SUPERCONDUCTIVITY
Superconducting spin smecticity evidencing the
Fulde-Ferrell-Larkin-Ovchinnikov state in Sr 2 RuO 4
K. Kinjo^1 , M. Manago^1 †, S. Kitagawa, Z. Q. Mao^2 , S. Yonezawa^1 , Y. Maeno^1 ‡, K. Ishida^1
Translational symmetry breaking is antagonistic to static fluidity but can be realized in superconductors,
which host a quantum-mechanical coherent fluid formed by electron pairs. A peculiar example of
such a state is the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, induced by a time-reversal
symmetry–breaking magnetic field applied to spin-singlet superconductors. This state is intrinsically
accompanied by the superconducting spin smecticity, spin density–modulated fluidity with spontaneous
translational-symmetry breaking. Detection of such spin smecticity provides unambiguous evidence for
the FFLO state, but its observation has been challenging. Here, we report the characteristic“double-horn”
nuclear magnetic resonance spectrum in the layered superconductor Sr 2 RuO 4 near its upper critical field,
indicating the spatial sinusoidal modulation of spin density that is consistent with superconducting spin
smecticity. Our work reveals that Sr 2 RuO 4 provides a versatile platform for studying FFLO physics.
C
onventional superconductivity, regarded
as a quantum-mechanical coherent fluid
formed by electron pairs, is spatially ho-
mogeneous in ideal situations and re-
tains translational symmetry. However,
such a homogeneous fluid state may become
unstable, and the translational symmetry can be
broken upon the application of time-reversal
symmetry–breaking magnetic fields. One triv-
ial example is quantized vortex lattices formed
in any type II superconductors. A more exotic
state may emerge under sufficient spin polar-
ization: With spin imbalance, electron pairs
with a finite total momentum are formed,
resulting in spatial oscillation of the super-
conducting (SC) order parameter. This state is
known as the Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) state ( 1 , 2 ); the concept has been ex-
tended to other systems such as cold atoms
( 3 ) and neutron stars ( 4 ). In the FFLO state
of a spin-singlet superconductor, the mod-
ulated SC order parameter is intrinsically
accompanied by both fluidity and spatial spin
modulation—namely“spin smecticity,”which
is analogous to liquid-crystal smectic phases.
Moreover, this spin smecticity causes non-
trivial local spin-density enhancement: Spin
density near the nodes of the order param-
eter oscillation becomes even higher than that
of the normal state (Fig. 1). An observation of
this characteristic spin smecticity would pro-
vide indisputable evidence of the FFLO state.
Demonstrating unambiguously the existence
of the FFLO state has proven difficult. This is
because superconductivity is in most cases
destroyed by vortex kinetic energy (orbital
limiting) before sufficient spin polarization is
induced by the Zeeman effect (Pauli limiting).
A first-order transition (FOT) between the
normal and SC states is a hallmark of sizable
spin polarization. Moreover, FFLO states are
quite fragile against impurities; only a few
clean superconductors with large effective elec-
tron mass ( 5 ), such as quasi–two-dimensional
(2D) organics ( 6 ) and heavy-fermion CeCoIn 5
( 7 ), exhibit the FOT and subphases within the
SC state. For CeCoIn 5 , the observed high-field
subphase (Q-phase) ( 8 ) is not a simple FFLO
phase because it is intertwined with a mag-
netic order. In ( 6 – 8 ), bulk anomalies related
to the FFLO state have been detected, but evi-
dence more closely linked to the FFLO modu-
lation, such as the SC spin smecticity, has not
been reported. Nuclear magnetic resonance
(NMR) spectrum is a promising probe with
which to observe spatial spin modulation be-
cause NMR spectrum can detect the local spin
susceptibility with high sensitivity ( 9 – 12 ).
In this work, we focused on the layered
perovskite Sr 2 RuO 4 , which is an extremely
clean and archetypal unconventional super-
conductor with the SC critical temperature
(Tc)of1.5K( 13 – 15 ). Its normal state is well
understood as a strongly correlated Fermi-
liquid state. However, the nature of the SC
pairing state has not been elucidated. Re-
cently, a decrease in the in-plane spin suscep-
tibility in the SC state was observed ( 16 ) and
later confirmed by other groups ( 17 , 18 ), ruling
out the chiral spin-triplet scenario. The un-
changed spin susceptibility reported by pre-
vious NMR measurements ( 19 )isascribed
to the instantaneous destruction of super-
conductivity caused by overheating by NMR
pulses. The full elucidation of the SC state
of Sr 2 RuO 4 is entering a new stage ( 20 – 25 ).
One of the characteristic features observed
in Sr 2 RuO 4 is the FOT between the SC and
normal states in the in-plane field ( 26 ). In
addition, Sr 2 RuO 4 has an exceedingly longmean-free-pathlof ~3mmforthein-plane
direction, which is much longer than the in-
plane SC coherence length of 0.066mm( 14 , 15 ).
These features favor the emergence of the
FFLO state.
Moreover, Sr 2 RuO 4 has a crystallographic
advantage for detecting the SC spin smecticity.
There are two inequivalent oxygen sites in
Sr 2 RuO 4 : the planar-oxygen [O(1)] and the
apical-oxygen [O(2)] sites (Fig. 2A). The smaller
hyperfine coupling and higher local symmetry
attheO(2)sitemakethelocalNMRspectrum
much sharper and stronger than that at the
O(1) site. Thus, the NMR spectrum at the O(2)
site provides an ideal probe for the spatial
variation of spin density. In other FFLO can-
didates, the anomalies in the spectral width
and the enhancement of the nuclear spin-
lattice relaxation rate 1/T 1 suggesting the local
enhancement in the density of states were
reported in the FFLO state ( 9 – 11 ). Such an
enhancement of 1/T 1 is considered to be a
decisive dynamical signature of the FFLO state
originating from the spin-polarized Andreev
bound state ( 9 – 12 ). Nevertheless, previous
attempts to probe the static FFLO spin modu-
lations by the spectral peak splitting have not
yielded a clear and unambiguous observa-
tion, mainly owing to unfavorable crystal struc-
ture and complications caused by magnetic
ordering.
We used^17 O-NMR measurements with a
very precise field-direction control to observe
that the spin density ascribed to the SC pairs
becomes spatially inhomogeneous, featuring
spin-“dense”and spin-“thin”portions. Details
of the experimental methods are described in
( 27 ). This spin modulation emerges in a lim-
ited field region near SC upper critical field
Hc2and only when magnetic fieldHis parallel
to theabplane. Our observations indicate the
existence of SC spin smecticity, providing un-
ambiguous evidence for the FFLO state.
The in-plane field variation of the^17 O-NMR
spectrum is shown in Fig. 2B at the O(2) site
nearHc2, measured at 70 mK. All spectra were
obtained by using the free-induction-decay
method with the radio frequency (RF)–pulse
energy small enough to avoid overheating
(figs. S2 and S3). Together with time-resolved
ac susceptibility measurements by using the
identical sample and setup, the data confirm
that the sample is globally in the SC state
belowHc2(fig. S2) ( 16 , 17 ). In the following,
NMR spectra are presented as a function of
the Knight shiftK, which is defined asK=
(f–f 0 )/f 0 , wheref 0 is the bare NMR frequency
evaluated byf 0 =gH/2p,withthenuclear
gyromagnetic ratiog/2p= 5.7718 MHz/T of(^17) O nuclei.Kat the peak (Kpeak) is propor-
tional to the electronic spin susceptibility at the
nuclear site. The NMR spectra abovem 0 Hc2=
1.4 T, where the sample is in the normal state,
are almost symmetric. The spectra below 1.1 T
SCIENCEscience.org 22 APRIL 2022•VOL 376 ISSUE 6591 397
(^1) Department of Physics, Graduate School of Science, Kyoto
University, Kyoto, Japan.^2 Department of Physics,
Pennsylvania State University, State College, PA, USA.
*Corresponding author. Email: [email protected]
(K.K.); [email protected] (K.I.)
†Present address: Department of Physics and Material Science,
Shimane University, Shimane, Japan.
‡Present address: Toyota Riken–Kyoto University Research Center
(TRiKUC), Kyoto, Japan.
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