# Unconventional gap structures and the intermediate mixed state: a vortex
lattice study of the noncentrosymmetric superconductor BiPd

A. S. Cameron  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK.  Institut für Festkörper- und
Materialphysik, Technische Universität Dresden, D-01069 Dresden, Germany.  L.
Lemberger  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK.  Institut Laue Langevin, 71 Avenue des
Martyrs, F-38000 Grenoble cedex 9, France.  R. Riyat  School of Physics and
Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK.  A.
T. Holmes  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK.  European Spallation Source ERIC, P.O. Box
176, SE-221 00, Lund, Sweden.  Y. S. Yerin  CNR-SPIN, via del Fosso del
Cavaliere, 100, 00133 Roma, Italy.  A. D. Hillier  ISIS Pulsed Neutron and
Muon Source, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot OX11
0QX, United Kingdom.  B. Joshi  Tata Institute of Fundamental Research,
Mumbai-40005, India.  S. Ramakrishnan  Tata Institute of Fundamental Research,
Mumbai-40005, India.  J. Gavilano  Laboratory for Neutron Scattering and
Imaging (LNS), Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland.
C. D. Dewhurst  Institut Laue Langevin, 71 Avenue des Martyrs, F-38000
Grenoble cedex 9, France.  E. M. Forgan  School of Physics and Astronomy,
University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK.  E. Blackburn
<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Birmingham, Edgbaston, Birmingham, B15 2TT, UK.  Division of Synchrotron
Radiation Research, Lund University, SE-22100 Lund, Sweden.

###### Abstract

We report on neutron scattering measurements on the vortex lattice of the
noncentrosymmetric superconductor BiPd. We observe the existence of the
intermediate mixed state, a region where Meissner and vortex lattice phases
coexist, which is a feature of low $\kappa$ Type-II superconductors. Following
this, we obtain an estimate of the value of $\kappa$ using the extended London
model, which confirms the expectation that $\kappa$ should be small. Finally,
we find that the temperature dependence of the vortex lattice form factor fits
well to a model designed to describe singlet-triplet mixing in non-
centrosymmetric superconductors, which may shed light on the question of the
gap structure in BiPd.

High Tc superconductivity, Vortex lattice, flux lines, d-wave

## Introduction

Superconductivity in non-centrosymmetric (NCS) materials is of particular
interest, as it has been predicted that the breaking of spatial inversion
symmetry by the crystal lattice could lead to unconventional pairing
symmetries where the mixing of singlet and triplet order parameters is
possible Sigrist and Ueda (1991); Sigrist  _et al._ (2007); Gor’kov and Rashba
(2001); Bauer  _et al._ (2004); Frigeri  _et al._ (2004); Bauer and Sigrist
(2012). Following the discovery of superconductivity in the NCS system CePt3Si
Bauer  _et al._ (2004), investigations into these materials have flourished,
and many more NCS superconductors have now been reported Smidman  _et al._
(2017). Broken inversion symmetry leads to anisotropic spin-orbit coupling
(ASOC), which removes the spin degeneracy of the electronic bands Rashba
(1960), leading to spin-split Fermi surfaces. Combined with their unique
pairing state, which may itself generate anisotropic gaps and accidental nodes
Hayashi  _et al._ (2006), this results in a variety of unusual phenomena Karki
_et al._ (2010); Chen  _et al._ (2011); Takimoto and Thalmeier (2009);
Samokhin  _et al._ (2004); Yuan  _et al._ (2006), including spontaneous
magnetisation at twin boundaries Arahata  _et al._ (2013) and helical phases
of the order parameter Mineev and Samokhin (1994); Kaur  _et al._ (2005). From
the perspective of the vortex lattice (VL), there is a predicted stabilisation
of the Fulde-Ferrell-Larkin-Ovchinnikov state Hiroko Tanaka and Hasegawa
(2007), a weakened paramagnetic limiting response and the emergence of the
anomalous magnetoelectric effect Edelstein (1995); Mineev (2005); Fujimoto
(2005); Mineev (2011); Hiasa and Ikeda (2008). Theoretical studies of the
structure of vortices and the VL have predicted an altered structure Yip
(2005); Hayashi  _et al._ (2006) and the emergence of tangential components of
the magnetic field Oka  _et al._ (2006); Lu and Yip (2009); Kashyap and
Agterberg (2013). Furthermore, experimental evidence for unusual behaviour of
the VL in the NCS material Ru7B3 has previously been reported Cameron  _et
al._ (2019, 2023), as well as evidence for singlet-triplet mixing in the order
parameter of this and several other NCS superconductors Cameron  _et al._
(2022); Yuan  _et al._ (2006), indicating this is a rich field of study with
many interesting phenomena predicted.

Figure 1: (Color online) (a-c) Diffraction patterns at around 200 G for fields
applied along different crystallographic directions. (a) 220 G with
$B\parallel(010)$, at 0.05 K. (b) 200 G with $B\parallel\sim$(101), at 0.1 K.
(c) 210 G with $B\parallel(101)$, at 0.1 K. (d) A photograph of the samples
with the $\sim$(101) (top) and (010) (bottom) orientations perpendicular to
the sample holder. The angle $\alpha$ between two scattering vectors of the
vortex lattice is given in panel (b).

$\alpha$-BiPd, which we refer to in this paper simply as BiPd, is an NCS
material, forming a monoclinic structure below 210∘C, with space group P21
Bhatt and Schubert (1979). It undergoes a superconducting transition at 3.8 K,
but there is some contention over the upper critical field, as different
measurements have produced wildly different results. Electrical resistivity
measurements by Joshi _et al._ Joshi  _et al._ (2011) found a lower critical
field of 123 G and an upper critical field of 7000 G for fields applied along
the (010) direction, with a moderate anisotropy of not greater than $30\%$ for
fields applied along the other crystallographic axes. From this, the GL
parameter was estimated to be $\kappa\sim 7.6$ Joshi  _et al._ (2011), which
would make this a moderate Type-II superconductor. However, magnetisation and
heat capacity measurements Peets (2014); Sun  _et al._ (2015) gave values of
the upper critical field an order of magnitude smaller than this. To date,
there is not a conclusive consensus on the origin of this discrepancy,
although the presence of filamentary superconductivity has been proposed Peets
(2014); Ramakrishnan  _et al._ (2017).

Fermi surface measurements point to the existence of multiple bands
Ramakrishnan  _et al._ (2016), which is not surprising as at the very least
NCS materials have spin-split bands due to the ASOC. The gap structure of BiPd
is the subject of a complex debate. Some heat capacity measurements by
Ramakrishnan et al. Ramakrishnan  _et al._ (2017) were best fit to a two-gap
model; an anisotropic single-gap model did not appear to fit the data. Point
contact spectroscopy measurements Mondal  _et al._ (2012) also supported a
two-gap model. The presence of a conductance peak at zero bias in these data
was attributed to Andreev bound states Mondal  _et al._ (2012); Kashiwaya and
Tanaka (2000), which indicated an unconventional nature for the order
parameter. However, bulk measurements of magnetisation, resistivity and
specific heat reported by Peets Peets (2014) can all be explained using
conventional s-wave behaviour, although Peets notes that the upward curvature
of the upper critical field is suggestive of possible multi-band
superconductivity. An STM study by Sun _et al._ Sun  _et al._ (2015) saw no
evidence for anisotropy in the gap Sun  _et al._ (2015), but other STM
measurements show signs of unconventional superconductivity Ramakrishnan  _et
al._ (2017). Tunnel diode oscillator measurements of the penetration depth
Jiao  _et al._ (2014) along different crystal axes were best fit to a two-band
BCS model where one of the bands is anisotropic, although the authors
concluded that such an order parameter could be due to the singlet-triplet
mixing predicted for NCS superconductors. In short, while there is reasonable
support for two-gap scenario, the question of anisotropy in the gap is
contested. Clearly, the situation is complex and requires further study. In
this paper, we provide additional evidence from values of the penetration
depth, obtained via small angle neutron diffraction measurements of the vortex
lattice.

## Experimental Details

Samples of BiPd with a superconducting transition temperature of $T_{\rm
c}=3.8$ K were grown by the process detailed in Ref. Joshi  _et al._ (2011).
They were cut into plates which were several mm along each side, but less than
a mm thick, with chosen crystal directions perpendicular to the plane of the
plates. These were then glued with Cytop® onto flat copper sample holders,
with cadmium used to shield from the neutron beam those parts of the holder
not containing the sample. Copper sample holders were chosen, instead of the
aluminium typically used in neutron scattering, to avoid the aluminium
becoming superconducting below 100 G Cochran and Mapother (1958). Several
samples were cut such that the (010) and (101) axes were accessible either
with the sample plate perpendicular to the applied magnetic field and the
neutron beam (Fig.1(d)), or where only a small rotation of the plate was
needed. For one sample, the cut was $13^{\circ}$ away from the (101) axis; we
refer to this as $\sim$(101) in the rest of the article.

To measure the vortex lattice, small angle neutron scattering (SANS)
measurements were carried out. A vortex lattice (VL) was created by applying a
magnetic field to the BiPd sample in the normal state and then cooling into
the superconducting mixed state. If the field (and the vortices) are nearly
parallel to the neutron beam, the neutrons can diffract off the vortices, and
we measure the first order Bragg peaks arising from the VL.

The SANS measurements were carried out on the SANS-I instrument Kohlbrecher
and Wagner (2000) at the SINQ facility at the Paul Scherrer Institute (PSI) in
Villigen, Switzerland, and the D33 instrument Dewhurst (2008) at the Institut
Laue-Langevin (ILL) in Grenoble, France. Incoming neutrons were velocity
selected to have a wavelength between 10 and 22 Å, depending on the lattice
spacing of the vortex lattice which changes as a function of field, with a
$\Delta\lambda/\lambda$ of around $10\%$. Diffracted neutrons were detected by
a two-dimensional position-sensitive detector, and incoming neutrons were
collimated such that the collimation length matched the sample-detector
distance. Low temperatures were achieved using a dilution refrigerator insert
in top-loading cryomagnets, which provide the horizontal field applied nearly
parallel to the neutron beam.

Measurements were taken by holding the applied field fixed with respect to the
sample, and then rocking the sample and field together throughout the angles
that fulfilled the Bragg condition for the first-order VL diffraction spots;
examples of the resulting diffraction patterns are shown in Fig. 1.
Temperature scans were taken on warming, after cooling the sample from above
$T_{\rm c}$ in the required field. For these temperature scans, full rocking
scans were not taken at each point, but instead extended counts at a rocking
angle giving peak intensity at chosen Bragg reflections were carried out. This
was done to save time and maximise the signal-to-noise ratio. To check that
the angles chosen remained valid over the changes in temperature, at a few
select temperatures full rocking scans were done. This also allowed us to
confirm that the widths of the rocking curves did not change as a function of
temperature. The samples were cooled in a constant field, rather than
oscillating the field value during cooling, after initial tests found a
superior VL quality with simple field cooling. The cooling rate was $\sim$ 0.1
K per minute. Background measurements were taken above $T_{\rm c}$, using the
same set of rocking angles as the foreground measurements, and were then
subtracted from the foregrounds measured inside the superconducting phase to
leave only the signal due to the vortex lattice. Data reduction was performed
using the GRASP GRA  software from the ILL, and diffraction patterns were
treated with a Bayesian method for maximising the signal-to-noise of the
small-angle diffraction data Holmes (2014).

## Results and Discussion

Figure 1 shows a representative selection of VL diffraction patterns obtained
from fields applied along the (010), $\sim$(101), and (101) directions. These
three diffraction patterns were taken at similar fields of 220 G, 200 G, and
210 G respectively, at a base temperature of between 0.05 K and 0.1 K. Each
shows a slightly distorted hexagonal lattice, with the ratios of largest to
smallest $q$ being 1.11, 1.09, and 1.12, respectively. Panel (d) is a picture
of two of the samples used in these measurements, oriented such that the top
sample in the picture has the magnetic field applied along the $\sim$(101)
direction and the bottom sample has the field applied along the (010)
direction. While there are measurable differences in the vortex structure
dependent on the direction of applied field, as indicated by the different
degrees of anisotropy, we did not observe a statistically significant
variation in the vortex structure as a function of field or temperature.

###  .1 Intermediate mixed state

Firstly, we report the observation of the intermediate mixed state (IMS) in
part of the superconducting phase of BiPd. Within Ginzburg-Landau (GL) theory,
the transition between Type-I and Type-II superconductors takes place when
$\kappa=\lambda/\xi=1/\sqrt{2}$. If $\kappa$ is below this threshold, no mixed
state exists, and only the Meissner state is observed in applied magnetic
field (Type I). For high-$\kappa$ superconductors (Type II), there is a
clearly defined vortex lattice in the superconducting state between the
Meissner state and the upper critical field. However, superconductors with
$\kappa\gtrsim 1/\surd 2$ have an additional phase between the Meissner and
vortex-lattice phases. This is the intermediate mixed state, in which the
superconducting state separates spatially into domains of vortex lattice state
and Meissner state. This was first observed by Bitter decoration Träuble and
Essmann (1967), and has been well studied in niobium using neutron scattering
Christen  _et al._ (1977); Backs  _et al._ (2019), and for example more
recently in ZrB12 by Biswas et al. Biswas  _et al._ (2020).

The unusual behavior of vortex matter and the correspondingly more complex
phase diagram of low-$\kappa$ superconductors has been predicted by various
theoretical approaches as well. In particular, for the $s$-wave superconductor
near the point $\kappa=1/\sqrt{2}$, the $B$-$T$ phase diagram was studied in
the framework of the GL theory, various tricritical points on it were
determined, and regions with non-monotonic character of the inter-vortex
interaction were characterized Luk’yanchuk (2001). A similar broadening of the
region in the phase diagram between the Meissner and the mixed phases was also
found in the framework of numerical solutions of microscopic Eilenberger
equations for the isotropic $s$-wave gap Miranovic and Machida (2003).

In the mixed state, each vortex carries a quantum of flux, and so the density
of vortices must be such as to match the magnetic flux density of the average
induction in the sample. As the vortices are subject to a repulsive inter-
vortex interaction, the unit cell size of the vortex lattice is set by the
applied field. However, in the IMS, this relationship between the applied
field and the separation of the vortices is broken. The plane spacings become
fixed as a function of applied field, being instead determined by the distance
at which the inter-vortex interaction changes from repulsive to attractive
Brandt and Das (2011). Hence in the IMS the VL regions have a flux density
$B_{0}$ determined by their fixed spacing. These regions occupy a fraction of
the sample equal to $B_{0}\ /<B>$. The characteristic signature of the
intermediate mixed state in neutron scattering is therefore a low-field region
in which the scattering vector is independent of magnetic field. Figure 2 is a
plot of the magnitude of the scattering vector, $q$, for the stronger Bragg
reflections in Fig. 1, in all three orientations of the magnetic field
measured here. There is clearly a flat region at low field where the
scattering vector is independent of magnetic field, and the high-field data
follow the typical field-dependent relationship for a VL,

$q=2\pi\sqrt{\frac{1}{\mathrm{sin(\alpha)}}\frac{B}{\Phi_{0}}},$ (1)

which arises geometrically from the constraint of one flux quantum per unit
cell, where in this case $q$ is the length of the scattering vector of the
four stronger spots in Fig. 1, $\alpha$ is the angle between two of them,
illustrated in Fig. 1(b), and $\Phi_{0}$ is the flux quantum. Leaving $\alpha$
as a free parameter to fit the high-field data of Fig. 2 returns opening
angles between $70^{\circ}$ and $75^{\circ}$, which is consistent with the
diffraction patterns of Fig. 1. If we define the critical field of the
intermediate mixed state, $B_{\mathrm{IMS}}$, as the point where the flat
region of $q$ (determined from the average of the low-field data points) meets
the field-dependent region, we have $B_{\mathrm{IMS}}=230$ G for
$H\parallel(010)$, $B_{\mathrm{IMS}}=170$ G for $H\parallel\sim$(101), and
$B_{\mathrm{IMS}}=190$ G for $H\parallel(101)$.

Figure 2: (Colour online) Length of the scattering vector, $q$, as a function
of applied field along the (a) (010), (b) $\sim$(101), and (c) (101)
directions respectively. The flat lines (black) indicate the fixed vortex
spacing characteristic of the intermediate mixed state. The field-dependent
lines (red) are fits to the relationship described in the text.

Figure 3 is a phase diagram of the vortex lattice states in BiPd for field
applied along the $(010)$ direction, which illustrates where the IMS sits in
relation to the rest of the superconducting state, and is constructed from
data in this study. Upper critical field points are determined from fits to
the temperature dependence of the form factor, which will be discussed in a
later section, and are shown by square markers. From this we estimate an upper
critical field line, which is a guide to the eye. The transition points
between the mixed state and intermediate mixed state were determined from SANS
data by measuring the scattering vector of data taken within the IMS, at 100
G, and then calculating the corresponding mixed-state field in equation 1, and
are shown in black circles. The data point at 0.1 K is also shown with a red
circle, and was taken from Fig. 2, and an estimate for the transition line
from these data is shown with a dashed red line, which is a guide to the eye.
Below the IMS the system will transition into the Meissner state, however we
do not have direct information on $H_{\rm c1}$ from this study as the thin-
plate geometry of this sample will lead to a large demagnetising effect. From
values of $\kappa$ and $\lambda$ we determine later in this study, one can
estimate that $H_{\rm c1}$ should be of order 100 G.

Figure 3: Vortex phase diagram of the vortex lattice phases in BiPd for
magnetic field oriented along the $(010)$ direction. The upper critical field
is indicated by squares, with the blue dashed line being a guide for the eye.
The transition between the mixed state (MS) and intermediate mixed state (IMS)
is indicated by black and red circles and the red dashed line, is labelled
$B_{\rm IMS}$, and was determined by data from this study in a manner
described in the text.

The presence of an intermediate mixed state means that BiPd must be an
intermediate-kappa superconductor. Calculations find that the intermediate
mixed state may appear in superconductors with $0.71\leqslant\kappa\leqslant
1.5$ Jacobs (1971); Brandt (1975, 1976). For example, niobium has a GL
parameter of 0.77 near $T_{\rm c}=9.27$ K Laver  _et al._ (2006). In the next
section, we will use measurements of the VL form factor in the mixed state to
obtain a value for $\kappa$ for BiPd.

###  .2 Field dependence of the form factor

We can express the local field within the VL as a sum over the spatial Fourier
components at the reciprocal lattice vectors q, and the magnitudes of the
Fourier components $F(\mathbf{q})$ are known as the VL form factor. The
magnitudes of these can be calculated from the integrated intensities
$I_{\mathbf{q}}$ of Bragg reflections using the Christen formula Christen  _et
al._ (1977):

$I_{\mathbf{q}}=2\pi
V\phi(\frac{\gamma}{4})^{2}\frac{\lambda_{n}^{2}}{\Phi_{0}^{2}q}|F(\mathbf{q})|^{2},$
(2)

where V is the illuminated sample volume, $\phi$ is the neutron flux, $\gamma$
is the magnetic moment of the neutron, $\lambda_{n}$ is the neutron
wavelength, and $\Phi_{0}=h/2e$ is the flux quantum. In our experiment, the
integrated intensities $I_{\mathbf{q}}$ of the first-order Bragg reflections
were calculated by fitting the rocking curves of the reflections to a
Lorentzian line-shape. These intensities were then corrected by the Lorentz
factor, the cosine of the angle between the perpendicular to the rocking axis
and q.

Figure 4 shows the form factor as a function of field for fields applied along
the (010), $\sim$(101) and (101) directions, as measured at 0.1 K. The data
points are an average of the form factor for all six first-order Bragg
reflections of the VL. The dotted vertical line indicates the transition line
between the intermediate mixed state, as determined from the scattering vector
(Fig. 2), and the solid line is a fit to the extended London model, which is
described below. Above the transition line, in the mixed state, the VL form
factor behaves in a typical manner, decreasing with increasing field as the
density of vortices increases and their spacing is reduced. However, below the
transition line a completely different behaviour is observed. In this case,
the intensity is rising in a reasonably linear fashion from zero. This is
because the inter-vortex spacing is independent of applied field in the
intermediate mixed state, but the volume-fraction of the sample occupied by
vortex matter increases with field. Therefore, as the reflectivity of the
vortex lattice remains constant while its volume increases, the intensity of
the Bragg reflections also increases. However, this means that mixed-state
methods of analysing the field-dependence of the VL form factor are no longer
valid, so we discount these data from our numerical analysis.

We note that for field applied along the $\sim(101)$ and $(101)$ directions,
the transition between the IMS and VL states as determined by the scattering
vector and the down-turn of the form factor are separated slightly in field,
by up to 50 G. The situation for field applied parallel to the (010) direction
is less clear, due to a lack of data just above the IMS transition. This may
be an intrinsic effect which has been seen before in niobium Mühlbauer  _et
al._ (2009), where a hysteresis in the scattered intensity from the flux
lattice was observed about the IMS transition. This has been attributed to
different processes of flux intrusion and flux island formation in the
presence of pinning Essmann (1972). Our data was taken after field-cooling,
unlike the hysteretic data from niobium, which was taken with the field
changed while below $T_{\rm c}$. We might expect our field-cooled data to be
equivalent to the decreasing-field data, as both are approaching the
measurement point from outside the superconducting phase, but we find that it
infact resembles the increasing-field data.

Figure 4: (Color online) (a) Form factor for field applied parallel to (010).
(b) Form factor for field applied parallel to the $\sim$(101) direction. (c)
Form factor for field applied parallel to (101). The solid red lines are fits
to the extended London model, and the dashed lines indicate the transition
line to the intermediate mixed state, as determined in Fig. 2.

We fit the data within the mixed state to the ‘extended London model’ i.e. the
London model with core corrections, given by

$F(\textbf{q})=\frac{\langle
B\rangle\exp(-cq^{2}\xi^{2})}{1+q^{2}\lambda^{2}},$ (3)

where $\langle B\rangle$ is the average internal induction, $q$ is the
magnitude of the reciprocal space lattice vector for the first order Bragg
reflection being measured, $\xi$ is the coherence length, $\lambda$ is the
penetration depth, and $c$ is a parameter which accounts for the finite size
of the vortex cores, with a suitable value being 0.44 Bowell (2008); White
_et al._ (2011); Kawano-Furukawa  _et al._ (2011); Cameron  _et al._ (2022);
Campillo  _et al._ (2022). There are some complications regarding the use of
the London model to fit the data, which is that it was constructed assuming
the system was far from $H_{\rm c2}$, and in the limit that the
electrodynamics could be considered local in nature, restricting $\kappa\gg
1$. Since we are fitting data at low temperature, the first consideration is
met, but we know that we are straining the latter as the intermediate mixed
state implies that $\kappa\sim 1$. Models based on the GL theory are valid for
low $\kappa$, but GL theory is only accurate close to $T_{\rm c}$, so we will
retain the extended London model and consider the results to be estimates. The
data were fitted using experimentally measured values of $q$, and the smooth
curves displayed in the figure were calculated from a fit to these
experimentally measured values of $q$ using the same relationship as in Fig.
2. We have chosen to fit to average values of the form factor, rather than
trying to use the anisotropic London model to attempt to separate out values
of the coherence length and penetration depth along each crystallographic
axis, as out of the twelve different vortex nearest neighbour directions found
within the three separate orientations of applied field, only one lies
parallel to a crystal axis. This is undoubtedly due to the complications of
the monoclinic structure, and prohibits the normal strategies used to fit the
anisotropic London model. The fits returned values of the coherence length and
penetration depth at base temperature of $\xi=79$ nm and $\lambda=80$ nm for
$\mathbf{B}\parallel(010)$, $\xi=73$ nm and $\lambda=92$ nm for
$\mathbf{B}\parallel\sim$(101), and $\xi=65$ nm and $\lambda=120$ nm for
$\mathbf{B}\parallel(101)$. This gives average values of $\xi=72$ nm and
$\lambda=98$ nm, which result in $\kappa=1.4$. This value of $\kappa$ is
consistent with the expectation, discussed in the section on the intermediate
mixed state, that it should fall within the region of
$0.71\leqslant\kappa\leqslant 1.5$. However, this is much smaller than a
previous result of $\kappa=7.6$ Joshi  _et al._ (2011); we note that this
value was an order of magnitude estimate derived from the higher estimate of
$H_{\rm c2}$ discussed in the introduction. The presence of the IMS constrains
$\kappa$, and so this indicates that the higher estimate of $H_{\rm c2}$
cannot be correct.

###  .3 Temperature dependence of the form factor

The temperature dependence of the superfluid density, $\rho_{s}(T)$, which is
accessible through the form factor, is one method to test the gap structure
and thus gain information about the pairing state. In this case, the amount of
thermal energy required to break Cooper pairs is determined by the magnitude
of the gap, and so the form of the curve $\rho_{s}(T)$ is fundamentally
determined by the gap. Typically, superconductors which have a full gap
arising from s-wave pairing have a constant low-temperature value of the
superfluid density as the thermal energy falls below the gap value on approach
to absolute zero. Conversely, superconductors which have nodes in the gap can
experience quasiparticle excitations even at the lowest temperatures, and so
have a finite slope of $\rho_{s}(T)$ at zero temperature. One of the key
predictions of the possible mixing of s\- and p-wave pairing states in NCS
superconductors is the emergence of either anisotropic gaps or accidental
nodes, which would modify the form of $\rho_{s}(T)$ away from that expected
for centrosymmetric superconductors. A theory to model this has been
constructed by Hayashi et al. Hayashi  _et al._ (2006). This has been used
successfully to identify mixed gaps in Li2Pd3B and Li2Pt3B Yuan  _et al._
(2006), and Ru7B3 Cameron  _et al._ (2022). The data presented here
demonstrate neither the flat region at low temperatures characteristic of
s-wave superconductors nor the pronounced finite slope at zero temperature of
nodal superconductors Prozorov and Giannetta (2006), but rather a slight slope
down to the lowest temperatures in a similar manner to the data from Ru7B3
Cameron  _et al._ (2022). We therefore employ the same model here in order to
determine if the data supports NCS gap mixing in BiPd, and if so, to what
extent.

In the mixed state, the magnitude of the scattering vector, which contributes
to the form factor, is determined solely by the applied field and the vortex
lattice structure. Therefore, if measuring at a fixed field, and in the
absence of any VL structure transitions, the change in the form factor comes
only from changes in the superfluid density. These are the conditions to be
used to obtain $\rho_{s}(T)$.

However, in the IMS the scattering vector is not determined by the applied
field, but rather the balance between the attractive and repulsive parts of
the inter-vortex interaction. This balance can change with temperature, and
modifies both q and the scattered intensity from the vortex lattice, both of
which contribute to the form factor. This has been seen quite clearly in, for
instance, a SANS study on the vortex lattice in niobium Backs  _et al._
(2019). We do see a change in the scattering vector with temperature in the
IMS of BiPd, which is shown in Fig. 5(d). While the scattering vector of the
data from the mixed state remains constant as a function of temperature, the
scattering vector of the data taken within the intermediate mixed state
decreases as the temperature increases. This is due to decreasing $B_{\rm 0}$
within the vortex islands, as the vortices increase in size with increasing
temperature. In order to avoid the complications of multiple contributing
parameters varying within a single measurement we confine ourselves to
temperature scans of the VL in the mixed state.

Figure 5: (Colour online) Temperature dependence of the form factor,
normalised to its low-temperature value, for several different orientations
and temperatures. (a) Field parallel to the (010) direction at 250 G. (b)
Field parallel to the $\sim$(101) direction at 200 G. (c) Field parallel to
the $\sim$(101) direction at 400 G. Solid lines are a fit to the NCS model
described in the text, and panel (a) also has plots of s-wave and p-wave
models. (d) Temperature dependence of the wavevector $q$ of one of the first
order Bragg reflections from the vortex lattice, as measured at a field above
$B_{\rm IMS}$ at 200 G (red), and below $B_{\rm IMS}$ at 150 G (black), with
the field applied along the $\sim$(101) direction.

Equation 3 may be used to give the form factor as a function of temperature at
a fixed field. Two parameters in this relationship have a temperature
dependence: the penetration depth $\lambda$ and the coherence length $\xi$, so
we can use this to fit the temperature dependence of the form factor. The NCS
model of Hayashi et al. gives the superfluid density as a function of
temperature, which enters the form factor through the penetration depth as per
the relationship $1/\lambda^{2}\>\propto\>\rho_{s}$. The gap also contributes
to the form factor through the coherence length $\xi$, which is inversely
proportional to the superconducting gap given by the model. This involves
Cooper pairing including both spin singlet and spin triplet contributions to
the order parameter. From this, an expression for the superfluid density was
calculated, which describes the system in terms of the ratio of the singlet
($\Psi$) and triplet ($\Delta$) order parameters, $\nu=\Psi/\Delta$. It also
involves the difference of the density of states of the spin-split Fermi
surfaces, labelled I and II, $\delta=(N_{\rm I}-N_{\rm II})/(N_{\rm I}+N_{\rm
II})$, and finally the coupling constant $\lambda_{\rm m}$ which describes
Cooper pair scattering between the two channels. We calculate the temperature
dependence of the superfluid density and the gap by the method described by
Hayashi _et al._ Hayashi  _et al._ (2006).

For comparison, we also fitted the data to both fully gapped and nodal models
appropriate for a superconductor with no order parameter mixing, in the same
manner as was done in previous studies of the VL form factor by SANS Kawano-
Furukawa  _et al._ (2011); White  _et al._ (2011); Morisaki-Ishii  _et al._
(2014); Cameron  _et al._ (2022); Campillo  _et al._ (2022). We have chosen to
use s-wave and p-wave models as these are the two potential components of the
order parameter in NCS superconductors. For these cases, the superfluid
density in the clean limit is given by the expression

$\begin{split}\rho_{s}(T)&=1-\frac{1}{4\pi k_{B}T}\\\
&\int_{FS}\int_{0}^{\infty}\cosh^{-2}\left(\frac{\sqrt{\varepsilon^{2}+\Delta_{\textbf{k}}^{2}(T,\theta,\phi)}}{2k_{B}T}\right)\mathrm{d}\theta\mathrm{d}\phi\mathrm{d}\varepsilon.\end{split}$
(4)

Here $\Delta_{\textbf{k}}$ describes the gap function, $\theta$ and $\phi$ are
the polar and azimuthal angles about the Fermi surface (FS) and
$\sqrt{\varepsilon^{2}+\Delta_{\textbf{k}}^{2}(T,\theta,\phi)}$ gives the
excitation energy spectrum. The gap is separated such that its momentum and
temperature parts can be described by
$\Delta_{\textbf{k}}(T,\theta,\phi)=g_{\textbf{k}}(\theta,\phi)\Delta_{0}(T)$,
where $g_{\textbf{k}}(\theta,\phi)$ is the momentum dependence of the gap as a
function of angle around the Fermi surface. The temperature dependence of the
gap can be approximated by

$\Delta_{0}(T)=\Delta_{0}(0)\tanh\left(\frac{\pi T_{\rm
c}}{\Delta_{0}(0)}\sqrt{a\bigg{(}\frac{T_{c}}{T}-1\bigg{)}}\right),$ (5)

where ${\Delta_{0}}$ is the magnitude of the gap at zero temperature and $a$
is a parameter related to the pairing state Prozorov and Giannetta (2006). For
s-wave pairing, we took $g_{\textbf{k}}(\theta,\phi)=1$,
$\Delta_{0}(0)=1.76T_{\rm c}$ and $a=1$, while for p-wave pairing
$g_{\textbf{k}}(\theta,\phi)=\cos(\phi)$, $\Delta_{0}(0)=2.14T_{\rm c}$ and
$a=4/3$. These latter values of $\Delta_{0}(0)$ and $a$ are those used for
d-wave pairing Prozorov and Giannetta (2006), and for
$g_{\textbf{k}}(\theta,\phi)$ we have used a two-dimensional approximation.
These were chosen to capture the essential features of the nodal pairing
state. In the NCS case, the fitting parameters were given by the theory
Hayashi  _et al._ (2006).

In Fig. 5, we present the temperature dependence of the form factor, alongside
fits to the noncentrosymmetric model of Hayashi et al. Hayashi  _et al._
(2006) for several different magnitudes and orientations of the applied
magnetic field. In panel (a), the field is parallel to the (010) direction at
250 G, and in panels (b) and (c) are fields of 200 G and 400 G respectively
parallel to the $\sim$(101) direction. In panel (a) we show examples of the
the pure $s$\- and $p$-wave models; neither model describes the data well. We
found no set of parameters for which either of these models could describe the
data, indicating that neither a fully gapped nor nodal description is
suitable. For the fit to the non-centrosymmetric model, we held the parameter
$\lambda_{\rm m}$ constant at 0.2, as suggested by Hayashi _et al._ Hayashi
_et al._ (2006). This value has been used in other examples of the
implementation of this model Yuan  _et al._ (2006); Cameron  _et al._ (2022),
as it was determined in the initial publication that varying this parameter
does not have any strong quantitative or qualitative effects on the model.
Following this, we found a solution with common values for $\nu$ and $\delta$
for all of the data sets. The fits returned values of $\nu=1.45\pm 0.1$ and
$\delta=0.75\pm 0.05$.

These values clearly indicate that the mixed order parameter of the NCS model
is an appropriate description for these data, while neither a single-gap
s-wave or nodal order parameter would fit. A value of $\delta\neq 1$ indicates
a difference in the density of states between the spin-split bands of an NCS
superconductor, and the value of $\nu=1.45$ indicates that a reasonable amount
of order-parameter mixing is taking place, leading to a nodeless but
anisotropic gap. This result is in reasonable consistency with the published
data, and offers a possible resolution to the conflict between different
results discussed in the introduction. We have a multi-gap scenario due to
spin-split bands from the ASOC, with anisotropic but nodeless gaps from
singlet-triplet mixing with the singlet component dominant. The gap on each of
the spin split bands is also different, with one being described by
$\mid\psi+\Delta{\rm sin}\theta\mid$, and the other by $\mid\psi-\Delta{\rm
sin}\theta\mid$ Hayashi  _et al._ (2006). This is consistent with the multiple
reports of a two-gap order parameter Mondal  _et al._ (2012); Ramakrishnan
_et al._ (2017); Jiao  _et al._ (2014), but may also account for results which
reported conventional behaviour Peets (2014); Sun  _et al._ (2015), as we
still have a finite gap over all the Fermi surface.

Figure 6: (Color online) Plot of the gap values on two spin-split Fermi
surfaces from the NCS fit described in the text.

Using the value of $\nu=1.45$ from the fits in Fig. 5, we plot the modulus of
the gap values on each of the spin-split Fermi surfaces, I and II, in Fig. 6.
This was done using the method described in Ref. Hayashi  _et al._ (2006). The
gap on Fermi surface I is given by $\mid\psi+\Delta{\rm sin}\theta\mid$, and
the gap on Fermi surface II is given by $\mid\psi-\Delta{\rm sin}\theta\mid$,
where $\theta$ is the azimuthal angle in momentum space. The gaps are
reasonably anisotropic, as expected for a value of $\nu$ close to 1, but at no
point do they fall to zero as the singlet component is larger in magnitude
than the triplet component.

In Table 1 we present values of $\nu$ and $\delta$ from fitting the
temperature dependence of the superfluid density to the NCS model, alongside
values from other NCS superconductors where this model has also been used. The
values of the energy of the ASOC are also presented, following Smidman et al.
Smidman  _et al._ (2017). We see some general trends in the data with regards
to the ratio of singlet to triplet gap, $\nu$, where systems with larger ASOC
tend to have more prevalent triplet components. It was noted in the paper on
Li2Pd3B and Li2Pt3B that the ratio of the density of states, $\delta$, was
expected to be proportional to the ASOC Yuan  _et al._ (2006), and for the two
materials they investigate this is true, although it does not hold for the
others as they have larger values of $\delta$ with a significantly lower ASOC
than the platinum compound. However, we note that unlike the platinum and
palladium compounds the other materials investigated are not related to each
another.

Material | $\nu$ | $\delta$ | EASOC (meV)
---|---|---|---
Li2Pd3B Yuan  _et al._ (2006) | 4 | 0.1 | 30
Li2Pt3B Yuan  _et al._ (2006) | 0.6 | 0.3 | 200
Ru7B3 Cameron  _et al._ (2022) | 2.5(5) | 0.70(7) | 20
BiPd | 1.45(1) | 0.75(5) | 50
Table 1: Values for the singlet-to-triplet gap ratio, $\nu=\Psi/\Delta$, and
the density of states ratio, $\delta$, as measured from NCS materials whose
superfluid density as a function of temperature has been fitted to the model
described in the text. Values for the energy of the ASOC are also given, taken
from Ref. Smidman  _et al._ (2017).

## Conclusions

We have observed the presence of the intermediate mixed state in BiPd, which
means that the material must be a low-$\kappa$ Type-II superconductor, with a
low value of the upper critical field in the bulk of the sample. The values of
the coherence length and penetration depth that we estimate from a London-
model fit to the field dependence of the VL form factor conform to this
expectation, giving a value of $\kappa\sim 1.4$. Further, we fitted the
temperature dependence of the VL form factor to a model derived from the mixed
singlet-triplet order parameter of non-centrosymmetric superconductors, and
find the data to be in good agreement with the model. This work may also
motivate a thorough theoretical study of the richness of the vortex-matter
phase diagram and the structure of the VL in non-centrosymmetric
superconductors with spin-triplet mixing.

## Acknowledgements

ASC, ATH, EMF and EB acknowledge support from the UK Engineering and Physical
Sciences Research Council grant numbers EP/J016977/1 and EP/G027161/1. LL
acknowledges financial support from the Institut Laue-Langevin and the
University of Birmingham. ASC acknowledges support from Lund University during
the preparation of the manuscript. This work is based on experiments performed
at the Swiss spallation neutron source SINQ, Paul Scherrer Institute,
Villigen, Switzerland. Experiments at the ILL gave data available at DOI
10.5291/ILL-DATA.5-42-374 ILL experiment 5-42-374  and 10.5291/ILL-
DATA.5-42-390 ILL experiment 5-42-390 .

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