# Magnon-magnon coupling in synthetic ferrimagnets

A. Sud<EMAIL_ADDRESS>WPI Advanced Institute for Materials
Research, Tohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan  K.
Yamamoto  Advanced Science Research Center, Japan Atomic Energy Agency, 2-4
Shirakata, Tokai 319-1195, Japan  K. Z. Suzuki  WPI Advanced Institute for
Materials Research, Tohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan
Advanced Science Research Center, Japan Atomic Energy Agency, 2-4 Shirakata,
Tokai 319-1195, Japan  S. Mizukami  WPI Advanced Institute for Materials
Research, Tohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan  Center
for Science and Innovation in Spintronics, Tohoku University, Sendai,
980-8577, Japan  H. Kurebayashi  WPI Advanced Institute for Materials
Research, Tohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan  London
Centre for Nanotechnology, University College London, London WC1H 0AH, United
Kingdom  Department of Electronic and Electrical Engineering, University
College London, London, WC1E 7JE, United Kingdom

###### Abstract

Magnetic multilayers with interlayer exchange coupling have been widely
studied for both static and dynamic regimes. Their dynamical responses depend
on the exchange coupling strength and magnetic properties of individual
layers. Magnetic resonance spectra in such systems are conveniently discussed
in terms of coupling of acoustic and optical modes. At a certain value of
applied magnetic field, the two modes come close to being degenerate and the
spectral gap indicates the strength of mode hybridisation. In this work, we
theoretically and experimentally study the mode hybridisation of interlayer-
exchange-coupled moments with dissimilar magnetisation and thickness of two
ferromagnetic layers. In agreement with symmetry analysis for eigenmodes, our
low-symmetry multilayers exhibit sizable spectral gaps for all experimental
conditions. The spectra agree well with the predictions from the Landau-
Lifshitz-Gilbert equation at the macrospin limit whose parameters are
independently fixed by static measurements.

††preprint: AIP/123-QED

##  I Introduction

In two magnetic layers separated by a thin nonmagnetic spacer, conduction
electrons in the spacer magnetically couple two spatially separated moments,
via the so-called Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [1, 2, 3].
This interlayer exchange coupling arises from coherent propagation of electron
spin across the spacer layer [4, 5, 6]. Due to the Friedel-like oscillation of
the electron phase, the exchange coupling changes its sign as a function of
the interlayer distance, switching between ferromagnetic and antiferromagnetic
ordering of the two magnetic layers [7, 8, 9, 10]. The antiferromagnetically
ordered states of two identical magnetic layers, often called synthetic
antiferromagnets (SyAFs), have served as a testbed for studying
antiferomagnetism where SyAFs’ relatively weak RKKY exchange coupling,
comparable to the strength of magnetic fields achievable in laboratories,
helps realise experiments otherwise difficult in atomically-ordered,
crystalline antiferromagnets [11, 12, 13]. One such property is the magnetic
resonances in SyAFs whose typical frequency resides within a range of GHz that
is readily accessible by modern microwave techniques [14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27]. In a canted static state in SyAFs under an
applied magnetic field, the low-lying, spatially-uniform resonance modes are
usually called acoustic and optical modes where the precessions of two
exchange-coupled moments are primarily in- and out-of-phase, respectively
[28]. The resonance frequencies of these modes exhibit different magnetic
field dependence, allowing them to come almost degenerate in a certain field
range. Unless some symmetry conditions are satisfied, there is no exact
degeneracy and the spectrum is gapped as a function of magnetic field [29,
23]. In the regime where the two modes have well-defined in- and out-of-phase
characters away from the degenerate range, the gap, defined as the minimum
split with respect to the field between the two resonance frequencies,
represents the coupling strength of the in- and out-of-phase oscillations and
when it is greater than the linewidths, it means that the energy transfer
between them takes place more frequently before the excited state is lost,
termed as the strong coupling regime [30]. A large coupling strength has been
favoured for potential magnonic applications in which such energy transfer
might play a crucial role [31, 30, 32, 33].

Figure 1: (a) In the laboratory frame, we define the $z$ direction normal to
the plane, and $x$ direction such that the static external magnetic field lies
in the $x$-$z$ plane. In the canted regime when applying the field ($\bm{H}$)
in-plane, two sub-lattice moments ($\bm{M}_{A}$ and $\bm{M}_{B}$) reside
within the plane, canted towards $\bm{H}$. For general static states, we
introduce new coordinate axes $\bm{X},\bm{Y},\bm{Z}$ adapted to the two-fold
rotation $\mathcal{C}_{2}$ that brings the unit vector along $\bm{M}_{A}$ to
that along $\bm{H}$. See Eq. (9) for the concrete definition. For $\bm{H}$ in-
plane and identical magnetic layers, $\mathcal{C}_{2}$ combined with
interchanging A and B layers is a symmetry of the system. (b) and (c) When we
apply $\bm{H}$ with the polar angle $\theta\neq 90\degree$ or two magnetic
moments are not identical, $\mathcal{C}_{2}$ followed by the magnetic layer
interchange ceases to be a symmetry. This impacts on the mode coupling as
discussed in this study.

In the most commonly studied setup, however, the in- and out-of-phase
oscillations are seen to remain eigenmodes for any magnetic field value and
become fully degenerate, where symmetry of the system plays a crucial role for
the decoupling [34, 35, 25, 36]. MacNeill et al. present that the coupled
Landau-Lifshitz-Gilbert (LLG) equations due to the interlayer exchange
interaction are symmetric under twofold rotation around the applied field
direction combined with a layer swap, as long as the field is within the film
plane (Fig. 1(a)) [34]. The acoustic and optical modes are odd and even upon
the symmetry operation respectively and therefore unable to hybridise with
each other, leading to the mode degeneracy at a resonance point [14, 28]. This
specific symmetry can be broken in several ways, for example, tilting the
external magnetic field towards out-of-plane direction (Fig. 1(b)) [34, 23,
35]. For general expressions of spin-wave mode frequencies in interlayer
exchange-coupled systems, Layadi presented analytical solutions with a
particular focus on the effect of the biquadratic exchange coupling and in-
plane anisotropy on the spectra for in-plane magnetised cases [37]. While
spectroscopic measurements of interlayer exchange-coupled tri-layers with
different magnetic-layer thicknesses were reported by some groups in the past
[38, 18, 39, 22, 40], there seem no study fully dedicated to quantitative
discussions of the mode hybridisation in such asymmetric interlayer-exchange-
coupled systems.

In this paper, we present our detailed experimental and theoretical study of
the magnon-magnon coupling phenomena in synthetic ferrimagnets where two
magnetically coupled layers are not identical (Fig. 1(c)). We systematically
compare spin-wave spectra measured by broadband ferromagnetic resonance (FMR)
experiments and calculated using magnetic parameters deduced from static
magnetometry. In all cases examined, we find excellent agreement between
experiment and theory, suggesting that the coupled LLG equations at the
macrospin limit are indeed a reliable tool for designing and analysing the
spectral properties of the magnetic multilayers. Our calculations further
reveal dissimilar roles of quadratic and biquadratic exchange interactions for
the size of the gap. Our results help design and control magnetic resonance
spectra in exchange-coupled magnetic moments that can be synthetic
antiferro(ferri)magnets, van der Waals antiferromagnets [34, 41, 42] and
ferromagnetic bi-layers [43, 44, 45].

##  II A macrospin model of synthetic ferrimagnet

For our purposes, a theoretical model that extends the result of Ref. [37] for
arbitrary direction of the static magnetic field is required, which we present
in this section with an emphasis on breaking of the two-fold rotation
symmetry. Let $\bm{M}_{A},\bm{M}_{B}$ be the magnetisations of the two
ferromagnetic layers. We are interested in the situations where the two
magnetic materials are not identical $\left|\bm{M}_{A}\right|\equiv
M_{A}\neq\left|\bm{M}_{B}\right|\equiv M_{B}$, and the two layers have
different thicknesses $d_{A}\neq d_{B}$. The film normal is chosen $z$ axis
and the film is regarded infinitely extended in the $x,y$ directions, as shown
in Fig. 1(a).

The static state of the magentisations corresponds to the minimum of free
energy per unit area $W$. We include the external magnetic field $\bm{H}$,
demagnetising field, and biquadratic as well as the usual quadratic interlayer
exchange interactions:

$\displaystyle W=$ $\displaystyle\
d_{A}\left\\{-\mu_{0}M_{A}\bm{H}\cdot\bm{n}_{A}+\frac{\mu_{0}M_{A}^{2}}{2}\left(n_{A}^{z}\right)^{2}\right\\}$
$\displaystyle+d_{B}\left\\{-\mu_{0}M_{B}\bm{H}\cdot\bm{n}_{B}+\frac{\mu_{0}M_{B}^{2}}{2}\left(n^{z}_{B}\right)^{2}\right\\}$
$\displaystyle+J_{1}\bm{n}_{A}\cdot\bm{n}_{B}+J_{2}\left(\bm{n}_{A}\cdot\bm{n}_{B}\right)^{2}.$
(1)

Here we have normalised the magnetisations
$\bm{n}_{A(B)}=\bm{M}_{A(B)}/M_{A(B)}$, and introduced the phenomenological
exchange energies per unit area $J_{1}$ and $J_{2}$. Without loss of
generality, with the weak crystalline anisotropy being ignored, the magnetic
field can be taken
$\bm{H}=H\left(\hat{\bm{x}}\sin\theta+\hat{\bm{z}}\cos\theta\right)$. We
determine the static state $\bm{n}_{A(B)}^{0}$ by numerical minimisation of
$W$, which is parameterised by

$\bm{n}^{0}_{A(B)}=\begin{pmatrix}\sin\theta_{A(B)}\cos\phi_{A(B)}\\\
\sin\theta_{A(B)}\sin\phi_{A(B)}\\\ \cos\theta_{A(B)}\\\ \end{pmatrix}.$ (2)

If the magnetic field is in-plane $\theta=90\degree$ and $0<2J_{2}<J_{1}$, the
static state undergoes two phase transitions at $H_{\rm sf}$ and $H_{\rm ff}$
as $\left|H\right|$ is increased from zero, where

$\displaystyle H_{\rm sf}=$
$\displaystyle\left|\frac{1}{d_{B}M_{B}}-\frac{1}{d_{A}M_{A}}\right|\frac{J_{1}-2J_{2}}{\mu_{0}},$
(3) $\displaystyle H_{\rm ff}=$
$\displaystyle\left|\frac{1}{d_{B}M_{B}}+\frac{1}{d_{A}M_{A}}\right|\frac{J_{1}+2J_{2}}{\mu_{0}}.$
(4)

Below $H_{\rm sf}$, the static state is antiferromagentic
$\bm{n}^{0}_{B}=-\bm{n}^{0}_{A}$ with $\bm{n}^{0}_{A}\cdot\bm{H}\gtrless 0$
according to $d_{A}M_{A}\gtrless d_{B}M_{B}$. Above $H_{\rm ff}$, the system
is in a forced ferromagnetic state
$\bm{n}^{0}_{A}=\bm{n}^{0}_{B}=\bm{H}/\left|H\right|$. In between lies the
spin-flop, or canted, state where
$H\cos\phi_{A,B}>0,\sin\phi_{A}\sin\phi_{B}<0$.

To calculate the magnetic resonance frequencies, let us introduce the linear
perturbation $\bm{n}_{A(B)}\approx\bm{n}^{0}_{A(B)}+\bm{n}^{1}_{A(B)}$ where
$\bm{n}^{0}_{A(B)}\cdot\bm{n}^{1}_{A(B)}=0$. The Landau-Lifshitz equations
follow from the free energy $W$ through the usual procedure [46]. Although one
can press on using $\bm{n}_{A(B)}^{1}$ as the dynamical variables [35], we
normalise them so as to make them canonical in the sense of Hamiltonian
mechanics [46], which ensures that the resulting eigenvalue problem retains
the correct Bogoliubov form [47]:

$\bm{\delta}_{A}=\sqrt{\frac{Sd_{A}M_{A}}{\hbar\left|\gamma_{A}\right|}}\bm{n}_{A}^{1},\quad\bm{\delta}_{B}=\sqrt{\frac{Sd_{B}M_{B}}{\hbar\left|\gamma_{B}\right|}}\bm{n}_{B}^{1},$
(5)

where $S$ denotes the area of the film, and $\gamma_{A(B)}<0$ are the
gyromagnetic ratios. The linearised equations of motion read

$\displaystyle\bm{n}_{A}^{0}\times\frac{d\bm{\delta}_{A}}{dt}=$
$\displaystyle\
\gamma_{A}\mu_{0}\left[\left\\{\bm{H}\cdot\bm{n}_{A}^{0}-M_{A}\left(\hat{\bm{z}}\cdot\bm{n}_{A}^{0}\right)^{2}\right\\}\bm{\delta}_{A}+M_{A}\left(\hat{\bm{z}}\cdot\bm{\delta}_{A}\right)\left\\{\hat{\bm{z}}-\left(\hat{\bm{z}}\cdot\bm{n}_{A}^{0}\right)\bm{n}_{A}^{0}\right\\}\right]$
$\displaystyle-\frac{\gamma_{A}}{d_{A}M_{A}}\left\\{J_{1}+2\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)J_{2}\right\\}\left[\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)\bm{\delta}_{A}-\sqrt{\frac{\gamma_{B}dAM_{A}}{\gamma_{A}d_{B}M_{B}}}\left\\{\bm{\delta}_{B}-\left(\bm{n}_{A}^{0}\cdot\bm{\delta}_{B}\right)\bm{n}_{A}^{0}\right\\}\right]$
$\displaystyle+\frac{2\gamma_{A}}{d_{A}M_{A}}J_{2}\left\\{\bm{n}_{B}^{0}-\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)\bm{n}_{A}^{0}\right\\}\left(\bm{n}_{B}^{0}\cdot\bm{\delta}_{A}+\sqrt{\frac{\gamma_{B}d_{A}M_{A}}{\gamma_{A}d_{B}M_{B}}}\bm{n}_{A}^{0}\cdot\bm{\delta}_{B}\right),$
(6) $\displaystyle\bm{n}_{B}^{0}\times\frac{d\bm{\delta}_{B}}{dt}=$
$\displaystyle\
\gamma_{B}\mu_{0}\left[\left\\{\bm{H}\cdot\bm{n}_{B}^{0}-M_{B}\left(\hat{\bm{z}}\cdot\bm{n}_{B}^{0}\right)^{2}\right\\}\bm{\delta}_{B}+M_{B}\left(\hat{\bm{z}}\cdot\bm{\delta}_{B}\right)\left\\{\hat{\bm{z}}-\left(\hat{\bm{z}}\cdot\bm{n}_{B}^{0}\right)\bm{n}_{B}^{0}\right\\}\right]$
$\displaystyle-\frac{\gamma_{B}}{d_{B}M_{B}}\left\\{J_{1}+2\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)J_{2}\right\\}\left[\left(\bm{n}_{A}^{0}\cdot\bm{\delta}_{B}\right)\bm{\delta}_{B}-\sqrt{\frac{\gamma_{A}d_{B}M_{B}}{\gamma_{B}d_{A}M_{A}}}\left\\{\bm{\delta}_{A}-\left(\bm{n}_{B}^{0}\cdot\bm{\delta}_{A}\right)\bm{n}_{B}^{0}\right\\}\right]$
$\displaystyle+\frac{2\gamma_{B}}{d_{B}M_{B}}J_{2}\left\\{\bm{n}_{A}^{0}-\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)\bm{n}_{B}^{0}\right\\}\left(\bm{n}_{A}^{0}\cdot\bm{\delta}_{B}+\sqrt{\frac{\gamma_{A}d_{B}M_{B}}{\gamma_{B}d_{A}M_{A}}}\bm{n}_{B}^{0}\cdot\bm{\delta}_{A}\right).$
(7)

Equations (6) and (7) describe two coupled harmonic oscillators, i.e. there
are four independent real functions of time to be determined. We are
interested in the resonance properties, which can be analyzed in terms of any
consistent choice of the four independent variables. Had it not been for the
shape anisotropy and the asymmetry between $d_{A},M_{A},\gamma_{A}$ and
$d_{B},M_{B},\gamma_{B}$, two-fold rotation around $\bm{H}$ would have mapped
$\bm{n}_{A}^{0}$ to $\bm{n}_{B}^{0}$ and the symmetry-adapted variables would
have been convenient. Following MacNeil _et al._ [34], let $\mathcal{C}_{2}$
denote the two-fold rotation that brings $\bm{n}_{A}^{0}$ to $\bm{n}_{B}^{0}$
whose axis coincides with $X$ direction in Fig. 1. Algebraically the action of
$\mathcal{C}_{2}$ on an arbitrary vector $\bm{v}$ is given by

$\mathcal{C}_{2}\bm{v}=\frac{\left(\bm{n}_{A}^{0}+\bm{n}_{B}^{0}\right)\cdot\bm{v}}{1+\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}}\left(\bm{n}_{A}^{0}+\bm{n}_{B}^{0}\right)-\bm{v}.$
(8)

Although $\mathcal{C}_{2}$ is not in general a symmetry of the problem, it
helps make sense of the results in terms of the familiar notions used in
previous studies [34, 23]. The definition of $\mathcal{C}_{2}$ becomes
ambiguous for $\left|H\right|<H_{\rm sf}$ and what follows does not work for
$\left|H\right|>H_{\rm ff}$ either, but these collinear cases are simple and
separately handled in the Appendix. Focusing on the spin-flop phase, we
introduce
$\bm{\delta}_{\pm}=\left(\bm{\delta}_{A}\pm\mathcal{C}_{2}\bm{\delta}_{B}\right)/\sqrt{2}$
that are even and odd eigenvectors of
$\mathcal{C}_{2}\times\\{A\leftrightarrow B\\}$. To pick out two independent
components each for $\bm{\delta}_{\pm}$, we define a new coordinate frame
$XYZ$ (Fig. 1) given by

$\hat{\bm{X}}=\frac{\bm{n}_{A}^{0}+\bm{n}_{B}^{0}}{\sqrt{2+2\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}}},\quad\hat{\bm{Y}}=\frac{\bm{n}_{A}^{0}-\bm{n}_{B}^{0}}{\sqrt{2-2\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}}},$
(9)

and $\hat{\bm{Z}}=\hat{\bm{X}}\times\hat{\bm{Y}}$. By construction,
$\bm{n}_{A}^{0}\cdot\bm{\delta}_{\pm}=0$ so that one may write
$\bm{\delta}_{\pm}=\delta_{\pm}^{\perp
Z}\hat{\bm{Z}}\times\bm{n}_{A}^{0}+\delta_{\pm}^{\parallel Z}\hat{\bm{Z}}$. As
is usual in cavity magnonics, we work with the complex variables
$\alpha=\delta_{-}^{\perp Z}-i\delta_{-}^{\parallel Z},\beta=\delta_{+}^{\perp
Z}-i\delta_{+}^{\parallel Z}$ that would represent annihilation operators in
the quantum regime. This change of variables brings Eqs. (6) and (7) into

$i\frac{d}{dt}\begin{pmatrix}\alpha\\\ -\overline{\alpha}\\\ \beta\\\
-\overline{\beta}\\\
\end{pmatrix}=\begin{pmatrix}f_{1}-h_{1}&f_{2}-h_{2}-if_{3}&g_{1}&g_{2}-ig_{3}\\\
f_{2}-h_{2}+if_{3}&f_{1}-h_{1}&g_{2}+ig_{3}&g_{1}\\\
g_{1}&g_{2}-ig_{3}&f_{1}+h_{1}&f_{2}+h_{2}-if_{3}\\\
g_{2}+ig_{3}&g_{1}&f_{2}+h_{2}+if_{3}&f_{1}+h_{1}\\\
\end{pmatrix}\begin{pmatrix}\alpha\\\ \overline{\alpha}\\\ \beta\\\
\overline{\beta}\\\ \end{pmatrix},$ (10)

where overbars denote complex conjugation. Note that the equation is in the
Bogoliubov form with the matrix on the right-hand-side being Hermitian. For
succinct expressions of the matrix coefficients, let us introduce two distinct
orthogonal decompositions of the film normal
$\hat{\bm{z}}=z_{A}\bm{n}_{A}^{0}+z_{\perp
A}\hat{\bm{Z}}\times\bm{n}_{A}^{0}+z_{Z}\hat{\bm{Z}}=z_{B}\bm{n}_{B}^{0}+z_{\perp
B}\hat{\bm{Z}}\times\bm{n}_{B}^{0}+z_{Z}\hat{\bm{Z}}$, where
$z_{A}=\bm{n}_{A}^{0}\cdot\hat{\bm{z}},z_{\perp
A}=\left(\hat{\bm{Z}}\times\bm{n}_{A}^{0}\right)\cdot\hat{\bm{z}},z_{Z}=\hat{\bm{Z}}\cdot\hat{\bm{z}}$
and similarly for the $B$ layer. The coefficients are then given by

$\displaystyle f_{1}=$ $\displaystyle\
\mu_{0}\bm{H}\cdot\frac{\left|\gamma_{A}\right|\bm{n}_{A}^{0}+\left|\gamma_{B}\right|\bm{n}_{B}^{0}}{2}-\frac{1}{2}\left(\frac{\left|\gamma_{A}\right|}{d_{A}M_{A}}+\frac{\left|\gamma_{B}\right|}{d_{B}M_{B}}\right)\left[J_{1}\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}+J_{2}\left\\{3\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)^{2}-1\right\\}\right]$
$\displaystyle+\left|\gamma_{A}\right|\mu_{0}M_{A}\frac{z_{\perp
A}^{2}+z_{Z}^{2}-2z_{A}^{2}}{4}+\left|\gamma_{B}\right|\mu_{0}M_{B}\frac{z_{\perp
B}^{2}+z_{Z}^{2}-2z_{B}^{2}}{4}$ (11) $\displaystyle f_{2}=$ $\displaystyle\
\left|\gamma_{A}\right|\mu_{0}M_{A}\frac{z_{\perp
A}^{2}-z_{Z}^{2}}{4}+\left|\gamma_{B}\right|\mu_{0}M_{B}\frac{z_{\perp
B}^{2}-z_{Z}^{2}}{4}+\frac{1}{2}\left(\frac{\left|\gamma_{A}\right|}{d_{A}M_{A}}+\frac{\left|\gamma_{B}\right|}{d_{B}M_{B}}\right)J_{2}\left\\{1-\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)^{2}\right\\},$
(12) $\displaystyle f_{3}=$ $\displaystyle\
\frac{\left|\gamma_{A}\right|\mu_{0}M_{A}z_{\perp
A}+\left|\gamma_{B}\right|\mu_{0}M_{B}z_{\perp B}}{2}z_{Z},$ (13)
$\displaystyle g_{1}=$ $\displaystyle\
\mu_{0}\bm{H}\cdot\frac{\left|\gamma_{A}\right|\bm{n}_{A}^{0}-\left|\gamma_{B}\right|\bm{n}_{B}^{0}}{2}-\frac{1}{2}\left(\frac{\left|\gamma_{A}\right|}{d_{A}M_{A}}-\frac{\left|\gamma_{B}\right|}{d_{B}M_{B}}\right)\left[J_{1}\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}+J_{2}\left\\{3\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)^{2}-1\right\\}\right]$
$\displaystyle+\left|\gamma_{A}\right|\mu_{0}M_{A}\frac{z_{\perp
A}^{2}+z_{Z}^{2}-2z_{A}^{2}}{4}-\left|\gamma_{B}\right|\mu_{0}M_{B}\frac{z_{\perp
B}^{2}+z_{Z}^{2}-2z_{B}^{2}}{4}$ (14) $\displaystyle g_{2}=$ $\displaystyle\
\left|\gamma_{A}\right|\mu_{0}M_{A}\frac{z_{\perp
A}^{2}-z_{Z}^{2}}{4}-\left|\gamma_{B}\right|\mu_{0}M_{B}\frac{z_{\perp
B}^{2}-z_{Z}^{2}}{4}+\frac{1}{2}\left(\frac{\left|\gamma_{A}\right|}{d_{A}M_{A}}-\frac{\left|\gamma_{B}\right|}{d_{B}M_{B}}\right)J_{2}\left\\{1-\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)^{2}\right\\},$
(15) $\displaystyle g_{3}=$ $\displaystyle\
\mu_{0}\frac{\left|\gamma_{A}\right|M_{A}z_{\perp
A}-\left|\gamma_{B}\right|M_{B}z_{\perp B}}{2}z_{Z},$ (16) $\displaystyle
h_{1}=$ $\displaystyle\
-\sqrt{\frac{\gamma_{A}\gamma_{B}}{d_{A}M_{A}d_{B}M_{B}}}\left[\frac{1+\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}}{2}J_{1}+\left\\{2\left(\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)^{2}+\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}-1\right\\}J_{2}\right],$
(17) $\displaystyle h_{2}=$ $\displaystyle\
\sqrt{\frac{\gamma_{A}\gamma_{B}}{d_{A}M_{A}d_{B}M_{B}}}\frac{1-\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}}{2}\left\\{J_{1}+2\left(1+2\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}\right)J_{2}\right\\}.$
(18)

The eigenfrequencies of Eq. (10) can be calculated as

$\displaystyle\omega^{2}$ $\displaystyle=$ $\displaystyle
f_{1}^{2}-f_{2}^{2}-f_{3}^{2}+g_{1}^{2}-g_{2}^{2}-g_{3}^{2}+h_{1}^{2}-h_{2}^{2}$
$\displaystyle\pm
2\sqrt{\left(f_{1}g_{1}-f_{2}g_{2}-f_{3}g_{3}\right)^{2}+\left(f_{1}h_{1}-f_{2}h_{2}\right)^{2}-\left(g_{1}h_{2}-g_{2}h_{1}\right)^{2}-g_{3}^{2}\left(h_{1}^{2}-h_{2}^{2}\right)}.$

One can observe that the “couplings” $g_{1,2,3}$ between $\alpha$ and $\beta$
all vanish if the two layers are identical and $\bm{H}$ is in the plane. For
identical layers with $\theta\neq 90\degree$, $g_{1}=g_{2}=0,g_{3}\neq 0$ due
to $z_{A\perp}=-z_{B\perp}$ and the problem reduces to that of Refs. [34, 23].
The variables $\alpha,\beta$ represent oscillations that are odd and even
under $\mathcal{C}_{2}\times\\{A\leftrightarrow B\\}$, and can be considered
generalisations of the acoustic and optical modes in SyAFs, respectively. When
$g_{1,2,3}$ become comparable with $f_{1,2,3},h_{1,2}$, however, $\alpha$ and
$\beta$ evenly contribute to the eigenmodes for all values of $H$. This makes
it meaningless to talk about hybridisation between odd and even modes, which
would require the modes be weakly coupled away from a resonance region and
come almost degenerate upon tuning some parameters. Indeed, there is no simple
relation between $g_{1,2,3}$ and the spectral gap in general.

Figure 2: (a-e) Normalised $M$-$H$ loops for different set of samples (a)
NiFe(5)/Ru(0.4)/NiFe(3) (b) NiFe(3)/Ru(0.4)/NiFe(5) (c)
CoFeB(3)/Ru(0.45)/NiFe(3) (d) CoFeB(3)/Ru(0.5)/NiFe(3) and (e)
CoFeB(3)/Ru(0.55)/NiFe(3). The field is applied along the in-plane easy axis.
The solid lines are fit obtained by the theoretical static state calculations
based on Eq. (1). (f-j) Static state angles of magnetisation for two FM layers
calculated for the best fit parameters corresponding to (a-e) respectively.
(k-o) Angle between the two magnetisations.

##  III Sample growth and magnetometry characterisation

Samples used in this study were grown by using magnetron sputtering techniques
inside a chamber at a base pressure better than 5$\times$10-6 Pa. As
summarised in Table 1, we studied five different multi-layers
Ta(5)/FM1($d_{\textrm{A}}$)/Ru/FM2($d_{\textrm{B}}$)/Ta(5)/thermally oxidized
Si substrate (numbers in the brackets represent layer thickness in nm) after
optimising growth conditions [48, 23, 49]. Figure 2 shows normalised
hysteresis loops for the samples measured for static external field in the
plane by vibrating sample magnetometer (VSM) techniques. Three regions
distinguished by the alignment of magnetisations of the two layers are
indicated in different colours. As explained in the previous section, due to
the competition between the exchange and Zeeman energies, our samples undergo
two phase transitions. In the small magnetic field limit $H<H_{\rm sf}$
(shadowed in pink), the exchange energy dominates and the two moments are
aligned antiferromagnetically. As the field is increased, the spin-flop
transition takes place, after which the two moments tilt away from the field
in a canted state. Finally, at higher field values $H>H_{\rm ff}$, the Zeeman
energy prevails and the magnetic moments point along the field direction
entering the forced ferromagnetic regime as indicated in green for each plot.

Equation (1) was used for fitting to determine the static states of each
moment. For obtaining the ground state, we find the values of
$\cos\phi_{\rm{A,B}}$ that minimise Eq. (1) for
$\theta_{A}=\theta_{B}=90\degree$ in an iterative manner for each magnetic
field. The orange curves in the first row of Fig. 2 are the normalised
magnetisation calculated for each field value as:

$\frac{M(H)}{M_{s}}=\frac{d_{A}M_{A}\cos\phi_{A}+d_{B}M_{B}\cos\phi_{B}}{d_{A}M_{A}+d_{B}M_{B}}.$
(20)

where $M_{s}$ is the total saturation magnetisation of the sample.
Optimisation with respect to the experimental curves yielded the best-fit
values of $M_{A},M_{B}$ as well as the quadratic
($\mu_{0}H_{\rm{ex}}=J_{1}/\sqrt{d_{A}d_{B}M_{A}M_{B}}$) and biquadratic
exchange fields ( $\mu_{0}H_{2\rm{ex}}=J_{2}/\sqrt{d_{A}d_{B}M_{A}M_{B}}$),
which are summarised in the Table 1.

Table 1:  Summary of the VSM magnetometry parameters used to obtain the theoretical magnetisation curves shown in Fig. 2 according to Eqs. (1) and (20). The left column represents the sample geometry where $".."$ indicates the thermally oxidized Si substrate and the FM near to the substrate is the second FM layer referred to as $B$ layer. $\mu_{0}H_{\rm{ex}},\mu_{0}H_{\rm 2ex}$ are the quadratic and biquadratic exchange fields respectively and $M_{A(B)},d_{A(B)}$ are the magnetisation and thickness for the two ferromagnetic layers (NiFe/CoFeB).  Sample | $\mu_{0}M_{A}$ | $\mu_{0}M_{B}$ | $\mu_{0}H_{\rm{ex}}$ | $\mu_{0}H_{2\rm{ex}}$ | $d_{A}$ | $d_{B}$
---|---|---|---|---|---|---
geometry | (T) | (T) | (T) | (T) | (nm) | (nm)
Ta/NiFe/Ru(0.4)/NiFe/Ta/.. | 0.95 | 0.9 | 0.145 | 0.03 | 5 | 3
Ta/NiFe/Ru(0.4)/NiFe/Ta/.. | 0.95 | 0.9 | 0.1 | 0.02 | 3 | 5
Ta/CoFeB/Ru(0.45)/NiFe/Ta/.. | 1.5 | 1.0 | 0.048 | 0.005 | 3 | 3
Ta/CoFeB/Ru(0.5)/NiFe/Ta/.. | 1.5 | 1.0 | 0.02 | 0.003 | 3 | 3
Ta/CoFeB/Ru(0.55)/NiFe/Ta/.. | 1.5 | 1.0 | 0.03 | 0.002 | 3 | 3
Figure 3: (a) Schematic of the sample structure. (b)-(c) Absorption spectra
for the sample NiFe(5)/Ru(0.4)/NiFe(3) at (b) low and (c) high field for
$\theta$=90∘. (d) Microwave transmission as a function of frequency and field
for the sample NiFe(5)/Ru(0.4)/NiFe(3). The field is applied within the plane
$\theta$=90∘. A clear avoided-crossing gap is visible at field $\mu_{0}H=0.25$
T. (e) Fitting results for data as in (d). The solid lines are fitted curves
obtained from macrospin model.  Figure 4: (a)-(d) Microwave transmission as a
function of frequency and applied field for the sample NiFe(5)/Ru(0.4)/NiFe(3)
for different $\theta$. The angle $\theta$ is defined as in Fig. 1. (e)-(h)
Resonance frequency as a function of field obtained by derivative Lorentzian
fitting of the experimental data. The solid lines in the figure are
theoretical results obtained from the macrospin model. (i) Spectral gap as a
function of $\theta$ obtained from theoretical model calculations. It can be
seen that a maximum gap of $\approx$ 4.5 GHz is achieved. The spectral gap is
defined as the minimum of the difference between the upper ($f_{\textrm{u}}$)
and lower ($f_{\textrm{l}}$ ) resonance frequencies as a function of
$\mu_{0}H$ as shown by the dotted line in inset for the sample
NiFe(5)/Ru(0.4)/NiFe(3).  Figure 5: (a)-(d) Microwave transmission as a
function of frequency and applied field for the sample
CoFeB(3)/Ru(0.45)/NiFe(3) for different $\theta$. The spectral gap increases
as $\theta$ is decreased. (e)-(h) Resonance frequency as a function of field
obtained by derivative Lorentzian fitting of the experimental data. The solid
lines in (e)-(h) are theoretical results obtained from the macrospin model.
(i) The spectral gap as a function of $\theta$ for different Ru thicknesses,
which shows a gradual increase as $\theta$ is decreased. Inset shows the
variation of $\Delta$ as a function of $\mu_{0}H_{\textrm{ex}}$. (j) Spectral
gap as a function of $\theta$ for sample with Ru thickness 0.45 nm at
$\mu_{0}H_{\textrm{ex}}$ = 0.1 T and 0.05 T. The gap shows an increase as
$\mu_{0}H_{\textrm{ex}}$ is increased. The spectra used for extracting the
spectral gap is given in Appendix C.

##  IV Spin dynamics

High frequency responses of the coupled moment systems were characterised by
broadband on-chip microwave absorption techniques. As illustrated in Fig.
3(a), each sample chip was placed face-down on a coplanar waveguide [50]. For
each measurement, we fixed the frequency $f$ and swept a dc external magnetic
field $\mu_{0}H$ while applying an ac magnetic field at 12 Hz for lock-in
detection techniques. Here we show our measurements on the samples
NiFe(5)/Ru(0.4)/NiFe(3) and CoFeB(3)/Ru/NiFe(3), both showing avoided crossing
[23, 34, 29] due to the asymmetry of thickness and magnetic moment size,
respectively.

###  IV.1 NiFe(5 nm)/Ru(0.4 nm)/NiFe(3 nm)

Figures 3(b) and (c) represent individual measurement curves targeted at two
resonance modes in the sample NiFe(5 nm)/Ru(0.4 nm)/NiFe(3 nm) for
$\theta=90^{\circ}$ and different frequencies. These individual scans are used
to produce a $f$-$\mu_{0}H$ two-dimensional plot as shown in Fig. 3(d) to
capture the absorption spectrum. At $\mu_{0}H\approx$ 0.25 T, instead of mode
degeneracy, we observe the avoided crossing, suggesting that the in- and out-
of-phase oscillations are strongly hybridised [23, 29]. Figure 3(e) plots peak
positions extracted by individual curve fittings using derivative Lorentzian
functions [51, 52, 53]. Equation (II) with material parameters independently
extracted in the static VSM measurements (Table 1) generates curves that are
in reasonable agreement with experiment. This displays the applicability of
the macro-spin model with the minimal set of phenomenological parameters for
this type of experiments. To highlight the role of thickness asymmetry for gap
opening, we also show two additional sets of model calculations for
($d_{A}$,$d_{B}$)=(3 nm, 3 nm) and (3 nm, 4 nm). The model shows that the
spectral gap widens as the thickness asymmetry is increased. Due to the
asymmetry $d_{A}\neq d_{B}$, some of the coupling parameters in the off-
diagonal blocks in Eq. (10), i.e. $g_{1}$ and $g_{2}$, become non-zero, for
instance through the prefactor
$\left|\gamma_{A}\right|/\mu_{0}M_{A}-\left|\gamma_{B}\right|/\mu_{0}M_{B}$).
Therefore, even for the case of $\theta=90\degree$, the thickness asymmetry
generates the hybridisation of in- and out-of-phase oscillations.

We can further increase the gap size by tilting the moments towards the out-
of-plane, as we previously demonstrated in the symmetric cases [23]. Figures
4(a-d) summarise the experimentally-measured $\theta$ dependence of the
magnetic resonances. We performed peak position analysis for these
$\theta$-dependent results as shown in Fig. 4(e-h), together with
$\Delta$-$\theta$ relationship plotted in Fig. 4(i). Here $\Delta$ is defined
as the minimum of the difference between the upper and lower resonance
frequencies as shown in the Fig. 4(i) inset. Our theoretical curves
successfully reproduce the experimental results without any tunable
parameters. As the out-of-plane field increases, the gap is enhanced in
comparison with that due to the thickness asymmetry alone and might be
attributed to an increase of $g_{3}$ (Eq. (16)) with reducing $\theta$. The
observed trend is further supported by repeated experiments with a sample with
the inverted growth order, i.e. NiFe(3 nm)/Ru(0.4 nm)/NiFe(5 nm),
demonstrating approximately the same quantitative behaviour as shown in Figs.
4(i).

###  IV.2 CoFeB(3 nm)/Ru/NiFe(3 nm)

Figure 6: Spectral gap obtained from simulations by varying (a) $M_{B}$ and
(b) $\mu_{0}H_{\textrm{ex}}$. The other fixed parameters used for the
simulations are indicated on the plot. As the asymmetry is increased, a very
large spectral gap $\approx$ 12 GHz is obtained for $\mu_{0}H_{\textrm{ex}}$ =
0.15 T as shown in (b). (c) Spectral gap as a function of $\theta$ for
different biquadratic exchange field values $\mu_{0}H_{2\textrm{ex}}$ for the
sample with Ru thickness of 0.45 nm. The other parameters used for the
simulation are the same as given in Table 1. For low $\mu_{0}H_{2\textrm{ex}}$
values, the increase in gap size is not prominent as $\theta$ is varied.

In order to experimentally demonstrate the effect of symmetry breaking due to
the asymmetry in magnetic moments ($M_{\rm A}\neq M_{\rm B}$) [35], we grew
multilayers of CoFeB/Ru/NiFe where the thickness of the two FM materials was
kept fixed at 3 nm. Figure 5 summarises the spectral measurements/analysis for
the sample CoFeB(3 nm)/Ru(0.45 nm)/NiFe(3 nm) for different values of
$\theta$. A clear avoided-crossing gap is visible in the spectra shown in Fig.
5(a) for $\theta=\pi/2$ and the model calculations (solid curves) reproduce
the dispersion curves with the degree of moment asymmetry fixed by the static
VSM measurements in this stack as shown in Fig. 5(e). This is because $g_{1}$
and $g_{2}$ become non-zero when $M_{A}\neq M_{B}$ (see Eqs. (14)-(15)).
$g_{3}$ further adds to the coupling when the two moments have out-of-plane
components and this tendency is experimentally demonstrated as shown in Figs.
5(a)-(h).

Figure 5(i) displays the gap size $\Delta$ as a function of $\theta$ for the
samples CoFeB(3 nm)/Ru($t$)/NiFe(3 nm) with three different Ru thicknesses,
i.e. $t=0.45,0.50$ and 0.55 nm; the magnetic parameters of these samples
extracted from VSM measurements are listed in Table 1. The Ru thickness does
not directly enter the free energy equation or LLG equation, instead mostly
influencing the interlayer exchange coupling strength $\mu_{0}H_{\text{ex}}$.
Hence, comparing these three samples can be a good experimental demonstration
of the effect of the exchange coupling strength on GHz spectra for the coupled
moments. There is indeed direct correlation between $\Delta$ and
$\mu_{0}H_{\text{ex}}$ as shown in the inset of Fig. 5(i) for $\theta$ = 90∘.
We also perform further simulations using the same parameters in the sample
CoFeB(3 nm)/Ru(0.45 nm)/NiFe(3 nm), except for $\mu_{0}H_{\text{ex}}$ being
0.1 T. $\Delta$ of this simulation as a function of $\theta$ is plotted in
Fig. 5(j), supporting our claim.

We have so far shown the reliability of our macrospin model in reproducing the
experimental results of magnetic resonance spectra in coupled moments via the
interlayer exchange interaction. Here we present our theoretical predictions
to discuss the magnetic-parameter dependence of $\Delta$. The asymmetry of
coupled moments, i.e. $M_{A}$ and $M_{B}$, can be further enhanced in
simulation and we find that $\Delta$ is monotonically increased by enlarging
the difference between $M_{A}$ and $M_{B}$ for a fixed value of $\mu_{0}H_{\rm
ex}$ as shown in Fig. 6(a), reaching up to approximately 7.5 GHz with
$\mu_{0}M_{A}=1.5$ T and $\mu_{0}M_{B}=0.4$ T. This might be achieved by
selecting low-moment magnets as a counterpart of CoFeB to form a stack of
synthetic ferrimagnet. Our model simulations also suggest that in such
synthetic ferrimagnets with large moment asymmetry, $\mu_{0}H_{\textrm{ex}}$
that can be tuned by the thickness of the intermediate layer can act as a knob
to further enhance $\Delta$ as presented in Fig. 6(b). See Appendix for
individual spectra for extracting $\Delta$. Finally, the $\theta$ dependence
of $\Delta$ for different values of $\mu_{0}H_{2\textrm{ex}}$ is plotted in
Fig. 6(c). For these simulations, an increase of $\mu_{0}H_{2\textrm{ex}}$
decreases $\Delta$, which is qualitatively different from the role of
$\mu_{0}H_{\textrm{ex}}$ on $\Delta$, e.g. in Fig. 5(j). This is partially
because of the general competition between $J_{1}$ and $J_{2}$ which prefer
different static state configurations and therefore combine to soften the
order and decrease the scale of resonance frequency. While
$\mu_{0}H_{2\textrm{ex}}$ is not a material parameter that can be easily tuned
by growth conditions, it is interesting to notice that the biquadratic nature
enters the spectral responses very differently from the quadratic counterpart.
In general, when the off-diagonal block elements $g_{1},g_{2},g_{3}$ become
comparable with the diagonal block ones as in the present case, the notion of
coupling between acoustic and optical modes becomes inappropriate, leading to
the complex dependence of $\Delta$ on not only the asymmetry related
parameters but also the symmetry-respecting ones such as $\mu_{0}H_{\rm ex}$
and $\mu_{0}H_{2{\rm ex}}$.

##  V Conclusion

We studied the dynamics of synthetic ferrimagnets and theoretically and
experimentally showed their magnon-magnon coupling with dissimilar material
and thickness of two ferromagnetic layers. We presented analytical expressions
of the coupled mode resonance frequencies and used them to quantitatively
discuss experimental results. Using the rich and controllable spin-wave
spectra in interlayer-coupled magnetic moments, these materials might find
their important use for future magnonic/spintronic applications [54, 55, 31,
30, 32, 33].

## Acknowledgements

A. S. thanks JSPS Postdoctoral fellowship for research in Japan (P21777) and
EPSRC for their supports through NPIF EPSRC Doctoral studentship
(EP/R512400/1) during her PhD at UCL. K. Y. is supported by JST PRESTO Grant
No. JPMJPR20LB, Japan and JSPS KAKENHI (No. 21K13886). SM thanks to CSRN in
CSIS at Tohoku Univ. and to JSPS KAKENHI (No. 21H04648, 22F21777, 22KF0030)

##  Appendix A Collinear ground states

The coordinate axes we used in the main text, Eq. (9) are not well-defined for
$\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}=\pm 1$, namely when the two magnetisations
are collinear in the static state. This happens for $H\leq H_{\rm sf}$ and
$H\geq H_{\rm ff}$ if the magnetic field is in-plane $\theta=90\degree$, and
more generally at high fields if the two layers are identical.

Let us first discuss the antiferromagnetic state
$\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}=-1$, for which we can assume
$\bm{H}=H\hat{\bm{x}}$. In place of $\mathcal{C}_{2}$ given in Eq. (8), the
static state satisfies $\mathcal{C}_{2}^{\prime}\bm{n}_{A}^{0}=\bm{n}_{B}^{0}$
where

$\mathcal{C}_{2}^{\prime}\bm{v}=\left(\hat{\bm{y}}\cdot\bm{v}\right)\hat{\bm{y}}-\bm{v}.$
(21)

One may still then define
$\bm{\delta}_{\pm}=\left(\bm{\delta}_{A}\pm\mathcal{C}_{2}^{\prime}\bm{\delta}_{B}\right)/\sqrt{2}$
and decompose them as $\bm{\delta}_{\pm}=\delta_{\pm}^{\perp
z}\hat{\bm{z}}\times\bm{n}_{A}^{0}+\delta_{\pm}^{\parallel z}\hat{\bm{z}}$.
The rest does not have to be changed with $z_{A}=z_{\perp A}=z_{B}=z_{\perp
B}=0,z_{Z}=1$ and $\bm{n}_{A}^{0}=-\bm{n}_{B}^{0}=\pm\hat{\bm{x}}$ according
to $d_{A}M_{A}\gtrless d_{B}M_{B}$.

For the ferromagnetic state $\bm{n}_{A}^{0}\cdot\bm{n}_{B}^{0}=1$,
$\hat{\bm{X}}=\bm{n}_{A}^{0}$ is well-defined and one may redefine
$\hat{\bm{Y}}=\hat{\bm{y}}$. With this provision, $\mathcal{C}_{2}$ is simply
a two-fold rotation about $\hat{\bm{X}}$ and
$\bm{\delta}_{\pm}=\bm{\delta}_{A}\mp\bm{\delta}_{B}$. Again nothing needs to
be modified in Eq. (10) and beyond with $z_{\perp A}=z_{\perp B}=0$.

##  Appendix B Additional magnetisation-dynamics results for other samples
measured in this study

Figure 7: (a)-(b) Extra plots of microwave transmission as a function of
frequency and applied field for CoFeB(3)/Ru(0.45)/NiFe(3) for different
$\theta$ values. The spectral gap increases as $\theta$ is decreased. Figures
(c-d) shows resonance frequency obtained using derivative Lorentzian fitting
of the experimental data and the solid lines are the theoretical curves
obtained from macrospin model. Figure 8: (a)-(b) Extra plots of microwave
transmission as a function of frequency and applied field for
NiFe(5)/Ru(0.4)/NiFe(3) for different $\theta$ values. Figures (c-d) shows
resonance frequency obtained using derivative Lorentzian fitting of the
experimental data and the solid lines are the theoretical curves obtained from
macrospin model for the experimental data as in (a-b).  Figure 9: (a)-(d)
Microwave transmission as a function of frequency and applied field for
CoFeB(3)/Ru(0.5)/NiFe(3) for different $\theta$. The gap opening is smaller as
compared to sample with Ru thickness 0.45 nm due to smaller
$\mu_{0}H_{\textrm{ex}}$ of this sample. Figure 10: (a)-(d) Resonance
frequency extracted from derivative Lorentzian fitting of experimental data as
a function of applied field along with theoretical prediction for
CoFeB(3)/Ru(0.5)/NiFe(3). These correspond to the data shown in Fig. 9.
Figure 11: (a)-(d) Microwave transmission as a function of frequency and
applied field for CoFeB(3)/Ru(0.55)/NiFe(3) for different $\theta$ values. A
small variation in spectral gap is seen as the $\theta$ varied.  Figure 12:
(a)-(d) Resonance frequency extracted from fitting of experimental data as a
function of applied field along with theoretical prediction for
CoFeB(3)/Ru(0.55)/NiFe(3). These correspond to the data shown in Fig. 11.

This section provides supplementary results for samples measured in this
study, which further supports the observations and claims as described in the
main text. Top panels (a-b) in Figs. 7 and 8 show measurements for some
remaining angles not shown in the main text for the samples with stacking
pattern CoFeB/Ru(0.45)/NiFe and NiFe(5)/Ru(0.4)/NiFe(3) respectively. The
fittings produced by our macrospin model are shown in bottom panel, which
agree well with the experimental data.

The measurements were repeated for other sets of samples following the
procedure outlined in the main text and we saw similar behaviours of spectral
gap variation with change in applied field angle towards out-of-plane as shown
in Fig. 9 for sample CoFeB/Ru(0.5)/NiFe and Fig. 11 for sample
CoFeB/Ru(0.55)/NiFe. The resonance frequency obtained by fitting of
experimental data using derivative of Lorentzian function along with the
theoretical predictions are plotted in Figs. 10 and 12 corresponding to Figs.
9 and 11 respectively. For samples with Ru thickness 0.5 and 0.55 nm the gap
opening is smaller than the sample with Ru thickness 0.45 nm as shown in Fig.
9 and Fig. 11. This is due to the lower value of $\mu_{0}H_{\textrm{ex}}$ in
these samples. These results further support our observation of spectral gap
controlled by the out-of-plane angle $\theta$ and exchange field strength
$\mu_{0}H_{\textrm{ex}}$ as mentioned in the main text.

##  Appendix C Numerical simulations to study the impact of varying parameters
on coupling gap

Using numerical simulations, we explored different parameter regimes beyond
the experimental conditions. In an effort to understand the magnetic-parameter
dependence of $\Delta$ we performed numerical simulations by varying different
parameters $\mu_{0}M_{\textrm{A}}$, $\mu_{0}H_{\textrm{ex}}$ and $\theta$ as
shown in Fig. 13.

$\Delta$ corresponding to Fig. 13 is shown in Fig. 6 given in the main text.
Our numerical simulations suggest that we can tune $\Delta$ by varying
different parameters.

Figure 13: Resonance frequency simulation as a function of field obtained by
varying different parameters; (a-d) $\mu_{0}M_{\textrm{A}}$, (e-h)
$\mu_{0}H_{\textrm{ex}}$, and (i-l) $\theta$. The other parameters which are
kept fixed are $\mu_{0}M_{\textrm{A}}$ = 1.5 T for all cases. For (a-d)
$\mu_{0}H_{\textrm{ex}}$ = 0.048 T, $\theta$ = 90∘, (e-h)
$\mu_{0}M_{\textrm{B}}$ = 0.5 T, $\theta$ = 90∘ and (i-j)
$\mu_{0}M_{\textrm{B}}$ = 0.5 T, $\mu_{0}H_{\textrm{ex}}$ = 0.12 T. A maximum
spectral gap of 12.9 GHz is obtained for (l).

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# Induced color charges in QGP at Polyakov’s loop and chromomagnetic fields

V. Skalozub, I. Gamolsky
Oles Honchar Dnipro National University, 49010 Dnipro, Ukraine e-mail:
<EMAIL_ADDRESS>
( June 24, 2023; )

###### Abstract

In quark-gluon plasma (QGP), at high temperature the spontaneous generation of
color magnetic fields, $H^{3}(T),H^{8}(T)\not=0$, and usual magnetic field
$H(T)\not=0$ is realized. Also, classical field - $A_{0}(T)$ condensate
directly related to the Polyakov loop - is spontaneously created. The common
generation of both of them within the two loop effective potential was
investigated recently for SU(2) gluodynamics in [1], [2]. The values of the
field strengths and the mechanism of the magnetic fields stabilization due to
$A_{0}(T)$ have been discovered.

In the present paper, we develop further these investigations in order to find
new effects of QGP. First of all, we generalize the $SU(2)$ consideration to
the full QCD case. Then we calculate the induced color charge $Q_{ind}^{3}$ at
the background magnetic field $H(T)\not=0$ and the condensates $A_{0}^{3}(T)$.
The $A_{0}^{8}(T)$ is zero in used two-loop approximation. So, $Q_{ind}^{8}$
is zero also. We conclude that magnetic field essentially influences the
induced charge compared to the zero field case investigated in [3]. The
extension to other type magnetic fields is given.

Key words: spontaneous magnetization, high temperature, asymptotic freedom,
effective potential, $A_{0}$ condensate, effective charge, effective vertexes.

##  1 Introduction

Quark-gluon plasma (QGP) is the state of matter consisting of quarks and
gluons liberated form nuclei at high temperature due to asymptotic freedom of
non-Abelian gauge fields. A classical background of QGP is formed out of two
type condensates - the $A_{0}(T)$ (Polyakov loops (PL)) and the spontaneously
generated temperature dependent chromomagnetic $H^{3}(T),H^{8}(T)$, where 3
and 8 are color indexes of $SU(3)$ group, and usual magnetic $H(T)$ fields.
The first kind condensate results in the color $Z(3)$ symmetry breaking and
the Furry’s theorem violation. The second ones considerably change the spectra
of quarks and gluons as well. So, new phenomena have to be realized. In
particular, the induced color charges $Q^{3}_{ind},Q^{8}_{ind}$ are expected.
The PL and $A_{0}(T)$ are the order parameters for the phase transition. At
low temperature, PL and $A_{0}$ equal zero. At high temperature they become
nonzero. The same concerns the magnetic fields. Recently, on the principles of
the Nielsen’s identity method and new type integral and sum representations we
derived the gauge invariant expression for the $A_{0}$ condensate in the
magnetic fields in two-loop approximation [1], [2]. This (in particular) opens
a possibility for calculating the induced color charges for this case.

To realize that, we have to calculate the contribution of the diagram depicted
in Figure 1. Here, the solid line presents the quark propagator in the $A_{0}$
and magnetic fields and the wavy line presents the zero component of gluon
fields $G_{0}^{3}$ or $G_{0}^{8}$. At finite temperature, in the Matsubara
formalism one has to calculate the temperature sum over discrete momenta
$p_{4}=2\pi T(l+1/2),l=0,\pm 1,...$, integrate over momentum component $p_{3}$
oriented along the space field direction, calculate the sum over spin variable
$\sigma=\pm 1$ and sum up over n = 0, 1, 2,… , in correspondence to the
fermion spectrum in magnetic field H:
$(p_{4}+gA_{0})^{2}=m^{2}+p_{3}^{2}+(2n+1)gH-\sigma gH$. Here we write $gH$ as
a general expression corresponding to each of the fields. For instance, this
is $eH$ for usual magnetic field, $gH^{3},gH^{8}$ \- for color fields.

In what follows, we apply the low level approximation, $n=0,\sigma=+1$ giving
a leading contribution for strong external fields. We obtain that the induced
color charge $Q^{3}_{ind}$ is nonzero. The presence of the magnetic fields
changes the values of it compared to the zero field case. So that the QGP has
to be magnetized and color charged.

Figure 1: Tadpole diagram

##  2 Induced Color charge

In this section, we calculate the induced color charge generated by the
tadpole diagram of Fig. 1. In charged basis, we have two components of the
induced charge for the shifts $A_{0}^{3}$ and $A_{0}^{8}$. But accounting for
the result [3] $A_{0}^{8}=0$ , we have to calculate the contribution for the
case $(A_{0})_{\mu}^{a}=A_{0}\delta_{\mu 4}\delta^{a3}$. The explicit form in
the Euclid space-time is $Q_{4}^{3}Q^{3}_{ind}$, and we have

$Q_{ind}^{3}=\frac{g}{\beta}\sum_{p_{4}}\int\frac{d^{3}p}{(2\pi)^{3}}Tr\left[\frac{\lambda^{3}}{2}\gamma_{4}\frac{\hat{p}_{\sigma}\gamma_{\sigma}+m}{\hat{p}^{2}+m^{2}}\right],$
(1)

where $\hat{p}=(p_{4}=p_{4}\pm A_{0},\mathbf{p}),\,p_{4}=2\pi
T(l+1/2),\,l=0,\pm 1,\dots$. The trace is calculated over either space-time or
color variables. $\lambda^{3}$ is Gell-Mann matrix. Here also we noted as
$A_{0}$ the value $A_{0}=\frac{gA_{0}}{2}$.

Calculating the traces over the space and the internal indexes we get,

$Q_{ind}^{3}=\frac{4g}{\beta}\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{p_{4}}\frac{(p_{4}+A_{0})}{(p_{4}+A_{0})^{2}+\epsilon_{p}^{2}},$
(2)

where $\epsilon_{p}^{2}=\vec{p}^{2}+m^{2}$. In case of nonzero field,
$\epsilon_{p}^{2}={p_{3}^{2}+m^{2}+(2n+1)gH-gH\sigma}$.

To calculate the temperature sum we use the following representation

$Q_{ind}^{3}=4g\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\beta}{\pi}\oint_{C}\tan\frac{\beta\omega}{2}\frac{(\omega+A_{0})}{(\omega+A_{0})^{2}+\epsilon^{2}_{p}}d\omega.$
(3)

This is in the case of zero field. If the field is nonzero, we have to replace
$\frac{d^{3}p}{(2\pi)^{3}}->\frac{dp_{3}}{2\pi}\frac{gH}{(2\pi)^{2}}$ in
correspondence to the particle spectrum. The integrand function has two
imaginary poles of the first order. We use the residues to find the charge
value.

The result, after transformation into spherical coordinates and angular
integration, is

$Q_{ind}^{3}=\frac{g\sin{(A_{0}\beta)}}{\pi^{2}}\int_{0}^{\infty}p^{2}dp\frac{1}{\cos\beta
A_{0}+\cosh(\beta\epsilon_{p})}.$ (4)

In what follows, we calculate the integral in the high-temperature limit
$T\to\infty$. In this case we use

$\epsilon_{p}=\sqrt{\mathbf{p}^{2}+m^{2}}\approx|\mathbf{p}|+\frac{1}{2}\frac{m^{2}}{|\mathbf{p}|}$
(5)

because large values of momentum give dominant contribution.

After integration over momentum we obtain at zero field [3]

$Q_{ind}^{3}=gA_{0}^{3}\bigl{[}\frac{T^{2}}{3}-\frac{m^{2}}{2\pi^{2}}\bigr{]}.$
(6)

As we see, the first term depends on temperature as $\sim T^{2}$. The second
one depends on mass, only. At high-temperature, the first term is dominant and
the plasma acquires the spontaneous induced charge in the case $m=0$, also.

Now, we turn to nonzero $H$. Using the low Landau level approximation,
$\sigma=+1,n=0$, we get after integration over $p_{3}$ momentum

$Q_{ind}^{3}(H,T)=g\frac{gH}{2\pi^{3}}\frac{\sin(A_{0}^{3}\beta)}{\beta}(1+7\beta^{2}m^{2}Zeta^{{}^{\prime}}(-2)).$
(7)

At high temperature, this expression reads

$Q_{ind}^{3}(H,T)=\frac{gH}{2\pi^{3}}(gA_{0}^{3})(1+7\beta^{2}m^{2}Zeta^{{}^{\prime}}(-2)).$
(8)

Note that numerically $Zeta^{{}^{\prime}}(-2)=-0.03044485$. Thus, one of the
consequences of the $A_{0}$ condensate presence is the $Z(3)$ symmetry and the
$C$-parity violation, which leads to the induction of color charge in the
plasma.

As we noted in Introduction, the factor $gH$ presents different kinds of
magnetic fields - usual magnetic field $eH$, color magnetic fields
$gH^{3},gH^{8}$ or even some combination of them. Remind also that in the
above formulas $A_{0}^{3}$ has to be substituted by $\frac{gA_{0}^{3}}{2}.$

##  3 Discussion

Let us compare the values of induced color charges given by formulas (6) and
(8). The first leading in temperature terms are of interest, now. Both
expressions have the factor $gA_{0}^{3}$ but a different temperature
dependence. In former case, the factor $\sim T^{2}$ stands and determines high
temperature behavior. In latter case, it is determined by the temperature
dependence of the magnetic field $H(T)$. This behavior has been investigated
recently in two-loop approximation [4]

$gH=\frac{1}{16}\frac{(2a_{2}\alpha_{s})^{2/3}}{\pi^{2/3}}T^{2}.$ (9)

Here, $a_{2}=1.856$ is number,
$\alpha_{s}=g^{2}/(1+\frac{11}{12}\frac{g^{2}}{\pi^{2}}\log(T/\mu))$ is
running coupling constant, $\mu$ is a normalization point for temperature. So,
at reference temperature $\alpha_{s}=g^{2}$. Because of this factor, the value
of the field strength is always smaller compared to $T^{2}$. As a result, the
induced color charge (8) in the magnetic field is also smaller compered to
eq.(6).

On the other hand, during carried out calculations we have taken the field
strength as a given number which is arbitrary. So that it can be the field
produced by some external current. In this case the induced color charge will
be completely determined by external field. Such a situation is expected and
discussed for heavy ion collision experiments.

## References

  * [1] V. V. Skalozub, Phys. Part. Nucl. Letts 18 (7), 738 (2021). [ArxiV: 2006.05737 hep-ph]
  * [2] M. Bordag, V. Skalozub, Eur. Phys. J. C 82, 390 (2022). DOI:10.1140/epjc/s10052-022-10339-4.
  * [3] V. Skalozub, A. Turinov, J. Phys. Electon. 30(1), 3 (2022).
  * [4] V. Skalozub, ArxiV: 2305. 00757 17 May v 2 2023 [hep-ph].


# Absence of magnetic order and emergence of unconventional fluctuations in
$J_{\rm eff}=1/2$ triangular lattice antiferromagnet YbBO3

K. Somesh  S. S. Islam  S. Mohanty  School of Physics, Indian Institute of
Science Education and Research Thiruvananthapuram-695551, India  G. Simutis
<EMAIL_ADDRESS>Laboratory for Neutron and Muon Instrumentation, Paul
Scherrer Institut, CH-5232 Villigen PSI, Switzerland  Department of Physics,
Chalmers University of Technology, SE-41296 Göteborg, Sweden  Z. Guguchia  Ch.
Wang  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, Villigen
PSI, Switzerland  J. Sichelschmidt  M. Baenitz  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  R.
Nath<EMAIL_ADDRESS>School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India

###### Abstract

We present the ground state properties of a new quantum antiferromagnet YbBO3
in which the isotropic Yb3+ triangular layers are separated by a non-magnetic
layer of partially occupied B and O(2) sites. The magnetization and heat
capacity data establish a spin-orbit entangled effective spin $J_{\rm
eff}=1/2$ state of Yb3+ ions at low temperatures, interacting
antiferromagnetically with an intra-layer coupling $J/k_{\rm B}\simeq 0.53$ K.
The absence of oscillations and a $1/3$ tail in the zero-field muon
asymmetries rule out the onset of magnetic long-range-order as well as spin-
freezing down to 20 mK. An anomalous broad maximum in the temperature
dependent heat capacity with a unusually reduced value and a broad anomaly in
zero-field muon depolarization rate centered at $T^{*}\simeq
0.7\frac{J}{k_{\rm B}}$ provide compelling evidence for a wide fluctuating
regime ($0.182\leq T/J\leq 1.63$) with slow relaxation. We infer that the
fluctuating regime is a universal feature of a highly frustrated triangular
lattice antiferromagnets while the absence of magnetic long-range-order is due
to perfect two-dimensionality of the spin-lattice protected by non-magnetic
site disorder.

Frustrated magnets are a special class of systems where the ground state
degeneracy leads to captivating low temperature properties. One such emergent
phenomenon is the quantum spin liquid (QSL), a disordered state characterized
by fractionalized excitations, quantum entanglement, and absence of magnetic
long-range-order (LRO) [1, *Broholmeaay0668]. The quest for this exotic phase
has escalated remarkably since Anderson’s proposal of resonating valence bond
model, a prototype of QSL in spin-$1/2$ triangular lattice antiferromagnets
(TLAFs) [3]. However, the subsequent theoretical studies revealed three-
sublattice Néel order as the real ground state for Heisenberg TLAFs [4,
*Capriotti3899]. On a pragmatic point of view, the real materials are often
plagued with certain degree of perturbations such as exchange anisotropy,
couplings beyond nearest-neighbor (NN), structural disorder etc that may alter
the actual ground state and pave the way even for richer physics [6, 7, 8, 9].
A prominent manifestation of such effects is the broad fluctuating regime with
slow dynamics observed in TLAFs (Na,K,H)CrO2 in low temperatures [10, 11].
Indeed, spin-$1/2$ TLAFs with above perturbations are recently perceived as an
ideal host for gapped/gapless QSL and other exotic pahses [12, 13].

Rare-earth ($4f$) based TLAFs, mainly with ytterbium (Yb3+) set a new platform
to explore non-trivial phases of matter. Here, the interplay of strong spin-
orbit coupling (SOC) and non-cubic crystal electric field (CEF) lead to
Kramers doublet with an effective $J_{\rm eff}=1/2$ ground state. Moreover,
the strong SOC is expected to induce anisotropic exchange interactions, which
is proposed to be an ingredient in stabilizing QSL state [14]. Further,
disorder in the frustrated magnets is believed to destroy the states that are
stabilized in disorder-free compounds and concedes various fascinating ground
states [15]. For instance, recently, it was found that structural disorder can
trigger bond randomness and promote QSL like states [7, 6]. The most
celebrated compounds in this category are Yb(Mg,Zn)GaO4 where disorder due to
site mixing of Ga3+ and Mg2+/Zn2+ leads to randomized interactions which
apparently forbids LRO and results highly fluctuating moments [16, 17, 18, 19,
20, 21, 22]. This opens up a new paradigm for realizing randomness induced
non-trivial states that mimic QSL [23, 24, 25, 26]. The disorder free
chalcogenides (Na,Cs,Li)Yb$X_{2}$ ($X=$ O, S, and Se) featuring triangular
lattice also show signature of QSL in zero-field and field induced transition
in higher fields [27, 28, 29, *Ding144432, 31, 32, 33, 34, *Dai021044, 36].
Though, no LRO is detected down to 50 mK in another disorder-free TLAF
Ba3Yb(BO3)3 but the dominant interaction is found to be long-range dipole-
dipole coupling rather than exchange interactions [37, 38].

Herein, we report a comprehensive study of a new TLAF YbBO3 (hexagonal,
$P6_{3}/m$) with partial non-magnetic site-disorder. The magnetic Yb3+ ions
fully occupy the regular triangular sites and are free from any sort of
magnetic site-disorder (see Fig. 1) [39]. One of the oxygen site [O(2)] out of
two and the only one boron site are 33% occupied each. The triangular layers
are separated by a layer of these partially occupied B and O(2) sites. We
choose YbBO3 in order to investigate how the non-magnetic site disorder
affects the local environment of Yb3+ and hence the ground state. We
illustrate that the Yb3+ ions form Kramers doublet with effective spin $J_{\rm
eff}=1/2$ at low temperatures. Despite a relatively large Curie-Weiss
temperature $\theta_{\rm CW}\simeq-0.8$ K, magnetic LRO is forbidden down to
20 mK. Intriguingly, the $\mu$SR measurements reveal a broad low temperature
fluctuating regime with slow dynamics. These features are attributed to the
combined effects of magnetic frustration and non-magnetic site disorder.

Figure 1:  (a) Crystal structure of YbBO3 showing triangular Yb3+ sheets
separated by a layer of B and O atoms. The O(1) site is fully occupied while
O(2) and B sites are only occupied 33% each. (b) A section of the Yb3+ layer
with equilateral triangles.

Polycrystalline sample of YbBO3 was synthesized via the conventional solid-
state reaction technique. The phase purity and crystal structure [hexagonal,
$P6_{3}/m$ (No. 176)] of the compound were confirmed from powder x-ray
diffraction [see the Supplementary Material (SM) for details [40]] [39].
Magnetization ($M$) was measured using a SQUID magnetometer (MPMS-3, Quantum
Design) down to 0.4 K using a 3He (iHelium3, Quantum Design Japan) attachment
to the MPMS. Heat capacity [$C_{\rm p}(T)$] measurement was performed on a
piece of pellet using a PPMS (Quantum Design). For measurements down to 0.4 K,
a 3He attachment was used in PPMS. The $\mu$SR measurement was performed at
the S$\mu$S muon source at Paul Scherrer Institute using a combination of two
spectrometers (GPS and HAL) down to 20 mK in zero-field. The details of the
$\mu$SR experiment is described in the SM[40].

Figure 2:  (a) $\chi(T)$ of YbBO3 measured in various applied magnetic fields.
Inset: The low-temperature $\chi(T)$ data (after subtracting $\chi_{\rm VV}$)
along with the CW fit for $H=0.01$ T. (b) Effective moment ($\mu_{\rm eff}$)
as a function of $T$. (c) $M$ vs $H$ measured at different temperatures along
with the Brillouin fits (solid lines). The horizontal dashed line marks the
Van-Vleck contribution. (d) $M$ vs $H/T$ to visualize the scaling of
magnetization curves in the correlated regime.

Magnetic susceptibility $\chi~{}[\equiv M/H]$ as a function of temperature in
different applied fields ($H$) is depicted in Fig. 2(a). $\chi(T)$ increases
monotonically as the temperature is lowered without showing any signature of
magnetic long-range-order (LRO) down to 0.4 K, a possible finger print of
disorder ground state. Above 150 K, the inverse susceptibility ($1/\chi$) for
$H=0.01$ T could be fitted well by $\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm
CW}}$, where $\chi_{0}$ is the $T$-independent susceptibility and the second
term is the Curie-Weiss (CW) law. The fit yields $\chi_{0}\simeq 6.3\times
10^{-4}$ cm3/mol, a high-$T$ effective moment $\mu_{\rm eff}^{\rm HT}$
$[=\sqrt{(3k_{\rm B}C/N_{\rm A}\mu_{\rm B}^{2})}$, where $C$, $k_{\rm B}$,
$N_{\rm A}$, and $\mu_{\rm B}$ are the Curie constant, Boltzmann constant,
Avogadro’s number, and Bohr magneton, respectively] $\simeq 4.53~{}\mu_{\rm
B}$, and a high-$T$ CW temperature $\theta_{\rm CW}^{\rm HT}\simeq-62.6$ K.
This value of $\mu_{\rm eff}^{\rm HT}$ is in good agreement with the expected
value of $4.54~{}\mu_{\rm B}$ for Yb3+ ($J=7/2$, $g=8/7$) in the
${}^{4}f_{13}$ configuration.

$1/\chi$ is found to change the slope below 80 K and displays a perfect linear
behaviour [inset of Fig. 2(a)] after substraction of the Van-Vleck
susceptibility $\chi_{\rm VV}\simeq 3.32\times 10^{-3}$ cm3/mol, obtained from
the $M$ vs $H$ analysis, discussed later. A CW fit below 50 K yields $\mu_{\rm
eff}\simeq 3.2~{}\mu_{\rm B}$ and $\theta_{\rm CW}\simeq-0.8$ K. This
experimental $\mu_{\rm eff}$ value is reminiscent of an effective spin $J_{\rm
eff}=1/2$ with an average $g\simeq 3.6$. Further, this value of $g$ is
significantly larger than the free-electron $g$-value of 2.0 due to the spin-
orbit coupling and is consistent with the one extracted from our ESR
experiments ($g\simeq 3.4$) [40]. Typically, at low temperatures, the magnetic
behavior of Yb3+-based compounds is governed by the Kramers doublets
($m_{J}=\pm\frac{1}{2}$) triggered by the CEF. In this case, the low-lying
ground-state is described by an effective pseudo-spin $J_{\rm eff}=1/2$
Hamiltonian and the higher-lying levels produce a sizable Van-Vleck
contribution ($\chi_{\rm VV}$) to $\chi$ [14, 16, 41]. Indeed, several
Yb3+-based compounds have witnessed $J_{\rm eff}=1/2$ ground state [28, 29,
*Ding144432, 42]. The negative sign of $\theta_{\rm CW}$ indicates dominant
AFM intra-plane coupling. Taking the number of NN spins $z=6$ for a TLAF and
the experimental value of $\theta_{\rm CW}$, the NN AFM exchange coupling
[$\theta_{\rm CW}=-zJS(S+1)/3k_{\rm B}$] is estimated to be $J/k_{\rm B}\simeq
0.53$ K.

In order to establish the correlated behaviour at low temperatures, the
effective magnetic moment $\mu_{\rm eff}$ [$=\\{(3k_{B}/N_{A}\mu_{\rm
B}^{2})\chi T\\}^{1/2}=2.8284\sqrt{\chi T}$] vs $T$ in different applied
fields is plotted in Fig. 2(b). At high-$T$s, $\mu_{\rm eff}$ for all the
fields approaches the free ion value of $\sim 4.53~{}\mu_{\rm B}$ and then
falls to a plateau of $\sim 3.3$ $\mu_{\rm B}$ in the $T$-range $4-20$ K
corresponding to $J_{\rm eff}=1/2$. At low temperatures ($T<4$ K), the curves
deviate significantly from the plateau value. In low fields ($H<1$ T)
$\mu_{\rm eff}(T)$ tends to show an upward curvature while in high fields
($H\geq 1$ T) it shows a downward curvature. This downward curvature in high-
fields is a clear testimony of the development of AFM correlation at low
temperatures [43].

Figure 2(c) presents the $M$ vs $H$ curves measured at various temperatures in
the low-$T$ regime. At $T=0.4$ K, $M$ saturates at around $H_{\rm S}\simeq
0.7$ T and then increases weakly for higher fields due to the Van-Vleck
contribution. The value of $H_{\rm S}$ exactly reproduces the intra-plane
exchange coupling $J/k_{\rm B}=g\mu_{\rm B}H_{\rm S}/4.5k_{\rm B}\simeq 0.53$
K, taking $g\simeq 3.4$. This indicates that the inter-layer and/or second NN
interactions are negligible and do not contribute to $H_{\rm S}$ [28]. The
slope of the linear fit for $H>3$ T results $\chi_{\rm VV}\simeq 3.32\times
10^{-3}$ cm3/mol and the $y$-intercept corresponds to the saturation
magnetization $M_{\rm sat}\simeq 1.6~{}\mu_{\rm B}$. This value of $M_{\rm
sat}$ provides $g\simeq 3.2$ for $J_{\rm eff}=1/2$ ($M_{\rm sat}=gJ_{\rm
eff}~{}\mu_{\rm B}$) which is very close to our ESR value ($g=3.4$) [28] (see
SM [40]). In the absence of exchange interaction, i.e. in the paramagnetic
(PM) state, a magnetic isotherm can be modeled by $M(H)=fgJ_{\rm
eff}N_{A}\mu_{\rm B}B_{J_{\rm eff}}(H)$, where $f$ is the molar fraction of
free spins and $B_{J_{\rm eff}}(H)$ is the Brillouin function for $J_{\rm
eff}=1/2$ [44]. Figure 2(c) presents the isotherm fits at different
temperatures, fixing $g=3.4$. The value of $f$ is estimated to be about $\sim
0.97$ for all the temperatures. For $T>1.8$ K, the isotherms are well fitted
by the Brillouin function, which comprehends the uncorrelated spins. However,
for $T\leq 1.8$ K, the fit deviates significantly from the experimental data
reflecting the growth of AFM correlation. In order to visualize this striking
feature, we scaled the $T$-axis by dividing $H$ and plotted in Fig. 2(d).
Above 1.8 K, all the curves collapse on to a single curve, implying the PM
nature of the spins while below 1.8 K, the curves progressively deviate from
the scaling, ascertaining the emergence of AFM correlation.

Figure 3:  (a) $C_{\rm p}$ vs $T$ for YbBO3 measured in different applied
magnetic fields. The solid line represents $C_{\rm p}(T)$ of non-magnetic
analog YBO3. The horizontal dashed line guides the Dulong-Petit value. (b)
Entropy change ($\Delta S$) vs $T$ for different magnetic fields. Inset:
$C_{\rm mag}/R$ vs $T/J$ in zero field. (c) Schottky contribution [$C_{\rm
p}(H)-C_{\rm p}(0)$] vs $T$ along with the fit using Eq. (1). (d)
$\Delta/k_{\rm B}$ and $f$ vs $H$ in the right and left $y$-axes,
respectively. The solid line represents the straight line fit to
$\Delta/k_{\rm B}(H)$.

Heat capacity ($C_{\rm p}$) measured as a function of temperature down to 0.4
K in different magnetic fields is shown in Fig. 3(a). At high temperatures,
$C_{\rm p}(T)$ is completely dominated by phononic contribution ($C_{\rm
ph}$). Below about 4 K, $C_{\rm p}$ in zero-field increases towards low
temperatures as the magnetic correlation sets in and then levels off below
$\sim 0.5$ K, similar to NaYb(O,S)2 [28, 27]. Absence of any sharp peak rules
out the onset of magnetic LRO down to 0.4 K in zero-field. An external
magnetic field suppresses the magnetic correlation and gives rise to a broad
anomaly which moves towards higher temperatures with field. This broad feature
portrays the Schottky anomaly due to the Zeeman splitting of the ground state
Kramers doublet. The magnetic heat capacity ($C_{\rm mag}$) is obtained by
subtracting the heat capacity ($C_{\rm ph}$) of the non-magnetic analogue YBO3
from the total $C_{\rm p}$ of YbBO3. The change in magnetic entropy ($\Delta
S$) calculated by integrating $C_{\rm mag}/T$ with respect to $T$ is plotted
in Fig. 3(b) for different magnetic fields. In zero field, $\Delta S$ attains
a saturation value of $\sim 2.76$ J/mol-K which is almost 50% of $R\ln 2$,
expected for $J_{\rm eff}=1/2$. This suggests that the remaining 50% entropy
is released below 0.4 K due to the persistence of strong magnetic correlation.
The saturation value of $\Delta S$ increases with $H$ and is almost recovered
to $R\ln 2$ above 2 T ($>H_{\rm S}$).

To probe the Schottky contribution, the zero field data $C_{\rm p}(T,H=0)$ are
subtracted from the high field data $C_{\rm p}(T,H)$ [i.e. $C_{\rm
Sch}(T,H)=C_{\rm p}(T,H)-C_{\rm p}(T,H=0)$]. Figure 3(c) shows the fit of
$C_{\rm p}(T,H)-C_{\rm p}(T,H=0)$ data using the two level Schottky function
[44]

$C_{\rm Sch}(T)=fR\left(\frac{\Delta}{k_{\rm
B}T}\right)^{2}\frac{e^{(\Delta/k_{\rm B}T)}}{\left[1+e^{(\Delta/k_{\rm
B}T)}\right]^{2}}.$ (1)

Here, $\Delta/k_{\rm B}$ is the crystal field gap between the ground state and
the first excited state doublets and $R$ is the gas constant. The estimated
$f$ and $\Delta/k_{\rm B}$ values are plotted as a function of $H$ in the
right and left $y$-axes, respectively in Fig. 3(d). $f$ increases with $H$ and
then saturates to a value of about $\sim 1$ for $H>2$ T. This confirms that
magnetic field splits the energy levels and excites the free Yb3+ spins to the
higher energy levels. For $H<H_{\rm S}$, a fraction of spins are correlated
which is reduced with increasing field and above $H_{\rm S}$, all the free
spins are excited. This explains why $\Delta S$ increases with $H$ and then
attains a saturation value $Rln2$ for $H>H_{\rm S}$ [38]. Similarly, the peak
maximum of the $C_{\rm p}(H)-C_{\rm p}(0)$ curves remains unchanged for $H>2$
T, suggesting that $\sim 100$% spins become free in higher fields [see Fig.
3(c)]. In Fig. 3(d), $\Delta/k_{\rm B}$ increases linearly with $H$ and a
straight line fit results the zero-field energy-gap $\Delta/k_{\rm B}(0)\simeq
1.5$ K which possibly indicates an intrinsic field in the system [45]. From
the value of $\Delta/k_{\rm B}\simeq 20.6$ K at 9 T, one can estimate the
$g$-value as $\Delta/k_{B}=g\mu_{\rm B}H/k_{B}$ that yields $g\simeq 3.3$,
consistent with the ESR value.

Furthermore, $C_{\rm mag}(T)$ in the inset of Fig. 3(b) shows a plateau at
$\sim 0.5$ K with a maximum value $C_{\rm mag}^{\rm max}/R\simeq 0.137$. For a
frustrated spin-$1/2$ TLAF one expects such a broad maximum at $T/J\simeq
0.84$ but with a reduced value $C_{\rm mag}/R\simeq 0.223$ [46] compared to a
non-frustrated 2D antiferromagnet (0.44) [47, *Kim2705]. Though, our maximum
appears nearly at the anticipated position ($T/J\simeq 0.9$) but the absolute
value is significantly lower than that expected for a TLAF. Similar peak type
feature in YbMgGaO4 and NaYbS2 is ascribed to the disorder induced and
disorder free QSLs, respectively [17, 27]. In order to examine this peculiar
feature, $\mu$SR experiments are carried out in zero field (ZF).

Muon spin is susceptible to extremely small magnetic field ($\sim 10^{-5}$ T)
and because of its much shorter time window (10 ns–15 $\mu$s), $\mu$SR is an
excellent probe in tracing the dynamics of slowly fluctuating magnets. In
general, the ZF $\mu^{+}$ spin depolarization rate ($\lambda$) is related to
spin-spin correlation function $\lambda_{\rm
ZF}=\gamma_{\mu}^{2}\int_{0}^{\infty}\langle B_{\rm loc}^{\perp}(t)\cdot
B_{\rm loc}^{\perp}(0)\rangle\,dt\propto S_{\omega\to 0}^{\perp}$, where
$S_{\omega\to 0}^{\perp}=\int_{0}^{\infty}\langle s_{i}^{\perp}(t)\cdot
s_{i}^{\perp}(0)\rangle\,dt$ is the static spin structure factor [49, 25].
Hence, by measuring muon asymmetry as a function of temperature one can
comprehend the correlated behaviour of a spin system. Figure 4(a) depicts the
muon asymmetry curves at different temperatures down to 20 mK which are fitted
well by a stretched exponential function, $P(t)=P(0)e^{-(\lambda t)^{\beta}}$.
Here, $P(0)$ ($\sim 0.118$ and 0.213 for HAL and GPS, respectively) is the
initial asymmetry. The introduction of stretching parameter $\beta$ suggests
that there is a distribution of the relaxation rates. The obtained $\lambda$
and $\beta$ as a function of $T$ are summarized in Fig. 4(b) and (c),
respectively.

Figure 4: (a) Muon decay asymmetry as a function of time, with solid lines
representing the fit using stretched exponential function. Temperature
dependence of (b) depolarization rate ($\lambda$) and (c) stretching parameter
($\beta$). The shaded portion highlights the fluctuating regime.

At high temperatures, the magnetic moments fluctuate very fast and only mildly
depolarize the implanted muon before it decays and the dominant depolarization
channel is due to nuclear spins. This gives rise to a temperature independent
$\lambda$, as we observed in Fig. 4(b). Similarly, $\beta$ remains temperature
independent in high temperatures and it’s value is slightly below 1,
indicating a distribution likely due to weak disorder in the O-site (muon
stopping site) or a few muon sites. As the temperature is lowered below $\sim
0.9$ K which is close to $\theta_{\rm CW}$, $\beta$ drops and $\lambda$
increases rapidly exhibiting a broad minima and maxima, respectively centered
at $T^{*}\simeq 0.4$ K. The rapid increase in $\lambda$ suggest the slowing
down of the fluctuating moments as the correlation sets in and replicates the
behaviour of $C_{\rm p}(T)$. Upon further lowering the temperature, the values
of $\beta$ and $\lambda$ recover back to their initial values below $\sim 0.1$
K, clearly suggesting an exotic dynamical regime (0.1 K $\leq T\leq 0.9$ K)
with slow fluctuations, marked by shading in Fig. 4(b) and (c) [50]. In the PM
regime ($T>1$ K) where $\lambda_{\rm ZF}$ is temperature independent, the spin
fluctuation rate ($\nu$) can be calculated as $\nu=\sqrt{z}JS/\hbar\sim
8.542\times 10^{10}$ Hz taking $z=6$, $S=1/2$, and $J/k_{\rm B}\simeq 0.53$ K
[51, 52]. In the fast fluctuation limit, using the relation $\lambda_{\rm
ZF}=2\Delta_{\mu}^{2}/\nu$, the distribution width of the local magnetic
fields is estimated to be $\Delta_{\mu}\simeq 254$ MHz $\ll\nu$ considering
$\lambda_{\rm ZF}~{}(T>1~{}{\rm K})\simeq 1.5~{}\mu{\rm s}^{-1}$.

In case of a conventional magnetic LRO, the muon decay asymmetry should
exhibit an oscillating signal while for a spin-glass (SG) transition, it
should feature a non-oscillating and undamped $1/3$ tail for the
polycrystalline samples [25]. Further, for a SG transition, $\beta$ is
expected to drop to $\sim 0.33$ as $T$ tends to $T_{\rm g}$ and $\lambda$ to
show a peak or divergence [53, 54, 55]. Absence of any such signatures in Fig.
4 rules out both a static magnetic LRO and a SG transition. We also do not
observe any saturation of the increased $\lambda$ at low temperatures, in
contrast to that reported for many QSL candidates due to persistent spin
fluctuations [56, 25, 52]. Additionally, though, $\beta$ value drops to $\sim
0.4$ at $T^{*}$ but it recovers at low temperatures without settling to a
particular value as empirically observed in QSL systems. These observations
exclude the QSL scenario and imply that the slowly fluctuating regime is
restricted to $0.182\leq T/J\leq 1.63$ which could be due to short-range
correlations.

The observed peculiar behaviour prompted us to compare the structural aspects
of the Yb3+ based TLAFs. The ratio of interlayer to intralayer spacing $d_{\rm
inter}/d_{\rm intra}\simeq 1.17$ in YbBO3 is smaller compared to NaYb(O,S,Se)2
($\sim 1.64-1.71$) and YbGaMgO4 ($\sim 2.4$) but almost equal to NaBaYb(BO3)2
($\sim 1.1$) [57]. In NaYb(O,S,Se)2, the Yb3+ layers follow $ABC$ stacking.
This staggered arrangement generates inter-layer frustration which possibly
forbids magnetic LRO in zero-field and foster QSL state [28, 31, 27, 33].
Similarly, for YbGaMgO4, large $d_{\rm inter}$ leads to pronounced two-
dimensionality, $ABC$ stacking produces inter-layer frustration, and non-
magnetic Mg/Ga site mixing results bond-randomness [7]. Consequently, the
dominant effect from randomness along with magnetic frustration is responsible
for curbing of magnetic LRO and establishing QSL-like behaviour in YbGaMgO4
[7, 18, 16]. On the other hand, NaBaYb(BO3)2 which has $ABC$ stacking, does
not possess any structural disorder and undergoes a magnetic LRO at $T_{\rm
N}\simeq 0.4$ K [42, 42, 58]. Despite the same $d_{\rm inter}/d_{\rm intra}$
ratio as that of NaBaYb(BO3)2 and with $AAA$ stacking, YbBO3 manifests no
magnetic LRO and exhibits a unconventional fluctuating regime.

Interestingly, the behaviour of $\lambda(T)$ resembles with that of the
$S=3/2$ TLAFs $\alpha$-HCrO2 and NaCrO2 [10, 11]. In these compounds, there
exists a broad fluctuating regime with slow dynamics at low temperatures
followed by the formation of a static order at very low temperatures. The
$\lambda$ vs $T/J$ plot divulges the broad maximum centered at the same
position ($T/J\simeq 0.6-0.7$) for both compounds, establishing that the broad
dynamical regime is universal to TLAFs. These chromates belong to the
delafossite family where the interlayer frustration due to $ABC$-stacking of
the magnetic layers was believed to be the main reason for this unconventional
dynamics. For YbBO3, $C_{\rm p}(T)$ shows a very broad peak and the center
position of the broad maxima in $\lambda$ corresponds to $T/J\simeq 0.7$ with
$J/k_{\rm B}\simeq 0.53$ K, quite similar to the chromates. However, unlike
the delafossites, the Yb3+ layers in YbBO3 are arranged in a $AAA$-stacking
sequence which is expected to produce a non-frustrated interlayer geometry and
should favour magnetic LRO at a finite temperature depending on the strength
of the inter-layer coupling [33].

A closer inspection of the crystal structure revealed that the Yb3+ ions are
eight-fold coordinated with oxygen [six O(1) and six O(2)] atoms forming YbO8
polyhedra [39]. The polyhedra of two adjacent layers are connected only via
partially occupied O(2) and B sites which provide the inter-layer interaction
path. This positional disorder of O(2) and B atoms adversely affects the local
environment of Yb3+ and may induce bond randomness, similar to YbMgGaO4.
Secondly, it also weakens the inter-layer interaction which is responsible for
the onset of three-dimensional (3D) magnetic LRO. As a consequence, the 3D
magnetic LRO is prevented, rendering the spin-lattice an ideal 2D system.
Thus, the absence of magnetic LRO can be accredited to the non-magnetic site
disorder and the intermediate fluctuating regime with slow dynamics could be a
universal feature of isotropic TLAFs arising due to strong intra-layer
frustration.

In summary, we performed a comprehensive study of the static and dynamic
properties of a new quantum magnet YbBO3 possessing isotropic Yb3+ triangular
lattice. The low temperature properties are described well by $J_{\rm
eff}=1/2$ Kramers doublets of the Yb3+ ions. Neither magnetic LRO nor spin
freezing are witnessed down to 20 mK. setting the lower limit of the
frustration parameter $f(=|\theta_{\rm CW}|/T_{\rm N})\simeq 40$, a factual
hallmark of a highly frustrated magnet. The analysis of $\chi(T)$ and magnetic
isotherms suggests the emergence of AFM correlation below $\sim 0.9$ K. The
$\mu$SR depolarization rate together with heat capacity data demonstrate an
extended fluctuating regime with slow dynamics at low temperatures. This
unusual dynamics and the absence of magnetic LRO are ascribed to the effects
of intra-layer frustration of a perfect 2D TLAF and positional disorder,
respectively. With these features YbBO3 stands out as an exception among the
Yb3+-based TLAFs.

##  I Acknowledgments

We would like to acknowledge SERB, India for financial support bearing
sanction Grant No. CRG/2019/000960. Part of this work is based on experiments
performed at the Swiss Muon Source S$\mu$S, Paul Scherrer Institute, Villigen,
Switzerland. The project has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 884104 (PSI-FELLOW-III-3i). We thank Robert
Scheuermann for his support with the HAL instrument.

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