# Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in CeCoIn5
revealed by high-resolution magnetostriction measurements

Shunichiro Kittaka  Department of Physics, Chuo University, Bunkyo-ku, Tokyo
112-8551, Japan  Yohei Kono  Department of Physics, Chuo University, Bunkyo-
ku, Tokyo 112-8551, Japan  Kaito Tsunashima  Department of Physics, Chuo
University, Bunkyo-ku, Tokyo 112-8551, Japan  Daisuke Kimoto  Department of
Physics, Chuo University, Bunkyo-ku, Tokyo 112-8551, Japan  Makoto Yokoyama
Faculty of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan and
Institute of Quantum Beam Science, Ibaraki University, Mito, Ibaraki 310-8512,
Japan  Yusei Shimizu  Institute for Materials Research (IMR), Tohoku
University, Oarai, Ibaraki 311-1313, Japan  Toshiro Sakakibara  Institute for
Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Minoru Yamashita  Institute for Solid State Physics, University of Tokyo,
Kashiwa, Chiba 277-8581, Japan  Kazushige Machida  Department of Physics,
Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

###### Abstract

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is an exotic superconducting
phase formed by Cooper pairs with finite center-of-mass momentum $\bm{q}$. On
theoretical grounds, the superconducting order parameter in the FFLO state is
spatially modulated along the $\bm{q}$ vector, and the emergence of an
associated anisotropy is expected at the phase transition from the Abrikosov
state to the FFLO state. Here, we report the results of high-resolution
magnetostriction measurements for a single crystal of CeCoIn5 around
$B\parallel c$. We find two anomalies in the magnetostriction along the $c$
axis, parallel to the magnetic-field orientation. In sharp contrast, this
$B_{\rm K}$ anomaly disappears in the magnetostriction along the $a$-axis
direction, perpendicular to the magnetic-field orientation. To explain this
uniaxial expansion, we suggest a possibility that the FFLO transition occurs
slightly below the upper critical field, and the FFLO modulation vector
parallel to the applied magnetic field gives rise to the anisotropic response.

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state Fulde and Ferrell (1964);
Larkin and Ovchinnikov (1964) is a prototype example of the so-called pair
density wave superconductivity (inset of Fig. 1) Agterberg et al. (2020);
Agterberg and Tsunetsugu (2008) where translational symmetry is spontaneously
broken in addition to $U(1)$ gauge symmetry. It is a topologically interesting
object Mizushima et al. (2005a) and has been long sought in a wide range of
research fields, from condensed matter Matsuda and Shimahara (2007) to cold
neutral atoms.Zwierlein et al. (2006); Partridge et al. (2006); Mizushima et
al. (2005b) In order to realize the FFLO state, several strict conditions are
required to be fulfilled.Matsuda and Shimahara (2007) For example, the system
must be very clean because the FFLO state is easy to be broken by a small
amount of impurities and/or defects. Furthermore, the orbital pair-breaking
effect needs to be sufficiently weaker than the Pauli-paramagnetic effect.
Therefore, for a long period of time, there were few experimental reports on
the FFLO state. However, in recent years, there has been a gradual increase in
the number of materials that are expected to realize the FFLO state, such as
quasi-low-dimensional organic superconductors,Singleton et al. (2000); Uji et
al. (2001, 2006); Shinagawa et al. (2007); Yonezawa et al. (2008a, b);
Mayaffre et al. (2014); Imajo et al. (2021, 2022) iron-based
superconductors,Burger et al. (2013); Cho et al. (2017); Kasahara et al.
(2020) and Sr2RuO4.Kittaka et al. (2018); Kinjo et al. (2022)

The heavy-fermion superconductor CeCoIn5 is also a good candidate realizing
the FFLO state. It has been well established as a spin-singlet
$d_{x^{2}-y^{2}}$-wave superconductor.Curro et al. (2001); Kohori et al.
(2001); Izawa et al. (2001); An et al. (2010); Allan et al. (2013); Zhou et
al. (2013) In the heavy-fermion system, the orbital pair-breaking effect is
sufficiently weak due to the large effective mass of heavy quasiparticles.
Indeed, the emergence of a first-order superconducting transition at the upper
critical field $B_{\mathrm{c2}}$ below 0.7 K, as well as the suppression of
$B_{\mathrm{c2}}$ at low temperatures, have been reported in CeCoIn5,Bianchi
et al. (2002); Tayama et al. (2002); Murphy et al. (2002) indicating that the
Pauli-paramagnetic effect overcomes the orbital pair-breaking effect.
Moreover, a specific-heat anomaly has been found inside the superconducting
phase under an in-plane magnetic field.Bianchi et al. (2003); Radovan et al.
(2003) This anomaly may be attributed to a transition from the uniform
superconducting state, i.e., Abrikosov vortex state, to the FFLO state.Ichioka
et al. (2007); Suzuki et al. (2020) The double-peak spectral structure
detected slightly below $B_{\mathrm{c2}}$ from nuclear magnetic resonance
(NMR) experiments also supports the presence of a high-field phase.Kakuyanagi
et al. (2005); Kumagai et al. (2006, 2009, 2011) Thus, CeCoIn5 has a high-
field superconducting phase, particularly for $B\parallel ab$.

However, it remains controversial whether this high-field phase is the FFLO
phase. From NMR and neutron-scattering experiments, it has been revealed that,
in the high-field phase for $B\parallel ab$ (the so-called $Q$ phase), a spin
density wave (SDW) order coexists with superconductivity.Young et al. (2007);
Kenzelmann et al. (2008) Because the magnetic structure in this $Q$ phase is
independent of the direction of the in-plane magnetic field, it was suggested
that the $Q$ phase is not driven by the FFLO state.Kenzelmann et al. (2010)
Recent theoretical studies and thermal conductivity measurements have also
suggested that the FFLO state competes with the SDW phase.Lin et al. (2020)
The coexistence of SDW order and superconductivity complicates the
interpretation of the high-field phase in CeCoIn5 for $B\parallel ab$.

Then, it seems reasonable to turn our attention to a possible high-field phase
in CeCoIn5 for $B\parallel c$ because the magnetic ordering is quickly
suppressed by tilting the magnetic field from the $ab$ plane.Blackburn et al.
(2010) Indeed, no spectrum broadening is reported in the recent NMR
measurements for $B\parallel c$,Taniguchi et al. (2020) showing the absence of
the magnetic instability around $B_{\mathrm{c2}}$. Therefore, if the high-
field phase exists in $B\parallel c$, it is expected to be a pure FFLO phase.
From NMR experiments, double-peak structure, similar to the one for
$B\parallel ab$,Kakuyanagi et al. (2005); Kumagai et al. (2011) has been
observed in the range $4.7\ {\rm T}\lesssim B\lesssim B_{\mathrm{c2}}$ for
$B\parallel c$.Kumagai et al. (2006, 2009) However, there is no other crucial
evidence for the occurrence of the high-field phase in $B\parallel c$. In
order to provide further experimental evidence for the FFLO transition in
$B\parallel c$, we have performed high-resolution magnetostriction
measurements.

High-quality single crystals of CeCoIn5 were grown by the self-flux method.
This paper focuses on the results of one of three samples used; the others are
shown in Secs. I and II of the Supplemental Material (SM).SM  The isothermal
magnetostriction, $\Delta L_{i}(B)=L_{i}(B)-L_{i}(B_{0})$, and thermal
expansion, $\Delta L_{i}(T)=L_{i}(T)-L_{i}(T_{0})$, along the $i$ ($i=a$ or
$c$) axis were measured using a homemade capacitance dilatometer in a dilution
refrigerator (Kelvinox AST Minisorb, Oxford). Here, $B_{0}$ ($T_{0}$)
represents a selected constant field (temperature). The sample lengths were
1.34 and 0.9 mm along the $a$ and $c$ axes, respectively. The crystalline axes
were determined using an x-ray back-scattering Laue camera (RASCO-BL II,
Rigaku). A magnetic field $B$ was generated using a 7-T split-pair magnet in
the horizontal $x$ direction. For $L_{c}$ ($L_{a}$) measurements, the
measurement direction of the compact dilatometer was set to be parallel to the
$x$ ($z$) axis, so that the magnetic field can be applied along the $c$ axis.
The field-angle $\phi$ dependences of $L_{c}$ and $L_{a}$ were investigated by
rotating the sample together with the refrigerator around the vertical $z$
direction using a stepper motor at the top of the magnet Dewar, where $\phi$
denotes the field angle measured from the $c$ axis to the $b$ axis. Although,
due to the non-rectangular shape of the sample, the $a$ axis was not perfectly
aligned to the measurement direction of $L_{a}$, this misalignment does not
affect the conclusion of this paper (see Sec. IV of SM SM  for more details).

Figure 1:  The $c$-axis magnetostriction coefficient $\lambda_{c}(B)=(\partial
L_{c}/\partial B)/L_{c}$ at several temperatures for $B\parallel c$. Each set
of data is shifted vertically by $10^{-4}$ T-1 for clarity. Closed (open)
circles represent the data taken in the field-increasing (-decreasing)
processes. Inset shows a schematic image of the spatial modulation of the
order parameter $\Delta(\bm{r})$ in the FFLO state along the modulation vector
$\bm{q}$.

Figure 1 presents the field derivative data, $\lambda_{c}(B)=(\partial
L_{c}/\partial B)/L_{c}$ at several temperatures in the field-increasing and
-decreasing processes. As shown in Fig. 1, $|\lambda_{c}(B)|$ shows a sharp
peak at $B_{\mathrm{c2}}$ with a hysteresis that develops below 0.6 K,
demonstrating the first-order superconducting transition at
$B_{\mathrm{c2}}$.Bianchi et al. (2002); Tayama et al. (2002) The sharp peak
of $|\lambda_{c}(B)|$ at $B_{\mathrm{c2}}$ demonstrates the high quality of
the present sample. Most remarkably, slightly below $B_{\mathrm{c2}}$, a kink
anomaly appears at $B_{\rm K}$ concomitantly with the development of the
hysteresis at low temperatures. We confirm the reproducibility of this $B_{\rm
K}$ anomaly in the measurement of another sample (see Sec. II of SM SM ).
Indeed, the specific-heat anomaly close to $B_{\mathrm{c2}}$ is suggested in
the previous report.Bianchi et al. (2003) It should be noted that our high-
resolution magnetostriction measurements performed at small steps of $\sim
0.002$ T near $B_{\mathrm{c2}}$ allow us to clearly detect the $B_{\rm K}$
anomaly in $B\parallel c$ that was not reported in the previous work.Correa et
al. (2007); Takeuchi et al. (2002)

Figure 2:  Thermal expansion $\Delta L_{c}/L_{c}$ measured at 5, 4.89, 4.85,
4.8, 4.7, 4.5, 4.3, and 4 T (from bottom to top) for $B\parallel c$. Closed
and open circles represent the data taken in the warming and cooling
processes, respectively. Inset shows temperature dependence of the thermal
expansion coefficient $\alpha_{c}=(\partial L_{c}/\partial T)/L_{c}$ along the
$c$ axis at 0 T.

The inset of Fig. 2 displays the linear thermal expansion coefficient
$\alpha_{c}$ in zero field, where $\alpha_{c}$ denotes $(\partial
L_{c}/\partial T)/L_{c}$. A sharp anomaly is seen in $\alpha_{c}$ at
$T_{\mathrm{c}}=2.25$ K in zero field; $\alpha_{c}$ exhibits no sign of
$T_{\mathrm{c}}$ distribution within the resolution limit. Figure 2 shows the
thermal expansion $\Delta L_{c}(T)/L_{c}$ at several magnetic fields for
$B\parallel c$. The hysteresis behavior in $\Delta L_{c}(T)$ becomes prominent
above 4.8 T in the superconducting state, while no difference was found
between cooling and warming in the $\Delta L_{c}(T)$ data at 4.7 T. The
$B_{\rm K}$ anomaly is less clearly detected from the thermal expansion
(temperature scan) because the superconducting transition at $T_{\rm c}$
affects the thermal expansion in wide temperature range (see Sec. III of SM SM
).

Figure 3:  (a) Field-temperature phase diagram of CeCoIn5 for $B\parallel c$.
(b) Field dependence of $d\lambda_{c}/dB$ at several temperatures. Both
increasing and decreasing field data are shown above 0.2 K (only the former at
0.11 K). Each set of data in (b) is shifted vertically by $5\times 10^{-3}$
T-2 for clarity. The positions of $B_{\mathrm{c2}}$ [squares in (a)] and
$B_{\rm K}$ (triangles) are determined by large and small peaks in
$d\lambda_{c}/dB$, respectively, as indicated by solid arrows in (b). The
symbols shown in red (blue) are determined from the field-increasing
(-decreasing) data. Crosses represent the field above which $\lambda_{c}(B)$
starts to change markedly toward $B_{\rm K}$ with increasing $B$, possibly
corresponding to a boundary between the Abrikosov state and the FFLO state.

Figure 3(a) shows the field-temperature phase diagram of CeCoIn5 for
$B\parallel c$, in which the positions of $B_{\mathrm{c2}}$ and $B_{\rm K}$
are determined by the two peaks in $d\lambda_{c}/dB$ as indicated by arrows in
Fig. 3(b). The $B_{\rm K}$ anomaly becomes indistinguishable above 0.6 K due
to the broadening of the $B_{\mathrm{c2}}$ transition. The tricritical point
between the FFLO, homogeneous Abrikosov vortex, and paramagnetic normal states
can be determined by the onset critical field above which the superconducting
transition becomes first order. To determine this critical magnetic field, we
estimate the temperature dependence of the magnetic hysteresis at
$B_{\mathrm{c2}}$ between the field increasing and decreasing measurements
(Fig. 3). The magnetic hysteresis between these measurements appears at around
0.6 K and 4.7 T for $B\parallel c$, which is suggested as the tricritical
point (see also Fig. S4 of SM SM ).

Figure 4:  Field angle $\phi$ dependence of (a) $\Delta L_{i}/L_{i}$ at 0.11 K
and 5 T, and (b) their field-angle derivatives, $\lambda_{\phi,i}=(\partial
L_{i}/\partial\phi)/L_{i}$ ($i=a$ or $c$). Closed (open) symbols represent the
data taken in the $\phi$-increasing (decreasing) process. The gray solid line
in (b) represents the simulation spectrum consisting of two Gaussian functions
(dashed and dash-dotted lines). Inset in (a) shows a schematic view of the
flux lines and the FFLO modulation. The periodic nodal planes, separated by
$L=2\pi/q$, run perpendicular to the flux lines that form a vortex lattice in
the $ab$ plane. Inset in (b) shows a wider view of $\lambda_{\phi,i}(\phi)$ in
the $\phi$-increasing (decreasing) process for $\phi>0$ ($\phi<0$).

Due to the anisotropy in $B_{\mathrm{c2}}$ of CeCoIn5
($B_{\mathrm{c2}}^{\parallel ab}/B_{\mathrm{c2}}^{\parallel c}\sim 2.4$), the
superconducting transition can be induced by rotating a magnetic field from
the $c$ axis when $B_{\mathrm{c2}}^{\parallel c}<B<B_{\mathrm{c2}}^{\parallel
ab}$. Figure 4(a) shows the field-angle $\phi$ dependence of $\Delta
L_{c}(\phi)=L_{c}(\phi)-L_{c}(\phi_{0})$ measured at 0.11 K under a magnetic
field of 5 T rotated within the $bc$ plane. Here, $\phi_{0}$ is a selected
constant field angle. A first-order superconducting transition with clear
hysteresis has been observed in $\Delta L_{c}(\phi)$ at $\phi\sim 14^{\circ}$.
The field-angle derivative data, $\lambda_{\phi,c}(\phi)=(\partial
L_{c}/\partial\phi)/L_{c}$, are presented in Fig. 4(b). In
$\lambda_{\phi,c}(\phi)$, the $B_{\rm K}$ anomaly can be seen at $\phi\sim
16^{\circ}$, which is compatible with the one observed in $\lambda_{c}(B)$
(Fig. 1). The height of the $B_{\rm K}$ anomaly does not depend on the
direction of the field angle sweep, whereas the $B_{\mathrm{c2}}$ anomaly at
$\phi\sim 14^{\circ}$ is more prominent for the transition from normal to
superconducting state [closed symbols in Fig. 4(b)] than for the transition
from superconducting to normal state [open symbols in Fig. 4(b)]. Thus, two
anomalies exhibit qualitatively different features, likely stemming from
different origins.

Furthermore, the $a$-axis magnetostriction $\Delta
L_{a}(\phi)=L_{a}(\phi)-L_{a}(\phi_{0})$ of the same sample was measured under
a rotating magnetic field within the plane normal to the measurement
direction, i.e., the approximate $bc$ plane. Figures 4(a) and 4(b) show the
$\phi$ dependences of $\Delta L_{a}(\phi)$ and its angle derivative
$\lambda_{\phi,a}(\phi)=(\partial L_{a}/\partial\phi)/L_{a}$, respectively.
The change in $\Delta L_{a}/L_{a}$ at $B_{\mathrm{c2}}$ is $2\times 10^{-6}$,
about half of the change in $\Delta L_{c}/L_{c}$. These results are consistent
with the previous report.Takeuchi et al. (2002); Correa et al. (2007) In sharp
contrast to $\lambda_{\phi,c}(\phi)$, $\lambda_{\phi,a}(\phi)$ shows one sharp
transition without the second anomaly at $B_{\rm K}$. The observed peak in
$|\lambda_{\phi,a}(\phi)|$ has similar features to the main peak in
$|\lambda_{\phi,c}|$ (see Sec. IV of SM SM ).

As shown in the inset of Fig. 4(b), both $\lambda_{\phi,c}$ and
$\lambda_{\phi,a}$ are symmetric with respect to the angle direction. This
symmetric angle dependence in $\lambda_{\phi,c}$ eliminates the possibility
that the $B_{\rm K}$ anomaly is caused by a domain with a tilted $c$ axis in
the sample, because such a domain should show the $B_{\rm K}$ anomaly at a
smaller or larger field angle when the magnetic field is rotated in the other
direction. The results of the Gaussian fits to $\lambda_{\phi,c}$ [a solid
line in Fig. 4(b)] reveal that a full width at half maximum (FWHM) of the
$B_{\rm K}$ anomaly ($\sim 1.42^{\circ}\pm 0.05^{\circ}$) [a dash-dotted line
in Fig. 4(b)] is narrower than the FWHM of the $B_{\rm c2}$ anomaly ($\sim
1.65^{\circ}\pm 0.02$) [a dashed line in Fig. 4(b)] (see Sec. IV of SM SM  for
more details). From this fact, the $B_{\rm K}$ anomaly is unlikely to be
caused by sample inhomogeneities that exist in regions not detected from the
$L_{a}$ measurements because suppression of $B_{\mathrm{c2}}$ by impurities
and/or defects usually results in a wider distribution of $B_{\mathrm{c2}}$.

Let us discuss why the $B_{\rm K}$ anomaly exists only in $L_{c}$ when
$B\parallel c$. The most plausible origin is the formation of the FFLO state
with $\bm{q}\parallel B$, as detailed below.

The FFLO state is characterized by a periodic spatial modulation of the
superconducting order parameter $\Delta(\bm{r})$ with the wave vector
$\bm{q}$. The direction of the $\bm{q}$ vector is determined by the
combination of the relative stabilities between the vortex lattice
configuration and the nesting condition, where the Zeeman-split Fermi surfaces
are maximally touched under the translation by $\bm{q}$. When the FFLO state
is realized in a superconductor with an isotropic spherical Fermi surface, the
$\bm{q}$ vector directs parallel to the field direction;
$\bm{q}\parallel\bm{B}$. This is because the vortex lattice configuration is
least perturbed by the formation of the FFLO state, otherwise the two factors
of the relative stabilities, i.e., vortex lattice configuration and nesting
condition, interfere each other [see the inset of Fig. 4(a)]. For CeCoIn5, it
is necessary to consider the nesting condition based on the actual band
structure.Shishido et al. (2003); Settai et al. (2001); Hall et al. (2001) The
main Fermi surfaces of CeCoIn5 are the two heavy electron bands ($\alpha$ and
$\beta$ bands) with the warped cylindrical shape, open along the $c$ axis at
the four corners of the tetragonal Brillouin zone. For $B\parallel c$, the
optimal $\bm{q}$ direction is parallel to the $c$ axis under this uniaxial
symmetry situation because the two Fermi surfaces are nested circularly around
the warped neck region, as confirmed by a model calculation.Shimahara (2021)

Figure 5:  (a) The wave number $q$ normalized by the coherent length and (b)
the normalized entropy $S=S_{\rm s}(T)/S_{\rm n}(T_{\mathrm{c}})$ as a
function of $h=H/H_{\mathrm{c2}}$, taken from Ref. Suzuki et al., 2020. Here,
$S_{\rm s}$ ($S_{\rm n}$) is the entropy in the superconducting (normal)
state. (c) $|\lambda_{c}|$ as a function of $B$ at $T=0.06$ K for $B\parallel
c$. (d) $|\alpha_{c}|$ as a function of $T$ at $B=4.73$ T for $B\parallel c$.
(e) $|\lambda_{\phi,c}|$ as a function of the angle $\phi$ at $B=4.94$ T and
$T=0.11$ K. The yellow and the pink colored regions represent the Abrikosov
state and the FFLO state, respectively; the boundary in $|\lambda_{c}|$ and
$|\lambda_{\phi,c}|$ ($|\alpha_{c}|$) is determined by the onset of the
$B_{\rm K}$ anomaly [inferred from the phase diagram in Fig. 3(a)].

In the FFLO state, $\Delta(\bm{r})$ has the nodal planes perpendicular to the
$\bm{q}$ direction whose separation is given by $L=2\pi/q$, where
$q=|\bm{q}|$. Thus, the flux lines intersect perpendicular to the nodal
planes, as shown in the inset of Fig. 4(a). The nodal plane accommodates the
excess imbalanced spin component produced by the applied magnetic field
through the Pauli paramagnetic effect because the sign change of
$\Delta(\bm{r})$ allows the Andreev bound state at the zero energy
position.miz  The nodal plane or domain wall contains the excess paramagnetic
moment, accompanying the longitudinal strain wave.Ichioka et al. (2007);
Suzuki et al. (2020)

Upon entering the FFLO state from the Abrikosov vortex state with increasing
$B$, the wave number $q$ quickly increases from $q=0$ at the transition field
as shown in Fig. 5(a). This abrupt increase is a universal feature,
independent of theoretical frameworks.Machida and Nakanishi (1984) This rapid
growth of $q$ results in an upward field dependence common to other physical
quantities such as the entropy [Fig. 5(b)], the density of state, and the
paramagnetic moment. Therefore, $L_{c}$ is expected to show a similar
dependence on $B$, $T$ and $\phi$ near the FFLO transition. Indeed, field,
temperature, and field-angle derivatives of the change in $\Delta
L_{c}/L_{c}$, corresponding to $\lambda_{c}$, the linear thermal expansion
coefficient $\alpha_{c}$, and $\lambda_{\phi,c}$ respectively, exhibit quick
change at the Abrikosov-FFLO boundary along the three different paths ($B$,
$T$ and $\phi$), as shown in Figs. 5(c)-5(e). In this phase transition, the
phase boundary between the Abrikosov and FFLO states corresponds to the lower
onset field as shown in Fig. 5(b). Therefore, the phase boundary is determined
by the lower onset of the $B_{\rm K}$ anomaly [crosses in Fig. 3(a)], rather
than its peak position.

The absence of the $B_{\rm K}$ anomaly in the $L_{a}$ measurements is also
consistent with the FFLO scenario, because the spatial modulation brought by
the FFLO transition runs only along $\bm{q}\parallel c$ with keeping the
uniformity in the $ab$ plane [see the inset of Fig. 4(a)]. Indeed, even if the
$a$-axis length is maximally distorted according to the Poisson’s ratio ($0.3$
for typical metals), the $B_{\rm K}$ anomaly in $|\lambda_{\phi,a}|$ would be
at most $0.15\times 10^{-6}$/deg, which should be less prominent than the one
observed in $|\lambda_{\phi,c}|$. This anisotropic magnetostriction response
of the $B_{\rm K}$ anomaly is reminiscent of a FFLO transition suggested in
$\kappa$-(BEDT-TTF)2Cu(NCS)2,Imajo et al. (2022) whose acoustic response
depends on the sound propagation direction. Moreover, the observed field range
$\Delta B_{\rm LO}$, which is estimated by the width of the magnetic field
area between the onset of the $B_{\rm K}$ anomaly and $B_{\mathrm{c2}}$ [the
pink region in Fig. 5(c)], relative to $B_{\mathrm{c2}}$, $\Delta B_{\rm
LO}/B_{\mathrm{c2}}\sim 2.4$%, is comparable to the theoretical calculation
($\sim 2.7\%$).Suzuki et al. (2020) Thus, the $B_{\rm K}$ anomaly can be well
understood if the FFLO state with $\bm{q}\parallel c$ is formed in CeCoIn5 for
$B\parallel c$, which should be scrutinized by further measurements in future
to directly detect the spatial modulation of the superconducting gap
structure.

In summary, we have performed high-resolution magnetostriction measurements on
CeCoIn5 along the $c$ and $a$ axes in the magnetic field applied around the
$c$ axis. We find a double superconducting transition at $B_{\rm K}$ and
$B_{\mathrm{c2}}$ in the $c$-axis magnetostriction. On the other hand, this
$B_{\rm K}$ anomaly is absent in the $a$-axis magnetostriction of the same
sample. To explain this anisotropic expansion, we suggest a possibility that
the anisotropic length changes of $L_{c}$ and $L_{a}$ are a direct
manifestation of the FFLO formation, and that the modulation vector $\bm{q}$
points parallel to the field direction along the $c$ axis. The characteristic
changes in the $|\lambda_{c}(B)|$ curves with an upward curvature are
consistent with the theoretical prediction assuming the FFLO formation. This
study paves the way for determining the $\bm{q}$-vector orientation of the
FFLO state from the magnetostriction in different measurement directions.

###### Acknowledgements.

We thank Kitami Tsuji and Hirohiko Sato for their support. A part of this work
was carried out as joint research in ISSP. This work was also supported by
KAKENHI (JP20K20893, JP23H04868, JP17K05553, JP21K03455, JP17K05529,
JP20K03852, JP23H01116) from JSPS and Chuo University Grant for Special
Research.

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Supplemental Material for
Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in CeCoIn5
revealed by high-resolution magnetostriction measurements
Shunichiro Kittaka1, Yohei Kono1, Kaito Tsunashima1, Daisuke Kimoto1, Makoto
Yokoyama2, Yusei Shimizu3,

Toshiro Sakakibara4, Minoru Yamashita4, and Kazushige Machida5

1Department of Physics, Chuo University, Bunkyo-ku, Tokyo 112-8551, Japan
2Faculty of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan and
Institute of Quantum Beam Science, Ibaraki University, Mito, Ibaraki 310-8512,
Japan
3Institute for Materials Research (IMR), Tohoku University, Oarai, Ibaraki
311-1313, Japan
4Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba
277-8581, Japan
5Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
(Dated: )

## I. Experimental method

In this magnetostriction study, we used three samples: two samples (one
sample) grown at Ibaraki (Chuo) University by the self-flux method, which are
referred to as the samples s1 and s2 (sample s3). The thicknesses of the
samples s1, s2, and s3 along the $c$ axis are 0.50, 0.51, and 0.90 mm,
respectively. The main text reports the results obtained by using the sample
s3. We developed three homemade capacitance dilatometers with a resolution
better than 1 pm. The $L_{c}$ measurements on the samples s1 and s2 (sample
s3) were performed at University of Tokyo (Chuo University) in Oxford dilution
refrigerator Kelvinox 100 and 25 (AST minisorb), respectively, using a
standard (first-developed compact) dilatometer. The isothermal
magnetostriction roughly parallel to the $a$ axis $\Delta
L_{a}(B)=L_{a}(B)-L_{a}(B_{0})$, for the sample s3 was measured using a
second-developed compact dilatometer whose resolution is slightly higher than
the first one. The frame diameter and height of a standard (compact)
dilatometer are approximately 25 (20) and 45 (25) mm, respectively. For
magnetostriction measurements on the samples s1 and s2 (sample s3), a magnetic
field $B$ was generated using 15-T and 9-T solenoid magnets (a 7-T split-pair
magnet) in the vertical $z$ (horizontal $x$) direction, respectively.

## II. Sample dependence of the $c$-axis magnetostriction

Figure S1(a) compares a change in $\Delta L_{c}(B)/L_{c}$ of the samples s1,
s2, and s3 at 0.2 K for $B\parallel c$ in the field-increasing process. In the
field range $4.4\ {\rm T}\leq B\leq 5$ T, $\Delta L_{c}(B)$ of all samples
decreases with increasing $B$ in the superconducting state. The change of
$\Delta L_{c}/L_{c}\sim 5\times 10^{-6}$ in the vicinity of $B_{\mathrm{c2}}$
is consistent with the previous reports.Correa et al. (2007); Takeuchi et al.
(2002) These results reinforce the reliability of our measurements. A sharp
drop at $B_{\mathrm{c2}}$ in $\Delta L_{c}(B)$ for the samples s2 and s3 is a
sign of the occurrence of a first-order superconducting transition. By
contrast, $\Delta L_{c}$ of the sample s1 decreases gradually even near
$B_{\mathrm{c2}}$. These results indicate that the samples s2 and s3 are in
higher quality than the sample s1. The field derivative data of Fig. S1(a),
$\lambda_{c}(B)=(\partial L_{c}/\partial B)/L_{c}$, are shown in Fig. S1(b).
The broad anomaly for the sample s1 can be attributed to wide
$B_{\mathrm{c2}}$ distribution originating from sample inhomogeneity. A
relatively sharp dip can be seen at $B_{\mathrm{c2}}$ in $\lambda_{c}(B)$ of
the samples s2 and s3.

Figures S2 and S3 show the magnetostriction [(a)] and its field derivative
[(b)] for the samples s3 and s2, respectively, in both field-increasing and
decreasing processes at several temperatures. Hysteresis behavior can be seen
roughly below 0.6 (0.4) K for the sample s3 (s2), and the results of these two
samples are in reasonable agreements (Fig. S4). A tricritical point between
the FFLO, homogeneous Abrikosov vortex, and paramagnetic normal states is
suggested to be present around 0.6 K and 4.7 T because the superconducting
transition at $B_{\mathrm{c2}}$ becomes a first-order transition when the
temperature (magnetic field) is below (above) the tricritical point. A double
superconducting transition can be seen at the lowest temperature of roughly
0.1 K for both samples. The second anomaly at $B_{\rm K}$ slightly below
$B_{\mathrm{c2}}$ can be seen in the wider temperature range for the sample s3
than for the sample s2; the $B_{\rm K}$ anomaly seems to depend on the sample
quality.

## III. Additional thermal expansion data

Figure S5 shows the thermal expansion coefficient $\alpha_{c}=(\partial
L_{c}/\partial T)/L_{c}$ at several magnetic fields for $B\parallel c$. The
$B_{\rm K}$ anomaly is less clearly detected by thermal expansion measurements
than by magnetostriction measurements. This is caused by the broadening of the
superconducting transition in the temperature scan owing to the small slope of
the $B_{\mathrm{c2}}(T)$ phase boundary, $dB_{\mathrm{c2}}/dT$, at low
temperatures. Indeed, the $B_{\rm K}$ anomaly is only weakly detected from our
thermal expansion measurements in $4.73\ {\rm T}\lesssim B\lesssim 4.8$ T [see
Figs. 5(d), S5(c), and S5(d)], and is not well resolved at the magnetic fields
in the previous work [Fig. S5(e)].

## IV. Field-angle dependence of the change in the sample length

Figure S6(b) represents the field-angle $\phi$ dependence of the field-angle
derivative $\Delta L_{c}$ data, i.e., $\lambda_{\phi,c}=(\partial
L_{c}/\partial\phi)/L_{c}$, for the sample s3 at 0.11 K in the field range
$4.83~{}{\rm T}\leq B\leq 5~{}{\rm T}$. Here, $\phi$ denotes the field angle
measured from the $c$ axis, and the magnetic field is rotated within the $bc$
plane. Above 4.89 T, the two anomalies are clearly observed in
$\lambda_{\phi,c}(\phi)$, and the transition field angle becomes larger with
increasing magnetic field, as shown in Fig. S6(a).

Figure S7(a) compares the field-angle $\phi^{\ast}$ dependences of the
$c$-axis and $a$-axis magnetostrictions measured under a magnetic field
rotated around the $a$ axis and the measurement direction (roughly parallel to
the $a$ axis), respectively. Here, $\phi^{\ast}$ is the azimuthal angle in the
field-rotational plane measured from the direction in which the rotating
magnetic field is closest to the $c$ axis. It was found that the
superconducting transition field angle in $\Delta L_{a}(\phi^{\ast})$ does not
match with that in $\Delta L_{c}(\phi^{\ast})$. If we assume $\phi={\rm
acos}[\cos\phi^{\ast}\cos(10.6^{\circ})]$ ($\phi=\phi^{\ast}$) for the $L_{a}$
($L_{c}$) measurements, an onset field angle of the superconducting transition
becomes consistent between $L_{c}(\phi)$ and $L_{a}(\phi)$ [see Fig. S7(b)].
Therefore, the measurement direction for $L_{a}$ seems to be unexpectedly
tilted from the $a$ to $c$ axis by 10.6∘, possibly due to the irregular shape
of the sample; in this study, minimally polished as-grown samples were used to
avoid cracking and distortion of the samples that could easily induce the
$B_{\mathrm{c2}}$ distribution.

As already explained in the main text, the second anomaly is absent in
$L_{a}(\phi)$. The same conclusion can be obtained when the magnetic field is
increased up to 5.2 T (Fig. S8). Similar to the large anomaly in
$\lambda_{\phi,c}$ (Fig. S6), the anomaly in $\lambda_{\phi,a}$ depends on the
direction of the rotating field, supporting that the small second anomaly
observed in $\lambda_{\phi,c}$ is absent in $\lambda_{\phi,a}$.

To characterize the two anomalies at $B_{\rm K}$ and $B_{\mathrm{c2}}$, we fit
to the data of $\lambda_{\phi,c}$ by using a double-Gaussian function

$f(\phi)=A_{1}\exp\biggl{[}-\frac{(\phi-\phi_{1})^{2}}{2c_{1}^{2}}\biggl{]}+A_{2}\exp\biggl{[}-\frac{(\phi-\phi_{2})^{2}}{2c_{2}^{2}}\biggl{]}+f_{0}$
(1)

in the range $12^{\circ}\leq\phi\leq 18^{\circ}$, where a double peak is
observed. We also fit to the data of $\lambda_{\phi,a}$ by using the same
function with $A_{2}=0$ in the range $13^{\circ}\leq\phi\leq 15^{\circ}$,
where a single peak is observed. The fitting results are represented by solid
lines in Fig. S9. Here, for $\lambda_{\phi,c}$, the first- and second-term
contributions are represented by dashed and dotted lines. From these fits, the
full width at half maximum (FWHM) can be estimated to be $2\sqrt{2\ln 2}c_{j}$
($j=1$ or $2$). For $\lambda_{\phi,c}$, the FWHM of the $B_{\rm K}$ anomaly is
$1.42^{\circ}\pm 0.05^{\circ}$ whereas that of the $B_{\mathrm{c2}}$ anomaly
is $1.65^{\circ}\pm 0.02^{\circ}$. It should be noted that the former value is
smaller than the latter one. This result eliminates a possibility that the
$B_{\rm K}$ anomaly is caused by sample inhomogeneities that exist in regions
not detected from the $L_{a}$ measurements because a decrease of
$B_{\mathrm{c2}}$ by impurities and/or defects is accompanied by a broadening
of the $B_{\mathrm{c2}}$ distribution as well as suppression of the first-
order nature. For $\lambda_{\phi,a}$, the FWHM of the single anomaly is
$1.67^{\circ}\pm 0.03^{\circ}$, which matches well with the FWHM of the
$B_{\mathrm{c2}}$ anomaly in $\lambda_{\phi,c}$, further evidencing the
absence of the $B_{\rm K}$ anomaly in $\lambda_{\phi,a}$.

Due to the sample misalignment, the magnetic field cannot be applied along the
$c$ axis precisely during the $L_{a}$ measurements. However, the $c$-axis
component of $B$, i.e., $B_{\parallel c}=B\cos\phi$, may be predominant to
determine the magnetostriction in this field-angle range because of the large
anisotropy in $B_{\mathrm{c2}}$ of CeCoIn5. Figure S10 shows
$\lambda^{\ast}_{\phi,i}=[\partial L_{i}(\phi)/\partial B_{\parallel
c}]/L_{i}$ ($i=c$ or $a$) as a function of $B_{\parallel c}$ by using the data
of Fig. 4(a) of the main text for $\phi>0$. Here, the data of $\lambda_{c}(B)$
taken at 0.11 K in the field-increasing process (Fig. 1 of the main text) are
also plotted. As shown in Fig. S10, $\lambda^{\ast}_{\phi,c}(B_{\parallel c})$
essentially reproduces the field dependence of $\lambda_{c}(B)$ at the same
temperature, except for the smaller $B_{\mathrm{c2}}$ caused by the in-plane
magnetic field component in $\lambda^{\ast}_{\phi,c}(\phi)$ measurements. This
qualitative agreement between $\lambda^{\ast}_{\phi,c}(\phi)$ and
$\lambda_{c}(B)$ shows that the $B_{\rm K}$ anomaly is also absent in
$\lambda_{a}(B)=(\partial L_{a}/\partial B)/L_{a}$ for $B\parallel c$. Thus,
the sample misalignment does not affect our key finding that the $B_{\rm K}$
anomaly is not detected from the magnetostriction measurements when the
measurement direction is perpendicular to the applied magnetic field near
$B\parallel c$.

## References

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Figure S1:  Field dependences of (a) the magnetostriction along the $c$ axis,
$\Delta L_{c}/L_{c}$, of the samples s1, s2, and s3 on increasing-field
process at 0.2 K for $B\parallel c$ and (b) their field derivatives
$\lambda_{c}=(\partial L_{c}/\partial B)/L_{c}$.  Figure S2:  Field
dependences of (a) $\Delta L_{c}/L_{c}$ and (b) $\lambda_{c}$ of the samples
s3 for $B\parallel c$ at several temperatures. Each set of data in (a) [(b)]
is shifted vertically by $1\times 10^{-5}$ ($1\times 10^{-4}$/T) for clarity.
Figure S3:  Field dependences of (a) $\Delta L_{c}/L_{c}$ and (b)
$\lambda_{c}$ of the samples s2 for $B\parallel c$ at several temperatures.
Each set of data in (a) [(b)] is shifted vertically by $1\times 10^{-5}$
($1\times 10^{-4}$/T) for clarity.  Figure S4:  Temperature dependence of the
difference between $B_{\mathrm{c2}}^{\rm u}$ and $B_{\mathrm{c2}}^{\rm d}$,
where $B_{\mathrm{c2}}^{\rm u}$ and $B_{\mathrm{c2}}^{\rm d}$ are
$B_{\mathrm{c2}}$ in the field-increasing and decreasing processes,
respectively. $B_{\mathrm{c2}}$ of the sample s2 (s3) is defined from the
high-field peak in $|\lambda_{c}(B)|$ [$|d\lambda_{c}/dB|$]. The shaded areas
are guides to the eye.  Figure S5:  (a)-(d) Temperature dependence of the
thermal expansion coefficient, $\alpha_{c}=(\partial L_{c}/\partial T)/L_{c}$,
at various magnetic fields along the $c$ axis. (e) The data taken from the
previous report Bianchi et al. (2002).  Figure S6:  (a) $B$–$\phi$ phase
diagram of CeCoIn5. (b) Field-angle $\phi$ dependence of $\lambda_{\phi,c}$ of
the sample s3 at 0.11 K in several magnetic fields rotated within the $bc$
plane. Closed (open) symbols in (a) represent anomalies in (b)
[$\lambda_{c}(B)$ at 0.11 K and $\phi=0$]. Crosses indicate an onset of the
$\Delta L_{c}$ anomaly. Each set of data in (b) is shifted vertically by
$2\times 10^{-6}$/deg for clarity.  Figure S7:  (a) Field-angle $\phi^{\ast}$
dependence of $\Delta L_{i}/L_{i}$ ($i=a$ or $c$) of the sample s3 at 0.11 K
and 5 T, where $\phi^{\ast}$ is the azimuthal angle in the field-rotational
plane. (b) The same data obtained by converting $\phi^{\ast}$ to $\phi$ (see
text), where $\phi$ is the azimuthal angle in the $bc$ plane.  Figure S8:
Field-angle $\phi$ dependences of (a) $\Delta L_{a}/L_{a}$ and (b)
$\lambda_{\phi,a}$ of the sample s3 at 0.11 K and 5.2 T.  Figure S9:  Field-
angle $\phi$ dependence of $\lambda_{\phi,i}$ ($i=a$ or $c$) of the sample s3
at 0.11 K and 5 T in the $\phi$-increasing process, compared with the Gaussian
fits (see text).  Figure S10:  Field-angle-resolved $L_{i}(\phi)$ data of the
sample s3 plotted as a function of the $c$-axis component of $B$, i.e.,
$B_{\parallel c}=B\cos\phi$. Large and small closed circles are
$\lambda^{*}_{\phi,i}=[\partial L_{i}(\phi)/\partial B_{\parallel c}]/L_{i}$
($i=a$ or $c$) for $\phi>0$ in the $\phi$-increasing and decreasing processes
(i.e., $B_{\parallel c}$-decreasing and increasing processes), respectively.
Open circles in grey are $\lambda_{c}(B)$ for $B\parallel c$ at 0.11 K in the
field-increasing process (Fig. 1 in the main text) for comparison.