# Damping in yttrium iron garnet film with an interface

Ravinder Kumar<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Kanpur 208016, India.  Institute of Physics, Bhubaneswar 751005,
India.  B. Samantaray  Department of Physics, Indian Institute of Technology,
Kanpur 208016, India.  Shubhankar Das  Institute of Physics, Johannes
Gutenberg-University Mainz, 55099 Mainz, Germany.  Kishori Lal  Department of
Physics, Indian Institute of Technology, Kanpur 208016, India.  D. Samal
Institute of Physics, Bhubaneswar 751005, India.  Homi Bhabha National
Institute, Anushakti Nagar, Mumbai 400085, India.  Z. Hossain
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur
208016, India.  Institute of Low Temperature and Structure Research, 50-422
Wroclaw, Poland.

###### Abstract

We report strong damping enhancement in a 200 nm thick yttrium iron garnet
(YIG) film due to spin inhomogeneity at the interface. The growth-induced thin
interfacial gadolinium iron garnet (GdIG) layer antiferromagnetically (AFM)
exchange couples with the rest of the YIG layer. The out-of-plane angular
variation of ferromagnetic resonance (FMR) linewidth $\Delta H$ reflects a
large inhomogeneous distribution of effective magnetization $\Delta 4\pi
M_{eff}$ due to the presence of an exchange springlike moments arrangement in
YIG. We probe the spin inhomogeneity at the YIG-GdIG interface by performing
an in-plane angular variation of resonance field $H_{r}$, leading to a
unidirectional feature. The large extrinsic $\Delta 4\pi M_{eff}$
contribution, apart from the inherent intrinsic Gilbert contribution,
manifests enhanced precessional damping in YIG film.

##  I Introduction

The viability of spintronics demands novel magnetic materials and YIG is a
potential candidate as it exhibits ultra-low precessional damping, $\alpha\sim
3\times 10^{-5}$Sparks (1964). The magnetic properties of YIG thin films
epitaxially grown on top of Gd3Ga5O12 (GGG) vary significantly due to growth
tuningHauser  _et al._ (2016); Kumar  _et al._ (2017), film thicknessWang  _et
al._ (2014), heavy metals substitutionHelseth  _et al._ (2001); Kumar  _et
al._ (2019); Onbasli  _et al._ (2016) and coupling with thin metallic
layersSun  _et al._ (2013); Wang  _et al._ (2015); Chang  _et al._ (2017). The
growth processes may also induce the formation of a thin interfacial-GdIG
layer at the YIG-GGG interfaceJakubisova  _et al._ (2016); Gomez-Perez  _et
al._ (2018); Kumar  _et al._ (2021). The YIG-GdIG heterostructure derived out
of monolithic YIG film growth on GGG exhibits interesting phenomena such as
all-insulating equivalent of a synthetic antiferromagnetGomez-Perez  _et al._
(2018) and hysteresis loop inversion governed by positive exchange-bias Kumar
_et al._ (2021). The radio frequency magnetization dynamics on YIG-GdIG
heterostructure still remains unexplored and need a detailed FMR study.

The relaxation of magnetic excitation towards equilibrium is governed by
intrinsic and extrinsic mechanisms, leading to a finite $\Delta H$Yuan  _et
al._ (2003); Zakeri  _et al._ (2007). The former mechanism dictates Gilbert
type relaxation, a consequence of direct energy transfer to the lattice
governed by both spin-orbit coupling and exchange interaction in all magnetic
materialsYuan  _et al._ (2003); Zakeri  _et al._ (2007). Whereas, the latter
mechanism is a non-Gilbert-type relaxation, divided mainly into two
categoriesYuan  _et al._ (2003); Zakeri  _et al._ (2007)\- (i) the magnetic
inhomogeneity induced broadening: inhomogeneity in the internal static
magnetic field, and the crystallographic axis orientation; (ii) two-magnon
scattering: the energy dissipates in the spin subsystem by virtue of magnon
scattering with nonzero wave vector, $k\neq 0$, where, the uniform resonance
mode couples with the degenerate spin waves. The angular variation of $H_{r}$
provides information about the presence of different magnetic anisotropiesWang
_et al._ (2014); Kumar  _et al._ (2019). Most attention has been paid towards
the angular dependence of $H_{r}$Wang  _et al._ (2014); Kumar  _et al._
(2019), whereas, the angular variation of the $\Delta H$ is sparsely
investigated. The studies involving angular dependence of $\Delta H$ may help
to probe different contributions to the precessional damping.

In this paper, the effects of intrinsic and extrinsic relaxation mechanisms on
precessional damping of YIG film is studied extensively using FMR technique.
An enhanced value of $\alpha\sim 1.2\times 10^{-3}$ is realized, which is
almost two orders of magnitude higher than what is usually seen in YIG thin
films, $\sim 6\times 10^{-5}$Sparks (1964); Hauser  _et al._ (2016). The out-
of-plane angular variation of $\Delta H$ shows an unusual behaviour where spin
inhomogeneity at the interface plays significant role in defining the $\Delta
H$ broadening and enhanced $\alpha$. In-plane angular variation showing a
unidirectional feature, demands the incorporation of an exchange anisotropy to
the free energy density, evidence of the presence of an AFM exchange coupling
at the YIG-GdIG interface. The AFM exchange coupling leads to a Bloch domain-
wall-like spiral moments arrangement in YIG and gives rise to a large $\Delta
4\pi M_{eff}$. This extrinsic $\Delta 4\pi M_{eff}$ contribution due to spin
inhomogeneity at the interface adds up to the inherent Gilbert contribution,
which may lead to a significant enhancement in precessional damping.

##  II Sample and Measurement Setups

We deposit a $\sim 200$ nm thick epitaxial YIG film on GGG(111)-substrate by
employing a KrF Excimer laser (Lambda Physik COMPex Pro, $\lambda$ = 248
$\mathrm{nm}$) of 20 $\mathrm{ns}$ pulse width. A solid state synthesized
$Y_{3}Fe_{5}O_{12}$ target is ablated using an areal energy of 2.12 J.cm-2
with a repetition frequency of 10 $\mathrm{Hz}$. The GGG(111) substrate is
placed 50 $\mathrm{mm}$ away from the target. The film is grown at 800 oC
temperature and in-situ post annealed at the same temperature for 60 minutes
in pure oxygen environment. The $\theta-2\theta$ X-ray diffraction pattern
shows epitaxial growth with trails of Laue oscillations (Fig. 3(a) of refKumar
_et al._ (2017)). FMR measurements are performed using a Bruker EMX EPR
spectrometer and a broadband coplanar waveguide (CPW) setup. The former
technique uses a cavity mode frequency $f\approx 9.60$ GHz, and enables us to
perform FMR spectra for various $\theta_{H}$ and $\phi_{H}$ angular
variations. The latter technique enables us to measure frequency dependent FMR
spectra. We define the configurations H parallel ($\theta_{H}=90^{o}$) and
perpendicular ($\theta_{H}=0^{o}$) to the film plane for rf frequency and
angular dependent measurements. The resultant spectra are obtained as the
derivative of microwave absorption w.r.t. the applied field H.

##  III Results and Discussion

###  III.1 Broadband FMR

Fig. 1(a) shows typical broadband FMR spectra in a frequency $f$ range of 1.5
to 13 $\mathrm{GHz}$ for 200 nm thick YIG film at temperature $T=300$ K and
$\theta_{H}=90^{o}$. The mode appearing at a lower field value is the main
mode, whereas the one at higher field value represents surface mode. We
discuss all these features in detail in the succeeding subsection III.2. We
determine the resonance field $H_{r}$ and linewidth (peak-to-peak linewidth)
$\Delta H$ from the first derivative of the absorption spectra. Fig. 1(b)
shows the rf frequency dependence of $H_{r}$ at $\theta_{H}=90^{o}$ and
$0^{o}$. We use the Kittel equation for fitting the frequency vs. $H_{r}$ data
from the resonance condition expressed asChang  _et al._ (2017),
$f=\gamma\left[{H_{r}}({H_{r}}+4\pi{M_{eff}})\right]^{1/2}/(2{\pi})$ for
$\theta_{H}=90^{o}$ and $f=\gamma({H_{r}}-4\pi{M_{eff}})/(2{\pi})$ for
$\theta_{H}=0^{o}$. Where, $\gamma=g\mu_{B}/\hslash$ is the gyromagnetic
ratio, $4\pi M_{eff}=4\pi M_{S}-H_{ani}$ is the effective magnetization
consisting of $4\pi M_{S}$ saturation magnetization (calculated using M(H))
and $H_{ani}$ anisotropy field parametrizing cubic and out-of-plane uniaxial
anisotropies. The fitting gives $4\pi M_{eff}\approx 2000$ Oe, which is used
to calculate the $H_{ani}\approx-370$ Oe.

Fig. 1(c) shows the frequency dependence of $\Delta H$ at $\theta_{H}=90^{o}$.
The intrinsic and extrinsic damping contributions are responsible for a finite
width of the FMR signal. The intrinsic damping $\Delta H_{int}$ arises due to
the Gilbert damping of the precessing moments. Whereas, the extrinsic damping
$\Delta H_{ext}$ exists due to different non-Gilbert-type relaxations such as
inhomogeneity due to the distribution of magnetic anisotropy $\Delta
H_{inhom}$, or two-magnon scattering (TMS) $\Delta H_{TMS}$. The intrinsic
Gilbert damping coefficient ($\alpha$) can be determined using the Landau-
Liftshitz-Gilbert equation expressed asChang  _et al._ (2017), $\Delta
H=\Delta H_{in}+\Delta H_{inhom}=(4\pi\alpha/\sqrt{3}\gamma)f+\Delta
H_{inhom}$. Considering the above equation where $\Delta H$ obeys linear $f$
dependence, the slope determines the value of $\alpha$, and $\Delta H_{inhom}$
corresponds to the intercept on the vertical axis. We observe a very weak non-
linearity in the $f$ dependence of $\Delta H$, which is believed to be due to
the contribution of TMS to the linewidth $\Delta H_{TMS}$. The non-linear $f$
dependence of $\Delta H$ in Fig. 1(c) can be described in terms of TMS,
assuming $\Delta H=\Delta H_{in}+\Delta H_{inhom}+\Delta H_{TMS}$. We put a
factor of $1/\sqrt{3}$ to $\Delta H$ due to the peak-to-peak linewidth value
extractionYuan  _et al._ (2003). The TMS induces non-linear slope at low
frequencies, whereas a saturation is expected at high frequencies. TMS is
induced by scattering centers and surface defects in the sample. The defects
with size comparable to the wavelength of spin waves are supposed to act as
scattering centres. The TMS term at $\theta_{H}=90^{o}$ can be expressed
asJermain  _et al._ (2017)-

$\Delta
H_{TMS}(\omega)=\Gamma\sin^{-1}\sqrt{\frac{\sqrt{\omega^{2}+(\omega_{0}/2)^{2}}-\omega_{0}/2}{\sqrt{\omega^{2}+(\omega_{0}/2)^{2}}+\omega_{0}/2}},$
(1)

with $\omega=2\pi f$ and $\omega_{0}=\gamma 4\pi M_{eff}$. The prefactor
$\Gamma$ defines the strength of TMS. The extracted values are as follows:
$\alpha=1.2\times 10^{-3}$, $\Delta H_{0}=13$ Oe and $\Gamma=2.5$ Oe. The
Gilbert damping for even very thin YIG film is extremely low, $\sim 6\times
10^{-5}$. Whereas, the value we achieved is higher than the reported in the
literature for YIG thin filmsHauser  _et al._ (2016). Also, the value of
$\Gamma$ is insignificant, implying negligible contribution to the damping.

Figure 1: Room temperature frequency dependent FMR measurements. (a)
Representative FMR derivative spectra for different frequencies at
$\theta_{H}=90^{o}$. (b) Resonance field vs. frequency data for
$\theta_{H}=90^{o}$ and $\theta_{H}=0^{o}$ are represented using red and blue
data points, respectively. The fitting to both the data are shown using black
lines. (c) Linewidth vs. frequency data at $\theta_{H}=90^{o}$. The solid red
circles represent experimental data, whereas the solid black line represents
$\Delta H$ fitting. Inhomogeneous ($\Delta H_{inhom}$), Gilbert ($\Delta
H_{\alpha}$) and two-magnon scattering ($\Delta H_{TMS}$) contributions to
$\Delta H$ are shown using dashed green, solid yellow and blue lines,
respectively.

###  III.2 Cavity FMR

Fig. 2(a) shows typical $T=300$ K cavity-FMR ($f\approx 9.6$ GHz) spectra for
YIG film performed at different $\theta_{H}$. The FMR spectra exhibit some
universal features: (i) Spin-Wave resonance (SWR) spectrum for
$\theta_{H}=0^{o}$; (ii) rotating the H away from the $\theta_{H}=0^{o}$, the
SWR modes successively start diminishing, and at certain critical angle
$\theta_{c}$ (falls in a range of $30-35^{o}$; shaded region in Fig. 2(b)),
all the modes vanish except a single mode (uniform FMR mode). Further rotation
of H for $\theta_{H}>\theta_{c}$, the SWR modes start re-emerging. We observe
that the SWR mode appearing at the higher field side for
$\theta_{H}>\theta_{c}$, represents an exchange-dominated non-propagating
surface modeVittoria and Schelleng (1977); Yu  _et al._ (1975); Liu  _et al._
(2007). The above discussed complexity in $H_{r}$ vs $\theta_{H}$ behaviour
has already been realized in some material systemsLiu  _et al._ (2007),
including a $\mu$-thick YIG filmYu  _et al._ (1975). The localized mode or
surface spin-wave mode appears for $\textbf{H}\parallel$ but not $\perp$ to
the film-planeVittoria and Schelleng (1977); Yu  _et al._ (1975); Liu  _et
al._ (2007). We assign the SWR modes for the sequence $n=1,2,3,....$, as it
provides the best correspondence to $H_{ex}\propto n^{2}$, where,
$H_{ex}=H_{r}(n)-H_{r}(0)$ defines exchange fieldKlingler  _et al._ (2014).
The exchange stiffness can be obtained by considering the modified Schreiber
and Frait classical approach using the mode number $n^{2}$ dependence of
resonance field (inset Fig. 3(c))Klingler  _et al._ (2014). For a fixed
frequency, the exchange field $H_{ex}$ of thickness modes is determined by
subtracting the highest field resonance mode ($n=1$) from the higher modes
($n\neq 1$). In modified Schreiber and Frait equation, the $H_{ex}$ shows
direct dependency on the exchange stiffness $D$:
$\mu_{0}H_{ex}=D\frac{\pi^{2}}{d^{2}}n^{2}$ (where $d$ is the film thickness).
The linear fit of data shown in the inset of Fig. 2(b) gives $D=3.15\times
10^{-17}$ T.m2. The exchange stiffness constant $A$ can be determined using
the relation $A=D\ M_{S}/2$. The calculated value is $A=2.05$ pJ.m-1, which is
comparable to the value calculated for YIG, $A=3.7$ pJ.m-1Klingler  _et al._
(2014).

YIG thin films with in-plane easy magnetization exhibit extrinsic uniaxial
magnetic and intrinsic magnetocrystalline cubic anisotropiesLee  _et al._
(2016). The total free energy density for YIG(111) is given byLandau and
Lifshitz (1984); Lee  _et al._ (2016):

$\begin{array}[]{c}F=-H{M_{S}}\left[\begin{array}[]{l}\sin{\theta_{H}}\sin{\theta_{M}}\cos\left({{\phi_{H}}-{\phi_{M}}}\right)\\\
+\cos{\theta_{H}}\cos{\theta_{M}}\end{array}\right]\\\ +2\pi
M_{S}^{2}{\cos^{2}}{\theta_{M}}-{K_{u}}{\cos^{2}}{\theta_{M}}\\\
+\frac{{{K_{1}}}}{{12}}\left(\begin{array}[]{l}7{\sin^{4}}{\theta_{M}}-8{\sin^{2}}{\theta_{M}}+4-\\\
4\sqrt{2}{\sin^{3}}{\theta_{M}}\cos{\theta_{M}}\cos
3{\phi_{M}}\end{array}\right)\\\
+\frac{{K_{2}}}{{108}}\left(\begin{array}[]{l}-24{{\sin}^{6}}{\theta_{M}}+45{{\sin}^{4}}{\theta_{M}}-24{{\sin}^{2}}{\theta_{M}}+4\\\
-2\sqrt{2}{{\sin}^{3}}{\theta_{M}}\cos{\theta_{M}}\left({5{{\sin}^{2}}{\theta_{M}}-2}\right)\cos
3{\phi_{M}}\\\ +{{\sin}^{6}}{\theta_{M}}\cos
6{\phi_{M}}\end{array}\right)\end{array}\ $ (2)

The Eq. 2 consists of the following different energy terms; the first term is
the Zeeman energy, the second term is the demagnetization energy, the third
term is the out-of-plane uniaxial magnetocrystalline anisotropy energy
$K_{u}$, and the last two terms are the first and second order cubic
magnetocrystalline anisotropy energies ($K_{1}$ and $K_{2}$), respectively.
The total free energy density equation is minimized by taking partial
derivatives w.r.t. to $\theta_{M}$ and $\phi_{M}$ to obtain the equilibrium
orientation of the magnetization vector $\bf{M(H)}$, i.e., $\partial
F/\partial\theta_{M}=\partial F/\partial\phi_{M}=0$. The resonance frequency
of uniform precession at equilibrium condition is expressed asLee  _et al._
(2016); Suhl (1955); Smit and Beljers (1955):

${\omega_{res}}=\frac{\gamma}{{{M_{S}}\sin{\theta_{M}}}}{\left[{\frac{{{\partial^{2}}F}}{{\partial\theta_{M}^{2}}}\frac{{{\partial^{2}}F}}{{\partial\phi_{M}^{2}}}-{{\left({\frac{{{\partial^{2}}F}}{{\partial{\theta_{M}}\partial{\phi_{M}}}}}\right)}^{2}}}\right]^{{\raise
2.1097pt\hbox{$1$}\\!\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}\\!\lower
2.1097pt\hbox{$2$}}}}\ $ (3)

Mathematica is used to numerically solve the resonance condition described by
Eq. 3 for the energy density given by Eq. 2. The solution for a fixed
frequency is used to fit the angle dependent resonance data ($H_{r}$ vs.
$\theta_{H}$) shown in Fig. 2(b). The main mode data simulation is shown using
a black line. The parameters obtained from the simulation are
$K_{u}=-1.45\times 10^{4}$ erg.cm-3, $K_{1}=1.50\times 10^{3}$ erg.cm-3, and
$K_{2}=0.13\times 10^{3}$ erg.cm-3. The calculated uniaxial anisotropy field
value is $H_{u}\sim-223$ Oe.

Figure 2: Room temperature out-of-plane angular $\theta_{H}$ dependence of
FMR. (a) Derivative FMR spectra shown for different $\theta_{H}$ performed at
$\approx 9.6$ GHz. (b) $\theta_{H}$ variation of uniform mode and SWR modes of
resonance field $H_{r}$. Inset: Exchange field ($H_{ex}$) vs mode number
square ($n^{2}$). (c) $\theta_{H}$ variation of the linewidth ($\Delta H$),
where, the experimental and simulated data are represented by solid yellow
circles and black line, respectively. The different contributions $\Delta
H_{\alpha}$, $\Delta 4\pi M_{eff}$, $\Delta\theta_{H}$ and $\Delta H_{TMS}$
are represented by gray, purple, green and red lines, respectively.

The $\Delta H$ manifests the spin dynamics and related relaxation mechanisms
in a magnetic system. The intrinsic contribution to $\Delta H$ arises due to
Gilbert term $\Delta H_{int}\approx\Delta H_{\alpha}$, whereas, the extrinsic
contribution $\Delta H_{ext}$ consists of line broadening due to $\Delta
H_{inhom}$ and $\Delta H_{TMS}$. The terms representing the precessional
damping due to intrinsic and extrinsic contributions can be expressed in
different phenomenological forms. Figure 2(c) shows $\Delta H$ as a function
of $\theta_{H}$. The $\theta_{H}$ variation of $\Delta H$ shows distinct
signatures due to different origins of magnetic damping. We consider both
$\Delta H_{int}$ and $\Delta H_{ext}$ magnetic damping contributions to the
broadening of $\Delta H$, $\Delta H=\Delta H_{\alpha}+\Delta H_{inhom}+\Delta
H_{TMS}$. The first term can be expressed asYuan  _et al._ (2003)-

$\Delta
H_{\alpha}=\frac{\alpha}{M_{S}}\left[\frac{\partial^{2}F}{\partial\theta^{2}_{M}}+\frac{1}{\sin^{2}\theta_{M}}\frac{\partial^{2}F}{\partial\phi^{2}_{M}}\right]\bigg{|}\frac{\partial(\frac{2\pi
f}{\gamma})}{\partial H_{r}}\bigg{|}^{-1}.$ (4)

The second term $\Delta H_{inhom}$ has a formYuan  _et al._ (2003)-

$\Delta H_{inhom}=\bigg{|}\frac{dH_{r}}{d4\pi M_{eff}}\bigg{|}\Delta 4\pi
M_{eff}+\bigg{|}\frac{dH_{r}}{d\theta_{H}}\bigg{|}\Delta\theta_{H}.$ (5)

Where, the dispersion of magnitude and direction of the $4\pi M_{eff}$ are
represented by $\Delta 4\pi M_{eff}$ and $\Delta\theta_{H}$, respectively. The
$\Delta H_{inhom}$ contribution arises due to a small spread of the sample
parameters such as thickness, internal fields, or orientation of crystallites
within the thin film. The third term $\Delta H_{TMS}$ can be written
asKurebayashi  _et al._ (2013)-

$\begin{array}[]{c}\Delta
H_{TMS}=\sum\limits_{i=1}\frac{\Gamma_{i}^{out}f_{i}(\phi_{H})}{\mu_{0}\gamma\Phi}\sin^{-1}\sqrt{\frac{\sqrt{\omega^{2}+(\omega_{0}/2)^{2}}-\omega_{0}/2}{\sqrt{\omega^{2}+(\omega_{0}/2)^{2}}+\omega_{0}/2}},\\\
\Gamma_{i}^{out}=\Gamma_{i}^{0}\Phi
A(\theta-\pi/4)\frac{dH_{r}(\theta_{H})}{d\omega(\theta_{H})}\bigg{/}\frac{dH_{r}(\theta_{H}=0)}{d\omega(\theta_{H}=0)}\end{array}$
(6)

The prefactor $\Gamma^{out}_{i}$ defines the TMS strength and has a
$\theta_{H}$ dependency in this case. The type and size of the defects
responsible for TMS is difficult to characterize which makes it non-trivial to
express the exact form of $\Gamma^{out}_{i}$. Although, it may have a
simplified expression given in Eq. 6, where, $\Gamma^{0}_{i}$ is a constant;
$A(\theta-\pi/4)$, a step function which makes sure that the TMS is
deactivated for $\theta_{H}<\pi/4$; and
$dH_{r}(\theta_{H})/d\omega(\theta_{H})$, a normalization factor responsible
for the $\theta_{H}$ dependence of the $\Gamma^{out}_{i}$. In fig. 2(c)the
solid dark yellow circles and black solid line represent the experimental and
simulated $\Delta H$ vs $\theta_{H}$ data, respectively. We also plot
contributions of different terms such as $\Delta H_{\alpha}$ (blue color
line), $\Delta 4\pi M_{eff}$ (purple color line), $\Delta\theta_{H}$ (green
color line) and $\Delta H_{TMS}$ (red color line). The fitting provides
following extracted parameters, $\alpha=1.3\times 10^{-3}$, $\Delta 4\pi
M_{eff}=58$ Oe, $\Delta\theta_{H}=0.29^{o}$ and $\Gamma_{i}^{0}=1.3$ Oe. The
precessional damping calculated from the $\Delta H$ vs. $\theta_{H}$
corroborate with the value extracted from the frequency dependence of $\Delta
H$ data (shown in Fig. 1(c)); $\alpha=1.2\times 10^{-3}$. The $\Delta H$
broadening and the overwhelmingly enhanced precessional damping are the direct
consequence of contributions from intrinsic and extrinsic damping. Usually,
the Gilbert term and the inhomogeneity due to sample quality contribute to the
broadening of $\Delta H$ and enhanced $\alpha$ in YIG thin films. If we
interpret the $\Delta H$ vs $\theta_{H}$ data, it is clear that damping
enhancement in YIG is arising from the extrinsic magnetic inhomogeneity.

The role of an interface in YIG coupled with metals or insulators leading to
the increments in $\Delta H$ and $\alpha$ has been vastly explored. Wang et.
al. Wang  _et al._ (2015) studied a variety of insulating spacers between YIG
and Pt to probe the effect on spin pumping efficiency. Their results suggest
the generation of magnetic excitations in the adjacent insulating layers due
to the precessing magnetization in YIG at resonance. This happens either due
to fluctuating correlated moments or antiferromagnetic ordering, via
interfacial exchange coupling, leading to $\Delta H$ broadening and enhanced
precessional damping of the YIGWang  _et al._ (2015). The impurity relaxation
mechanism is also responsible for $\Delta H$ broadening and enhanced magnetic
damping in YIG, but is prominent only at low temperaturesJermain  _et al._
(2017). Strong enhancement in magnetic damping of YIG capped with Pt has been
observed by Sun et. al. Sun  _et al._ (2013). They suggest ferromagnetic
ordering in an atomically thin Pt layer due to proximity with YIG at the YIG-
Pt interface, dynamically exchange couples to the spins in YIGSun  _et al._
(2013). In recent years, some research groups have reported the presence of a
thin interfacial layer at the YIG-GGG interfaceJakubisova  _et al._ (2016);
Gomez-Perez  _et al._ (2018); Kumar  _et al._ (2021). The 200 nm film we used
in this study is of high quality with a trails of sharp Laue oscillations [see
Fig 3(a) in ref.Kumar  _et al._ (2017)]. Thus it is quite clear that the
observed $\Delta H$ broadening and enhanced $\alpha$ is not a consequence of
sample inhomogeneity. The formation of an interfacial GdIG layer at the YIG-
GGG interface, which exchange couples with the YIG film may lead to $\Delta H$
broadening and increased $\alpha$. Considering the above experimental
evidences leading to $\Delta H$ broadening and enhanced Gilbert damping due to
coupling with metals and insulatorsWang  _et al._ (2015); Sun  _et al._
(2013), it is safe to assume that the interfacial GdIG layer at the interface
AFM exchange couples with the YIGJakubisova  _et al._ (2016); Gomez-Perez  _et
al._ (2018); Kumar  _et al._ (2021), and responsible for enhanced $\Delta H$
and $\alpha$.

Figure 3: (a) In-plane angular $\phi_{H}$ variation of $H_{r}$. The
experimental data are represented by solid grey circles. Whereas, the
simulated data for total and exchange (unidirectional) anisotropy are
represented by black and red solid lines, respectively. (a) 200 nm thick YIG
sample. (b) 100 nm thick YIG sample.

Fig. 3 shows in-plane $\phi_{H}$ angular variation of $H_{r}$. We simulate the
in-plane $H_{r}$ vs $\phi_{H}$ angular variation using the free energy
densities provided in ref. Wang  _et al._ (2017) and an additional term,
$-{K_{EA}}.{\sin}{\theta_{M}}.{\cos}{\phi_{M}}$, representing the exchange
anisotropy ($K_{EA}$). Even though $\phi_{H}$ variation of $H_{r}$ shown in
Fig. 3(a) is not so appreciable as the film is 200 nm thick, a very weak
unidirectional anisotropy trend is visible, suggesting an AFM exchange
coupling between the interface and YIG. It has been shown that the large
inhomogeneous $4\pi M_{eff}$ is a direct consequence of the AFM exchange
coupling at the interface of LSMO and a growth induced interfacial
layerGhising  _et al._ (2020). The YIG thin film system due to the presence of
a hard ferrimagnetic GdIG interfacial layer possesses AFM exchange
couplingJakubisova  _et al._ (2016); Gomez-Perez  _et al._ (2018); Kumar  _et
al._ (2021). A Bloch domain-wall-like spiral moments arrangement takes place
due to the AFM exchange coupling across the interfacial GdIG and top bulk YIG
layerJakubisova  _et al._ (2016); Gomez-Perez  _et al._ (2018); Kumar  _et
al._ (2021). An exchange springlike characteristic is found in YIG film due to
the spiral arrangement of the magnetic moments Jakubisova  _et al._ (2016);
Gomez-Perez  _et al._ (2018); Kumar  _et al._ (2021). The FMR measurement and
the extracted value of $\Delta 4\pi M_{eff}$ reflect inhomogeneous
distribution of $4\pi M_{eff}$ in YIG-GdIG bilayer system. The argument of
Bloch domain-wall-like spiral arrangement of moments is conceivable, as this
arrangement between the adjacent layers lowers the exchange interaction
energyGhising  _et al._ (2020). To further substantiate the presence of an
interfacial AFM exchange coupling leading to spin inhomogeneity at YIG-GdIG
interface, we performed in-plane $\phi_{H}$ variation of $H_{r}$ on a
relatively thin YIG film ($\sim 100$ nm with growth conditions leading to the
formation of a GdIG interfacial layerKumar  _et al._ (2021)). Fig. 3(b) shows
prominent feature of unidirectional anisotropy due to AFM exchange coupling in
100 nm thick film. It is evident that the interfacial layer exchange couples
with the rest of the YIG film and leads to a unidirectional anisotropy. We
observe that the interfacial exchange coupling may cause $\Delta H$ broadening
and enhanced $\alpha$ due to spin inhomogeneity at the YIG-GdIG interface,
even in a 200 nm thick YIG film.

##  IV Conclusions

The effects of spin inhomogeneity at the YIG and growth-induced GdIG interface
on the magnetization dynamics of a 200 nm thick YIG film is studied
extensively using ferromagnetic resonance technique. The Gilbert damping is
almost two orders of magnitude larger ($\sim 1.2\times 10^{-3}$) than usually
reported in YIG thin films. The out-of-plane angular dependence of $\Delta H$
shows an unusual behaviour which can only be justified after considering
extrinsic mechanism in combination with the Gilbert contribution. The
extracted parameters from the $\Delta H$ vs $\theta_{H}$ simulation are, (i)
$\alpha=1.3\times 10^{-3}$ from Gilbert term; (ii) $\Delta 4\pi M_{eff}=58$ Oe
and $\Delta\theta_{H}=0.29^{o}$ from the inhomogeneity in effective
magnetization and anisotropy axes, respectively; (iii) $\Gamma_{i}^{0}=1.3$ Oe
from TMS. The TMS strength $\Gamma$ is not so appreciable, indicating high
quality thin film with insignificant defect sites. The AFM exchange coupling
between YIG and the interfacial GdIG layer causes exchange springlike
behaviour of the magnetic moments in YIG, leading to a large $\Delta 4\pi
M_{eff}$. The presence of large $\Delta 4\pi M_{eff}$ impels the quick
dragging of the precessional motion towards equilibrium. A unidirectional
behaviour is observed in the in-plane angular variation of resonance field due
to the presence of an exchange anisotropy. This further reinforces the spin
inhomogeneity at the YIG-GdIG interface due to the AFM exchange coupling.

ACKNOWLEDGEMENTS

We gratefully acknowledge the research support from IIT Kanpur and SERB,
Government of India (Grant No. CRG/2018/000220). RK and DS acknowledge the
financial support from Max-Planck partner group. ZH acknowledges financial
support from Polish National Agency for Academic Exchange under Ulam
Fellowship. The authors thank Veena Singh for her help with the angular
dependent FMR measurements.

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the specific parameter combination embodied by the equilibrium
pressure has a special physical significance.  By referring explicitly
to pressure in the “PRFM” moniker, we underscore this point.

Even though the dependence of $\SSFR$ is
on the combination of variables in $\PDE$ rather than as a power law $\SSFR \propto \Sgas^{1+p}$, an apparent power law relation can arise observationally for a number of reasons.  For example, if the vertical gravity is dominated
by the gas and kinetic turbulence dominates the pressure, as may be the case in starburst regions, $\SSFR = \PDE/\Upstot \rightarrow \pi G \Sgas^2/(2 \Upsturb)$ for $\Upsturb$ of a few $100 \kms$ (center panel of <ref>). As previously noted in [Ostriker & Shetty, 2011] and Narayanan et al., 2012, the approximate relation $\SSFR \propto \Sgas^2$ in this regime could appear as a slope between 1 and 2 in $\log \SSFR$  vs. $\log W_{\rm CO}$, since the decrease of $\alpha_{\rm CO}\equiv \Sigma_{\rm mol}/W_{\rm CO}$ in higher-excitation gas <cit.> means that $\Sgas$ increases sublinearly  with $W_{\rm CO}$.  This may explain
the power law with $p=0.4$ identified
by Kennicutt, 1998, which adopted constant $\alpha_{\rm CO}$ <cit.>.  Similarly, a power law dependence
$\SSFR \propto \Sgas \Sigma_{*}^{0.5}$
could describe normal galaxies if their
vertical gravity is dominated by
the stellar component and there is
limited variation in $h_*$.

Enhanced observational testing
Observational tests to date at $\sim\kpc$-scale resolution, as summarized from previous work in
<ref> and directly compared with our new simulation results in
<ref>, show good agreement with the PRFM model.  For the future,
it will be especially valuable to refine the empirical measurements of
parameters that enter in $\PDE$.  As noted above, the midplane density of
the old stellar disk, $\rho_*$, is needed to obtain $\PDE$. This requires
knowledge of both the total surface density of old stars,
$\Sigma_*$, and the effective half-thickness $h_*$ of the stellar layer.
For face-on galaxies, $h_*$ cannot be directly
measured, and a common practice for normal spiral galaxies <cit.>
has been to assume a
constant stellar disk thickness proportional to the radial exponential
scale length $R_s$ of the old stellar disk, with $\rho_*=\Sigma_*/(0.54 R_s)$
<cit.>.  However, as noted in [Ostriker et al., 2010], the choice $h_*=0.27 R_s$ may in fact
overestimate $h_*$, leading to an underestimate for $\rho_*$.
Also, stellar disks may flare with radius
[de Grijs & Peletier, 1997, Narayan & Jog, 2002, Momany et al., 2006, López-Corredoira & Molgó, 2014, López-Corredoira et al., 2020].
An alternative to adopting an empirical relationship between $h_*$ and $R_s$ is to estimate $h_*$ from dynamical considerations.  For example, a very simple estimate adopts $h_*=\sigma_{*,z}^2/(\pi G \Sigma_*)$ for $\sigma_{*,z}$ the effective vertical velocity dispersion of the old stellar disk, based on the Spitzer, 1942 solution for an isothermal plane-parallel system; in this case (and assuming the dark matter contribution is negligible) the second term in <ref> becomes $\sqrt{\pi} G \Sgas \Sigma_* \sigma_{\rm eff}/\sigma_{*,z}$. However, improved estimates for the vertical stellar distribution can be obtained by including  the influence of gas and dark matter gravity in confining the stellar disk and solving the coupled system of Jeans equation and hydrostatic equation for stars and gas, respectively, and allowing for radial and vertical gradients in velocity dispersions <cit.>.
A path forward to more accurate values of $h_*$
would be to seek statistical relationships between observed measures of the stellar disk thickness and
other stellar properties from edge-on disk galaxies, while simultaneously using
synthetic observations of simulated edge-on disks to calibrate the
true $h_*$ in terms of observables (including testing sensitivity to
dust extinction).

Empirical measurements of $\PDE$ require the total gas surface density
$\Sgas$ and the effective vertical velocity dispersion $\sigma_{\rm eff}$.
Especially in the central regions of galaxies, improved calibrations of
$\alpha_{\rm CO}$, e.g. making use of two or more rotational
lines to allow for varying
excitation [Gong et al., 2020], will aid in obtaining more accurate $\Sgas$
from CO emission.
The effective vertical velocity dispersion $\sigma_{\rm eff}$ includes magnetic contributions
that are difficult to measure empirically; since these are likely to
scale with the (more easily observable) kinetic turbulence, however,
this can be accounted for at lowest order with a simple multiplicative factor.
It is important to note that
the closest observable proxy of
the kinetic contribution to $\sigma_{\rm eff}$
is a mass-weighted value.  Thus, it must be derived from observations
of the atomic and molecular gas that together comprise most of the ISM's mass.
While ionized gas velocity dispersions are sometimes
more readily available, especially
for high-redshift galaxies, these sample expanding H2 regions
and diffuse ionized gas; since the motions of ionized gas are not
in general representative of the neutral ISM
<cit.>, linewidths of
H$\alpha$ or other tracers of ionized gas  should not be used as a proxy to estimate
the kinetic contribution to $\sigma_{\rm eff}$.
Even when CO and H1 lines are spectrally resolved,
it is difficult to correct for spiral-arm streaming
and other in-plane motions that can contaminate measurement of the vertical
velocity dispersion, so face-on systems provide the most reliable targets.

Further model development
The suite of TIGRESS simulations analyzed here represents a significant advance
in resolved modeling of the star-forming, multiphase, magnetized ISM.
Nevertheless, the TIGRESS implementation employed for this suite
(as described in Kim & Ostriker, 2017) has
limitations that could affect our results quantitatively, if not qualitatively.
First, a simple fitted cooling function and fixed FUV heating efficiency
are adopted.  Rather than
directly following the FUV photons responsible for photoelectric heating
via radiative transfer, we also adopt a simple analytic attenuation formula.
In addition, we do not follow ionizing radiation (or other
“early feedback”) from young clusters,
which is known to strongly affect the immediate environment of forming stars.
Extending beyond these simplifying assumptions is an important direction
for future work, and efforts along these lines have already begun.
Namely, we have extended our TIGRESS framework to include adaptive ray tracing
radiative transfer <cit.> in  order
to follow FUV as well as ionizing radiation produced
by the star cluster particles that form, and we have also
implemented more sophisticated heating and cooling functions based on
nonequilibrium photochemistry
(J.-G. Kim et al 2022, submitted).  Initial simulations with this new
framework produce results that are generally quite
consistent with those reported here, and in particular turning photoionizing
radiation on or off changes $\SSFR$ by $\lesssim 30\%$ (C.-G. Kim et al 2022, in preparation).
Another potential concern is that with our current implementation
of sink particles – representing stellar
clusters with a coeval stellar population,
star formation may be somewhat more correlated in space and time than
is realistic, which could quantitatively affect certain results
Testing sensitivity
to this correlation, as well as exploring alternatives to the
current sink particle approach, are also
important directions for future work.

The TIGRESS simulations analyzed here assume solar metallicity, but
it is of much interest to investigate how higher or lower
metallicity would affect the results.
We have every expectation that the PRFM theory will hold in some generality,
but quantitative calibration of feedback yields at
low metallicity are needed for realistic application to high-redshift
galaxies.  By extending the range of simulated systems, not
just in metallicity but to environments with much higher and lower gas
surface density, with deeper and shallower stellar potentials,
with global galactic as well as local frameworks (while still resolving
all phases of the ISM and feedback effects), it will be possible to test
the general validity of the PRFM theory.
In strongly disturbed systems (tidal encounter and merger systems), as
well as galaxies at early stages before disk rotation dominates other
kinetic and thermal energy <cit.>,
it remains to be seen whether a modified version of the PRFM theory
(generalizing to quasi-spherical rather than disk geometry) can be
used to predict the SFR. This could only be expected if feedback is the main
source resupplying losses of the forms of energy that combine to support against gravity.
From dimensional analysis, the expected gas depletion time would be $\tdep \sim
\Upstot /g$ for $g$ the
relevant gravitational field that is supported by terms with rapid losses ($g_z$ for a disk, or $g_r$ for a quasi-spherical system).
Our own analyses of TIGRESS model extensions –
with spiral arm potential perturbations in mid-disk environments
[Kim et al., 2020], and of bar-fed star-forming rings in galactic
center regions [Moon et al., 2021, Moon et al., 2022] – have already corroborated
the PRFM principles. It would be straightforward for other groups
to apply the same kind of analysis to their own simulations with
physics implementations similar to TIGRESS and a fully resolved multiphase ISM,
in order to test these principles further.

This work was supported in part by NASA ATP grant No. NNX17AG26G and by grant No. 510940 from  the Simons Foundation to E. C. Ostriker.
We are grateful to Jiayi Sun and Jorge Barrera-Ballesteros for sharing
observational data sets, and to the referee for helpful comments on the

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