In a clean s-wave
superconductor, the NLME is absent at $T\ll T_{c}$ as
$\Upsilon\propto\exp(-\Delta/k_{B}T)$ Bardeen , but occurs in the dirty limit
$(\xi\lesssim l)$ in which $\Upsilon\sim 1$ even at $T\to 0$ Maki . In
multiband superconductors the NLME could probe the proliferation of interband
phase textures GV  or the line nodes and interband sign change in the order
parameter or mixed $s-d$ pairing symmetries in iron pnictides FBS .

So far the observations of the NLME in high-$T_{c}$ cuprates have been
inconclusive nme1 ; nme2 ; nme3 ; nme4 ; nme5 ; oates ; carr , mostly because
of a small field region of the Meissner state in high-$\kappa$ type-II
superconductors and contributions of extrinsic materials factors, such as
grain boundaries or local nonstoichiometry. Since the NLME becomes essential
in fields $B$ of the order of $B_{c}$, penetration of vortices above the lower
critical field $B_{c1}=(\phi_{0}/4\pi\lambda^{2})(\ln\kappa+0.5)\ll B_{c}$ ehb
limits the nonlinear correction in Eq. (2) to
$(B_{c1}/B_{c})^{2}/8\simeq\ln\kappa/16\kappa^{2}\ll 1$. Yet even small NLME
terms in Eq. (1) causes intermodulation effects Dahm1 ; Dahm2  under strong ac
fields, as it was observed in YBa2Cu3O7-x nme5 ; oates .

In this paper we investigate the NLME in a thin film Nb3Sn coplanar resonator
in a parallel dc magnetic field, which mitigates the problems of vortex
penetration and nonequilibrium effects. We used the method of Ref. ggc,  in
which the resonant frequency $\omega=(LC)^{-1/2}$ of a coplanar resonator is
measured as a function of a parallel dc field $B$. Here $C$ is the strip-to-
ground capacitance and $L=L_{g}+L_{k}$ is the total inductance containing both
the geometrical inductance $L_{g}$, and the field-dependent kinetic inductance
of the superconducting condensate, $L_{k}(B)\propto\lambda^{2}(B)$ OD . To
extend the field region of the NLME, we performed our measurements on thin
films of thickness $d<\lambda$ for which $B_{c1}=(2\phi_{0}/\pi
d^{2})\ln(d/\xi)$ can be much higher than the bulk $B_{c1}$ Abrikos . Rotating
the field in the plane of the film gives rise to an orientational dependence
of $\lambda(B)$ ggc . Unlike the standard quadratic field correction to
$\lambda(B)$ observed on Nb ggc  and Al Al  thin film resonators, we observed
a nearly linear field dependence of $\lambda(B)$ in polycrystalline Nb3Sn
films. The fact that the NLME in the s-wave superconductor Nb3Sn exhibits the
behavior expected from a clean d-wave superconductor YS1 ; YS2  shows the
importance of materials factors, particularly local nontoichiometry and weakly
coupled grain boundaries characteristic of Nb3Sn, cuprates and pnictides gb ;
grb .

The paper is organized as follows. In Sec. II we describe the experimental
setup and Nb3Sn coplanar resonator used in the measurements of NLME. Sec. III
contains the main experimental results. Sec. IV summarizes the essential
mechanisms of NLME and theoretical results necessary for the comparison of
theory with experiment. Sec. V contains discussion of our results.

##  I Experimental

###  I.1 Film Deposition and Patterning

The coplanar resonator was fabricated from a 50 nm thick Nb3Sn film on a 10 mm
$\times$ 10 mm $\times$ 1 mm Al2O3 substrate. The film was prepared with
magnetron co-sputtering using both Nb and Sn targets in a growth chamber at
University of Wisconsin-Madision, as described in Refs. chris, ; ml, . Figure
1 shows the resistive transition in a film grown under a similar condition.
The film had a midpoint $T_{c}\approx 17.2$ K, normal state sheet resistance
of $5.1\,\Omega$, and a residual resistance ratio (RRR),
$R_{s}(300\,K)/R_{s}(18\,K)\approx 3.2$.

Figure 1: Temperature dependence of the resistance of the film $R_{s}(T)$ with
a midpoint critical temperature $T_{c}=17.2$ K.

The film has a polycrystalline structure with rigid grains along the [-1011]
direction of the Al2O3 substrate as revealed by the atomic force microscopy
shown in Fig. 2. Those grains contributed to an RMS roughness of approximately
10 nm chris ; ml .

Figure 2: AFM image showing a polycrystalline structure of our films.

The sample was patterned into a half-wave coplanar waveguide resonator using
contact lithography followed by Ar ion milling. The optical image of the
resonator is shown in Fig. 3. The meandered resonator has a total length
$l\approx 24.6$ mm corresponding to the fundamental resonant frequency
$f_{0}=2.236$ GHz. The center conductor has a width $w=15\,\mu$m, and a gap
width $s=8.8\,\mu$m between the center strip and the ground plane. The $s/w$
ratio was set to achieve a characteristic line impedance $50\,\Omega$. The
resonator is coupled to input and output RF probes by interdigital capacitors
patterned on the strip. At the ends of the transmission line, landings pads
for ground-signal-ground (GSG) probes were fabricated, shown as the lightly
shaded region in Fig. 3(a). These were made by first removing few nanometers
of oxide layers on the surface of Nb3Sn using Ar ion milling and then
depositing a 20 nm thick layer of Pd in-situ using a lift-off technique. The
landing pads made of Pd serve to prevent oxidation and damage of the film from
repeated touchdown of the probes and ensure ohmic contact between the probe
and the sample.

Figure 3: (a) The image of the Nb3Sn coplanar half-wave resonator with
$f_{0}=2.236$ GHz. The meandered resonator in the center is terminated
capacitively on both ends which tapers out to the input and output landing
pads. (b) Zoomed in section of the coplanar resonator where the width of the
strip is $w=15\,\mu\mathrm{m}$ and the gap between the signal strip and the
ground is $s=8.8\,\mu\mathrm{m}$.

###  I.2 Measurement Setup

The patterned sample was mounted inside a cryogenic probe station equipped
with a closed cycle cryocooler probe , as shown in Fig. 4. The complex
transmission coefficient $S_{21,12}$ was measured as functions of temperature
and the external in-plane magnetic field $B$. The resonant frequencies and the
London penetration depth were extracted by fitting the dependence of the phase
on frequency of the transmission spectra pp ; anlage ; porch ; SL ; collin ;
hein ; mohebbi . The temperature of the sample was varied using a resistive
heater underneath the sample stage, and a parallel dc field up to 200 mT was
produced with a NbTi superconducting magnet. This magnet was mounted on a six-
motor hexapod system that allowed for fine tuning of magnet orientation by
$\pm 7^{\circ}$ in three axes while taking the sample measurements. The
temperature of the sample was measured using a calibrated Cernox (CX-1050-CU-
HT, Lakeshore Cryotronics) resistance-temperature device fixed on the stage
next to the sample. The output port of a Vector Network Analyzer (VNA)
provided rf power that was delivered to the resonator by landing two GSG
probes to the contacts. These probes and the cables connecting to the network
analyzer were calibrated using a Short-Open-Load-Through calibration substrate
mounted on the sample stage at 7 K. The drive power of VNA was selected to be
-30 dBm to maximize the signal-to-noise ratio while avoiding distortion of the
Lorentzian shape in transmission signal observed at higher power due to
nonlinear heating effects SL ; abdo  as shown in Fig. 5. To minimize the
number of vortices trapped in the sample during its cooldown through $T_{c}$,
we used three pairs of Helmholtz coils to reduce the ambient field $B_{a}$.
The magnitude of $B_{a}$ was measured by a magnetometer while adjusting the
coil currents to achieve $B_{a}<$4 mG in an optimum configuration.

Figure 4: A setup of the sample stage with two GSG probes. The sample is
mounted using silver paint, and the thermometer and the OFHC bobbin for
thermal anchoring of the lead wires are screwed onto the sample stage.  Figure
5: A typical transmission spectrum near resonance displays a symmetric
Lorentzian line shape with VNA power output of -30 dBm. Higher input power
distorts the Lorentzian shape at -5 dBm.

The resonance frequency $f_{0}=(CL)^{-1/2}/2\pi$ is determined by the ground
capacitance $C$ and the resonator inductance $L$. Here $L=L_{g}+L_{k}$
contains a geometrical inductance $L_{g}$ and a kinetic inductance
$L_{k}(T,B)$ associated with the inertia of supercurrents. The geometric
inductance for the parameters of our sample $L_{g}=420.5$ nH/m was calculated
in Appendix A following Ref. collin, . The kinetic inductance for a thin film
strip of thickness $d<\lambda(T)$ and width $w$ is given by OD ; porch ;
anlage

$L_{k}\approx\frac{\mu_{0}\lambda(T)}{w}\coth\left[\frac{d}{\lambda(T)}\right]\approx\frac{\mu_{0}\lambda^{2}(T)}{wd}.$
(3)

Temperature dependencies of $L_{k}(T)$ and $\lambda(T)$ were inferred from the
measured frequency shift $\delta f/f_{0}=[f_{0}(T)-f_{0}(7K)]/f_{0}(7K)$:

$\frac{\delta
f(T)}{f_{0}(7K)}=\frac{\sqrt{L_{g}+L_{k}(7K)}}{\sqrt{L_{g}+L_{k}(T)}}-1,$ (4)

where $C$ and $L_{g}$ are assumed independent of $T$. By fitting the observed
$\delta f(T)/f_{0}(7K)$ to Eqs. (3) and (4), we obtained $\lambda(T)$ as
described in the next subsection.

For the NLME measurements, the alignment of the dc field ${\bf B}$ to the
plane of the strip is crucial to keep the superconductor in the Meissner state
and avoid perpendicular vortices penetrating from the film edges. These
vortices caused by the misaligned field reduce the quality factor and give
rise to an additional field dependence of $\delta f(B,T)$ unrelated to the
NLME. To find the orientation of the magnet which produces ${\bf B}$ parallel
to the film plane and the minimum amount of trapped flux, the loaded quality
factor $Q_{L}({\bf B})$ and $\delta f({\bf B})$ were measured as functions of
the out-of-plane field angle $\zeta$. We first measured the initial values of
$f_{0i}$ and $Q_{Li}$ at zero field, ramped the field up to 60 mT and down to
zero, and measured the resulting values of $f_{0a}$ and $Q_{La}$ affected by
the number of vortices trapped in the process. The sample was then thermal
cycled above $T_{c}=17.2$ K at zero field to flush out trapped vortices.
Measurements were repeated after the magnet was adjusted to a new angle. Shown
in Fig. 6 are the normalized shifts $\delta f_{0}/f_{0}=(f_{0a}-f_{0i})/f_{0}$
and $\delta Q_{L}/Q_{L}=(Q_{La}-Q_{Li})/Q_{L}$ as functions of the magnet
angle $\zeta$. Both $\delta f_{0}(\zeta)/f_{0}$ and $\delta
Q_{L}(\zeta)/Q_{L}(0)$ peaked at $\zeta=3.8^{\circ}$ which we adopted as a
magnet orientation producing the dc field parallel to the plane of the film.
This procedure is similar to that which was used in Ref. ggc, .

Figure 6: Normalized shifts in (a) resonant frequencies and (b) loaded quality
factors after cycling $B$ from $0$ to $60$ mT and back to $0$ as a function of
the offset angle $\zeta$. Both $\delta f(\zeta)$ and $\delta Q_{L}(\zeta)$ are
peaked at $\zeta=3.8^{\circ}$.

Having aligned the magnet, we measured $f_{0}(T,B)$ as a function of in-plane
dc field up to 200 mT at parallel $\varphi=0^{\circ}$ and perpendicular
$\varphi=90^{\circ}$ field orientations with respect to the strip indicated by
Fig. 7 at temperatures between 7 K and 12 K. After measurements at a given
temperature were completed, the sample was warmed up above $T_{c}$ to expel
any trapped vortices. For each field and temperature point, we repeated the
measurement over 50 times, and the average $f_{0}(B,T)$ was calculated.

Figure 7: The coordinate system for the magnet orientation with respect to the
direction of the rf current on the coplanar resonator.

###  I.3 Temperature Dependence

The temperature-dependence of $\lambda(T)$ can be obtained from the
measurements of the resonance frequency $f_{0}(T)$. Using Eqs. (3) and (4) we
extracted $\lambda(T)$ from the measured $f_{0}(T)$ by fitting the temperature
dependence of a relative frequency shift $\delta
f/f_{0}=[f_{0}(T)-f_{0}(7K)]/f_{0}(7K)$ with the conventional two-fluid
approximation of $\lambda(T)=\lambda(0)[1-(T/T_{c})^{4}]^{-1/2}$. Shown in
Fig. 8 is the temperature dependent part of $f_{0}(T)$ along with the fit with
Eq. 4. The fit gives $\lambda(0)=353$ nm, well above the London penetration
depth $\lambda(0)\approx 90$ nm for a clean stoichiometric Nb3Sn hein . The
latter may result from nonstoichimetric inclusions which cause a slight
reduction of $T_{c}$ in our films chris ; ml . For $\lambda(0)=352$ nm,
$w=15\,\mu$m and $d=50$ nm, the kinetic inductance
$L_{k}=\mu_{0}\lambda^{2}(T)/wd\simeq 200$ nH/m at $9$ K accounts for about
$1/3$ of the total inductance $L=L_{g}+L_{k}$ with $L_{g}=420.5$ nH/m.

Another factor contributing to the large value of $\lambda(0)$ is the grain
boundary structure of our Nb3Sn films shown in Fig. 2. It has been well-
established that Sn depletion at GBs gb1 ; gb2 ; gb3 ; gb4  results in weak
Josephson coupling of crystalline grains, which has been used to optimize
pinning of vortices by GBs in Nb3Sn conductors arno ; pin . In turn, the
weakly-coupled GBs facilitate preferential penetration of the magnetic field
along the GB network causing an increase of the global $\lambda$, as is
characteristic of many superconductors with short coherence length, including
Nb3Sn, cuprates and pnictides grb ; gb .

Figure 8: The normalized temperature-dependent part of the resonant frequency.
The fit of the data to Eq. 4 gives $\lambda(0)=353$ nm.

###  I.4 Field Dependence

Figure 9: Normalized frequency shift $\delta f_{0}(T,B)/f_{0}(T,0)$ as a
function of the in-plane dc magnetic field (a) parallel and (b) perpendicular
to the strip.

Shown in Fig. 9 are the observed field dependencies of the frequency shifts
for in-plane ${\bf B}$ parallel and perpendicular to the strip. In both cases
$\delta f(B)$ decreases nearly linearly with $B$ above $30-40$ mT but flattens
at lower fields. Here the slope of $\delta f(B)$ for in-plane ${\bf B}$
parallel to the strip is about twice of the slope of $\delta f(B)$ for in-
plane ${\bf B}$ perpendicular to the strip. In the field range $0<B<200$ mT of
our measurements the Nb3Sn film of thickness $50$ nm is in the Meissner state
as the parallel field $B$ remains below the nominal lower critical field of a
vortex in a thin film. Indeed, an estimate of $B_{c1}=(2\phi_{0}/\pi
d^{2})\ln(d/\xi)$ in the London approximation Abrikos  with $d=50$ nm and
$\xi=5$ nm orlando ; arno  yields $B_{c1}\simeq 1.17$ tesla exceeding
$B_{c}=0.54$ tesla of Nb3Sn orlando ; arno . The slope of $\delta f(B)$ in
Fig. 9 increases with increasing temperature, consistent with the temperature
dependence of $\lambda(T)$.

Our $\delta f(B)$ data exhibit significant scatter, as has also been observed
in NLME experiments on cuprate nme3 . The error bars in Fig. 9 represent a
standard deviation from repeated measurements at each data points, $\delta
f(B)$ data for ${\bf B}\|y$ having larger error bars as compared to ${\bf
B}\|z$. The main contribution to the error bars comes from vibrations of the
sample stage and GSG probes originating from the cryocooler. For ${\bf B}$
perpendicular to the strip, longer probe arms had to be used in the NLME
measurements, which increases the amplitude of vibrations.

##  II Contributions to NLME

In this section we consider different contributions to the resonant frequency
shift $\delta f(T,B)$ caused by the in-plane dc magnetic field $B$.

###  II.1 Meissner current pairbreaking

We start with the calculation of the contribution of pairbreaking Meissner
currents to $\delta f(H)$ using the TDGL equations for a dirty s-wave
superconductor Kopnin ; tdgl :

$\displaystyle\tau_{GL}(1+4\tau_{E}^{2}\Delta^{2})^{-1/2}\left(\frac{\partial}{\partial
t}+2ie\Phi+2\tau_{E}^{2}\frac{\partial\Delta^{2}}{\partial t}\right)\Psi$
$\displaystyle=\left(1-\frac{\Delta^{2}}{\Delta_{0}^{2}}\right)\Psi+\xi^{2}\left(\mathbf{\nabla}-2ie\mathbf{A}\right)^{2}\Psi,$
(5)
$\displaystyle\mathbf{J}=-\frac{\pi\sigma_{0}}{4eT_{c}}\Delta^{2}\mathbf{Q}-\sigma_{0}\left(\mathbf{\nabla}\Phi+\frac{\partial\mathbf{A}}{\partial
t}\right).$ (6)

Here $\tau_{GL}=\pi\hbar/8k_{B}(T_{c}-T)$, $\xi=[\pi\hbar
D/8k_{B}(T_{c}-T)]^{1/2}$ is the coherence length, $D$ the electron
diffusivity, $\Phi$ is a scalar potential, $\tau_{E}$ is an energy relaxation
time due to electron-phonon scattering Kopnin ,
$\Delta_{0}^{2}=8\pi^{2}k_{B}^{2}T_{c}(T_{c}-T)/7\zeta(3)$,
$\sigma_{0}=2e^{2}DN(0)$ is the normal state conductivity, $N(0)$ is the
density of states at the Fermi surface, and $-e$ is the electron charge.
Equations (5) and (6) were derived from the kinetic BCS theory assuming that
$\mathbf{Q}(\mathbf{r},t)$ and $\Delta(\mathbf{r},t)$ vary slowly over
$\xi_{0}\simeq(\hbar D/k_{B}T_{c})^{1/2}$, the diffusion length
$L_{E}=(D\tau_{E})^{1/2}$ and $\tau_{E}$ tdgl ; Kopnin , where

$\tau_{E}=\frac{8\hbar}{7\pi\zeta(3)\gamma
k_{B}T_{F}}\left(\frac{c_{s}}{v_{F}}\right)^{2}\left(\frac{T_{F}}{T}\right)^{3}.$
(7)

Here $c_{s}$ is the speed of longitudinal sound, $v_{F}$ and
$T_{F}=\epsilon_{F}/k_{B}$ are the Fermi velocity and temperature,
respectively, and $\gamma$ is a dimensionless electron-phonon coupling
constant. For $c_{s}/v_{F}\simeq 10^{-3}$, $T_{F}\sim 10^{5}$ K, $T_{c}=17$ K
and $\gamma\simeq 1.5$ orlando , Eq. (7) yields $\tau_{E}(T_{c})\sim 10$ ps.

For a wide film in a parallel magnetic field, $Q(x)$ and $\Delta(x)$ depend
only on the coordinate $x$ across the film, and the TDGL equations in the
gauge $\Phi=0$ can be written in the dimensionless form:

$\displaystyle(1+g^{2}\psi^{2})^{1/2}\dot{\psi}=(1-q^{2})\psi+\psi^{\prime\prime}-\psi^{3},$
(8) $\displaystyle j=-u\psi^{2}q-\dot{q},$ (9)

where $\psi=\Delta/\Delta_{0}$, $q=Q\xi$, $g=2\Delta_{0}\tau_{E}/\hbar$,
$j=J/J_{0}$, $t$ is in units of $\tau_{GL}$,
$J_{0}=\sigma_{0}/2e\xi\tau_{GL}$, $x$ is in units of $\xi$, the prime and
overdot denotes differentiation with respect to $x$ and $t$, respectively, and
$u=\pi^{4}/14\zeta(3)\approx 5.79$.

For a coplanar resonator of thickness $d<\lambda$ and width $w\gg\lambda$ in a
parallel dc field ${\bf B}$ inclined by the angle $\varphi$ to the $z$-axis
along the strip, we have:

$q_{z}=-hx\sin\varphi+a_{\omega}e^{i\omega t},\qquad q_{y}=hx\cos\varphi,$
(10)

where $h=B/B_{c2}$, $B_{c2}=\phi_{0}/2\pi\xi^{2}$,
$a_{\omega}=A_{\omega}e^{i\omega t}/A_{0}$, $A_{\omega}$ is a small rf vector
potential excited along the strip, $A_{0}=\phi_{0}/2\pi\xi$, $x=0$ is taken in
the middle of the film and the London screening at $d\ll\lambda$ is
disregarded.

At $a_{\omega}=0$ the dc field causes a reduction of $\psi(x)=1-\psi_{1}(x)$.
The equation for a field-induced correction $\psi_{1}$ is obtained from Eq.
(8) in the first order in $h^{2}\ll 1$:

$\psi_{1}^{\prime\prime}-2\psi_{1}=-h^{2}x^{2},\qquad\psi_{1}^{\prime}(\pm
d/2)=0.$ (11)

where $d$ is in units of $\xi$. The squared order parameter averaged over the
film thickness, $\bar{\psi}^{2}\simeq 1-2d^{-1}\int_{-d/2}^{d/2}\psi_{1}dx$,
is obtained by integrating Eq. (11) over $x$:

$\bar{\psi}^{2}=1-\frac{d^{2}h^{2}}{12}.$ (12)

To calculate a linear response current $\delta I_{\omega}e^{i\omega t}$
induced by a weak $a_{\omega}e^{i\omega t}$ in the presence of a parallel dc
field, we linearize Eq. (9) in $a_{\omega}$:

$\frac{\delta
I_{\omega}}{wJ_{0}}=-ud\bar{\psi}^{2}a_{\omega}+2hu\sin\varphi\\!\int_{-d/2}^{d/2}\\!\\!x\delta\psi(x)dx-
id\omega a_{\omega}.$ (13)

This shows that $\delta\psi$ is coupled with the dc Meissner current flowing
along the strip due to the field component $B_{y}=B\sin\varphi$. Here
$\delta\psi(x)\propto a_{\omega}$ satisfies the following equation obtained
from Eq. (8) linearized with respect to small $\delta\psi$ and $a_{\omega}$:

$\displaystyle\delta\psi^{\prime\prime}-k_{\omega}^{2}\delta\psi=-2a_{\omega}hx\sin\varphi,\qquad\delta\psi^{\prime}(\pm
d/2)=0,$ (14) $\displaystyle
k_{\omega}^{2}=2+i\omega\tau,\qquad\tau=\tau_{GL}\sqrt{1+(2\tau_{E}\Delta/\hbar)^{2}}.$
(15)

The solution of Eq. (14) is given by:

$\displaystyle\delta\psi(x)=\sum_{n=0}^{\infty}A_{n}\sin q_{n}x,\qquad
q_{n}=\frac{\pi}{d}(2n+1),$ (16) $\displaystyle
A_{n}=\frac{8(-1)^{n}a_{\omega}h\sin\varphi}{dq_{n}^{2}(k_{\omega}^{2}+q_{n}^{2})}.$
(17)

Inserting $\delta\psi(x)$ in Eq. (13) and integrating over $x$ yields:

$\frac{\delta
I_{\omega}}{dwJ_{0}}=-\biggl{[}u\bar{\psi}^{2}-\sum_{n=0}^{\infty}\frac{32ud^{2}h^{2}\sin^{2}\varphi}{\pi^{4}(2n+1)^{4}(2+i\omega
r+q_{n}^{2})}+i\omega\biggr{]}a_{\omega}$ (18)

The sum in Eq. (18) converges rapidly, so $q_{n}^{2}\propto(\xi/d)^{2}\ll 1$
in the denominator can be neglected in films with $\xi\ll d\ll\lambda$. Using
$\sum_{n=0}^{\infty}(2n+1)^{-4}=\pi^{4}/96$ and restoring the original units,
we obtain the linear response current $\delta I_{\omega}$ induced by the ac
vector potential $A_{\omega}$:

$\delta
I_{\omega}=-\left(\frac{1}{\mu_{0}\tilde{\lambda}_{\omega}^{2}}+i\omega\sigma_{0}\right)dwA_{\omega},$
(19)

where the complex London penetration depth is given by:

$\displaystyle\frac{1}{\tilde{\lambda}_{\omega}^{2}}=\frac{1}{\lambda^{2}}\bigg{[}1-\frac{1}{3}\left(\frac{2\pi\xi
dB}{\phi_{0}}\right)^{2}\left(\frac{1}{4}+\frac{\sin^{2}\varphi}{2+i\omega\tau}\right)\bigg{]}.$
(20)

Here the NLME term $\propto B^{2}$ depends on $\omega$ and the field
orientation angle $\varphi$. The maximum NLME contribution to
$\lambda_{\omega}$ occurs at $\varphi=\pi/2$ when ${\bf B}\perp z$ and the rf
current is parallel to the dc Meissner current. At $\omega\tau\ll 1$ the
angular dependence of $\lambda_{\omega}(\varphi)$ in Eq. (20) reduces to that
was obtained previously in a quasi-static limit ggc .

The imaginary part of $\tilde{\lambda}_{\omega}^{2}$ contributes to the
dynamic conductivity $\sigma_{\omega}$. Denoting
$\lambda_{\omega}^{2}=\mbox{Re}\tilde{\lambda}_{\omega}^{2}$, and separating
real and imaginary parts in Eqs. (19) and (20), yields:

$\displaystyle\lambda_{\omega}^{2}=\biggl{[}1+\frac{1}{3}\left(\frac{2\pi\xi
dB}{\phi_{0}}\right)^{2}\left(\frac{1}{4}+\frac{2\sin^{2}\varphi}{4+\omega^{2}\tau^{2}}\right)\biggr{]}\lambda^{2},$
(21)
$\displaystyle\sigma_{\omega}=\sigma_{0}+\frac{1}{3\mu_{0}\lambda^{2}}\left(\frac{2\pi\xi
dB}{\phi_{0}}\right)^{2}\frac{\tau\sin^{2}\varphi}{4+\omega^{2}\tau^{2}}.$
(22)

The NLME field correction to the kinetic inductance
$L_{k}^{M}=\mu_{0}\lambda^{2}(B)/dw$ is given by:

$\delta L_{k}^{M}=\frac{\mu_{0}\lambda^{2}}{3dw}\left(\frac{\pi\xi
dB}{\phi_{0}}\right)^{2}\left[1+\frac{2\sin^{2}\varphi}{1+(\omega\tau/2)^{2}}\right].$
(23)

As follows from Eqs. (22) and (23), the NLME correction to $\sigma_{\omega}$
remains finite at $T_{c}$, while $\delta L_{k}^{M}\propto(T_{c}-T)^{-2}$
increases stronger than the zero-field kinetic inductance
$L_{k}^{M}=\lambda^{2}/d\propto(T_{c}-T)^{-1}$ as $T\to T_{c}$. The
dependencies of $\lambda_{\omega}$ and $\sigma_{\omega}$ on the orientation of
${\bf B}$ persist as long as $\omega\tau\lesssim 1$ and disappear at
$\omega\tau\gg 1$. The latter occurs both at $T\to T_{c}$ where $\tau_{GL}(T)$
diverges and at low temperatures where $\tau_{E}(T)\propto T^{-3}$ increases
strongly.

We estimate $\delta\lambda(B)=\lambda(B)-\lambda=(\pi\xi dB/\phi_{0})^{2}/2$
at $\varphi=\pi/2$ and $\omega\tau\ll 1$ for $d=50$ nm and $\xi=5$ nm. Here
$\delta\lambda/\lambda=7.7\times 10^{-4}$ at $B=100$ mT, which translates to
$\delta\lambda=0.27$ nm at $\lambda=350$ nm. If the total $L$ is dominated by
the kinetic inductance, Eq. (23) yields the maximum NLME frequency shift:

$\frac{\delta f}{f}=-\frac{1}{6}\left(\frac{\pi\xi
dB}{\phi_{0}}\right)^{2}\left[1+\frac{2\sin^{2}\varphi}{1+(\omega\tau/2)^{2}}\right].$
(24)

In superconductors with $\kappa\gg 1$ the maximum $\delta f(B_{c1})/f_{0}$ in
a thin film is much greater than the maximum NLME bulk shift
$\delta\lambda/\lambda=(B_{c1}/B_{c})^{2}/8=(\ln\kappa/4\kappa)^{2}$ which
follows from Eq. (2). For $d=50$ nm $\xi=5$ nm, $\varphi=\pi/2$ and $B=100$
mT, we obtain $\delta f(B)/f_{0}=(\pi\xi dB/\phi_{0})^{2}/2\simeq 8\times
10^{-4}$, well below the observed $\delta f/f_{0}$. Taking the geometrical
inductance into account further reduces $\delta f/f_{0}$ by a factor $\simeq
3$. Moreover, according to Eq. (21) the field-induced shift $\delta\lambda$ at
$\varphi=\pi/2$ for which the rf currents are parallel to the dc Meissner
currents is 3 times larger than $\delta\lambda$ at $\varphi=0$. This is
inconsistent with the experimental data shown in Fig. 9 where the slope of
$\delta f(B)/f_{0}$ at $\varphi=0$ is about 2 times larger than for $\delta
f(B)/f_{0}$ at $\varphi=\pi/2$. Thus, not only is the Meissner pairbreaking
too weak to account for the observed $\delta f/f_{0}$ but it yields the field
and orientational dependencies of $\delta f({\bf B})/f_{0}$ inconsistent with
our experimental data on Nb3Sn.

###  II.2 Grain boundary contribution

A significant contribution to $L_{k}$ can come from local non-stoichiometry,
strains, and grain boundaries (GBs) in polycrystalline Nb3Sn. Particularly,
the well-known Sn depletion at GBs gb1 ; gb2 ; gb3 ; gb4  results in weak
Josephson coupling of grains in Nb3Sn. Our polycrystalline films have lateral
grain sizes $l_{2}\sim 0.1-1~{}\mu$m (see Fig. 2) and local nonstoichiometry
causing inhomogeneities of superconducting properties on the same length
scales chris ; ml . If weakly coupled GBs are regarded as planar Josephson
junctions (JJs), each GB has a kinetic inductance BP

$L_{J}=\frac{\phi_{0}}{2\pi I_{c}\cos\theta},$ (25)

where $I_{c}(T)$ is a critical current of the JJ. The field dependence of
$L_{J}(B)$ is determined by the phase difference $\theta({\bf r})$ induced by
the dc field on a GB. Because GBs in Nb3Sn can have broad distributions of
sizes and $I_{c}$ values gb , low-$I_{c}$ GBs can significantly increase
$L_{J}$ if they form interfaces blocking the cross-section of the film.

An array of weakly-coupled GBs can be modeled by a Hamiltonian of a granular
superconductor net ; stroud1 ; stroud2

${\cal H}=-\sum_{ij}J_{ij}\cos(\chi_{i}-\chi_{j}-A_{ij}),$ (26)

where the coupling energies $J_{ij}=\hbar I_{c}^{ij}/2e$ of the $i$-th and
$j$-th grains are proportional to the respective intergrain Josephson critical
currents $I_{c}^{ij}$, $\chi_{i}$ and $\chi_{j}$ are phases of the
superconducting order parameters in the grains, and the magnetic phase factors
$A_{ij}$ are given by:

$A_{ij}=\frac{2\pi}{\phi_{0}}\int_{i}^{j}{\bf A}\cdot d{\bf l}.$ (27)

Monte-Carlo simulations of the Hamiltonian (26) of a disordered XY model have
shown that the helicity moduli and the global kinetic inductance change nearly
linearly with the dc magnetic field at $B\lesssim\phi_{0}/l_{2}^{2}$ and
exhibit strong fluctuations as functions of $B$ due to transitions between
many metastable states in finite JJ arrays stroud1 ; stroud2 . These results
appear qualitatively consistent with the observed field dependence of $\delta
f(B)/f_{0}$ shown in Fig. 9.

To get an insight into the nearly linear decrease of $\delta f(B)$ with $B$
shown in Fig. 9, we use a mean-field model in which
$\cos(\chi_{i}-\chi_{j}-A_{ij})$ is replaced with an averaged value
$\langle\cos\theta\rangle$. Then a disturbance $\delta\theta$ induced by a
weak rf current $J_{\omega}e^{i\omega t}$ on an overdamped GB is described by
the dynamic equation of the RSJ model

$\tau_{J}\delta\dot{\theta}+\langle\cos\theta\rangle\delta\theta=(J_{\omega}/J_{c})e^{i\omega
t},$ (28)

where $\tau_{J}$ is a relaxation time caused by ohmic quasiparticle current
through JJ, and $\langle...\rangle$ denotes averaging over orientations of the
GB planes:

$\langle\cos\theta\rangle=\frac{1}{NS}\sum_{s}n_{zs}^{2}\int_{S}\cos\theta_{s}({\bf
r})dS$ (29)

Here the summation goes over all GBs, and $\theta_{s}({\bf r})$ on a planar GB
smaller than the Josephson length
$\lambda_{J}=(\phi_{0}/2\pi\mu_{0}dJ_{c})^{1/2}$ is determined by BP

$\nabla\theta=\frac{2\pi\Lambda}{\phi_{0}}[{\bf B}\times{\bf n}],$ (30)

where ${\bf n}$ is a unit vector normal to the GB surface, and $\Lambda\sim d$
for a JJ in a thin film $(d\lesssim\lambda)$ in a parallel magnetic field BP ;
jj1 ; jj2 . The factor $n_{z}^{2}$ in Eq. (29) takes into accounts that only
the perpendicular component of the rf current
$n_{z}J_{\omega}=J_{c}\delta\theta$ causes the Josephson voltage
$V_{\omega}\hbar\delta\dot{\theta}n_{z}/2e$ along $J_{\omega}$. From Eq. (28),
we obtain $\delta\theta=J_{\omega}e^{i\omega
t}/(\langle\cos\theta\rangle+i\omega\tau_{J})J_{c}$ and the impedance
$Z_{J}=NV_{\omega}/I_{\omega}$ of $N$ grain boundaries:

$Z=\frac{i\omega NR\langle L_{J}\rangle}{R+i\omega\langle
L_{J}\rangle},\qquad\langle L_{J}\rangle=\frac{\phi_{0}}{2\pi
I_{c}\langle\cos\theta\rangle}$ (31)

Figure 10: Geometry of a rectangular tilted GB (yellow). The red arrow shows
the normal to the GB plane.

We calculate $\langle\cos\theta\rangle$ for randomly-oriented planar GBs
parameterized by the Euler angles $\alpha$ and $\beta$ shown in Fig. 10. As
shown in Appendix B, averaging $\cos\theta$ over the area $S$ of a tilted GB
in a magnetic field gives:

$\displaystyle\bar{c}=\frac{1}{S}\int\cos\theta
dS=\frac{\sin(q_{1}l_{1})\sin(q_{2}l_{2})}{q_{1}l_{1}q_{2}l_{2}},$ (32)

where

$\displaystyle q_{1}=\frac{\pi B}{\phi_{0}}\Lambda\sin\alpha,\quad
q_{2}=\frac{\pi B}{\phi_{0}}\Lambda\sin\beta\cos\alpha,\quad{\bf B}\|z$ (33)
$\displaystyle q_{1}=\frac{\pi B}{\phi_{0}}\Lambda\cos\alpha,\quad
q_{2}=\frac{\pi B}{\phi_{0}}\Lambda\sin\beta\sin\alpha,\quad{\bf B}\|y$ (34)

There is a significant difference of GB lengths $l_{1}$ and $l_{2}$ in our
Nb3Sn coplanar resonator with $d\ll w$. Here $l_{1}$ is smaller or of the
order of the film thickness, $l_{1}\lesssim d\simeq 50$ nm, whereas lateral GB
lengths $l_{2}$ are in a submicron range $l_{2}\sim 0.1-1\,\mu$m (see Fig. 2)
so that $l_{1}\sim(10^{-2}-10^{-1})l_{2}$. As a result,
$\sin(q_{1}l_{1})/l_{1}q_{1}$ in Eq. (32) remains close to $1$ in the field
region $B\lesssim\phi_{0}/\pi\Lambda l_{1}\sim\phi_{0}/\pi d^{2}\sim B_{c1}$
of our measurements, while $\sin(q_{2}l_{2})/l_{2}q_{2}$ has a strong field
dependence at $B>B_{0}\simeq\phi_{0}/\pi dl_{2}$. Here $B_{0}\simeq 25$ mT at
$l_{2}=0.5\,\mu$m and $d=50$ nm. As a result, Eq. (32) at $B<B_{c1}$ simplify
to

$\displaystyle\bar{c}=\frac{\sin(b\cos\alpha\sin\beta)}{b\cos\alpha\sin\beta},\qquad{\bf
B}\|z$ (35)
$\displaystyle\bar{c}=\frac{\sin(b\sin\alpha\sin\beta)}{b\sin\alpha\sin\beta},\qquad{\bf
B}\|y$ (36) $\displaystyle b=B/B_{0},\qquad B_{0}=\phi_{0}/\pi\Lambda l_{2}.$
(37)

Here only tilted GBs which are not perpendicular to the film plane $(\beta\neq
0)$ contribute to the strong field dependence of $\bar{c}(B)$. The average
$\langle\cos\theta\rangle$ over all GB orientations in Eq. (29) can be written
in the form

$\langle\cos\theta\rangle=\sum_{i,j}P_{ij}\bar{c}_{ij}\cos^{2}\alpha_{i}\cos^{2}\beta_{j},$
(38)

where $\bar{c}_{ij}$ is given by either Eq. (35) or (36), depending on the
direction of ${\bf B}$, $i$ and $j$ label different GBs,
$n_{z}^{ij}=\cos\alpha_{i}\cos\beta_{j}$, and $P_{ij}$ is a probability
distribution of $\alpha_{i}$ and $\beta_{j}$ normalized by
$\sum_{ij}P_{ij}=1$.

The angular distributions of GBs is affected by the crystalline texturing
during the film growth and other materials factors. We consider here a simple
case of randomly-oriented GBs with equal probabilities of all $\alpha_{i}$ and
$\beta_{j}$. Then Eq. (38) reduces to the integrals (60) and (61) given in
Appendix B. Numerical calculation of these integrals for
$\langle\cos\theta\rangle$ yields the field dependencies of $\langle
L_{J}\rangle$ shown in Fig. 11.

Figure 11: The field-dependencies of the kinetic inductance $\langle
L_{J}(B)\rangle$ calculated from Eqs. (31), (60) and (61) for random
orientation of GB planes, and the dc magnetic field applied parallel $({\bf
B}\|z)$ and perpendicular $({\bf B}\|y)$ to the strip line.

As follows from Fig. 11, grain boundaries can radically change the field
dependence of the kinetic inductance as compared to the NLME caused by the
Meissner pairbreaking. First, the GB contribution $\langle L_{J}(B)\rangle$ is
quadratic in $B$ only at very low fields $B\lesssim B_{0}\ll B_{c}$ and
exhibits a nearly linear field dependence at $B\gtrsim B_{0}\ll B_{c}$,
whereas the Meissner pairbreaking gives $\delta L_{k}\propto B^{2}$ all the
way to $B=B_{c1}$. Second, the field ${\bf B}\|z$ applied along the strip
causes stronger increase of $\langle L_{J}(B)\rangle$ than the transverse
field ${\bf B}\|y$. This is the opposite of the orientational field dependence
of $\delta L_{k}^{M}(B)$ described by Eq. (23) and observed on Nb coplanar
resonator ggc . Yet both features of $\langle L_{J}(B)\rangle$ are in
agreement with our experimental data on polycrystalline Nb3Sn shown in Fig. 9.

##  III Discussion

To compare the contributions of Meissner pairbreaking and weakly-coupled GBs
to $\delta f(B)/f_{0}$, we evaluate the GB kinetic inductance $L_{k}^{J}$ per
unit length of the strip in the above mean-field model. If GBs have the same
critical current density $J_{c}$, dimensions $d\times l_{2}$ but different
orientations, $L_{k}^{J}\sim\langle L_{J}\rangle/l_{2}$, where $\langle
L_{J}\rangle$ is given by Eq. (25) with $I_{c}\sim dwJ_{c}$. As a result,

$L_{k}^{J}\sim\frac{\phi_{0}}{2\pi J_{c}wdl_{2}\langle\cos\theta\rangle}\sim
L_{k}^{M}\times\frac{\xi J_{d}}{l_{2}J_{c}}$ (39)

Here $J_{d}=\phi_{0}/2\pi\mu_{0}\lambda^{2}\xi$ is of the order of the GL
depairing current density. The kinetic inductance is dominated by weakly-
coupled GBs if $J_{c}\ll\xi J_{d}/l_{2}$. For our Nb3Sn films with $\xi\simeq
5$ nm and $l_{2}\simeq 200$ nm, the GB contribution dominates if
$J_{c}\lesssim 10^{-2}J_{d}$.

The effective penetration depth $\lambda$ extracted from the measured kinetic
inductance is affected by GBs. The Meissner contribution $L_{k}^{M}$ is
determined by the London penetration depth $\lambda$ in the dirty limit Kopnin
:

$L_{k}^{M}=\frac{\mu_{0}\lambda^{2}}{dw}=\frac{\hbar\rho_{s}}{\pi
dw\Delta(T)}\coth\frac{\Delta(T)}{2k_{B}T},$ (40)

where $\rho_{s}$ is the normal state resistivity. If GBs can be modeled as
S-I-S Josephson junctions BP , their contribution to $L_{k}$ can be evaluated
from Eq. (39):

$L_{k}^{J}\sim\frac{\hbar}{eJ_{c}wdl_{2}}\simeq\frac{\hbar
R_{\perp}}{dwl_{2}\Delta(T)}\coth\frac{\Delta(T)}{2k_{B}T},$ (41)

where $R_{\perp}$ is the tunneling resistance of the JJ per unit area.
Defining the effective penetration depth $\tilde{\lambda}$ in the total
kinetic inductance $L_{k}=L_{k}^{M}+L_{k}^{J}=\tilde{\lambda}^{2}/dw$ and
combining Eqs. (40) and (41), we obtain:

$\tilde{\lambda}^{2}(T)=\frac{\hbar}{\pi\Delta(T)}\left[\rho_{s}+C_{1}\frac{R_{\perp}}{l_{2}}\right]\coth\frac{\Delta(T)}{2k_{B}T},$
(42)

where the factor $C_{1}\sim 1$ accounts for details of the shape and angular
distributions of GBs. In the S-I-S model the temperature dependencies of the
GB and Meissner contributions to $\tilde{\lambda}$ are the same. This is no
longer the case if GBs are proximity-coupled S-N-S junctions BP  for which
$J_{c}\propto(1-T/T_{c})^{2}$. The S-N-S scenario may be more relevant for
Nb3Sn in which strongly-coupled GBs can transmit high current densities which
are still well below the depairing limit arno ; gb1 ; gb2 ; gb3 ; gb4 ; pin .

The above estimate pertains to $B=0$ but the field-induced frequency shift
$\delta f/f_{0}$ is determined by small field - dependent corrections $\delta
L_{k}^{M}$ and $\delta L_{k}^{J}$. Here $\delta L_{k}^{M}$ is given by Eq.
(23) and $\delta L_{k}^{J}\sim(\mu_{0}\lambda^{2}/dw)(\xi
J_{d}/l_{2}J_{c})(B/B_{0})$ at $B\gtrsim B_{0}\simeq\phi_{0}/3\pi dl_{2}$, as
shown in Fig. 11. Hence,

$\frac{\delta L_{k}^{M}}{\delta L_{l}^{J}}\sim\frac{B\xi
dJ_{c}}{\phi_{0}J_{d}},\qquad B\gtrsim B_{0}$ (43)

This ratio is independent of the lateral GB sizes and is much smaller than 1
since $J_{c}\ll J_{d}$, and $Bd\xi/\phi_{0}<2\times 10^{-2}$ at $B<200$ mT,
$\xi=5$ nm and $d=50$ nm. Thus, the NLME field-dependent frequency shift in
our polycrystalline Nb3Sn coplanar resonators is dominated by grain
boundaries, even if their contribution to the total kinetic inductance at
$B=0$ is much smaller than that of the Meissner currents. This conclusion is
consistent not only with the observed nearly linear field dependence of
$\delta f/f_{0}$ but also with the fact that the slope of $\delta f/f_{0}$ at
${\bf B}$ along the strip is larger than the slope of $\delta f/f_{0}$ at
${\bf B}$ perpendicular to the strip.

The fit of the observed $\delta f(T,B)/f_{0}(T,0)$ to the GB model depends on
many uncertain parameters such as distribution of orientations and local
$J_{c}$ values of GBs, their geometrical sizes and mechanisms of current
transport through GBs. Shown in Fig. 12 is an example of $\delta
f(T,B)/f_{0}(T,0)$ calculated for uniform distributions of the Euler angles of
GBs:

$\frac{\delta
f(T,B)}{f_{0}(T,0)}=\left[\frac{1+a\epsilon(T)/\langle\cos\theta(0)\rangle}{1+a\epsilon(T)/\langle\cos\theta(B)\rangle}\right]^{1/2}-1,$
(44)

where $a=L_{k}^{J}(0,0)/L$, the factor $\epsilon(T)=[1-(T/T_{c})^{4}]^{-2}$
approximates the S-N-S temperature dependence of $J_{c}(T)$ in Eq. (39) at
$7K<T<T_{c}$, and $\langle\cos\theta(B)\rangle$ is given by Eqs. (60) and
(61). We took $a=3.5\times 10^{-4}$, $B_{0}=10$ mT and different temperatures
corresponding to those in Fig. 9. As follows from Fig. 12, the model captures
the observed features of $\delta f(T,B)/f_{0}(T,0)$ for both orientations of
${\bf B}$ shown in Fig. 9, although one can hardly expect a perfect fit from
such a crude model. For instance, the difference between the slopes of $\delta
f(T,B)/f_{0}(T,0)$ for two field orientations in Fig. 12 is about 30$\%$
higher than in Fig. 9, which can occur because the GB orientations shown in
Fig. 2 are not completely random. Yet the GBs can dominate the field-induced
frequency shift even if they only contribute less than $10^{-3}$ to the total
inductance of the strip.

Figure 12: Normalized shift in $\delta f_{0}(T,B)/f_{0}(T,0)$ as a function of
field calculated from Eqs. (44), (60) and (61) at $B_{0}=10$ mT,
$L_{k}^{J}(0,0)/L=3.5\times 10^{-4}$ and different temperatures corresponding
to those in Fig. 9 for: (a) ${\bf B}\|z$ and (b) ${\bf B}\|y$.

Grain boundaries can also contribute to the field-dependent resistance
$R_{J}(B)$. Indeed, the coplanar resonator impedance per unit length,
$Z_{g}(B)=R_{J}+iL_{k}^{J}$ estimated from Eq. (31) is given by:

$Z_{g}\sim\frac{i\omega\bar{L}_{\square}\bar{R}_{\square}}{wdl_{2}(i\omega\bar{L}_{\square}+\bar{R}_{\square})},$
(45)

where $\bar{L}_{\square}=\phi_{0}/2\pi J_{c}\langle\cos\theta\rangle$, and
$\bar{R}_{\square}$ is the GB quasiparticle resistance per unit area averaged
over the GB orientations. The field dependencies of $R_{J}(B)$ and
$L_{k}^{J}(B)$ can be strongly affected by $\omega$ and $T$. At high
frequencies, $\omega\gg\omega_{c}=\bar{R}_{\square}/\bar{L}_{\square}$ the
impedance $Z_{g}\sim\bar{R}_{\square}/wd$ becomes independent of $B$, that is,
GBs do not contribute to the NLME. For S-I-S GBs, the crossover frequency
$\omega_{c}$ is,

$\omega_{c}\sim\langle\cos\theta\rangle\frac{\Delta}{\hbar}\tanh\left(\frac{\Delta}{2k_{B}T}\right)$
(46)

Here $\omega_{c}\sim\Delta/\hbar$ if $T$ is not very close to $T_{c}$ and
$B\lesssim B_{0}$, but $\omega_{c}(T,B)$ decreases as $B$ increases beyond
$B_{0}$ and $T$ approaches $T_{c}$. For proximity-coupled GBs, $\omega_{c}$
can be smaller than $\Delta/\hbar$ even at $T\ll T_{c}$ and $B<B_{0}$ since
$J_{c}R_{\square}$ for S-N-S JJs can be much smaller than $\Delta/e$ BP . For
Nb3Sn in which $\Delta/\hbar\sim 1$ THz is some 2 orders of magnitude higher
than the frequency $2\pi f_{0}$ of our resonator, it appears that the observed
behavior of $\delta f(T,B)$ at $7<T<12$ K is consistent with the low-frequency
limit $\omega\ll\omega_{c}(T,B)$ in which Eq. (45) gives:

$R_{J}(B)\sim\frac{\phi_{0}^{2}\omega^{2}}{4\pi^{2}R_{\square}J_{c}^{2}wdl_{2}\langle\cos\theta\rangle^{2}}$
(47)

For $\langle L_{J}(B)\rangle$ calculated above (see Fig. 11), $R_{J}(B)\propto
B^{2}$ in Eq. (47) increases quadratically with $B$ at $B\gtrsim B_{0}$. This
field dependence is the same as for the Meissner contribution to
$\sigma_{\omega}(B)$ in Eq. (22), but the orientational dependence of
$R_{J}(B)$ is opposite to that of $\sigma_{\omega}(B)$. Thus, GBs can give
rise to a strong nonlinearity of the electromagnetic response of
polycrystalline films, which can be essential for Nb3Sn thin film coatings of
high-Q resonator cavities in particle accelerators ml ; gb3 ; gb4 ; ml1 .

In conclusion, grain boundaries and local nonstoichiometry on nanometer scales
can significantly contribute to the NLME in polycrystalline Nb3Sn.
Particularly, GBs can cause the linear field-dependence of the magnetic
penetration depth expected from a clean d-wave superconductor at low
temperatures. By contrast, for elemental superconductors such as Nb ggc  and
Al Al  with large coherence lengths, $\delta\lambda(T,B)\propto B^{2}$ is
described well by the Meissner pairbreaking. However, extended crystalline
defects in superconductors with short coherence lengths can radically change
the field-dependence of $\delta\lambda(T,B)$, even if their contribution to
the kinetic inductance at zero field is small. This feature can impose
stringent requirements for the quality of single crystals used for the
observation of manifestations of d-wave pairing in $\lambda(T,B)$,
particularly in cuprates and pnictides which are prone to the weak-link
behavior of grain boundaries.

##  IV ACKNOWLEDGMENTS

This work was supported by DOE under Grant DE-SC 100387-020.

##  Appendix A Geometric inductance of the coplanar waveguide

The geometric inductance per unit length $L_{g}$ of the coplanar resonator is
calculated by conformal mapping of the cross section of the coplanar resonator
into parallel plates collin . For a thin film with $d\ll w$, this yields

$L_{g}=\frac{\mu_{0}K(k^{\prime})}{4K(k)},$ (48)

where $k=s/(2w+s)$, $k^{\prime}=\sqrt{1-k^{2}}$, and $K(k)$ is a complete
elliptic integral of the first kind. For $w=15\,\mu\mathrm{m}$ and
$s=8.8\,\mu\mathrm{m}$, Eq. (48) gives $L_{g}=420.5$ H/m.

##  Appendix B tilted GB

For a rectangular JJ of width $l_{1}$ and length $l_{2}$ shown in Fig. 10, the
solution of Eq. (30) is

$\displaystyle\theta=\frac{2\pi
B\Lambda}{\phi_{0}}\left(n_{x}y-n_{y}x\right),\qquad{\bf B}\|z$ (49)
$\displaystyle\theta=\frac{2\pi
B\Lambda}{\phi_{0}}\left(n_{z}x-n_{x}z\right),\qquad{\bf B}\|y$ (50)

We rotate the coordinate system by the Euler angles $\alpha$ and $\beta$ about
the $x$ and the $y$ axes to a new cartesian system $(X,Y,Z)$ in which $Z$ is
perpendicular to the GB plane and $X$ and $Y$ are directed along the GB sides
of lengths $l_{1}$ and $l_{2}$, respectively. Here $X_{i}=R_{ij}x_{j}$, where
the rotation matrix $R_{ij}=R^{y}_{ik}R^{x}_{kj}$ is given by:

$R_{ij}=\left(\begin{array}[]{ccc}\cos\beta&0&\sin\beta\\\ 0&1&0\\\
-\sin\beta&0&\cos\beta\end{array}\right)\\!\cdot\\!\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\alpha&-\sin\alpha\\\ 0&\sin\alpha&\cos\alpha\end{array}\right)\\!.$
(51)

The transpose matrix $R^{T}_{ij}$ is then:

$R^{T}_{ij}=\left(\begin{array}[]{ccc}\cos\beta&0&-\sin\beta\\\
\sin\alpha\sin\beta&\cos\alpha&\sin\alpha\cos\beta\\\
\cos\alpha\sin\beta&-\sin\alpha&\cos\alpha\cos\beta\end{array}\right)\\!.$
(52)

Using $x_{i}=R^{T}_{ij}X_{j}$ we obtain:

$\displaystyle x=X\cos\beta-Z\sin\beta,$ (53) $\displaystyle
y=X\sin\beta\sin\alpha+Y\cos\alpha+Z\cos\beta\sin\alpha,$ (54) $\displaystyle
z=X\sin\beta\cos\alpha-Y\sin\alpha+Z\cos\beta\cos\alpha,$ (55)
$\displaystyle\\!\\!\\!n_{x}=-\sin\beta,\quad n_{y}=\cos\beta\sin\alpha,\quad
n_{z}=\cos\beta\cos\alpha$ (56)

Equations (49)-(50) and (53)-(55) with $Z=0$ give:

$\displaystyle\theta=-\frac{2\pi
B\Lambda}{\phi_{0}}\left[X\sin\alpha+Y\sin\beta\cos\alpha\right],\quad{\bf
B}\|z$ (57) $\displaystyle\theta=\frac{2\pi
B\Lambda}{\phi_{0}}\left[X\cos\alpha-Y\sin\beta\sin\alpha\right],\quad{\bf
B}\|y.$ (58)

Next, we calculate $\bar{c}=S^{-1}\int\cos\theta(X,Y)dS$:

$\bar{c}=\frac{1}{l_{1}l_{2}}\mbox{Re}\int_{-l_{1}/2}^{l_{1}/2}\\!dX\int_{-l_{2}/2}^{l_{2}/2}\\!dYe^{i\theta(Y,Z)},$
(59)

which leads to Eqs. (32)-(34).

For uniform distributions of the GB orientation angles, Eqs. (35) and (38)
yield at ${\bf B}\|z$:

$\displaystyle\langle\cos\theta\rangle=$
$\displaystyle\frac{1}{\pi^{2}b}\int_{0}^{\pi}d\alpha\\!\int_{0}^{\pi}d\beta\sin(b\cos\alpha\sin\beta)\frac{\cos\alpha\cos\beta}{\tan\beta}.$
(60)

Likewise, we get from Eqs. (36) and (38) at ${\bf B}\|y$:

$\displaystyle\langle\cos\theta\rangle=$
$\displaystyle\frac{1}{\pi^{2}b}\int_{0}^{\pi}d\alpha\\!\int_{0}^{\pi}d\beta\sin(b\sin\alpha\sin\beta)\frac{\cos\alpha\cos\beta}{\tan\alpha\tan\beta}.$
(61)

At $b=B/B_{0}\ll 1$ we obtain from Eqs. (60)-(61):

$\displaystyle\langle\cos\theta\rangle=\frac{1}{4}\left(1-\frac{b^{2}}{32}\right),\quad{\bf
B}\|z,$ (62)
$\displaystyle\langle\cos\theta\rangle=\frac{1}{4}\left(1-\frac{b^{2}}{96}\right),\quad{\bf
B}\|y.$ (63)

The field ${\bf B}\|z$ applied along the strip causes stronger reduction of
$\langle\cos\theta\rangle$ than the transverse field ${\bf B}\|y$.

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