Theoretical evaluation on the temperature dependence of magnetic anisotropy constants of Nd2Fe14B - Effects of exchange field and crystal field strength -

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by  Ryo Sasaki

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Theoreticalevaluationon the temperaturedependenceof magneticanisotropy
constantsof Nd Fe B: Effectsof exchangefield and crystalfield strength–
2 14
RyoSasaki,DaisukeMiura,andAkimasaSakuma
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1 DepartmentofAppliedPhysics,TohokuUniversity,Aoba-ku,Sendai980-8579,Japan
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2

n
a
Toidentifythepossiblemechanismofcoercivity(H )degradationofNd-Fe-Bsinteredmagnets,westudy
J c
8 therolesoftheexchangefieldactingonthe4felectronsinNdionsandtheoreticallyinvestigatehowthe

] variationoftheexchangefieldaffectsthevaluesofthemagneticanisotropyconstantsK1 andK2.Wefind
i
c that,withdecreasingexchangefieldstrength,bothvaluesdecreaseasaresultofthelowerasphericityof
s
the4felectroncloud,indicatingthatthelocalanisotropyconstantsmightbecomesmallaroundthegrain
-
l
r boundarieswheretheexchangefieldsaredecreasedowingtothesmallercoordinationnumber.
t
m
.
t
a
m
Nd-Fe-Bsintered magnets1–3) havethelargest maximumenergy product amongthecurrent mag-
-
d netsandhavebeenwidelyusedformagneticdevicessuchasvoicecoilmotorsinmagneticrecording
n
o systems. Recently, because of the rapidly growing interest in electric vehicles, much effort has been
c
[ made to suppress the degradation of the coercivity (Hc) of Nd-Fe-B magnets. However, from an in-

1 dustrial viewpoint, reduction in the usage of Dy is strongly desired, because Dy is a rare metal and
v
2 themagnetizationoftheNd-Fe-BmagnetsdecreasesbysubstitutingDywithNdowingtotheantipar-
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allel coupling between Dy and Fe moments. Realizing Dy-free high-performance Nd-Fe-B magnets
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1 requiresafurtherincreaseofH intheNd-Fe-Bsystembymicrostructureoptimization,4–11)andthere-
0 c
. fore,establishingthemicroscopicfoundationforthecoercivitymechanismisdesired.Fromatheoret-
1
0
icalviewpoint,manyworks12–16)havefocusedonthechangeofmagneticanisotropyconstantsaround
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1 the grain boundary surfaces as a result of the stresses, defects, and change of spatial symmetry. In
:
v
addition, micromagnetic modelcalculations haveshown thatthesurface c-plane anisotropy candras-
i
X
ticallydecreasethecoercivity.17) Forthesereasons,evaluationofthelocalanisotropyconstantsaround
r
a
the grain boundaries and determining their temperature dependence are important to investigate the
degradation ofthecoercivity.
With regard to the magnetic anisotropy of rare earth (RE) transition metal compounds, it is be-
lievedthatthe4felectronsinREionsareresponsibleforthemainpartofthemagneticanisotropyand
thatthecrystallineelectricfield(CEF)actingonthe4felectronsdominatesthisproperty.18) Underthe
assumption that the exchange field on the 4f electrons is strong enough, by using the CEFparameter
A0,theleadinganisotropyconstantK canbeapproximatelydescribedbyK = −3J(J−1)αhrˆ2iA0N ,
2 1 1 2 R
whereαistheStevensfactor, Jisthetotalangularmomentum,hrˆ2iistheaverageofrˆ2 overtheradial
wave function of the 4f electrons, and N is the density of RE ions. Note that the CEF parameter
R
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is easily affected by circumstances especially around the grain boundaries. Actually, Moriya et al.15)
predictedbyusingafirst-principles calculationthattheCEFparameter A0 exhibitsanegativevalueat
2
the(001)surfaceofNd Fe B.According toRef.17,thismayleadtoadrasticdegradation of H .
2 14 c
Worth mentioning here is that the exchange field acting on the 4f electrons from surrounding Fe
spins can also bechanged asaresult of thesituation surrounding the REion. Itisnatural to consider
that the exchange fields are weak around the grain boundaries compared to those in the bulk. In this
case,theaboveexpressionforK nolongerappliesbecausetheprecondition thattheexchangefieldis
1
strongenoughnolongerholds.Especially,thedecreaseoftheexchangefieldpossiblyhasasignificant
influence on the temperature dependence of the anisotropy constants because of thermal fluctuations
of the 4f moments. Based on this viewpoint, we focus on the roles of the exchange field on the 4f-
related anisotropy constants, andwetheoretically investigate howthemagnetic anisotropy isaffected
byavariationofexchangefieldsandtheCEFactingonthe4felectrons. Tothisaim,thetemperature-
dependent anisotropy constants K (T)and K (T)arecalculated basedoncrystalfieldtheory.
1 2
The calculation method we use here is based on conventional crystal field theory for the 4f elec-
tronic system in the RE ions. The total Hamiltonian for 4f electrons is given as Hˆ = Hˆ +Hˆ +
ion CEF
Hˆ , where Hˆ describes the intra-atomic interactions in an RE ion, Hˆ denotes the CEF part
mag ion CEF
representing the electrostatic field acting on the 4f electrons from the surrounding charge, and Hˆ
mag
denotes the effective exchange interactions between the 4f and Fe moments. When Nd ions are in-
volved, only the ground J multiplet can be considered as the 4f states because 4felectrons ofthe Nd
ion have large spin-orbit coupling compared to the CEF splitting. In this case, Hˆ can be written
CEF
as Hˆ = BmOˆm, where Bm = θhrliAm is the CEF coefficient and Oˆm is the Stevens operator
CEF l,m l l l l l l
expressed byPthe multinomial of the orbital angular momentum operators. θ is the Stevens factor α,
l
β, or γ for l = 2, 4, or 6, respectively, and Am are the CEF parameters. In this work, we take into
l
account only B0, B0,and B0,forsimplicity. Within theground J multiplet, Hˆ canbeexpressed as
2 4 6 mag
Hˆ = 2(g −1)Jˆ·H ,whereg istheLandefactorand H istheeffectiveexchangefieldreflecting
mag J ex J ex
theexchangeinteractionwithsurrounding Fespins.Inprinciple, H dependsontemperaturethrough
ex
a self-consistent equation for a molecular field approximation in the full system of Nd Fe B. How-
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ever, in the present work, we assume that H is proportional to a molecular field of Fe spins hS i
ex Fe
and that the molecular field is described by the Heisenberg model for the Fe-spin system. Solving
the self-consistent equation for hS i under the condition that the spin of the Fe ion is 1 and that the
Fe
Curietemperature T is583K,wehave confirmedthat hS ireproduces wellthetemperature depen-
c Fe
dence of the magnetization of Y Fe B.19) Thus we have introduced temperature dependency in H
2 14 ex
by H (T) = H hS i,where H isthestrength parameter ofthe effectiveexchange field. Here, the
ex ex Fe ex
direction oftotalmagnetization isassumed tocoincide withthatof H ,because thetotal magnetiza-
ex
tion is mostly governed by the Fe moments even when the direction of 4f moments differs from that
oftheFemoments.
By defining θ as the angle between H (T) (total magnetization) and the c axis of the crystal
ex
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lattice,thefreeenergyofthe4felectronic system canbewrittenas
Hˆ(θ,T)
F(θ,T) = −k T lnTrexp − ,
B
 k T 
B
wherethe eigenvalues arecalculated bydiagonalizing Hˆ(θ,T)within the J = 9/2 subspace (10×10
matrix),and F(θ,T)isnumerically obtained.
The4f-related anisotropy constants K and K are coefficients ofsin2θ and sin4θ in F(θ,T). For
1 2
smallθ, F(θ,T)canbeexpanded as
θ2 θ4 θ6
F(θ,T) = F(0,T)+ F(2)(0,T)+ F(4)(0,T)+ F(6)(0,T)+···
2 4! 6!
1
≡ F(0,T)+K (T)θ2 + − K (T)+K (T) θ4+··· .
1 1 2
" 3 #
Therefore, K and K canbeexpressed by
1 2
1
K (T) = F(2)(0,T),
1
2
1 1
K (T) = K + F(4)(0,T),
2 1
3 4!
wherewecalculate F(2)(0,T)and F(4)(0,T)fromfinitedifferences.
Toperformaquantitative analysisonthemagneticanisotropy constants ofNd Fe B,westartby
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reproducing the experimental results of K and K of bulk Nd Fe B by using the present approach.
1 2 2 14
Figure 1 shows the calculated results obtained for K (T) and K (T) using A0 = 450 K/a 2, A0 =
1 2 2 0 4
−45 K/a 4, A0 = −0.1 K/a 6, and H = 364 K,20) together with the experimental data.21) Here, a
0 6 0 ex 0
is the Bohr radius. In calculating K (T), we add the Fe sublattice contribution KFe(T) deduced from
1 1
experimental data for Y Fe B.19) This has an almost constant value of ∼1 MJ/m3 below 300 K and
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decreases to zero when the temperature approaches T .The difference between two Ndsublattices is
c
not taken into account here. The inset in Fig. 1 shows the tilting angle of the magnetization vector
calculatedfrom ∂F(θ,T) = 0.Includingthisspinreorientationbehavior,onecanseeagreementbetween
∂θ
calculated dataandexperimental onestoacertaindegree.
It should be noted here that the experimental values of K (T) and K (T) are not obtained by the
1 2
direct measurement but are usually deduced from magnetization curves or torque curves under finite
appliedfields.Furthermore,inournumericalanalysis,wemadesomeapproximationstoexpressK (T)
1
and K (T). In this sense, we believe that the CEF and H parameters reported by Yamada et al. are
2 ex
morereliable,becauseintheiranalysistheZeemantermisconsideredinadditiontotheHamiltonianto
ascertainthemagnetizationcurvesandtomakeadirectcomparisonwiththemeasuredones.Reflecting
thisdifference,theCEFandH parametersobtainedbyYamadaetal.areconsiderably differentfrom
ex
ours. To examine the availability of the present method and reliability of the parameters obtained
above,wealsocalculated the K (T)and K (T)curvesusingtheparameters givenbyYamadaetal.18)
1 2
Except for the behavior of K (T) below 70 K, we confirmed that the calculated data agree with both
2
the experimental and our calculated results. These results lead us to believe that the present analysis
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50
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3400 gle [deg]12235050
3m] 20 K2 An1005
J/ 0 40 80 120 160
M 10 Temperature [K]
[
Ki
0
-10 K
1
-20
0 100 200 300 400 500 600
Temperature [K]
Fig. 1. Calculatedanisotropyconstants K and K (solid lines)as a functionof temperatureobtainedusing
1 2
A0 = 450K/a 2, A0 = −45 K/a 4, A0 = −0.1 K/a 6, and H = 364K. The closed and opencircles are the
2 0 4 0 6 0 ex
measuredvalues21)ofK andK ,respectively.Theinsetshowsthetiltinganglesofthemagnetizationfromthe
1 2
caxisasafunctionoftemperaturetogetherwiththeexperimentalresults(closedcircles).21)
20
K
15 2
] 10
3
m
/
J 5
M
[
Ki 0
-5
K
1
-10
0 100 200 300 400 500 600
Temperature [K]
Fig. 2. Calculatedanisotropyconstants K and K (solid lines)as a functionof temperatureobtainedusing
1 2
thesameparametersasinFig.1(dashedlines)butwiththehalfvalueofH .
ex
andthe parameters obtained here canprovide anappropriate characterization for K (T)and K (T)at
1 2
thequantitative levelatleastabove100K.
Next we proceed to examine how K (T) and K (T) are affected by the variation of amplitude of
1 2
H (T). Figure 2 shows the calculated K (T) and K (T) using the half value of H with the other
ex 1 2 ex
parameters unchanged. The results in Fig. 1 are also shown for comparison. One can see a steep
reductioninbothK (T)andK (T)nearatemperatureof70K.Thisinturnleadstotheresultthat,for
1 2
temperaturesabove200K,both K andK valuesbecomesmallerthanthoseinFig.1.ToseetheH
1 2 ex
dependence of K and K values, we plot in Fig. 3 these values as a function of H at T = 300 K.
1 2 ex
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300K
12 K
2
]
3
m
8
J/
M
[
Ki 4 K
1
0
-4
0 200 400 600 800 1000
Hex [K]
Fig. 3. H dependenceof the anisotropyconstants K and K at T = 300 K. The CEF parametersare the
ex 1 2
sameasinFig.1.
Inthesecalculations, weneglect theFecontribution KFe(T)tofocusononlytheeffectsof H onthe
1 ex
4f-related anisotropy constants. Onefindsthat K exhibitsamonotonic increasewith H whereas K
2 ex 1
hasapeakataround H = 300Kandgoesintothenegativeregionabove H = 800K.Thenegative
ex ex
valueofK forthehigh-H regionisnaturally understood byconsidering thefactthat K startsfrom
1 ex 1
anegativevalueatT = 0;thatis,thehighlimitof H effectivelycorresponds tothelow-temperature
ex
limit. Further attention should be paid tothe low-H region below H = 300 K,where both K and
ex ex 1
K values degrade with decreasing H . This is because the 4f electron cloud approaches a spherical
2 ex
one, in accordance with the decrease of H , resulting in an insensitivity to the CEF and a decrease
ex
of the anisotropy energy. From this behavior, one should recognize that the temperature dependence
oftheanisotropy constants K and K comesnotonlyfromthethermalfluctuation ofthe4fmoments
1 2
but also from the strength of H (T) through its temperature dependence as hS i. This can explain
ex Fe
the vanishing of K , K , and hence H at the Curie temperature. Further, even at lower temperature,
1 2 c
the anisotropy constants may become small around the grain boundaries where the exchange fields
becomesmallowingtothesmallercoordination numberortosomeotherfactors suchasstresses and
lattice defects. Forthese reasons, ataround the operating temperature of electric vehicle motors (500
K),thedecrease intheamplitudeof H inaddition tothereduction ofhS iowingtoitstemperature
ex Fe
dependence maysignificantly influencemagnetperformance.
Finally, we investigate the effects of variation of the CEF parameters on K (T) and K (T). As
1 2
pointed out in our previous work,15) there may be a possibility that A0 exhibits a negative value at
2
the grain surfaces, depending on whether their characteristics differ from those of the bulk. Figure
4 shows the behaviors of K (T) and K (T) using the parameters A0 = −820 K/a 2, A0 = A0 = 0,
1 2 2 0 4 6
and H = 364 K. Clearly seen is that K (T) exhibits a negative value for any temperature and that
ex 1
K (T) vanishes at temperatures above 200 K, which implies that the system favors planar anisotropy
2
above200 K.Thustheprediction of K < 0when A0 < 0reported intheprevious study isjustified in
2
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500
400
] 300
3
m
J/ 200
M
K[i 100 K2
0
K
1
-100
0 100 200 300 400 500 600
Temperature [K]
Fig. 4. Calculated anisotropy constants K and K as a function of temperature obtained using A0 =
1 2 2
−820K/a 2,A0 = A0 =0,andH =364K.
0 4 6 ex
the room-temperature region, from which Mitsumata et al.17) suggested that H drastically degrades
c
owingtothenegative K atthesurface.
To conclude this paper, we summarize our study as follows. To identify the possible mechanism
of coercivity (H ) degradation of Nd-Fe-B sintered magnets, we have investigated how the magnetic
c
anisotropy is affected by a variation of exchange fields and the CEF acting on the 4f electrons. To
this aim, the temperature-dependent anisotropy constants K (T) and K (T) were calculated based
1 2
on crystal field theory. Using a certain set of CEF parameters and exchange field strength, we can
obtain K (T) and K (T) curves that reproduce well the experimental data for bulk Nd Fe B. It was
1 2 2 14
found that, with decreasing exchange field strength, both values decrease as a result of the lower
asphericity of the 4f electron cloud. This feature makes K (T), K (T), and hence H vanish at the
1 2 c
Curie temperature, where the exchange field strength tends to zero. Even at a lower temperature, the
anisotropyconstantsmaybecomesmallaroundthegrainboundarieswheretheexchangefieldsbecome
smallowingtothesmallercoordinationnumber.Forthesereasons,ataroundtheoperatingtemperature
of electric vehicle motors (500 K), the decrease in the amplitude of H in addition to the reduction
ex
of hS i owing to its temperature dependence may significantly influence magnet performance. It is
Fe
alsoconfirmedthatthenegativevalueofA0,theleadingCEFparameter, resultsinaplanaranisotropy
2
at room temperature. Thus the prediction of K < 0 when A0 < 0 reported in the previous study is
2
justifiedattheroom-temperature region.
Acknowledgment ThisworkwassupportedbyJST–CREST.
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ex
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