A unified description of collective magnetic excitations
Benjamin W. Zingsem Michael Winklhofer Ralf Meckenstock Michael Farle





Faculty of Physics and Center for Nanointegration (CENIDE),

	University Duisburg-Essen, 47057 Duisburg, Germany
Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons and Peter Grünberg Institute,

	Forschungszentrum Jülich GmbH, 52425 Jülich, Germany


Faculty of Physics and Center for Nanointegration (CENIDE),

	University Duisburg-Essen, 47057 Duisburg, Germany
IBU/School of Mathematics and Science, University of Oldenburg,

	Carl-von-Ossietzky-Strasse 9-11, D-26129 Oldenburg, Germany
	

Faculty of Physics and Center for Nanointegration (CENIDE),

	University Duisburg-Essen, 47057 Duisburg, Germany
	

Faculty of Physics and Center for Nanointegration (CENIDE),

	University Duisburg-Essen, 47057 Duisburg, Germany
Center for Functionalized Magnetic Materials, Immanuel Kant Baltic Federal University,

	236041 Kaliningrad, Russian Federation



In this work, we define a set of analytic tools to describe the dynamic response of the magnetization to small
perturbations, which can be used on its own or in combination with micromagnetic 
simulations and does not require saturation. We present a general analytic description of the ferromagnetic high
frequency susceptibility tensor to describe angular
as well as frequency dependent ferromagnetic resonance spectra and account for asymmetries
in the line shape. Furthermore, we expand this model to reciprocal
space and show how it describes the magnon dispersion. Finally
we suggest a trajectory dependent solving tool to describe the equilibrium
states of the magnetization. 


A unified description of 
collective magnetic excitations
    Michael Farle
    Received: date / Accepted: date
=========================================================






§ INTRODUCTION

Many solutions of the Ferromagnetic high frequency susceptibility
(Polder-) tensor <cit.> 
have been formulated. Mostly these solutions represent simplified versions to
suit particular problems, such as a certain energy landscape, a certain
kind of coupling or a specific symmetry. In this work we formulate
a generalized linearization of the Landau-Lifshitz-Gilbert <cit.> equations
(LLG), which does not require symmetry assumptions and is applicable regardless
of the coupling as well as the types of damping present in the system. It allows one to start with a general formulation of the free energy density of ferromagnets, including all magnetic inderactions which might be present in the magnetic material, e.g. exchange, dipole-dipole, Dzyaloshinskii-Moriya interaction, anisotropies, etc. and can be expanded to antiferromagnetis and multilayer artificial antiferromagnets in the usual way <cit.>.
The conventional approaches mostly involve solving large systems of
equations, linearizing at different points in the calculation
in order to formulate the high frequency susceptibility <cit.>. This is avoided
here, by applying a straight forward linearization through series
expansion of the LLG. Furthermore this algorithm is formulated to
cover the entire magnon dispersion, including ferromagnetic resonance
modes as well as traveling waves with non-zero wave-vectors.

In the second part we present a model that can be used to calculate
the equilibrium orientations of the magnetization, using an algorithm
that closely resembles the actual measurement procedures used in ferromagnetic
resonance measurements. Following a gradient of the energy landscape imposed on the magnetization, this model can be used
to describe meta-stable and stable equilibrium states of the magnetization even for fields that are not applied along symmetry directions.

Neglecting thermal fluctuations, a combination of those models can
be used to make accurate predictions about the magnetodynamic properties
of ferromagnetic systems.




§ ANALYTIC MODEL




 §.§ The ferromagnetic high frequency susceptibility tensor


In the derivation presented here we assume a system that is described
by one macro-spin <cit.> M⃗ 
which is subjected to one effective magnetic
field B⃗ yielding one high frequency susceptibility tensor
χ_hf. This model therefore
as derived here is designed to describe a fully saturated sample.
It is not limited to a single magnetization though and can be applied
to a set of macro-spins where the local effective magnetic field is known at each site. In that case the total high frequency susceptibility would be given as χ=n∑χ_n where χ_n is
the high frequency susceptibility of the n^th macro-spin
M⃗_n due to the field B⃗_n it is exposed to. This
can be used for non saturated systems and samples with inhomogeneous
magnetization or magnetic nanoparticle configurations.

In order to derive the full tensor we start from the Landau-Lifshitz-Gilbert
Equation <ref> using the Polder-Ansatz <cit.>
as shown in eq. <ref> 

    L⃗-γM⃗×B⃗-α/MM⃗×Ṁ⃗̇-Ṁ⃗̇=0


    [                    M⃗(t)                          M⃗(M,θ_M,φ_M)+m⃗exp(ω t);                    B⃗(t)                          B⃗(B,θ_B,φ_B)+b⃗exp(ω t) ]

Considering the dynamic excitation and response quantities m⃗
and b⃗ to be sufficiently small, the ferromagnetic high-frequency
susceptibility χ_hf can be
expressed as a linear tensor 

    m⃗=χ_hf·b⃗

where linear means, that χ_hf
does not depend on m⃗ and b⃗. This is usually the
case for microwave fields ‖⃗ ⃗b⃗‖⃗<[1]mT.
To obtain the magnetic flux that the magnetization is exposed to,
we consider the magnetic contribution to the free energy per unit volume
F(B⃗_appl,M⃗) where B⃗_appl
corresponds to the applied magnetic field and M⃗ is the magnetization
vector as discussed in the literature (See for example <cit.>).
The Helmholtz free energy density F usually contains an anisotropic
contribution due to the crystal lattice, particularly spin orbit interaction,
as well as several other contributions that arise from surfaces/interfaces,
the shape of the sample and the Zeeman-Energy. In this generalized
approach the nature of these magnetic energies almost does not matter. The only
necessary requirement is that the first and second derivatives used
in eq. <ref> exist. The total magnetic flux
is then given as 

    [             B⃗(t)=∇_M⃗F(B⃗_appl,M⃗); +J_M⃗(∇_M⃗F(B⃗_appl,M⃗))·m⃗exp(ω t);                         +b⃗exp(ω t) ]

where ∇_M⃗F(B⃗_appl,M⃗)
is the anisotropy-field and J_M⃗(∇_M⃗F(B⃗_appl,M⃗))
the response function that accounts for a field caused by a precessing
m⃗, where ∇_M⃗ is the gradient in M⃗
and J_M⃗ the Jacobian matrix in
M⃗. Using this we can now go back to eq. <ref> and
obtain


    [   L⃗→L⃗(b⃗,m⃗)=-γM⃗(t)×B⃗(t); -α/MM⃗(t)×Ṁ⃗̇(t)-Ṁ(t)!=0 ∀ t ]

 which defines the hyper-plane in which all dynamic motion of the
magnetization takes place. Since m⃗ and b⃗ are small,
as defined in <ref> we can now approximate L⃗(b⃗,m⃗)
by using a Taylor-expansion around L⃗(b⃗=0⃗,m⃗=0⃗)
to obtain

    L⃗(b⃗,m⃗)≈0⃗L⃗(0⃗,0⃗)+J_b⃗,m⃗·(b_x,b_y,b_z,m_x,m_y,m_z)^⊤

This leads to the system of equations <ref>, 

    0⃗!=J_b⃗,m⃗·(b_x,b_y,b_z,m_x,m_y,m_z)^⊤

where J_b⃗,m⃗=J_(b_x,b_y,b_z,m_x,m_y,m_z)^T(L(b⃗,m⃗))
is the Jacobian matrix of L⃗ in b⃗ and m⃗.
Eq. <ref> can then be further decomposed into 


    0⃗=    J_b⃗·b⃗+J_m⃗·m⃗
    m⃗=     -((J_m⃗)^-1·J_b⃗)·b⃗

where J_m⃗ and J_b⃗
are the Jacobian matrices in m⃗ and b⃗ respectively.
By comparison to eq. <ref> we find 

    χ_hf=-((J_m⃗)^-1·J_b⃗)

which we refer to as the complete analytic solution of the ferromagnetic
high-frequency susceptibility. Note that this approach is independent
of the form of the free energy functional. Since we obtain the full
tensor without assumptions regarding its entries, we have to project
it on the unit vectors u⃗_b and u⃗_m that represent
an excitation-measurement-pair of observables to obtain a representative spectrum.
In a typical numerical evaluation one would assume u⃗_b to
be parallel to the unit-vector in ϕ direction of the applied
field B⃗ and u⃗_m to be parallel to the unit-vector
in ϕ direction of the magnetization vector in spherical coordinates.
Nonparallel unit-vectors u⃗_b and u⃗_m can be
used to account for nonuniform microwave fields. The angle between u⃗_b
and u⃗_m represents an effective phase shift Δ between
the excitation and the response. This is illustrated in the inset
in fig. <ref>. Such a phase shift can be created
for instance by having the sample covered by a conductive layer
in which the microwave creates an eddy current that in turn creates
a phase shifted microwave signal that superimposes with the original
one as described in <cit.>. The approach presented here
was used in <cit.> to calculate asymmetric line
shapes. 





 §.§ Extension to reciprocal-(k⃗-)space and description of the magnon dispersion


The model presented above can be extended to reciprocal space in order
to obtain the magnon dispersion. Accordingly, the spatial contributions
to the energy landscape are included in the energy density
formulation. Also the Ansatz has to be changed such that the dynamic
magnetization has a spatial dependence. We imagine that the spatial
distribution of the magnetization can be described as a constant part
and a dynamic part where the dynamic part is a Fourier series. In
contrast to the description by Suhl, where this Ansatz appears <cit.>
we consider the amplitude for every k⃗ to be small, such that
we can perturb the system with a single k⃗ at a time, yielding
an Ansatz of the form

    M⃗(t,x)M⃗(M,θ_M,φ_M)+m⃗_⃗k⃗exp(ω t-k⃗·x⃗)

where k⃗ is the reciprocal vector for which the susceptibility
is being calculated and x⃗ is the spatial coordinate at which
the wave is observed.

For example we can consider exchange energy contribution in a continuum
model 

    F_ex=d^2B_ex/‖M⃗‖(M⃗(t,x)·M⃗(t,x))

and a k⃗ dependent dipolar coupling
to include dynamic aspects of dipolar interactions

    F_Demag=1/2μ_0‖M⃗‖m⃗_k·k⃗/‖m⃗‖ ^2‖k⃗‖ ^2k⃗·M⃗(t,x)

where d is the distance between two neighboring spins, B_ex
is the exchange field they exert on each other and  is
the Laplace operator in real space.

 Adding this contribution to the Helmholtz energy density we can proceed
as before and calculate the susceptibility for every k⃗ in
the Brillouin zone as shown exemplary in fig. <ref>. The results agree with the literature (see for instance <cit.>).
For other spatial contributions such as anisotropic exchange and chiral
coupling , the model can be applied in the same way.



 §.§ The equilibrium position of the magnetization


In order to use the result in eq. <ref> to obtain
the susceptibility it is necessary for M⃗(θ_M,ϕ_M)
to locally minimize the free energy density. The orientation of the
magnetization vector has to be determined form the shape of the free
energy landscape including an applied magnetic field. In the following
we present our recursive method to efficiently find these minima.
In terms of infinitesimals this method can be viewed 
as a trajectory depended analytic solution. Due to its infinite recursion
along a chosen trajectory however it resembles a second order newton
algorithm, which is a numerical tool, and we therefore tend to call
it a semi-analytic trajectory dependent solution of the equilibrium
states of the magnetization.

For certain paths in the applied field space, where the trajectory passes sufficiently far beyond a hard direction, the equilibrium angles
are discontinuous if the Zeeman energy does not overcome the anisotropic
contributions of this hard direction. This can lead to a hysteretic
behavior of the magnetization depending on the trajectory of B⃗(τ):=B⃗(B(τ),θ_B(τ),φ_B(τ)).
To account for this behavior a solution representing the equilibrium
angles must depend on the trajectory B⃗(τ) and
not only on a momentary configuration of B⃗. Without loss
of generality we will only consider the equilibrium angles {θ_M,φ_M}
of the magnetization in spherical coordinates to minimize the free
energy, since in many applications the norm of the magnetization may
be considered constant. Once a minimizer Ω⃗(B⃗(0))={θ_M,φ_M} _0
of the free energy F(B⃗,M⃗) is known for a
certain starting configuration B⃗(0), a small change
in B⃗→B⃗(0+δ) that yields a
small change in the position of the minimum of F(B⃗,M⃗)
can be accounted for by calculating a series expansion of F(B⃗(0+δ),M⃗)
at the position {θ_M,φ_M} _0 to
the second order. 

The position of the minimum of this parabola will be close to the
minimum {θ_M,φ_M} _0+δ of F(B⃗(0+δ),M⃗).
In fact as δ decreases the solution obtained this way will
get closer to the exact minimum. This procedure is illustrated in
fig. <ref>, where the free energy was defined to be F=sin^2(2ϕ_M)-5cos(ϕ_B-ϕ_M).
Since the function obtained from the series expansion is of quadratic
order it can always be written in a form such that the vertex can
be directly extracted from the function. Therefore a recursive function
of the form <ref>

    Ω⃗(B⃗(0-δ))=Ω⃗(B⃗(0))-
    .H_F^-1|_Ω⃗(B⃗(0-δ))·.∇⃗F|_Ω⃗(B⃗(0-δ))
    Ω⃗(B⃗(0-2δ))=Ω⃗(B⃗(0-δ))-
    .H_F^-1|_Ω⃗(B⃗(0-2δ))·.∇⃗F|_Ω⃗(B⃗(0-2δ))
    
    ...

 can be derived to describe the position of a minimum for certain
trajectories B⃗(τ), where H_F
is the Hessian Matrix of the free energy density that described the
curvature and ∇⃗F the gradient that describes the slope
of the free energy. Conceptually this can be considered a second order
Newton algorithm with the exception that it starts from a known position
making the number of iterations required tend towards 1 as δ
gets small. To determine a minimizer that can be used as a starting
point in eq. <ref> the easiest approach in a numerical
calculation is to start at a field value sufficiently higher than
the field at which the Zeeman energy fully overcomes the anisotropy
energy – in the sense that there is only one minimum and
one maximum left in the energy landscape – and to assume
that the magnetization is parallel to the applied field in this configuration.
This approach was implemented and found to be very accurate in <cit.>
for fitting FMR spectra recorded at different microwave frequencies.
Figure <ref> shows some calculated spectra
using the solution presented above, with the corresponding equilibrium
angles calculated with this trajectory dependent algorithm. The overall
calculation time was about five minutes for 540180 data points.


Equation <ref> in combination with eq. <ref>
describe a very fast algorithm to calculate the complete susceptibility
for any given free energy density and any measurement trajectory.
This algorithm however will not always align the magnetization in
the absolute minimum of the free energy, in fact it will fall into
meta-stable states if for instance a fourfold crystalline anisotropy
is considered and the applied field is swept along the field angle
rather than the field amplitude, predicting the occurrence of ferromagnetic
resonance in meta-stable states. 



§ SUMMARY


We have devised a versatile analytic model, capable of accurately
describing FMR experiments as well as modeling the full magnon dispersion.
The model is simple in that it requires only derivatives. Condensed
into a single operator χ_hf,
it is compact and thus easy to use in analytic and numeric applications.
The formulation through an energy density allows for easy modification
of the model to adapt different types of interactions, such as dipole-dipole-interaction,
spin-spin-interactions like the Dzyaloshinskiǐ-Moriya interaction
and spin-orbit interactions. It can also be applied directly to spatial
dependent spin configurations obtained from micromagnetic simulations
to retrieve information about the magnetodynamic properties of spin
textures. The model is not restricted to evaluating the magnon dispersion
as a function ω(k) but instead yields the magnonic
response amplitude χ(ω,k) as a Green's function.
In addition to this, the algorithm described in sec. <ref>
makes it possible to apply the model on orientations of the magnetization
which are non collinear with the symmetry directions of the system
or the applied magnetic field. This can be used to calculate angular
dependent spectra, as well as identify meta-stable states and describe
their magnetodynamic behavior.



