††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Chemical disorder effects on the Gilbert damping of FeCo alloys

Zhiwei Lu  Department of Applied Physics, School of Engineering Sciences, KTH
Royal Institute of Technology, AlbaNova University Center, SE-10691 Stockholm,
Sweden<EMAIL_ADDRESS>I. P. Miranda  Department of Physics and Astronomy,
Uppsala University, Box 516, SE-75120 Uppsala, Sweden
<EMAIL_ADDRESS>Simon Streib  Department of Physics and Astronomy,
Uppsala University, Box 516, SE-75120 Uppsala, Sweden  Qichen Xu  Department
of Applied Physics, School of Engineering Sciences, KTH Royal Institute of
Technology, AlbaNova University Center, SE-10691 Stockholm, Sweden  Rajgowrav
Cheenikundil  School of Science and Technology, Örebro University, SE-701 82,
Örebro, Sweden  Manuel Pereiro  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-75120 Uppsala, Sweden  Erik Sjöqvist  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
Olle Eriksson  Department of Physics and Astronomy, Uppsala University, Box
516, SE-75120 Uppsala, Sweden  School of Science and Technology, Örebro
University, SE-701 82, Örebro, Sweden  Wallenberg Initiative Materials Science
for Sustainability (WISE), Uppsala University, Box 516, SE-75120 Uppsala,
Sweden  Anders Bergman  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-75120 Uppsala, Sweden  Danny Thonig  School of Science
and Technology, Örebro University, SE-701 82, Örebro, Sweden  Department of
Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden
Anna Delin  Department of Applied Physics, School of Engineering Sciences, KTH
Royal Institute of Technology, AlbaNova University Center, SE-10691 Stockholm,
Sweden  SeRC (Swedish e-Science Research Center), KTH Royal Institute of
Technology, SE-10044 Stockholm, Sweden  Wallenberg Initiative Materials
Science for Sustainability (WISE), KTH Royal Institute of Technology, SE-10044
Stockholm, Sweden

###### Abstract

The impact of the local chemical environment on the Gilbert damping in the
binary alloy Fe100-xCox is investigated, using computations based on density
functional theory. By varying the alloy composition $x$ as well as Fe/Co atom
positions we reveal that the effective damping of the alloy is highly
sensitive to the nearest neighbor environment, especially to the amount of Co
and the average distance between Co-Co atoms at nearest neighbor sites. Both
lead to a significant local increase (up to an order of magnitude) of the
effective Gilbert damping, originating mainly from variations of the density
of states at the Fermi energy. In a global perspective (i.e., making a
configuration average for a real material), those differences in damping are
masked by statistical averages. When low-temperature explicit atomistic
dynamics simulations are performed, the impact of short-range disorder on
local dynamics is observed to also alter the overall relaxation rate. Our
results illustrate the possibility of local chemical engineering of the
Gilbert damping, which may stimulate the study of new ways to tune and control
materials aiming for spintronics applications.

##  I Introduction

Magnetic damping, also known as Gilbert damping ($\alpha$) in the context of
the Landau-Lifshitz-Gilbert (LLG) equation [1], plays a crucial role in
determining the rate at which energy and angular momentum dissipate in a
magnetic system. It thus generally dictates the timescale of processes
involving a magnetic transition from an excited state to an equilibrium state,
magnetic domain wall motion, and spin-wave propagation. These are the
fundamental processes that govern the performance of spintronics [2] and
magnonics [3] devices. Reduced magnetic damping can lead to higher efficiency
and faster operations in these devices by minimizing energy dissipation during
their operation. Hence, a detailed investigation into the fundamental physics
involved in the mechanism of damping is of great interest [4, 5, 2, 6].
Previous theoretical and experimental research has established that $\alpha$
is not a simple scalar quantity (as is usually assumed), but instead a
tensorial quantity [7, 8, 9, 10, 11, 12] that is temperature-dependent [13,
14, 15], non-local in real-space [16, 17, 7, 8, 18, 19, 20, 21, 22], and
presents an anisotropic behavior – as outlined in various studies [17, 23, 24,
25, 26]. Therefore, fundamental understanding and materials engineering at the
local (atomistic) scale compose, in principle, a possible route to optimum
properties with desired damping magnitude.

Apart from their exceptional properties, including high Curie temperature,
high saturation magnetization and high permeability [27], Fe-Co alloys
constitute also a suitable platform to the fundamental investigation of the
magnetic damping. It was experimentally demonstrated that the Fe100-xCox
alloys (up to $x=60\%$) can hold an ultra-low intrinsic damping, of the order
of $10^{-4}$, at $x\sim 25\%$ [4], despite being metallic. This surprising
result, comparable to the values found in ferrimagnetic insulators (such as
yttrium iron garnet [28]), was explained by the existence, at this particular
value of $x$, of a sharp minimum density of states at the Fermi level, and
corroborated by different methods and levels of theory [29, 14, 10, 19, 30,
31]. In addition, Fe-Co has recently attracted attention with the observation
of a giant anisotropy effect in $\alpha$ [24] in Fe50Co50 thin films,
initially attributed to the variation of spin-orbit coupling due to local
distortions in the alloy. This was also experimentally verified for other
similar alloys involving Fe and Co [32, 33], but the exact mechanisms
responsible for this effect is still an open question [23, 34, 32, 35].
Additionally, at the theoretical level, Lu et al. [19] suggested that for
$x\in\\{30\%,50\%\\}$ the explicit treatment of damping as a nonlocal
parameter in spin-dynamics have the effect of increased magnon lifetime
predictions (for specific wave vectors $\mathbf{q}$). Together, those
observations clearly consolidate FeCo as a benchmark for Gilbert damping
investigations, encompassing many hidden features across the alloy series.

Despite the progress in studying these interesting phenomena in FeCo, other
potential sources of hidden features are still not quite well-explored in the
scope of Gilbert damping. Some have been recently highlighted as effects that
are or might be relevant, also accounting for site-nonlocal damping [36, 23].
They are often related to changes in the local environment: reduced
dimensionality, chemical disorder, and thermal fluctuations in the lattice
(atomic displacements) and in the magnetic state. Although the former remains
as the least discussed among the known effects [36], the other two, which
constitute causes for electron scattering processes, are generally addressed
by effective alloy analogy models (see, e.g., Refs. [37, 29]). In particular,
an explicit atomistic picture is still missing, and here we choose to address
this knowledge gap in the context of the chemical disorder.

In this sense, the alloy models generally used – the virtual crystal
approximation [38] or the coherent potential (CPA) approximation [39] – were
originally developed to describe chemical disorder in materials [40] via an
effective medium. The effective medium is a result of locally averaging
possible clusters with different configurations and compositions within the
alloy, and the CPA is designed to give the scattering properties of electron
states that correspond to a proper configuration average of the alloy. This
theoretical description works reasonably well to reproduce experimentally
measured electronic structure properties of many alloys [41, 42, 43, 44] and
related materials properties, including Gilbert damping [14, 31, 45, 19].
Nevertheless, they frequently overlook the impacts of the local environment,
including atom clustering, among others – precisely the conditions under which
phenomena like damping anisotropy were originally suggested to arise [24].

The local chemical environment has been shown to modify the magnetic
properties of a given material, such as the atomic magnetic moment, magnetic
exchange interactions [46, 47, 48, 49], as well as magnetic anisotropy [50].
For instance, the presence of heterogeneous easy-axis orientations in clusters
leads to a vanishing low magnetic anisotropy in permalloy (Ni80Fe20), although
the favored easy-axis orientations at each site, dependent on the local atomic
arrangement and the cluster’s composition, can be several times larger than
that of elemental Fe or Ni [50]. As a result, the spin Hamiltonian and, thus,
the energy landscape is modified, impacting the dynamics of the local magnetic
moments in a system described by the LLG equation [51]. In the context of
FeCo, early calculations of ordered bcc Fe50Co50 structures have demonstrated
that the magnetic anisotropy can change by some two orders of magnitude when
particular Fe/Co configurations are considered [52]. Also, experimental
studies have shown that the magnetic properties are influenced by various
local environment factors, including composition, morphology [53, 54, 55], as
well as the size of Fe-Co alloy nanoclusters.

In this study, we utilize Fe-Co alloys as a platform to investigate the impact
of local compositions and atom arrangements on $\alpha$ via ab-initio
calculations. Here, we suggest the existence of hidden features (e.g.,
enhanced or decreased damping values) when an explicit atomistic picture is
considered. As a consequence, local chemical engineering may be considered as
a possible route to manipulate the Gilbert damping of such materials.

The paper is organized as follows. In Sec. II, we provide details on the
density functional theory (DFT) calculations, the embedded cluster virtual
crystal approximation model, along with the considered atomic configurations
with different Co concentrations. Section III presents the results of the
local environment effects on the Gilbert damping. In Sec. IV, we give a
summary and an outlook.

##  II Method

###  II.1 Computational details

The electronic structure and, from it, the magnetic properties of the Fe-Co
alloys were obtained using the fully self-consistent real-space linear muffin-
tin orbital in the atomic sphere approximation (RS-LMTO-ASA) [56, 57]. In this
first-principles scheme, the eigenvalue problem is solved directly in the
real-space by employing Haydock recursion method [58] (for $LL$ recursion
steps) together with Beer-Pettifor terminator [59], suitable for metallic
systems. Here, the local spin density approximation (LSDA), with
parametrization by von Barth and Hedin [60], was used for the exchange-
correlation functional (XC). To evaluate the Gilbert damping in the torque-
correlation method, the spin-orbit coupling (SOC) term,
$\xi\hat{L}\cdot\hat{S}$, was included self-consistently at each variational
step [61]. The LMTO-ASA [62] is a linear method that gives precise results
around a given energy $E=E_{\nu}$, often chosen as the center of gravity of
the $s$, $p$ and $d$ bands. Thus, as in the previous study [19], we consider
an expression accurate to $(E-E_{\nu})^{2}$ starting from the orthogonal
representation of LMTO-ASA – given that nonlocal damping parameters are fine
quantities. For all cases investigated, we considered $LL=31$, which produces
reliable results for the damping of Fe-Co alloys when compared to available
experimental and theoretical results in the literature [19].

In order to simulate alloys, we designed the following simulation protocol: we
performed ab-initio calculations on a virtual-crystal-approximation (VCA)
medium of bcc FeCo alloys to account for the chemical disorder. In general,
this approximation is well-known to be reasonable for systems where the energy
bands behave more or less rigidly with changes in the alloy concentration –
which is exactly the case of FeCo, neighboring elements in the periodic table
(for a more detailed discussion on the VCA for FeCo, see Appendix D). Here,
the medium matrix contains $\sim 55000$ atoms. The Fe-Co alloys still maintain
a body-centred-cubic (bcc) phase for the range of Co concentration from 0% to
60% [4] and the lattice parameter $a_{0}=2.87$Å was used for all values of
$x$. Although the FeCo alloys present a slight variation of $\sim 1\%$ in
$a_{0}$ for $x\in[0\%,60\%]$ [63, 64], we keep $a_{0}$ fixed to concentrate on
the effect coming solely from the chemical disorder. It also should be noticed
that the composition of $x=60\%$ Co is at the edge of the transition to body-
centred tetragonal (bct) phase [65], but for simplicity, we remain here in the
bcc phase. As for the VCA approach we replace the nuclear charge ($Z$) and the
number of valence electrons ($n_{v}$) by the alloy average based on the
concentration, e.g., in the case of Fe60Co40 we consider $n_{v}=8.4$ and
$Z=26.4$.

Figure 1: (Color online) Schematic representation of the EC-VCA model in the
bcc structure. The Fe and Co atoms are depicted by yellow and blue spheres,
respectively, forming an explicit Fe53Co47 embedded cluster with radius $R$.
In turn, the gray spheres symbolize the VCA medium mimicking the same alloy
concentration, that extends significantly beyond the depicted limits (the full
VCA matrix has $\sim 55000$ atoms). Image produced with the VESTA software
[66].

After the self-consistent procedure on the pure VCA structure, $15$-atom
clusters composed of different Fe and Co compositions as well as atomic
arrangements were embedded into the center of the corresponding VCA medium as
local impurities (see Figure 1 for a schematic representation).

The impurity region was self-consistently treated while the potential
parameters and Fermi energy of the pure VCA matrix host were kept fixed. This
is a particularly suitable approach and certainly an improvement with respect
to the simple VCA approximation, because the short-range disorder (which pure
VCA neglects) is made explicit, while the more distant sites are replaced by
the virtual atoms with long-range disorder properties. Naturally, this model
tends to become more accurate whenever the radius $R$ of the explicit region
increases, but with two main limitations: (i) $R$ cannot be too close from the
full medium matrix radius, to avoid any surface effects in real space; and
(ii) $R$ should be maximum to still be considered as a local perturbation, in
which no appreciable change in the Fermi level of the crystalline host can be
observed. As our intention here is to analyze the implications of very short
range chemical disorder in the torque-correlation theory for the Gilbert
damping (in its nonlocal formulation), the $15$-atom cluster ($R=a_{0}$) is a
simple and useful model. From now on, this model will be referred to as
embedded cluster VCA (or, in short, EC-VCA).

From the electronic structure of this spatially disordered alloy, we determine
both the onsite (for $i=j$) and the non-local (for $i\neq j$) contributions of
the Gilbert damping $\alpha_{ij}$ from the following expression,

$\begin{split}\alpha_{ij}^{\mu\nu}=\frac{g}{m_{i}\pi}\int_{-\infty}^{\infty}\eta(\epsilon)\textnormal{Tr}\left(\hat{T}_{i}^{\mu}\hat{A}_{ij}(\epsilon_{+})(\hat{T}_{j}^{\nu})^{\dagger}\hat{A}_{ji}(\epsilon_{+})\right)d\epsilon,\end{split}$
(1)

where $\epsilon_{+}=\epsilon+i\delta$ for some small positive value $\delta$
(in energy units), Tr denotes the trace, the hat notation (e.g., $\hat{A}$) is
used for operators (or tensors), and
$\hat{A}_{ij}(\epsilon_{+})=\frac{1}{2i}(\hat{G}_{ij}(\epsilon_{+})-\hat{G}_{ji}^{\dagger}(\epsilon_{+}))$
is the $ij$ block of the spectral function at the energy $\epsilon$. The
imaginary part $\delta$ describes the electron band broadening and, thus, is
related to the electron lifetime associated with the relaxation from an
excited state to the ground state due to intrinsic scattering events. This
involves electro-magnon and electron-phonon scattering as well as electron
correlation.

Typically $\delta$ serves as the input, as seen in Ref. [67] or is calculated
as a self-energy by dedicated methods, such as the alloy analogy model used in
[29]. However, in our approach, it is determined from the terminated continued
fractions in the Haydock recursion method. It has been shown previously that
this procedure results in a good agreement of the Gilbert damping for several
materials to experimental measurements [19]. The term
$\hat{T}^{\mu}_{i}=\left[\hat{\sigma}^{\mu}_{i},\hat{\mathcal{H}}_{so}\right]$
is the so-called torque operator [17] at site $i$, $\mu,\nu\in\\{x,y,z\\}$
represent the Cartesian coordinates, and $\eta(\epsilon)=-\frac{\partial
f(\epsilon)}{\partial\epsilon}$ is the derivative of the Fermi-Dirac
distribution $f(\epsilon)$ with respect to the energy $\epsilon$. In all our
simulations, the electron temperature is here taken to be zero, so the energy
integral is only performed at the Fermi level. The total magnetic moment at a
site $i$, which is equal to the sum of orbital moment
$m^{\textnormal{orb}}_{i}$ and spin moment $m^{\textnormal{spin}}_{i}$, is
represented by $m_{i}$. The $g$-factor is given by
$g=2\left(1+\frac{m^{\textnormal{orb}}}{m^{\textnormal{spin}}}\right)$ and
$\hat{\sigma}^{\mu}_{i}$ is the $\mu$-component of the Pauli matrices vector
(at site $i$).

Altogether, Eq. 1 implies that the non-local Gilbert damping is generally non-
symmetric with respect to the $ij$ pairs, as also pointed out by Ref. [36]:

$\alpha_{ij}^{\mu\nu}=\left(\frac{m_{j}^{\textnormal{spin}}}{m_{i}^{\textnormal{spin}}}\right)\alpha_{ji}^{\nu\mu},$
(2)

which should hold because of the trace properties applied to Eq. 1. Naturally,
Eq. 1 results in a $3\times 3$ damping tensor $\alpha_{ij}^{\mu\nu}$. From
having the spin quantization axis along the $\mathbf{z}$ ($[001]$) direction,
we then can define a scalar damping value $\alpha_{ij}$ as the average
$\alpha_{ij}=\frac{1}{2}(\alpha_{ij}^{xx}+\alpha_{ij}^{yy})$ (for a discussion
about the validity of this definition in a chemical disorder environment,
please see Section III.1).

From the non-local Gilbert damping, we define the effective damping of atomic
site $i$, $\alpha_{i}$, as

$\begin{split}\alpha_{i}=\sum_{j}\alpha_{ij}.\end{split}$ (3)

In practice, $\alpha_{i}$ is the cumulative non-local damping summation inside
a given cutoff radius $r_{c}$ from the reference ($i$-th) site:
$\alpha_{i}=\sum_{r_{ij}\leq r_{c}}\alpha_{ij}$. Since $\alpha_{ij}$ is
typically long-range and scales asymptotically (i.e., for large $r_{ij}$) as
$1/r_{ij}^{2}$ [17, 19, 36], we consider the sum not only over the 15 atoms in
the embedded impurity cluster, but also into the VCA medium in the EC-VCA
model. For $r_{c}=6a_{0}$, $\alpha_{i}$ is approximately converged, involving
$1836$ neighboring atoms in the damping calculations plus the onsite term.

Experiments often measure damping as a single parameter from ferromagnetic
resonance (FMR) techniques [68, 69], in which the magnetic moments are excited
in a coherent mode ($\mathbf{q}=\mathbf{0}$). As demonstrated in Appendix E,
the measured value (effective damping, $\alpha^{\textnormal{eff}}$) can be
related to the site-resolved effective damping $\alpha_{i}$ (Eq. 3) as

$\alpha^{\textnormal{eff}}=\frac{1}{\sum_{i}m_{i}}\sum_{i}\alpha_{i}m_{i},$
(4)

where the index $i$ runs over all atoms in the unit cell. For the general
case, however, the relation is unfortunately not so straightforward; a
correspondent $\alpha^{\textnormal{eff}}$ should be possible to be extracted
by comparing how the magnetization behaves dynamically in the non-local
damping environment and in a local, single-parameter $\alpha$ candidate. We
note that previous investigations have engaged with the aspect of an averaged
$\alpha$ from element- or site-specific dampings, albeit without using the
moments $m_{i}$ as weights (e.g., Ref. [70]). For an alloy, several
configurations can be constructed for a given concentration of elements. In
that case, the average of all obtained $\alpha^{\textnormal{eff}}$ values (for
that concentration and chosen $\delta$) is the best theoretical estimation for
a measured intrinsic $\alpha$. This quantitative analysis for all
concentrations of the FeCo system is not the primary aim here, although we
investigate the full scenario for an Fe-rich (Fe87Co13) alloy in III.2.
Instead, our emphasis is directed towards the impact of the chemical
environment on the $\alpha_{i}$, aiming for the report of locally hidden
features of that quantity.

###  II.2 Atomistic spin dynamics with an explicit embedded cluster

The generalized LLG equation [36, 17, 19, 71], which incorporates the nonlocal
damping, is solved by using an implicit midpoint solver as implemented in the
UppASD code [72], with a time step of $dt=10^{-17}$ s; for more details on the
solver and the spin Hamiltonian used see Ref. [19]. In each simulation, a
$40\times 40\times 40$ spin lattice with perfect bcc crystal symmetry is
modelled with periodic boundary conditions. To isolate the effects of distinct
magnetic interactions and damping parameters, all calculations start from the
same random (noncollinear) spin state. The spin-spin exchange interactions and
the nonlocal damping parameters are considered up to a cutoff radius of
$r_{c}=6a_{0}$, in compliance with the convergence criteria established in
Section II.1. As in Ref. [19], the fluctuation-dissipation relation in the
presence of nonlocal damping is considered up to the approximation
$\alpha_{ij}^{\mu\nu}\rightarrow\frac{1}{3}\textnormal{Tr}(\hat{\alpha}_{ij})\delta_{ij}\delta_{\mu\nu}$,
which is strictly reasonable in the low-temperature limit ($T\rightarrow 0$),
i.e., for vanishing stochastic fields. To break the rotational invariance of
the Heisenberg exchange interaction term, a small cubic magnetic anisotropy
was also added to the spin Hamiltonian. This term is characterized by an
experimental constant $K_{1}$ of the order of $\sim 3$ $\mu$eV/atom and easy
axis $[100]$, approximately correspondent to the Fe87Co13 composition [73]. As
all calculations include such anisotropy value, it thus does not contribute to
the final differences observed.

When an embedded cluster is incorporated (i.e., beyond just VCA-type
interactions), its 15-atom center coincides with the spin lattice center. Both
$J_{ij}$ and $\alpha_{ij}$ parameters correspondent to each explicit Fe or Co
sites (up to a distance of $6a_{0}$) are included in the Hamiltonian,
considering the symmetry rules $J_{ij}=J_{ji}$ and the one defined in Eq. 2.
Of course, when short-range disorder is explicitly considered, the inversion
symmetry is also broken, and Dzyaloshinskii-Moriya interactions can emerge as
a consequence. As Fe and Co are elements with relatively weak spin-orbit
coupling (which is also a cause of the $\sim 10^{-3}$ intrinsic damping
values), we choose to disregard that term in the Hamiltonian and concentrate
on the effects of the inhomogeneous nonlocal damping field.

###  II.3 Selected clusters and notation

In our study, the embedded nano-clusters within the EC-VCA model are tailored
to match the composition of the surrounding VCA medium. It is important to
realize, however, that in a real sample this boundary is less sharp, and local
configurations of solely Fe or Co (alloy clustering) are possible, albeit
being less likely; this scenario is deliberately avoided here. Due to the vast
number of possible configurations, our study specifically investigates a total
of 64 unique configurations across all the compositions. The study is
particularly concentrated on investigating the influence of neighboring Co
atoms’ positions on the effective damping of the central atom within each
cluster configuration. For simplicity, the $\alpha_{i}$ refers to the
effective damping of the central (reference) atom $i$ in each cluster in the
following discussion.

As we deal with several configurations, an appropriate notation system is
imperative. Figure 2 illustrates the set of investigated Co-centered clusters
in the Fe53Co47 composition, in which the horizontal axis depicts, from left
to right, the embedded clusters with increasing number of Co atoms at the
first neighboring shell (1st NN) of the reference atom $i$. Whenever possible,
the notation will be exemplified with that particular composition, even though
it is transferable to other Fe100-xCox alloys. Therefore, we adopt the
following definitions and conventions:

  * •

$n$ denotes the number of Co atoms positioned at the first neighboring shell
of the reference (central) site. For Fe53Co47, $n=\\{0,1,\ldots,6\\}$.
However, it may exceed these values for Co-enhanced compositions (e.g.,
Fe40Co60);

  * •

$d$ symbolizes the average interatomic distance among the Co atoms in the
first neighboring shell (normalized by lattice constant $a_{0}$), calculated
as

$d=\frac{1}{n(n-1)}\sum_{ij}\frac{|\bm{r}_{ij}|}{a_{0}},$ (5)

where $\bm{r}_{ij}$ is the distance vector between Co atoms at sites $i$ and
$j$;

  * •

$m=\\{1,2,3\\}$ groups the clusters based on the value of their averages
distances $d$. Thus, $m=1$ and $m=3$ correspond to the clusters with the
smallest and largest $d$, respectively, whereas $m=2$ identifies clusters that
fall in between;

  * •

C$n$ labels the cluster with $n$ Co atoms in the first neighboring shell, for
a given alloy composition;

  * •

C$n$-$m$ categorizes the cluster C$n$, identifying it with the proper Co
average distance group $m$ (not used for the cases C0, C1, C7, and C8 by
construction).

It should be noted that the possible number of average distances groups $m$ is
larger than the 3 shown here. We have confined the number of clusters due to
considerations of computational efficiency. Despite this constraint, the
selected cases can adequately demonstrate the underlying physics. The
investigated clusters can have the same 1st NN Co arrangement but a different
2nd NN environment in different compositions, attributed to the varying Co
amounts. The total number of Co atoms and the maximum/minimum $n$ of each
composition are shown in Table 2 in Appendix A.

Figure 2:  Schematic representation of some possible embedded Fe53Co47
clusters in the VCA medium. Within this composition, there are 8 Fe atoms
(yellow spheres) and 7 Co atoms (blue spheres) in total. The horizontal line
organizes, from left to right, the embedded clusters with increasing amount on
nearest-neighbor cobalt $n$ (see text). Within each column, the clusters have
the same $n$ but an increasing average Co interatomic distance $d$ defined in
Eq. 5(in units of $a_{0}$) from the bottom to the top side.

##  III RESULTS

###  III.1 Investigation of off-diagonal elements

Before proceeding to the analysis of local chemical disorder effects on the
magnetic damping, a natural question that arises – linked to the tensorial
character of $\alpha$ – is about the practical contribution coming from the
off-diagonal elements in a heterogeneus environment. In pure bcc or fcc $3d$
ferromagnets, when the spin quantization axis (SQA) aligns with a fourfold or
threefold symmetry axis (e.g., in $\mathbf{z}$ direction), only diagonal
elements are expected to be non-zero in the Gilbert damping tensor, as
discussed in Refs. [1, 7, 17] and corroborated by [19]. This situation becomes
ideal to extract a scalar $\alpha_{ij}$, which can be defined as
$\alpha_{ij}=\frac{1}{2}\left(\alpha_{ij}^{xx}+\alpha_{ij}^{yy}\right)$, and
to directly compare to what is typically measured in experiments [19].
However, when considering an alloy with explicit chemical disorder, the
intrinsic symmetry is broken, rendering non-zero off-diagonal elements
plausible, even with SQA$\parallel\mathbf{z}$.

To address this problem, we considered the effective off-diagonal terms
$\alpha^{\textnormal{eff}}_{xy}$ and $\alpha^{\textnormal{eff}}_{yx}$ (see Eq.
14) from a bcc special quasirandom structure containing 16 atoms (SQS-16)
[74], obtained for a particularly simple FeCo concentration (Fe50Co50). The
SQS-16 cell was created using the MCSQS algorithm within the framework of the
Alloy Theoretic Automated Toolkit (ATAT) [75]; more details on the method
inputs to create the cell can be found in Ref. [19].

Clearly, the off-diagonal terms do not vanish (see Fig. 10 in the Appendix C),
but they represent a very small fraction of the diagonal terms ($\sim 4-5\%$).
Also, the asymmetry is reduced between the diagonal terms ($\sim 2\%$), which
indicates that at large scales the chemical disorder plays a smaller role on
the format of the $\hat{\alpha}^{\textnormal{eff}}$ tensor than the symmetry
of the bcc structure in connection with the SQA itself. In other words, it is
a decent approximation to consider
$\alpha^{\textnormal{eff}}\rightarrow\frac{1}{2}(\alpha_{xx}^{\textnormal{eff}}+\alpha_{yy}^{\textnormal{eff}})\sim\alpha_{xx}^{\textnormal{eff}}$.
Therefore, although locally off-diagonal elements maybe not negligible, we can
consider in our investigation only the average
$\alpha_{ij}=\frac{1}{2}\left(\alpha_{ij}^{xx}+\alpha_{ij}^{yy}\right)$, that
will effectively contribute to $\alpha^{\textnormal{eff}}$ in the bulk scale.

###  III.2 Local effects: the Fe87Co13 case

Contrary to the Heisenberg exchange, which shows an isotropic nature in the
collinear state (i.e., dependent on the atomic pair distance, and without the
SQA as a symmetry reducer), the Gilbert damping is anisotropic with respect to
the SQA – even within the same neighboring shell. Formally, while the former
is explained by a simultaneous conservation of the trace of the block Green’s
functions $\hat{G}_{ij}$ (and, consequently, also $\hat{A}_{ij}$) and
$\ell$-dependent diagonal exchange-splitting matrices (see, e.g., [76, 77]),
the later is explained by the non-diagonal character of the torque operators
$\hat{T}_{i}^{\mu}$ (Eq. 1). In other words, not only the distance and
chemical classification of the $ij$-pair is important for the composition of
$\alpha_{ij}$ values, but also their position with respect to the SQA
(throughout the text we consider $\textnormal{SQA}\parallel\mathbf{z}$ –
unless otherwise noted). Physically, this can be understood by two connected
facts: (i) for small rotations of the spin about the SQA ($\mathbf{z}$), if it
is a high symmetry axis, the change in the magnetization will stay confined in
the other two independent reference orientations ($\mathbf{x}$-$\mathbf{y}$
plane in this case, so $\mu\in\\{x,y\\}$), and (ii) the spin-orbit torques
will recruit different contributions from the orbitals in the inter-site
electronic spectral functions (i.e., through $\hat{A}_{ij}$) to participate in
the dissipation process, obeying their symmetries in real-space. This
anisotropic behavior of $\alpha_{ij}$ has been observed and reported in Refs.
[23, 19, 17], where sometimes more than a single value of non-local damping is
found for the same inter-site distance $r_{ij}$, even for single-element
materials. As an example, we can analyze the second neighboring shell of
(pure) bcc Fe, which presents two distinct $\alpha_{ij}$ values [19]. In this
case, the orbital symmetry of the sites localized at positions $(\pm
a_{0},0,0)$ and $(0,\pm a_{0},0)$ ensure a swap between $\alpha_{ij}^{xx}$ and
$\alpha_{ij}^{yy}$ in the respective $\hat{\alpha}_{ij}$ tensors, resulting in
the same $\alpha_{ij}$ parameters according to the definition in Section II.
However, the sites localized at positions $(0,0,\pm a_{0})$ are characterized
by inter-site propagators $\hat{G}_{ij}$ with completely different (diagonal
and non-diagonal) orbital contributions, resulting also in distinct
$\hat{\alpha}_{ij}$ tensors when compared to the previous 4 sites.
Unfortunately, the non-diagonal character of the $\hat{T}_{i}^{\mu}$ operators
prevents us from discretizing the diagonal $\alpha_{ij}^{\mu\nu=\mu}$ elements
in orbital contributions in a simple and analogous way as for the Heisenberg
exchange interactions ($J_{ij}$’s, see Refs. [78, 79]).

Given the complexity associated with the set of degrees of freedom that
potentially influence the real-space $\alpha_{ij}$’s (chemical classification,
distance, and position around a SQA), the best possible scenario to
investigate local effects is to consider all configurations for the particular
alloy concentration of interest. In this context, as a bonus, not only local
effects can be studied, but also explicit alloying disorder (short-range
disorder) can be indirectly considered as one source of electronic finite
lifetime in these materials [14] – generally treated by effective medium
approaches, such as the simple VCA [19] or the more accurate CPA [14, 30, 80,
31]. In a 15-atom cluster, with $p$ number of Co atoms inside the cluster,
this means a total number of configurations equal to
$C^{15}_{p}=\binom{15}{p}=\frac{15!}{p!(15-p)!}$ for a given concentration
Fe${}_{{100-\frac{20p}{3}}}$Co${}_{\frac{20p}{3}}$.

As this result scales quickly to thousands of possible configurations, we can
choose an Fe-rich alloy to be our proof-of-concept, i.e., we choose $p=2$ (or
Fe87Co13). In this case, the 105 total configurations can be reduced to 16
nonequivalent situations, following the symmetry of the Gilbert damping
parameters in the bcc structure. However, before we analyze the average on
this alloy, we discuss in detail a single case among the 16 nonequivalent
situations.

Figure 3: (Color online) Schematic representation of a 15-atom embeddedcluster
with alloy concentration Fe87Co13. Within the EC-VCA model, this figure
depicts only the explicit part, which in the calculations is surrounded by the
corresponding VCA medium. The color scheme of the spheres is consistent with
Fig. 1. Bonds are drawn to guide the eye. Image produced with the VESTA
software [66].

As an example, consider the cluster displayed in Figure 3. In principle, due
to the cubic environment, each of the $\\{\mathbf{x},\mathbf{y},\mathbf{z}\\}$
axes present a 4-fold ($C_{4}$) symmetry with respect to the $\alpha_{ij}$
values, representing a total of 24 non-equivalent configurations (from the
initial 105). However, in view of the reasons discussed above, when a Co atom
occupies any site within the SQA ($\mathbf{z}$-axis), the
$\alpha_{\textnormal{Fe-Co}}$ value is distinct, reducing the multiplicity of
this example to 16. In that case, the variation of the nearest-neighbor Co
position w.r.t. the central Fe atom do not change the $\alpha_{ij}^{\mu\nu}$
components, but the second nearest-neighbors (NNN), even when occupied by Co
atoms, follow the same symmetry rules as the pure bcc Fe (swap
$\alpha_{ij}^{xx}\leftrightarrow\alpha_{ij}^{yy}$ for NNN in the $xy$-plane,
different components for NNN in $\mathbf{z}$).

We can quantify the difference for this example. Taking into account the
multiplicity, the variation in $\alpha_{i}$ (Eq. 3, $r_{c}=6a_{0}$) of having
Co in the $\mathbf{z}$-axis or in the $xy$-plane amounts to $\sim 1\%$ –
similarly to other quasi-equivalent cases among the 16 configurations. That is
not a purely non-local effect, as the onsite contribution also varies with
about the same intensity. Although small for Fe-Co alloys, given their
relatively weak spin-orbit strength, we anticipate that the variation in
$\alpha_{i}$ is larger for environments with higher spin-orbit coupling,
$\xi$, in the limit where the adiabatic and mean-field approximations (assumed
for torque-correlation formalism used here) are still reasonable [81].

Figure 4: (a) $\sum_{r_{ij}\leq r_{c}}\alpha_{ij}$ up to a given cutoff radius
($r_{c}$) of all nonequivalent configurations of Fe87Co13 in the EC-VCA model.
The different curves shown in colors with different shades of blue represent
different Fe-centered clusters, while the curves shown with different shades
of red show different Co-centered ones. (b) Comparison of the effective
damping obtained from all Fe87Co13 configurations in the EC-VCA model (blue
point) with theoretical values obtained by the use of effective medium
approaches (VCA [19], and CPA [31, 14, 10]) and with the experimental values
of intrinsic damping [69]. In (a), the yellow and green areas represent the
explicit alloy and VCA medium regions of the EC-VCA model, respectively.

It is instructive to analyze the damping that can be calculated from all
possible nonequivalent configurations of Fe87Co13. The results of both
individual cumulative damping summations (Eq. 3), and comparison between the
effective damping of Fe87Co13 and previous literature results are shown in
Figure 4. We first notice from Fig. 4(a) that generally, the values of
$\alpha_{i}$ for Co-centered clusters in the explicit alloy region are higher
(scaling with $\frac{1}{m_{i}}$) than the Fe-centered ones, and also subject
to more oscillations, especially throughout the VCA medium. More importantly,
even within each set of Fe- or Co-centered configurations, we note the
presence of distinct curves, which is a direct consequence of the different
spatial distributions of Fe/Co inside the embedded cluster region – or, in
other words, a direct influence of the local environment. Although the onsite
contribution ($\alpha_{ii}$) drives the majority of the effect, essentially
due to changes in the local density of states around $E_{F}$ because of the
charge transfer between species, the non-local contribution cannot be
disregarded. This is partially in line with the importance of site-resolved
non-local damping suggested in Ref. [36], although the authors indicate it in
the context of reduced dimensionality – and here we argue its relevance in the
presence of short-range chemical disorder. Explicitly, the largest variation
of the $\alpha_{ii}$ values among the Fe- and Co-centered clusters are $\sim
4.7\times 10^{-4}$ and $\sim 3.4\times 10^{-4}$, respectively, while the
(pure) nonlocal contributions vary $\sim 0.9\times 10^{-4}$ ($\sim 1.3\times
10^{-4}$) and $\sim 0.6\times 10^{-4}$ ($\sim 0.9\times 10^{-4}$) for
$r_{c}=a_{0}$ ($r_{c}=6a_{0}$), where $r_{c}=a_{0}$ represents the limit of
the embedded cluster region.

With the definition given by Eq. 4, the effective Gilbert damping can be
obtained. The comparison of $\alpha^{\textnormal{eff}}$ for Fe87Co13 with
other literature results is shown in Fig. 4(b). The error bar, here identified
as the precision of $\alpha^{\textnormal{eff}}$, is calculated by the standard
deviation and evaluated as $\sim 11\%$. This percentage will be used in a
later stage as the approximate error of other Co concentrations averages. The
final result of the EC-VCA model, $\alpha^{\textnormal{eff}}=(1.52\pm
0.17)\times 10^{-3}$, can then be compared to the pure VCA calculation,
$\alpha^{\textnormal{eff}}=1.27\times 10^{-3}$. Despite the consideration of
almost the same chosen broadening parameter, $\delta$, the effective damping
for the EC-VCA model is enhanced in comparison with the pure VCA value. This
is partially related to the fact that the configuration average indirectly
introduces the chemical disorder as a cause for change in $\delta$ (electronic
lifetime), and the randomly arranged atoms as centers for electronic
scattering. Another obvious reason associated with this difference, that
cannot be neglected, is the relative small size of the embedded cluster.
However, as Fig. 4(b) shows, both the VCA and EC-VCA calculatons capture most
experimental results fairly well.

####  III.2.1 Spin dynamics: remagnetization

The calculation of explicit $\alpha_{ij}$ parameters in the EC-VCA model
allows for the inspection of the local environment effects on dynamical
processes, in a fully atomistic picture. Although the impact of a 15-atom
explicit region on the overall energy dissipation rate is expected to be
small, the spin dynamics within this region can significantly differ from that
obtained considering an array consisting solely of VCA sites. In this sense,
the remagnetization stands as a suitable process to investigate; its relevance
becomes evident in the context of, e.g., ultrafast pump-probe experiments and
magnetization switching [82, 83, 84], even though care should be taken in the
direct comparison to those experimental results as variations in the spin
moment length are not accounted for in the generalized LLG equation [36, 17,
19, 71].

With this aim, we assume minimal influence from off-diagonal elements (see
Section III.1) and employ the method outlined in Ref. [19]. As a test case, we
examine the remagnetization dynamics with an embedded cluster as shown in Fig.
3, starting from a random noncollinear state (same for all simulations). This
dynamics with an explicit cluster can then be compared to the Fe87Co13 fully
VCA model to account for the impact of the locally distinct nonlocal Gilbert
dampings. In this context, the $J_{ij}$ set also differs. Thus, a third model
in which all exchange interactions are artificially replaced by the
corresponding VCA values is considered, to isolate the effect of
$\alpha_{ij}$’s. All sets of computed $J_{ij}$ parameters, correspondent to
both the pure VCA calculation and the EC-VCA model, are depicted in Appendix
B.

Figure 5(a) shows the obtained average magnetic moments, considering both
nonlocal and site-resolved effective damping descriptions, for all three
models (pure VCA, EC-VCA, and EC-VCA with the VCA $J_{ij}$ set). The first
point to be noted is the global decrease of the energy dissipation rate by the
nonlocal damping formulation of the equation of motion ($\alpha_{ij}$, full
lines) compared to the effective damping ($\alpha_{i}$, dashed lines).
Moreover, all three models show distinct remagnetization times, as highlighted
by the Insets. For example, among the three models, we obtain maximum
differences of $\sim 600$ fs and $\sim 200$ fs to reach $\sim 77\%$ of the
fully saturated magnetization in the $\alpha_{i}$ and $\alpha_{ij}$
simulations, respectively; as these differences occur between the pure VCA and
the EC-VCA with the same exchanged $J_{ij}$ set, they can be ascribed solely
to the $\alpha_{ij}$’s. Interestingly, here the contrast between the nonlocal
and effective descriptions is also evident: the dissipation rate of the pure
VCA model is the highest in the former and the lowest in the latter.

Altogether, those results clearly demonstrate how the short-range disorder
influence on the local dynamics can modify the overall relaxation rate.
However, here the impact is reduced due to two main factors: (i) the
previously mentioned small replaced region in the VCA matrix (15-atom cluster
in a $40\times 40\times 40$ lattice); and (ii) the choice of a cluster whose
effective damping (central atom $\alpha_{i}=1.32\times 10^{-3}$) does not
considerably deviate from the VCA value ($\alpha_{i}=1.27\times 10^{-3}$).
This suggests promising prospects for enhanced effects in larger embedded
regions or in scenarios where damping varies significantly (up to an order of
magnitude), as explored in the next section. It also corroborates with the
assertion from Ref. [36] that considering inhomogeneous damping can improve
the accuracy of spin dynamics simulations.

To better illustrate the influence of chemical disorder on the local dynamics,
in Fig. 5(b) we show the trajectories of both the central spin, which
coincides with the center of the embedded cluster, and a peripheral spin
located more than $6a_{0}$ from the center, encompassing only VCA-type
interactions. Concerning the former, it is possible to see that the
trajectories largely deviate in all three models from the beginning. In turn,
for the latter, the dynamics only start to differ after a time interval of
$\Delta t\sim 0.13$ ps, by the perturbation of spin waves coming from the
cluster region.

Figure 5: (a) Remagnetization process for Fe87Co13 simulated with atomistic
spin dynamics, considering both the fully nonlocal Gilbert damping
($\alpha_{ij}$, solid lines) and the site-resolved effective damping
($\alpha_{i}$, Eq. 3, dashed lines). Green, orange and blue lines symbolize
the simulations performed with purely VCA, EC-VCA, and EC-VCA parameters
considering the VCA $J_{ij}$ set, respectively. Insets: a zoom-in of the
normalized magnetization curves. (b) Spin trajectories lying on the unit
sphere at $\Delta t\leq 0.13$ ps for (b.1) a peripheral and (b.2) the central
spin, which coincides with the center of the embedded cluster. The red dot in
(b.1) and (b.2) marks the initial position of the spin (at $t=0$), equivalent
for the three models considered.

###  III.3 Short-range disorder impact on damping: a view throughout distinct
alloy concentrations

Having previously demonstrated and quantified the impact of short-range
disorder on the effective Gilbert damping and spin dynamics by fully analyzing
a particular FeCo concentration (Fe87Co13), in this section we extend our
investigation to include all concentrations of Co (up to $60\%$), with a focus
on the damping behaviour with respect to average Co interatomic distances,
$d$.

The interatomic separation within clusters has a considerable influence on
magnetic properties, such as magnetic exchange and magnetic moment, as already
identified in previous studies [85, 86, 87], but its impact on the effective
damping is still poorly understood. To further explore the impact of Co
interatomic distances on $\alpha_{i}$, we keep a constant number of Co atoms
at the 1st NN sites($n$) along with a consistent 2nd NN atomic environment
across each cluster group. We then vary the arrangement of the Co atoms at the
nearest neighborhood around the central (or reference) atom, thereby altering
the average distance $d$ defined in Section II.3. It is important to note that
multiple cluster configurations, not necessarily equivalent in terms of
damping values, can share the same $n$ and $d$ values. For instance, fixing
the 2nd NN environment for C2-1 results in 12 possible clusters per
composition. However, a single cluster is randomly selected from these as a
representative to analyze its damping. Consequently, our analysis is limited
to these selected clusters. It should be further noted that as the cluster
groups C0, C1, C7, and C8 are comprised of a single cluster each (i.e., no
variation in $d$), our investigation predominantly employs the C2 to C6
cluster groups as a platform.

To analyze the effective damping variation among clusters, we introduce two
quantities: $\Delta\alpha^{\textnormal{eff}}$, defined, for each concentration
$x$, by the clusters within the same C$n$ group with the smallest ($m=1$) and
largest ($m=3$) $d$, set as

$\Delta\alpha^{\textnormal{eff}}=\frac{(\alpha_{i}^{m=3}-\alpha_{i}^{m=1})}{\alpha_{i}^{m=1}}\times
100\%\>\>,$ (6)

and $\Delta\alpha^{\textnormal{max}}$, which represents the highest variation
among all clusters investigated within each alloy concentration (not
necessarily with same $n$ or $m$ indexes).

Figure 6:  Site-resolved damping parameter, $\alpha_{i}$ as a function of the normalized average Co interatomic distance $d$ as defined in Eq. 5, in units of $a_{0}$. The filled circles in (a) to (e) show the effective damping of the clusters within C2-C6 groups for different FeCo alloy compositions. As a reference, the red dashed line shows the damping computed considering purely the VCA model. Table 1: Effective Gilbert damping variation $\Delta\alpha^{\text{eff}}$ as a function of $n$ and composition $x$ of Fe100-xCox. $\Delta\alpha^{\text{max}}(\%)$, $\Delta n({E_{f}})^{\text{max}}(\%)$ represent the maximum variation of damping as well as density of states at the Fermi level between clusters within each Co composition, respectively. $n$ |  |  |  $x$ $(\%)$ |  |
---|---|---|---|---|---
| 20 | 26 | 40 | 47 | 60
2 | 8 | 14 | 34 | 58 | 46
3 | – | 15 | 30 | 41 | 50
4 | – | – | 68 | 83 | 161
5 | – | – | 27 | 24 | 24
6 | – | – | – | 25 | 39
$\Delta\alpha^{\text{max}}$ | 8 | 47 | 387 | 1122 | 685
$\Delta n({E_{f}})^{\text{max}}$ | 3 | 22 | 115 | 195 | 150

Figure 6 shows $\alpha_{i}$ as a function of $d$, in different cluster groups
C$n$ and alloy concentrations $x$. From those results, two observations can be
made: first, $\Delta\alpha^{\textnormal{eff}}$ is in all cases positive and
can reach values up to $161\%$ in C4 cluster group of Fe40Co60. Consequently,
damping tends to increase with $d$ (see Table 1).

Second, the damping parameter depends on $n$, indicating a shift towards
higher damping magnitudes as more Co atoms occupy 1st NN sites. Comparing the
lowest and highest damping value among $n$ and $m$ for a fixed composition
$x$, we observe a variation of up to $1122\%$ (or one order of magnitude) for
$x=47$ between the configurations C2-1 and C6-3 (see Table 1, bottom row, and
Fig. 2 for an illustration of the referred clusters). This corroborates with
the significance of the 1st NN Co atoms on the effective Gilbert damping in
the central (or reference) site of the embedded cluster.

To delve deeper into damping’s dependence on $d$, and to understand the
relevant enhancement of damping in terms of the number of 1st NN Co atoms, we
analyze the on-site $\alpha^{\textnormal{onsite}}$ (see Fig. 7, left column)
and the nonlocal $\alpha^{\textnormal{nonlocal}}$ contributions. We obtain
these two parameters from decomposing Eq. 3 into
$\alpha^{\textnormal{onsite}}=\alpha_{ii}$ and
$\alpha^{\textnormal{nonlocal}}=\sum_{i\neq j}\alpha_{ij}$. In agreement with
previous investigations [23, 17, 19], $\alpha^{\textnormal{onsite}}$ is
positive and typically larger than $\alpha^{\textnormal{nonlocal}}$ for all
clusters; for $3d$ materials such as FeCo, each $\alpha_{ij}$ is one order of
magnitude lower than $\alpha_{ii}$, or more. Analogously to the effective
damping, $\alpha^{\textnormal{onsite}}$ tends to increase with $d$ as well as
$n$ and with similar variations. Surprisingly,
$\alpha^{\textnormal{nonlocal}}$ changes sign and starts to be negative at
$x>40\%$ for all $n$ and $m$. Different to the damping from pure VCA, where
$\alpha^{\textnormal{nonlocal}}$ turns negative only at $x=60\%$, the nearest
neighbor $\alpha_{ij}$’s drive the most significant contribution to the
negative $\alpha^{\textnormal{nonlocal}}$ at $x>40\%$ (data not shown).
Furthermore, $\alpha^{\textnormal{onsite}}$ seems to be more sensitive (by a
factor of $\sim 10$) in variations of $n$, $d$, and $x$ compared to
$\alpha^{\textnormal{nonlocal}}$ and, thus, serves as the dominant component
in the effective damping trends. The only exception to this general
observation is the C2 cluster group, where onsite and nonlocal dampings are
equally sensitive in variations of $d$ and for all Co concentrations $x$.

Figure 7: Onsite damping $\alpha_{ii}$ (left panel) and density of states at
the Fermi level $n(E_{f})$ (right panel) as a function of the normalized
average Co interatomic distance $d$ as defined in Eq. 5, in units of $a_{0}$,
for different FeCo alloy compositions. The colors follow the scheme of Fig. 6.

To further identify the electronic structure origin of the results in the
effective, onsite, and nonlocal dampings, we link the first principles
determined dampings to the approximate (or simplified) Kambersky’s formula for
the relaxation of spin-dependent electronic populations [88],

$\begin{split}\alpha=\frac{\gamma}{4M_{s}}n({E_{f}})\xi^{2}(\delta
g)^{2}\tau,\end{split}$ (7)

where $\gamma$ is the gyromagnetic ratio, $M_{s}$ denotes the saturation
magnetization, $n({E_{f}})$ represents the total density of states at the
Fermi level, $\xi$ is the strength of spin-orbit coupling (SOC), $\delta g$ is
the deviation of the $g$-factor from the free-electron value, and $\tau$ is
the electron lifetime related to intrinsic scattering events with other
degrees of freedom.

The variation of the damping with respect to possible changes in $\xi$ is
negligible across all cluster groups, with the largest difference of less than
$1\%$ found for the $3d$ states contribution to the SOC constant. Regarding
$\tau$, its relation to the terminator and recursion levels of the Haydock
method shows minor differences among clusters [23], although the precise
calculation of $\tau$ is beyond the scope of this study. In turn, the damping
of different ferromagnetic materials have been associated with $(\delta
g)^{2}$ before [89], but the
$\frac{m^{\textnormal{orb}}}{m^{\textnormal{spin}}}$ relation, within Kittel’s
first-order perturbation picture [90], only changes subtly in each
concentration $x$ (and specially within each C$n$ group). Explicitly, in all
considered clusters, the central Co exhibits a spin moment of around
$1.7\>\mu_{B}$ and an orbital moment of about $0.1\>\mu_{B}$, being in
agreement with FeCo alloys and nanoclusters from previous works [91, 47, 92].
No clear trend is observed in spin and orbital moments with the average
interatomic distance $d$. Nevertheless, an increase is noted with $n$,
featuring the largest spin moment variation at $\sim 1.8\%$ and the largest
orbital moment variations at $\sim 30-38\%$ for $x\geq 40\%$, to be compared
with $\Delta\alpha^{\textnormal{max}}$ — i.e., giving somewhat a contribution
but still not sufficient to explain the values shown in Table 1.

Therefore, in the scope of Eq. 7, the only quantity that remains to be
analyzed is the total density of states (DOS) at the Fermi level. Significant
correlation between $\alpha_{i}$ and $n({E_{f}})$ has been noted in this study
and already in literature [45]. It is shown in the second column of Fig. 7
that $n({E_{f}})$ exhibits a consistent enhancement trend with $d$ and $n$.
The largest $\Delta n({E_{f}})$ found in each composition is shown in the
bottom row of Table 1, with qualitative agreement to the trends we observed
for the effective damping. In detail, the local environment has a negligible
impact on the spin-up channel’s local DOS (LDOS), while it induces a shift in
the spin-down channel towards lower energies, thus enhancing the corresponding
LDOS at the Fermi level (data not shown).

Lastly, our findings reveal that the $\alpha_{i}$ is comparatively less
sensitive on changes of $n$ for the Fe-centered clusters as well as on the
local environment variation of the second nearest neighboring sites. Although
the damping values of these clusters differ when compared to the pure VCA
values, they are substantially lower than those observed in Co-centered cases
and the cases varying in $d$ and $n$, respectively. For instance, when
replacing the central Co atom with an Fe atom in Fe53Co47, and comparing the
damping of clusters with different $n$, the maximum observed variation in
damping – between the lowest and highest values – is $27\%$ (far less than in
Co-centered cases). Additionally, our tests on several clusters with different
second nearest neighbor environments show that the damping variations are
negligible (less than $\sim 5\%$). Furthermore, neither of these cases exhibit
a clear trend in damping variation. The underlying reason of this behavior is
that the local environment impacts on $n({E_{f}})$ of Fe center clusters are
negligible.

###  III.4 Effective damping: towards the global frame

After analyzing the impact of specific Fe/Co configurations on Gilbert damping
within a localized frame, we now discuss the EC-VCA results from a broader,
more global perspective – towards a random alloy model with short-range
structure resolution. To this end, we proceed to calculate the average damping
for each Co concentration. Obviously, as detailed in Section III.2, as we move
away from Co-poor alloys approaching Fe50Co50, the number of possible
nonequivalent configurations greatly increases, which prevents a brute-force
calculation of the precise average values for all $x$. As an illustration, we
can briefly analyze the cases of $\rm Fe_{53}Co_{47}$ and $\rm
Fe_{40}Co_{60}$. By utilizing a cluster expansion approach similar to that
described in Refs. [93, 94, 95], 197 and 159 non-equivalent clusters are found
for these compositions, respectively, without considering magnetic symmetry
operations (SQA effect). Thus, a limited set has to be selected. We here
choose to investigate one Fe-centered and one Co-centered cluster for each
C$n$-$m$ group, including C0 and C1, for $0\%<x\leq 40\%$; an example of this,
applied to Co-centered clusters was already depicted in Fig. 2. This approach
yields a simplified and unbalanced average, which we argue is sufficiently
representative to replicate the experimental trend.

Figure 8 shows the average effective damping, obtained by using Eq. 4,
considering both the Co-centered and Fe-centered clusters among the different
concentrations. On one hand, quantitatively we observe that the average
damping shown in Fig. 8 is enhanced in comparison with the experimental
results and even with VCA calculations. This, however, is expected from the
unbalanced average model we choose and has two simple explanations: (i) the
averages are taken from incomplete sets of clusters; and (ii) the
high-$\alpha_{i}$ clusters are included. This can be clearly seen by
incorporating the complete set value obtained in Section III.2 for $\rm
Fe_{87}Co_{13}$, denoted by a green star in Figure 8, and comparing it with
the average value obtained by our established selection criteria. As Figure 8
shows, the average represented by the green star is rather close to the
experimentally observed value.

On the other hand, qualitatively we note that the experimental trend (with a
minimum around $x\sim 20\%$) and order of magnitude are well reproduced by the
calculated values shown by black dots, despite the limited set and
high-$\alpha_{i}$ clusters included. The error bars follow the percentage
($\sim 11\%$) found in Section III.2, although we expect increased standard
deviations as $x\rightarrow 50\%$, specially because outliers with
considerable $\Delta\alpha^{\textnormal{eff}}$ become more common in those
alloy concentrations. Therefore, although unusually high $\alpha_{i}$ clusters
can emerge as a result of chemical disorder, those cases are hidden in real
intrinsic damping measurements by statistical averages. This makes the local
scenario described in Section III.3 consistent with FMR experiments again.

Figure 8: The effective intrinsic damping as a function of Co concentration.
The averages (black dots) are performed over all considered Fe-centered and
Co-centered clusters within each C$n$-$m$ group, and calculated in the FMR
regime by Eq. E. The blue and red dots show, respectively, the damping
obtained from pure VCA calculations and experimental results from Ref. [69].
The average value for $\rm Fe_{87}Co_{13}$ obtained in Section III.2 is
depicted by the green star. Error bars are calculated as $\sim 11\%$ of each
value, as a result of the standard deviation obtained in the $\rm
Fe_{87}Co_{13}$ case (see text). Lines are guides for the eyes.

##  IV Conclusions and outlook

In this study, we theoretically explored the role of local environments (or
the short-range chemical disorder), specifically atomic configurations and
compositions, on the intrinsic Gilbert damping. Our investigation is focused
on various bcc FeCo alloys, namely Fe100-xCox, with $x\in[0\%,60\%]$. With
that perspective, a hybrid explicit/effective medium model (EC-VCA) was
introduced. From the analysis of the complete set of configurations in the
Fe87Co13 concentration, it was demonstrated that different spatial
distributions of Fe/Co within the embedded cluster region can markedly affect
the damping. Unlike other quantities such as the Heisenberg exchange
interactions, damping is anisotropic with respect to the spin quantization
axis, which further contributes (albeit slightly in FeCo) to the effects of
chemical disorder. The computed average damping aligns well with both
experiments and the pure VCA calculation. When low-temperature explicit
atomistic spin dynamics simulations are performed, the influence of short-
range disorder on local dynamics is observed to alter the overall relaxation
rate in both nonlocal and effective damping formulations of the generalized
Landau-Lifshitz-Gilbert (LLG) equation, even when such a small (15-atom)
embedded region is considered.

Transitioning to configurations with higher Co content across the FeCo alloy
series, for which inevitably only representative sets can be analyzed, the
effective damping is found to be notably influenced by the atomic
configuration at the nearest neighborhood. In the most extreme cases, a
variation of about one order of magnitude is found. A direct correlation
between effective damping and average the Co interatomic distance ($d$), as
well as the quantity of Co atoms occupying the first neighborhood ($n$) were
found. This local environment-dependent damping behavior was explained in
light of the simplified Kambersky’s formula (Eq. 7), demonstrating a
consistent correlation between damping and the local density of states at the
Fermi level. In a global perspective (i.e., performing a configuration
average), those differences in damping are masked by statistical averages,
reconciling the local picture to what is typically observed in FMR experiments
for FeCo.

Considering damping as a nonlocal (and inhomogeneous) quantity reveals several
interesting characteristics, such as a pronounced dependence on the local
environment, which are overlooked by effective medium theories. Such local
environment-dependent damping may have implications, in real materials, on
phenomena like the damping anisotropy observed in Fe50Co50, which was
originally attributed to short-range configurational effects [24]. Moreover,
our results naturally encourage further exploration of the potential for local
engeneering of damping, and the investigation into how defects affect both
damping and the magnetization dynamics at local and global scales.

##  V Acknowledgments

Financial support from Vetenskapsrådet (grant numbers VR 2016-05980, VR
2019-05304, VR 2019-03666, and VR 2023-04239), and the Knut and Alice
Wallenberg foundation (grant numbers 2018.0060, 2021.0246, 2022.0108, and
2022.0079) is acknowledged. Support from the Swedish Research Council (VR),
the Swedish Energy Agency (Energimyndigheten), the European Research Council
(854843-FASTCORR), eSSENCE and STandUP is acknowledged by O.E. Support from
the Swedish Research Council (VR) is acknowledged by D.T., O.E. and A.D. O.E.
and A.D. acknowledge support from WISE, Wallenberg Initiative Materials
Science, funded by the Knut and Alice Wallenberg Foundation. The China
Scholarship Council (CSC) is acknowledged by Z.L. The computations/data
handling were enabled by resources provided by the National Academic
Infrastructure for Supercomputing in Sweden (NAISS), partially funded by the
Swedish Research Council through grant agreement no. 2022-06725.

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##  Appendix A Number of Co atoms in each composition

In Table 2 we enumerate the total number of Co atoms in the 15-atom cluster
that corresponds to each alloy concentration ($N_{\textnormal{Co}}$), as well
as the maximum/minimum number of Co atoms that can be placed within the
nearest neighboring shell ($n_{\textnormal{max}}$ and $n_{\textnormal{min}}$
in their Co-centered configurations, respectively – see Section II.3 for
definitions).

##  Appendix B ASD simulations $J_{ij}$ parameters

Figure 9 depicts the $J_{ij}$ parameters used to obtain the atomistic spin
dynamics (ASD) results of Section III.2.1. The plots correspond to the
computed $J_{ij}$ values for each of the Fe and Co atoms of the embedded
cluster of Fig. 3 taken as the reference site, as well as for the Fe87Co13 VCA
(Inset, in gray).

Figure 9: Calculated Heisenberg exchange interaction parameters for each Fe
(shades of blue) or Co (shades of red) atom within the embedded cluster
illustrated in Fig. 3, with each atom taken as a reference site. The Inset
highlights, in gray, the $J_{ij}$ parameters computed for the VCA Fe87Co13.
Lines are guides for the eyes. Table 2:  The total number of Co atoms
($N_{\textnormal{Co}}$) and the maximum (minimum) amount of Co in the nearest-
neighborhood $n_{\textnormal{max}}$ ($n_{\textnormal{min}}$) within the
cluster and in each composition. All these clusters refer to Co-centered
configurations.

Composition | $N_{\textnormal{Co}}$ | $n_{\textnormal{max}}$ | $n_{\textnormal{min}}$ |
---|---|---|---|---
Fe | 0 | 0 | 0 |
Fe87Co13 | 2 | 1 | 0 |
Fe80Co20 | 3 | 2 | 0 |
Fe74Co26 | 4 | 3 | 0 |
Fe60Co40 | 6 | 5 | 0 |
Fe53Co47 | 7 | 6 | 0 |
Fe40Co60 | 9 | 8 | 2 |

##  Appendix C Convergence of off-diagonal damping terms

Figure 10 shows the comparison of $\alpha^{\textnormal{eff}}_{xy}$ and
$\alpha^{\textnormal{eff}}_{yx}$ with the diagonal terms, as well as their
asymmetry
$\left(\frac{\alpha^{\textnormal{eff}}_{xx}}{\alpha^{\textnormal{eff}}_{yy}}\right)$.

Figure 10: Relative values of the off-diagonal terms
$\alpha^{\textnormal{eff}}_{xy}$ (blue dots) and
$\alpha^{\textnormal{eff}}_{yx}$ (orange dots) in the damping tensor
$\hat{\alpha}^{\textnormal{eff}}$ for a SQS-16 cell of bcc Fe50Co50. The
values were obtained using the formulation of Eq. 14. Inset: asymmetry between
the diagonal terms $\alpha^{\textnormal{eff}}_{xx}$ and
$\alpha^{\textnormal{eff}}_{yy}$, defined as
$\frac{\alpha^{\textnormal{eff}}_{xx}}{\alpha^{\textnormal{eff}}_{yy}}$. The
lines are guides for the eyes.

##  Appendix D VCA approach in FeCo systems

In the proposed EC-VCA model, the VCA was used to describe the external (non-
explicit) medium. Despite its simple nature, the VCA approach has been
demonstrated to be a reasonably valid method for investigating the properties
of FexCo1-x alloys. This has been discussed in the context of: (i) magnetic
moments (both spin and orbital) as corroborated by the replication of the
well-known Slater-Pauling curve (see Fig. 11) [38, 96]; (ii) transport
properties [97]; (iii) a qualitatively correct behavior of the magnetic
anisotropy energy, as discussed in Refs. [98, 99, 100, 63]; and (iv) a
qualitatively accurate trend of the total intrinsic Gilbert damping, as shown
in Ref. [19], for which VCA is able to capture the remarkable low value at
$\sim 20-30\%$ of Co concentration, physically attributed to a pronounced
minimum in the density of states at the Fermi level [69]. Although a non-
negligible disorder is present in the spin-minority channel of bcc FeCo alloys
[100, 101], in an effective picture (in the context of Eq. 7), the average
density of states of an explicit supercell is well reproduced by VCA [19].
This trend, initially obtained via CPA analysis [29, 14, 31], was later
confirmed by experiments [69].

Figure 11: Total magnetic moments (spin + orbital) as a function of the Co
concentration, obtained for the FeCo alloys, exhibiting the well-known Slater-
Pauling behavior. The calculations in this work are performed using 3
different models: VCA (green circles), SQS-16 (black star), and ordered B2
structure (black triangle). The blue dots represent the data obtained by using
the CPA-LMTO method (extracted from Ref. [102]). Experimental data from Ref.
[103] are depicted by the red and purple circles, representing the
measurements for disordered and ordered FeCo alloys, respectively. Lines are
guides for the eyes.

To give a more comprehensive relation between VCA, CPA and experimental
results, we present a comparison of the Curie temperature ($\rm T_{C}$), spin
wave stiffness ($\rm D_{ex}$) and Gilbert damping of Fe and Fe1-xCox alloys in
Tables 3-5. We found that the $T_{C}$, $\rm D_{ex}$ and dampings obtained from
VCA and CPA show a good agreement, with acceptable discrepancies comparing to
experiments.

Table 3:  The Curie temperature ($\rm T_{C}$) of the pure bcc Fe and Fe1-xCox
alloys. In this work, whenever $x>0$, the electronic structure (and derived
magnetic parameters) are calculated considering the VCA approach. The Curie
temperature is evaluated from mean-field approximation ($\rm T_{C}^{MFA}$),
random phase approximation ($\rm T_{C}^{RPA}$), and Monte Carlo simulations
($\rm T_{C}^{MC}$). For comparison, the theoretical and experimental value of
$\rm T_{C}$ is shown. In other theoretical works, the CPA approach is adopted.

This work | Other works
---|---
|  $\rm T_{C}^{MFA}$(K) |  $\rm T_{C}^{RPA}$(K) |  $\rm T_{C}^{MC}$(K) | Expt. | Theory
bcc Fe | 1274 | 919 | 960 | 1044[104] | 900$-$1238[105, 106, 107, 91, 108]
bcc Fe90Co10 | 1712 | 1086 | 1300 | 1164[109] | 1069[110]
bcc Fe80Co20 | 2064 | 1426 | 1570 | 1225[109] | 1369[110]
bcc Fe70Co30 | 2256 | 1667 | 1740 |  $\sim 1260$[109, 111] | 1490$-$1656[91, 108, 110]
bcc Fe60Co40 | 2304 | 1790 | 1800 | 1268[109] | 1547[110]
bcc Fe50Co50 | 2230 | 1782 | 1780 | 1253$–$1370[109, 112] | 1568$-$1634[110, 91, 108]
bcc Fe40Co60 | 2067 | 1670 | 960 | 1211[111] | $-$

Table 4:  The comparison of exchange stiffness ($\rm D_{ex}$) in this work and
other works for the pure bcc Fe and Fe1-xCox alloys.

| This work | Other works
---|---|---
|  $\rm D_{ex}$ (meV$\cdot$Å2) | Expt. | Theory
bcc Fe | 290 | 266$-$314[113, 114, 115] | 247$-$410[106, 110]
bcc Fe90Co10 | 330 |  $\sim 345$ [116] | 405[110]
bcc Fe80Co20 | 411 |  $\sim 360$ [116] | 477[110]
bcc Fe70Co30 | 493 |  $\sim 410$$-$470[117, 116] | 567[110]
bcc Fe60Co40 | 561 |  $\sim 445$$-$530[117, 116] | 624[110]
bcc Fe50Co50 | 592 | 800[117] | 526$-$677[108, 110]
bcc Fe40Co60 | 578 |  $\sim 480$ [116] | 466[108]

Table 5:  The comparison of damping ($\alpha$) in this work and other works
for the pure bcc Fe and Fe1-xCox alloys.

| This work | Other works
---|---|---
|  $\rm\alpha$ ($\times 10^{-3}$) | Expt. | Theory
bcc Fe | 2.1 | 1.9$-$7.2[69, 89, 14, 118, 119, 120, 121, 122] | 1.3$-$3.2[14, 67, 15]
bcc Fe90Co10 | 1.4 | 1.2[69] | 0.5$-$0.8[31, 14, 10]
bcc Fe80Co20 | 1.0 | 0.8[69] | 0.4$-$0.6[31, 14, 10]
bcc Fe70Co30 | 0.9 | 0.5$-$1.7[69, 89, 123] | 0.5$-$0.8[31, 14]
bcc Fe60Co40 | 1.1 |  $\sim 1.1$111bcc Fe65Co35 [122] | 0.8$-$1.1[31, 14, 10]
bcc Fe50Co50 | 1.6 | 2.0$-$3.2[69, 89, 24] | 1.0$-$1.3[31, 14, 10]
bcc Fe40Co60 | 1.6 | 1.3$-$2.5[69] | 1.6[31, 14]

##  Appendix E  Effective damping

As mentioned in Section II, in the FMR experiments the magnetic moments are
excited in a coherent mode. Thus, a macrospin description with the summed
total effective magnetization $\bm{M}^{\textnormal{eff}}_{i}$ reads

$\begin{split}\bm{M}^{\textnormal{eff}}_{i}=\sum_{i}\bm{M}_{i}.\end{split}$
(8)

The dynamics of this macrospin is well described by the Landau-Lifshitz-
Gilbert equation

$\frac{\partial\bm{M}^{\textnormal{eff}}}{\partial
t}=\bm{M^{\textnormal{eff}}}\times\left(-\gamma\bm{B}+\frac{\alpha^{\textnormal{eff}}}{\left|\bm{M}^{\textnormal{eff}}\right|}\frac{\partial\bm{M}^{\textnormal{eff}}}{\partial
t}\right),$ (9)

where $\gamma$ is the gyromagnetic ratio, $\bm{B}$ is the effective field
acting on the macrospin and
$\left|\bm{M}^{\textnormal{eff}}\right|=\left|\sum_{i}\bm{m}_{i}\right|\leq\sum_{i}\left|\bm{m}_{i}\right|$.
Since we aim to describe a coherent motion of the individual spins, i.e., a
ferromagnetic collinear state at all times $t$, by construction we have
$\left|\bm{M}^{\textnormal{eff}}\right|=\sum_{i}\left|\bm{m}_{i}\right|$ and
$\bm{B}$ isotropic with respect to all atomic sites $i$ ($\bm{B}_{i}=\bm{B}$).
A general form of LLG equation incorporating the non-local damping (first
assuming it to be a scalar parameter) is:

$\frac{\partial\bm{m}_{i}}{\partial
t}=\bm{m}_{i}\times\left(-\gamma\left[\bm{B}_{i}+\bm{b}_{i}(t)\right]+\sum_{j}\frac{\alpha_{ij}}{m_{j}}\frac{\partial\bm{m}_{j}}{\partial
t}\right).$ (10)

Working with Eq. 9 we have:

$\displaystyle\frac{\partial\bm{M}^{\textnormal{eff}}}{\partial t}$
$\displaystyle=\sum_{i}\frac{\partial\bm{m}_{i}}{\partial t}$
$\displaystyle=\sum_{i}\left[\bm{m}_{i}\times(-\gamma\bm{B}_{i}+\sum_{j}\frac{\alpha_{ij}}{m_{j}}\frac{\partial\bm{m}_{j}}{\partial
t})\right]$
$\displaystyle=-\gamma\left[\sum_{i}\bm{m}_{i}\right]\times\bm{B}+\left[\sum_{i}{\bm{m}_{i}}\right]\times\left[\sum_{j}\frac{\alpha_{ij}}{m_{j}}\frac{\partial\bm{m}_{j}}{\partial
t}\right].$ (11)

Therefore, in the FMR regime, one can compare Eqs. 9 and E, resulting in

$\bm{M}^{\textnormal{eff}}\times\frac{\alpha^{\textnormal{eff}}}{\left|\bm{M}^{\textnormal{eff}}\right|}\frac{\partial\bm{M}^{\textnormal{eff}}}{\partial
t}=\sum_{i}\bm{m}_{i}\times\sum_{j}\frac{\alpha_{ij}}{m_{j}}\frac{\partial\bm{m}_{j}}{\partial
t}.$ (12)

Let us write
$\bm{M}^{\textnormal{eff}}=\left|\bm{M}^{\textnormal{eff}}\right|\bm{e}^{\textnormal{eff}}$
and $\bm{m}_{i}=\left|\bm{m}_{i}\right|\bm{e}_{i}$, where
$\bm{e}^{\textnormal{eff}}$ and $\bm{e}_{i}$ are the spin moment orientation
of the macrospin and the atomic spin at site $i$, respectively, at a given
time $t$. Since we describe a coherent precession, then it is straightforward
that $\frac{\partial\bm{e}_{i}}{\partial
t}=\frac{\partial\bm{e}^{\textnormal{eff}}}{\partial t}$ and
$\bm{e}_{i}=\bm{e}^{\textnormal{eff}}$. Hence, by Eq. 12, we obtain

$\displaystyle\alpha^{\textnormal{eff}}$
$\displaystyle=\frac{1}{M^{\textnormal{eff}}}\sum_{ij}\left|\bm{m}_{i}\right|\alpha_{ij}$
$\displaystyle=\frac{1}{\sum_{i}\left|\bm{m}_{i}\right|}\sum_{ij}\left|\bm{m}_{i}\right|\alpha_{ij}=\frac{1}{\sum_{i}\left|\bm{m}_{i}\right|}\sum_{i}\left|\bm{m}_{i}\right|\alpha_{i}.$
(13)

Here, we define the effective damping of spin moment $i$ as
$\alpha_{i}=\sum_{j}{\alpha_{ij}}$ [19]. For the general case where the
Gilbert damping is expressed as a tensor $\bm{\alpha}_{ij}$, it becomes harder
to disentangle a closed form for $\alpha^{\textnormal{eff}}$, and the
effective damping parameter should be extracted directly from simulations
containing the explicit $\bm{\alpha}_{ij}$ expression. However, if we assume
that the macrospin is also subjected to an effective damping in the tensor
form, as suggested by previous works (e.g., Ref. [7, 124]), so that the term
$\frac{\alpha^{\textnormal{eff}}}{\left|\bm{M}^{\textnormal{eff}}\right|}\frac{\partial\bm{M}^{\textnormal{eff}}}{\partial
t}$ is replaced by
$\frac{\bm{\alpha}^{\textnormal{eff}}}{\left|\bm{M}^{\textnormal{eff}}\right|}\cdot\frac{\partial\bm{M}^{\textnormal{eff}}}{\partial
t}$, then for each element of $\bm{\alpha}^{\textnormal{eff}}$ we have

$\alpha^{\textnormal{eff}}_{\mu\nu}=\frac{1}{M^{\textnormal{eff}}}\sum_{ij}\left|\bm{m}_{i}\right|\alpha_{ij}^{\mu\nu},$
(14)

where $\mu,\nu\in\\{x,y,z\\}$. This result can be derived by applying the same
reasoning of Eqs. E-E, however considering that both effective and atomistic
LLG equations assume the general form of a tensorial Gilbert damping
($\bm{\alpha}^{\textnormal{eff}}$ and $\bm{\alpha}_{ij}$).