Charge screening and magnetic anisotropy in metallic rare-earth systems

7 Pages·0.13 MB·English

by  V. Yu. Irkhin

Charge screening and magnetic anisotropy in metallic rare-earth systems
∗
V.Yu.Irkhin and Yu.P.Irkhin
Institute of Metal Physics, 620219 Ekaterinburg, Russia
Weconsiderthemagneticionatthepointr=0inthe
The calculation of magnetic anisotropy constants is per-
crystal field of the surrounding charges. The spherically
formedbeyondthepointchargemodelforacontinuouscharge
symmetric potentialV , whichis inducedby a centerat
cf
densitydistributionofscreeningconductionelectrons. Anim-
the pointr=Randoriginatesfromthe pointchargeQ
0
portantroleofthenon-uniformelectrondensity,inparticular,
and conduction electron charge density Z(r R), has
of the Friedel oscillations, in the formation of crystal field is | − |
the form
demonstrated. Such effects can modify strongly the effective
8 ion (impurity) charge and even change its sign. This enables Q +Q (r R)
9 one to justify the anion model, which is often used at dis- Vcf(r)= 0 rel R| − | (1)
9 cussing experimental data on hydrogen-containing systems. | − |
1
Possibleapplications tothepurerare-earthmetalsandRCo5 where

n compoundsarediscussed. Thedeformationofmagneticstruc-
a ture near the interstitial positive muon owing to the strong r
J local anisotropy, and the corresponding contribution to the Qel(r)=4πZ0 ρ2dρZ(ρ) (2)
6 dipole field at themuon are considered.
2 is the conduction electron charge inside the sphere with
the radius r, Q′ (r) = 4πr2Z(r), the system of units
1 el
where the electron charge e = 1 is used. At large r
v −
complete screening take place, so that Q ( )= Q .
9 el 0
The old problem of strong magnetic anisotropy in the ∞ −
6 The anisotropyconstants are determined from the an-
rare-earth-based intermetallic systems [1], which has a
2 gle dependence of the energy of the magnetic ions in the
great practical importance, is up to now extensively in-
1 crystal field
0 vestigated. Recently the anisotropy induced by intersti-
8 tialhydrogenatoms[2,3]andpositivemuons[4]hasbeen δ = K cos2θ+... (3)
cf 1
9 discussed. E −
t/ It is accepted now that the main mechanism of mag- Firstwediscussthepointchargemodel. Thedependence
a netic anisotropy origin in the rare-earth systems is the (3)iscalculatedwiththeuseoftheexpansioninr,which
m
crystalfieldone. Estimationsoftheanisotropyconstants corresponds to the expansion in small radius of f-shell:
- are usually performed in the point-charge model. At the
d
same time, this model leadsfrequently to difficulties and 1 1 r r2
n = 1+ cosθ+ (3cos2θ 1)+...
o contradictionswithexperimentaldata. Forexample,the r R R(cid:20) R 2R2 − (cid:21)
c calculated anisotropy constant K1 in RCo5 compounds | − | e e
(4)
: turns out to be verylarge andhave an incorrectsign [5].
v
Thus screening of the crystalfield should play animpor-
i
X tant role. This screening is often taken into account by Here θ is the angle between the vectors R and r, which
∗ can be expressed in the spherical coordinate system of
r introducing the effective ion charge Q which can differ e
a the crystal as
considerablyfromthe bare ioncharge. In particular,the
fittedchargeofhydrogenionsfortheRCo -basedsystems
2
turns out to be negative (the anion model [2,3]). Physi- cosθ =cosθcosθR sinθsinθRcos(φ φR) (5)
− −
cal explanation of such situations is not simple. Shield-
where θRe and φR are the polar and azimuthal angles of
ing of crystalfields inionic solidswas treatedin Ref. [6].
the vector R. Substituting (5) into (4) and picking out
Screening by conduction electrons, which form some lo-
the term, which is proportionalto cos2θ, one obtains for
calized levels near the rare earth ion, was discussed in
the point-charge contribution (cf. Refs. [7,3])
the paper [5]. However, these effects turn out to be in-
sufficient for explaining experimental data. Kpc = 3Λ r2 α J(J 1/2)Q e2 (6)
In the present paper we perform calculations of mag- 1 − h fi J − 0
netic anisotropy constants with account of continuous where r2 is the average square of the f-shell radius, J
charge-density distribution in a metal. We shall demon- h fi
is the total angular momentum of rare-earth ions, α is
J
stratethatthemagneticanisotropyisstronglyinfluenced
the Stevens factor,
notonlybythe totalchargeinducedbysurroundingions
orimpuritycentres,butalsobyaconcreteformofscreen- 3cos2θR 1
ing electron density, in particular, by the Friedel oscilla- Λ= − (7)
R3
tions. XR
1
Forthehcplatticewiththeparameterscanda(purerare Then we derive
earth metals) we derive after summation over nearest-
neighbor magnetic ions the standard result (see Ref. [5]) K1/K1pc =Q∗/Q0 (14)
y2 2/3 with the effective charge
Λ=6a−3 16 − 1 2.44a−3( 8/3 y)
(cid:18) (4/3+y2)5/2 − (cid:19)≃ −
p Q∗ =Q +Q (R) 4πR3[Z(R) RZ′(R)] (15)
(8) 0 el − 3 −
where y = c/a 1.57 1.59 for heavy rare earths [9]. Thus we have obtained the expression for the effective
≃ ÷ ∗
Thesmallgeometricalfactor 8/3 y =1.633 y 0.05 charge Q , which should be used at calculating the ob-
occurs due to that the contrpibutio−ns from the−ma∼gnetic servable anisotropy constant K1. We see that the ratio
ions in the same plane and in neighbor planes have op- (14)dependsexplicitly,besidesthetotalchargeinsidethe
posite signs and almost cancel each other. spherewiththeradiusR,alsoonthechargedensityZ(R)
′
For the RCo compounds, where R = a/√3 for six andthederivativeZ (R).Thelatterquantitycanbelarge
5
neighbor Co ions in the same plane and R = 1√a2+c2 provided that the rare-earth ion lies in the region where
2 thechargedensitychangessharply. Notethatforthecon-
forsixCoionsintheupperandlowerplanes,wehave[5]
stantchargedensityZ (Z′(R)=0,Q (R)= 4πR3Z)we
el 3
Λ=6a−3(cid:18)16(12+y2y−2)15/2 −33/2(cid:19)≃−2a33.4 (9) whaitvheKth1at=tKhe1pcchaanrdgescirneetnhiengspishaebreseinntcr(etahsisesisacsoRnn3e,cbteudt
the quadrupole interaction decreases as R−3).
∗
with y =c/a 0.8, a 5˚A.In this casethe cancellation To obtain the value of Q , one has to investigate
of contributio≃ns from≃different planes does not play a the charge screening for a concrete electronic spectrum,
crucialrole,sothattheexpressioninthebracketsisnever which is, generally speaking, a very difficult task. We
small. discussthe one-centrescreeningproblemwithin asimple
Consider the case where anisotropy is induced by ran- model of free conduction electrons (E = k2/2) in the
dom next-neighbor substitutional or interstitial impuri- impurity-induced rectangular potential well which has
ties. For an impurity in the octahedralinterstitial of the thewidth danddepthE0 =k02/2[8]. Thismodelenables
hcplattice[10]wehaveR=a/√2,cosθR =1/√3.Then oneto calculate the chargedistributionofscreeningcon-
Λ=0fortheidealhcplattice,andtheimpuritycontribu- duction electrons in terms of the scattering phase shifts
tiontoK1vanishes,aswellas(8)(notethatthesituation ηl. The value of k0 should be determined for given kF
isdifferentinthehydrogen-containingRCo systems[3]). and d from the Friedel sum rule
2
However, there occurs the local anisotropy term ∞
2
Q = (2l+1)η (k ) (16)
δ = K cos2θ (10) 0 π l F
Ecf − 1loc Xl=0
with e The phase shifts are calculated as
Kpc =12√2 r2 α J(J 1/2)Q e2/a3. (11)
1loc h fi J − 0 kdj1(kd) β0j0(kd) j1(kd)
tanη (k)= − , β =kd (17)
0 0
Note that for the positive charges Q0 the signs of K1loc kdn1(kd)−β0n0(kd) j0(ked)
and K coincide. e
1 kdjl−1(kd) βljl(kd) jl−e1(kd)
Now we treat the case of a continuous charge distri- tanηl>0(k)= − , βl =kd (18)
bution of the screening conduction electrons. Similar to kdnl−1(kd)−βlnl(kd) jl(ked)
e
(4), the potential (1) can be also expanded at small r.
where j(x) and n (x) are the spherical Bessel andeNeu-
Performing the expansion of the integral (2) in l l
mann functions, k2 = k2 +k2. The disturbance of the
0
r R R= rcosθ+(r2/2R)sin2θ+... charge density is given by
e
| − |− −
∞
we obtain up to r2 e e
δZ(r)= kdkδρ(k,r) (19)
−Z
0
Q (r R)=Q (R) 4πR2Z(R)rcosθ+2πRr2
el el
| − | − For r > d one obtains
Z(R)+[Z(R)+RZe′(R)]cos2θ (12)
×{ }
δρ(k,r)/ρ (k)= (2l+1) [n2(kr) j2(kr)]sin2η
Taking into account the expansion (4) we have e 0 { l − l l
Xl
Q (r R) Q (R) r r2 jl(kr)nl(kr)sin2ηl (20)
el | − | = el [1 cosθ+ (3cos2θ 1)] − }
|r−R| R − R 2R2 − with ρ0(k) = k/π2. For r < d, joining of the wavefunc-
2π[Z(R)(2rRcosθ r2sin2θ) eRZ′(R)r2cos2eθ] (13) tions at the boundary of the potential well r = d yields
− − −
e e e
2
δρ(k,r)/ρ (k)= (2l+1)[j (kd)cosη Table 1. The total angular momenta J, Stevens fac-
0 l l
Xl torsαJ,averagesquaresofthef-shellradiushrf2i(atomic
n (kd)sinη ]2[j (kr)/j (kd)]2 1 (21) units) and c/a ratios for rare-earthelements; the experi-
l l l l
− − mentalvaluesofK (K/rare-earthion)forthepureheavy
1
The parameter d should bee deteremined by the geom- rare earths [1] and RCo systems (the contribution of
5+x
etry of the lattice near the impurity. In Ref. [8], where rare-earth sublattice [11]), and the corresponding calcu-
impurities in the Ag host were considered, d was chosen latedvaluesofQ∗.ThevaluesofKexp forErandTmare
1
to be equal to the Wigner-Seitz radius, so that kFd=2. obtained from the anisotropy of magnetic susceptibility
Inthe casewhere the impurity is localizedinanintersti- (see Ref. [12]). The values of the local anisotropy con-
tial the choice of d may be different. stant K for an impurity in the octahedral interstitial
1loc
We have performed the calculations for kFd = 2 and of a rare-earthmetal are calculated for Q∗ =1.
kFd=3.AtQ0 =1Eq.(16)yieldsk0d=1.46andk0d= R Tb Dy Ho Er Tm
1.235 respectively. With increasing kF, the number of l J 6 15/2 8 15/2 6
values to be taken into account in the sums grows; for α 100 -1.0 -0.63 -0.22 0.25 1.01
J
kFd = 3 the contributions up to l = 3 are appreciable. r2× 0.76 0.73 0.69 0.67 0.64
The results are presented in Figs.1-3. Note that a weak h fi
c/a 1.58 1.57 1.57 1.57 1.57
singularity occurs at the joining point r=d. Kexp -123 -114 -50 58 100
1
One can see that at R<d, except for the case of very ∗
Q 1.3 1.1 1.2 1.45 1
∗
small R where Q (R) slightly decreases, the derivative Kcalc/Q∗ -2100 –2050 -800 800 2000
term results in that Q∗(R) grows (despite an increase 1loc
of Q (R)). For R d, where Z′ is maximum, the RCo5+x TbCo5.1 DyCo5.2 HoCo5.5 ErCo6
| el | ≃ Kexp -96 -211 -203 80
non-uniformdistributionofelectrondensityleadstothat 1
∗
∗ Q -0.02 -0.03 -0.09 -0.04
the effective ion charge Q is positive and exceeds con-
Another interesting example is the case of imbed-
siderably its bare value Q (Q = 1 in our case). At
0 0
ded positive muons [10,13] which induce a strong lo-
the same time, with further increasing R the situation
′ cal anisotropy. This leads to strong deformation of the
changes drastically: Z decreases and becomes negative,
magnetic structure near the muon and to a considerable
so that “overscreening”of the ion charge occurs. Such a
contribution to the dipole field at the point of its loca-
situation corresponds to the “anion model”. In the sim-
∗
tion [14,4]. According to Table 1, for Q 1 the local
plemodelunderco∗nsideration,thevaluesabout−0.5can anisotropy constants make up about 103K∼and are large
be obtained for Q (the value of -1 was assumed for the
∗ in comparisonwith the values of the molecular field act-
hydrogenionsinRef.[3]). AtlargedistancesQ tendsto
ing on the rare-earthions, λ 100K.Therefore a strong
zero, but considerable oscillations of the effective charge ∼
cast of magnetic moments should take place in the limit
sign take place, which attenuate rather slowly. One can
of large K . In the case K > 0, K > 0 for the
see from Fig.2 that the contribution of the derivative 1loc 1 1loc
ferromagnetic ordering along the z-axis we have for the
term in (15) predominates at large R too. Of course,
dipole field
more complicated models are required for a regular lat-
tice ofscreenedcharges,buttheconclusionaboutanim-
∆Bz =4√3M/a3 25kG (22)
portant role of the non-uniform distribution of electron dip ∼
charge should hold also in that case.
where M J(J +1)µ is the magnetic moment of
The effects of screening under consideration can be ≃ B
important for the values of the anisotropy constants in rare-earth ionp. In the case K1 < 0, K1loc < 0 the direc-
tion and value of the dipole field depend on the relation
both stoichiometric rare-earth compounds and systems
between K and λ (see Ref. [4]), but ∆B has the
containing hydrogen and other impurities. The experi- 1 | dip|
same order of magnitude. Thus the local distortion of
mental data onthe firstmagnetic anisotropyconstantof
the magneticstructure yieldsthe contributionto ∆B ,
the heavy rare-earth metals and corresponding RCo dip
5+x
which is of the same order as the sum over the regular
compounds at low temperatures are presented in Table
∗ lattice (cf. the calculation for holmium [15]; note that
1. ThevaluesoftheeffectivechargeQ arecalculatedby
∗ the nearest-neighbor ions give zero contribution to this
using(14),(6). Onecanseethatthe valuesofQ forthe
sumin the absence ofthe distortion). Phenomenawhich
pure rare earths are of order of unity. At the same time,
occur in ferromagnetic and helical phases at intermedi-
theeffectivechargesoftheCoionsintheRCo systems
5+x
ate values of K with changing the relation between
turn out to be negative and very small (a similar situa- 1loc
K , K andλ(e.g.,withthechangeoftemperatureor
tion takes place in the case where R is a light rare earth 1loc 1
external magnetic field) are discussed in Ref. [4].
[5]), sothat a weak“overscreening”occurs. Note thatin
To conclude, the effects of screening of ion (impu-
facttheeffectivechargesmaybedifferentfortwokindsof
rity) charge by conduction electrons turn out to be very
theCoionswhichcontribute(9),sincethecorresponding
strong: the anisotropy constants can be strongly modi-
values of R are different. Besides that, a possible role of
fied and even change their signs. More quantitative in-
theneighborrare-earthionsinthecrystalfieldformation
vestigations with account of a real electronic structure,
should be considered in such a situation.
3
multi-centereffectsandnon-sphericalcharge-densitydis-
tribution seem to be important for the problem of mag-
netic anisotropy in the systems under consideration.
The research described was supported in part by the
Grant N 96-02-16999-afromthe Russian Basic Research
Foundation.
Figure captions
Fig.1. The distance dependence of the function
4πR3δZ(R)fork d=2(solidline)andk d=3(dashed
F F
line).
Fig.2. The dependence 4πR4δZ′(R) for the same pa-
rameter values as in Fig.1.
Fig.3. The distance dependence ofthe effective charge
∗
Q (R) for the same parameter values as in Fig.1.
∗ Electronic address: [email protected]
[1] Yu.P.Irkhin,Usp.Fiz.Nauk, 151, 321 (1988).
[2] W.E. Wallace et al, J.Appl.Phys.49(3), 1486 (1978).
[3] N.V.Mushnikov,V.S. Gaviko, A.V. Korolyov, and N.K.
Zaikov,J.Alloys and Compounds 218, 165 (1993).
[4] Yu.P.Irkhinand V.Yu.Irkhin,to bepublished.
[5] Yu.P. Irkhin, E.I. Zabolotsky, E.V. Rosenfeld, and V.P.
Karpenko,Fiz.Tverd.Tela15,2963(1973); Yu.P.Irkhin,
Physica B86-88, 1434 (1977).
[6] R.M. Sternheimer, Phys.Rev.127, 812 (1962); 146, 140
(1966); R.M. Sternheimer, M. Blume, and R.F. Peierls,
Phys.Rev.173, 376 (1968).
[7] T.Kasuya,In: Magnetism,Vol.2B,AcademicPress,New
York,1966, p.212.
[8] E. Daniel, J.de Phys.et Radium, 20, 769 (1959).
[9] K.N.R. Taylor and M.I. Darby, Physics of Rare Earth
Solids, Chapman and Hall, London, 1972.
[10] H. Kronmueller, H.R. Hilzinger, P. Monachesi, and A.
Seeger, Appl.Phys.18, 183 (1979).
[11] A.S.Ermolenko,Fiz.MetallovMetalloved.53,706(1982);
A.S. Ermolenko and A.F. Rozhda, Fiz.Metallov Met-
alloved.54, 697 (1982); 55, 267 (1983).
[12] B.Coqblin,TheElectronic Structure ofRare-Earth Met-
als and Alloys: the Magnetic Heavy Rare-Earths, Aca-
demic Press, 1977.
[13] W. Hofmann et al, Phys.Lett.A65, 343 (1978).
[14] I.A.Campbell, J. dePhys.Lett.45, L27 (1984).
[15] I.A. Krivosheev, A.A. Nezhivoj, B.A. Nikolsky, A.I.
Ponomarev, V.N. Duginov, V.G. Olshevsky, and V.Yu.
Pomyakushin,Pis’ma ZhETF, 65, 77 (1997).
4
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5
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5
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