# Optical anisotropy of the kagome magnet FeSn: Dominant role of excitations
between kagome and Sn layers

J. Ebad-Allah  These authors contributed equally to this work.
Experimentalphysik II, Institute for Physics, Augsburg University, D-86135
Augsburg, Germany  Department of Physics, Tanta University, 31527 Tanta, Egypt
M.-C. Jiang  These authors contributed equally to this work. Department of
Physics and Center for Theoretical Physics, National Taiwan University, Taipei
10617, Taiwan  RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako
351-0198, Japan  R. Borkenhagen  Experimentalphysik II, Institute for Physics,
Augsburg University, D-86135 Augsburg, Germany  F. Meggle  Experimentalphysik
II, Institute for Physics, Augsburg University, D-86135 Augsburg, Germany  L.
Prodan  Experimentalphysik V, Center for Electronic Correlations and
Magnetism, Institute for Physics, Augsburg University, D-86135 Augsburg,
Germany  Institute of Applied Physics, Moldova State University, MD-2028
Chisinau, Republic of Moldova  V. Tsurkan  Experimentalphysik V, Center for
Electronic Correlations and Magnetism, Institute for Physics, Augsburg
University, D-86135 Augsburg, Germany  Institute of Applied Physics, Moldova
State University, MD-2028 Chisinau, Republic of Moldova  F. Schilberth
Experimentalphysik V, Center for Electronic Correlations and Magnetism,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany
Department of Physics, Institute of Physics, Budapest University of Technology
and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  G.-Y. Guo
Department of Physics and Center for Theoretical Physics, National Taiwan
University, Taipei 10617, Taiwan  Physics Division, National Center for
Theoretical Sciences, Taipei 10617, Taiwan  R. Arita  RIKEN Center for
Emergent Matter Science, 2-1 Hirosawa, Wako 351-0198, Japan  Research Center
for Advanced Science and Technology, University of Tokyo, 4-6-1 Meguro-ku,
Tokyo 153-8904, Japan  I. Kézsmárki  Experimentalphysik V, Center for
Electronic Correlations and Magnetism, Institute for Physics, Augsburg
University, D-86135 Augsburg, Germany  C. A. Kuntscher  Experimentalphysik II,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany

###### Abstract

Antiferromagnetic FeSn is considered to be a close realization of the ideal
two-dimensional (2D) kagome lattice, hosting Dirac cones, van Hove
singularities, and flat bands, as it comprises Fe3Sn kagome layers well
separated by Sn buffer layers. We observe a pronounced optical anisotropy,
with the low-energy optical conductivity being surprisingly higher
perpendicular to the kagome planes than along the layers. This finding
contradicts the prevalent picture of dominantly 2D electronic structure for
FeSn. Our material-specific theory reproduces the measured conductivity
spectra remarkarbly well. A site-specific decomposition of the optical
response to individual excitation channels shows that the optical conductivity
for polarizations both parallel and perpendicular to the kagome plane is
dominated by interlayer transitions between kagome layers and adjacent Sn-
based layers. Moreover, the matrix elements corresponding to these transitions
are highly anisotropic, leading to larger out-of-plane conductivity. Our
results evidence the crucial role of interstitial layers in charge dynamics
even in seemingly 2D systems.

###### pacs:

The kagome lattice is a two-dimensional (2D) hexagonal network of corner-
sharing triangles and is heavily investigated recently due to the variety of
emergent quantum phases such as quantum spin liquid, Dirac or magnetic Weyl
fermions, and magnetic skyrmions Yan  _et al._ (2011); Broholm  _et al._
(2020); Mazin  _et al._ (2014); Kuroda  _et al._ (2017); Hou  _et al._ (2017).
In its tight-binding band structure, the kagome lattice hosts a saddle point,
leading to van Hove singularities in the density of states, two linearly
dispersing Dirac-bands around the $K$ point, that can give rise to
topologically nontrivial Weyl points and nodal lines, and a flat band,
emerging from destructive interference of electronic wavefunctions. Due to
this interference, the electronic states become geometrically confined within
a kagome hexagon Xu  _et al._ (2020); Yin  _et al._ (2019); Kang et al.
(2019); Ye  _et al._ (2018); Armitage  _et al._ (2018); Bergholtz and Liu
(2013); Kang  _et al._ (2020); Meier  _et al._ (2020). This localization and
the related quench of kinetic energy amplify electronic correlations, making
the kagome lattice an ideal playground to study the interplay between
geometry, topology, and correlation physics.

One fundamental question is how these prototypical states evolve upon the
stacking of kagome layers in three-dimensional (3D) crystals. In other words,
whether the characteristic features of the kagome band structure are still
manifested in real materials. For example, it was shown for the bilayer kagome
material Fe3Sn2 that the interplane hopping causes a double Dirac structure
near $K$, with Dirac points extending to helical nodal lines, and the flat
bands acquire dispersion Ye  _et al._ (2018); Fang  _et al._ (2022);
Schilberth  _et al._ (2022). As was recently shown for the sister compound
Co3Sn2S2, such gapped nodal lines can leave fingerprints in the optical
response and dominate the anomalous Hall conductivity Okamura  _et al._
(2020); Schilberth  _et al._ (2023). Also, in 3D flat band materials like the
pyrochlore metal CaNi2, containing stacked kagome layers, deviations from
flatness occur due to multiple vectorial interferences of several Ni $3d$
orbitals Wakefield  _et al._ (2023). Therefore, sensitive spectroscopic probes
are required both for the identification of these states and for determining
their impact on the materials response, e. g., transport or optical
properties.

Our target material is the kagome metal FeSn with antiferromagnetic (AFM)
order below $T_{\text{N}}$=365 K Yamamoto (1966); Sales  _et al._ (2019). It
has been suggested to host Dirac fermions and flat bands, exhibiting
characteristic features of an individual kagome lattice despite the interlayer
interactions inevitably present in 3D crystals. Kang et al. (2019); Multer
_et al._ (2023); Han  _et al._ (2021). It crystallizes in the hexagonal space
group P6/$mmm$, where Fe3Sn kagome layers in the $ab$ plane are alternatingly
stacked with Sn honeycomb layers along the $c$ direction Kang et al. (2019).
FeSn forms a kagome metal series together with Fe3Sn2 and Fe3Sn. In this
order, the electronic structure is claimed to turn from dominantly 2D to 3D,
with FeSn being the closest realization of the independent kagome layers Kang
et al. (2019).

Electronic band structure calculations of FeSn indeed predicted flat bands
around 0.5 eV above the Fermi energy $E_{\text{F}}$ and a Dirac cone located
at the $K$ point around 0.4 eV below $E_{\text{F}}$, together with a
corresponding nodal line along $K-H$ Lin  _et al._ (2020); Fang  _et al._
(2022); Zhang  _et al._ (2022); Kang et al. (2019). Furthermore, flat bands in
limited regions of the Brillouin zone were predicted Lin  _et al._ (2018);
Fang  _et al._ (2022), as well as observed by angle resolved photoemission
spectroscopy (ARPES) and scanning tunneling spectroscopy, in both FeSn and
Fe3Sn2 Kang et al. (2019); Multer  _et al._ (2023). ARPES measurements also
revealed the predicted Dirac cone(s) in FeSn and Fe3Sn2, respectively Kang et
al. (2019); Ye  _et al._ (2018). For Fe3Sn2 it was shown that electronic
correlation effects cannot be neglected, since they cause a reorganization of
the energy bands close to $E_{\text{F}}$ Schilberth  _et al._ (2022). It
should also be noted that there are several additional conventional energy
bands in the vicinity of $E_{\text{F}}$ according to electronic band structure
calculations Schilberth  _et al._ (2022). Nevertheless, recent optical
spectroscopy studies on Fe3Sn2 observed features in the low-energy optical
conductivity spectrum, which were interpreted as excitations of Dirac cones,
helical nodal lines and flat bands close to $E_{\text{F}}$ Biswas  _et al._
(2020); Schilberth  _et al._ (2022).

In this Letter, we report the anisotropic optical conductivity of the kagome
magnet FeSn based on a combined experimental and theoretical study. We find a
pronounced anisotropy in the low-energy optical conductivity, with a higher
magnitude perpendicular to the kagome plane. This is opposite to the general
expectations, where the reduced dimensionality of the crystal structure is
directly manifested in the optical anisotropy Kézsmárki  _et al._ (2006). By
comparing the experimental optical conductivity with material-specific theory
calculated in the framework of density functional theory we can attribute the
spectral features to specific intra- and interlayer excitations, revealing
significant contributions of the interstitial Sn layers to both in- and out-
of-plane conductivity. No clear sign of flat band transitions and Dirac
electron excitations is detected in the optical conductivity of FeSn.

A comparison of the experimental optical conductivity spectra of single-
crystalline FeSn for the polarization directions E$\parallel$$a$ (parallel to
the kagome plane) and E$\parallel$$c$ (perpendicular to the kagome plane) is
presented in Fig. 1 (a) and (b), respectively. The experimental details about
the crystal growth, the reflectivity measurements and the corresponding
polarization- and temperature-dependent reflectivity spectra can be found in
the supplemental material (SM) Sup  (see also references Ebad-Allah  _et al._
(2019); Köpf  _et al._ (2020); Ebad-Allah  _et al._ (2021); Tanner (2015);
Kresse and Furthmüller (1996, 1996); Kresse and Joubert (1999); Perdew  _et
al._ (1996); Pizzi  _et al._ (2020); Schilberth  _et al._ (2022); Biswas  _et
al._ (2020); Li  _et al._ (2023) therein). For E$\parallel$$a$ we observe a
pronounced absorption band at around 3100 cm-1 ($\sim$0.38 eV) and a dip in
the frequency range 1000 - 1300 cm-1 ($\sim$0.12 - 0.16 eV) followed by the
low-frequency Drude contribution. The metallic Drude response is also resolved
for E$\parallel$$c$, in fact, it is stronger than in the in-plane
conductivity. The out-of-plane conductivity spectrum also shows an absorption
band around 3100 cm-1 and an additional one around 800 cm-1. For both
polarization directions the observed absorption bands hardly shift with
decreasing temperature but only sharpen. Most interestingly, for energies
below 9000 cm-1 ($\sim$1.1 eV) the optical conductivity perpendicular to the
kagome layers is higher than along the kagome plane, as illustrated in the
inset of Fig. 1(b). These results clearly demonstrate a strong anisotropy of
charge excitations in FeSn, but in the opposite way as one would expect for
independent kagome layers.

This behavior contradicts the frequently made assumption that transport in
layered systems is mostly confined to the individual layers. In fact, similar
behavior has been recently reported for the dc transport characteristics in
the sister compound CoSn Huang  _et al._ (2022). Huang et al. attributed the
observed anomalous dc transport in CoSn to the unique properties of flat band
electrons with large in-plane effective mass. In addition, a larger dc
conductivity along the $c$ direction was also reported for other layered
kagome magnets such as HoAgGe Zhao  _et al._ (2020), YCr6Ge6 Yang  _et al._
(2022), Fe3Sn Prodan et al. , Fe3Sn2 Du  _et al._ (2022), and recently
suggested for FeSn Sales  _et al._ (2019). The last three examples show that
this “reversed” conductivity anisotropy is a general attribute of iron-tin
kagome magnets.

Figure 1: Experimental optical conductivity spectra $\sigma_{1}$ of FeSn for
polarization direction (a) E$\parallel$$a$ and (b) E$\parallel$$c$ as a
function of temperature. Inset of (b): Comparison of E$\parallel$$a$ and
E$\parallel$$c$ optical conductivity at 6 K. Figure 2: Comparison between the
experimental and theoretical interband optical conductivity spectra of FeSn
for polarization direction (a) E$\parallel$$a$ and (b) E$\parallel$$c$. Panels
(c) and (d) show the site-specific contributions to the calculated
$\sigma_{xx}$ and $\sigma_{zz}$, namely intralayer excitations within the Fe
kagome plane (Fe-Fe), intralayer excitations within the Sn plane (Sn-Sn), and
interlayer excitations between the Fe kagome plane and the Sn plane (Fe-Sn).
The intralayer and interlayer excitations are illustrated in (e) showing the
crystal structure of FeSn. The Fe3Sn kagome layers and the Sn buffer layers
are highlighted in green and blue, respectively. The two black arrows indicate
the A-type antiferromagnetic order of Fe magnetic moments, as described in the
text.

For the separation of intra- and interband transitions in the optical
conductivity spectra, we carried out simultaneous fittings of reflectivity and
optical conductivity data using the Drude-Lorentz model. Before investigating
the interband excitations, utilizing the results of the intraband excitations,
we will briefly comment on the estimation of the electronic correlation
strength, which is often discussed for kagome materials Schilberth  _et al._
(2022). From optical conductivity, we can estimate the electronic correlation
strength by calculating the ratio between the electronic kinetic energy
derived from the experimentally measured Drude weight and the calculated
plasma frequency from the band theory ($\omega_{\text{pl},ii}$) Dagotto
(1994); Millis  _et al._ (2005); Qazilbash  _et al._ (2009). The plasma
frequency is directly linked to the Drude spectral weight as described in Ref.
Dressel and Grüner (2002). The results of the $\omega_{\text{pl},ii}$ from the
experiments (calculations) are 1.62 eV (2.40 eV) and 2.38 eV (2.06 eV) for
$\omega_{\text{pl},xx}$ and $\omega_{\text{pl},zz}$, respectively. Details are
given in the Supplemental Material (SM) Sup . We can see that in both the in-
plane and out-of-plane direction, the plasma frequency is in the same order of
magnitude, making the electron kinetic energy ratio expected to fall around 1.
This means that FeSn is a conventional metal, i.e., it is weakly correlated.

Here, we refocus back on the contributions of interband excitations to the
optical conductivity and therefore subtract the Drude contributions from the
measured conductivity spectra. The resulting interband optical conductivity
$\sigma_{1,\text{interband}}$ at 6 K is depicted in Figs. 2(a) and 2(b) for
E$\parallel$$a$ and E$\parallel$$c$, respectively (the fit of the
$\sigma_{1}$-spectra with all fitting contributions can be found in the SM Sup
). For both polarization directions, we find two distinct absorption bands in
the energy range below $\sim$0.6 eV, centered at around 80 meV (105 meV) and
380 meV (380 meV) for E$\parallel$$a$ (E$\parallel$$c$), respectively. The
higher-energy transitions cause a broad excitation continuum in the
conductivity profile for both polarizations.

To understand the pronounced anisotropy in the optical response of FeSn and to
specify the excitations contributing to its interband optical conductivity
(IOC), we carried out ab initio calculations in the framework of density
functional theory. Note that the calculations are conducted for an A-type AFM
FeSn [i.e., magnetic moments of the Fe atoms are ferromagnetically aligned
within each kagome plane but AFM-coupled along the $c$ axis, see Fig. 2(e)]
without considering spin-orbit coupling. Fig. S4 in the SM Sup  displays the
results with and without the spin-orbit coupling, and we can see that the
spin-orbit coupling has a minor effect on the diagonal part of IOC in FeSn.
The IOC can be obtained from the Kubo-Greenwood formula written as follows
Pizzi  _et al._ (2020):

$\noindent\sigma_{\textbf{k},\alpha\beta}(\hbar\omega)=\frac{ie^{2}}{\hbar
V}\sum_{\textbf{k},n,m}(f_{m,\textbf{k}}-f_{n,\textbf{k}})\cdot\\\
\frac{\epsilon_{m,\textbf{k}}-\epsilon_{n,\textbf{k}}}{\epsilon_{m,\textbf{k}}-\epsilon_{n,\textbf{k}}-(\hbar\omega+i\eta)}A^{\rm(H)}_{nm,\alpha}(\textbf{k})A^{\rm(H)}_{nm,\beta}(\textbf{k})$
(1)

where

$A^{\rm(H)}_{nm,\alpha}=\left\langle
u_{n,\textbf{k}}\middle|i\nabla_{\textbf{k}_{\alpha}}\middle|u_{m,\textbf{k}}\right\rangle$
(2)

is the Berry connection written in the Hamiltonian gauge. The $\alpha$ and
$\beta$ are the indices in the Cartesian coordinates, $V$ is the cell volume,
$f_{n,\textbf{k}}=f(\epsilon_{n,\textbf{k}})$ is the Fermi-Dirac distribution
function, $\omega$ is the optical frequency and $\eta>0$ is the smearing
parameter. Under the Wannier interpolation, every object inside Eqn. 2 should
consistently be in either the Hamiltonian gauge ($A^{\rm(H)}$) or the Wannier
gauge ($A^{\rm(W)}$) Wang  _et al._ (2006). The Hamiltonian gauge labels $n,m$
as band indices of the projected band structures in the Hilbert space, while
the Wannier gauge labels $n,m$ as state vectors in the “tight-binding space”
defined by the real-space Wannier functions Wang  _et al._ (2006). A unitary
rotation matrix connects these two gauges. Importantly, this allows us to
locate particular transitions not only in and/or associate them with by energy
bands in $k$-space (after the similarity transformation), but also by the
real-space Wannier functions (before the similarity transformation).

First of all, we highlight that the IOCs, including their spectral shape and
anisotropy, are nicely captured by the theoretical calculations, as shown in
Figs. 2(a), (b). Peaks at 0.05, 0.34 eV for $\sigma_{xx}$ (in-plane
polarization) and 0.1, 0.36 eV for $\sigma_{zz}$ (out-of-plane polarization)
coincide well with the experimental data, respectively. The largest predicted
optical anisotropy $\sigma_{zz}/\sigma_{xx}$ in the low-frequency region is
7.37 at 0.13 eV. The ratio $\sigma_{zz}/\sigma_{xx}$ persists to be greater
than one up to 1.67 eV.

In order to assign spectral features and to evaluate the origin of the optical
anisotropy, we select excitations by real-space orbital sites, as there is an
overwhelming amount of possible transitions in the metallic bands of FeSn (see
Fig. S7 in the SM Sup ). Therefore, we can distinguish intralayer excitations
between sites within the Fe-kagome- and within the Sn-planes from interlayer
excitations between the Fe-kagome-plane and the adjacent Sn-buffer plane, as
illustrated in Fig. 2(e). Note that the long-range interlayer excitations
between the two distinct kagome planes and between the two distinct Sn planes
[labeled ”Long-range” in Fig. 2(e)] are relatively small. The results are
summarized in Figs. 2(c) and (d), where the black line denotes the overall IOC
for $\sigma_{xx}$ and $\sigma_{zz}$ calculated via Eqn. 1 and the others are
the site-selected IOC. In certain frequency windows, the individual
contributions, especially for the Fe-Sn excitation, are larger than the total
IOC. We would like to address this issue before looking into the site-selected
IOC in more detail.

Instead of a truncated integration by omitting some band indices, what we
manipulate during the site selection is the Berry connection
$A^{\rm(W)}_{nm,\alpha}$ itself. Since the Berry connection is a complex
matrix, after the unitary transformation and the complex conjugation of
$A^{\rm(H)}_{nm,\alpha}A^{\rm(H)}_{mn,\beta}$, matrix elements may cancel each
other. Therefore, blocking certain terms during the site selection may lead to
an optical spectrum larger than the total spectrum. Alternatively, we could
picture the optical excitations in real space. Excitation due to a certain
frequency photon excites not just one but various sites simultaneously. The
final measurement is the result of the interference between all these signals.
Blocking out terms before the complex conjugation
$A^{\rm(H)}_{nm,\alpha}A^{\rm(H)}_{mn,\beta}$ ignores such interference. Thus,
site-dependent IOC can be larger than the total IOC. Such a scenario works
better in low-energy regions where possible excitations are limited, as we
compare the data of $\sigma_{zz}$ below and above 1.4 eV.

With this clarification, we notice that for $\sigma_{zz}$ the interlayer
excitation Fe-Sn dominates the optical spectrum below 1.5 eV, as would be
expected for transitions for this polarization. Surprisingly, this
contribution is also the largest for the in-plane conductivity $\sigma_{xx}$:
The interlayer excitations are dominant between 0.45 and 1.0 eV and are still
larger below 0.45 eV, where the intralayer excitations are of the same order.
This indicates that in-plane polarization would excite all possible
transitions, even favoring Fe-Sn interlayer transitions in certain energy
windows, while $z$-polarized light (E$\parallel$$c$) would highly favor the
transitions between the Fe-kagome plane and the Sn-buffer plane. While most
investigations focus on the kagome plane, this finding clearly reveals the
crucial role of interstitial layers in the optical properties of kagome
materials.

Next, we discuss whether the characteristic signatures of interband
transitions of Dirac cones or excitations between flat bands can be observed
in the profile of our optical conductivity spectra. Interband transitions of
linearly dispersing bands close to the Dirac/Weyl nodes are expected to cause
a power-law behavior in the optical conductivity according to
$\sigma_{1}$$\sim$$\omega^{d-2}$, where $d$ is the dimensionality of the
system Tabert  _et al._ (2016). Flat band excitations should lead to sharp
peaks in the $\sigma_{1}$ spectrum. Indeed, the 3D Weyl semimetal Co3Sn2S2
with Co3Sn kagome layers shows both a linear-in-frequency behavior and a sharp
low-energy peak in its optical conductivity Xu  _et al._ (2020); Yang  _et
al._ (2020); Schilberth  _et al._ (2023). For the kagome magnet TbMn6Sn6, a
flat behavior of $\sigma_{1}$ due to quasi-2D Dirac bands was observed Li  _et
al._ (2023). For Fe3Sn2, signatures of interband transitions of the double
Dirac structure and flat band excitations in the optical conductivity were
reported Biswas  _et al._ (2020) but could not be confirmed Schilberth  _et
al._ (2022).

For FeSn, the most pronounced contributions to the measured IOC are two
absorption peaks below 0.6 eV present for both polarization directions [see
Figs. 2(a) and (b)]. It is important to note that a linear-in-frequency (3D
Weyl/Dirac cone) or frequency-independent (2D Weyl/Dirac cone) behavior of
$\sigma_{1}$ is not observed in our data. According to the electronic band
structure of FeSn depicted in Fig. S7(a) in the SM Sup , a Dirac nodal line is
located at 0.4 eV below $E_{\text{F}}$ at the $K-H$ line. Furthermore, flat
bands occur in only limited $k$-space regions at -0.2 eV or considerably
further away from $E_{\text{F}}$, consistent with earlier reports Lin  _et
al._ (2020). In Fig. S7(b) Sup , we show the band-selected IOC of various band
sets. We highlight the band-selected IOC from band 30 to band 32 to address
the excitation between Dirac nodal lines. We notice a flat band-selected IOC
for $\sigma_{xx}$ from 0.42 eV to 0.61 eV, similar to the feature reported in
Li  _et al._ (2023), which could be related to the transitions between Dirac
nodal lines and linearly dispersed bands around the $H$ point and along the
$\Gamma-A$ line. Nevertheless, neither $\sigma_{xx}$ nor $\sigma_{zz}$ from
band 30 to 32 contributes significantly to the spectrum. In addition, the
results with and without spin-orbit coupling are similar, meaning that the gap
opening of Dirac nodal lines has minor effects on the interband optical
transition (Fig. S6). We thus conclude that no dominant optical signature of
flat bands and Dirac fermions are found in the optical spectra of FeSn as they
are overwhelmed by transitions between trivial bands.

Finally, we remark on the influence of AFM order on the optical conductivity
by comparing the electronic structures and IOC between nonmagnetic and AFM
FeSn. While transitioning from nonmagnetic to AFM FeSn, band splitting happens
due to the magnetic moments, as pointed out in Fig. S8(a) Sup . Spin channels
of the same kagome layer lose spin-degeneracy but form degeneracy with the
neighboring kagome layer of the opposite spin to maintain zero net
magnetization, shown in Fig. S8(b) Sup . IOC of nonmagnetic FeSn is given in
Fig. S9 Sup . The mentioned band splitting is the main factor of the changes
in the IOC as we observe similar spectral patterns but peak shifts for
$\sigma_{xx}$ and $\sigma_{zz}$. Nevertheless, despite the peak shifting, the
optical anisotropy of $\sigma_{zz}>\sigma_{xx}$ is still observed in the
nonmagnetic phase, consistent with the fact of overwhelming trivial bands plus
the dominant role of kagome to Sn excitations in our target system FeSn.

In conclusion, our combined experimental and theoretical study of the kagome
magnet FeSn reveals a pronounced anisotropy in the optical response, with a
surprisingly higher optical conductivity for the polarization perpendicular to
the kagome layers than along the layers for energies below $\sim$1.1 eV.
According to our material-specific calculations, the main contributions to the
low-energy optical conductivity stem from interlayer transitions between the
Fe-kagome layers and the adjacent Sn-buffer layers for both polarization
directions parallel and perpendicular to the kagome plane. The optical
conductivity does not reveal the typical signatures of interband transitions
of linearly dispersing bands close to the Dirac/Weyl nodes and flat band
excitations.

Our study shows that the nature of charge dynamics in FeSn, as revealed in the
real compound, is in strong contrast to what is expected for this material on
the kagome-model-system level. In general, one should be cautious when
deriving or interpreting material properties based on certain structural
motifs, such as the kagome layers, instead of investigating the system in its
full complexity.

###### Acknowledgements.

The authors are grateful to Joachim Deisenhofer for fruitful discussions. This
work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – TRR 360 – 492547816. VT acknowledges the the support
via Project No. ANCD 20.80009.5007.19. M.-C. J. was supported by RIKEN’s IPA
Program. M.-C. J. and G.-Y. G. acknowledge the support from the Ministry of
Science and Technology and the National Center for Theoretical Sciences (NCTS)
of the R.O.C. R. A. was supported by a Grant-in-Aid for Scientific Research
(No. 19H05825) from the Ministry of Education, Culture, Sports, Science and
Technology.

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# Anisotropic Superconducting Spin Transport at Magnetic Interfaces

Yuya Ominato1, Ai Yamakage2, and Mamoru Matsuo1,3,4,5 1Kavli Institute for
Theoretical Sciences, University of Chinese Academy of Sciences, Beijing
100190, China  2Department of Physics, Nagoya University, Nagoya 464-8602,
Japan  3CAS Center for Excellence in Topological Quantum Computation,
University of Chinese Academy of Sciences, Beijing 100190, China  4Advanced
Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan
5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198,
Japan

###### Abstract

We present a theoretical investigation of anisotropic superconducting spin
transport at a magnetic interface between a $p$-wave superconductor and a
ferromagnetic insulator. Our formulation describes the ferromagnetic resonance
modulations due to spin-triplet current generation, including the frequency
shift and enhanced Gilbert damping, in a unified manner. We find that the
Cooper pair symmetry is detectable from the qualitative behavior of the
ferromagnetic resonance modulation. Our theory paves the way toward
anisotropic superconducting spintronics.

Introduction.—Use of spin-triplet Cooper pairs as carriers for spin currents
in the emergent field of superconducting spintronics is challenging Linder and
Robinson (2015); Han  _et al._ (2020). Previous studies have demonstrated spin
transport mediated by spin-triplet Cooper pairs that formed at the $s$-wave
superconductor (SC)/ferromagnet interfaces of Josephson junctions. The spin-
singlet pairs in SCs are converted into spin-triplet pairs in half-metallic
$\mathrm{CrO}_{2}$ Keizer  _et al._ (2006). However, previous studies on spin-
triplet pairs at magnetic interfaces have been limited to cases induced by the
proximity effect.

One promising candidate material system for investigation of spin-triplet
currents to enable more active use of spin-triplet pairs is the $p$-wave
SC/ferromagnetic insulator (FI) bilayer thin film system Tanaka  _et al._
(2009); Brydon (2009). Tunneling of the spins is driven by the magnetization
dynamics excited by ferromagnetic resonance (FMR) in the ferromagnetic
material via interfacial exchange coupling between the magnetization in the FI
and the electron spins in the $p$-wave SC, and a spin-triplet current is
expected to be generated. Furthermore, as a back action of spin injection,
both the FMR frequency and the Gilbert damping of the FI should be modulated
Tserkovnyak  _et al._ (2002, 2005); Šimánek and Heinrich (2003). Although
similar scenarios have already been studied vigorously in $s$-wave
SC/ferromagnet systems, most previous studies have focused on the Gilbert
damping modulation due to spin injection Inoue  _et al._ (2017); Kato  _et
al._ (2019); Silaev (2020a, b); Bell  _et al._ (2008); Jeon  _et al._ (2018);
Yao  _et al._ (2018); Li  _et al._ (2018); Jeon  _et al._ (2019a, 2018, b);
Golovchanskiy  _et al._ (2020); Zhao  _et al._ (2020). To gain an in-depth
understanding of the spin-triplet transport mechanisms, the FMR modulation
processes, including both the frequency shift and the enhanced Gilbert
damping, should be formulated microscopically in a systematic manner.
Determination of the pairing symmetry of the spin-triplet $p$-wave SCs within
the same framework is also desirable. Despite many years of research based on
several experimental techniques that reflect the spin susceptibility,
including nuclear magnetic resonance Leggett (1975), polarized neutron
scattering Shull and Ferrier (1963); Shull (1963); Shull and Wedgwood (1966),
and muon-spin resonance techniques Luke  _et al._ (1993), there are few
established candidate systems for spin-triplet SCs Saxena  _et al._ (2000);
Aoki  _et al._ (2001); Huy  _et al._ (2007).

Figure 1: Mechanism of FMR modulation due to anisotropic superconducting spin
transport at magnetic interfaces. (a) FMR signal modulation in the SC/FI
bilayer system compared with the signal in the FI monolayer. (b) Two types of
spin-triplet Cooper pair considered in this work. (c) Precession axis located
on the $x$-$z$ plane, where the angle between the precession axis and the
$z$-axis is $\theta$ (where $0\leq\theta\leq\pi/2$)

In this Letter, we investigate anisotropic superconducting spin transport at
the magnetic interfaces of hybrid systems composed of $p$-wave SC/FI thin
films theoretically, as illustrated in Fig. 1 (a). The FMR frequency and
linewidth are both modulated because of the spin transfer occurring at the
interface. Figure. 1 (b) shows a schematic image of the spin-triplet Cooper
pairs for the chiral and helical $p$-wave SCs considered in this work. Using
the nonequilibrium Green’s function method, we formulate the FMR modulations
due to the back action of the spin-triplet transport process systematically.
The frequency shift and the enhanced Gilbert damping are both described in a
unified manner. Additionally, it is shown that the symmetry of the spin-
triplet pairs can be extracted from the FMR modulations. The results presented
here offer a pathway toward development of anisotropic superconducting
spintronics.

Model Hamiltonian.—The FMR modulation due to the SC material located adjacent
to the FI material is calculated microscopically using the spin tunneling
Hamiltonian method Ohnuma  _et al._ (2014, 2017); Tatara and Mizukami (2017);
Matsuo  _et al._ (2018); Ominato and Matsuo (2020); Ominato  _et al._ (2020);
Inoue  _et al._ (2017); Kato  _et al._ (2019); Silaev (2020a). The total
Hamiltonian $H(t)$ comprises three terms:

$\displaystyle H(t)=H_{\rm FI}(t)+H_{\rm SC}+H_{\rm ex}.$ (1)

The first term $H_{\rm FI}(t)$ describes the bulk FI, as follows:

$\displaystyle H_{\rm FI}(t)=$
$\displaystyle\sum_{\bm{k}}\hbar\omega_{\bm{k}}b^{\dagger}_{\bm{k}}b_{\bm{k}}-h_{\rm
ac}^{+}(t)b^{\dagger}_{{\bm{k}}=\bm{0}}-h_{\rm ac}^{-}(t)b_{{\bm{k}}=\bm{0}},$
(2)

where $b^{\dagger}_{\bm{k}}$ and $b_{\bm{k}}$ denote the creation and
annihilation operators of magnons with the wave vector ${\bm{k}}$,
respectively. We assume the parabolic dispersion
$\hbar\omega_{\bm{k}}=Dk^{2}-\hbar\gamma H$, where $\gamma(<0)$ is the
electron gyromagnetic ratio. The coupling between the microwave radiation and
the magnons is given by $h_{\rm ac}^{\pm}(t)=\hbar\gamma h_{\rm
ac}\sqrt{SN/2}e^{\mp i\omega t}$, where $h_{\rm ac}$ and $\omega$ are the
amplitude and the frequency of the microwave radiation, respectively, and $S$
is the magnitude of the localized spin in the FI. The Hamiltonian given above
is derived from a ferromagnetic Heisenberg model using the Holstein-Primakoff
transformation and the spin-wave approximations
$S^{Z}_{\bm{k}}=S-b^{\dagger}_{\bm{k}}b_{\bm{k}}$,
$S_{{\bm{k}}}^{+}=\sqrt{2S}b_{{\bm{k}}}$, and
$S_{-{\bm{k}}}^{-}=\sqrt{2S}b_{{\bm{k}}}^{\dagger}$, where $\bm{S}_{\bm{k}}$
represents the Fourier transform of the localized spin in the FI. Note that
the precession axis for the localized spin is fixed along the $Z$-axis (see
Fig. 1 (c)).

The second term $H_{\rm SC}$ describes the two-dimensional bulk SCs as
follows:

$\displaystyle H_{\rm
SC}=\frac{1}{2}\sum_{{\bm{k}}}\bm{c}_{{\bm{k}}}^{\dagger}H_{\rm
BdG}\bm{c}_{{\bm{k}}},$ (3)

where the four-component notations
$\bm{c}_{{\bm{k}}}^{\dagger}=(c_{{\bm{k}}\uparrow}^{\dagger},c_{{\bm{k}}\downarrow}^{\dagger},c_{-{\bm{k}}\uparrow},c_{-{\bm{k}}\downarrow})$
and
$\bm{c}_{{\bm{k}}}=(c_{{\bm{k}}\uparrow},c_{{\bm{k}}\downarrow},c_{-{\bm{k}}\uparrow}^{\dagger},c_{-{\bm{k}}\downarrow}^{\dagger})^{\rm
T}$ are used. Here, $c_{{\bm{k}}\sigma}^{\dagger}$ and $c_{{\bm{k}}\sigma}$
denote creation and annihilation operators, respectively, of electrons with
the wave vector ${\bm{k}}=(k_{x},k_{y})$ and the $z$-component of the spin
$\sigma=\uparrow,\downarrow$. The Bogoliubov-de Gennes Hamiltonian $H_{\rm
BdG}$ is a $4\times 4$ matrix given by

$\displaystyle H_{\rm
BdG}=\begin{pmatrix}\xi_{{\bm{k}}}\sigma_{0}&\Delta({\bm{k}})\\\
-\Delta^{\ast}(-{\bm{k}})&-\xi_{{\bm{k}}}\sigma_{0}\end{pmatrix},$ (4)

where $\xi_{\bm{k}}$ represents the energy of the electrons as measured from
their chemical potential, $\sigma_{0}$ is a $2\times 2$ unit matrix, and the
pairing potential $\Delta({\bm{k}})$ is also a $2\times 2$ matrix. We consider
three pairing potential types, including the spin-singlet $s$-wave pairing
$\Delta({\bm{k}})=\Delta_{0}i\sigma^{y}$ and two spin-triplet $p$-wave
pairings $\Delta({\bm{k}})=(\bm{d}_{\bm{k}}\cdot\bm{\sigma})i\sigma^{y}$,
where their $d$-vectors are given by:

$\displaystyle\bm{d}_{{\bm{k}}}=\begin{cases}\Delta_{0}(0,0,e^{i\phi_{\bm{k}}})&:{\rm
Chiral}{~{}}p{\rm-wave}\\\
\Delta_{0}(-\sin\phi_{\bm{k}},\cos\phi_{\bm{k}},0)&:{\rm Helical}{~{}}p{\rm-
wave}\end{cases}$ (5)

where $\phi_{\bm{k}}=\arctan(k_{y}/k_{x})$ is an azimuth angle. The
phenomenological form of the gap function is assumed here, where
$\Delta_{0}(T)=1.76k_{\rm B}T_{c}\tanh\left(1.74\sqrt{T_{c}/T-1}\right)$.

The third term $H_{\rm ex}$ represents the proximity exchange coupling that
occurs at the interface, which describes the spin transfer between the SC and
the FI Ohnuma  _et al._ (2014); Kato  _et al._ (2019):

$\displaystyle H_{\rm
ex}=\sum_{{\bm{q}},{\bm{k}}}\left(J_{{\bm{q}},{\bm{k}}}\sigma^{+}_{\bm{q}}S^{-}_{\bm{k}}+{\rm
h.c.}\right),$ (6)

where $J_{{\bm{q}},{\bm{k}}}$ is the matrix element for the spin transfer
processes, and $\sigma^{\pm}_{{\bm{q}}}=\sigma^{X}_{{\bm{q}}}\pm
i\sigma^{Y}_{{\bm{q}}}$ represent the spin-flip operators for the electron
spins in the SCs. The relationships between the spin operators in the
($x,y,z$) coordinates and the ($X,Y,Z$) coordinates are given by
$\sigma^{X}_{{\bm{q}}}=\cos\theta\sigma^{x}_{{\bm{q}}}-\sin\theta\sigma^{z}_{{\bm{q}}}$,
$\sigma^{Y}_{{\bm{q}}}=\sigma^{y}_{{\bm{q}}}$, and
$\sigma^{Z}_{{\bm{q}}}=\sin\theta\sigma^{x}_{{\bm{q}}}+\cos\theta\sigma^{z}_{{\bm{q}}}$,
where the angle $\theta$ varies from $0$ to $\pi/2$.

Modulation of FMR.—The FMR modulation can be read from the retarded component
of the magnon Green’s function Ohnuma  _et al._ (2014), which is given by:

$\displaystyle
G_{\bm{k}}^{R}(\omega)=\frac{2S/\hbar}{\omega-\omega_{\bm{k}}+i\alpha\omega-(2S/\hbar)\Sigma^{R}_{\bm{k}}(\omega)},$
(7)

where the Gilbert damping constant $\alpha$ is introduced phenomenologically
Kasuya and LeCraw (1961); Cherepanov  _et al._ (1993); Jin  _et al._ (2019).
In the second-order perturbation calculation with respect to the matrix
element $J_{{\bm{q}},{\bm{k}}}$, the self-energy caused by proximity exchange
coupling is given by:

$\displaystyle\Sigma^{R}_{{\bm{k}}}(\omega)=-\sum_{{\bm{q}}}|J_{{\bm{q}},{\bm{k}}}|^{2}\chi_{{\bm{q}}}^{+-}(\omega),$
(8)

where the dynamic spin susceptibility of the SCs is

$\displaystyle\chi_{{\bm{q}}}^{+-}(\omega):=\int
dte^{i(\omega+i0)t}\frac{i}{\hbar}\theta(t)\langle[\sigma_{{\bm{q}}}^{+}(t),\sigma_{-{\bm{q}}}^{-}(0)]\rangle.$
(9)

Then, by solving the equation $\omega-\omega_{{\bm{k}}=\bm{0}}-(2S/\hbar){\rm
Re}\Sigma^{R}_{{\bm{k}}=\bm{0}}(\omega)=0$ at a fixed microwave frequency
$\omega$, we obtain the magnetic field at which the FMR occurs. Consequently,
the frequency shift is given by:

$\displaystyle\delta
H=-\frac{2S}{\gamma\hbar}\sum_{{\bm{q}}}|J_{{\bm{q}},{\bm{k}}=\bm{0}}|^{2}{\rm
Re}\chi^{+-}_{{\bm{q}}}(\omega).$ (10)

Similarly, the modulation of the Gilbert damping is given by:

$\displaystyle\delta\alpha=\frac{2S}{\hbar\omega}\sum_{{\bm{q}}}|J_{{\bm{q}},{\bm{k}}=\bm{0}}|^{2}{\rm
Im}\chi^{+-}_{{\bm{q}}}(\omega).$ (11)

The relationship presented above means that the FMR modulation provides
information about both the interface coupling properties and the dynamic spin
susceptibility of the SCs. Using the system’s symmetry, the dynamic spin
susceptibility $\chi_{\bm{q}}^{+-}(\omega)$ can be written as:

$\displaystyle\chi_{\bm{q}}^{+-}(\omega)$
$\displaystyle=\cos^{2}\theta\chi_{\bm{q}}^{xx}(\omega)+\chi_{\bm{q}}^{yy}(\omega)+\sin^{2}\theta\chi_{\bm{q}}^{zz}(\omega),$
(12)

which means that both $\delta H$ and $\delta\alpha$ show a dependence on
$\theta$ when the dynamic spin susceptibility is anisotropic.

To evaluate $\delta H$ and $\delta\alpha$, two models for the matrix elements
of the spin transfer processes $J_{{\bm{q}},{\bm{k}}}$ are considered. In the
first model, the matrix elements are constant, i.e.,
$J_{{\bm{q}},{\bm{k}}=\bm{0}}=J_{1}$. In this model, $\delta H$ and
$\delta\alpha$ are written as: $\delta H=-\frac{2SJ^{2}_{1}}{\gamma\hbar}{\rm
Re}\chi^{+-}_{\rm loc}(\omega)$ and
$\delta\alpha=\frac{2SJ^{2}_{1}}{\hbar\omega}{\rm Im}\chi^{+-}_{\rm
loc}(\omega)$, respectively, where the local spin susceptibility is defined as
$\chi^{+-}_{\rm loc}(\omega):=\sum_{{\bm{q}}}\chi^{+-}_{{\bm{q}}}(\omega)$.
The physical meaning of this model is that the SC/FI interface is rough, and
that there is no conservation of momentum in the spin transfer processes. In
the second model, the matrix element is only finite for $|{\bm{q}}|=0$ and
$J_{{\bm{q}},{\bm{k}}=\bm{0}}=\delta_{{\bm{q}},\bm{0}}J_{2}$. In this model,
$\delta H$ and $\delta\alpha$ are written as: $\delta
H=-\frac{2SJ^{2}_{2}}{\gamma\hbar}{\rm Re}\chi^{+-}_{\rm uni}(\omega)$ and
$\delta\alpha=\frac{2SJ^{2}_{2}}{\hbar\omega}{\rm Im}\chi^{+-}_{\rm
uni}(\omega)$, respectively, where the uniform spin susceptibility is defined
as $\chi^{+-}_{\rm uni}(\omega):=\lim_{|{\bm{q}}|\to
0}\chi^{+-}_{{\bm{q}}}(\omega)$. The physical meaning of this model is that
the SC/FI interface is flat, and that the spin transfer processes conserve
momentum.

In the remainder of the paper, we mainly present the results for the $p$-wave
SCs, and also explain the qualitative differences among the three types of
SCs: the $s$-wave, chiral $p$-wave, and helical $p$-wave SCs. In the
Supplementary Material, we provide the results that are not given in the main
text. For the rough and flat interface models, we calculate $\chi^{+-}_{\rm
loc}(\omega)$ and $\chi^{+-}_{\rm uni}(\omega)$, respectively. In the results
that follow, constant level broadening $\Gamma$ is introduced to describe the
finite quasi-particle lifetime.

Rough SC/FI interface.—The local spin susceptibility $\chi^{+-}_{\rm
loc}(\omega)$ is independent of $\theta$ because of the isotropy of the local
spin susceptibilities, where $\chi^{xx}_{\rm loc}(\omega)=\chi^{yy}_{\rm
loc}(\omega)=\chi^{zz}_{\rm loc}(\omega)$; this means that $\chi^{+-}_{\rm
loc}(\omega)$ is written as $\chi^{+-}_{\rm loc}(\omega)=2\chi^{xx}_{\rm
loc}(\omega)$ for all the SCs considered here. Furthermore, it can be shown
that there are no differences between the chiral and helical $p$-wave SCs.

Figure 2 shows (a) the frequency shift and (b) the modulation of the Gilbert
damping for the chiral and helical $p$-wave SCs as a function of both
frequency and temperature. Figure 2 (a) shows that a resonance peak occurs at
$\hbar\omega=2\Delta_{0}(T)$. This type of resonance peak does not appear in
the $s$-wave SC (see the Supplementary Material). In Fig. 2 (b), two important
properties are shown. First, there is a coherence peak in the low frequency
region, where $\hbar\omega/k_{\rm B}T_{c}\leq 1$, but this coherence peak is
smaller than the coherence peak calculated for the $s$-wave SC (see the
Supplementary Material). Second, $\delta\alpha$ drops abruptly at
$\hbar\omega=2\Delta_{0}(T)$. This is in contrast to the kink structure that
was derived for the $s$-wave SC Kato  _et al._ (2019). Note that the cutoff
energy $E_{c}$ was introduced here to cause the integral for ${\rm
Re}\chi^{+-}_{\rm loc}(\omega)$ to converge. Although ${\rm Re}\chi^{+-}_{\rm
loc}(\omega)$ is approximately proportional to $E_{c}$, the two qualitative
properties that were explained above are independent of $E_{c}$.

The FMR modulation properties of the three SCs are summarized in Table 1. In
the case of the rough SC/FI interface, the pairing symmetry can be detected
from either the absence or the existence of the resonance peak of $\delta H$.
The pairing symmetry can also be detected from the properties of
$\delta\alpha$, the height of the coherence peak, and the structure at
$\hbar\omega=2\Delta_{0}(T)$. When compared with the resonance peak for
$\delta H$, however, these properties are too ambiguous to allow the pairing
symmetry to be distinguished clearly.

Figure 2: Rough interface. (a) Frequency shift and (b) modulation of the Gilbert damping as a function of both frequency $\omega$ and temperature $T$ for the chiral and helical $p$-wave SCs. The terms $\delta H_{1}$ and $\delta\alpha_{1}$ are given by $\delta H_{1}=-\frac{2SJ_{1}^{2}}{\gamma\hbar}4\pi D^{2}_{F}k_{B}T_{c}$ and $\delta\alpha_{1}=8\pi SJ_{1}^{2}D^{2}_{F}$, respectively, where $D_{F}$ is the density of states at the Fermi level. The cutoff energy is set at $E_{c}/k_{\rm B}T_{c}=10$.  Table 1:  Rough SC/FI interface FMR modulation properties.  Pairing symmetry | $s$ | Chiral | Helical
---|---|---|---
Resonance peak of $\delta H$ | – | $\checkmark$ | $\checkmark$
Coherence peak of $\delta\alpha$ | large | small | small
$\delta\alpha$ at $\hbar\omega=2\Delta_{0}(T)$ | kink | jump | jump

Flat SC/FI interface.—In both the chiral and helical $p$-wave SCs, the uniform
spin susceptibility $\chi^{+-}_{\rm uni}(\omega)$ is dependent on $\theta$
because of the anisotropy of the uniform spin susceptibilities, where
$\chi^{xx}_{\rm uni}(\omega)=\chi^{yy}_{\rm uni}(\omega)\neq\chi^{zz}_{\rm
uni}(\omega)$. Using these relationships, we can obtain $\chi^{+-}_{\rm
uni}(\omega)=2\chi^{xx}_{\rm uni}(\omega)+\sin^{2}\theta\left[\chi^{zz}_{\rm
uni}(\omega)-\chi^{xx}_{\rm uni}(\omega)\right]$. We found that the
temperature and frequency dependences of the chiral and helical $p$-wave SCs
at a fixed value of $\theta$ were the same, and thus we only present the
results for the chiral $p$-wave SC here. The results for both the chiral and
helical $p$-wave SCs are shown when we discuss the $\theta$ dependence.

Figure 3 shows the frequency shift and modulation characteristics of the
Gilbert damping for the chiral $p$-wave SC as a function of temperature at
several frequencies. In Fig. 3, we set $\theta=0$. In the low frequency
region, where $\hbar\omega/k_{\rm B}T_{c}\leq 1$, $\delta H$ decreases with
increasing $\omega$ near the transition temperature $T_{c}$. In contrast,
$\delta H$ remains almost independent of $\omega$ near the zero temperature
limit. $\delta\alpha$ decreases monotonically and no coherence peak is
observed. This contrasts with the behavior of the local spin susceptibility.
In the high frequency region, where $\hbar\omega/k_{\rm B}T_{c}\geq 1$, a
resonance peak occurs at $\hbar\omega=2\Delta_{0}(T)$ for both $\delta H$ and
$\delta\alpha$. This behavior also differs from that of the rough interface
model, in which a resonance peak only occurs for $\delta
H\propto\mathrm{Re}\chi^{+-}_{\mathrm{loc}}(\omega)$.

Figure 3: Flat interface. Frequency shift and modulation characteristics of
the Gilbert damping as a function of temperature at several frequencies. The
upper panels (a) and (b) show the behavior in the low frequency region, in
which $\hbar\omega/k_{\rm B}T_{c}\leq 1$, and the lower panels (c) and (d)
show the behavior in the high frequency region, where $\hbar\omega/k_{\rm
B}T_{c}\geq 1$. The terms $\delta H_{2}$ and $\delta\alpha_{2}$ are given by
$\delta H_{2}=-\frac{2SJ_{2}^{2}}{\gamma\hbar}\frac{2D_{F}}{A}$ and
$\delta\alpha_{2}=\frac{2SJ_{2}^{2}}{k_{B}T_{c}}\frac{2D_{F}}{A}$,
respectively, where $A$ is the SC/FI interface area.

Figure 4 shows both $\delta H$ and $\delta\alpha$ for the chiral and helical
$p$-wave SCs at several values of $\theta$. In both cases, we set
$\hbar\omega/k_{\rm B}T_{c}=3.0$ as the typical values at high frequencies. In
the chiral $p$-wave SC, $\delta H$ and $\delta\alpha$ tended to decrease when
$\theta$ increased from $0$ to $\pi/2$. Conversely, in the helical $p$-wave
SC, the qualitative behavior showed the opposite trend. $\delta H$ and
$\delta\alpha$ both tended to increase when $\theta$ increased from $0$ to
$\pi/2$. Therefore, the pairing symmetry can be detected from the $\theta$
dependence data for the FMR modulation.

Figure 4:  Flat interface. Frequency shift and modulation characteristics of
the Gilbert damping as a function of temperature at angles of
$\theta=0,\pi/4,\pi/2$. The upper and lower panels show the characteristics
for the chiral and helical $p$-wave SCs, respectively.

The FMR modulation properties of the three SCs are summarized in Table 2. All
SCs considered here can be distinguished based on two properties: the presence
of their resonance peak and their $\theta$ dependence. For the $s$-wave SC,
$\delta H$ and $\delta\alpha$ show no resonance and no $\theta$ dependence,
while for the chiral and helical $p$-wave SCs, both $\delta H$ and
$\delta\alpha$ exhibit a resonance at $\hbar\omega=2\Delta_{0}(T)$ and a
$\theta$ dependence. In addition, these two $p$-wave SCs can be distinguished
from their $\theta$ dependences of $\delta H$ and $\delta\alpha$, which are
characterized by $\partial_{\theta}(\delta H)$ and
$\partial_{\theta}(\delta\alpha)$, respectively.

Table 2:  Flat FI/SC interface FMR modulation properties.  Pairing symmetry | $s$ | Chiral | Helical
---|---|---|---
Resonance peak of $\delta H$, $\delta\alpha$ | – | $\checkmark$ | $\checkmark$
$\partial_{\theta}(\delta H)$, $\partial_{\theta}(\delta\alpha)$ | $0$ | negative | positive

Spin-triplet current generation.—The relationship between the modulation of
the Gilbert damping discussed above and the spin-triplet current generation
must also be discussed. The enhancement of the Gilbert damping is known to
originate from the spin current generation at the magnetic interface
Tserkovnyak  _et al._ (2002); Ohnuma  _et al._ (2014). In our setup, the
Gilbert damping $\delta\alpha$ is proportional to the generated spin-triplet
current, for which the spin polarization can be controlled by varying the
magnetization direction of the FI.

Discussion.—In this paper, we have considered two models for the interface
spin transfer processes. The first models a rough interface, representing
atomic-scale roughness, while the second models a flat interface. We expect
the results from these two models to be valid in the dirty limit and the clean
limit, respectively. Calculations for intermediate surface roughness levels
and thus more realistic models would be useful when numerous FMR measurements
of a variety of materials have been made. However, this is beyond the scope of
this paper and will be left as a subject for future study.

We should emphasize two important aspects of the FMR method presented here:
the spectroscopic probe method for the $p$-wave SC thin films and the
versatile spin injection method. First, the FMR measurement procedure can
provide a new spin-sensitive measurement method that will complement other
measurement methods to enable a breakthrough in the discovery of spin-triplet
SCs. Second, the FMR method represents a promising way to generate spin-
triplet currents in $p$-wave SC thin films.

Conclusions.—We have investigated the anisotropic superconducting spin
transport at magnetic interfaces composed of a $p$-wave SC and an FI based on
a microscopic model Hamiltonian. The FMR signal in these $p$-wave SC/FI
bilayer systems is modulated via spin transfer at the interface, which
generates spin-triplet currents. We have shown that the pairing symmetry of
the SCs can be extracted from the FMR modulation characteristics. Our approach
provides a unique way to explore anisotropic superconducting spintronics,
which will be useful for application to emerging device technologies.

Note added.— After the submission of this manuscript, we became aware of a
closely related work, where a way to converting spin-triplet currents to
magnon spin currents in SC/FI bilayer systems is discussed Johnsen  _et al._
(2021).

We thank R. Ohshima, M. Shiraishi, H. Chudo, G. Okano, K. Yamanoi, and Y.
Nozaki for helpful discussions. This work was supported by the Priority
Program of the Chinese Academy of Sciences under Grant No. XDB28000000, and by
JSPS KAKENHI under Grants Nos. JP20K03835 and JP20H04635.

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##  I Supplemental Material

In the Supplemental Material, we explain the FMR modulations, which are not
described in the main text.

###  I.1 Rough interface: $s$-wave SC

Figure 5 shows (a) the frequency shift $\delta H$ and (b) the modulation of
the Gilbert damping $\delta\alpha$ for the $s$-wave SC. At a fixed frequency,
$\delta H$ decreases monotonically. There is no resonance peak which occurs in
the $p$-wave SCs. $\delta\alpha$ exhibits a coherence peak in the low
frequency region. The coherence peak in the $s$-wave SC is higher than the
corresponding coherence peak in the $p$-wave SCs.

Figure 5: Rough interface. (a) The frequency shift and (b) the modulation of
Gilbert damping as a function of both frequency $\omega$ and temperature $T$
for the $s$-wave SCs. The terms $\delta H_{1}$ and $\delta\alpha_{1}$ are
given by $\delta H_{1}=-\frac{2SJ_{1}^{2}}{\gamma\hbar}4\pi
D^{2}_{F}k_{B}T_{c}$ and $\delta\alpha_{1}=8\pi SJ_{1}^{2}D^{2}_{F}$, where
$D_{F}$ is the density of states at the Fermi level. The cutoff energy is set
at $E_{c}/k_{\rm B}T_{c}=10$.

###  I.2 Flat interface: $s$-wave and helical $p$-wave SCs

Figures 6 and 7 show the characteristics of the frequency shift $\delta H$ and
the modulation of the Gilbert damping $\delta\alpha$ for the $s$-wave and
helical $p$-wave SCs, respectively. In the $s$-wave SC, the qualitative
properties of $\delta H$ and $\delta\alpha$ are similar to each other. In the
low frequency region, both $\delta H$ and $\delta\alpha$ decrease
monotonically with decreasing temperature. In the high frequency region, both
$\delta H$ and $\delta\alpha$ are almost zero at any temperature. In the
helical $p$-wave SC, the qualitative properties of $\delta H$ and
$\delta\alpha$ are the same as the corresponding properties in the chiral
$p$-wave SC.

Figure 6: Flat interface. Frequency shift and modulation of the Gilbert
damping as a function of temperature at several frequencies for the $s$-wave
SC. Upper panels (a) and (b) show the behavior in the low frequency region,
where $\hbar\omega/k_{\rm B}T_{c}\leq 1$, and the lower panels (c) and (d)
show the corresponding behavior in the high frequency region, where
$\hbar\omega/k_{\rm B}T_{c}\geq 1$. The terms $\delta H_{2}$ and
$\delta\alpha_{2}$ are given by $\delta
H_{2}=-\frac{2SJ_{2}^{2}}{\gamma\hbar}\frac{2D_{F}}{A}$ and
$\delta\alpha_{2}=\frac{2SJ_{2}^{2}}{k_{B}T_{c}}\frac{2D_{F}}{A}$,
respectively, where $A$ is the area of the SC/FI interface.  Figure 7: Flat
interface. Frequency shift and the modulation of the Gilbert damping as a
function of temperature at several frequencies for the helical $p$-wave SC.