First-principles calculation of the Gilbert damping parameter via the linear response formalism with application to magnetic transition-metals and alloys

0.26 MB·English

by  S. Mankovsky

APS/123-QED
First-principles calculation of the Gilbert damping parameter via the linear response
formalism with application to magnetic transition-metals and alloys
S. Mankovsky1, D. K¨odderitzsch1, G. Woltersdorf2, and H. Ebert1
1 University of Munich, Department of Chemistry,
Butenandtstrasse 5-13, D-81377 Munich, Germany and
2 Department of Physics, Universita¨t Regensburg, 93040 Regensburg, Germany
(ΩDated: January 11, 2013)
A method for the calculations of the Gilbert damping parameter α is presented, which based on
the linear response formalism, has been implemented within the fully relativistic Korringa-Kohn-
3
Rostoker band structure method in combination with the coherent potential approximation alloy
1
theory. Toaccountforthermaldisplacementsofatomsasascatteringmechanism,analloy-analogy
0
model is introduced. This allows the determination of α for various types of materials, such as
2
elemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for

n disordered magnetic alloys at T =0 K and above. The effects of spin-orbit coupling, chemical and
a temperature induced structural disorder are analyzed. Calculations have been performed for the
J 3d transition-metals bcc Fe, hcp Co, and fcc Ni, their binary alloys bcc Fe1−xCox, fcc Ni1−xFex,
0 fcc Ni1−xCox and bcc Fe1−xVx, and for 5d impurities in transition-metal alloys. All results are in
1 satisfying agreement with experiment.

] PACSnumbers: 72.25.Rb71.20.Be71.70.Ej75.78.-n
r
e
h
t I. INTRODUCTION been discussedextensively inthe literature7–13. The line
o
width that is observed in ferromagnetic resonance spec-
.
t tra is usually caused by intrinsic and extrinsic relax-
a Duringthelastdecadesdynamicalmagneticproperties
m haveattractedalotofinterestduetotheirimportancein ation effects. The extrinsic contributions are a conse-
quence of spatially fluctuating magnetic properties due
- the development of new devices for spintronics, in par-
d ticular, concerning their miniaturization and fast time to sample imperfections. Short range fluctuations lead
n to two magnon scattering while long range fluctuations
scaleapplications. Adistinctive propertyofsuchdevices
o lead to an inhomogeneous line broadening due a super-
is the magnetization relaxation rate characterizing the
c position of local resonances14. In order to separate the
[ time scale on which a system being deviated from the
intrinsic Gilbert damping from the extrinsic effects it is
equilibrium returns to it, or how fast the device can be
1 necessary to measure the frequency and angular depen-
switched from one state to another. In the case of dy-
v dence of the ferromagnetic resonance line width, e. g.
namics of a uniform magnetization /vecM this property
4
1 is associatedwith the Gilbert damping parameterG˜(M) two magnon scattering can be avoided when the mag-
netization is aligned along the film normal11 (perpen-
1 usedfirstinthephenomenologicalLandau-Lifshitz(LL)1
2 and Landau-Lifshitz-Gilbert (LLG) theory2 describing dicular configuration). Usually one finds a linear fre-
. quency dependence with a zero frequency offset and one
1 the magnetization dynamics processes by means of the
can write ∆H(ω) = αω +∆H(0). When such measure-
0 equation: γ
3 ments are performed over a wide frequency range the
1 1dM G˜(M)dM slope of ∆H as a function of frequency can be used
v: γ dτ =−M×Heff +M×"γ2Ms2 dτ # , (1) tmoeteaxltlircacfterrtohme aingtnreitnssiGcilGbielrbterdtamdapminpginisgmcoonstsltyancta.useInd
i
X by electron magnon scattering. In addition Gilbert-like
where M is the saturation magnetization, γ the gy-
r s damping can be caused by eddy currents. The magni-
a romagnetic ratio and Heff = −∂MF[M(r)] being the tude of the eddy current damping is proportional to d2,
effective magnetic field. Sometimes it is more conve-
where d is the sample thickness10. In sufficiently thin
nient to use a dimensionless Gilbert damping parame-
magnetic films (d 10 nm) the eddy current damping
ter α given by α = G˜/(γM ) (see, e.g.3–5). Safonov ≤
s can be neglected10. However, for very thin films relax-
has generalized the Landau-Lifshitz equation by intro-
ation mechanisms that occur at the interfaces can also
ducing atensorialformforthe Gilbert dampingparame-
increase and even dominate the damping. Such effects
terwiththediagonaltermscharacterisingmagnetization
are spin pumping15,16 and the modified electronic struc-
dissipation6. Beingintroducedasaphenomenologicalpa-
ture at the interfaces. In the present work spin pumping
rameter, the Gilbert damping is normally deduced from
and the modified interface electronic structure are not
experiment. In particular, it can be evaluated from the
considered and we assume that bulk-like Gilbert damp-
resonant line width in ferromagnetic-resonance (FMR)
ing dominates.
experiments. The difficulty of these measurements con-
sists in the problem that there exist several different Muchunderstandingofdynamicalmagneticproperties
sources for the broadening of the line width, which have could in principle be obtained from the simulation of
2
these processes utilizing time-dependent first-principles potential scattering caused by chemical disorder5.
electronicstructurecalculations,thatinturnwouldpave Theoretical investigations of the magnetization dissi-
the way to developing and optimizing new materials for pation by means of first-principles calculations of the
spintronic devices. In spite of the progress in the de- Gilbert damping parameter already brought much un-
velopment of time-dependent density functional theory derstanding of the physical mechanisms responsible for
(TD-DFT)duringthelastdecades17thatallowstostudy this effect. First of all, key roles are played by two ef-
variousdynamicalprocessesinatomsandmoleculesfrom fects: the SOC of the atomic species contained in the
firstprinciples,applicationstosolidsarerare. Thisisdue system and scattering on various imperfections, either
to a lack of universally applicable approximationsto the impurities or structural defects, phonons, etc. Account-
exchange-correlationkernelofTD-DFT forsolids. Thus, ing for the crucial role of scattering processes respon-
atthemoment,atractableapproachconsistsintheuseof sible for the energy dissipation, different types of scat-
the classical LLG equations, and employing parameters tering phenomena have to be considered. One can dis-
calculated within a microscopic approach. Note however tinguishbetweenthe ordered-compoundorpure-element
that this approach can fail dealing with ultrafast mag- systems for which electron-phonon scattering is a very
netization dynamics, which is discussed, for instance, in important mechanism for relaxation, and disordered al-
Refs. [18 and 19], but is not considered in the present loys with dominating scattering processes resulting from
work. randomly distributed atoms of different types. In the
Most of the investigations on the magnetization dissi- first case, the Gilbert damping behavior is rather differ-
pation havebeen carriedout within model studies. Here ent at low and high temperatures. At high temperature
one has to mention, in particular, the so-called s-d or atomic displacements create random potentials leading
p-d exchange model20–23 based on a separate considera- toSOC-inducedspin-flipscattering. Atlowtemperature,
tion of the localized ’magnetic’ d-electrons and delocal- wherethe magnetizationdissipationiswelldescribedvia
ized s- and p-electrons mediating the exchange interac- the BFS (Breathing Fermi-surface) mechanism25,26, the
tions between localizedmagnetic moments and responsi- spin-conservingelectron-phononscatteringisrequiredto
ble for the magnetization dissipation in the system. As bring the electronic subsystem to the equilibrium at ev-
waspointedoutbySkadsemet al.24,thedissipationpro- ery step of the magnetization rotation, i.e. to reoccupy
cess in this case can be treated as an energy pumping the modified electronic states.
out of the d-electron subsystem into the s-electron bath In this contribution we describe a formalism for the
followed by its dissipation via spin-flip scattering pro- calculation of the Gilbert damping equivalent to that
cesses. This model gave a rather transparent qualita- of Brataas et al.4, however, based on the linear re-
tive picture for the magnetization relaxation in diluted sponce theory28 as implemented in fully relativistic mul-
alloys, e.g. magnetic semiconductors such as GaMnAs. tiple scattering based Korringa-Kohn-Rostoker (KKR)
However,it fails to give quantitative agreement with ex- formalism. It will be demonstrated that this allows to
periment in the case of itinerant metallic systems (e.g. treat elegantly and efficiently the temperature depen-
3d-metalalloys),wherethed-statesareratherdelocalised dence of α in pure crystals as well as disordered alloys.
and strongly hybridized with the sp-electrons. As a con-
sequence the treatment of all valence electrons on the
same footing is needed, which leads to the requirement
offirst-principlescalculationsoftheGilbertdampinggo-
ing beyond a model-based evaluation. II. THEORETICAL APPROACH
Various such calculations of the Gilbert damping pa-
rameter are already present in the literature. They usu- To have direct access to real materials and to obtain
ally assume a certain dissipation mechanism, like Kam- a deeper understanding of the origin of the properties
bersky’sbreathingFermisurface(BFS)25,26,ormoregen- observed experimentally, the phenomenological Gilbert
eraltorque-correlationmodels (TCM)3,27. These models damping parameter has to be treated on a microscopic
include explicitly the spin-orbit coupling (SOC), high- level. This implies to deal with the electrons responsible
lightingitskeyroleinthemagnetizationdissipationpro- fortheenergydissipationinthemagneticdynamicalpro-
cesses. However, the latter methods used for electronic cesses. Thus, one equates the corresponding expressions
structurecalculationscannottakeexplicitlyintoaccount for the dissipation rate obtained in the phenomenologi-
disorder in the system that in turn is responsible for the cal and microscopic approaches E˙mag = E˙dis. Although
aforementioned spin-flip scattering process. Therefore, a temporal variation of the magnetization is a required
this has to be simulated by using external parameters conditionforthe energydissipationtooccur,the Gilbert
characterizing the finite lifetime of the electronic states. damping parameter is defined in the limit ω 0 (see
→
ThisweakpointwasrecentlyaddressedbyBrataasetal.4 e.g.,Ref.[24])andthereforecanbecalculatedwithinthe
who described the Gilbert damping by means of scatter- adiabatic approximation.
ing theory. This development supplied the formal basis In the phenomenological LLG theory the time depen-
for the first parameter-free investigations on disordered dent magnetization M(t) is described by Eq. (1). Ac-
alloys for which the dominant scattering mechanism is cordingly, the time derivative of the magnetic energy is
3
given by: with ... indicating a configurationalaveragein case of
c
h i
a disordered system.
dM 1
E˙ =H = (mˆ˙ )T[G˜(M)mˆ˙ ] (2) The most reliable way to account for spin-orbit cou-
mag eff · dτ γ2
pling as the source of Gilbert damping is to evaluate
where mˆ = M/M denotes the normalized magnetiza- Eq. (7) using a fully relativistic Hamiltonian within the
s
tion. To represent the Gilbert damping parameter in framework of local spin density formalism (LSDA)30:
terms of a microscopic theory, following Brataas et al.4,
Hˆ =cα p+βmc2+V(r)+βσ mˆB(r). (8)
the energy dissipation is associated with the electronic
· ·
subsystem. The dissipation rate upon the motion of the Here α and β are the standard Dirac matrices, σ de-
i
magnetizationE˙ = dHˆ ,isdeterminedbytheunder- notes the vector of relativistic Pauli matrices, and p is
dis dτ
lying Hamiltonian Hˆ(Dτ). AEssuming a small deviation of the relativistic momentum operator31. The functions
V(r) and B = σ mˆB(r) are the spin-averaged and
the magnetic moment from the equilibrium the normal- ·
spin-dependent parts, respectively, of the LSDA poten-
ized magnetization mˆ(τ) can be expanded around the
tial. The spin density m (r) as well as the effective ex-
equilibrium magnetization mˆ s
0 change field B(r) are assummed to be collinear within
mˆ(τ)=mˆ +u(τ), (3) the unit cell and aligned along the z-direction in the
0
equilibrium (i. e. m (r) = m (r)mˆ = m (r)e and
resulting in the expressionfor the linearizedtime depen- s,0 s 0 s z
B (r) = B(r)mˆ = B(r)e ). Tilting of the magnetiza-
dent Hamiltonian for the system brought out of equilib- 0 0 z
tion direction by the angle θ according to Eq. (3), i.e.
rium:
m (r) = m (r)mˆ = m (r)(sinθcosφ,sinθsinφ,cosθ)
s s s
∂
Hˆ =Hˆ (mˆ )+ u Hˆ(mˆ ). (4) and B(r) = B(r)mˆ leads to a perturbation term in the
0 0 µ 0
∂uµ Hamiltonian
µ
X
Duetothesmalldeviationfromtheequilibrium,E˙ can ∆V(r)=βσ (mˆ mˆ0)B(r)=βσ uB(r), (9)
dis · − ·
beobtainedwithinthelinearresponseformalism,leading
with (see Eq. (4))
to the expression4:
∂
E˙ = π~ u˙ u˙ ψ ∂Hˆ ψ ψ ∂Hˆ ψ ∂uµHˆ(mˆ0)=βσµB(r). (10)
dis µ ν i j j i
− ij µν * (cid:12)∂uµ(cid:12) +* (cid:12)∂uν(cid:12) + TheGreen’sfunctionG+ inEq.(7)canbeobtainedin
XX (cid:12) (cid:12) (cid:12) (cid:12)
δ(E E(cid:12) )δ(E(cid:12) E )(cid:12), (cid:12) (5) a very efficient way by using the spin-polarizedrelativis-
× F − (cid:12)i (cid:12)F − j (cid:12) (cid:12) tic version of multiple scattering theory30 that allows us
where E is the Fermi energy and the sums run over
F to treat magnetic solids:
all eigenstates of the system. As Eq. (5) characterizes
the rate of the energy dissipation upon transition of the G+(r,r′,E)= ZΛn(r,E)τΛnΛm′(E)ZΛm′×(r′,E)
system from the tilted state to the equilibrium, it can
ΛΛ′
X
be identified with the corresponding phenomenological
quantity in Eq. (2), E˙mag = E˙dis. This leads to an ex- −δnm ZΛn(r,E)JΛn′×(r′,E)Θ(rn′ −rn)
epqliuciivtaelexnptrleyssfioornthfoerdtahmepGinilgbepratradmametpeirngα t=enGs˜o/r(γG˜Mosr) XΛ +(cid:2)JΛn(r,E)ZΛn′×(r′,E)Θ(rn−rn′) .(11)
(Ref. [4]): Here r,r′ refer to site n and m, respectively, w(cid:3)here
Zn(r,E) = Z (r ,E) = Z (r R ,E) is a function
~γ ∂Hˆ ∂Hˆ Λ Λ n Λ − n
αµν = ψi ψj ψj ψi centeredatsiteRn. The four-componentwavefunctions
−πMs Xij Xµν * (cid:12)(cid:12)∂uµ(cid:12)(cid:12) +* (cid:12)(cid:12)∂uν(cid:12)(cid:12) + ZΛn(r,E) (JΛn(r,E)) are regular (irregular) solutions to
δ(E (cid:12) E )(cid:12)δ(E E(cid:12) ), (cid:12) (6) the single-site Dirac equation labeled by the combined
× F (cid:12)− i(cid:12) F − (cid:12)j (cid:12) quantum numbers Λ (Λ = (κ,µ)), with κ and µ being
where the summation is running over all states at the the spin-orbit and magnetic quantum numbers31. The
Fermi surface EF. superscript indicates the left hand side solutionof the
In full analogy to the problem of electric Dirac equat×ion. τnm(E) is the so-called scattering path
ΛΛ′
conductivity29, the sum over eigenstates ψi may be operator that transfers an electronic wave coming in at
| i
expressedintermsoftheretardedsingle-particleGreen’s sitem intoa wavegoingoutfromsite nwithallpossible
function ImG+(EF) = −π i|ψiihψi|δ(EF −Ei). This intermediate scattering events accounted for.
leads for the parameter α to a Kubo-Greenwood-like Using matrix notation with respect to Λ, this leads to
P
equation: the following expression for the damping parameter:
~γ g
α = Trace α = Trace T0µτ˜0nTnµτ˜n0 (12)
µν −πM µµ πµ c
s tot
n
∂Hˆ ∂Hˆ X (cid:10) (cid:11)
ImG+(EF) ImG+(EF) (7) with the g-factor 2(1+µorb/µspin) in terms of the spin
*∂uµ ∂uν +c and orbital moments, µspin and µorb, respectively, the
4
total magnetic moment µ = µ + µ , τ˜0n = transformation matrices37,39
tot spin orb ΛΛ′
1(τ0n τ0n ) and with the energy argument E omit-
t2eid.ΛΛT′h−e mΛ′aΛtrix elements in Eq. (12) are idenFtical to ULnL′(sν,E)=4π il′′+l−l′
those occurring in the context of exchange coupling32: L′′
X
×CLL′L′′jl′′(snν√E)YL′′(ˆsnν). (14)
∂
Tnµ = d3r Zn×(r) Hˆ(mˆ ) Zn(r)
Λ′Λ Z Λ′ (cid:20)∂uµ 0 (cid:21) Λ Here me is the electron mass, jl a spherical Bessel func-
tion, CLL′L′′ stands for the Gaunt coefficients, and a
= d3r ZΛn′×(r) [βσµBxc(r)]ZΛn(r). (13) non-relativistic angular momentum representation with
Z L=(l,ml)hasbeenused. PerformingaClebsch-Gordon
The expression in Eq. (12) for the Gilbert-damping transformation for the transformation matrix Un to
LL′
parameterα is essentiallyequivalentto the one obtained the relativistic Λ representation, the t matrix tn for the
within the torquecorrelationmethod(see e.g. Refs.[33– shifted atom can be obtained from the non-shifted one
35]). However, in contrast to the conventional TCM the tn from the expression
0
electronicstructureishererepresentedusingtheretarded
electronic Green function giving the present approach
much more flexibility. In particular, it does not rely on tn =(Un)−1tnUn . (15)
a phenomenological relaxation time parameter. ν ν 0 ν
The expression Eq. (12) can be applied straightfor-
wardly to disordered alloys. This can be done by de- Treating for a discrete set of displacements sn each
ν
scribing in a first step the underlying electronic struc- displacement as an alloy component, we introduce an
ture (for T = 0 K) on the basis of the coherent po- alloy-analogy model to average over the set sn that
ν
tential approximation (CPA) alloy theory. In the next is chosen to reproduce the thermal root mean square
step the configurational averagein Eq. (12) is taken fol- average displacement u2 T for a given temperature
lowing the scheme worked out by Butler29 when dealing T. This in turn may hbeiset according to u2 T =
with the electrical conductivity at T = 0 K or residual 1 3h2 [Φ(ΘD/T) + 1p] with Φ(Θ /T) the Debhyeifunc-
4π2mkΘD ΘD/T 4 D
resistivity, respectively, of disordered alloys. This im- tion, h the Planck constant, k the Boltzmann constant
pliesinparticularthatso-calledvertexcorrectionsofthe andΘ theDebyetemperature40. Ignoringthezerotem-
D
type hTµImG+TνImG+ic −hTµImG+ichTνImG+ic that perature term 1/4 and assuming a frozen potential for
accountforscattering-inprocessesinthelanguageofthe theatoms,thesituationcanbedealtwithinfullanalogy
Boltzmann transport formalism are properly accounted to the treatment of disordered alloys on the basis of the
for. CPA (see above).
One has to note thatthe factor g inEq.(12)is sep-
µtot For small displacements the transformation Eq. (14)
arated from the configurational average ... , although can be expanded with respect to sn (see Ref. [39]) re-
h ic ν
bothvalues,g andµtot,havetorepresenttheaverageper sulting in a linear dependence on snν for non-vanishing
unit cell doing the calculations for compounds and al- contributions with ∆l = l l′ = 1. This leads, in
loys. Thisapproximationisratherreasonableinthecase particular, in the presence| o−f at|omic±displacements for
of compounds or alloys where the properties of the ele- transition-metals (TM), for which an angular momen-
ments of the system are similar (e.g. 3d-element alloys), tum cut-off of l = 2 in the KKR multiple scattering
max
butcanbequestionableinthecaseofsystemscontaining expansion is in general sufficient for an undistorted lat-
elementsexhibitingsignificantdifferences(3d-5d-,3d-4f- tice, to an angular momentum expansion up to at least
compounds, etc), or in the case of non-uniform systems l =3. However,thisiscorrectonlyundertheassump-
max
as discussed by Nibarger et al36. tionof verysmalldisplacements allowinglinearisationof
Thermal vibrations as a source of electron scattering the transformation U with respect to the displacement
can in principle be accounted for by a generalization of amplitude s. Thus, since the temperature increase leads
Eqs. (7) – (13) to finite temperatures and by including toamonotonousincreaseofs,thecut-offl shouldalso
max
the electron-phonon self-energy Σel−ph when calculating be increased.
the Green’sfunctionG+. Herewerestrictourconsidera-
tiontoelasticscatteringprocessesbyusingaquasi-static
representationofthethermaldisplacementsoftheatoms
III. MODEL CALCULATIONS
from their equilibrium positions. The atom displaced
from the equilibrium position in the lattice results in a
corresponding variation ∆tn = tn tn of the single-site In the following we present results of calculations for
scatteringmatrixinthe globalfram−eo0freference37,38. A which single parameters have artificially been manipu-
single-site scattering matrix tn (the underline denotes a lated in the first-principles calculations in order to sys-
matrix in an angular momentum representation Λ) for tematically reveal their role for the Gilbert-damping.
the atom n displaced by the value sn from the equilib- This approach is used to disentangle competing influ-
ν
rium position in the lattice can be obtained using the ences on the Gilbert-damping parameter.
5
A. Vertex corrections
Fe V
1-x x
40
The impact of vertex corrections is shown in Fig. 1
without vertex corrections
for two different cases: Fig. 1(a) represents the Gilbert with vertex corrections
dampingparameterforanFe1−xVx disorderedalloyasa
function of concentration, while Fig. 1(b) gives the cor- 30
1
responding value for pure Fe in the presence of temper- α × 20
ature induced disorder and plotted as a function of tem-
perature. Both figures show results calculated with and
withoutvertexcorrectionsallowingforcomparison. First
of all, a significant effect of the vertex corrections is no-
ticeableinbothcases,althoughthevariationdependson 0
increasingconcentrationofVinthebinaryFe1−xVxalloy 0 0.1 0.2 0.3 0.4 0.5
and the temperature in the case of pure Fe, respectively.
concentration x
Some differences in their behavior can be explained by V (a)
30
the differencesofthe systemsunderconsideration. Deal-
ingwithtemperatureeffectsviathealloyanalogymodel, bcc Fe
the system is considered as an effective alloy consisting 25
without vertex corrections
of a fixed number of components characterizingdifferent with vertex corrections
types of displacements. Thus, in this case the tempera- 20
ture effect is associated with the increase of disorder in
3
0
the system caused only by the increase of the displace- 1 15
ment amplitude, or, in other words – with the strength α ×
of scattering potential experienced by the electrons rep- 10
resented by tn(T) tn. In the case of a random alloy
− 0
the A1−xBx variation of the scattering potential, as well 5
as the difference tn tn, upon changing the concentra-
A − B
tions is less pronounced for small amounts of impurities
0
Bandtheconcentrationdependenceisdeterminedbythe 0 100 200 300 400 500
amount of scatterers of different types. However, when temperature (K) (b)
the concentration of impurities increases, the potentials
FIG.1. TheGilbertdampingparameterfor(a)bccFe1−xVx
of the components are also modified (this is reflected, (T =0K)asafunctionofVconcentrationand(b)forbcc-Fe
e.g., in the shift of electronic states with respect to the
asafunctionoftemperature. Full(open)symbolsgiveresults
Fermi level, that will be discussed below) and this can with (without) the vertexcorrections.
lead to a change of the concentration dependence of the
vertex corrections. An important issue which one has to 0.4
stress that neglect of the vertex corrections may lead to
Py+15%Os
the unphysical result, α < 0, as is shown in Fig. 1(a).
In terms of the Boltzmann transport formalism, this is 0.3
because of the neglect of the scattering-in term41 lead-
ing obviously to an incomplete description of the energy
transfer processes. 1/2 0.2
α

0.1
B. Influence of spin-orbit coupling
Aswasalreadydiscussedabove,thespin-orbitcoupling
0
for the electrons of the atoms composing the system is
the main driving force for the magnetization relaxation, 0 0.5 1 1.5 2
SOC scaling parameter
resultingintheenergytransferfromthemagneticsubsys-
temtothecrystallattice. Thus,theGilbertdampingpa-
rameter should approach zero upon decreasing the SOC
FIG. 2. The Gilbert damping parameter for Py+15%Os as
in the system. Fig. 2 shows the results for Py+15%Os,
afunction ofthescaling parameterofspin-orbitcouplingap-
where √α is plotted as a function of the scaling param-
plied to all atoms contained in the alloy. Red dashed line in
eter of the spin-orbit coupling42 applied to all atoms in plot – linear fit. The values 0 and 1 for the SOC scaling pa-
the alloy. As one can see, √α has a nearly linear depen- rameter correspond to the scalar-relativistic Schr¨odinger-like
dence on SOC implying that α varies in second order in and fully relativistic Dirac equations, respectively.
the strength of the spin-orbit coupling43.
6
IV. RESULTS AND DISCUSSIONS chemical disorder. Large discrepancies between the lat-
ter experimental data47 and theoretical results of the α
A. 3d transition-metals calculationsforbccFearerelatedtotheverysmallthick-
nessofthefilminvestigatedexperimentally,thatleadsto
an increase of spin-transfer channels for magnetization
We have mentioned above the crucial role of scatter-
dissipation as was discussed above.
ing processes for the energy dissipation in magnetiza-
Results for the temperature dependent Gilbert-
tion dynamic processes. In pure metals, in the absence
damping parameter α for hcp Co are presented in Fig.
of any impurity, the electron-phonon scattering mecha-
3(b)whichshows,despitecertainunderestimation,area-
nism is of greatimportance,althoughit plays a different
sonable agreement with the experimental results45. The
role in the low- and high-temperature regimes. This was
demonstrated by Ebert et al.28 using the alloy analogy general trends at low and high temperatures are similar
approach,aswellasbyLiu44etal. usingthe’frozenther- to those seen in Fe.
mal lattice disorder’ approach. In fact both approaches TheresultsforpureNiaregiveninFig.3(c)thatshow
are based on the quasi-static treatment of thermal dis- in full accordance with experiment a rapid decrease of α
placements. However, while the average is taken by the with increasing temperature until a regime with a weak
CPA within the alloy analogy model the latter requires variation of α with T is reached.
a sequence of super-cell calculations for this purpose. Note that in the discussions above we have treated α
As a first example bcc Fe is considered here. The cal- asa scalarinsteadofa tensorialquantity ignoringa pos-
culations have been performed accounting for the tem- sible anisotropy of the damping processes. This approx-
perature induced atomic displacements from their equi- imation is reasonable for the systems considered above
librium positions, according to the alloy analogy scheme with the magnetizationdirections along a three- or four-
describedinsectionII. Thisleads,evenforpuresystems, fold symmetry axis (see, e.g., the discussions in Ref. [48
to a scattering process and in this way to a finite value and 49]). For a more detailed discussion of this issue
for α (see Fig. 3(a)). One can see that the experimen- the anisotropy of the Gilbert damping tensor α(M) has
tal results available in the literature are rather different, beeninvestigatedfor bcc Fe. To demonstratethe depen-
depending on the conditions of the experiment. In par- dence of α on the magnetization direction M, the cal-
ticular, the experimental results Expt. 2 (Ref. [45]) and culations have been performed for M = zˆM with the
| |
Expt.3(Ref.[46])correspondtobulkwhilethemeasure- zˆ axis taken along the 001 , 111 and 011 crystallo-
h i h i h i
mentsExpt.1(Ref.[47])havebeendoneforanultrathin graphic directions. Fig. 4 presents the temperature de-
film with 2.3 nm thickness. The Gilbert damping con- pendenceofthediagonalelementsαxx andαyy. Astobe
stant obtainedwithin the present calculationsfor bcc Fe expectedforsymmetryreasons,αxx differsfromαyy only
(circles, a = 5.44 a.u.) is compared in Fig. 3(a) with in the case of zˆ 011 . One can see that the anisotropic
kh i
the experiment exhibiting rather good agreement at the behavior of the Gilbert damping is pronounced at low
temperature above 100 K despite a certainunderestima- temperatures. With an increase of the temperature the
tion. One can also see a rather pronounced increase of anisotropynearly disappears,because of the smearingof
the Gilbert damping observed in the experiment above theenergybandscausedbythermalvibrations49. Asim-
400 K (Fig. 3(a), Expt. 2 and Expt. 3), while the theo- ilar behavior is caused by impurities with a random dis-
reticalvalueshowsonlylittletemperaturedependentbe- tribution,aswasobservedforexamplefortheFe0.95Si0.05
havior. Nevertheless, the increase of the Gilbert damp- alloy system. The calculations of the diagonal elements
ing with temperature becomes more pronounced when αxx and αyy for two different magnetization directions
the temperature induced lattice expansion is taken into along 001 and 011 axes give αxx = αyy = 0.00123 in
h i h i
account, that can be seen from the results obtained for the first case and αxx = 0.00123 and αyy = 0.00127 in
a = 5.45 a.u. (squares). Thick lines are used to stress the second, i.e. the damping is nearly isotropic.
the temperature regions for which corresponding lattice The damping parameter α increases very rapidly with
parameters are more appropriate. At low temperatures, decreasing temperature in the low temperature regime
below 100 K, the calculated Gilbert damping parameter (T 100 K) for all pure ferromagnetic 3d metals, Fe,
≤
goes up when the temperature decreases, that was ob- Co, and Ni (see Fig. 3), leading to a significant discrep-
served only in the recent experiment47. This behavior is ancybetweentheoreticalandexperimentalresultsinthis
commonly denoted as a transitionfromlow-temperature regime. The observed discrepancy between theory and
conductivity-like to high-temperature resistivity-like be- experimentcanbe relatedtothe exactlimitω =0taken
havior reflecting the dominance of intra- and inter-band intheexpressionfortheGilbertdampingparameter. Ko-
transitions, respectively3. The latter are related to the renmannandPrange13haveanalyzedthemagnondamp-
increaseof the smearingof electronenergy bands caused ing in the limit of small wave vector of magnons q 0,
→
by the increase of scattering events with temperature. assuming indirect transitions in the electron subsystem
Note that even a small amount of impurities reduces andtaking into accountthe finite lifetime τ of the Bloch
strongly the conductivity-like behavior28,45, leading to states due to electron-phonon scattering. They discuss
the more pronounced effect of impurity-scattering pro- the limiting cases of low and high temperatures showing
cesses due to the increase of scattering events caused by the analogy of the present problem with the problem of
7
10 extreme cases for the conductivity leading to the normal
bcc Fe and anomalous skin effect. On the basis of their result,
Theory: a = 5.42 a.u.
the authorspointout thatthe expressionfor the Gilbert
8 Theory: Fe+0.1% Vac.
damping obtained by Kambersky25, with α τ is cor-
Theory: a = 5.45 a.u. ∼
Expt. 1 rectin the limit ofsmall lifetime (i.e. qvFτ 1, in their
≪
6 Expt. 2 model consideration, where q is a magnon wave vector
3
α 10× Expt. 3 atenmdpveFraitsuraeFleimrmititvheelolcifietytimofetτheineclercetarsoens)w. iItnh tdheecrleoaws--
4
ing T and one has to use the expression corresponding
to the ’anomalous’ skin effect for the conductivity, i.e.
2 α tan−1(qv τ)/qv , leadingto a saturationofα upon
F F
∼
the increase of τ.
Another possible reason for the low-temperature be-
0
0 200 400 600 havior of the Gilbert damping observed experimentally
Temperature (K) (a) can be structural defects present in the material. To
20 simulatethiseffect,calculationshavebeenperformedfor
fcc Ni and bcc Fe with 0.1% of vacancies and for hcp
hcp Co
Co with 0.1% and 0.03% of vacancies. Fig. 3(a)-(c)
15 shows the corresponding temperature dependence of the
Expt Gilbertdampingparameterapproachingafinitevaluefor
Theory T 0. The remainingdifference inthe T-dependentbe-
30 Theory: Co + 0.03% Vac ha→viorcanbe attributedtothe non-lineardependence of
α 1× 10 Theory: Co + 0.1% Vac thescatteringcrosssectionatlowtemperaturesasisdis-
cussedintheliteraturefortransportpropertiesofmetals
and is not accounted for within the present approxima-
5
tion.
0
0 100 200 300 400 500 600 B. 3d Transition-metal alloys
Temperature (K) (b)
0.25 As is mentioned above, the use of the linear response
formalism within multiple scattering theory for the elec-
fcc Ni
tronicstructurecalculationsallowsustoperformthenec-
0.2
essary configurational averaging in a very efficient way
Expt avoiding supercell calculations and to study with mod-
0.15 Theory erate effort the influence of varying alloy composition
Theory: Ni + 0.1%Vac
α on α. The corresponding approach has been applied to
0.1
30
0.05 bcc Fe
25
Theory: z||<001>: α =α
xx yy
0 Theory: z||<011>: α
0 100 200 300 400 500 20 Theory: z||<011>: αxx
Temperature (K) (c) 30 Theory: z||<111>: αyy=α
1 15 xx yy
α ×
FIG. 3. Temperature variation of the Gilbert damping pa-
10
rameter of pure systems. Comparison of theoretical results
withexperiment: (a)bcc-Fe: circlesandsquaresshowthere-
sultsforidealbccFefortwolatticeparameters,a=5.42a.u. 5
and a = 5.45 a.u.; stars show theoretical results for bcc Fe
(a = 5.42 a.u.) with 0.1% of vacancies (Expt. 1 - Ref. [47], 0
0 100 200 300 400 500
Expt. 2 - Ref. [45], Expt. 3 - Ref. [46]); (b) hcp-Co: circles
Temperature (K)
show theoretical results for ideal hcp Co, stars - for Co with
0.03% of vacancies, and ’pluses’ - for Co with 0.1% of vacan- FIG. 4. Temperature variation of the αxx and αyy com-
cies (Expt.Ref. [45]); and (c) fcc-Ni (Expt. Ref. [45]).
ponents of the Gilbert damping tensor of bcc Fe with the
magnetizationdirectiontakenalongdifferentcrystallographic
directions: M = zˆ|M|kh001i (circles), Mkh011i (squares),
Mkh111i (diamonds).
8
6 Fe Co 60 6
1-x x Fe Co
1-x x
5 bcc, CPA 50
Exp.
3104 CbcscC, lN: CLP CAP A(p a(ortrida)l ord) 40 s./Ry) 304 Theory
α ×3 bnc(Ec, )NL CPA (disord) 30 ) (stF α 1×
2 F 20 E
n( 2
n(E )
1 F 10
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
concentration x 0 0.2 0.4 0.6
Co (a)
concentration x
Co (a)
V) 2 EF 40
e
sts./ 1.5 xx == 00..051 Ni1-xCox
) ( 1 30 Exp.
E
↓n(tot0.5 3 Theory (Starikov et al.)
0 Theory
0 120
V) 2 α ×
e
sts./ 1.5 10
E) ( 1
↑(ntot0.5 0
0 0.2 0.4 0.6
0 -8 -6 -4 -2 0 2 concentration xCo (b)
energy (eV) (b) 20
Ni Fe
1-x x
FIG. 5. (a) Theoretical results for the Gilbert damping pa- Theory (Starikov et al.)
rameter of bcc Fe1−xCox as a function of Co concentration: Theory
CPA results for the bcc structure (full circles) describing 3 Expt 1
0
tsyhsetermand(oompenaeldloysqsuyasrtee)m,forresxult=s fo0r.5th(ie.e.partfoiarllyFeo1−rdxeCroedx α 1×10 Expt 2
alloy with CsCl structure and alloy components randomly
distributed in two sublattices in the following proportions:
(Fe0.9Co0.1)(Fe0.1Co0.9), the NL-CPA results for random al-
loy with bcc structure (opened circles) and the NL-CPA re-
sults for the the system with short-range order within the 0
0 0.2 0.4 0.6
first-neighbor shell (opened diamonds). The dashed line rep- concentration x
resentstheDOSattheFermienergy,EF,asafunctionofCo 5 Fe (c)
concentration. (b) spin resolved DOS for bcc Fe1−xCox for
x=0.01 (dashed line) and x=0.5 (solid line). Fe1-xVx
4
Expt.
Theory, T = 0 K
3
3 Theory, T = 300 K
the ferromagnetic 3d-transition-metal alloy systems bcc 0
1
Fe1−xCox,fccNi1−xFex,fccNi1−xCox andbccFe1−xVx. α ×2
Fig. 5(a) shows as an example results for the Gilbert
damping parameter α(x) calculated for bcc Fe1−xCox 1
for T = 0 K at different conditions. Full circles rep-
resent the results of the single-cite CPA calculations
0
characterizing the random Fe-Co alloy. These results 0 0.2 0.4 0.6
are compared to those obtained employing the non-local concentration xV (d)
CPA52,53 (NL-CPA) assuming no short-range order in
the system(opened circles). Dealing in both cases(CPA FIG. 6. The Gilbert damping parameter for Fe1−xCox (a)
and NL-CPA), with completely disordered system, the Ni1−xCox (b) and Ni1−xFex (c) as a function of Co and Fe
concentration, respectively: present results within CPA (full
NL-CPA maps the alloy problem on that of an impurity circles), experimental data by Oogane50 (full diamonds). (d)
cluster embedded in a translational invariant effective Results for bcc Fe1−xVx as a function of V concentration:
medium determined selfconsistently, thereby accounting T =0K (fullcircles) andT =300K(open circles). Squares:
experimental data51. Open circles: theoretical results by
Starikov et al.5.
9
for nonlocal correlations up to the range of the cluster resultofadifferentvariationoftheDOSn(E )causedby
F
size. The present calculations have been performed for a concentrationdependent modification of the electronic
thesmallestNL-CPAclusterscontainingtwositesforbcc states and shift of the Fermi level.
basedsystem,accountingfortheshort-rangeorderinthe
first-neighborshell. Asonecansee,thisresultsinasmall
decrease of the α value in the region of concentrations C. 5d impurities in 3d transition metals
aroundx=0.5 (opened diamonds), that is in agreement
with the results obtained for partially ordered system
As discussed in our recent work28 investigating the
(opened square)for x=0.5. The latter havebeen calcu-
temperature dependent Gilbert damping parameter for
lated for the Fe1−xCox alloy having CsCl structure and pure Ni and for Ni with Cu impurities, α is primarily
alloycomponentsrandomlydistributedintwosublattices
determinedbythe thermaldisplacementintheregimeof
in the following proportions: (Fe Co )(Fe Co ).
0.9 0.1 0.1 0.9 smallimpurityconcentrations. Thisbehaviorcanalsobe
Becausethemomentsandspin-orbitcouplingstrength seeninFig. 7,wheretheresultsforFewith5d-impurities
do not differ very much for Fe and Co, the variation of are shown. Solid lines represent results for T = 0 K for
α(x) should be determined in the concentrated regime an impurity concentration of 1% (full squares) and 5%
essentially by the electronic structure at the Fermi en- (full circles). As one can see, at smaller concentrations
ergy EF. As Fig. 5(a) shows, there is indeed a close the maximum of the Gilbert damping parameter occurs
correlationwiththe densityofstatesn(EF)thatmaybe for Pt. With increasing impurity content the α parame-
seen as a measure for the number of available relaxation ter decreases in such a way that at the concentration of
channels. The change of α(x) due to the increase of the 5% a maximum is observed for Os.
Co concentration is primarily determined by an appar- The reason for this behavior lies in the rather weak
ent shift of the Fermi energy also varying with concen- scattering efficiency of Pt atoms due to a small DOS
tration (Fig. 5(b)). The alloy systems considered have n(E ) of the Pt states when compared for example for
F
the common feature that the concentration dependence Osimpurities(seeFig. 9). Thisresultsinaslowdecrease
of α is governedby the concentration dependent density ofαatsmallPtconcentrationwhenthe BFSmechanism
of states n(EF). is mostly responsible for the energy dissipation. A con-
A comparison of theoretical α values with the experi- sequence of this feature can be seen in the temperature
mentforbccFe1−xCoxisshowninFig. 6(a),demonstrat- dependence of α (T = 300 K, opened squares): a most
ing satisfying agreement. In the case of Ni1−xFex and pronouncedtemperature induceddecreaseofthe αvalue
Ni1−xCox alloysshowninFig. 6,(b)and(c),theGilbert is observed for Pt and Au. When the concentration of
dampingdecreasesmonotonouslywiththeincreaseofthe 5d-impurities is increased up to 5%, the maximum in α
Fe andCoconcentration,inlinewithexperimentaldata. occurs for the element with the most efficient scatter-
Atallconcentrationsthe experimentalresultsareunder- ing potential resulting in spin-flip scattering processes
estimatedby theoryapproximatelyby afactor of2. The responsible for dissipation. The temperature effect at
calculateddampingparameterα(x)isfoundinverygood this concentration is very small.
agreementwiththeresultsbasedonthescatteringtheory Consideringinmoredetailthetemperaturedependent
approach5 demonstrating numerically the equivalence of behavior of the Gilbert damping parameter for Fe with
the two approaches. An indispensable requirement to Os and Pt impurities, shown in Fig. 8, one can also ob-
achieve this agreement is to include the vertex correc- serve the consequence of the features mentioned above.
tions mentioned above. As suggested by Eq. (12) the At 1% of Pt impurities α decreases much steeper upon
variation of α(x) with concentration x may also reflect increasing the temperature, as compared to the case of
tosomeextentthevariationoftheaveragemagneticmo- Os impurities. Therefore,in the first case the role of the
ment of the alloy, µtot. scatteringprocessesduetoatomicdisplacementsismuch
ThepeculiarityoftheFe1−xVxalloywhencomparedto more pronounced than in the second case with rather
those discussed above is that V is a non-magnetic metal strong scattering on the Os impurities. When the con-
and has only an induced spin magnetic moment. De- centration increases to 5% the dependence of α on the
spite that, the concentration dependence of the Gilbert temperature in both cases becomes less pronounced.
damping parameter at T = 0 K for small amounts of Thepreviousresultscanbecomparedtotheresultsfor
V (see Fig. 6(d)) displays the same trend as the pre- the 5d-impurities in the permalloy Fe Ni (Py), which
80 20
viously discussed alloys shown in Fig. 6(a)-(c). Taking hasbeeninvestigatedalsoexperimentally54. Thissystem
into account a finite temperature of T = 300 K changes shows some difference in the concentration dependence
α value significantly at small V concentrations leading whencomparedwithpureFe, becausePyis adisordered
to an improved agreement with experiment for pure Fe, alloy with a finite value of the α parameter. Therefore,
whileitstillcomparespoorlywiththeexperimentaldata a substitution of 5d impurities leads to a nearly linear
atx =0.27. Oneshouldstressonce morethat the con- increase of the Gilbert damping with impurity content,
V
centration dependent behavior of the Gilbert damping just as seen in experiment54.
parameter of the alloys discussed above is different for The total damping for 10% of 5d-impurities shown in
anincreasedamountofimpurities(morethan10%),asa Fig.10(a)variesroughlyparabolicallyoverthe5dTMse-
10
10 5
x = 0.05; T = 0 K
8 x = 0.05; T = 300 K Pt Fe Me
0.99 0.01
x = 0.01; T = 0 K
4
3 6 x = 0.01; T = 300 K
0
1
α × 4 30
1 3
α ×
2
Os
0 2
Ta W Re Os Ir Pt Au
FIG. 7. Gilbert damping parameter for bcc Fe with 1% 1
(squares) and 5% (circles) of 5d impurities calculated for 0 100 200 300 400 500
T =0K (full symbols) and for T =300K (opened sysmbols). temperature (K) (a)
4
Fe Me
ries. This variationof α with the type of impurity corre- 0.95 0.05
lateswellwiththedensityofstatesn5d(EF)(Fig. 10(b)). 3.5 Os
Again the trend of the experimental data is well repro-
duced by the calculated values that are however some-
3
0
what too low. 1 3
α ×
V. SUMMARY
2.5
Pt
Insummary,aformulationfortheGilbertdampingpa-
rameterα interms of linearresponsetheory wasderived
2
thatledtoaKubo-Greenwood-likeequation. Thescheme 0 100 200 300 400 500
was implemented using the fully relativistic KKR band temperature (K) (b)
structuremethodincombinationwiththeCPAalloythe-
ory. This allows to account for various types of scat-
tering mechanisms in a parameter-free way, that might FIG. 8. Gilbert damping parameter for bcc Fe1−xMx with
M =Pt(circles) andM =Os(squares)impuritiesasafunc-
be either due to chemical disorder or to temperature-
tion of temperature for 1% (a) and 5% (b) of theimpurities.
induced structural disorder (i.e. electron-phonon scat-
tering effect). The latter has been described by using
the so-called alloy-analogy model with the thermal dis-
placement of atoms dealt with in a quasi-static manner.
Corresponding applications to pure metals (Fe, Co, Ni) 3.2 E
aswellastodisorderedtransition-metalalloysledtovery V) F
e2.4
goodagreementwithresultsbasedonthescatteringthe- s./ Pt
eoxrypearpimpreonatcahl roefsuBlrtas.taaTsheetcarlu.4ciaanldrowleelolfrveperrtoedxucceosrrtehce- E) (st1.6 Os
↓( 0.8
n
tions for the Gilbert damping is demonstrated both in
the case of chemical as well as structural disorder and 0
3.2 E
the accuracyoffinite-temperatureresultsisanalyzedvia V) F
test calculations. e2.4
s./
ofFtuhrethperrmesoenret, stchheemfleexiwbailsitydeamnodnsnturmateerdicablyeffia csiteundcyy E) (st1.6
on metallic systems on a series of binary 3d-alloys ↑n( 0.8
(Fe1−xCox, Ni1−xFex, Ni1−xCox and Fe1−xVx), 3d 5d 0
− -8 -6 -4 -2 0 2
TM systems, the permalloy-5d TM systems. The agree-
energy (eV)
ment between the present theoretical and experimental
results is quite satisfying, although one has to stress
a systematic underestimation of the Gilbert damping
by the numerical results. This disagreement could be FFeIG1−.xO9.sx (DdOasShefdorlinPet) fionr xFe=1−0x.P01t.x (full line) and Os in
causedeitherbytheidealizedsystemconsideredtheoret-
ically (e.g., the boundary effects are not accounted for