A unified description of collective magnetic excitations

1.8 MB·

by  Benjamin W. Zingsem

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A unified description of magnetic field, angular,
(cid:126)
frequency and k-dependent collective magnetic
excitations
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Benjamin W. Zingsem, Michael Winklhofer, Ralf Meckenstock, Michael FarleFaculty of Physics and Center for
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2 Nanointegration (CENIDE), University Duisburg-Essen, 47057 Duisburg, Germany

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J

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Abstract—We Present a general analytic description of straight forward linearization through series expansion of the

]
r the ferromagnetic high frequency susceptibility tensor that can LLG. Furthermore this algorithm is formulated to cover the
e
h be used to describe angular as well as frequency dependent
entire magnon dispersion, including ferromagnetic resonance
t
o FMR spectra and account for asymmetries in the line shape.
. modesaswellastravelingwaveswithnon-zerowave-vectors.
t Furthermoreweexpandthismodeltoreciprocalspaceandshow
a
m how it describes the magnon dispersion. Finally we suggest a In the second part we present a model that can be used
-
d trajectory dependent solving tool to describe the equilibrium to calculate the equilibrium orientations of the magnetiza-
n
statesofthemagnetization.Thuswedefineasetofanalytictools tion, following an algorithm the closely resembles the ac-
o
c to describe the dynamic response of the magnetization to small
[ tualmeasurementproceduresusedinferromagneticresonance
perturbations, which can be used on its own or in combination

1 measurements. Following a gradient, this model can be used
v with micromagnetic simulations.
8 to describe meta-stable and stable equilibrium states of the
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0 magnetization.
9 I. INTRODUCTION
0 Neglecting thermal fluctuations, a combination of those
.
1
ManysolutionsoftheFerromagnetichighfrequencysuscep- models can be used to make accurate predictions about the
0
7
tibility tensor have been formulated. Mostly these solutions magnetodynamic properties of ferromagnetic systems.
1
:
v a formulated to suit particular problems, such a a certain
i
X II. ANALYTICMODEL
energy landscape, a certain kind of coupling or a specific
r
a A. The ferromagnetic high frequency susceptibility tensor
symmetry. In this work we formulate a generalized lineariza-
tion of the Landau-Lifshitz-Gilbert equations (LLG), which In the derivation presented here we assume a system that
does not require symmetry assumptions and is independent is described by one macro-spin M(cid:126) that is subjected to one
of the coupling as well as the types of damping present effective magnetic field B(cid:126) yielding one high frequency sus-
in the system. The conventional approaches mostly involve ceptibility tensor χ . This model therefore as derived here is
hf
solving large systems of equations, arbitrarily linearizing at designed to describe a fully saturated sample. It is not limited
differentpointsinthecalculationinordertoformulatethehigh to a single magnetization though and can be applied to a set
frequency susceptibility. This is avoided here, by applying a of macro-spins where the local field is known for each one
2
of them. In that case the total high frequency susceptibility flux is then given as
(cid:80)
would be given as χ= χ where χ is the high frequency (cid:16) (cid:17)
n n B(cid:126)(t)=∇ F B(cid:126) ,M(cid:126)
n M(cid:126) appl
susceptibility of the nth macro-spin M(cid:126) due to the field B(cid:126) (cid:16) (cid:16) (cid:17)(cid:17)
n n +J ∇ F B(cid:126) ,M(cid:126) ·m(cid:126) exp(ıωt) (4)
M(cid:126) M(cid:126) appl
it is exposed to. This can be used for non saturated systems
+(cid:126)bexp(ıωt)
and samples with inhomogeneous magnetization.
(cid:16) (cid:17)
where ∇ F B(cid:126) ,M(cid:126) is the anisotropy-field and
M(cid:126) appl
(cid:16) (cid:16) (cid:17)(cid:17)
J ∇ F B(cid:126) ,M(cid:126) the response function that accounts
M(cid:126) M(cid:126) appl
for a field caused by a precessing m(cid:126), where ∇ is the
M(cid:126)
gradient in M(cid:126) and J the Jacobian matrix in M(cid:126). Using this
M(cid:126)
In order to derive the full tensor we start from Landau-
we can now go back to eq. 1 and obtain
Lifshitz-Gilbert Equation 1 using the Polder-Ansatz [1] as
(cid:16) (cid:17)
shown in eq. 2 L(cid:126) →L(cid:126) (cid:126)b,m(cid:126) =−γM(cid:126) (t)×B(cid:126)(t)
(5)
−αM(cid:126) (t)×M(cid:126)˙ (t)−M˙ (t)=! 0∀t
L(cid:126) :=−γM(cid:126) ×B(cid:126) − α M(cid:126) ×M(cid:126)˙ −M(cid:126)˙ =0 (1) M
M
which defines the hyper-plane in which all dynamic motion
M(cid:126) (t) := M(cid:126) (M,θ ,ϕ )+m(cid:126) exp(ıωt) of the magnetization takes place. Since m(cid:126) and(cid:126)b are small, as
M M
(2)
(cid:16) (cid:17)
B(cid:126) (t) := B(cid:126) (B,θ ,ϕ )+(cid:126)bexp(ıωt) defined in 3 we can now approximate L(cid:126) (cid:126)b,m(cid:126) by using a
B B
(cid:16) (cid:17)
Taylor-expansion around L(cid:126) (cid:126)b=(cid:126)0,m(cid:126) =(cid:126)0 to obtain
Considering the dynamic excitation and response quantities
m(cid:126) and (cid:126)b to be sufficiently small, the ferromagnetic high- (cid:16) (cid:17) (cid:16) (cid:17)
L(cid:126) (cid:126)b,m(cid:126) ≈L(cid:126) (cid:126)0,(cid:126)0 +J ·(b ,b ,b ,m ,m ,m )(cid:62) (6)
(cid:126)b,m(cid:126) x y z x y z
frequencysusceptibilityχ canbeexpressedasalineartensor (cid:124) (cid:123)(cid:122) (cid:125)
hf (cid:126)0
m(cid:126) =χ ·(cid:126)b (3) This leads to the system of equations 7,
hf
where linear means, that χ does not depend on m(cid:126) and (cid:126)b. (cid:126)0=! J(cid:126)b,m(cid:126) ·(bx,by,bz,mx,my,mz)(cid:62) (7)
hf
(cid:126)
This is usually the case for microwave fields (cid:107)b(cid:107)<1mT. To (cid:16) (cid:16) (cid:17)(cid:17)
where J = J L (cid:126)b,m(cid:126) is the Ja-
(cid:126)b,m(cid:126) (bx,by,bz,mx,my,mz)T
obtain the magnetic flux that the magnetization is exposed to,
cobian matrix of L(cid:126) in (cid:126)b and m(cid:126). Eq. 7 can then be further
we look at the magnetic contribution to the free energy per
decomposed into
(cid:16) (cid:17)
unit volume F B(cid:126) ,M(cid:126) where B(cid:126) corresponds to the
appl appl
applied magnetic field and M(cid:126) is the magnetization vector as (cid:126)0=J ·(cid:126)b+J ·m(cid:126)
(cid:126)b m(cid:126)
discussed in various literature [2]. The Helmholtz free energy (cid:18)(cid:16) (cid:17)−1 (cid:19) (8)
m(cid:126) =− J ·J ·(cid:126)b
m(cid:126) (cid:126)b
density F usually contains an anisotropic contribution due to
thecrystallattice,particularlyspinorbitinteraction,aswellas where Jm(cid:126) and J(cid:126)b are the Jacobian matrices in m(cid:126) and (cid:126)b
several other contributions that arise from surfaces/interfaces, respectively. By comparison to eq. 3 we find
the shape of the sample and the Zeeman-Energy. In this (cid:18)(cid:16) (cid:17)−1 (cid:19)
χ =− J ·J (9)
generalized approach the nature of these contributions does hf m(cid:126) (cid:126)b
not matter. The only necessary requirement is that the first which we refer to as the complete analytic solution of the
and second derivatives used in eq. 4 exist. The total magnetic ferromagnetic high-frequency susceptibility. Note that this ap-
3
proachisindependentoftheformofthefreeenergyfunctional. Amplitude [Arb. U.]
Since we obtain the full tensor without assumptions regarding -1. -0.5 0. 0.5 1.
its entries we have to project it in the unit vectors (cid:126)u and (cid:126)u
b m
that represent an excitation-measurement-projection to obtain
2π
a representative spectrum. In a typical numerical evaluation
3π
we would assume (cid:126)ub to be parallel to the unit-vector in φ Δ
2
t
direction of the applied field B(cid:126) and (cid:126)um to be parallel to hif
s π
e
the unit-vector in φdirection of the magnetization vector in
s
a
h π
sphericalcoordinates. Nonparallelunit-vectors(cid:126)u and(cid:126)u can
b m P
2
beusedtoaccountfornonuniformmicrowavefields,wherethe
0
field is not symmetric across the sample. The angle between 20 25 30 35 40 45 50
(cid:126)u and (cid:126)u represents an effective phase shift ∆ between the
b m Applied Field [mT]
excitation and the response. This is illustrated in the inset in 1.0
.] Δ=0
fig.1.Suchaphaseshiftcanbecreatedforinstancebyhaving U 0.8
b
the sample encompassed in a conductive layer in which the Ar 0.6 Δ=π
[ 2
microwavecreatesaneddycurrentthatinturncreatesaphase de 0.4
u
t 0.2
shifted microwave signal that superimposes with the original pli
m 0.0
oneasdescribedin[3].Theapproachpresentedherewasused
A
-0.2
in [4] to calculate asymmetric line shapes.
20 25 30 35 40 45 50
Applied Field [mT]
Figure1. Top:Amplitudeofthesusceptibilityasafunctionoftheapplied
magnetic field and the phase-shift ∆. At the angles π/2 and 3π/2 the line-
B. Extension to the k-space and description of the magnon
shapeisfullyanti-symmetricsimilartothederivativeofaLorenzline-shape.
Inthevicinityofthoseanglesthesignalisasymmetric.Attheangles0and
dispersion 2π the signal is symmetric with positive amplitude and at π the signal is
symmetric with negative amplitude. The inset illustrates the phase shift ∆
inducedbychoosinganonparalleltupleofexcitationmeasurementprojection
vectors (cid:126)ub and (cid:126)um shown in the precession cone of the time dependent
The model presented above can be extended to reciprocal magnetization. For simplicity the the precession is indicated as a circular
motion normalized to the length of the unit vectors perpendicular to M(cid:126). In
space in order to obtain the magnon dispersion. To achieve generalitwouldbeapproximatedtobeellipticalandtheopeningofthecone
ismuchsmallercomparedtotheMagnetizationvector.Bottom:Selectedlines
that, the spatial contributions to the energy landscape have to atspecificangles0and π2.
beincludedintheenergydensityformulation.AlsotheAnsatz
an Ansatz of the form
has to be changed such that the dynamic magnetization has
a spatial dependence. We imagine that the spatial distribution (cid:16) (cid:17)
M(cid:126) (t,x):=M(cid:126) (M,θ ,ϕ )+m(cid:126) exp ıωt−(cid:126)k·(cid:126)x (10)
M M k
of the magnetization can be described as a constant part and
a dynamic part where the dynamic part is a Fourier series. In where(cid:126)k isthereciprocalvectorforwhichthesusceptibilityis
contrast to the description by Suhl, where this Ansatz appears being calculated and (cid:126)x is the spatial coordinate at which the
[5]weconsidertheamplitudeforevery(cid:126)ktobesmall,suchthat wave is observed.
we can perturb the system with a single(cid:126)k at a time, yielding For example we can consider exchange energy contribution
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log(Amplitude)[Arb.U.] log(Amplitude)[Arb.U.]
it resembles a second order newton algorithm which is a
0. 0.5 1.0. 0.5 1.
numerical tool and we therefore tend to call it a semi-analytic
31.5 31.5
trajectory dependent solution of the equilibrium states of the
z]) 29. z]) 29. magnetization.
H H
[ [
nf( 26.5 nf( 26.5
l l
24. 24.
Γ H P N Γ P N H Γ H P N Γ P N H
Figure2. Themagnondispersioncalculatedforabccstructureincludinga
cubicanisotropyexchangeanddipolarcoupling(left)andforasimilarsystem
withstrongchiralcoupling(right).
For certain paths in the applied field space the equilib-
in a continuum model
rium angles are discontinuous if the Zeeman energy does
B (cid:16) (cid:17)
Fex =d2(cid:13) ex(cid:13) M(cid:126) (t,x)·(cid:52)M(cid:126) (t,x) (11) not overcome the anisotropic contributions to the energy
(cid:13)M(cid:126)(cid:13)
(cid:13) (cid:13)
landscape. This can lead to a hysteretic behavior of the
anda(cid:126)kdependentdipolarcouplingtoincludedynamicaspects magnetization depending on the trajectory of B(cid:126) (τ) :=
of dipolar interactions B(cid:126) (B(τ),θ (τ),ϕ (τ)). To account for this behavior a so-
B B
1 (cid:13) (cid:13) m(cid:126) ·(cid:126)k lution representing the equilibrium angles must depend on the
F = µ (cid:13)M(cid:126)(cid:13) k (cid:126)k·M(cid:126) (t,x) (12)
Demag 2 0(cid:13) (cid:13)(cid:107)m(cid:126)(cid:107)2(cid:13)(cid:13)(cid:126)k(cid:13)(cid:13)2 trajectory B(cid:126) (τ) and not only on a momentary configuration
(cid:13) (cid:13)
of B(cid:126). Without loss of generality we will only consider the
where d is the distance between two neighboring spins, B
ex
equilibrium angles {θ ,ϕ } of the magnetization in spher-
is the exchange field they exert on each other and (cid:52) is the M M
ical coordinates to minimize the free energy, since in many
Laplace operator in real space. Adding this contribution to
applicationsthenormofthemagnetizationmaybeconsidered
the Helmholtz energy density we can proceed as before and
(cid:16) (cid:17)
calculate the susceptibility for every (cid:126)k in the Brillouin zone constant. Given a minimizer Ω(cid:126) B(cid:126) (0) = {θM,ϕM}0 of
(cid:16) (cid:17)
the free energy F B(cid:126),M(cid:126) is known for a certain starting
as shown exemplary in fig. 2. For other spatial contributions
configuration B(cid:126) (0), a small change in B(cid:126) → B(cid:126) (0+δ)
such as anisotropic exchange and chiral coupling , the model
that yields a small change in the position of the minimum
can be applied in the same way.
(cid:16) (cid:17)
of F B(cid:126),M(cid:126) can be accounted for by calculating a series
(cid:16) (cid:17)
C. The equilibrium position of the magnetization expansion of F B(cid:126) (0+δ),M(cid:126) at the position {θ ,ϕ }
M M 0
In order to use the result in eq.9 to obtain the susceptibility to the second order. The position of the minimum of this
it is necessary for M(cid:126) (θ ,φ ) to locally minimize the free parabola will be close to the minimum {θ ,ϕ } of
M M M M 0+δ
(cid:16) (cid:17)
energydensity.Theorientationofthemagnetizationvectorhas F B(cid:126) (0+δ),M(cid:126) . In fact as δ decreases the solution ob-
to be determined form the shape of the free energy landscape tained this way will get closer to the exact minimum. This
including an applied magnetic field. In the following we procedure is illustrated in fig. 3, where the free energy was
present our recursive method to efficiently find these minima. described as a F =sin2(2φ )−5cos(φ −φ ). Since the
M B M
We are not opposed to viewing this method in terms of function obtained from the series expansion is of quadratic
infinitesimals as a trajectory depended analytic solution. Due orderitcanalwaysbewritteninaformsuchthatthevertexcan
to its infinite recursion along a chosen trajectory however be directly extracted from the function. Therefore a recursive
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U] Amplitude[Arb.U.]
. -1. -0.5 0. 0.5 1.
b
r
A
[ π π
sity gleϕB 34π gleϕB 34π
n An π An π
e e 2 e 2
D an π an π
y Inpl 4 Inpl 4
g 0 0
r 0 50 100 150 200 250 300 0 50 100 150 200 250 300
e
n AppliedField[mT] AppliedField[mT]
E π
FreeϕM0-2δ ϕMMa0g-nδetizaϕtioMn0 AngϕleM0ϕ+Mδ ϕM0+2δ InplaneAngleϕB 34πππ420 0π agnetizationAngleϕM
0 50 100 150 200 250 300 -π M
AppliedField[mT]
Figure 3. The environment of a minimum of the energy Landscape as a
functionoftheMagnetizationangle(blue)andthesameenergylandscapeafter Figure4. CalculatedspectraattypicalX-Bandfrequencies:10GHz(topleft),
changing the applied field angle φB by a small quantity δ (black) together and18GHz(topright)andthecorrespondingsolutionsforthemagnetization
with a Taylor expansion of the changed energy landscape around the the angles (bottom). The model that has been used for the free energy here
positionφM0 (reddashed) ipserapecnudbiciculaarnitsoottrhoepyaziKm4uth=alp4l.a8ne·(1θ04=J/πm)3togwethheerrewthiteh1an10i-nd-iprelacntieonuniis-
2
axialanisotropyKu=−7.5·104J/m3anda2-foldoutofplaneanisotropy
K2=0.3·104J/m3accordingto[2]withthedemagnetizingtensorofathin
function of the form 13 film. The g-factor was set to 2.09 and the damping constant of α=0.004
wasused.
(cid:16) (cid:17) (cid:16) (cid:17)
Ω(cid:126) B(cid:126) (0−δ) =Ω(cid:126) B(cid:126) (0) −
(cid:12) (cid:12)
H−1(cid:12) · ∇(cid:126)F(cid:12)
F (cid:12)Ω(cid:126)(B(cid:126)(0−δ)) (cid:12)Ω(cid:126)(B(cid:126)(0−δ))
and to assume that the magnetization is parallel to the applied
(cid:16) (cid:17) (cid:16) (cid:17)
Ω(cid:126) B(cid:126) (0−2δ) =Ω(cid:126) B(cid:126) (0−δ) − (13)
field in this configuration. This approach was implemented
(cid:12) (cid:12)
H−1(cid:12) · ∇(cid:126)F(cid:12)
F (cid:12)Ω(cid:126)(B(cid:126)(0−2δ)) (cid:12)Ω(cid:126)(B(cid:126)(0−2δ)) and found to be very accurate in [4] for fitting FMR spectra
... recorded at different microwave frequencies. Figure 4 shows
some calculated spectra using the solution presented above,
can be derived to describe the position of a minimum for
withthecorrespondingequilibriumanglescalculatedwiththis
certain trajectories B(cid:126) (τ), where H is the Hessian Matrix
F trajectory dependent algorithm. The overall calculation time
of the free energy density that described the curvature and
was about five minutes for 540180 data points.
∇(cid:126)F the gradient that describes the slope of the free energy.
Conceptually this can be considered a second order Newton Equation 9 in combination with eq. 13 describe a very
algorithm with the exception that it starts from a known fast algorithm to calculate the complete susceptibility for any
positionmakingthenumberofiterationsrequiredtendtowards givenfreeenergydensityandanymeasurementtrajectory.This
1 as δ gets small. To determine a minimizer that can be used algorithm however will not always align the magnetization in
asastartingpointineq.13theeasiestapproachinanumerical the absolute minimum of the free energy, in fact it will fall
calculation is to start at a field value sufficiently higher into meta-stable states if for instance a fourfold crystalline
than the field at which the Zeeman energy fully overcomes anisotropy is considered and the applied field is swept along
the anisotropy energy – in the sense that there is only one the field angle rather than the field amplitude, predicting the
minimum and one maximum left in the energy landscape – occurrence of ferromagnetic resonance in meta-stable states.
6
III. SUMMARY [4] R. Salikhov, L. Reichel, B. Zingsem, F. M. Römer, R.-M. Abrudan,
J.Rusz,O.Eriksson,L.Schultz,S.Fähler,M.Farleetal.,“Enhancedand
We have devised a versatile analytic model, capable of tunablespin-orbitcouplingintetragonallystrainedfe-co-bfilms,”arXiv
accurately describing FMR experiments as well as modeling preprintarXiv:1510.02624,2015.
[5] H.Suhl,“Thetheoryofferromagneticresonanceathighsignalpowers,”
the full magnon dispersion. The model is simple in that it
JournalofPhysicsandChemistryofSolids,vol.1,no.4,pp.209–227,
requires only derivatives. Condensed into a single operator
1957. [Online]. Available: http://www.sciencedirect.com/science/article/
χ ,itiscompactandthuseasytouseinanalyticandnumeric pii/0022369757900100
hf
applications. The formulation through an energy density al-
lowsforeasymodificationofthemodeltoadaptdifferenttypes
of interactions, such as dipole-dipole-interaction, spin-spin-
interactions like the Dzyaloshinskiˇı-Moriya interaction and
spin-orbitinteractions.Itcanalsobeapplieddirectlytospatial
dependent spin configurations obtained from micromagnetic
simulationstoretrieveinformationaboutthemagnetodynamic
propertiesofspintextures.Themodelisnotrestrictedtoeval-
uating 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.
II-C 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
identifymeta-stablestatesanddescribetheirmagnetodynamic
behavior.
REFERENCES
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[2] M.Farle,“Ferromagneticresonanceofultrathinmetalliclayers,”Reports
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http://stacks.iop.org/0034-4885/61/i=7/a=001
[3] V. Flovik, F. Macià, A. D. Kent, and E. Wahlström, “Eddy
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