††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>kassel.de

# Polycrystalline exchange-biased bilayers:
magnetically effective vs. structural antiferromagnetic grain volume
distribution

Maximilian Merkel  Meike Reginka  Rico Huhnstock  Arno Ehresmann  Institute of
Physics and Center for Interdisciplinary Nanostructure Science and Technology
(CINSaT), University of Kassel, Heinrich-Plett-Strasse 40, 34132 Kassel,
Germany

###### Abstract

The magnetic characteristics of polycrystalline exchange-biased
antiferromagnet/ferromagnet-bilayers are determined by a complex interplay of
parameters, describing structural and magnetic properties of the material
system, including in particular the grain volume distribution of the
antiferromagnet. An ideal characterization of such systems would be a non-
destructive determination of the relevant parameters for each individual
grain. This is in most cases not feasible, since typical characterization
methods do average over larger areas. Here, we show that it is however
possible to determine averaged microscopic parameters from averaged
macroscopic magnetic quantities measured by vectorial Kerr magnetometry in
comparison to an elaborate model. In particular, we estimate the magnetically
effective antiferromagnetic grain size distribution, being essential for the
interface exchange coupling to the ferromagnetic layer. We found that the
distribution of magnetically active grain sizes differs from the structural
one, indicating that the antiferromagnetic order, relevant for the exchange
bias, extends only over a part of the grains’ structural volumes.

††preprint: APS/123-QED

##  I Introduction

An antiferromagnet (AF) and a ferromagnet (F) sharing an interface in a thin
bilayer system commonly results in a horizontal shift of the ferromagnetic
hysteresis loop accompanied by an additional modification of its coercivity as
compared to loops of a pure F [1, 1, 2, 3]. With the effect arising from
exchange interaction across the common interface, the phenomenon has been
named exchange bias (EB) and is a key effect for of the development of data
storage and magnetic sensor technology [4, 5, 6, 7]. Further, domain
engineering of polycrystalline EB thin films [8, 9, 10, 11, 12, 13] has proven
to be an important ingredient of lab-on-a-chip devices [14] enabling the
actuation of magnetic particles in liquid media [15, 16, 17].

The exchange bias shift $H_{\mathrm{EB}}$ is caused by an interaction of the F
magnetic moments with uncompensated interface moments of the AF layer. It is
interpreted as a consequence of a macroscopic unidirectional magnetic
anisotropy (UDA), resulting from an aligned pinning of the F spins to the AF
ones [1, 18]. The modification of the coercivity $H_{\mathrm{C}}$ is
attributed to rotatable AF moments resulting in a dynamic rotatable magnetic
anisotropy (RMA) [18, 19]. In polycrystalline systems, these anisotropies are
- apart from to the AF/F-interface [20, 21] and the AF crystal structure [22,
23] \- mainly determined by the grain volume distribution (GVD) of the AF [3,
24, 25, 26]. A general description of the EB in polycrystalline systems
solidified over the last decades [3, 27, 28, 25, 8], connecting the AF GVD
with the macroscopically observable features by subdividing the AF ensemble
into classes of grains differently responding to rotating F magnetic moments
during their reversal. The grains exhibit a varying thermal stability with
respect to the reorientation of their uncompensated magnetic interface moment
upon the F layer’s remagnetization. For given temperature and duration of
observation, AF grains can be classified to contribute to the macroscopic UDA
or RMA [3, 25, 28]. In addition to the grain-specific magnetic anisotropy and
exchange coupling constant, the interaction of an AF grain with the F magnetic
moments is determined by the ratio of the interface area, mediating the
coupling, and the magnetically effective AF grain volume [27, 8]. For columnar
AF grains and assuming that the magnetic anisotropy extends over their
complete structural volumes, this relates to the grain-specific aspect ratio
of cylindrical AF grains, scaling directly with the AF layer thickness for
thin layers [29, 3].

One - to the knowledge of the authors - hitherto unanswered question is,
whether the structural GVD is identical to the distribution of the
magnetically active AF grain volumes. Therefore, a quantitative link between
the magnetic characteristics of polycrystalline AF/F-bilayers and their
microstructure is crucial, even if the connection between thickness-dependent
relations of $H_{\mathrm{EB}}$ and $H_{\mathrm{C}}$ and the AF layer’s
granular characteristic, or the nature of the EB as an interface effect
itself, can be reasoned [30, 29, 31, 32, 3, 33].

We present systematic thickness-dependent investigations of $H_{\mathrm{EB}}$
and $H_{\mathrm{C}}$ in the case of columnar grain growth, which we could
validate by grain size analysis by means of atomic force microscopy. A
quantitative connection between the averaged macroscopic magnetic
characteristics and averaged microscopic material parameters is established by
comparing thickness-dependent measurements with model calculations utilizing
an extended time-dependent Stoner-Wohlfarth (SW) approach [28, 26]. In
conjunction, analytic expressions for the thickness-dependent relations were
derived in the context of a generalized description of polycrystalline EB
systems [3, 25, 28], which further solely depend on measurement conditions and
parameters characterizing the averaged microscopic properties of the system.

##  II Experimental

Prototypical AF/F-bilayer systems of the type
Si(100)/Cu${}^{5\text{nm}}$/Ir17Mn83${}^{t_{\mathrm{AF}}}$/Co70Fe30${}^{t_{\mathrm{F}}}$/Si${}^{20\text{nm}}$
were fabricated on naturally oxidized Si by rf-sputter deposition at room
temperature of alloy targets with the same compositions. Different nominal
thicknesses $t_{\mathrm{AF}}$ between 2.5 and 50 nm with fixed
$t_{\mathrm{F}}=10$ nm and different $t_{\mathrm{F}}$ between 5 and 30 nm at
fixed $t_{\mathrm{AF}}=5$ and 30 nm have been prepared. Deposition rates have
been $\eta_{\mathrm{Cu}}=(6.7\pm 0.2)$ nm/min, $\eta_{\mathrm{IrMn}}=(5.5\pm
0.8)$ nm/min, $\eta_{\mathrm{CoFe}}=(3.40\pm 0.13)$ nm/min and
$\eta_{\mathrm{Si}}=(3.84\pm 0.96)$ nm/min for all layers, determined as
described in Ref. [26]. Furthermore, an unbiased F sample with
$t_{\mathrm{F}}=10$ nm was fabricated as a reference by omitting the AF layer
($t_{\mathrm{AF}}=0$ nm). The base pressure was $<10^{-6}$ mbar, the working
pressure $\approx 10^{-2}$ mbar and during deposition, an in-plane magnetic
field set to 28 kA/m was applied. While the 20 nm Si capping layer serves as
oxidation protection and further enhances the contrast in the magneto-optical
measurements [28], the 5 nm Cu buffer layer induces the desired (111) texture
in the IrMn layer [22, 26].

For the determination of the distribution $\tilde{\varrho}(r_{\mathrm{AF}})$
of AF surface grain radii $r_{\mathrm{AF}}$ at thicknesses $t_{\mathrm{AF}}$
between 5 and 100 nm, the F and capping layer have been omitted. Similarly,
for samples with $t_{\mathrm{F}}=10$ nm as well as $t_{\mathrm{AF}}=5$ and 30
nm, the capping layer has been omitted to determine the F grain surface radii
distribution $\tilde{\varrho}(r_{\mathrm{F}})$. For tuning the average aspect
ratio of AF grains, the layer stack has been fabricated with different AF
deposition rates from 0.9 to 6.8 nm/min for samples with $t_{\mathrm{AF}}$
between 2.5 and 50 nm at fixed $t_{\mathrm{F}}=10$ nm.

The distributions of grain radii were determined by atomic force microscopy in
contact mode measuring several spots on the samples’ surface with a nominal
resolution of 0.49 nm/pixel. Utilizing the Watershed algorithm provided by the
evaluation software Gwyddion (V.2.51) [34], the surface topography was
analyzed applying the same evaluation procedure as described in Ref. [26].

The samples were magnetically characterized by vectorial magneto-optical Kerr
magnetometry as described in Ref. [28]. Magnetization reversal curves were
obtained for angles between $\varphi=0^{\circ}$ and 360∘ with an increment of
1∘, where $\varphi$ is the angle between the magnetic field applied during
layer growth and the field applied during the measurements. $\varphi$ has been
corrected by considering $H_{\mathrm{C}}(\varphi)$ to be largest at
$\varphi=0^{\circ}$ and 180∘ with an accuracy of 1∘ in accordance to Refs.
[26, 28, 35]. The magnetization curves shared a sweep rate of $\nu\approx
7.27$ kA/m/s and a resolution of $\Delta H\approx 0.53$ kA/m resulting in a
measurement time of $t_{\mathrm{Hys}}\approx 44$ s.

##  III Polycrystalline Model

###  III.1 General description

The physical interpretation of the experimental results will be performed
within the model for polycrystalline EB systems [27, 8, 18, 3, 28, 26, 36,
25]. The phenomenon is condensed down to the interaction between a uniform F
layer and a granular AF [3]. An individual AF grain i with a magnetically
effective volume $V_{\mathrm{AF,i}}$, not necessarily identical to the actual
physical volume, and an anisotropy constant $K_{\mathrm{AF,i}}$ interacting
with the F at the shared interface $A_{\mathrm{AF,i}}$ via exchange
interaction described by the microscopic exchange energy area density
$J_{\mathrm{EB,i}}$ possesses an energy barrier [27, 8]

$\displaystyle\Delta E_{\mathrm{AF,i}}=\,$ $\displaystyle
K_{\mathrm{AF,i}}V_{\mathrm{AF,i}}\,-$ $\displaystyle
J_{\mathrm{EB,i}}A_{\mathrm{AF,i}}\left(1-\frac{J_{\mathrm{EB,i}}A_{\mathrm{AF,i}}}{4K_{\mathrm{AF,i}}V_{\mathrm{AF,i}}}\right)$
(1)

between two energy minima corresponding to the parallel (global minimum) and
antiparallel (local minimum) alignment of the grain-averaged uncompensated AF
interface magnetic moment $\vec{m}_{\mathrm{AF,i}}$ with respect to
$\vec{M}_{\mathrm{F}}$ representing the F magnetization. Eq. (III.1) is in
first order given by $\Delta E_{\mathrm{AF,i}}\approx
K_{\mathrm{AF,i}}V_{\mathrm{AF,i}}$ [3, 8, 27]. This allows for a connection
of the AF GVD $\varrho(V_{\mathrm{AF}})$ with the distribution of relaxation
times $\tau_{\mathrm{AF,i}}=\tau_{0}\mathrm{exp}\left\\{\Delta
E_{\mathrm{AF,i}}/k_{\mathrm{B}}T\right\\}$ with $\nu_{0}=1/\tau_{0}$ as the
characteristic frequency for spin reversal of the AF grains, $T$ representing
the observation temperature and $k_{\mathrm{B}}$ as Boltzmann’s constant [28,
3].

Figure 1:  (a) Distributions $\varrho\left(V_{\mathrm{AF}}\right)$ of AF grain
volumes $V_{\mathrm{AF}}$ schematically depicted for a thin (green, continuous
line) and a thick AF layer (blue, dash-dotted line) assuming a constant
distribution of AF grain radii for different $t_{\mathrm{AF}}$, i.e.
homogeneous columnar grain growth. The distributions are divided into four
classes of different thermal stability by boundaries $V_{\mathrm{I/II}}$,
$V_{\mathrm{II/III}}$ and $V_{\mathrm{III/IV}}$ connected to material
parameters and conditions during observation and post-treatment. Colored
antiparallel arrows ($\leftrightarrows$ and $\rightleftarrows$) schematically
depict uncompensated AF grain-averaged magnetic moments of the respective
grain class interacting with the reversing F magnetization represented by
black arrows ($\leftarrow$ and $\rightarrow$). Inset: Schematic top view of
columnar grains with different sizes connected to the distribution
$\tilde{\varrho}\left(r_{\mathrm{AF}}\right)$ of AF grain radii
$r_{\mathrm{AF}}$ and the scaling of individual grain volumes for fixed
$r_{\mathrm{AF}}$ with $t_{\mathrm{AF}}$. (b) Vectors in the applied extended
SW approach and corresponding angles with respect to an arbitrary reference
frame. $\vec{H}$ is the external magnetic field with its azimuthal angle
$\varphi$, $\vec{M}_{\mathrm{F}}$ is the F magnetization with the angle
$\beta_{\mathrm{F}}$, $K_{\mathrm{F}}$ is the energy density of the
ferromagnetic uniaxial magnetic anistropy (FUMA) with its easy direction
defined by $\gamma_{\mathrm{F}}$, $\vec{M}_{\mathrm{C}}^{\mathrm{II}}$ and
$\vec{M}_{\mathrm{EB}}^{\mathrm{III}}$ are the superposed uncompensated
magnetic moments related to AF grains of classes II and III with
$\gamma_{\mathrm{C}}^{\mathrm{II}}$ and $\gamma_{\mathrm{EB}}^{\mathrm{III}}$
as the corresponding azimuthal angles connected to the RMA and the UDA,
respectively. (c) Illustration of the RMA during a magnetization reversal of
the F at time steps $t$ and $t-\Delta t$ visualizing the continuous relaxation
of $\vec{M}_{\mathrm{C}}^{\mathrm{II}}$ into a state parallel to
$\vec{M}_{\mathrm{F}}$. (b) and (c) are reprinted with permission from [36],
Copyright (2021) by the American Physical Society.

For given measurement and storage temperatures and times, AF grains can be
classified with respect to their thermal stability, i.e. their individual
energy barriers $\Delta E_{\mathrm{AF,i}}$ and consequently relaxation times
$\tau_{\mathrm{AF,i}}$, directly related to the individual grain volumes
$V_{\mathrm{AF,i}}$ for constant $K_{\mathrm{AF,i}}$ [Fig. 1(a)] [27, 28, 26,
3]. A constant $K_{\mathrm{AF,i}}=K_{\mathrm{AF}}$ $\forall i$ is an
assumption possibly not valid for very small grain sizes [37]. Thermally
unstable grains of classes I and II exhibit either superparamagnetic behavior
(class I) or have relaxation times in the order of the observation time (class
II), which is in typical experiments equivalent to the hysteresis duration
$t_{\mathrm{Hys}}$ [28, 19, 25]. Grains of class II are called rotatable,
describing a realignment of the grain-averaged uncompensated AF interface
magnetic moment during the F’s remagnetization, consequently having an effect
on the coercivity $H_{\mathrm{C}}$ [36, 19]. Grains of classes III and IV are
thermally stable with relaxation times larger than $t_{\mathrm{Hys}}$ on the
timescale of observation. Class III grains are the origin of the
macroscopically observable EB shift $H_{\mathrm{EB}}$ because they have been
aligned by an initialization procedure (e.g. post-annealing) [26, 3, 1], via
applying an adequately strong external magnetic field during deposition [25,
26] or during an ion bombardment [38, 8, 39, 40], by thermally assisted
scanning probe lithography [13] or via laser-based annealing [41, 12, 11]. The
orientation of the uncompensated AF moments of class IV grains cannot be set
by one of the named treatments and their pinning directions are assumed to be
randomly distributed [28, 3]. Grains of classes II and III are assumed to be
superposable with respect to their uncompensated interface moments,
macroscopically resulting in a RMA mediating $H_{\mathrm{C}}$ [28, 19, 25] and
an UDA mediating $H_{\mathrm{EB}}$ [3, 28], respectively.

###  III.2 Grain size distribution and class boundaries

The grain size distribution of sputtered polycrystalline thin films is
typically found to be lognormal [Fig. 1(a)] [42, 43, 24, 44, 3]. For a
polycrystalline AF layer of thickness $t_{\mathrm{AF}}$ we assume cylindrical,
homogeneously grown grains [26] with radius $r_{\mathrm{AF}}$ and volume
$V_{\mathrm{AF}}$ [Fig. 1(a)]. The AF GVD can be calculated from the
lognormally distributed grain radii
$\tilde{\varrho}(r_{\mathrm{AF}},\mu,\sigma)$ via a change of variables giving

$\displaystyle\varrho\left(V_{\mathrm{AF}},t_{\mathrm{AF}},\mu,\sigma\right)=\,$
$\displaystyle\tilde{\varrho}\left(r_{\mathrm{AF}}\left(V_{\mathrm{AF}}\right),\mu,\sigma\right)\partial_{V_{\mathrm{AF}}}r_{\mathrm{AF}}\left(V_{\mathrm{AF}}\right)$
$\displaystyle=\,$
$\displaystyle\frac{\tilde{\varrho}\left(\sqrt{V_{\mathrm{AF}}/\pi
t_{\mathrm{AF}}},\mu,\sigma\right)}{2\sqrt{\pi
V_{\mathrm{AF}}t_{\mathrm{AF}}}}$ (2)

with $r_{\mathrm{AF}}\left(V_{\mathrm{AF}}\right)=\sqrt{V_{\mathrm{AF}}/\pi
t_{\mathrm{AF}}}$ [45, 26, 3, 37]. $\mu$ and $\sigma$ represent the parameters
characterizing the lognormal distribution with respect to $r_{\mathrm{AF}}$.
The expectation value $\langle r_{\mathrm{AF}}\rangle$ of the grain radius as
well as the standard deviation $\mathrm{SD}$ are given by $\langle
r_{\mathrm{AF}}\rangle=\mathrm{exp}\left\\{\mu+\sigma^{2}/2\right\\}$ and
$\mathrm{SD}=\langle
r_{\mathrm{AF}}\rangle\sqrt{\mathrm{exp}\left\\{\sigma^{2}\right\\}-1}$ [24].
With Eq. (III.2) and as shown in Fig. 1(a),
$\varrho\left(V_{\mathrm{AF}},t_{\mathrm{AF}},\mu,\sigma\right)$ can be
modified with respect to the grain class boundaries by varying
$t_{\mathrm{AF}}$ for a fixed distribution
$\tilde{\varrho}(r_{\mathrm{AF}},\mu,\sigma)$.

The boundaries between the grain classes are functions of temperature and time
[3] and can be estimated via [37]

$V_{\mathrm{AF}}\left(T,\tau\right)=\frac{k_{\mathrm{B}}T}{K_{\mathrm{AF}}\left(T\right)}\mathrm{ln}\left\\{\frac{\tau}{\tau_{0}}\right\\}.$
(3)

For hysteresis curve measurements, the boundary $V_{\mathrm{II/III}}$ between
classes II and III [Fig. 1(a)] is determined by the measurement temperature
$T=T_{\mathrm{RT}}$ (here: room temperature being $T_{\mathrm{RT}}$) and the
hysteresis duration $\tau=t_{\mathrm{Hys}}$. $V_{\mathrm{III/IV}}$ is
determined by $T=T_{\mathrm{ini}}$ and $\tau=t_{\mathrm{ini}}$ of, e.g., the
field-cooling process, whereas $V_{\mathrm{I/II}}$ is defined by
$T=T_{\mathrm{RT}}$ and by a time $\tau=t_{\mathrm{spp}}$. The latter is
connected to the timescale on which very small thermally unstable AF grains
behave superparamagnetic. Assuming that the temperature dependence of
$K_{\mathrm{AF}}$ is
$K_{\mathrm{AF}}\left(T\right)=K_{\mathrm{AF}}\left(0\right)\left(1-T/T_{\mathrm{N}}\right)$
[42] with $T_{\mathrm{N}}\approx 650$ K for IrMn [1, 37, 42] , and using the
experimentally determined values
$K_{\mathrm{AF}}\left(T_{\mathrm{RT}}\right)=(5.5\pm 0.5)$ $10^{5}$ J/m3 [37]
and $\tau_{0}=1/(2.1\pm 0.4)$ $10^{-12}$ s-1 [42], the boundaries can be
estimated for given observation temperatures and times.

For calculating the contributions of the grain classes, the integrals between
the respective bounds [Fig. 1(a)] have to be determined. We define

$p=\int\limits_{V_{\mathrm{I/II}}}^{V_{\mathrm{III/IV}}}\varrho\left(V_{\mathrm{AF}}\right)\text{d}V_{\mathrm{AF}}$
(4)

as the percentage of grains contributing to the UDA and the RMA at all, i.e.
all grains of classes II and III in relation to the number of all grains of
the polycrystalline ensemble. Based on this, the contributions
$p_{\mathrm{II}}$ and $p_{\mathrm{III}}$ of class II and III grains,
respectively, are given as the weighted integrals between the respective
bounds

$p_{\mathrm{II}}=\int\limits_{V_{\mathrm{I/II}}}^{V_{\mathrm{II/III}}}\frac{\varrho\left(V_{\mathrm{AF}}\right)}{p}\,\text{d}V_{\mathrm{AF}}$
(5)
$p_{\mathrm{III}}=\int\limits_{V_{\mathrm{II/III}}}^{V_{\mathrm{III/IV}}}\frac{\varrho\left(V_{\mathrm{AF}}\right)}{p}\,\text{d}V_{\mathrm{AF}}=1-p_{\mathrm{II}}.$
(6)

Knowing the parameters characterizing the distribution of grain sizes, $p$ and
$p_{\mathrm{III}}=1-p_{\mathrm{II}}$ can be expressed as functions of
$t_{\mathrm{AF}}$, $\mu$, $\sigma$ and the respective grain class boundaries
by

$\chi\left(V_{\mathrm{AF}},t_{\mathrm{AF}},\mu,\sigma\right)=\mathrm{erf}\left\\{\frac{\mathrm{ln}\sqrt{V_{\mathrm{AF}}/\pi
t_{\mathrm{AF}}}-\mu}{\sqrt{2\sigma^{2}}}\right\\},$ (7)

where $(1+\chi)/2$, as the integral of the lognormal distribution, represents
the proportion of grains that are smaller or have the same size than
$V_{\mathrm{AF}}$. $p$ and $p_{\mathrm{III}}$ are given by

$\displaystyle p=\left\\{\right.$
$\displaystyle\chi\left(V_{\mathrm{III/IV}},t_{\mathrm{AF}},\mu,\sigma\right)-$
$\displaystyle\chi\left(V_{\mathrm{I/II}},t_{\mathrm{AF}},\mu,\sigma\right)\left.\right\\}/2$
(8) $\displaystyle p_{\mathrm{III}}=\left\\{\right.$
$\displaystyle\chi\left(V_{\mathrm{III/IV}},t_{\mathrm{AF}},\mu,\sigma\right)-$
$\displaystyle\chi\left(V_{\mathrm{II/III}},t_{\mathrm{AF}},\mu,\sigma\right)\left.\right\\}/2p.$
(9)

###  III.3 Thickness dependencies

Varying the layer thicknesses $t_{\mathrm{AF}}$ and $t_{\mathrm{F}}$ of an
AF/F-bilayer yields a very rich phenomenology with respect to the alteration
of $H_{\mathrm{EB}}$ and $H_{\mathrm{C}}$, due to the change of the AF GVD as
well as the coupling strength at the common interface [20, 1, 18, 29]. Based
on the intuitive SW approach introduced by Meiklejohn and Bean [2, 46, 18],
the absolute value of the EB shift and the coercive field are here assumed to
be given by [1, 20, 18, 47]

$\left|H_{\mathrm{EB}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)\right|=\frac{J_{\mathrm{eff}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)}{\mu_{0}\,M_{\mathrm{S}}\,t_{\mathrm{F}}}\,p_{\mathrm{III}}\left(t_{\mathrm{AF}}\right)$
(10)
$H_{\mathrm{C}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)=\frac{J_{\mathrm{eff}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)}{\mu_{0}\,M_{\mathrm{S}}\,t_{\mathrm{F}}}\,p_{\mathrm{II}}\left(t_{\mathrm{AF}}\right)+\frac{2\,K_{\mathrm{F}}}{\mu_{0}\,M_{\mathrm{S}}}$
(11)

with $H_{\mathrm{C}}$ being shifted by an offset determined by the F uniaxial
anisotropy constant $K_{\mathrm{F}}$ and saturation magnetization
$M_{\mathrm{S}}$ [18]. The exchange bias shift is scaled by the product
between the effective coupling constant
$J_{\mathrm{eff}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)$ and the
proportion $p_{\mathrm{III}}(t_{\mathrm{AF}})$ of $H_{\mathrm{EB}}$-mediating
grains of class III, which should by definition [Eq. (6)] only depend on
$t_{\mathrm{AF}}$ [25, 36]. Likewise, in the case of the coercivity,
$H_{\mathrm{C}}-2K_{\mathrm{F}}/\mu_{0}M_{\mathrm{S}}$ is scaled by the
product of $J_{\mathrm{eff}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ and the
proportion $p_{\mathrm{II}}(t_{\mathrm{AF}})$ of class II grains mediating
$H_{\mathrm{C}}$, since the rotatable anisotropy is time-dependent but of
unidirectional nature [28]. The effective coupling constant is given by

$J_{\mathrm{eff}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)=J_{\mathrm{EB}}\left(t_{\mathrm{F}}\right)\,p\left(t_{\mathrm{AF}}\right)$
(12)

assuming that $J_{\mathrm{EB}}(t_{\mathrm{F}})$ is constant for all AF grains
($J_{\mathrm{EB,i}}=J_{\mathrm{EB}}\,\forall\,\mathrm{i}$) and is already
reduced due to, e.g. interface roughness, compensated moments or
stoichiometric gradients [21, 18, 1]. It is further supposed that the coupling
itself is solely determined by the coupling interfaces and not the individual
volumes of the AF grains.
$J_{\mathrm{eff}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)$ is proportional
to $p(t_{\mathrm{AF}})$ ensuring that the grain class specific exchange
coupling constants
$J_{\mathrm{II/III}}=J_{\mathrm{EB}}\,p\,p_{\mathrm{II/III}}=J_{\mathrm{EB}}A_{\mathrm{II/III}}/A$
are determined by the scaling of the microscopic exchange energy area density
with the proportion of the area $A_{\mathrm{II/III}}$ accounted to the
corresponding grain class with respect to the whole AF/F-interface area $A$
[25, 36].

###  III.4 Time-dependent Stoner-Wohlfarth (SW) ansatz

For numerical calculations of magnetization curves and the determination of
$H_{\mathrm{EB/C}}\left(t_{\mathrm{F}},t_{\mathrm{AF}}\right)$ and for fitting
model calculations to angular-resolved $H_{\mathrm{EB/C}}(\varphi)$, the
extended time-dependent SW approach introduced in Refs. [28, 48, 26, 36] will
be utilized. During remagnetization, a uniform in-plane magnetized F with
magnetization $\vec{M}_{\mathrm{F}}$ and saturation magnetization
$M_{\mathrm{S}}$ is assumed to rotate coherently, where the azimuthal angle of
$\vec{M}_{\mathrm{F}}$ is given by $\beta_{\mathrm{F}}$ [Fig. 1(b)]. Using the
perfect delay convention [49, 28, 26, 36], the time-dependent F free energy
area density $E\left(\beta_{\mathrm{F}}(t)\right)/A$ is sequentially minimized
with respect to $\beta_{\mathrm{F}}(t)$ for varying external magnetic field
$H$.

$E\left(\beta_{\mathrm{F}}(t)\right)/A=e_{\mathrm{pot}}+e_{\mathrm{FUMA}}+e_{\mathrm{RMA}}+e_{\mathrm{UDA}}$
(13)

is composed of the F layer’s potential energy density in the external magnetic
field $e_{\mathrm{pot}}$, its intrinsic uniaxial anisotropy
$e_{\mathrm{FUMA}}$ (FUMA) and additional anisotropy terms $e_{\mathrm{RMA}}$
and $e_{\mathrm{UDA}}$ representing the interaction with superposed rotatable
and fixed uncompensated AF moments. The potential energy area density is given
by

$e_{\mathrm{pot}}=-\mu_{0}HM_{\mathrm{S}}t_{\mathrm{F}}\cos\left(\beta_{\mathrm{F}}\left(t\right)-\varphi\right)$
(14)

with $\mu_{0}$ as the magnetic permeability in vacuum and $\varphi$ as the
azimuthal angle of the external magnetic field with respect to an arbitrary
reference frame [Fig. 1(b)]. The uniaxial anisotropy energy area density is
given by

$e_{\mathrm{FUMA}}=K_{\mathrm{F}}t_{\mathrm{F}}\sin^{2}\left(\beta_{\mathrm{F}}\left(t\right)-\gamma_{\mathrm{F}}\right)$
(15)

with the energy density $K_{\mathrm{F}}$ and the azimuthal angle
$\gamma_{\mathrm{F}}$ [Fig. 1(b)] defining the F’s anisotropy axis parallel to
the external magnetic field applied during deposition [28, 26]. The
interaction of the uniform F with AF grains contributing to the RMA or the UDA
is broken down to the interaction of the F with the macroscopic uncompensated
interface moments
$\vec{M}_{\mathrm{C/EB}}^{\mathrm{II/III}}=\sum_{i}\vec{m}^{\mathrm{II/III}}_{\mathrm{AF},i}$,
with the azimuthal angles $\gamma_{\mathrm{C}}^{\mathrm{II}}$ and
$\gamma_{\mathrm{EB}}^{\mathrm{III}}$ [Fig. 1(b)], as the superposition of the
grain-averaged magnetic moments $\vec{m}^{\mathrm{II/III}}_{\mathrm{AF},i}$ of
classes II and III [28, 36]. The anisotropy area densities representing the
RMA and the UDA are given by [28, 36]

$e_{\mathrm{RMA}}=-J_{\mathrm{eff}}p_{\mathrm{II}}\cos\left(\beta_{\mathrm{F}}\left(t\right)-\gamma_{\mathrm{C}}^{\mathrm{II}}\left(t,\tau_{\mathrm{C}}^{\mathrm{II}}\right)\right)$
(16)
$e_{\mathrm{UDA}}=-J_{\mathrm{eff}}p_{\mathrm{III}}\cos\left(\beta_{\mathrm{F}}\left(t\right)-\gamma_{\mathrm{EB}}^{\mathrm{III}}\right)$
(17)

with prefactors $J_{\mathrm{II/III}}=J_{\mathrm{eff}}p_{\mathrm{II/III}}$ [36]
as in Eqs. (10) and (11). The time-dependent contribution of the dynamic RMA
is represented by its azimuthal angle

$\displaystyle\gamma_{\mathrm{C}}^{\mathrm{II}}\left(t,\tau_{\mathrm{C}}^{\mathrm{II}}\right)=$
$\displaystyle\,\beta_{\mathrm{F}}\left(t-\Delta
t\right)\left(1-\mathrm{exp}\left\\{-\Delta
t/\tau_{\mathrm{C}}^{\mathrm{II}}\right\\}\right)+$
$\displaystyle\,\gamma_{\mathrm{C}}^{\mathrm{II}}\left(t-\Delta
t,\tau_{\mathrm{C}}^{\mathrm{II}}\right)\mathrm{exp}\left\\{-\Delta
t/\tau_{\mathrm{C}}^{\mathrm{II}}\right\\}$ (18)

with the average relaxation time

$\tau_{\mathrm{C}}^{\mathrm{II}}=\frac{\int\limits_{V_{\mathrm{I/II}}}^{V_{\mathrm{II/III}}}\tau_{\mathrm{AF}}(V_{\mathrm{AF}})\varrho\left(V_{\mathrm{AF}}\right)\,\text{d}V_{\mathrm{AF}}}{\int\limits_{V_{\mathrm{I/II}}}^{V_{\mathrm{II/III}}}\varrho\left(V_{\mathrm{AF}}\right)\,\text{d}V_{\mathrm{AF}}}$
(19)

of all rotatable grains of class II [27, 36]. The dynamic realignment of the
RMA is visualized in Fig. 1(c) showing that for each step during the
remagnetization of the F, for which $\beta_{\mathrm{F}}\left(t\right)$ is
determined,
$\gamma_{\mathrm{C}}^{\mathrm{II}}\left(t,\tau_{\mathrm{C}}^{\mathrm{II}}\right)$
is derived from the history of the F and the RMA at $t-\Delta t$ [28, 26, 36].

Additionally, to consider a possible offset of $H_{\mathrm{EB}}(\varphi)$ due
to the measurement procedure, which is not related to training effects, an
additional magnetic anisotropy term
$e_{\mathrm{add}}=-J_{\mathrm{add}}\cos(\beta_{\mathrm{F}}(t)-\varphi)$ is
added to Eq. (13) [28]. This additional term incorporates the interaction of
the F with AF grains, which align to a direction parallel to the applied
external magnetic field, determined by the additional effective coupling
constant $J_{\mathrm{add}}$. Since in the experiment $\varphi$ is varied
successively and not randomly, AF grains which are in the vicinity of the
grain class boundary between class II and III having relaxation times larger
than or similar to $t_{\mathrm{Hys}}$, do not contribute on the timescale of
the hysteresis to $H_{\mathrm{C}}$, but to $H_{\mathrm{EB}}$ on the timescale
of the angular-resolved measurement.

##  IV Results and Discussion

###  IV.1 Surface topography

Figs. 2(a-b) show the surface topography of the IrMn layer for
$t_{\mathrm{AF}}=5$ and 30 nm, measured by atomic force microscopy. Figs.
2(c-d) show the AF layers of the same thicknesses covered by 10 nm CoFe. For
all $t_{\mathrm{AF}}$, the IrMn and the CoFe layer exhibit a similar
polycrystalline structure with almost circular base areas, indicating columnar
grain growth with cylindrically shaped grains [26]. The root-mean-square
surface roughness of the IrMn layer showed for 5 nm $\leq t_{\mathrm{AF}}\leq$
50 nm no significant trend and the average value could be determined to be
$(0.29\pm 0.04)$ nm, whereas for $t_{\mathrm{AF}}=100$ nm it was determined to
be $(0.48\pm 0.04)$ nm. In the case of the CoFe layer, the root-mean-square
surface roughness was determined to be $(0.41\pm 0.11)$ nm.

Figure 2:  Atomic force microscopy images of the IrMn surface for (a)
$t_{\mathrm{AF}}=5$ and (b) 30 nm and of the CoFe layer for
$t_{\mathrm{F}}=10$ nm deposited on IrMn for (c) $t_{\mathrm{AF}}=5$ and (d)
30 nm. Distributions (e) $\tilde{\varrho}\left(r_{\mathrm{AF}}\right)$ of the
grain radius $r_{\mathrm{AF}}$ for $t_{\mathrm{AF}}=5$, 30 and 100 nm with
corresponding lognormal fits and distributions (f)
$\tilde{\varrho}\left(r_{\mathrm{F}}\right)$ of $r_{\mathrm{F}}$ for
$t_{\mathrm{AF}}=5$ nm and 30 nm.

Histograms displaying the distribution $\tilde{\varrho}(r_{\mathrm{AF}})$ of
AF grain radii determined with the Watershed algorithm are depicted with
lognormal fits in Fig. 2(e) exemplarily for $t_{\mathrm{AF}}=5,30$, and 100
nm. $\tilde{\varrho}(r_{\mathrm{AF}})$ does not change significantly for
varying $t_{\mathrm{AF}}$. We conclude that the expectation value $\langle
r_{\mathrm{AF}}\rangle$ of the AF grain radius is constant for the
investigated thicknesses and the average value could be determined to be
$\overline{\langle r_{\mathrm{AF}}\rangle}=(7.0\pm 0.3)$ nm. In combination
with Ref. [26], this validates the assumption of a homogeneous columnar grain
growth for the used deposition parameters enabling a linear scaling of the
individual AF grain volumes
$V_{\mathrm{AF}}=\pi{r_{\mathrm{AF}}}^{2}t_{\mathrm{AF}}$ with
$t_{\mathrm{AF}}$.

In Fig. 2(f), the distribution $\tilde{\varrho}(r_{\mathrm{F}})$ of F grain
radii in case of $t_{\mathrm{F}}=10$ nm at $t_{\mathrm{AF}}=5$ and 30 nm
reveals, that the polycrystalline CoFe layer inherits the distribution of
grain interfaces from the underlying IrMn layer with a trend towards larger
$\langle r_{\mathrm{F}}\rangle$ for increasing $t_{\mathrm{AF}}$.

###  IV.2 Magnetic properties

In the following description of the determined thickness-dependent magnetic
properties, a series of fit procedures are performed based on the equations
introduced in Sec. III. A detailed overview of the different fit scenarios is
given in the Appendix in Tab. 3.

Ferromagnetic thickness dependence. Experimentally determined
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ are depicted in Figs. 3(a-d)
for $t_{\mathrm{AF}}=5$ and 30 nm. The inverse proportionality is obvious as
well as the offset for $H_{\mathrm{C}}$, with the coercivity not changing
significantly from $t_{\mathrm{F}}=20$ nm for both $t_{\mathrm{AF}}$. While
for $t_{\mathrm{AF}}=5$ nm, $H_{\mathrm{C}}$ decreases until
$t_{\mathrm{F}}=30$ nm down to $(5.5\pm 0.9)$ kA/m, for $t_{\mathrm{AF}}=30$
nm a reduction to $(3.1\pm 0.7)$ kA/m is observable. With Eq. (11), this
suggests an increase of $K_{\mathrm{F}}$ or a reduction of $M_{\mathrm{S}}$
for small $t_{\mathrm{AF}}$.

Figure 3:  $t_{\mathrm{F}}$-dependent (a-b) $|H_{\mathrm{EB}}^{\mathrm{exp}}|$
and (c-d) $H_{\mathrm{C}}^{\mathrm{exp}}$ as well as parameters (e)
$p_{\mathrm{III}}$, (f) $J_{\mathrm{eff}}$, (g) $J_{\mathrm{add}}$, (h)
$\tau_{\mathrm{C}}^{\mathrm{II}}$ and (i) $K_{\mathrm{F}}$ obtained by fitting
$H_{\mathrm{EB/C}}^{\mathrm{SW}}(\varphi)$ to
$H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ [Tab. 3 (SW)] for
$t_{\mathrm{AF}}=5$ and 30 nm. Fits in (a-b) are based on Eqs. (10) and (11),
proportional to $1/t_{\mathrm{F}}$ with fit parameters
$J_{\mathrm{II/III}}=J_{\mathrm{eff}}p_{\mathrm{II/III}}$ in the case of (A)
and proportional to $1/{t_{\mathrm{F}}}^{2}$ in the case of (B) with
$J_{\mathrm{II/III}}(t_{\mathrm{F}})=j_{\mathrm{II/III}}/t_{\mathrm{F}}$ [Tab.
3 (A) and (B)].

Fitting simulated $H_{\mathrm{EB/C}}^{\mathrm{SW}}(\varphi)$ to experimentally
determined $H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ [Tab. 3 (SW)] for
varying $t_{\mathrm{F}}$ at ${t_{\mathrm{AF}}=5}$ and 30 nm allowed for the
determination of model parameters. Angles $\gamma_{\mathrm{F}}$ and
$\gamma_{\mathrm{EB}}^{\mathrm{III}}$ were not fixed but showed no significant
trend. The saturation magnetization $M_{\mathrm{S}}=(1527\pm 25)$ kA/m of the
CoFe layer in contact with the AF was determined for $t_{\mathrm{F}}=10$ nm
and both $t_{\mathrm{AF}}$ by utilizing a vector network analyzer
ferromagnetic resonance spectrometer. The extracted optimum parameters are
displayed in Fig. 3(e-i) and will be discussed in the following:

  * $\bm{p_{\mathrm{III}}}$

The proportion $p_{\mathrm{III}}$ of class III grains [Fig. 3(e)] shows no
significant dependence on $t_{\mathrm{F}}$ for $t_{\mathrm{AF}}=30$ nm staying
constant at $\langle p_{\mathrm{III}}\rangle=(0.81\pm 0.08)$. For
$t_{\mathrm{AF}}=5$ nm and $t_{\mathrm{F}}\leq 15$ nm $\langle
p_{\mathrm{III}}\rangle=(0.25\pm 0.16)$ but for $t_{\mathrm{F}}>15$ nm the fit
procedure results in a suppression of $p_{\mathrm{III}}$ accompanied by a
large uncertainty due to the rising difficulty of extracting EB related
parameters for increasing $t_{\mathrm{F}}$. Since a larger F layer thickness
should not result in a change of $p_{\mathrm{III}}$ it has been set constant
to 0.25 at $t_{\mathrm{F}}=20$ and 30 nm for $t_{\mathrm{AF}}=5$ nm. The
scaling of grain number percentages of classes II or III with
$t_{\mathrm{AF}}$ reproduces the expectation of a reduced $p_{\mathrm{III}}$
at small $t_{\mathrm{AF}}$.

  * $\bm{J_{\mathrm{eff}}}$

The effective coupling constant $J_{\mathrm{eff}}$ [Fig. 3(d)] decreases for
increasing $t_{\mathrm{F}}$ for both $t_{\mathrm{AF}}$. Likewise to
$|H_{\mathrm{EB}}|\propto J_{\mathrm{eff}}/t_{\mathrm{F}}$, the satisfying fit
$\propto 1/t_{\mathrm{F}}$ with respect to $J_{\mathrm{eff}}(t_{\mathrm{F}})$
suggests $J_{\mathrm{EB}}\propto 1/t_{\mathrm{F}}$ by considering Eq. (12).
This is more apparent for $t_{\mathrm{AF}}=5$ nm with a reduction of
$J_{\mathrm{eff}}$ to $\approx 45$%, whereas for $t_{\mathrm{AF}}=30$ nm
$J_{\mathrm{eff}}$ reduces to $\approx 82$%. Hence, the AF layer thickness has
an impact on the $t_{\mathrm{F}}$-dependence of $J_{\mathrm{eff}}$. This is
qualitatively understandable since a stronger contribution of class II grains
is connected to a larger grain interface to grain volume ratio and a reduction
of $K_{\mathrm{AF}}$ at smaller grain sizes [37]. For smaller grain sizes,
where $\Delta E_{\mathrm{AF}}=K_{\mathrm{AF}}V_{\mathrm{AF}}$ starts to loose
its validity, this influences crucially how smaller grains interact with the F
[37, 25, 27].

  * $\bm{J_{\mathrm{add}}}$

The additional effective coupling constant $J_{\mathrm{add}}$ [Fig. 3(g)]
exhibits no dependence on $t_{\mathrm{F}}$ but a trend to be larger for
increasing $t_{\mathrm{AF}}$.

  * $\bm{\tau_{\mathrm{C}}^{\mathrm{II}}}$

The average relaxation time $\tau_{\mathrm{C}}^{\mathrm{II}}$ in units of
$t_{\mathrm{Hys}}$ [Fig. 3(h)] exhibits an overall reduction with increasing
$t_{\mathrm{F}}$ for both $t_{\mathrm{AF}}$, whereas for $t_{\mathrm{AF}}=30$
nm an antiproportional dependence on $t_{\mathrm{F}}$ similar to
$H_{\mathrm{C}}(t_{\mathrm{F}})$ in Fig. 3(d) is observable. In the case of
$t_{\mathrm{AF}}=5$ nm, no significant trend is observable for
$t_{\mathrm{F}}\geq 7.5$ nm. The alteration of
$\tau_{\mathrm{C}}^{\mathrm{II}}$ with $t_{\mathrm{F}}$ for fixed
$t_{\mathrm{AF}}$ can only be explained by a $t_{\mathrm{F}}$-dependent
variation of the interaction between the F and the polycrystalline AF caused
by a differing magnetization reversal for different $t_{\mathrm{F}}$ [50, 36].

  * $\bm{K_{\mathrm{F}}}$

The anisotropy constant $K_{\mathrm{F}}$ [Fig. 3(i)] exhibits no significant
trend for $t_{\mathrm{AF}}=30$ nm. In contrast, for $t_{\mathrm{AF}}=5$ nm an
antiproportional dependence on $t_{\mathrm{F}}$ is observable as for
$J_{\mathrm{eff}}$ in Fig. 3(f) and $H_{\mathrm{C}}$ in Fig. 3(c) approaching
$K_{\mathrm{F}}(t_{\mathrm{AF}}=30$ nm$)$ for large $t_{\mathrm{F}}$. The
intrinsic uniaxial anisotropy is probably overestimated by the fit [Tab. 3
(SW)] and is connected to the increase of $H_{\mathrm{C}}$ for small
$t_{\mathrm{F}}$. This entanglement of the FUMA with the RMA is further
emphasized by the fit $\propto 1/t_{\mathrm{F}}$ depicted in Fig. 3(i), which
is in satisfying agreement with $K_{\mathrm{F}}(t_{\mathrm{F}})$ for
$t_{\mathrm{AF}}=5$ nm.

Table 1:  Parameters obtained for the prototypical bilayer system Ir17Mn83($t_{\mathrm{AF}}$)/Co70Fe30($t_{\mathrm{F}}$) by fitting relations based on Eqs. (10) and (11) to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{F}})|$ and $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ [Fig. 3(a-d)] proportional to $1/t_{\mathrm{F}}$ (A) with fit parameters $J_{\mathrm{II/III}}=J_{\mathrm{eff}}p_{\mathrm{II/III}}$ or with relations proportional to $1/{t_{\mathrm{F}}}^{2}$ (B) with $J_{\mathrm{II/III}}(t_{\mathrm{F}})=j_{\mathrm{II/III}}/t_{\mathrm{F}}$ [Tab. 3 (A) and (B)]. Parameters [Fig. 4(b),(c) and (f)] determined by fitting $H_{\mathrm{EB/C}}^{\mathrm{SW}}(\varphi)$ to $H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ [Tab. 3 (SW)] for $t_{\mathrm{AF}}=5$ and 30 nm with $t_{\mathrm{F}}=10$ nm. Parameters are given obtained by fitting Eqs. (10) and (11) to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ as displayed in Fig. 4(a) [Tab. 3 (C)]. Further, optimum parameters are displayed extracted by fitting Eq. (12) to $J_{\mathrm{eff}}(t_{\mathrm{AF}})$ (SW) in Fig. 4(c) [Tab. 3 (J)] and finally, $p_{\mathrm{III}}^{\mathrm{max}}$ is presented obtained by fitting Eq. (III.2), linked to Eq. (III.2), to $p_{\mathrm{III}}(t_{\mathrm{AF}})$ (SW) in Fig. 4(b) [Tab. 3 (P)]. |  |  |  $t_{\mathrm{F}}=10$ nm |  $t_{\mathrm{AF}}=5,30$ nm |  $t_{\mathrm{F}}=10$ nm
---|---|---|---|---|---
|  Data to be fitted |  $\rightarrow$ |  $H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ |  $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{F}})|$ & $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ |  $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ |  $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ |  $J_{\mathrm{eff}}(t_{\mathrm{AF}})$ & $p_{\mathrm{III}}(t_{\mathrm{AF}})$ from (SW)
Parameter $\downarrow$ | Fit scenario | $\rightarrow$ | (SW) | (A) | (B) | (C) | (C) | (J) and (P)
| $J_{\mathrm{EB}}$
---
($10^{-5}$ J/m2)
$t_{\mathrm{F}}$ = 10 nm |  |  |  | $18.91\pm 12.58$ | $27.66\pm 8.89$ | $21.74\pm 0.61$
$J_{\mathrm{II}}$ ($10^{-5}$ J/m2) |  $t_{\mathrm{AF}}$ = 5 nm | $7.35\pm 3.67$ | $18.75\pm 2.50$ | $8.13\pm 0.25$ |  |  |
$t_{\mathrm{AF}}$ = 30 nm | $3.80\pm 6.65$ | $7.05\pm 0.90$ | $3.05\pm 0.08$ |  |  |
$J_{\mathrm{III}}$ ($10^{-5}$ J/m2) |  $t_{\mathrm{AF}}$ = 5 nm | $2.78\pm 2.24$ | $2.17\pm 0.35$ | $1.40\pm 0.06$ |  |  |
$t_{\mathrm{AF}}$ = 30 nm | $18.18\pm 8.92$ | $17.62\pm 1.20$ | $10.64\pm 0.97$ |  |  |
$K_{\mathrm{F}}$ (kJ/m3) |  $t_{\mathrm{F}}$ = 10 nm |  |  |  |  | $1.21\pm 0.51$ |
$t_{\mathrm{AF}}$ = 5 nm | $10.68\pm 5.74$ | $0.40\pm 1.41$ | $4.37\pm 0.23$ |  |  |
$t_{\mathrm{AF}}$ = 30 nm | $3.80\pm 4.37$ | $1.14\pm 0.50$ | $2.64\pm 0.07$ |  |  |
$\mu$ (nm) |  $t_{\mathrm{F}}$ = 10 nm |  |  |  | $1.12\pm 0.08$ | $1.08\pm 7.90$ | $1.08\pm 1.05$
$\sigma$ (nm) |  |  |  | $0.15\pm 0.01$ | $0.17\pm 0.07$ | $0.09\pm 0.56$
$\langle r_{\mathrm{AF}}\rangle$ (nm) |  |  |  | $3.09\pm 0.25$ | $2.97\pm 2.34$ | $2.94\pm 3.23$
SD (nm) |  |  |  | $0.47\pm 0.24$ | $0.50\pm 6.24$ | $0.25\pm 0.98$
$V_{\mathrm{I/II}}$ (nm3) |  |  |  | $8\pm 20$ | $112\pm 177$ | $130\pm 305$
$V_{\mathrm{II/III}}$ (nm3) |  |  |  | $217\pm 35$ | $255\pm 50$ |
$V_{\mathrm{III/IV}}$ (nm3) |  |  |  | $2589\pm 350$ | $1717\pm 271$ | $2242\pm 18$
$p_{\mathrm{III}}^{\mathrm{max}}$ |  |  |  | $0.96\pm 0.38$ | $0.70\pm 0.19$ | $0.85\pm 0.03$

It is now aimed at the extraction of parameters by fitting
$|H_{\mathrm{EB}}(t_{\mathrm{F}})|$ and $H_{\mathrm{C}}(t_{\mathrm{F}})$ given
by Eqs. (10) and (11) to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{F}})|$
and $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ as displayed in Fig.
3(a-d) [Tab. 3 (A) and (B)]. In the case of fit scenario (A),
$J_{\mathrm{II}}=J_{\mathrm{eff}}p_{\mathrm{II}}$ and
$J_{\mathrm{III}}=J_{\mathrm{eff}}p_{\mathrm{III}}$ have been used as fit
parameters scaling the contribution of the UDA and RMA, respectively.
Furthermore, the most important result, relating to the determined model
parameters shown in Fig. 3(e-i), is the observed additional antiproportional
$t_{\mathrm{F}}$-dependence of the effective coupling constant
$J_{\mathrm{eff}}$. Considering this, relations based on Eqs. (10) and (11)
are fitted to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ with
$J_{\mathrm{II/III}}(t_{\mathrm{F}})=j_{\mathrm{II/III}}/t_{\mathrm{F}}$ and
$j_{\mathrm{II/III}}$ as the proportionality factor of the respective
effective coupling constant in the case of fit scenario (B). The obtained
parameters are given in Tab. 1 for the two investigated $t_{\mathrm{AF}}=5$
and 30 nm in comparison to the parameters obtained by fitting model
calculations based on the time-dependent SW ansatz [Eq. (13)] for
$t_{\mathrm{F}}=10$ nm, presented in Fig. 3(e-i).

Comparing the effective coupling constants $J_{\mathrm{II}}$ and
$J_{\mathrm{III}}$ obtained using the SW ansatz with the parameters determined
from fits (A) or (B) it can be seen, that $J_{\mathrm{II}}$ is overestimated
by fit (A) but reproduced by fit (B). In contrast, for $J_{\mathrm{III}}$ it
is vice versa with fit (B) underestimating $J_{\mathrm{III}}$, especially in
the case of $t_{\mathrm{AF}}=30$ nm. Parameters $J_{\mathrm{II}}$ and
$J_{\mathrm{III}}$ obtained by fit (B) are in all cases in agreement with the
parameters determined via the SW ansatz within their ranges of uncertainties.
The average absolute deviation between data points and fit (B) is always
smaller than 10% of the deviation to fit (A), except for $J_{\mathrm{III}}$
and $t_{\mathrm{AF}}=30$ nm. This is in good agreement with the
antiproportional $t_{\mathrm{F}}$-dependence of $J_{\mathrm{eff}}$ for
$t_{\mathrm{AF}}=5$ nm [Fig. 3(f)]. As said above, it is expected that the
extended SW ansatz overestimates the intrinsic FUMA of the F due to an
entanglement with the RMA. Hence, the anisotropy constant $K_{\mathrm{F}}$
determined by both fits (A) and (B) and for both $t_{\mathrm{AF}}$ is smaller
than the values determined by the fit based on Eq. (13).

Eqs. (10) and (11) are, therefore, in good agreement with the time-dependent
SW approach when an antiproportional dependence of $J_{\mathrm{eff}}$ on
$t_{\mathrm{F}}$ [Fig. 3(f)] is introduced. The latter additionally depends on
$t_{\mathrm{AF}}$ and investigations presented in literature further suggest,
that in general $H_{\mathrm{C}}+\mathrm{const.}\propto 1/{t_{\mathrm{F}}}^{n}$
and $H_{\mathrm{EB}}\propto 1/{t_{\mathrm{F}}}^{m}$ with $1\leq n,m\leq 2$
[33]. Although the $1/t_{\mathrm{F}}$-dependence of the exchange bias shift
and the coercivity has been tested and validated for a variety of systems [1,
20, 51, 52], deviations from this with $n,m>1$ [33, 53, 54] should be
considered depending on measurement conditions as well as the microstructure
of the system [33].

Antiferromagnetic thickness dependence.
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ for $t_{\mathrm{F}}=10$ nm
are displayed in Fig. 4(a). The commonly observed dependence [1, 20, 51, 29]
is reproduced, where a significant EB shift starts to be observable for
$t_{\mathrm{AF}}\geq 5$ nm, increasing up to $t_{\mathrm{AF}}=12.5$ nm. The EB
shift stays constant at about $(9.3\pm 1.2)$ kA/m as the average absolute
value for $t_{\mathrm{AF}}\leq 12.5$ nm. The coercivity shows a significant
increase for 2.5 nm $<t_{\mathrm{AF}}<$ 5 nm over $\langle
H_{\mathrm{C}}^{\mathrm{exp}}\rangle=(1.7\pm 1.2)$ kA/m (average value for
$t_{\mathrm{AF}}\leq 2.5$ nm representing the coercive field of the sole F
layer) and exhibits a maximum value of
$H_{\mathrm{C}}^{\mathrm{exp}}=(12.4+1.1)$ kA/m at $t_{\mathrm{AF}}=7.5$ nm.
At this thickness, $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ has the
largest slope. For larger $t_{\mathrm{AF}}$ the coercivity decreases, as the
EB shift reaches its plateau, until it does not change significantly and stays
constant at $\langle H_{\mathrm{C}}^{\mathrm{exp}}\rangle=(5.0\pm 1.2)$ kA/m
for $t_{\mathrm{AF}}\geq 30$ nm.

Figure 4:  $t_{\mathrm{AF}}$-dependent (a) $|H_{\mathrm{EB}}^{\mathrm{exp}}|$
and $H_{\mathrm{C}}^{\mathrm{exp}}$ as well as parameters (b)
$p_{\mathrm{III}}=1-p_{\mathrm{II}}$ (c) $J_{\mathrm{eff}}$, (d)
$J_{\mathrm{add}}$, (e) $\tau_{\mathrm{C}}^{\mathrm{II}}$ and (f)
$K_{\mathrm{F}}$ obtained by fitting
$H_{\mathrm{EB/C}}^{\mathrm{SW}}(\varphi)$ to
$H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ [Tab. 3 (SW)] for
$t_{\mathrm{F}}=10$ nm. The fits (C), (P) and (J) in (a-c) are based on Eqs.
(10), (11), (III.2), (III.2) and (12) [Tab. 3 (C), (P) and (J)].

Also here, model calculations $H_{\mathrm{EB/C}}^{\mathrm{SW}}(\varphi)$ based
on Eq. (13) are fitted to $H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ [Tab. 3
(SW)] as functions of $t_{\mathrm{AF}}$ for $t_{\mathrm{F}}=10$ nm with
$M_{\mathrm{S}}=(1527\pm 25)$ kA/m and angles
$\gamma_{\mathrm{F}}\neq\gamma_{\mathrm{EB}}^{\mathrm{III}}\neq 0$, with the
latter exhibiting no significant dependence. Optimum parameters extracted for
$t_{\mathrm{AF}}\geq 5$ nm are shown in Fig. 4(b-f) and are discussed in the
following:

  * $\bm{p_{\mathrm{III}}}$

The percentages
$p_{\mathrm{III}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}(t_{\mathrm{AF}})$ are
given in Fig. 4(b) with $p_{\mathrm{III}}$ increasing with increasing
$t_{\mathrm{AF}}$ as $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ in
Fig. 4(a), reaching a constant value $\langle p_{\mathrm{III}}\rangle=(0.8\pm
0.2)$ as the average for $t_{\mathrm{AF}}\geq 12.5$ nm much alike the EB shift
as a function of $t_{\mathrm{AF}}$. This implies a gradual shift of the AF GVD
to larger AF grain volumes. As larger AF grains are more probable in the
thicker polycrystalline AF layers, the proportion of grains accounted to class
III increases while the proportion of grains accounted to class II decreases.
Since $p_{\mathrm{II}}$ and $p_{\mathrm{III}}$ are defined as the percentages
of grains accounted to the respective grain classes, $p_{\mathrm{II}}$ will
approach 1 for small $t_{\mathrm{AF}}$. Displayed in Fig. 4(b) are fits using
Eq. (III.2), extended by the multiplicative factor
$p_{\mathrm{III}}^{\mathrm{max}}$, to
$p_{\mathrm{III}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}(t_{\mathrm{AF}})$ [Tab. 3
(P)], considering that $p_{\mathrm{III}}$ does not approach exactly 1 for
increasing $t_{\mathrm{AF}}$. The non-zero percentage of grains belonging to
class II at large $t_{\mathrm{AF}}$ is caused by a non-ideal interrupted
columnar growth, where a certain percentage of grains will not grow over the
complete thickness of the layer, resulting in effectively smaller AF grains in
contact with the F. Consequently, there will be always a finite non-zero
amount of AF grains that can be associated to class II for increasing
$t_{\mathrm{AF}}$.

  * $\bm{J_{\mathrm{eff}}}$

The effective coupling constant [Fig. 4(c)] as defined in Eq. (12), increases
and stays constant within the margin of uncertainty at $\langle
J_{\mathrm{eff}}\rangle=(2.3\pm 0.6)$ $10^{-4}$ J/m2 for $t_{\mathrm{AF}}\geq
7.5$ nm, with a decreasing tendency for increasing $t_{\mathrm{AF}}$. Assuming
a constant microscopic coupling constant $J_{\mathrm{EB}}$, this suggests that
at $t_{\mathrm{AF}}=7.5$ nm most of the AF grains belong to classes II or III.
For increasing $t_{\mathrm{AF}}$ the percentage of class IV grains will
increase accompanied by a decease of $p(t_{\mathrm{AF}})$. Eq. (12),
describing $J_{\mathrm{eff}}(t_{\mathrm{AF}})$ linked to $p(t_{\mathrm{AF}})$
defined by Eq. (III.2), is fitted to the values presented in Fig. 4(c) [Tab. 3
(J)], yielding the microscopic coupling constant $J_{\mathrm{EB}}=(2.17\pm
0.06)$ $10^{-4}$ J/m2.

  * $\bm{J_{\mathrm{add}}}$

The additional effective coupling constant $J_{\mathrm{add}}$ [Fig. 4(d)]
shows a significant enhancement between $t_{\mathrm{AF}}=15$ and 30 nm. This
occurs in the thickness regime where the decrease of
$H_{\mathrm{C}}^{\mathrm{exp}}$ with $t_{\mathrm{AF}}$ slows down to a
constant value. $J_{\mathrm{add}}$ is, therefore, connected with AF grains in
the vicinity of the grain class boundary between classes II and III.

  * $\bm{\tau_{\mathrm{C}}^{\mathrm{II}}}$

In Fig. 4(e), the average relaxation time $\tau_{\mathrm{C}}^{\mathrm{II}}$ of
grains associated to class II at room temperature is displayed in units of
$t_{\mathrm{Hys}}$. It increases with increasing $t_{\mathrm{AF}}$ and reaches
a plateau with an average value
$\langle\tau_{\mathrm{C}}^{\mathrm{II}}\rangle=(9\pm 4)$ $10^{-3}$
$t_{\mathrm{Hys}}$ for $t_{\mathrm{AF}}\geq 15$ nm. With the average
hysteresis duration $t_{\mathrm{Hys}}\approx 44$ s of the angular-resolved
measurements, this gives an average relaxation time of
$H_{\mathrm{C}}$-mediating grains of $(390\pm 170)$ ms. The increase of
$\tau_{\mathrm{C}}^{\mathrm{II}}$ with $t_{\mathrm{AF}}$ and its saturation
for larger $t_{\mathrm{AF}}$ is in agreement with the general description of
polycrystalline EB systems as well as the definition given in Eq. (19) [28].
As the averaging of $\tau_{\mathrm{AF}}$ is performed within the boundaries of
class II [Eq. (19)], $\tau_{\mathrm{C}}^{\mathrm{II}}$ should increase for
increasing $t_{\mathrm{AF}}$ until the expectation value of the AF GVD passes
the class boundary $V_{\mathrm{II/III}}$. From there,
$\tau_{\mathrm{C}}^{\mathrm{II}}$ will not increase further.

  * $\bm{K_{\mathrm{F}}}$

The anisotropy constant $K_{\mathrm{F}}$ [Fig. 4(f)] decreases from
$K_{\mathrm{F}}=(11\pm 6)$ kJ/m3 and stays constant at $\langle
K_{\mathrm{F}}\rangle=(4\pm 4)$ kJ/m3 within the range of uncertainty for
$t_{\mathrm{AF}}\geq 30$ nm. As the course of
$K_{\mathrm{F}}(t_{\mathrm{AF}})$ is comparable to the one of
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ in Fig. 4(a), likewise to the
$t_{\mathrm{F}}$-dependence of $K_{\mathrm{F}}$ displayed in Fig. 3(i), the
observable increase for small $t_{\mathrm{AF}}\geq 5$ nm is linked to an
entanglement of the F’s intrinsic FUMA with the RMA [36].

The fits in Fig. 4(b) and (c) with respect to
$p_{\mathrm{III}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}(t_{\mathrm{AF}})$ and
$J_{\mathrm{eff}}(t_{\mathrm{AF}})$ [Tab. 3 (P) and (J)], respectively,
validate Eqs. (III.2) and (III.2) describing the $t_{\mathrm{AF}}$-dependence
of $p$ and $p_{\mathrm{III}}$. Consequently, by incorporating these equations
together with Eq. (12) into Eqs. (10) and (11), relations
$|H_{\mathrm{EB}}(t_{\mathrm{AF}})|$ and $H_{\mathrm{C}}(t_{\mathrm{AF}})$ can
be fitted to the experimentally determined
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ as displayed in Fig. 4(a)
[Tab. 3 (C)]. As Eqs. (III.2) and (III.2) additionally depend on the grain
class boundaries $V_{\mathrm{I/II}}$, $V_{\mathrm{II/III}}$ and
$V_{\mathrm{III/IV}}$ and on the parameters $\mu$ and $\sigma$ describing the
distribution of AF grain radii $r_{\mathrm{AF}}$, these parameters connect the
$t_{\mathrm{AF}}$-dependent relations of the EB shift and the coercive field
with the polycrystalline AF GVD and the measurement conditions. The determined
fit parameters are presented in Tab. 1.

Although the fit to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ in
Fig. 4(a) yields $J_{\mathrm{EB}}=(1.9\pm 1.3)$ $10^{-4}$ J/m2 and the fit to
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ gives
$J_{\mathrm{EB}}=(2.8\pm 0.9)$ $10^{-4}$ J/m2 displaying rather large
uncertainties, both values agree with $J_{\mathrm{EB}}=(2.17\pm 0.06)$
$10^{-4}$ J/m2, obtained from fitting $J_{\mathrm{eff}}(t_{\mathrm{AF}})$ in
Fig. 4(c) [Tab. 3 (J)], as estimates for the microscopic coupling constant
$J_{\mathrm{EB}}$. $K_{\mathrm{F}}=(1.2\pm 0.5)$ kJ/m3 determined by fitting
Eq. (11) to $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ is significantly
smaller than the values determined by fitting model calculations based on the
SW ansatz (SW) and by the $t_{\mathrm{F}}$-dependent fit (B) in Fig. 3(c) and
(d) but larger than the values obtained utilizing fit (A) (Tab. 1). This can
understood by on overestimation of $K_{\mathrm{F}}$ in the case of fitting
$H_{\mathrm{C}}^{\mathrm{SW}}(\varphi)$ to
$H_{\mathrm{C}}^{\mathrm{exp}}(\varphi)$ (SW) and by an underestimation of
$K_{\mathrm{F}}$ in the case of fitting Eq. (11) to
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ (A), because
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$-values at large
$t_{\mathrm{F}}$ are needed to accurately determine the offset
$2K_{\mathrm{F}}/\mu_{0}M_{\mathrm{S}}$. When fitting Eq. (11) to
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ (C), $K_{\mathrm{F}}$ is
determined by values at small $t_{\mathrm{AF}}\rightarrow 0$, which is more
explicit as $H_{\mathrm{C}}^{\mathrm{exp}}$ can be measured for
$t_{\mathrm{AF}}=0$ nm by omitting the AF layer.

Figure 5:  (a) Comparison of the structural ($t_{\mathrm{AF}}=5$ nm [Fig.
2(e)]) and magnetically effective AF grain radius distribution
$\tilde{\varrho}(r_{\mathrm{AF}})$ based on the experimentally performed grain
size analysis by atomic force microscopy and the extracted parameters $\mu$
and $\sigma$ given in Tab. 1 [Tab. 3 (C)], respectively. (b-c) Structural and
magnetically effective AF GVD $\varrho(V_{\mathrm{AF}})$ have been
subsequently derived by use of Eq. (III.2) for $t_{\mathrm{AF}}=5$ and 30 nm.
Notice that (b) and (c) display the same data but with different axis limits.
The expectation values of the AF grain radius and volume, $\langle
r_{\mathrm{AF}}\rangle$ and $\langle V_{\mathrm{AF}}\rangle$, are depicted and
lognormal fits are given with respect to the structural AF grain radius and
volume distributions.

Values for $\mu$ and $\sigma$ as well as the expectation value $\langle
r_{\mathrm{AF}}\rangle$ of the AF grain radius and the standard deviation SD
extracted by fitting Eqs. 10 and (11) to
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ in Fig. 4(a) and Eq. (12) to
$J_{\mathrm{eff}}(t_{\mathrm{AF}})$ in Fig. 4(c) [Tab. 3 (C) and (J)] are
listed in Tab. 1. These fit scenarios yield considerably smaller values for
$\langle r_{\mathrm{AF}}\rangle$ as the structural average AF grain radius
$(7.0\pm 0.3)$ nm determined by atomic force microscopy. Averaging the
obtained values of $\mu$ and $\sigma$ in the case of fit scenario (C) yields
$\langle r_{\mathrm{AF}}\rangle=(3.0\pm 0.6)$ nm representing $\approx(43\pm
10)$% of the experimentally determined value. This indicates that only about
$\approx(18\pm 8)$% of the structural AF grain volume is effectively
contributing to the interfacial exchange coupling. The latter is visualized by
comparing the structural ($t_{\mathrm{AF}}=5$ nm [Fig. 2(e)]) and the
magnetically effective distribution of AF grain radii in Fig. 5(a) and by
comparing the structural and magnetically effective AF GVD for
$t_{\mathrm{AF}}=5$ and 30 nm in Fig. 5(b) with the help of Eq. (III.2).

Finally, from the determined grain class boundaries listed in Tab. 1, also the
timescales determining these boundaries for fixed temperatures can be derived
by Eq. (3). For the grain class boundaries between classes I and II as well as
between II and III, the respective time scales have been determined to be
$\tau_{\mathrm{I/II}}=(2\pm 2)$ $10^{-9}$ s and $\tau_{\mathrm{II/III}}=(41\pm
30)$ s with $T=T_{\mathrm{RT}}\approx 293$ K based on the average values of
$V_{\mathrm{I/II}}$ and $V_{\mathrm{II/III}}$ obtained by fitting Eqs. (10)
and (11) to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ [Tab. 3 (C)]. Despite the
rather large uncertainty, the measurement time $t_{\mathrm{Hys}}\approx 44$ s
is reproduced by $\tau_{\mathrm{II/III}}$.

###  IV.3 Deposition rate dependent analysis

Eqs. (10) and (11) as relations that can be fitted to
$t_{\mathrm{AF}}$-dependent data of the EB shift and the coercivity represent
a powerful tool to retrieve parameters characterizing the microstructure of
the AF layer in a polycrystalline EB system. Therefore,
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ have been experimentally
determined for fixed $t_{\mathrm{F}}=10$ nm for different deposition rates
$\eta_{\mathrm{AF}}$ of the AF layer to modify the distribution of AF grain
radii [24, 26].
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}},\eta_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}},\eta_{\mathrm{AF}})$ are
displayed in Fig. 6(a) and (b), respectively, for different
$\eta_{\mathrm{AF}}$.
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}},\eta_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}},\eta_{\mathrm{AF}})$ are
equivalent to the dependencies depicted in Fig. 4(a) for large
$\eta_{\mathrm{AF}}$, whereas for decreasing $\eta_{\mathrm{AF}}$ a gradual
suppression of the EB shift and the coercivity can be observed [26].

Figure 6:  (a) $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and (b)
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ for different deposition
rates $\eta_{\mathrm{AF}}$ of the AF layer with corresponding fits using Eqs.
(10) and (11) [Tab. 3 (C)]. $\eta_{\mathrm{AF}}$-dependent (c) $\langle
r_{\mathrm{AF}}\rangle$, (d) $J_{\mathrm{EB}}$ and (e) $K_{\mathrm{F}}$ as
averages of parameters obtained from fitting Eqs. (10) and (11) to
$t_{\mathrm{AF}}$-dependent relations in (a) and (b). Average values in the
$\eta_{\mathrm{AF}}$-intervals indicated by dashed lines are additionally
displayed.

Parameters $\langle r_{\mathrm{AF}}\rangle$, $J_{\mathrm{EB}}$ and
$K_{\mathrm{F}}$ as functions of $\eta_{\mathrm{AF}}$ have been determined by
fitting Eqs. (10) and (11) in Fig. 6(a-b) [Tab. 3 (C)] and are presented in
Fig. 6(c-e). In the deposition rate interval in which an overall increase of
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}},\eta_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}},\eta_{\mathrm{AF}})$ can be
observed, $\langle r_{\mathrm{AF}}\rangle$ and $J_{\mathrm{EB}}$ increase
gradually until saturation with $\overline{\langle
r_{\mathrm{AF}}\rangle}=(3.0\pm 0.3)$ nm and $\langle
J_{\mathrm{EB}}\rangle=(2.2\pm 0.5)$ $10^{-4}$ J/m2 for
$\eta_{\mathrm{AF}}\geq 2.43$ nm/min. $K_{\mathrm{F}}$ stays constant for all
$\eta_{\mathrm{AF}}$ with an average value of $\langle
K_{\mathrm{F}}\rangle=(1.4\pm 0.7)$ kJ/m3. This implies a constant FUMA
independent of $\eta_{\mathrm{AF}}$ but a dependence of the average AF grain
radius $\langle r_{\mathrm{AF}}\rangle$ and the microscopic coupling constant
$J_{\mathrm{EB}}$ on the AF deposition rate for $\eta_{\mathrm{AF}}<2.43$
nm/min. A reduction of $J_{\mathrm{EB}}$ might be connected to a
$\eta_{\mathrm{AF}}$-dependence of the AF/F-interface structure or the AF
crystal texture and homogeneity of AF crystallites, crucially determining the
coupling strength between individual AF grains with the F [1, 3, 26, 55, 22].

###  IV.4 Simulations and cross check

Table 2:  Input parameters for calculated AF grain size distributions as well as simulated $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and $H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ displayed in Fig. 8, representing the average values of the respective parameters given in Tab. 1 obtained from fitting Eqs. (10) and (11) to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and $H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ [Tab. 3 (C)] as displayed in Fig. 4(a). Optimum parameters obtained from fitting Eqs. (10) and (11) to simulated $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and $H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ are given, reproducing the input parameters of the simulations using the time-dependent SW ansatz based on Eq. (13) [Tab. 3 (CC)]. |  |  $t_{\mathrm{AF}}=10$ nm |  $t_{\mathrm{F}}=10$ nm
---|---|---|---
Data to be fitted |  $\rightarrow$ |  $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}})|$ & $H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}})$ |  $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{AF}})|$ |  $H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{AF}})$
Parameter | Input | (AC) | (CC)
| $J_{\mathrm{EB}}$
---
($10^{-5}$ J/m2)
23.29 |  | $23.35\pm 14.53$ | $23.17\pm 1.65$
| $J_{\mathrm{II}}$
---
($10^{-5}$ J/m2)
6.70 | $5.97\pm 0.06$ |  |
| $J_{\mathrm{III}}$
---
($10^{-5}$ J/m2)
16.59 | $16.80\pm 0.04$ |  |
$K_{\mathrm{F}}$ (kJ/m3) | 1.21 | $0.34\pm 0.07$ |  | $1.21\pm 0.32$
$\mu$ (nm) | 1.10 |  | $1.10\pm 0.89$ | $1.10\pm 0.58$
$\sigma$ (nm) | 0.16 |  | $0.16\pm 0.01$ | $0.16\pm 0.03$
$\langle r_{\mathrm{AF}}\rangle$ (nm) | 3.03 |  | $3.05\pm 2.71$ | $3.05\pm 1.78$
SD (nm) | 0.48 |  | $0.49\pm 0.21$ | $0.48\pm 0.52$
$V_{\mathrm{I/II}}$ (nm3) | 60 |  | $60\pm 14$ | $60\pm 70$
$V_{\mathrm{II/III}}$ (nm3) | 236 |  | $237\pm 422$ | $235\pm 274$
$V_{\mathrm{III/IV}}$ (nm3) | 2153 |  | $2142\pm 381$ | $2142\pm 160$

In addition to the experimental approach discussed so far, Eqs. (10) and (11)
are fitted to
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ that have been
simulated by using the time-dependent SW ansatz given by Eq. (13) [Tab. 3
(CC)]. With Eqs. (10) and (11) reproducing the input parameters of the
simulations, the validity of named relations is evidenced (cross check) in the
context of the time-dependent SW approach introduced in Sec. III.4 and Refs.
[28, 48, 26, 36].

Figure 7:  Flowchart describing the cross check’s individual steps comparing
simulated $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$
and $H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$, using the
time-dependent SW ansatz [Eq. (13)], with the extended phenomenological
relations $|H_{\mathrm{EB}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ defined by Eqs. (10) and
(11). It is aimed at reproducing the input parameters by fitting the relations
to the simulated values [Tab. 3 (CC)] in order to check for the validity of
Eqs. (10) and (11) in the context of the time-dependent SW ansatz on the basis
of calculated AF GVDs.

The individual steps of the cross check are shown in Fig. 7 and will be
explained in the following. The input parameters used are listed in Tab. 2
based on the averaged parameters given in Tab. 1 obtained by fitting Eqs. (10)
and (11) to $|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ [Tab. 3 (C)]. Starting from
the input parameters $\mu$ and $\sigma$, the AF GVD
$\varrho(V_{\mathrm{AF}},t_{\mathrm{AF}},\mu,\sigma)$ can be calculated for
different $t_{\mathrm{AF}}$ [Eq. (III.2)]. In Fig. 8(a), calculated AF GVDs
are displayed for exemplary thicknesses $t_{\mathrm{AF}}$ visualizing the
tunability of the grain classes’ population with the AF layer thickness.
$J_{\mathrm{eff}}(t_{\mathrm{AF}})$,
$p_{\mathrm{III}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}(t_{\mathrm{AF}})$, and
consequently $J_{\mathrm{II/III}}(t_{\mathrm{AF}})$, as well as
$\tau_{\mathrm{C}}^{\mathrm{II}}(t_{\mathrm{AF}})$ are determined for fixed
$J_{\mathrm{EB}}$ and $K_{\mathrm{F}}$ by integration of the AF GVD
considering respective bounds $V_{\mathrm{I/II}}$, $V_{\mathrm{II/III}}$ and
$V_{\mathrm{III/IV}}$ given in Tab. 2. With $J_{\mathrm{add}}=0$ J/m2,
$\gamma_{\mathrm{F}}=\gamma_{\mathrm{EB}}^{\mathrm{II}}=0^{\circ}$ and
$M_{\mathrm{S}}=1527$ kA/m,
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ have been
simulated using the time-dependent SW ansatz based on Eq. (13) and are
displayed in Fig. 8(b) and (d) with fits of Eqs. (10) and (11) [Tab. 3 (CC)]
for the reproduction of the input parameters as depicted in Fig. 7.

Figure 8:  (a) Calculated AF GVDs for different $t_{\mathrm{AF}}$ based on
parameters $\mu$ and $\sigma$ from Tab. 2. (b) Simulated
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}})$ for $t_{\mathrm{AF}}=10$ nm in
(a) with corresponding fits $\propto 1/t_{\mathrm{F}}$ [Tab. 3 (AC)].
$t_{\mathrm{AF}}$-dependent (c) $p^{\mathrm{SW}}$ and
$p_{\mathrm{III}}^{\mathrm{SW}}=1-p_{\mathrm{II}}^{\mathrm{SW}}$ obtained by
integrating distributions in (a) and corresponding fits based on Eqs. (III.2)
and (III.2) [Tab. 3 (PC)]. (d) Simulated
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{AF}})$ using
$p^{\mathrm{SW}}(t_{\mathrm{AF}})$ and
$p_{\mathrm{III}}^{\mathrm{SW}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}^{\mathrm{SW}}(t_{\mathrm{AF}})$
displayed in (c) alongside fits using Eqs. (10) and (11) [Tab. 3 (CC)]. Input
parameters for (a-d) are given in Tab. 2 alongside extracted fit parameters.

Simulated $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}})$ are presented in Fig. 8(b) for
$t_{\mathrm{AF}}=10$ nm with fits using Eqs. (10) and (11) [Tab. 3 (AC)]. The
$t_{\mathrm{F}}$-dependent relations $|H_{\mathrm{EB}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}$ fit well to the simulated
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}})$ and the effective coupling
constants $J_{\mathrm{II}}$ and $J_{\mathrm{III}}$ of the respective grain
classes are reproduced with a deviation of $\lesssim$ 10% from the input
values [Tab. 2]. However, $K_{\mathrm{F}}=(0.34\pm 0.07)$ kJ/m3 is differing
from the input value $1.21$ kJ/m3 as the offset
$2K_{\mathrm{F}}/\mu_{0}M_{\mathrm{S}}$ in Eq. (11) is determined by values of
the coercive field at large $t_{\mathrm{F}}$.

Simulated $p^{\mathrm{SW}}(t_{\mathrm{AF}})$,
$p_{\mathrm{III}}^{\mathrm{SW}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}^{\mathrm{SW}}(t_{\mathrm{AF}})$
and $|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{AF}})$ are depicted in Fig. 8(c) and
(d), which qualitatively reproduce the experimentally determined
$t_{\mathrm{AF}}$-dependencies displayed in Fig. 4(a-c). Fits based on Eqs.
(III.2) and (III.2) [Tab. 3 (PC)] as well as Eqs. (10) and (11) [Tab. 3 (CC)]
agree with the simulated dependencies. Within the uncertainty margins, input
parameters used for the simulated dependencies are reproduced by the fits (PC)
and (CC) [Tab. 2].

The agreement of the the relations
$|H_{\mathrm{EB}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ as defined by Eqs. (10) and
(11) with the simulated relations
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ based on Eq.
(13) emphasizes the validity of the direct connection between the SW approach
and the presented analytic expressions of the EB shift and the coercivity.

##  V Conclusion

We conducted a systematic investigation of the ferromagnetic (F) as well as
the antiferromagnetic (AF) thickness dependence of the exchange bias (EB)
shift and the coercive field of the prototypical polycrystalline AF/F-bilayer
IrMn($t_{\mathrm{AF}}$)/CoFe($t_{\mathrm{F}}$). Thickness-dependent relations,
further depending on the conditions of observation and the parameters
characterizing the AF grain volume distribution (GVD), are introduced and
validated by the comparison with simulations based on an extended time-
dependent Stoner-Wohlfarth (SW) ansatz. These prove to interlink the averaged
microscopic material parameters with averaged macroscopic magnetic quantities,
representing the adequate tool to check for the equality of the magnetically
effective and the structural AF GVD.

In contrast to the average structural AF grain radius ${(7.0\pm 0.3)}$ nm,
experimentally determined by atomic force microscopy, fits to the measured
$t_{\mathrm{AF}}$-dependent EB shift and coercive field gave rise to a
significantly smaller value of ${(3.0\pm 0.6)}$ nm. This indicates that the
grains’ antiferromagnetic order extends only over ${(18\pm 8)}$% of the
structural volume.

For the investigated system, the microscopic coupling constant could be
determined to be ${J_{\mathrm{EB}}=(2.3\pm 1.7)}$ $10^{-4}$ J/m2 by fitting
$t_{\mathrm{AF}}$-dependent relations of the EB shift and the coercive field
to thickness-dependent experimental data, whereas fits based on the time-
dependent SW ansatz yielded ${J_{\mathrm{EB}}=(2.17\pm 0.06)}$ $10^{-4}$ J/m2.
Furthermore, the timescale of observation for measurements at room temperature
could be reproduced and the timescale below which thermally unstable AF grains
exhibit superparamagnetic behavior could be estimated to be
${\tau_{\mathrm{I/II}}=(2\pm 2)}$ $10^{-9}$ s. Introducing the AF layer’s
deposition rate as an additional parameter alongside its thickness allowed for
a systematic study of the EB shift and the coercive field in dependence on the
average aspect ratio of AF grains. The extracted averaged microscopic
parameters as functions of the deposition rate are in agreement with the
utilized model description.

Successfully interlinking analytic expressions describing
${|H_{\mathrm{EB}}(t_{\mathrm{F}},t_{\mathrm{AF}})|}$ and
$H_{\mathrm{C}}(t_{\mathrm{F}},t_{\mathrm{AF}})$ with averaged microscopic
material parameters in the context of a generalized model emphasizes the
consistency of the latter. The presented overall macroscopic approach for the
description of polycrystalline EB bilayers in dependence on their
microstructure shall represent a showcase example for the modeling of
polycrystalline systems in general and especially more complex
heterostructures composed of systems similar to the ones investigated.

###### Acknowledgements.

We acknowledge funding by the DAAD (Project ID 57392264). Further, we thank
Dennis Holzinger, André Knie, Feliks Stobiecki, Bogdan Szymański, Piotr Kuświk
and Hubert Głowinski for fruitful discussions and Adam Krysztofik for
performing VNA-FMR measurements.

*

##  Appendix A Fit scenarios

Throughout the manuscript different fits are performed, which are listed in
Tab. 3. For each individual scenario, the fit function, the data to be fitted
and the extractable parameters are given.

Table 3:  Overview of the different fit scenarios referred to throughout the
present study. For each case it is displayed, which fit functions with the
respective dependencies are used, to which type of data they are fitted and
which fit parameters are extracted. (A), (B), (C), (AC) and (CC) are based on
Eqs. (10) and (11). (P) and (PC) connected to Eqs. (III.2) and (III.2), while
(J) is based on Eq. (12). (SW) represents the fit of model calculations, using
the extended time-dependent SW ansatz based on Eq. (13) and introduced in Sec.
III.4, to experimentally determined angular-resolved data.

(A) and (B) represent relations based on Eqs. (10) and (11) introduced in Sec.
III.3 fitted to experimentally determined
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{F}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{F}})$ for fixed $t_{\mathrm{AF}}$,
whereas (C) is connected to the same equations, which are however fitted to
$|H_{\mathrm{EB}}^{\mathrm{exp}}(t_{\mathrm{AF}})|$ and
$H_{\mathrm{C}}^{\mathrm{exp}}(t_{\mathrm{AF}})$ for fixed $t_{\mathrm{F}}$.
(SW) describes the fit of angular-resolved model calculations
$H_{\mathrm{EB/C}}^{\mathrm{SW}}(\varphi)$ to
$H_{\mathrm{EB/C}}^{\mathrm{exp}}(\varphi)$ by minimization of Eq. (13)
representative for the time-dependent SW ansatz introduced in Sec. III.4,
aiming for the quantitative determination of model parameters. (P) and (J) are
$t_{\mathrm{AF}}$-dependent fits of Eqs. (III.2), (III.2) and (12) to
$p(t_{\mathrm{AF}})$,
$p_{\mathrm{III}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}(t_{\mathrm{AF}}$ and
$J_{\mathrm{eff}}(t_{\mathrm{AF}})$ obtained by (SW). (AC), (PC) and (CC) are
$t_{\mathrm{F}}$\- and $t_{\mathrm{AF}}$-dependent fits of Eqs. (10), (11),
(III.2), (III.2) to
$|H_{\mathrm{EB}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})|$,
$H_{\mathrm{C}}^{\mathrm{SW}}(t_{\mathrm{F}},t_{\mathrm{AF}})$,
$p^{\mathrm{SW}}(t_{\mathrm{AF}})$ and
$p_{\mathrm{III}}^{\mathrm{SW}}(t_{\mathrm{AF}})=1-p_{\mathrm{II}}(t_{\mathrm{AF}})$
obtained by model calculations based on the time-dependent SW ansatz given by
Eq. (13) and the calculation of the AF GVD for a specific set of input
parameters (Tab. 2). These fit scenarios are variations of (A), (P) and (C) as
they are performed for the cross check between the extended phenomenological
relations introduced in Sec. III.3 and the time-dependent SW ansatz explained
in Sec. III.4.

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