# Multi-condensate lengths with degenerate excitation gaps in BaNi2As2
revealed by muon spin relaxation study

Kaiwen Chen  State Key Laboratory of Surface Physics, Department of Physics,
Fudan University, Shanghai 200438, China  Zihao Zhu  State Key Laboratory of
Surface Physics, Department of Physics, Fudan University, Shanghai 200438,
China  Yaofeng Xie  Department of Physics and Astronomy, Rice University,
Houston, Texas 77005, USA  Adrian D. Hillier  ISIS Facility, STFC Rutherford
Appleton Laboratory, Harwell Science and Innovation Campus, Didcot OX11 0QX,
United Kingdom  James S. Lord  ISIS Facility, STFC Rutherford Appleton
Laboratory, Harwell Science and Innovation Campus, Didcot OX11 0QX, United
Kingdom  Pengcheng Dai  Department of Physics and Astronomy, Rice University,
Houston, Texas 77005, USA  Lei Shu<EMAIL_ADDRESS>State Key Laboratory
of Surface Physics, Department of Physics, Fudan University, Shanghai 200438,
China  Shanghai Research Center for Quantum Sciences, Shanghai 201315,
People’s Republic of China

###### Abstract

The recently discovered (Ba,Sr)Ni2As2 family provides an ideal platform for
investigating the interaction between electronic nematicity and
superconductivity. Here we report the muon spin relaxation ($\mu$SR)
measurements on BaNi2As2. Transverse-field $\mu$SR experiments indicate that
the temperature dependence of superfluid density is best fitted with a single-
band $s$-wave model. On the other hand, the magnetic penetration depth
$\lambda$ shows magnetic field dependence, which contradicts with the single-
band fully-gapped scenario. Zero-field $\mu$SR experiments indicate the
absence of spontaneous magnetic field in the superconducting state, showing
the preservation of time-reversal symmetry in the superconducting state. Our
$\mu$SR experiments suggest that BaNi2As2 is a fully-gapped multiband
superconductor. The superconducting gap amplitudes of each band are nearly the
same while different bands exhibit different coherence lengths. The present
work helps to elucidate the controversial superconducting property of this
parent compound, paving the way for further research on doping the system with
Sr to enhance superconductivity.

##  I $\rm{\@slowromancap i@}$. INTRODUCTION

One of the core issues in quantum materials is how superconductivity interacts
with the Fermi surface instabilities such as charge density wave (CDW) and
nematic order (NO) Fradkin  _et al._ (2015); Keimer  _et al._ (2015); Johnston
(2010). Although the CDWs in cuprates have been discovered for nearly three
decades Tranquada  _et al._ (1995), their contribution to anomalous
superconducting properties has not yet been thoroughly understood Miao  _et
al._ (2021); Proust and Taillefer (2019); Agterberg  _et al._ (2020). In the
Fe-based superconductors, the nematic order causing $C_{4}$-rotational
symmetry-breaking strongly intertwines with the spin density wave Dai (2015);
Fernandes  _et al._ (2014); Paglione and Greene (2010). By suppressing these
orders with pressure or doping, the superconductivity appears and is maximumly
enhanced near the nematic and/or magnetic quantum critical point (QCP) Marel
_et al._ (2003); Lederer  _et al._ (2015). However, the entanglement between
plural electronic instabilities complicates our understanding of the
relationship between enhanced superconducting pairing and such instabilities.

Enormous efforts have been made to elucidate the physics of Fe-based
superconductors. In this process, many other new superconducting systems were
discovered Wang  _et al._ (2012); Jiang  _et al._ (2015); Gu and Wen (2022).
Among them, the nickel-based BaNi2As2, which shares the same high-temperature
tetragonal structure with the intensively studied 122 iron arsenide
superconductors Pfisterer and Nagorsen (1980); Alireza  _et al._ (2008); Mani
_et al._ (2009); Gallais  _et al._ (2013), has attracted attention recently.
Compared to its iron-analog, which does not superconduct without doping at
ambient pressure, BaNi2As2 itself shows superconductivity below $T_{\rm{c}}$
$\sim$ 0.6 K Ronning  _et al._ (2008) and undergoes a structure transition
into a triclinic phase near $T^{*}\sim$ 130 K Sefat  _et al._ (2009). An
increasing number of recent studies have revealed an exotic normal state,
characterized by complex charge density instabilities Lee  _et al._ (2019,
2021); Guo  _et al._ (2022) and dynamic nematic fluctuations Yao  _et al._
(2022). Though early studies on electronic properties have led to conventional
BCS superconductivity Shein and Ivanovskii (2009); Subedi and Singh (2008),
substituting Ba with Sr can boost $T_{\rm{c}}$ from 0.6 K to 3.6 K with the
suppression of structural transition to zero temperature, which indicates the
possible nematic-enhanced superconductivity near the optimal doping Eckberg
_et al._ (2020); Lederer  _et al._ (2020). Therefore, BaNi2As2 provides an
ideal platform to study how nematic fluctuations itself can influence
superconductivity as there is no magnetic order or spin fluctuations in the
system.

Up to now, most of the studies on BaNi2As2 are focused on the driving force of
structure transition or the origin of CDWs. In fact, the nature of
superconductivity of BaNi2As2 is still unclear. On the one hand, early
$ab~{}initio$ band structure calculation Subedi and Singh (2008) and angle-
resolved photoemission spectroscopy (ARPES) measurements Zhou  _et al._ (2011)
have revealed the presence of multiple bands crossing the Fermi level. The
S-shaped field dependence of thermal conductivity suggested the multiple
nodeless superconducting gaps Kurita  _et al._ (2009). On the other hand, both
specific-heat and thermal conductivity studies exclude the appearance of a
second gap Kurita  _et al._ (2009); Ronning  _et al._ (2008). For
unconventional superconductors coexisting with several electronic
instabilities, the multiband property is a crucial ingredient since it is
closely related to the pairing mechanism Kittaka  _et al._ (2014). Therefore,
it is important to determine the superconducting property of BaNi2As2.

Muon-spin relaxation/rotation ($\mu$SR) is a powerful technique to study the
gap symmetry of superconductors by providing direct information on magnetic
penetration depth $\lambda$ Hillier  _et al._ (2022); Sonier  _et al._ (2000).
In this work, transverse-field (TF)-$\mu$SR experiments have been performed to
study the temperature and magnetic field dependence of $\lambda$ of BaNi2As2.
The temperature dependence of superfluid density $\rho_{\rm{s}}$ is well-
fitted with a single-band $s$-wave function with zero temperature gap
amplitude $\Delta_{0}=0.101(6)$ meV. On the other hand, the field dependence
of muon spin relaxation rates is well-fitted with a two-band model.
Furthermore, $\lambda$ showing a magnetic field dependence, contradicting with
single-band fully-gapped scenario. Our TF-$\mu$SR results suggest that
BaNi2As2 is a multiband fully-gapped superconductor with equal gap value but
different coherence lengths in each band. The zero-field (ZF)-$\mu$SR
experiments reveal that the time-reversal symmetry is preserved in the
superconducting state of BaNi2As2.

##  II $\rm{\@slowromancap ii@}$. EXPERIMENTAL DETAILS

High-quality single crystals of BaNi2As2 were synthesized at the Rice
University with the same procedure described in Ref Ronning  _et al._ (2008).
The single crystal X-ray diffraction (XRD) experiment suggested the high
purity of our samples.

The $\mu$SR experiments were carried out on the MuSR spectrometer at ISIS
Neutron and Muon Facility, Rutherford Appleton Laboratory, Chilton, UK.
Several pieces of single crystals are mounted on the high purity silver plate
with diluted GE varnish. The experiments were carried out in temperatures
ranging from 50 mK to 1.0 K for both ZF and TF $\mu$SR setup. TF-$\mu$SR
experiments were performed under external magnetic fields from 5 mT to 40 mT.
The sample was field-cooled down above $T_{\rm{c}}$ to base temperature in
order to form an ideal vortex lattice. All the $\mu$SR spectra were collected
upon warming and the data were analyzed with the MUSRFIT software package
Suter and Wojek (2012).

##  III $\rm{\@slowromancap iii@}$. RESULTS

###  III.1 A. Zero-field muon spin relaxation

The zero-field (ZF) $\mu$SR measurement was performed to figure out the
magnetic ground state of BaNi2As2 and to examine whether the superconducting
transition accompanied with an additional broken of time-reversal symmetry.
Two representative ZF-$\mu$SR spectra measured above and below $T_{\rm{c}}$
are shown in Fig. 1. No oscillations or loss of initial asymmetry were
observed, suggesting the absence of both long-range and short-range magnetic
order down to 70 mK. This is consistent with the previous neutron scattering
study Kothapalli  _et al._ (2010). ZF muon spin asymmetry spectra after
subtracting the background signal $A_{\rm ZF}$ can be well fitted with a
damped Gaussian Kubo-Toyabe (KT) function A. Yaouanc and P. Dalmas de Réotier
(2011):

$\displaystyle
A_{\rm{ZF}}(t)=A_{0}(T)G_{\rm{KT}}(\sigma_{\rm{ZF}},t)e^{-\lambda t}.$ (1)

The KT term describes a Gaussian distribution of randomly oriented static (or
quasi-static) local fields at muon sites with distribution width $\delta
B_{\rm G}=\sigma_{\rm ZF}/\gamma_{\mu}$, where $\gamma_{\mu}=2\pi\times
135.53$ MHz/T is the muon gyromagnetic ratio Hayano  _et al._ (1979):

$\displaystyle
G_{\rm{KT}}(\sigma_{\rm{ZF}},t)=\frac{1}{3}+\frac{2}{3}(1-\sigma_{\rm{ZF}}^{2}t^{2})\cdot\exp(-\frac{1}{2}\sigma_{\rm{ZF}}^{2}t^{2}),$
(2)

The exponential term represents additional electronic depolarization rate A.
Yaouanc and P. Dalmas de Réotier (2011). $\sigma_{\rm{ZF}}$ is found to show
little temperature dependence and is fixed to the average value of
$\sigma_{\rm{ZF}}=0.122\ \rm{\mu s^{-1}}$ during the fitting. The temperature
dependence of $\lambda$ is displayed in Fig. 1(b). Though $\lambda$ grows at
the lowest temperature, whose origin is not clear based on the present data,
it does not show obvious change crossing $T_{\rm{c}}$. The current ZF-$\mu$SR
result suggests there is no spontaneous time-reversal symmetry breaking exists
with superconducting transition.

Figure 1: (a) ZF muon decay positron asymmetry $A(t)$, proportional to the
muon spin polarization, measured in the normal state (black circles) and the
superconducting state (red circles) respectively. The solid curves are the
fits of data with Eq. (1). (b) Temperature dependence of muon spin relaxation
rate $\lambda$ down to 70 mK. The grey dashed line indicates the
superconducting critical temperature $T_{\rm{c}}$.

###  III.2 B. Transverse-field muon spin relaxation

To elucidate the pairing symmetry and the band character of superconductivity
in BaNi2As2, systematic temperature-dependent $\mu$SR experiments were
performed under several transverse magnetic fields from 5 mT to 40 mT. The
lower critical field $H_{c1}$ of BaNi2As2 is estimated with Brandt (2003):

$\displaystyle H_{c1}=H_{c2}\frac{\ln{\kappa}+0.497}{2\kappa^{2}}.$ (3)

Considering $\kappa\approx 11$ estimated from normal state thermal
conductivity Kurita  _et al._ (2009), Eq. (3) gives $H_{c1}\approx$ 1.3 mT.
Although this is only an estimation, the applied magnetic field of 5 mT is
large enough to form the uniform vortex lattice in BaNi2As2.

Two representative TF-$\mu$SR spectra above and below $T_{\rm{c}}$ are shown
in Fig. 2. The damping of asymmetry is enhanced after entering the
superconducting state, indicating the bulk superconductivity of our sample.
The superconducting volume fraction can be estimated from the difference of
long-time spectra Combley (1985). However, the relaxation of asymmetry spectra
is relatively slow in BaNi2As2, and its asymmetry spectra failed to fully
relax even when the data points started to diverge (after 8 $\mu$s, not
shown). Nevertheless, based on the asymmetry spectra at 8 $\mu$s, the lower
limit of superconducting volume fraction can be estimated to be about 75$\%$.

Figure 2: TF-$\mu$SR asymmetry spectra measured at $\mu_{0}H=5~{}{\rm mT}$.
The black and red circles represent the asymmetry spectra in the normal state
and in the superconducting state respectively. The solid curves are the fits
of data with Eq. (4).

The TF-$\mu$SR data is analyzed with the function:

$A(t)/A_{0}=[(1-f)\exp(-\sigma^{2}t^{2}/2)\cos(\gamma_{\mu}\mu_{0}H_{\rm{int}}+\phi)\\\
+f\exp(-\sigma_{\rm{bg}}^{2}t^{2})\cos(\gamma_{\mu}\mu_{0}H_{\rm{bg}}+\phi)],$
(4)

where $A_{0}$ stands for the initial asymmetry; $f$ represents the proportion
of the background signal and $(1-f)$ represents the proportion of sample
signal; $\sigma$ is the Gaussian relaxation rate, which is related to the
Gaussian field distribution in the sample; $\mu_{0}H_{\rm{int}}$ is the
average field sensed by muons stopping in the sample; $\sigma_{\rm{bg}}$ is
the relaxation rate of muon stopping in the silver, which is temperature
independent and is fixed during the fitting Bueno  _et al._ (2011); Zhang  _et
al._ (2018); $\mu_{0}H_{\rm{int}}$ is the field sensed by muon stopping in the
background; and $\phi$ is the shared initial phase.

The extracted relaxation rates $\sigma$ with Eq. (4) are displayed in Fig.
3(a). For $T>T_{\rm{c}}$, the muon relaxation rates are almost the same under
different temperatures and magnetic fields, as expected for muon
depolarization by the randomly oriented nuclear dipole moments in the sample.
The muon spin relaxation rates are greatly enhanced after entering the
superconducting state due to the inhomogenous magnetic field distribution of
the flux-line lattice Aeppli  _et al._ (1987). The contribution from the flux-
line lattice adds in quadrature to the contribution of nuclear dipole moment
of the sample:

$\displaystyle\sigma^{2}=\sigma_{\rm{dip}}^{2}+\sigma_{\rm{FLL}}^{2},$ (5)

where $\rm{\sigma_{dip}}$ is the contribution from the randomly oriented
nuclear dipolar moments and $\rm{\sigma_{FLL}}$ is the flux-line lattice (FLL)
contribution. The temperature dependence of the extracted $\rm{\sigma_{FLL}}$
using Eq. (5) are plotted in Fig. 3(b). The behavior that $\sigma$ does not
change from the base temperature to $T\sim T_{c}/3$ suggests the fully gap
superconducting gap.

Figure 3: Temperature dependence of muon spin relaxation rates (a) $\sigma$
and (b) $\sigma_{\rm{FLL}}$ under different external fields respectively. The
solid lines are guides for eyes.

###  III.3 C. Magnetic field and temperature dependence of magnetic
penetration depth

The internal field distribution of type-$\rm{\@slowromancap ii@}$
superconductors can be described by the penetration depth $\lambda$ and
coherence length $\xi$ Brandt (2003). For a type-$\rm{\@slowromancap ii@}$
superconductor with $\kappa\geq 5$ and a moderate reduced magnetic field
$h=H/H_{\rm{c2}}>0.25/\kappa^{1.3}$, $\lambda$ can be calculated with the
numerical Ginzburg-Landau model with less than 5% error Brandt (2003):

$\sigma_{\rm{FLL}}[\mu s^{-1}]=4.83\times 10^{4}(1-h)\\\
\times[1+1.21(1-\sqrt{h})^{3}]\lambda^{-2}[\rm{nm^{-2}}].$ (6)

Noting that the muon spin relaxation rate is nearly temperature independent
below $T=0.1$ K, it is reasonable to analyze the field dependence of
$\sigma_{\rm{FLL}}$ at base temperature with the zero-temperature upper
critical field $H^{ab}_{c2}(0)=0.11$ T taken from Ref Kurita  _et al._ (2009).
The superscript here indicates that the direction of the external field is in
$ab$ plane. For conventional single-band $s$-wave superconductors with medium
$\kappa$, the value of $\lambda^{-2}(T)$ and $H^{ab}_{c2}(T)$ can be obtained
by fitting the isothermal magnetic field dependence of $\sigma_{\rm{FLL}}$ at
different temperatures Barker  _et al._ (2015). The field dependence of
$\sigma_{\rm FLL}$ using Eq. (6) with fixed $H^{ab}_{c2}(0)$ at base
temperature ($T=50$ mK) is shown in the black dashed line in Fig. 4. It can be
seen that the experimentally measured $\sigma_{\rm{FLL}}$ is far from the
theoretical calculation, dropping much faster than expected for single-band
$s$-wave superconductivity. Noting that the field dependence of
$\sigma_{\rm{FLL}}$ described by Eq. (6) is only valid under the assumption of
a field-independent $\lambda$ Brandt (2003), the deviation of
$\sigma_{\rm{FLL}}$ from theoretical expectation indicates that $\lambda$
exhibits magnetic field dependence, which is expected for superconductors with
nodes or/and multiband superconductors Sonier  _et al._ (1997); Kadono (2004);
Weyeneth  _et al._ (2010).

Figure 4: The magnetic-field dependence of muon spin relaxation rate
$\sigma_{\rm{FLL}}$. The black dashed line is the theoretically expected
$\sigma_{\rm{FLL}}$ with Eq. (6). The red solid line exhibits the fits of data
with the two-component Ginzburg-Landau model Ser , using $\xi^{ab}_{1}=55.1$
nm, $\xi^{ab}_{2}=71.6$ nm deduced from Ref. Ronning  _et al._ (2008) and Ref.
Kurita  _et al._ (2009), resulting $\lambda=0.38(1)$ $\mu$m and $w=0.7(1)$.

Considering the multiband nature of BaNi2As2 Subedi and Singh (2008); Zhou
_et al._ (2011), the field dependence of $\sigma_{\rm{FLL}}$ is further
analyzed with the two-component modified London model Ser :

$\overline{\Delta B^{2}}=\overline{B}^{2}\sum_{\boldsymbol{q}\neq
0}[w\frac{e^{-\boldsymbol{q}^{2}\xi^{2}_{1}/2(1-h_{1})}}{1+\boldsymbol{q}^{2}\lambda^{2}/(1-h_{1})}+\\\
(1-w)\frac{e^{-\boldsymbol{q}^{2}\xi^{2}_{2}/2(1-h_{2})}}{1+\boldsymbol{q}^{2}\lambda^{2}/(1-h_{2})}],$
(7)

where
$\boldsymbol{q}=\frac{4\pi}{\sqrt{a}}(\frac{\sqrt{3}m}{2},n+\frac{m}{2})$ are
the reciprocal lattice vectors for triangular FLL
($a=1.075\sqrt{\frac{\Phi_{0}}{B}}$ is the intervortex distance), $\xi_{i}$ is
the coherence length for the $i$ th band and $w$ stands for the contribution
of the first band to the total superfluid density. The cut off function
$e^{-\boldsymbol{q}^{2}\xi^{2}/2}$ is a standard approximation for Ginzburg-
Landau equations introduced to account for the finite size of vortex cores
Sonier (2007). Since the four parameters are strongly coupled, it is generally
hard to obtain a unique set of solutions. This can be resolved by fixing
$\xi_{i}$ based on the reported properties. The shorter coherence length
$\xi^{ab}_{1}\approx 55.1$ nm can be estimated by the upper critical field
with $\xi^{ab}_{1}=(\Phi/2\pi H^{ab}_{c2}(0))^{1/2}$. Here $H^{ab}_{c2}(0)$ is
the zero temperature upper critical field for $H\parallel ab$. The ratio of
two coherence lengths $\xi^{ab}_{1}/\xi^{ab}_{2}=\sqrt{1/0.6}\approx 1.3$ can
be obtained from the thermal conductivity measurement Kurita  _et al._ (2009),
which will be discussed later. The fitting result with fixed $\xi^{ab}_{1,2}$
is shown by the solid red line in Fig. 4, giving $\lambda=0.38(1)$ $\mu$m and
$w=0.7(1)$.

Next, we discuss the temperature dependence of $\lambda^{-2}$. For single-band
$s$-wave superconductors with uncertain upper critical fields,
$\lambda^{-2}(T)$ and $H^{ab}_{c2}(T)$ can be obtained by fitting the magnetic
field dependence of muon relaxation rates under different temperatures Barker
_et al._ (2015), while such a method cannot be used when $\lambda$ exhibits
magnetic field dependence Brandt (2003). We first tried to extract
$\lambda(T)$ with Eq. (7). However, the limited data points and the complexity
of formula made it hard to get reasonable results in the whole temperature
range, especially when the temperatures approached $T_{c}$. To subtract the
$\lambda^{-2}(T)$, Eq. (6) was used under constant magnetic field $H$ with
fixed upper critical field $H^{ab}_{c2}(T)$ Khasanov  _et al._ (2014a). With
the known reduced magnetic field $h$, $\lambda^{-2}(T)$ can be directly
calculated from $\sigma_{\rm{FLL}}$.

Considering the two-band feature of BaNi2As2, the temperature dependence of
$H^{ab}_{c2}$ was estimated with the empirical two-band model Zhu  _et al._
(2008); Changjan and Udomsamuthirun (2011).

$\displaystyle H^{ab}_{c2}(T)=H^{ab}_{c2}(0)\frac{1-t^{2}}{1+t^{2}},$ (8)

where $t=T/T_{c}$ and $H^{ab}_{c2}(0)=0.11$ T was measured by heat capacity
Kurita  _et al._ (2009). Since the upper critical fields $H^{ab}_{c2}(T)$ here
are estimates, we only calculate $\lambda^{-2}(T)$ at the lowest magnetic
field.

###  III.4 D. Temperature dependence of superfluid density and gap symmetry

The superfluid density $\rho_{\rm{s}}$ is proportional to $\lambda^{-2}$ based
on the London approximation. The resulting normalized superfluid density is
plotted in Fig. 5.

Figure 5: Temperature dependence of the reduced superfluid density versus the
reduced temperature $T/T_{\rm{c}}$ measured at $\mu_{0}H=5$ mT parallel to the
$ab$ plane. The colored curves stand for the fitting of Eq. (10) with
different gap symmetry functions. The inset shows the field dependence of
magnetic penetration depth $\lambda$ at $T=50$ mK and 0.2 K. The connecting
lines in the inset are guides for eyes.

To further study the gap symmetry of BaNi2As2, the temperature dependence of
$\rho_{s}$ is fitted with different gap symmetry functions in both dirty
limit:

$\displaystyle\frac{\rho_{s}(T)}{{\rho_{s}(0)}}=\frac{\Delta(T,\phi)}{\Delta(0,\phi)}{\rm{tanh}}[\frac{\Delta(T,\phi)}{2k_{B}T}]$
(9)

and clean limit,

$\displaystyle\frac{\rho_{s}(T)}{{\rho_{s}(0)}}=1+\frac{1}{\pi}\int_{0}^{2\pi}d\phi\int_{\Delta(T,\phi)}^{\infty}\frac{\partial
f}{\partial E}\frac{E}{\sqrt{E^{2}-\Delta(T,\phi)^{2}}}dE,$ (10)

where $f=[1+\exp(E/k_{B}T)]^{-1}$ is the Fermi function.
$\Delta(T,\phi)=\Delta(\phi)\delta(T/T_{c})$ is the gap symmetry function. For
$s$-wave model, $\Delta_{\rm{s}}(\phi)=\Delta_{0}$; for $d$-wave model,
$\Delta_{\rm{d}}=\Delta_{0}\cos(2\phi)$. The temperature dependence
$\delta(T/T_{c})$ can be approximated as Carrington and Manzano (2003):

$\displaystyle\delta(T/T_{c})=\tanh\\{1.82[1.018(T_{c}/T-1)]^{0.51}\\},$ (11)

where $\rm{\Delta_{0}}$ is the maximum of zero temperature gap amplitude.

To examine the possible multi-gap nature inferred from fermi surface structure
calculation Subedi and Singh (2008); Zhou  _et al._ (2011) and thermal
conductivity measurement Kurita  _et al._ (2009), we also consider two weakly
coupled superconducting bands ($s+s$ or $s+d$), where the linear combination
of Eq.(11) can be used:

$\displaystyle\frac{\lambda^{-2}(T)}{\lambda^{-2}(0)}=w\frac{\lambda^{-2}(T,\Delta_{1}(T))}{\lambda^{-2}(0,\Delta_{1}(0))}+(1-w)\frac{\lambda^{-2}(T,\Delta_{2}(T))}{\lambda^{-2}(0,\Delta_{2}(0))}.$
(12)

Table 1: Fitted parameters to the $\sigma_{\rm{sc}}$ of BaNi2As2 with different models described in the text. Model |  Gap value |  $T_{\rm{c}}$ |  $2\Delta/k_{\rm{B}}T_{\rm{c}}$ |  $\chi^{2}_{r}$
---|---|---|---|---
|  (meV) |  (K) |  |
$s$ wave c.l. |  $\Delta=0.112(3)$ |  0.611(8) |  4.2(1) |  0.98
$s$ wave d.l. |  $\Delta=0.101(6)$ |  0.610(8) |  3.8(2) |  1.03
|  $\Delta_{\rm 1}=0.110(4)$, $\Delta_{\rm 2}=0.11(1)$ |  |  |
$s+s$ wave |  $w=0.95(51)$ |  0.603(6) |  4.2(4) |  1.21
$d$ wave |  $\Delta=0.217(9)$ |  0.574(3) |  8.8(3) |  6.15

The fitting results with different models are summarized in Fig. 5 and Table
1. The temperature dependence of the normalized superfluid density can be well
fitted with both single-gap and multi-gap $s$-wave model. However, the
introduction of a second gap does not seem to improve the fitting quality to a
great extent, and we can hardly get the second gap value with a reasonable
standard error. Therefore, there is no need to introduce more than one gap
amplitude to describe $\lambda^{-2}(T)$. The estimated electronic mean free
path in the normal state places BaNi2As2 in the dirty limit Kurita  _et al._
(2009). The obtained zero temperature superconducting gap
$\Delta_{0}=0.101(6)$ meV and the critical temperature $T_{\rm{c}}=0.610(8)$ K
give $2\Delta/k_{\rm{B}}T_{\rm{c}}=3.8(2)$, which is close to the BCS value of
3.54. The value of superconducting gap is close to the value of
$\Delta_{0}=0.0946$ meV obtained from heat capacity measurement by Kurita $et$
$al.$ Kurita  _et al._ (2009).

###  III.5 $\rm{\@slowromancap iv@}$. DISCUSSION

By analyzing the temperature dependence of penetration depth alone, one may
conclude that BaNi2As2 is a common $s$-wave superconductor, which is
consistent with previous heat capacity measurement. However, in single-band
$s$-wave superconductors, $\lambda$ should be independent of magnetic field
Kadono (2004); Khasanov  _et al._ (2005, 2008). As shown in the inset of Fig.
5, $\lambda$ is nearly doubled when the field increased from 5 mT to 40 mT.
Such a large change of $\lambda$ cannot be attributed to the error of
$H_{c2}(T)$. A field dependence of $\lambda$ is expected for superconductors
with nodes or/and multiband superconductors Sonier  _et al._ (1997); Kadono
(2004); Weyeneth  _et al._ (2010). The former case can be excluded since the
penetration depth below ${T_{c}}/3$ is temperature independent, which
indicates fully-gapped superconductivity. For the latter case, the carriers in
one of the bands is suppressed faster than other bands, causing the change of
penetration depth. Since $\sigma_{\rm{FLL}}(H)$ can be well described by two-
component modified London model, combined with single-gap nature revealed by
$\rho_{\rm{s}}(T)$, we suggest that BaNi2As2 is a multiband single-gap
superconductor. The superconducting gap amplitude of each band is nearly the
same while different bands exhibit diverse coherence lengths Khasanov  _et
al._ (2014a).

A key step in our $\lambda(H)$ analysis is to determine the ratio of different
coherence lengths, which is obtained from the thermal conductivity
measurement. The shoulder-like anomaly of $\kappa_{0}(H))/T$ curve was
initially attributed to the spread in $H_{\rm{c2}}$ Kurita  _et al._ (2009),
since there was no evidence pointing to a second superconducting band at that
time. However, such S-shaped $\kappa_{0}(H))/T$ curves were later also found
in several Ni2X2 based superconductors Kurita  _et al._ (2011); Hong  _et al._
(2014), which share the same ThCr2Si2 structure and similar electronic
structure Ronning  _et al._ (2009). Furthermore, the ratio of different
coherence lengths in different bands $\xi^{ab}_{1}/\xi^{ab}_{2}\approx 1.3$
has been confirmed by the calculated band structure Shein and Ivanovskii
(2009); Subedi and Singh (2008) and the ARPES experiment Zhou  _et al._
(2011). According to the BCS theory, the coherence length is proportional to
$\left<v_{F}\right>/\Delta_{0}$. Here $\left<v_{F}\right>$ is the averaged
value of the Fermi velocity, which can be estimated from the slope of bands
crossing the Fermi level. There are three bands crossing the Fermi level in
BaNi2As2. The ratio of the slopes of them is about 1 along $Z$-$A$ and $\sim$
1-3 along $\Gamma$-$M$ Shein and Ivanovskii (2009); Subedi and Singh (2008);
Zhou  _et al._ (2011). In consideration of the same $\Delta_{0}$ of the three
bands, $\xi^{ab}_{1}/\xi^{ab}_{2}=1.3$ obtained from thermal conductivity data
is reasonable.

Considering the uniform gap value in each band revealed by both $\mu$SR and
thermodynamic properties Kurita  _et al._ (2009), the multiband
superconductivity in BaNi2As2 is quite unique compared with other multi-gap
superconductors Khasanov  _et al._ (2009, 2010); Carrington and Manzano
(2003). It is worth noting that, for a long time, it has been widely believed
that the number of excitation gaps determines the number of contributing bands
needed in a theoretical model. However, there is no necessary connection
between the number of excitation gaps and the number of length scale.
Superconductors with multi-bands can be: i) single-gap and single length scale
Gupta  _et al._ (2020); ii) multi-gap with single length scale Fente  _et al._
(2016, 2018); iii) single-gap with multi lengths scales such as SrPt3P
Khasanov  _et al._ (2014b) and BaNi2As2 in the present study.

In most cases, strong electronic scattering in conventional dirty
superconductors weakens the difference between bands. As a result, each band
tends to exhibit identical superconducting behavior. A natural question here
is what causes the unconventional multiband behavior in BaNi2As2. Recent
density functional theory (DFT) results showed that the rebonding of As anions
can induce complex structural instabilities. These low-temperature structures
are similar in energy but are very different in Fermiology Lei  _et al._
(2022). The competition between these bonding arrangements can possibly
explain this unconventional superconducting behavior. However, such scenario
is hard to reconcile with the single gap amplitude revealed by heat capacity
Kurita  _et al._ (2009) and $\mu$SR experiments. On the other hand, recent
Raman scattering measurements revealed a strong dynamic nematic fluctuations
in BaNi2As2, which strongly supported the nematic enhanced superconductivity
upon doping in this material Yao  _et al._ (2022). Whether the unconventional
single-gap multiband superconducting in BaNi2As2 is a manifestation of the
interaction between superconducting and nematicity demands further
confirmation. Subsequent researches on superconducting band properties in
doped systems will be very helpful to elucidate how the CDW and charge induced
nematicity interact with superconductivity.

##  IV $\rm{\@slowromancap v@}$. CONCLUSIONS

To summarize, systematic $\mu$SR experiments have been performed to study the
temperature and magnetic field dependence of penetration depth $\lambda$ in
BaNi2As2. The temperature independent behavior of superfluid density
$\rho_{\rm{s}}$ at low temperature suggests its fully-gapped nature. The
fitting with single-gap $s$-wave model in dirty limit gives the zero
temperature gap amplitude $\Delta_{0}=0.101(6)$ meV. The deviation of
$\sigma_{\rm{FLL}}(H)$ from single-band scenario can be understood with a two-
component model with different coherence lengths in each band. Our TF-$\mu$SR
results reveal that BaNi2As2 is a rare single-gap multiband superconductor.
The ZF-$\mu$SR experiments confirmed the preservation of time-reversal
symmetry in BaNi2As2. This work provides new experimental evidence to help
understand the interplay between CDW, nematicity and superconductivity.

##  V ACKNOWLEDGMENTS

We are grateful to the ISIS cryogenics Group for their valuable help during
the $\mu$SR experiments (ISIS.RIB 2010283). This research was funded by the
National Natural Science Foundations of China, No. 12174065, the National Key
Research and Development Program of China, No. 2022YFA1402203, and the
Shanghai Municipal Science and Technology Major Project Grant, No.
2019SHZDZX01. The materials synthesis efforts at Rice are supported by the
U.S. Department of Energy (DOE), Basic Energy Sciences (BES), under Contract
No. DE-SC0012311, and the Robert A. Welch Foundation, Grant No. C-1839 (P.D.).

##  Appendix A APPENDIX: CHARACTERIZATION MEASUREMENTS ON SAMPLES POSSIBLY
EXPOSED TO THE AIR

BaNi2As2 samples are air sensitive Ronning  _et al._ (2008). After the $\mu$SR
experiments, single crystals of BaNi2As2 were stored in a glass bottle filling
with nitrogen, and were shipped from the $\mu$SR facility to our university.
Due to the epidemic prevention policy, our sample got stuck in customs for
several months. The single crystal XRD and electrical resistance experiments
were performed on samples which were possibly exposed to the air for some
time.

The single crystal XRD diffraction patterns were obtained by using a Bruker D8
advanced X-ray diffraction spectrometer ($\lambda=1.5418\ \rm{\r{A}}$) at room
temperature. Single crystal BaNi2As2 are naturally oriented with the $c$-axis
normal to the plate, and the XRD patterns are shown in Fig. A1. The
diffraction peaks of specific lattice planes are obvious. The lattice
parameter $c$ is determined via the liner fitting between the diffraction
pattern index $n$ and $\sin(\theta)$, which gives $c$ = 11.66(4) Å. For
comparison, $c$ = 11.54(2) Å in Ref Ronning  _et al._ (2008).

Figure A1: Single crystal X-ray diffraction patterns of BaNi2As2.

Temperature dependence of resistance $R(T)$ from 300 K to 0.3 K was measured
using standard four-probe method on a physical property measurement system
(PPMS) equipped with Helium3 refrigerator. The resistance was measured with
cooling. Fig. A2(a) shows the temperature dependence of resistance $R(T)$ from
300 K to 2 K. A broad hump is observed near 150 K, which is identified as the
structural phase transition Ronning  _et al._ (2008). The width of transition
is much broader compared to Ref Ronning  _et al._ (2008), which may due to the
damage of our sample after its exposure to the air. The RRR (=$\rho$(300
K)/$\rho$(4 K)) is 10. For comparison, RRR = 5 in Ref Ronning  _et al._
(2008). The resistance below 2 K is shown in Fig. A2(b). The direction of the
external magnetic field is parallel to $ab$-plane. The resistance of the
sample reaches zero at 0.63 K under zero field and the sample shows
superconductivity under the external field of 400 G. It should be noted that,
the measurements with field higher than 100 G was conducted after the sample
being exposed to the air for another 10 minutes. There are obvious differences
between $R-T$ curves over and below 100 G, which further illustrate that
BaNi2As2 is very air-sensitive.

Figure A2: Temperature dependence of resistance from 300 K to 2 K(a) , and
from 2.0 K to 0.3 K (b) with external magnetic field parallel to $ab$ plane.
There is an additional 10 minutes’ exposure to air between the 100 G and 150 G
measurements in (b).

## References

  * Fradkin  _et al._ (2015) E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  * Keimer  _et al._ (2015) B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature (London) 518, 179 (2015).
  * Johnston (2010) D. C. Johnston, The puzzle of high temperature superconductivity in layered iron pnictides and chalcogenides, Adv. Phys. 59, 803 (2010).
  * Tranquada  _et al._ (1995) J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Evidence for stripe correlations of spins and holes in copper oxide superconductors, Nature (London) 375, 561 (1995).
  * Miao  _et al._ (2021) H. Miao, G. Fabbris, R. J. Koch, D. G. Mazzone, C. S. Nelson, R. Acevedo-Esteves, G. D. Gu, Y. Li, T. Yilimaz, K. Kaznatcheev, E. Vescovo, M. Oda, T. Kurosawa, N. Momono, T. Assefa, I. K. Robinson, E. S. Bozin, J. M. Tranquada, P. D. Johnson, and M. P. M. Dean, Charge density waves in cuprate superconductors beyond the critical doping, npj Quantum Mater. 6, 31 (2021).
  * Proust and Taillefer (2019) C. Proust and L. Taillefer, The remarkable underlying ground states of cuprate superconductors, Annu. Rev. Condens. Matter Phys. 10, 409 (2019).
  * Agterberg  _et al._ (2020) D. F. Agterberg, J. S. Davis, S. D. Edkins, E. Fradkin, D. J. Van Harlingen, S. A. Kivelson, P. A. Lee, L. Radzihovsky, J. M. Tranquada, and Y. Wang, The physics of pair-density waves: Cuprate superconductors and beyond, Annu. Rev. Condens. Matter Phys. 11, 231 (2020).
  * Dai (2015) P. Dai, Antiferromagnetic order and spin dynamics in iron-based superconductors, Rev. Mod. Phys. 87, 855 (2015).
  * Fernandes  _et al._ (2014) R. M. Fernandes, A. V. Chubukov, and J. Schmalian, What drives nematic order in iron-based superconductors?, Nat. Phys. 10, 97 (2014).
  * Paglione and Greene (2010) J. Paglione and R. L. Greene, High-temperature superconductivity in iron-based materials, Nat. Phys. 6, 645 (2010).
  * Marel  _et al._ (2003) D. v. d. Marel, H. J. A. Molegraaf, J. Zaanen, Z. Nussinov, F. Carbone, A. Damascelli, H. Eisaki, M. Greven, P. H. Kes, and M. Li, Quantum critical behaviour in a high-Tc superconductor, Nature (London) 425, 271 (2003).
  * Lederer  _et al._ (2015) S. Lederer, Y. Schattner, E. Berg, and S. A. Kivelson, Enhancement of superconductivity near a nematic quantum critical point, Phys. Rev. Lett. 114, 097001 (2015).
  * Wang  _et al._ (2012) M.-X. Wang, C. Liu, J.-P. Xu, F. Yang, L. Miao, M.-Y. Yao, C. Gao, C. Shen, X. Ma, X. Chen, et al., The coexistence of superconductivity and topological order in the $\mathrm{Bi}_{2}\mathrm{Se}_{3}$ thin films, Science 336, 52 (2012).
  * Jiang  _et al._ (2015) H. Jiang, G. Cao, and C. Cao, Electronic structure of quasi-one-dimensional superconductor $\mathrm{K}_{2}\mathrm{Cr}_{3}\mathrm{As}_{3}$ from first-principles calculations, Sci. Rep. 5, 16054 (2015).
  * Gu and Wen (2022) Q. Gu and H.-H. Wen, Superconductivity in nickel-based 112 systems, The Innovation 3 (2022).
  * Pfisterer and Nagorsen (1980) M. Pfisterer and G. Nagorsen, Zur Struktur ternärer Übergangsmetallarsenide/On the Structure of Ternary Arsenides, Z. Naturforsch. B 35, 703 (1980).
  * Alireza  _et al._ (2008) P. L. Alireza, Y. T. C. Ko, J. Gillett, C. M. Petrone, J. M. Cole, G. G. Lonzarich, and S. E. Sebastian, Superconductivity up to 29 K in $\mathrm{SrFe}_{2}\mathrm{As}_{2}$ and $\mathrm{BaFe}_{2}\mathrm{As}_{2}$ at high pressures, J. Phys. Condens. Matter 21, 012208 (2008).
  * Mani  _et al._ (2009) A. Mani, N. Ghosh, S. Paulraj, A. Bharathi, and C. S. Sundar, Pressure-induced superconductivity in $\mathrm{BaFe}_{2}\mathrm{As}_{2}$ single crystal, Europhys. Lett. 87, 17004 (2009).
  * Gallais  _et al._ (2013) Y. Gallais, R. M. Fernandes, I. Paul, L. Chauvière, Y.-X. Yang, M.-A. Méasson, M. Cazayous, A. Sacuto, D. Colson, and A. Forget, Observation of Incipient Charge Nematicity in ${\mathrm{Ba}(\mathrm{Fe}{}_{1-X}\mathrm{Co}{}_{X})}_{2}\mathrm{As}{}_{2}$, Phys. Rev. Lett. 111, 267001 (2013).
  * Ronning  _et al._ (2008) F. Ronning, N. Kurita, E. D. Bauer, B. L. Scott, T. Park, T. Klimczuk, R. Movshovich, and J. D. Thompson, The first order phase transition and superconductivity in ${\mathrm{BaNi}}_{2}$${\mathrm{As}}_{2}$ single crystals, J. Phys.: Condens.Matter 20, 342203 (2008).
  * Sefat  _et al._ (2009) A. S. Sefat, M. A. McGuire, R. Jin, B. C. Sales, D. Mandrus, F. Ronning, E. D. Bauer, and Y. Mozharivskyj, Structure and anisotropic properties of ${\text{BaFe}}_{2-x}{\text{Ni}}_{x}{\text{As}}_{2}$ ($x=0$, 1, and 2) single crystals, Phys. Rev. B 79, 094508 (2009).
  * Lee  _et al._ (2019) S. Lee, G. de la Peña, S. X.-L. Sun, M. Mitrano, Y. Fang, H. Jang, J.-S. Lee, C. Eckberg, D. Campbell, J. Collini, J. Paglione, F. M. F. de Groot, and P. Abbamonte, Unconventional Charge Density Wave Order in the Pnictide Superconductor $\mathrm{Ba}({\mathrm{Ni}}_{1-x}{\mathrm{Co}}_{x})_{2}{\mathrm{As}}_{2}$, Phys. Rev. Lett. 122, 147601 (2019).
  * Lee  _et al._ (2021) S. Lee, J. Collini, S. X.-L. Sun, M. Mitrano, X. Guo, C. Eckberg, J. Paglione, E. Fradkin, and P. Abbamonte, Multiple Charge Density Waves and Superconductivity Nucleation at Antiphase Domain Walls in the Nematic Pnictide ${\mathrm{Ba}}_{1-x}{\mathrm{Sr}}_{x}{\mathrm{Ni}}_{2}{\mathrm{As}}_{2}$, Phys. Rev. Lett. 127, 027602 (2021).
  * Guo  _et al._ (2022) Y. Guo, M. Klemm, J. S. Oh, Y. Xie, B.-H. Lei, S. Gorovikov, T. Pedersen, M. Michiardi, S. Zhdanovich, A. Damascelli, J. Denlinger, M. Hashimoto, D. Lu, S.-K. Mo, R. G. Moore, R. J. Birgeneau, D. J. Singh, P. Dai, and M. Yi, Spectral Evidence for Unidirectional Charge Density Wave in Detwinned $\mathrm{BaNi}_{2}\mathrm{As}_{2}$, arXiv e-prints 10.48550/arXiv.2205.14339 (2022).
  * Yao  _et al._ (2022) Y. Yao, R. Willa, T. Lacmann, S.-M. Souliou, M. Frachet, K. Willa, M. Merz, F. Weber, C. Meingast, R. Heid, A.-A. Haghighirad, J. Schmalian, and M. Le Tacon, An electronic nematic liquid in $\mathrm{BaNi}_{2}\mathrm{As}_{2}$, Nat. Commun. 13, 4535 (2022).
  * Shein and Ivanovskii (2009) I. R. Shein and A. L. Ivanovskii, Electronic and structural properties of low-temperature superconductors and ternary pnictides $A{\mathrm{Ni}}_{2}P{n}_{2}$ ($A=\mathrm{Sr},\mathrm{Ba}$ and $Pn=\mathrm{P},\mathrm{As}$), Phys. Rev. B 79, 054510 (2009).
  * Subedi and Singh (2008) A. Subedi and D. J. Singh, Density functional study of ${\mathrm{BaNi}}_{2}{\mathrm{As}}_{2}$: Electronic structure, phonons, and electron-phonon superconductivity, Phys. Rev. B 78, 132511 (2008).
  * Eckberg  _et al._ (2020) C. Eckberg, D. J. Campbell, T. Metz, J. Collini, H. Hodovanets, T. Drye, P. Zavalij, M. H. Christensen, R. M. Fernandes, S. Lee, et al., Sixfold enhancement of superconductivity in a tunable electronic nematic system, Nat. Phys. 16, 346 (2020).
  * Lederer  _et al._ (2020) S. Lederer, E. Berg, and E.-A. Kim, Tests of nematic-mediated superconductivity applied to ${\mathrm{Ba}}_{1-x}{\mathrm{Sr}}_{x}{\mathrm{Ni}}_{2}{\mathrm{As}}_{2}$, Phys. Rev. Res. 2, 023122 (2020).
  * Zhou  _et al._ (2011) B. Zhou, M. Xu, Y. Zhang, G. Xu, C. He, L. X. Yang, F. Chen, B. P. Xie, X.-Y. Cui, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, X. Dai, and D. L. Feng, Electronic structure of ${\mathrm{BaNi}}_{2}$${\mathrm{As}}_{2}$, Phys. Rev. B 83, 035110 (2011).
  * Kurita  _et al._ (2009) N. Kurita, F. Ronning, Y. Tokiwa, E. D. Bauer, A. Subedi, D. J. Singh, J. D. Thompson, and R. Movshovich, Low-Temperature Magnetothermal Transport Investigation of a Ni-Based Superconductor ${\mathrm{BaNi}}_{2}{\mathrm{As}}_{2}$: Evidence for Fully Gapped Superconductivity, Phys. Rev. Lett. 102, 147004 (2009).
  * Kittaka  _et al._ (2014) S. Kittaka, Y. Aoki, Y. Shimura, T. Sakakibara, S. Seiro, C. Geibel, F. Steglich, H. Ikeda, and K. Machida, Multiband Superconductivity with Unexpected Deficiency of Nodal Quasiparticles in $\mathrm{CeCu}_{2}\mathrm{Si}_{2}$, Phys. Rev. Lett. 112, 067002 (2014).
  * Hillier  _et al._ (2022) A. D. Hillier, S. J. Blundell, I. McKenzie, I. Umegaki, L. Shu, J. A. Wright, T. Prokscha, F. Bert, K. Shimomura, A. Berlie, et al., Muon spin spectroscopy, Nat. Rev. Methods Primers 2, 5 (2022).
  * Sonier  _et al._ (2000) J. E. Sonier, J. H. Brewer, and R. F. Kiefl, $\mu$SR studies of the vortex state in type-II superconductors, Rev. Mod. Phys. 72, 769 (2000).
  * Suter and Wojek (2012) A. Suter and B. Wojek, Musrfit: A Free Platform-Independent Framework for $\mu$SR Data Analysis, Phys. Procedia 30, 69 (2012).
  * Kothapalli  _et al._ (2010) K. Kothapalli, F. Ronning, E. D. Bauer, A. J. Schultz, and H. Nakotte, Single-crystal neutron diffraction studies on Ni-based metal-pnictide superconductor $\mathrm{BaNi}_{2}\mathrm{As}_{2}$, J. Phys.: Conf. Ser. 251, 012010 (2010).
  * A. Yaouanc and P. Dalmas de Réotier (2011) A. Yaouanc and P. Dalmas de Réotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter (Oxford University Press, Oxford, UK, 2011).
  * Hayano  _et al._ (1979) R. S. Hayano, Y. J. Uemura, J. Imazato, N. Nishida, T. Yamazaki, and R. Kubo, Zero-and low-field spin relaxation studied by positive muons, Phys. Rev. B 20, 850 (1979).
  * Brandt (2003) E. H. Brandt, Properties of the ideal ginzburg-landau vortex lattice, Phys. Rev. B 68, 054506 (2003).
  * Combley (1985) F. H. Combley, Muon Spin Rotation Spectroscopy: Principles and Applications in Solid State Physics, Phys. Bull. 36, 430 (1985).
  * Bueno  _et al._ (2011) J. F. Bueno, D. J. Arseneau, R. Bayes, J. H. Brewer, W. Faszer, M. D. Hasinoff, G. M. Marshall, E. L. Mathie, R. E. Mischke, G. D. Morris, K. Olchanski, V. Selivanov, and R. Tacik, Longitudinal muon spin relaxation in high-purity aluminum and silver, Phys. Rev. B 83, 205121 (2011).
  * Zhang  _et al._ (2018) J. Zhang, Z. Ding, C. Tan, K. Huang, O. O. Bernal, P.-C. Ho, G. D. Morris, A. D. Hillier, P. K. Biswas, S. P. Cottrell, H. Xiang, X. Yao, D. E. MacLaughlin, and L. Shu, Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of $\mathrm{YBa}_{2}\mathrm{Cu}_{3}\mathrm{O}_{\mathrm{y}}$, Sci. Adv. 4, eaao5235 (2018).
  * Aeppli  _et al._ (1987) G. Aeppli, E. J. Ansaldo, J. H. Brewer, R. J. Cava, R. F. Kiefl, S. R. Kreitzman, G. M. Luke, and D. R. Noakes, Magnetic penetration depth and flux-pinning effects in high-${T}_{c}$ superconductor ${\mathrm{La}}_{1.85}$${\mathrm{Sr}}_{0.15}$CuO, Phys. Rev. B 35, 7129 (1987).
  * Barker  _et al._ (2015) J. A. T. Barker, D. Singh, A. Thamizhavel, A. D. Hillier, M. R. Lees, G. Balakrishnan, D. M. Paul, and R. P. Singh, Unconventional Superconductivity in ${\mathrm{La}}_{7}{\mathrm{Ir}}_{3}$ Revealed by Muon Spin Relaxation: Introducing a New Family of Noncentrosymmetric Superconductor That Breaks Time-Reversal Symmetry, Phys. Rev. Lett. 115, 267001 (2015).
  * Sonier  _et al._ (1997) J. E. Sonier, R. F. Kiefl, J. H. Brewer, J. Chakhalian, S. R. Dunsiger, W. A. MacFarlane, R. I. Miller, A. Wong, G. M. Luke, and J. W. Brill, Muon-Spin Rotation Measurements of the Magnetic Field Dependence of the Vortex-Core Radius and Magnetic Penetration Depth in ${\mathrm{NbSe}}_{2}$, Phys. Rev. Lett. 79, 1742 (1997).
  * Kadono (2004) R. Kadono, Field-induced quasiparticle excitations in novel type II superconductors, J. Phys.: Condens.Matter 16, S4421 (2004).
  * Weyeneth  _et al._ (2010) S. Weyeneth, M. Bendele, R. Puzniak, F. Murányi, A. Bussmann-Holder, N. D. Zhigadlo, S. Katrych, Z. Bukowski, J. Karpinski, A. Shengelaya, R. Khasanov, and H. Keller, Field-dependent superfluid density in the optimally doped $\mathrm{SmFeAsO}_{1-x}\mathrm{F}_{y}$ superconductor, Europhys. Lett. 91, 47005 (2010).
  * (48) Effect of Two Gaps on the Flux-Lattice Internal Field Distribution: Evidence of Two Length Scales in ${\mathrm{M}\mathrm{g}}_{1-x}{\mathrm{A}\mathrm{l}}_{x}{\mathrm{B}}_{2}$ from $\mu$SR, author = Serventi, S. and Allodi, G. and De Renzi, R. and Guidi, G. and Romanò, L. and Manfrinetti, P. and Palenzona, A. and Niedermayer, Ch. and Amato, A. and Baines, Ch.,, .
  * Sonier (2007) J. E. Sonier, Muon spin rotation studies of electronic excitations and magnetism in the vortex cores of superconductors, Rep. Prog. Phys. 70, 1717 (2007).
  * Khasanov  _et al._ (2014a) R. Khasanov, A. Amato, P. K. Biswas, H. Luetkens, N. D. Zhigadlo, and B. Batlogg, ${\mathrm{SrPt}}_{3}\mathrm{P}$: A two-band single-gap superconductor, Phys. Rev. B 90, 140507 (2014a).
  * Zhu  _et al._ (2008) X. Zhu, H. Yang, L. Fang, G. Mu, and H.-H. Wen, Upper critical field, Hall effect and magnetoresistance in the iron-based layered superconductor $\mathrm{LaFeAsO}_{0.9}\mathrm{F}_{0.1-\delta}$, Supercond. Sci. Technol. 21, 105001 (2008).
  * Changjan and Udomsamuthirun (2011) A. Changjan and P. Udomsamuthirun, The critical magnetic field of anisotropic two-band magnetic superconductors, Solid State Commun. 151, 988 (2011).
  * Carrington and Manzano (2003) A. Carrington and F. Manzano, Magnetic penetration depth of $\mathrm{MgB}_{2}$, Phys. C Supercond. 385, 205 (2003).
  * Khasanov  _et al._ (2005) R. Khasanov, D. G. Eshchenko, D. Di Castro, A. Shengelaya, F. La Mattina, A. Maisuradze, C. Baines, H. Luetkens, J. Karpinski, S. M. Kazakov, and H. Keller, Magnetic penetration depth in ${\mathrm{RbOs}}_{2}{\mathrm{O}}_{6}$ studied by muon spin rotation, Phys. Rev. B 72, 104504 (2005).
  * Khasanov  _et al._ (2008) R. Khasanov, P. W. Klamut, A. Shengelaya, Z. Bukowski, I. M. Savić, C. Baines, and H. Keller, Muon-spin rotation measurements of the penetration depth of the ${\text{Mo}}_{3}{\text{Sb}}_{7}$ superconductor, Phys. Rev. B 78, 014502 (2008).
  * Kurita  _et al._ (2011) N. Kurita, F. Ronning, C. F. Miclea, Y. Tokiwa, E. D. Bauer, A. Subedi, D. J. Singh, H. Sakai, J. D. Thompson, and R. Movshovich, Fully gapped superconductivity in $\mathrm{Ni}$-pnictide superconductors $\mathrm{BaNi}_{2}\mathrm{As}_{2}$ and $\mathrm{SrNi}_{2}\mathrm{P}_{2}$, J. Phys.: Conf. Ser. 273, 012097 (2011).
  * Hong  _et al._ (2014) X. C. Hong, Z. Zhang, S. Y. Zhou, J. Pan, Y. Xu, H. Wang, Q. Mao, M. Fang, J. K. Dong, and S. Y. Li, Multigap nodeless superconductivity in nickel chalcogenide $\mathrm{TlNi}{}_{2}\mathrm{Se}{}_{2}$, Phys. Rev. B 90, 060504 (2014).
  * Ronning  _et al._ (2009) F. Ronning, E. Bauer, T. Park, N. Kurita, T. Klimczuk, R. Movshovich, A. Sefat, D. Mandrus, and J. Thompson, $\mathrm{Ni}_{2}\mathrm{X}_{2}$ ($\mathrm{X}$=pnictide, chalcogenide, or B) based superconductors, Phys. C Supercond. 469, 396 (2009).
  * Khasanov  _et al._ (2009) R. Khasanov, D. V. Evtushinsky, A. Amato, H.-H. Klauss, H. Luetkens, C. Niedermayer, B. Büchner, G. L. Sun, C. T. Lin, J. T. Park, D. S. Inosov, and V. Hinkov, Two-Gap Superconductivity in ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_{x}{\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$: A Complementary Study of the Magnetic Penetration Depth by Muon-Spin Rotation and Angle-Resolved Photoemission, Phys. Rev. Lett. 102, 187005 (2009).
  * Khasanov  _et al._ (2010) R. Khasanov, M. Bendele, A. Amato, K. Conder, H. Keller, H.-H. Klauss, H. Luetkens, and E. Pomjakushina, Evolution of Two-Gap Behavior of the Superconductor ${\mathrm{FeSe}}_{1-x}$, Phys. Rev. Lett. 104, 087004 (2010).
  * Cavalcanti  _et al._ (2020) P. J. F. Cavalcanti, T. T. Saraiva, J. A. Aguiar, A. Vagov, M. D. Croitoru, and A. A. Shanenko, Multiband superconductors with degenerate excitation gaps, J. Phys.: Condens.Matter 32, 455702 (2020).
  * Gupta  _et al._ (2020) R. Gupta, C. Löhnert, C. Wang, D. Johrendt, H. Luetkens, S. Malick, T. Shiroka, Z. Hossain, and R. Khasanov, Isotropic $s$-wave superconductivity in the noncentrosymmetric charge density wave superconductor ${\mathrm{SrPt}}_{2}{\mathrm{As}}_{2}$, Phys. Rev. B 102, 144515 (2020).
  * Fente  _et al._ (2016) A. Fente, E. Herrera, I. Guillamón, H. Suderow, S. Mañas Valero, M. Galbiati, E. Coronado, and V. G. Kogan, Field dependence of the vortex core size probed by scanning tunneling microscopy, Phys. Rev. B 94, 014517 (2016).
  * Fente  _et al._ (2018) A. Fente, W. R. Meier, T. Kong, V. G. Kogan, S. L. Bud’ko, P. C. Canfield, I. Guillamón, and H. Suderow, Influence of multiband sign-changing superconductivity on vortex cores and vortex pinning in stoichiometric high-${T}_{\rm{c}}\ {\mathrm{CaKFe}}_{4}{\mathrm{As}}_{4}$, Phys. Rev. B 97, 134501 (2018).
  * Khasanov  _et al._ (2014b) R. Khasanov, A. Amato, P. K. Biswas, H. Luetkens, N. D. Zhigadlo, and B. Batlogg, ${\mathrm{SrPt}}_{3}\mathrm{P}$: A two-band single-gap superconductor, Phys. Rev. B 90, 140507 (2014b).
  * Lei  _et al._ (2022) B.-H. Lei, Y. Guo, Y. Xie, P. Dai, M. Yi, and D. J. Singh, Complex structure due to As bonding and interplay with electronic structure in superconducting ${\mathrm{BaNi}}_{2}{\mathrm{As}}_{2}$, Phys. Rev. B 105, 144505 (2022).