# Magnon influence on the superconducting DOS in FI/S bilayers

A.S. Ianovskaia  National Research University Higher School of Economics,
Moscow, 101000 Russia  Moscow Institute of Physics and Technology,
Dolgoprudny, 141700 Russia  A.M. Bobkov  Moscow Institute of Physics and
Technology, Dolgoprudny, 141700 Russia  I.V. Bobkova  Moscow Institute of
Physics and Technology, Dolgoprudny, 141700 Russia  National Research
University Higher School of Economics, Moscow, 101000 Russia

###### Abstract

Heterostuctures superconductor/ferromagnetic insulator (FI/S) are paradigmic
systems for studying mutual influence of superconductivity and magnetism via
proximity effects. In particular, spin-split superconductivity is realized in
such structures. Recent experiments and theories demonstrate a rich variety of
transport phenomena occurring in devices based on such heterostructures that
suggest direct applications in thermoelectricity, low-dissipative spintronics,
radiation detection and sensing. In this work we investigate the influence of
the electron-magnon interaction at the superconductor/ferromagnetic insulator
interface on the spin-split superconductivity. It is predicted that due to the
magnon-mediated electron spin-flip processes the spin-split quasiparticle
branches are partially mixed and reconstructed, and the BCS-like spin-split
shape of the superconducting DOS, which is typical for superconductors in the
effective exchange field, is strongly modified. An odd-frequency
superconducting order parameter admixture to the leading singlet order
parameter is also found. These findings expand the physical picture of spin-
split superconductivity beyond the mean-field description of the ferromagnet
exchange field.

###### pacs:

##  I Introduction

Long ago it was demonstrated that the exchange field of ferromagnetic
insulators (FIs), such as EuS and EuO, can spin-split the excitation spectrum
of an adjacent thin-film superconductor [1, 2, 3]. The spin splitting in the
DOS observed in those experiments resembles the spin splitting created by a
strong in-plane field applied to a thin superconducting film. This discovery
opened up the way for performing spin-polarized tunneling measurements without
the need of applying large magnetic fields. A renewed interest in studying
ferromagnetic/superconductor (F/S) structures came with active development of
superconducting spintronics [4, 5], caloritronics and spin caloritronics [6,
7]. In particular, in F/S structures with spin-split density of states (DOS) a
series of promising phenomena have been studied. Among them are giant
thermoelectric [8, 9, 10, 11, 12, 13, 14, 15, 16], thermospin effects [9, 17,
18], highly efficient thermally-induced domain wall motion [19], spin and heat
valves [20, 21, 22, 23, 24], cooling at the nanoscale [25, 26], low-
temperature thermometry and development of sensitive electron thermometers
[27].

The spin-split DOS in F/S structures has also been explored in the presence of
magnetic inhomogeneities, such as textured ferromagnets and domain walls [28,
29, 30, 24, 31, 32, 33, 34]. Characteristic signatures of equal-spin triplet
pairing were reported [32]. It was shown that the characteristic spatial and
energy dependence of the spin-dependent DOS allows to tomographically extract
the structure of the spin-triplet Cooper pairs [34]. Furthermore, the
influence of the domain structure on the position-averaged superconducting DOS
in FI/S bilayer was studied [24].

Another important direction in the field of F/S hybrid structures is
investigation of interplay between the superconducting state and ferromagnetic
excitations - magnons. A series of interesting results, presumably related to
the influence of the superconductor on the magnon spectrum have been reported.
In particular, it was found that the adjacent superconductor works as a spin
sink strongly influencing Gilbert damping of the magnon modes [35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45] and can result in shifting of k = 0 magnon
frequencies (Kittel mode) [40, 43, 44, 45]. The electromagnetic interaction
between magnons in ferromagnets and superconductors also results in appearance
of magnon-fluxon excitations [46] and efficient gating of magnons [47].
Further it was reported that the magnetic proximity effect in thin film F/S
hybrids results in appearing of magnon-cooparons, which are composed of a
magnon in F and an accompanying cloud of spinful triplet pairs in S [48].

Some aspects of back influence of magnons on superconducting state have
already been investigated. For example, a possible realization of the magnon-
mediated superconductivity in F/S hybrids has been proposed [49, 50, 51, 52,
53, 54]. At the same time, the influence of magnons via the magnetic proximity
effect on the superconducting DOS practically has not yet been studied,
although the electron-magnon interaction and influence of this interaction on
the DOS in ferromagnetic metals have been investigated long ago [55, 56]. Here
we consider how the effects of electron-magnon interactions in FI/S thin-film
hybrids manifest themselves in the superconducting DOS and quasiparticle
spectra of the superconductor. It is found that the magnon-mediated electron
spin-flip processes cause the interaction and mixing of the spin-split bands
resulting in their reconstruction, which is especially important near the edge
of the superconducting gap. We demonstrate that the classical BCS-like Zeeman-
split shape of the superconducting DOS can be strongly modified due to the
electron-magnon interaction and this modification is temperature-dependent.
The influence of magnons on the temperature dependence of the Zeeman splitting
of the DOS and relevance of our findings to existing and future experiments
are also discussed.

The paper is organized as follows. In Sec. II we describe the system under
consideration and the Green’s functions formalism taking into account magnon
self-energies. In Sec. III the modifications of the quasiparticle spectra in
the superconductor due to the electron-magnon coupling are discussed. In Sec.
IV we study signatures of the electron-magnon interaction in the Zeeman-split
superconducting DOS and their temperature dependence. Our conclusions are
summarized in Sec. V.

##  II System and formalism

We consider a thin-film bilayer as depicted in Fig. 1, in which a
ferromagnetic insulator FI is interfaced with a conventional spin-singlet
s-wave superconductor S. The thickness of the S layer $d_{S}$ is assumed to be
small as compared to the superconducting coherence length $\xi_{S}$. In this
case the S layer can be considered as homogeneous along the normal to the
interface plane. The FI layer in its ground state is magnetized in-plane,
along the $z$-direction.

Figure 1: Sketch of the FI/S thin-film bilayer.

The Hamiltonian of the system takes the form:

$\hat{H}=\hat{H}_{S}+\hat{H}_{FI}+\hat{H}_{ex},$ (1)

where $\hat{H}_{S}$ is the standard mean-field BCS Hamiltonian describing
electrons in the superconducting film:

$\begin{gathered}\hat{H}_{S}=\sum\limits_{\bm{k}\sigma}\xi_{\bm{k}}c_{\bm{k}\sigma}^{{\dagger}}c_{\bm{k}\sigma}-\sum\limits_{\bm{k}}\Delta
c_{\bm{k}\uparrow}^{{\dagger}}c_{-\bm{k}\downarrow}^{{\dagger}}-\sum\limits_{\bm{k}}\Delta^{*}c_{-\bm{k}\downarrow}c_{\bm{k}\uparrow}.\end{gathered}$
(2)

$\xi_{\bm{k}}=k^{2}/2m-\mu$ is the normal state kinetic energy of the
electrons in the S layer, counted from the chemical potential of the
superconductor $\mu$. $\Delta$ is the superconducting order parameter in S,
which assumed to be of conventional isotropic $s$-wave type.
$c_{\bm{k}\sigma}^{+}$ and $c_{\bm{k}\sigma}$ are creation and annihilation
operators of electrons with the wave vector $\bm{k}$ and spin $\sigma$.

$\hat{H}_{FI}$ describes magnons in the FI. Assuming easy-axis magnetic
anisotropy in the FI it can be written as

$\begin{gathered}\hat{H}_{FI}=\sum\limits_{\bm{q}}(\omega_{0}+D\bm{q}^{2})b_{\bm{q}}^{{\dagger}}b_{\bm{q}},\end{gathered}$
(3)

where $b_{\bm{q}}^{+}$ and $b_{\bm{q}}$ are creation and annihilation
operators of magnons in FI with wave vector $\bm{q}$,
$\omega_{0}=|\gamma|(\mu_{0}H_{0}+2K_{a}/M_{s})$ is the magnonic frequency at
$q=0$, $D$ is the magnon stiffness constant, $\gamma$ is the typically
negative gyromagnetic ratio, $M_{s}$ is the saturation magnetization,
$\mu_{0}$ is the permeability of free space, $K_{a}$ is the easy-axis
anisotropy constant and $H_{0}$ is the external field (can be equal to zero in
our consideration).

Electronic and magnonic wave vectors $\bm{k}$ and $\bm{q}$ are assumed to be
two-dimensional (2D), that is the electrons and magnons can only propagate in
plane of the FI/S interface. The wave functions along the $y$-direction,
perpendicular to the interface, are assumed to be quantized. For simplicity,
in the formulas we leave only one transverse magnon mode. In fact, we have
checked that different modes give quantitatively different, but qualitatively
the same contributions to considered self-energies. Their effect can be
accounted for by multiplying our results for the self-energy corrections by an
effective number of working transverse modes (see below).

$\hat{H}_{ex}$ accounts for the exchange interaction between S and FI:

$\begin{gathered}\hat{H}_{ex}=-J\int
d^{2}\bm{\rho}\bm{S}_{FI}(\bm{\rho})\bm{s}_{e}(\bm{\rho}),\end{gathered}$ (4)

where $\bm{\rho}$ is a two-dimensional radius-vector at the interface plane,
$\bm{S}_{FI}$ and $\bm{s}_{e}$ are the spin density operators in the FI and S,
respectively. $J$ is the interface exchange constant. By performing the
Holstein-Primakoff transformation to the second order in the magnonic
operators in Eq. (4) one obtains

$\begin{gathered}\hat{H}_{ex}=\hat{H}_{1}+\hat{H}_{2}+\hat{H}_{3},\end{gathered}$
(5)

with

$\begin{gathered}\hat{H}_{1}=\sum\limits_{\bm{k},\bm{k}^{\prime}}U_{\bm{k},\bm{k}^{\prime}}(c_{\bm{k},\uparrow}^{{\dagger}}c_{\bm{k}^{\prime},\uparrow}-c_{\bm{k},\downarrow}^{{\dagger}}c_{\bm{k}^{\prime},\downarrow}),\\\
U_{\bm{k},\bm{k}^{\prime}}=\frac{JM_{s}}{2|\gamma|}\int
d^{2}\rho\Psi_{\bm{k}}^{*}(\bm{\rho})\Psi_{\bm{k}^{\prime}}(\bm{\rho}),\end{gathered}$
(6)
$\displaystyle\hat{H}_{2}=\sum\limits_{\bm{k},\bm{k}^{\prime},\bm{q},\bm{q}^{\prime}}T_{\bm{k},\bm{k}^{\prime},\bm{q},\bm{q}^{\prime}}b_{\bm{q}}^{{\dagger}}b_{\bm{q}^{\prime}}(c_{\bm{k},\uparrow}^{{\dagger}}c_{\bm{k}^{\prime},\uparrow}-c_{\bm{k},\downarrow}^{{\dagger}}c_{\bm{k}^{\prime},\downarrow}),$
$\displaystyle
T_{\bm{k},\bm{k}^{\prime},\bm{q},\bm{q}^{\prime}}=-\frac{J}{2}\int
d^{2}\rho\Psi_{\bm{k}}^{*}(\bm{\rho})\Psi_{\bm{k}^{\prime}}(\bm{\rho})\phi_{\bm{q}}^{*}(\bm{\rho})\phi_{\bm{q}^{\prime}}(\bm{\rho}),$
(7)
$\displaystyle\hat{H}_{3}=\sum\limits_{\bm{k},\bm{k}^{\prime},\bm{q}}V_{\bm{k},\bm{k}^{\prime},\bm{q}}(b_{\bm{q}}c_{\bm{k},\uparrow}^{{\dagger}}c_{\bm{k}^{\prime},\downarrow}+b_{\bm{q}}^{{\dagger}}c_{\bm{k}^{\prime},\downarrow}^{{\dagger}}c_{\bm{k},\uparrow}),$
$\displaystyle
V_{\bm{k},\bm{k}^{\prime},\bm{q}}=J\sqrt{\frac{M_{s}}{2|\gamma|}}\int
d^{2}\rho\Psi_{\bm{k}}^{*}(\bm{\rho})\Psi_{\bm{k}^{\prime}}(\bm{\rho})\phi_{\bm{q}}(\bm{\rho}),$
(8)

where $\hat{H}_{1}$ describes a spin-splitting of the electronic energy
spectrum in S in the mean-field approximation. The second term $\hat{H}_{2}$
represents the Ising-term, which physically accounts for the renormalization
of the spin-splitting by magnonic contribution. Since the processes of the
spin transfer between electrons and magnons are of primary importance for our
consideration, when calculating the electronic Green’s function we simplify
this term by substituting the magnon operator $b_{\bm{q}}^{\dagger}b_{\bm{q}}$
by its averaged value $\langle
b_{\bm{q}}^{\dagger}b_{\bm{q}}\rangle=n_{\bm{q}}\delta_{\bm{q}\bm{q}^{\prime}}$,
where $n_{\bm{q}}$ is the density of magnons with wave vector $\bm{q}$. The
third term $\hat{H}_{3}$ transfers spin between electron and magnon operators
and will turn out to be the most significant for effects under consideration.

If we choose the wave functions of electrons $\Psi_{\bm{k}}(\bm{\rho})$ and
magnons $\phi_{\bm{q}}(\bm{\rho})$ at the interface in the form of plane waves
propagating along the interface, that is
$\Psi_{\bm{k}}(\bm{\rho})=(1/\sqrt{d_{S}})e^{i\bm{k}\bm{\rho}}$ and
$\phi_{\bm{q}}(\bm{\rho})=(1/\sqrt{d_{FI}})e^{i\bm{q}\bm{\rho}}$, then
$\hat{H}_{ex}$ can be simplified:

$\begin{gathered}\hat{H}_{ex}=\tilde{U}\sum\limits_{k}(c_{k,\uparrow}^{{\dagger}}c_{k,\uparrow}-c_{k,\downarrow}^{{\dagger}}c_{k,\downarrow})+\\\
V\sum\limits_{k,q}(b_{q}c_{k,\uparrow}^{{\dagger}}c_{k-q,\downarrow}+b_{q}^{{\dagger}}c_{k-q,\downarrow}^{{\dagger}}c_{k,\uparrow}),\end{gathered}$
(9)

where $\tilde{U}=-J(M_{s}-N_{m}|\gamma|)/(2|\gamma|d_{S})$ is the averaged
spin-splitting field in the superconductor renormalized by the magnon density
$N_{m}$, and $V=J\sqrt{M_{s}/2|\gamma|d_{FI}A}(1/d_{S})$ is the electron-
magnon coupling constant, where $A$ is the area of the FI/S interface.

Introducing the following Nambu-spinor
$\check{\Psi}_{\bm{k}}=(c_{\bm{k}\uparrow},c_{\bm{k}\downarrow},-c_{-\bm{k}\downarrow}^{\dagger},c_{-\bm{k}\uparrow}^{\dagger})^{T}$,
we define the Gor’kov Green’s function in the Matsubara representation
$\check{G}_{\bm{k}}(\tau)=-\langle
T_{\tau}\check{\Psi}_{\bm{k}}\check{\Psi}_{\bm{k}}^{\dagger}\rangle$, where
$\langle T_{\tau}...\rangle$ means imaginary time-ordered thermal averaging.
Turning to the Matsubara frequency representation the Green’s function obeys
the following equation:

$\displaystyle(i\omega-\xi_{k}\tau_{z}-\tilde{U}\sigma_{z}-\Delta\tau_{x}-\check{\Sigma}_{m})\check{{G}}_{\bm{k}}(\omega)=1,$
(10)

where $\omega$ is the fermionic Matsubara frequency, $\sigma_{i}$ and
$\tau_{i}$ ($i=x,y,z$) are Pauli matrices in spin and particle-hole spaces,
respectively. $\check{\Sigma}_{m}$ is the magnonic self-energy, which
describes corrections to the electronic Green’s function due to the electron-
magnon interaction and in the framework of the self-consistent Born
approximation takes the form:

$\displaystyle\check{\Sigma}_{m}=-V^{2}T\sum\limits_{\bm{q},\Omega}B_{\bm{q}}(\Omega)\left\\{\sigma_{+}\check{G}_{\bm{k}-\bm{q}}(\omega-\Omega)\sigma_{-}+\right.$
$\displaystyle\left.\sigma_{-}\check{G}_{\bm{k}+\bm{q}}(\omega+\Omega)\sigma_{+}\right\\},$
(11)

where $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})$, $\Omega$ is the bosonic
Matsubara frequency and
$B_{\bm{q}}(\Omega)=[i\Omega-(\omega_{0}+Dq^{2})]^{-1}$ is the magnonic
Green’s function. From the spin structure of Eq. (11) it is seen that
$\check{\Sigma}_{m}$ is diagonal in spin space. For this reason the electronic
Green’s function, which is given by the solution of Eq. (10) is also diagonal
matrix in spin space and Eq. (10) can be written for the both spin subbands
separately:

$\displaystyle(i\omega-\xi_{k}\tau_{z}-\sigma\tilde{U}-\Delta\tau_{x}-\hat{\Sigma}_{m,\sigma})\hat{{G}}_{\bm{k},\sigma}(\omega)=1,$
(12)

where $\hat{G}_{\bm{k},\sigma}$ is $2\times 2$ matrix in the particle-hole
space corresponding to the electron spin $\sigma=\uparrow,\downarrow$.
$\hat{\Sigma}_{m,\sigma}$ is also $2\times 2$ matrix in the particle-hole
space representing the magnonic self-energy for the given spin subband
$\sigma$:

$\displaystyle\hat{\Sigma}_{m,\sigma}=-V^{2}T\sum\limits_{\bm{q},\Omega}B_{\bm{q}}(\Omega)\hat{G}_{\bm{k}-\sigma\bm{q},\bar{\sigma}}(\omega-\sigma\Omega).$
(13)

As a factor in the expressions $\sigma$ means $\pm 1$ for the spin-up (spin-
down) subbands, and $\bar{\sigma}$ means the opposite spin subband. The
dimensionless coupling constant quantifying the strength of the electron-
magnon coupling is $K=V^{2}A/4\pi\hbar v_{F}\sqrt{D\Delta}$. Our numerical
estimates made for the parameters corresponding to EuS/Al or YIG/Nb structures
suggest that $K$ should be rather small, $K\ll 1$, for the detailed discussion
of the numerical estimates see Sec. IV. The smallness of the electron-magnon
coupling constant allows us to use non self-consistent Born approximation when
calculating magnon self-energy. That is, we substitute
$\hat{G}_{\bm{k}-\sigma\bm{q},\bar{\sigma}}$ by the bare superconducting
Green’s function obtained without taking into account the magnon self-energy
$\hat{G}_{\bm{k}-\sigma\bm{q},\bar{\sigma}}^{(0)}$ in Eq. (13). Then the
explicit solution of Eq. (12) takes the form:

$\begin{gathered}\hat{{G}}_{\bm{k},\sigma}(\omega)=\frac{i\widetilde{\omega}_{\bm{k},\sigma}+\widetilde{\xi}_{\bm{k},\sigma}\tau_{z}+\widetilde{\Delta}_{\bm{k},\sigma}\tau_{x}}{(i\widetilde{\omega}_{\bm{k},\sigma})^{2}-(\widetilde{\xi}_{\bm{k},\sigma})^{2}-(\widetilde{\Delta}_{\bm{k},\sigma})^{2}}.\end{gathered}$
(14)

where all the quantities marked by $\tilde{~}{}$ are renormalized by the
electron-magnon interaction as follows:

$\begin{gathered}\widetilde{\Delta}_{\bm{k},\sigma}(\omega)=\Delta+\delta\Delta_{\bm{k},\sigma}(\omega)=\Delta-\\\
-V^{2}T\sum\limits_{\bm{q},\Omega}B_{\bm{q}}(\Omega)\frac{\Delta}{(i\omega-i\sigma\Omega+\widetilde{U}\sigma)^{2}-\xi^{2}_{\bm{k}-\sigma\bm{q}}-|\Delta|^{2}},\end{gathered}$
(15)
$\begin{gathered}\widetilde{\xi}_{\bm{k},\sigma}(\omega)=\xi_{\bm{k}}+\delta\xi_{\bm{k},\sigma}(\omega)=\xi_{\bm{k}}-\\\
-V^{2}T\sum\limits_{\bm{q},\Omega}B_{\bm{q}}(\Omega)\frac{\xi_{\bm{k}-\sigma\bm{q}}}{(i\omega-i\sigma\Omega+\widetilde{U}\sigma)^{2}-\xi^{2}_{\bm{k}-\sigma\bm{q}}-|\Delta|^{2}},\end{gathered}$
(16)
$\begin{gathered}\widetilde{\varepsilon}_{\bm{k},\sigma}(\omega)=i\omega-\widetilde{U}\sigma+\delta\varepsilon_{\bm{k},\sigma}(\omega)=i\omega-\widetilde{U}\sigma+\\\
+V^{2}T\sum\limits_{\bm{q},\Omega}B_{\bm{q}}(\Omega)\frac{i\omega-i\sigma\Omega+\widetilde{U}\sigma}{(i\omega-i\sigma\Omega+\widetilde{U}\sigma)^{2}-\xi^{2}_{\bm{k}-\sigma\bm{q}}-|\Delta|^{2}}.\end{gathered}$
(17)

Figure 2: (a) ${\rm Re}[\delta\Delta_{\bm{k},\uparrow}]/K\Delta$ and (b) ${\rm
Im}[\delta\Delta_{\bm{k},\uparrow}]/K\Delta$ as functions of $\xi$ and
quasiparticle energy $\varepsilon$. $K$ is the dimensionless quantity
characterising the electron-magnon coupling strength, see text for its exact
definition. Here and below throughout the calculations we take
$\omega_{0}=0.03\Delta$, $\delta=0.01\Delta$,
$D_{m}/\xi_{S}^{2}=10^{-4}\Delta$.

For the problem under consideration all the in-plane directions of $\bm{k}$
are equivalent. For this reason the magnonic corrections only depend on the
absolute value $k$ of the wave vector. Further in order to study the
quasiparticle spectra and density of states we turn from Matsubara frequencies
to the real energies in the Green’s functions $i\omega\to\varepsilon+i\delta$,
where $\delta$ is an infinitesimal positive number. The magnonic corrections
for spin-up electrons $\delta\Delta_{\bm{k},\uparrow}$,
$\delta\xi_{\bm{k},\uparrow}$ and $\delta\varepsilon_{\bm{k},\uparrow}$ are
presented in Figs. 2-4 as functions of the quasiparticle energy $\varepsilon$
and $\xi_{\bm{k}}\equiv\xi$, which after linearization in the vicinity of the
Fermi surface takes the form
$\xi_{\bm{k}}\approx\bm{v}_{F}(\bm{k}-\bm{k}_{F})$.

Figure 3: (a) ${\rm Re}[\delta\xi_{\bm{k},\uparrow}]/K\Delta$ and (b) ${\rm
Im}[\delta\xi_{\bm{k},\uparrow}]/K\Delta$ as functions of $\xi$ and
$\varepsilon$. Figure 4: (a) ${\rm
Re}[\delta\varepsilon_{\bm{k},\uparrow}]/K\Delta$ and (b) ${\rm
Im}[\delta\varepsilon_{\bm{k},\uparrow}]/K\Delta$ as functions of $\xi$ and
$\varepsilon$.

The key features of the corrections, which can be see in the presented plots
are:

(i) The dependence of the corrections on $\xi$ is very weak. The reason is
that the most important range of the magnonic wave numbers contributing to the
corrections is $q\lesssim 1/\xi_{S}$, where $\xi_{S}=v_{F}/\Delta$ is the
superconducting coherence length. Then taking parameters of the magnon
spectrum corresponding to YIG $\omega_{0,YIG}\sim 10^{-1}\Delta$,
$D_{YIG}\approx 5*10^{-40}J*m^{2}$ or EuS $\omega_{0,EuS}\sim 10^{-2}\Delta$,
$D_{EuS}\approx 3*10^{-42}J*m^{2}$, we obtain that $Dq^{2}\ll\omega_{0}$ to
very good accuracy for all reasonable parameters. Consequently, one can
disregard $Dq^{2}$ with respect to $\omega_{0}$ in the magnonic Green’s
function $B_{\bm{q}}$ and after linearization of
$\xi_{\bm{k}-\sigma\bm{q}}\approx\bm{v}_{F}(\bm{k}-\sigma\bm{q}-\bm{k}_{F})$
in the vicinity of the Fermi surface we see that the dependence on $\bm{k}$
drops from Eqs. (15)-(17).

(ii) The correction to the normal state electron dispersion $\delta\xi$ is
small with respect to all other corrections and is neglected below.

(iii) The important corrections $\delta\Delta$ and $\delta\varepsilon$ have
peaks at the energies corresponding to the superconducting coherence peaks of
the opposite spin subbands. While the coherence peaks for the spin-up subband
are located at $\varepsilon=\pm\Delta+\tilde{U}$, the peaks of the corrections
are at $\varepsilon=\pm\Delta-\tilde{U}$. It is an obvious consequence of the
process of electron spin flip accompanied by emission or absorption of a
magnon.

(iv) Correction $\delta\Delta$ represents an effective contribution to the
superconducting order parameter induced from the pure singlet pairing $\Delta$
via the electron-magnon interaction. It depends on the Matsubara frequency and
contains both singlet and triplet components. As can be seen from Eq. (15),
the correction obeys the condition $\delta\Delta_{\uparrow}(\omega)$ =
$\delta\Delta_{\downarrow}(-\omega)$. It means that the triplet component
$\delta\Delta_{t}(\omega)=\delta\Delta_{\uparrow}(\omega)-\delta\Delta_{\downarrow}(\omega)=-\delta\Delta_{t}(-\omega)$
works as an effective odd-frequency superconducting order parameter. This
situation is rather unusual because typically in F/S hybrid systems we
encounter an odd-frequency anomalous Green’s function, but at the same time
the order parameter is still even frequency in the framework of the
conventional BCS weak coupling theory.

##  III Quasiparticle spectra

Now we turn to discussion of how quasiparticle spectra in the S layer are
modified by the electron-magnon interaction.

Figure 5: (a) Electronic spectral function in the S layer for the both spins
$A_{\uparrow,\downarrow}(\varepsilon,\xi)$ at $K=0.01$. Dashed lines represent
the spin-up and spin-down branches of the spectrum without taking into account
the electron-magnon interaction, that is at $K=0$. $T=0.1\Delta$,
$\tilde{U}=0.5\Delta$. (b) The same on a logarithmic scale. (c) Only spin-up
electronic spectral function in the S layer $A_{\uparrow}(\varepsilon,\xi)$ at
$K=0.01$. The spin-down spectral function can be found as
$A_{\downarrow}(\varepsilon,\xi)=A_{\uparrow}(-\varepsilon,-\xi)$. (d) The
same as in (c) on a logarithmic scale.

In Fig. 5(a) we present the spectral functions for the both spins in the S
layer calculated from the Green’s function (14) according to the relation

$\displaystyle A_{\sigma}(\varepsilon,\bm{k})=-\frac{1}{\pi}{\rm
Tr}\left\\{\frac{1+\tau_{z}}{2}{\rm
Im}[\hat{G}_{\bm{k},\sigma}^{R}(\varepsilon)]\right\\}.$ (18)

Figure 6: The same as in Figs. 5(a)-(b), but for lower temperature
$T=0.01\Delta$. The other parameters are the same. The reconstruction and
broadening of the spectra are much less pronounced.

The spectral function is isotropic in momentum space and for this reason we
plot it as a function of $\xi_{\bm{k}}\equiv\xi$. The electron-like and hole-
like quasiparticle branches are clearly seen at positive and negative
energies, respectively. Black dashed lines represent the quasiparticle spectra
in the absence of the electron-magnon interaction. The electron-magnon
interaction leads to the following main modifications of the quasiparticle
spectra:

(i) The Zeeman splitting of spin-up and spin-down quasiparticle branches is
reduced due to the magnon-mediated interaction between quasiparticles with
opposite spins.

(ii) For positive energy branches, corresponding to electron-like
quasiparticles, the lifetime of spin-up quasiparticles and quasiparticles at
the upper part of the spin-down branch is considerably suppressed, what is
seen as a broadening of the corresponding branches. For negative energies,
corresponding to hole-like quasiparticles, the situation is symmetric if we
interchange spins. The broadening of the spin-down branch only occurs in the
energy region, where the spin-up branch also exists. The physical reason is
that the spin-flip processes providing the broadening are nearly horizontal
due to the fact that $\omega_{0}+Dq^{2}\ll\Delta$, that is the magnon energies
are small as compared $\Delta$ in the whole range of $\xi$, considered in Fig.
(5). The lower (upper) part of the spin-down (up) positive (negative) energy
branch is not broadened because there are no available states for the opposite
spin quasiparticles at the appropriate energies and, consequently, the spin-
flip processes are not allowed.

(iii) In Fig. 5(a) we also see a reconstruction of the spin-down spectral
branch in the energy range of the bottom of the spin-up branch. In order to
investigate this effect in more detail we plot the same figure on a
logarithmic scale in Fig. 5(b), what allows to clearly see weak spectral
features. Figs. 5(c) and (d) represent the spectral functions for the spin-up
band on the normal and on the logarithmic scale, respectively. From Figs. 5(b)
and (d) it is seen that due to the electron-magnon interaction in the energy
region of the extremum of the spin-up (down) branch, a nonzero density of
states appears for the opposite spin branch. It looks like a horizontal line
starting from the bottom of the corresponding branch. This line is horizontal
due to the independence of the electron-magnon self-energy corrections (15)
and (17) on $\xi$. This mixing of the spin-up and spin-down bands resulting
from the magnon-mediated spin-flip processes is natural and exists at all
energies, but the spectral weight of the opposite spin branch is too small
except for the regions of the extrema of the bands corresponding to the
coherence peaks of the superconducting DOS. Intersection of the additional
lines with the original spin-down band results in its reconstruction, which
looks like an avoided crossing point.

The results for the spectral function presented and discussed above correspond
to $T=0.1\Delta$. This temperature is higher than the gap in the magnonic
spectrum $\omega_{0}=0.03\Delta$, which we take in our calculations.
Therefore, a large number of thermal magnons are excited at this temperature.
In Fig. 6 the spectral function is demonstrated for lower temperature
$T=0.01\Delta<\omega_{0}$. It is seen that the characteristic signatures of
the magnon-mediated spin-flip processes, that is the mixing, reconstruction
and broadening of the branches are much less pronounced due to the suppression
of the thermally excited magnons at such low temperatures.

##  IV DOS in the presence of magnons

Now we turn to discussion of the local density of states (LDOS) in the S
layer, which is calculated as the momentum integrated spectral function:

$\displaystyle
N(\varepsilon)=\int\frac{d^{2}k}{(2\pi)^{2}}A(\varepsilon,\bm{k}).$ (19)

Fig. 7(a) demonstrates the LDOS in the presence of electron-magnon interaction
(solid line) as compared to the LDOS calculated at $V=0$ (dashed line). The
LDOS at $V=0$, that is calculated assuming mean-field approximation for the
exchange field, takes the conventional BCS-like shape. It manifests Zeeman-
split coherence peaks, and the outer peak is always higher than the inner one.
The electron-magnon interaction inverts the relative ratio of the peak heights
and broadens the outer peaks, while the width of the inner peaks remains
unchanged. The reason is the same as for the broadening of the spectra in Fig.
5: electron spin-flip processes accompanied by a magnon emission or
absorption. The outer coherence peaks in Fig.7(a) correspond to the energy
regions of the bottom (top) of the positive(negative)-energy spin-up(down)
bands. This type of broadening, which only affects outer peaks, differs from
the other physical mechanisms resulting in the broadening of the coherence
peaks, such as the orbital effect of the magnetic field, inelastic scattering
or magnetic impurities, which affect all the peaks [57] and can be roughly
described by the Dynes parameter.

Figure 7: (a) LDOS in the S layer with (solid line, $K=0.01$) and without
(dashed, $K=0$) taking into account electron-magnon interaction. (b) Spin-
resolved LDOS $N_{\uparrow}$ (red) and $N_{\downarrow}$ (blue) for $K=0.01$.
The solid line in panel (a) is obtained by summing red and and curves from
panel (b). $T=0.1\Delta$, $\tilde{U}=0.5\Delta$.

The other important manifestation of the electron-magnon interaction is that
the shape of the LDOS strongly depends on temperature even at very low
temperatures $\sim\omega_{0}\ll\Delta$, in agreement with the discussed above
behavior of the spectral function. The temperature evolution of the LDOS is
presented in Fig. 8. It is seen that the broadening of the outer peak develops
with increasing temperature in the temperature range $\sim\omega_{0}$. It is
clear if we remember that the broadening is caused by the spin-flip processes,
which are mediated by the thermally excited magnons. We do not consider larger
temperatures $T\gg\omega_{0}$ comparable to the critical temperature of the
superconducting film because in this temperature range the temperature
dependence of the superconducting gap comes into play and the correct
consideration of the problem requires solving of the self-consistency equation
for the order parameter.

Now let us discuss numerical estimates of the dimensionless constant
$K=V^{2}A/4\pi\hbar v_{F}\sqrt{D\Delta}$, which controls the strength of the
electron-magnon coupling. Substituting
$V=J\sqrt{M_{s}/2|\gamma|d_{FI}A}(1/d_{S})$ and expressing the interface
exchange coupling constant via the experimentally accessible quantity
$\tilde{U}$ as $|J|=2|\gamma|\tilde{U}d_{S}/M_{s}$ (where to the leading
approximation we neglect magnonic contribution to the magnetization), we
obtain $K=\tilde{U}^{2}(2|\gamma|/M_{s})1/(4\pi\sqrt{D\Delta}v_{F}d_{FI})$ for
one transverse magnon mode. The effective number of working transverse modes
$N_{\perp}\sim d_{FI}/a$, where $a$ is the interatomic distance in the
ferromagnet. According to our estimates for $d_{FI}\approx 10nm$
$N_{\perp}\sim 2\div 5$. One can take the following parameters for YIG/Nb
heterostructures: $\tilde{U}/\Delta=0.5$, $v_{F}=10^{6}m/s$,
$\Delta_{Nb}=2.7*10^{-22}J$, $a=1.2m$, $2|\gamma|/M_{s}=3.3*10^{-27}m^{3}$,
$D=D_{bare,YIG}-\delta D_{YIG}$, where $D_{bare,YIG}=5*10^{-40}J*m^{2}$[58] is
the exchange stiffness of $YIG$ and $\delta D_{YIG}$ is the renormalization of
the stiffness in FI/S bilayers due to formation of magnon-Cooparon
quasiparticles [48]. As it was predicted [48], for the material parameters of
YIG/Nb heterostructures $\delta D_{YIG}$ can be $\sim(0.5\div 1)D_{YIG,bare}$
for $d_{FI}\sim(1\div 0.5)d_{S}$. Therefore, the electron-magnon coupling
constant for YIG/Nb heterostructures can vary in a wide range
$K_{YIG/Nb}\gtrsim 10^{-4}$. The considered here values $K\sim 0.01$ can be
realized in the regime of strong renormalization of the exchange stiffness
constant $D$.

Figure 8: Inner LDOS peak (left panel) and outer LDOS peak (right panel) as
functions of energy. Different curves correspond to different temperatures.
$K=0.01$, $\tilde{U}=0.5\Delta$.

For EuS/Al heterostructures one can take $\tilde{U}/\Delta=0.25$ [24],
$v_{F}=10^{6}m/s$, $\Delta_{Al}=3.5*10^{-23}J$, $a=10^{-10}m$,
$2|\gamma|/M_{s}=3.3*10^{-28}m^{3}$, $D=D_{bare,EuS}$, where
$D_{bare,EuS}=3*10^{-42}J*m^{2}$[59]. The superconducting renormalization of
the stiffness due to formation of magnon-Cooparon quasiparticles is predicted
to be small for the parameters corresponding to EuS/Al heterostructures at
reasonable thicknesses $d_{FI}$ due to smaller values of $\Delta$ and larger
$M_{s}$. Substituting these parameters to the expression for $K$ we come to
the conclusion that for EuS/Al heterostructures $K_{EuS/Al}\sim 10^{-7}\div
10^{-6}$, that is the electron-magnon effects unlikely to be observed in such
structures.

In general, the electron-magnon effects in the LDOS and quasiparticle spectra
should be more pronounced in ultra-thin superconducting films with high
critical temperatures, where large absolute values of the effective exchange
field $\tilde{U}$ can be realized. The smaller values of the exchange
stiffness of the ferromagnet will also enhance the effect. The manifestations
of the electron-magnon coupling become more pronounced at $T\gtrsim\omega_{0}$
and grow with temperature.

Now we discuss the influence of the electron-magnon interaction on the
effective Zeeman splitting, which is defined as the distance between the split
coherence peaks of the LDOS divided by 2. Experimentally, the low-temperature
reduction of the effective Zeeman splitting at $T\ll\Delta$ for EuS/Al
heterostructures has been reported [24]. It was ascribed to the presence of
weakly bound spins at the interface of the EuS/Al. The renormalization of the
effective exchange field in the superconductor by the thermal magnons can also
contribute to this effect. Indeed, the fit of experimentally observed
temperature dependence of the distance between the Zeeman-split coherence
peaks $\Delta V_{peak}(T)$ by
$2|\tilde{U}|=J(M_{s}-N_{m}|\gamma|)/(2|\gamma|d_{S})$ with the magnon density
$N_{m}=(1/Sd_{FI})\sum_{\bm{q}}\left\\{\exp[-(\omega_{0}+Dq^{2})/T]-1\right\\}^{-1}$
and $\omega_{0}\approx 0.03K$ is in reasonable agreement with the experimental
data.

In addition, the broadening of the outer coherence peaks, predicted in this
work, leads to enhancement of the distance between the spin-split coherence
peaks. The broadening becomes stronger with increasing temperature. This
effect leads to an apparent growth of the peaks splitting with temperature
and, therefore, acts opposite to the renormalization of the effective Zeeman
field by magnons. However, our numerical estimates suggest that the
temperature growth is unlikely to be observed, at least for heterostructures,
consisting of the materials discussed above, because the renormalization of
the effective Zeeman field by magnons dominates.

##  V Conclusions

In this work the influence of the electron-magnon interaction at the
superconductor/ferromagnetic insulator interface in thin-film FI/S
heterostructures on the spectrum of quasiparticles and the LDOS in the
superconducting layer is studied. It is predicted that due to the magnon-
mediated electron spin-flip processes the spin-split quasiparticle branches
are partially mixed and reconstructed. The reconstruction is the most
pronounced in the region of the bottom of the energetically unfavorable spin
band because of the enhanced density of the electronic states and existence of
the available states in the opposite-spin band. The BCS-like Zeeman-split
shape of the superconducting DOS, which is typical for superconductors in the
effective exchange field, is strongly modified due to the electron-magnon
interaction. The outer spin-split coherence peaks are broadened, and the inner
peaks remain intact. This type of broadening is a clear signature of the
magnon-mediated spin flips and strongly differs from other mechanisms of the
coherence peaks broadening, which usually influence all peaks. The broadening
grows with temperature due to the thermal excitation of magnons. The described
above features in the electronic DOS are mainly caused by diagonal in the
particle-hole space magnonic contributions to the electron self-energy, that
is by the quasiparticle processes. Besides that we have also found an off-
diagonal in the particle-hole space magnonic contribution to the electronic
self-energy. It mimics an odd-frequency superconducting order parameter
admixture to the leading singlet order parameter. The study of its influence
on the superconducting properties of the system may be an interesting
direction for future research.

##  VI Acknowledgments

We acknowledge the discussions of the exchange interaction hamiltonian with
Akashdeep Kamra. The work was supported by the Russian Science Foundation via
the RSF project No. 22-42-04408.

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