Theory of surface anisotropy and coercivity in ferrimagnetic γ-Fe2O3

Journal of Magnetism and Magnetic Materials 117 (1992) 368-378 North-Holland

Theory of surface anisotropy and coercivity in ferrimagnetic /-Fe203 J.C. S l o n c z e w s k i IBM Research Dit'ision, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA

Received 17 February 1992

We propose a quantum mechanism for surface magnetic anisotropy in a special class of spinel ferrite. One-ion anisotropy arises from the orbitally degenerate state of a 3de-electron bound to an Fe 3+ core on a spinel B-site near a (ll0)-facet. Dependence of electron energy on the orientation of the B-site trigonal symmetry axis with respect to the facet plane leads to a maximum in surface anisotropy versus surface charge. This explains published experimental dependences of the coercive field on the amount of polymeric (NaPO3)n.Na20 deposited onto chemically reduced acicular particles of ~-Fe20 3. The experimental magnitudes of coercivity change are understood if individual polyphosphate molecules oxidize such anisotropic subsurface sites with considerable quantum efficiency.

I. Introduction This article proposes a microscopic m e c h a n i s m for surface m a g n e t i c anisotropy in a ferrimagnetic spinel, such as partially r e d u c e d ~/-Fe20 3, which contains a m o d e s t a m o u n t of Fe 2+. T h e anisotropy is attributed to s p i n - o r b i t energy of electrons populating 3de orbitals b o u n d to Fe3+-ion cores in a plane lying at a certain distance from the crystal surface. T h e p r e d i c t e d surface anisotropy is limited in m a g n i t u d e by the s p i n orbit coefficient, the level splitting due to a trigonal crystal field, and the n u m b e r of such subsurface electron states allowed by the spinel crystal structure. A postulated d e p e n d e n c e of electron energy on direction of the trigonal axis naturally explains several features of the experimental coercivity changes created by deposition of certain p h o s p h a t e c o m p o u n d s o n t o partially r e d u c e d acicular -y-Fe20 3 particles. O u r m e c h a n i s m explains Correspondence to: Dr. J.C. Slonczewski, IBM Research Cen-

ter, P.O. Box 218, Yorktown Heights, NY 10598, USA. Tel.: + 1-914-945-1219; telefax: + 1-914-945-4506.

the observed magnitudes of coercivity change with some difficulty. A l t h o u g h they were first introduced m o r e than 40 years ago, materials derived from particulate ~ - F e 2 0 3 continue to be p r o m i n e n t a m o n g storage media [1]. T h e coercivity of initially stoichiometric ~/-Fe20 3 particles usually increases u p o n introduction of Fe 2+ by m e a n s of partial chemical reduction, and further u p o n subsequent surface t r e a t m e n t with Co 2+ [1-3]. A n alternative surface t r e a t m e n t of the r e d u c e d ~/-Fe20 3 with n o n - m a g n e t i c phosphates also markedly changes the coercivity [4-7]. It was suggested that surface magnetic anisotropy, which is expressed by a seco n d - d e g r e e spherical harmonic, might cause the coercivity changes induced by p h o s p h a t e treatments [4]. A microscopic m e c h a n i s m combining s p i n orbit and crystal-field energy exists for the contributions o f Fe 2+ and Co 2+ to the cubic volume magneto-crystalline anisotropy, generally represented by f o u r t h - d e g r e e spherical harmonics, in spinel ferrites [8]. It has also b e e n applied to the effect of bulk Co2+-doping on the coercivity in

0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic y-Fe203

~/-Fe203 [9]. However, no microscopic mechanism is known for any of the surface modifications of magnetic properties of 3,-FezO 3. Understanding of surface treatments is thought to be crucial to further advances of magnetic recording using particulate media of various oxide and metallic compositions [10]. The goal of this article is to provide a fundamental basis for this understaning in terms of a quantum theory of surface anisotropy.

2. Quantum theory of one-ion anisotropy The chemical formula for an oxide having the cubic spinel structure is AB204. On spinel B-sites, which have the octahedral oxygen coordination illustrated for the central B-site shown in fig. 1, the principal term in the crystalline electric potential q~(r) satisfies cubic point-group symmetry. A common feature of Fe 2+ (d 6) and Co 2÷ (d 7) on B-sites is the unquenched total orbital angular momentum L present in the ground state of the cubic crystal-field Hamiltonian. This momentum couples on the one hand to the atomic lattice via a secondary term in q~ having a trigonal symmetry axis. On the other hand, spin-orbit energy couples L to S. The combination of these interactions effectively couples S to the lattice, giving rise to unusually strong magnetic anisotropies

[ooQ

[1,]

369

when Fe 2÷ or Co 2+ are present in bulk spinel ferrites [8]. We denote unit vectors of the B-site trigonalsymmetry axes by a 1 = 3 - 1 / 2 ( 1 1 1 ) , a 2 = 3 -1/2 (111), a 3 = 3-1/2(111), a 4 = 3-t/2(111), as shown in fig. 2a. The ground one-electron 3d orbital eigenstates ~bm, with m = - 1, 0, + 1, in the cubic potential of the octahedral B-site constitute a triplet having symmetry e [11]. For example, at any B-site for which a 4 is the trigonal axis, they may be approximated conveniently in terms of free-ion d-orbitals by the formula 6m=(to--myz+tomxz+xy)f(r),

m=0,_

1,

(1) where f ( r ) is a real radial function, and to = exp(2~ri/3) [12]. The half-filled shell configuration 3d 5 of Fe 3+ (S = ~) is an orbital singlet with vanishing orbital-momentum quantum n u m b e r L = 0. Therefore eq. (1) effectively represents as well the total orbital state of the configuration 3d 6 of Fe 2+ (S = 2), which has one additional electron. Equation (1) serves also for the 3d 7 configuration of Co 2+, if we replace L with - L , because its two additional de electrons are equivalent to one hole in the threefold degenerate de subshell. Direct evaluation verifies that the submattrix of L = - i r × V in the triplet (1) is - i where 1 is the conventional 3 × 3 angular momentum matrix for a p-state quantized parallel to a 4 with eigenvalues 12, = 0, + 1. Thus the orbitals described by eq. (1) comprise an effective-p state and the 3 × 3 effective Hamiltonian for Fe z÷ or Co s÷ is written [8,12]. ,,~ =P@tl 2, - AI. S,

(2)

Table 1 Empirical trigonal crystal-field splitting q~t of 3de level for substituents on B-sites of spinel ferrites [8]

•

13 sire

Fig. 1. Environment of the octahedrally coordinated cation B-site in a cubic spinel crystal lattice.

Substituent

qbt / h c [cm - 1l

Host

Measured effect

Fe 2+ Co 2 + Co 2+

1300 1600 630

MnFe204 Fe 304 MnFe204

cubic anisotropy magnetostriction magnetostriction

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic T-ge 2O3

370

with p = + 1. H e r e h is the spin-orbit constant, and we take hF~2+/hc = - 8 0 cm-1 as determined from spin resonance of Fe 2+ impurities on octahedral sites in MgO [11]. (Here h is Planck's constant and c is the velocity of light.) We assume also that a strong exchange field polarizes S in the direction opposite to the spontaneous magnetization: S = - S a where a is the unit magnetization vector. In eq. (2), ~t is the one-electron trigonal crystal-field splitting parameter which has been determined by interpolation of magnetic measurements for Fe 2+ (with p = + 1) or Co 2+ (with p = - 1) substituted into each of three spinel-ferrite compositions, as summarized in table 1. At room temperature, Tr = 298 K, we have kTr/hC = 207 cm -1 > [ AS I / h c = 160 cm -1. Adoption of ~t > kT~, as supported by table 1, would imply that essentially only the ground state I z, = 0 of Fe 2÷ is occupied. The second-order perturbation-theory estimate of the ground-state energy due to the spin-orbit term in eq. (2) gives the one,ion anisotropy wi=r(ai'tlt)2=K

COS2 0 i

(~t>kT, IhSI),

(3) where x = A2S2/q~t, on the four types of octahedral site i = 1, 2, 3, 4. Note the counter-intuitive quantum-mechanical prediction that lowering crystal-field symmetry by increasing ~t decreases one-ion anisotropy in this case. Since ~t is sensitive to crystal structure, particularly to the oxygen u-parameter [13], we may also consider the possibility q~t < - k T , - I AS[. The ground orbital state is now doubly degenerate so, in first approximation, we may replace 1 with its expectation values +ai. From eq. (2), the first-order anisotropic term in the energy is then • ±= _+AS cos 0i. The free energy per ion, generally written w i = - k T In Xn e x p ( - • n / k T ) , becomes

wi = - k T

In[2 cosh(hS cos Oi/kT)]

(rb, < - k T , -

I ;~S I),

(4)

neglecting occupation of the state l~,--0. The anisotropy in this case approaches a quadratic

form in directional cosines only for sufficiently large k T / I A S [ . A more exact version of eq. (4) accounts for the cubic anisotropy of Co 2÷ substitutions in Fe304 and MnFe204 [8,12].

3. Model of magnitic particle We index a 'matchstick' model for the shape of the ~/-Fe20 3 particle suggested by one observation [14] with facet planes (110), (001) and (110). It is illustrated by the rectangular parallelapiped shown in fig. 2b. Electron microscopy and diffraction [15] do indeed indicate that -y particles are single crystals with a nearly spinel structure (we neglect the vacancy ordering). Most are elongated along one face-diagonal axis which we index as (110) in the figure. The particle length ranges between about 0.1 and 1 i~m. The mean length-to-width ratio is typically between 6 : 1 and 8 : 1. Unlike our rectangular parallelepiped, however, the observed shape is very irregular and the cross section is often more typically oval than rectangular [15]. The results of our simple model can apply only approximately to the general acicular shapes actually observed. In attributing a coercivity contribution to surface anisotropy, we may neglect the small (110) facets at each end of the particle in fig. 2b. In order to contribute positively to coercivity, the surface anisotropy must favor the (ll0)-axis. By symmetry, the quadratic anisotropy energy of the (001) facets has the uniaxial form const. × Or{001), and therefore cannot qualify. Thus only the (110) facets of the particle need to be considered, because their symmetry implies 3 orthogonal inequivalent principal axes (001), (110) and (1-i0). The phenomenological surface energy may be written in the triaxial for A a ~ l l 0 ) + Ba~]io ), which makes (110) easy if B < 0 and B < A are satisfied. A cubic unit cell of the spinel structure with lattice parameter a(--- 8.4 ~,) contains 8 chemical formula units and therefore 16 B-sites [13]. One can show that all of the B-sites lie on a set of parallel (ll0)-planes distant 2-5/2a from one another. One such plane contains B-sites of the types i = 1 and 2, designated II, whose trigonal

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic ~-Fe20s

(o)

371

~(~10) --)'(1 10)

(b)

.......i~3- ....... :.:.: ::.::.__.. "'-"---;-~i - - .,'" t

-

-

edge of ~.:'~errace ::i: i:l

plone

po.ic,e

oi:: /:il - - -oi!::!i!!(l -

Fig. 2. (a) Geometry of (ll0)-facet plane, local trigonal axes ai(i = 1, 2, 3, 4) of B-sites, and cubic crystalline axes. (b) 'Matchstick' model of ~-Fe203 particle with (110) the long axis.

lqe ,oo II

-

7-Fe203

spinel

axes lie parallel to the (110)-plane (see fig. 2a). T h e next B-site plane contains types i = 3 and 4, designated/__, whose trigonal axes are inclined at an angle s i n - 1 ( 2 / 3 ) 1 / 2 = 5 4 . 7 ° from the ( l l 0 ) plane. T h e alternation of B-site planes between II and Z types is illustrated by the z-plane projection shown in fig. 3. Figure 3 indicates schematically the relationship o f the B-site planes to the assumedly terraced surface o f one of the particle's (ll0)-facets. (Terraces are observed on surfaces of the related spinel magnetite ( F e 3 0 4 ) by m e a n s of scanningtunneling microscopy [16].) D e p e n d i n g on the pattern o f terraces, some portion of the (110) facet will have a II-plane nearest to the crystal surface and the r e m a i n d e r will have a /.--plane nearest to the surface. O n e partial region o f each type a p p e a r s in fig. 3. W e postulate the existence o f surface-electron states, including some subsurface states b o u n d to Fe 3÷ cores lying on these nearest B-site planes. These subsurface B-sites are indicated in fig. 3 by circles. T h e resulting density of state for stoichiometric ~/-FezO 3 is pictured in fig. 4a. T w o densities are shown: P B ( E ) for bulk states, and p s ( E ) for surface states comprising a n o n - m a g n e t i c b a c k g r o u n d plus the II and Z type subsurface states. In the Fermi statistics, an u n o c c u p i e d state c e n t e r e d on an Fe nucleus represents Fe 3+ and an occupied state Fe z+. All o f these F e - c e n t e r e d states are shown lying above the Fermi energy E F

®

reconstructed FezO onieotropic # II-site --/-.--slte (~(~ subsurface sites Fig. 3. Enlarged projection of B-site positions and their trigonal axes on the (001)-plane for a representative region of the ~t-Fe203-particle model (inset at right) near a terraced (ll0)facet. Circles mark subsurface B-sites which are considered to be locations of anisotropic states for 3d-electrons. They lie on (ll0)-planes, each containing exclusively II-type or /_-type B-sites. A surface-reconstructed layer of composition FezO (~ < z < 1) is indicated. Hypothetical orthogonal position of the chemically bound linear polyphosphate ionic complex is indicated.

(o)

(b)

stolch. 7-Fe20 s

reduced -y- Fe203

bulk

A.

E,J Er

phosphate conc.~

surface

Et •

=:Pb

!

:,,

,

P

o Pi o b~ Ps Fig. 4. Proposed energy-level density in v-Fe203 and its dependence on superficial chemical treatment. (a) Bulk- (PB) and surface-state (#~) densities at a (ll0)-facet for stoichiometric ~-Fe203. Ps includes peaked terms due to /_- and H-type subsurface B-sites and a constant (oh) background. (b) Effects of chemical reduction (stage B) followed by progressive surface charging attributed to phosphate treatment in stages CDEF.

372

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic )l-fe203

because stoichiometric F e 2 0 3 has no Fe 2+. Our substrate states provide anisotropy and thus contribute to coercivity if Fe z+ is introduced by a chemical reduction causing some of the subsurface states to move down below EF, as indicated at stage B in fig. 4b. Since many more states are available in the interior of the particle than on the surface, E F is essentially determined by the degree of reduction in the bulk. We assume that the reduction is sufficient to depress the peaks of Ps (whose nature we explain presently) below E v. Each of the subsurface B-sites contributing to surface anisotropy satisfies the condition that it must be an interior B-site exposed to some degree of electrostatic perturbation by the facet. By 'interior' we mean surrounded by a complete complement of 6 nearest-neighbor O z-, 6 occupied nearest-neighbor A-sites, and 6 occupied nearest-neighbor B-sites as illustrated in fig. 1. Any non-interior B-site, such that some of its near neighbors are severely displaced by surface reconstruction or other irregularities, will have lower crystal-field symmetry than trigonal. As the orbital splitting by crystal fields increases, so does the orbital momentum become quenched and the anisotropy decreases. Thus we picture in fig. 3 a hypothetical reconstructed iron-oxide FezO zone lying between the spinel and the external surface and satisfying ~2 < z _ < 1. Within this zone, the atomic structure is sufficiently different from spinel that it contributes negligibly to magnetic anisotropy. This anisotropy is small either because Fe z+ is absent or because the crystalline electric field of FeZ+-sites has low symmetry. An alternative possibility is considered in the appendix.

4. Surface magnetic anisotropy The aspect of surface anisotropy relevant to coercivity is the barrier against rotational reversal. Given our uniaxial energy expression (3) or (4), and assuming equal occupations of II sites i = 1 and 2 (or L-sites i = 3 and 4), the surface energy of any (ll0)-plane containing B-sites has triaxial magnetic anisotropy. Using eq. (3) or (4) one finds that the axes (110) and (110) are always

opposite extremals of the energy of a given plane. When (110) is the magnetically easy axis, the energy barriers for reversal between [110] and [110] directions lie on the saddle directions [001] and [001]. Therefore the mean contribution to the reversal barrier by one Fe 2÷ ion on any plane is /311,L = wit,z_(001) - WII,L(1~0) where wll = ½(wl + 1 w 2) and wL = 2(w3 + w4), respectively. Thus, for the orbital singlet case qbt > kT, I AS I we find from eq. (3) flZ =--/311=K/3=A2S2/3C19t = (8500/crPt)hc [cm-l]. For t~ t < kT, thermal occupation of the excited doublet state will markedly reduce I/3mZ I- On the other hand, for the orbital doublet case ~t < - k T , - I AS I with T = 300 K we find from eq. (4) /3113oo/hc = 19 cm -1 and flL 3o0/hc = - 20 c m - 1. The assumed relation flit = -/3L. The assumed relation /3it = -/3L holds only approximately in this case. We write U for the total barrier energy due to occupied subsurface states. Using the notation /3 = flit = --/3/-' we have

U= /3 f ~ d E ( P N - p

L),

(s)

PlI,L are the surface densities of the respective types of B-site. It is reasonable to assume, for a given distance of B-site plane from the facet surface, that the crystalline electric field due to the free crystal surface shifts the mean energies of [I and L types by different amounts (formally proportional to the zero-order electric m o n o p o l e coefficient) without substantially changing /3 (which depends through q~t on the second-order electric quadrupole coefficient). We assume for convenience /3 > 0 (orbital doublet ground state) in the remaining discussion because the unique value/3113 oo/hc = 19 c m - 1 (at T = 300 K) makes our predictions more concrete. (If the ground state is the orbital singlet and /3 < 0, the predictions depends on the parameter t~ t and we must assume the peak of PH lies above that of p~_ in fig. 4 to explain the coercive effects of phosphate treatment.) After chemical reduction (stage B in fig. 4b) comes the phosphate treatment itself [4-7]. Briefly put, this usually consists of 1) mix ~/-Fe20 3 particles with sodium polyphosphate (sometimes where

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic T-Fee0 ~

called metaphosphate) (NaPO3) . dissolved in water, 2) dry in reduced air pressure, 3) heat in vacuum to 100-300°C, and 4) then allow to cool in reduced pressure. The result is that the y particles are now thinly coated with a dry glassy phosphate compound. Magnetically, the effect is to change the coercivity [4-7], by as much as 1000 Oe. We assume a local electrostatic potential V which varies in proportion to the phosphate concentration which tends to remove electrons from the subsurface state as indicated by the progressive stages C D E F in fig. 4b. In chemical terms, this binding of the phosphate to the particle facet causes subsurface oxidation of the type FEZ+--+ Fe 3 +. In order to concretely relate U to electrostatic charge, we write p s = p L + P , + P b for the surface-state density illustrated in fig. 4. Here Pb is an assumed constant background density of states (e.g. due to defects, impurities, etc.) without significant magnetic anisotropy which are located in the reconstructed F%O surface layer. It includes Fe 2+ in strongly non-cubic crystal fields as discussed above. The principal predictions of the theory given below are not sensitive to the functional form of the terms in the state density, but depend primarily on the order of the levels and their width relative to their separation. For the B-site anisotropic densities, it is convenient to write normalized Gaussian distributions with variances o-:

PJI

~r(2¢r)'/2

(6) pL =

NL e x p { _ [ ( E _ EL + q V ) 2 / 2 o ' 2 ] } o.(2~.)1/2

(7) where Nil and Nz_ are the total numbers of states with trigonal axes parallel and inclined, respectively, to the (ll0)-plane. Note that the centers E = Ell - q V and E = E L - q V of the B-site densities, where - q is electron charge, shift with the potential V induced by surface treatment. The illustrative numerical parameters used in figs. 4

1.0

i

r

i

373

l

r

!

'-_0.8 ,m

E

kL-

o 0.4 k-

"~-o = 0.2 0

a c,'7 ,-" 1 . . . .

-3

N ~,,E"

-2 -1 surface charge O

0

Fig. 5. Calculated rotational reversal barrier U due to surface anisotropy versus surface charge density attributed to phosphate treatment of reduced ~/-F%O3, assuming q~ < - kT, - I,tSr. Two values of level width ~r are used. Units are arbitrary. Dependence of U on T is proportional to eq. (4). The point A represents the starting stoichiometric ~/-FezO 3. Stages A B C D E F correspond with those of fig. 4.

and 5 are Nz_ =0.8NjI, Pb-----0.2NJAE where AE = E , - E L is the peak separation, and o-= 0.3 A E except for the lower curve in fig. 5 where ~r -- 0.7 A E is used. The electrostatic charge stored in surface states is Ev

Q(EF)=-qL

dE(pL+PH+Pb)"

(8)

After carrying out numerically the integrals in eqs. (5) and (8), we may eliminate V between them to obtain U versus Q as shown in fig. 5 using arbitrary units. We assume the phosphorus concentration [P]cx Q - Q0 where Q0 ( = - 3 units) is the initial charge before phosphate treatment. As [P] increases, V decreases, the surface states rise in energy as indicated by the sequence C D E F in fig. 4 and lose electrons, so Q increases and causes U to vary as shown in fig. 5. The sequence of anisotropy changes marked by the flags A B C D E F in fig. 4 and the upper curve in fig. 5 is as follows: At stage A, for untreated stoichiometric y-F%O3, the Fermi energy of the insulator lies in the band gap well below the bulk conduction band and the anisotropic subsurface levels. We have U = 0 because no electrons are available to populate the anisotropic subsurface B-sites. Following chemi-

J.C. Slonczewski / Surface anisotropy and coerciuity in ferrimagnetic "y-Fe20 3

374

To calculate NIl and thus estimate Umax, w e note that the primitive cubic cell with the lattice parameter a ( = 8.4 ,~) contains 8 spinel chemical-formula units [13]. The (ll0)-planes containing II-sites alternate with those containing L-sites with spacing a/25/2. Any one such plane therefore has Nil + N~ = 2 x 8 a - 3 X a/25/2 = 23/2a-2 B-sites per unit area. The main prediction of first rising, then falling, anisotropy shown in fig. 5 is not sensitive to the functional forms of one-ion anisotropy (3) and (4) and state densities (6) and (7). The important premises are the presence of two signs of the subsurface one-ion anisotropy barrier and that they correlate, by virtue of the 4-way directed trigonal crystallographic deformations of the B-

cal reduction, at stage B, U > 0 because all magnetic surface states are occupied and the number of L-states is smaller (by assumption) than the number of II-states (N~ = 0.8NIl). Between stages B and C, U changes little because charges are leaving only the non-magnetic background states. Between stages C and D, the L-states with/3 L < 0 are emptying, causing U to rise. Between stages D and E, the H-states with/31~ > 0 are emptying so that E falls. Beyond E, only non-magnetic background states are emptying so U does not change further. With increasing (r, the curve U([P]) becomes flatter and the maximum U decreases as indicated by the lower curve for tr = 0.7 in fig. 5. For o--* 0 the maximum U approaches its limit

Um.x=/3N,.

900

-

7(30 800

-/ ]

700

-/,~

0) 60c

o

{I-,~

~J

X

50~

"~ ~

f \

o

T

/~ / \

-

~

o sodium metaph°sphole 700 " sodium fripolyphosphate 600

600 I I

40~

I 2

l 3

[P]fl[Fe]

£ 4

i 5

P205

SOC I I

40O 0

(molar)

i 2

4

8

6

800[--

(f)

900

1600

zC~"& (e)

I~00

o

700

8001

A A

I 2

I 3

I 4

I 5

I 6

a 0.15 9/ml, pH=9, stoich. ~ 0.15 g/ml, pH=9, reduced 0.05 g / m l , pH=9, reduced 0.05 g/ml, pH=7, reduced

600 . -(]) -x

1200

o t

24,

70£ lO00

60C

¢°

5007 < 400

8OO

600

stoich. -

500

(NQPO.~)n

forced 13

~

J.

30o,"---~----°--.......... # I

1"(300

1

4

5

6

7

4O00--.

I 1

I 2

I 3

I 4

5

..................

[(NoP%)o'No20]/[Fe2%] (by weiohO

Fig. 6. Survey of published experimental dependence of coercivity (vertical axis in Oe) for reduced ',/-Fe203 on phosphate concentration. The authors are (a) ltoh and Satou [4], (b) Itoh, Satou and Yamazaki [5], (c)-(e) Itoh and Satou [6], (f) Spada, Berkowitz and Prokey [7]. Concentration ratios (horizontal axis) are molar ratios except in (f) which shows weight ratio of sodium polyphosphate to iron oxide. W a t e r solution of sodium polyphosphate (or melaphosphate) (NaPO3) n with m e a n n -- 8 was used in every case except: (b) where the lower curve is for sodium tripolyphosphate, (c) where the solute is P205, and (f) which reports m e a n n - 14..The polyphosphate or m e t a p h o s p h a t e has straight chains while the tripolyphosphate has a ring structure. In plot (d) the treatment was altered by coating with hydrous (NaPO3) ., which was first preheated at reduced pressure. Plot (e) shows dependence on chemical pH of the phosphate solution. The lower curve in plot (f) shows that the coercivity of stoichiometric ~-Fe203 is unaffected by phosphate treatment.

J.C. Slonczewski / Surface anisotropy and coercicity in ferrimagnetic y-Fe20~

sites, with the level shifts caused by proximity to a crystallite facet. Indeed, the assumption of a rectangular particle cross section, made for the sake of concreteness, is not necessary. Since any facet plane containing the axis ('110), except (001) and (111) for which our anisotropy barrier must vanish by symmetry, will give rise to an anisotropic barrier, our predictions hold qualitatively for a generally shaped acicular crystallite whose long axis is (110).

5. Comparison with experiment For comparison with the theory of fig. 5, fig. 6 shows six plots of experimental coercivity versus the concentration ratio [P] - [Fe] of P in solution to Fe in the particles. Except where indicated otherwise in the figure caption, the phosphorus is in the form of sodium polyphosphate (or metaphosphate) (NaPO3) ~ with mean n -- 8. Figure 6 exhibits considerable experimental variation, due partly to variations of the degree of preliminary chemical reduction, details of the phosphate treatment, and the chemical composition of the phosphate solution [4-7]. The principal variations are indicated briefly in the figure caption. However, the data has certain recurrent features which the theory explains, given the assumptions that surface charge Q changes in proportion to the phosphorus concentration [P] and the H c changes in proportion to the surface-anisotropy barrier U: 1) The initial H~ ( = 450-500 Oe) for [P] = 0, just after chemical reduction, is greater by an amount of 120-170 Oe than for stoichiometric -y-FezO 3 (H~ = 330 Oe). This is connected with our ad hoc assumption N L
375

crystal structure. For if only one sign were present, a strictly monotonic dependence of H c on [P] up to some saturation value would follow. The predicted symmetry of H c versus [P] about the peak is fairly well corroborated by plot (f) and by (e) for some of the pH values used. In these cases, the phosphate molecular ion may attach orthogonally to the facet as indicated in fig. 3. The lack of this symmetry in plots (b) and (d) could arise from non-constant Pb- Another possibility is that the ions tend to lie flat, so that already deposited phosphate tends to obstruct further deposition, thus causing the charging rate and IdHc/d[P]l to decrease at large [P]. This would raise the right shoulder of the peak, as observed in (b) and (d). 3) From the particle dimensions one estimates that the peak concentration of = 0.4 g polyphosphate per gram -y-Fe20 3 appearing in fig. 6f is equivalent to -~ 5 molecules of (NaPO3)14 • N a 2 0 per anisotropic subsurface B-site, whose surface density on (ll0)-facets is 23/2a-2. (We assume equal mean coverage of (110)- and (001)-particle facets with phosphate.) If we momentarily abandon the matchstick model and base the site density on the physically measured specific surface area of 26 m2/g [18], we find, instead --1.5 molecules per subsurface B-site. The substantial excess of these estimates (5 or 1.5) to the expected -- 21 for the fraction of L-type subsurface sites would indicate that the 'quantum' efficiency is substantially less than 1. 4) Plots (b), (e) and (f) in fig. 6 show that, for sufficient [P], H c takes on values even lower than that at [P] = 0. In particular, plot (f) includes two experimental points having H~ = 330 and 340 Oe when reduced ~/-Fe20 3 is treated with large values of [P]. These correspond approximately to the H e = 330 Oe of stoichiometric ~/-Fe203, which is not affected by phosphate treatment as shown by the lower curve in plot (f). Since our model attributes the increase of Hc upon chemical reduction to filling of all the anisotropic subsurface states, it explains naturally this effect through the ultimate emptying by the phosphate of all the anisotropic surface states. The Hc of the stoiChiometric 3,-Fe203 is restored at large [P] ~even though its interior is presumably still in a chemi-

376

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic 'y-Fe20 3

cally reduced state. This agreement supports our tacit neglect of the coercive effect of Fe 2+ in the interior of the particle, where the quadratic oneion anisotropic cancel by virtue of cubic crystal symmetry. 5) When the phosphate is entirely removed from the phosphate-treated particles by rinsing with water and drying, H c returns to the original value ( = 5 0 0 Oe) of the reduced -y-Fe20 3 [7]. This agrees with our model, in which removal of the phosphate simply restores all of the surface charges to the initial condition at [P] = 0, since the shifting of energy levels by phosphate treatment does not involve irreversible changes in atomic structure of the crystallites. 6) Plots (c) and (f) show indications of an initial delay in the increase of H c with [P] which could be associated with the theoretical plateau between stages B and C in fig. 5. In this range of Q, only magnetically isotropic levels are being depopulated so that U is not changing. The extent of this effect will vary the mathematical parameters of the model. 7) The optimal experimental degree of reduction to maximize H c is observed to be F e 2 + / F e 3+--0.10-0.15 [5], which we express as x - 0.2 in the formula (Fe 3 + )[Fex2 + Fe~5-2x)/3 3+ [] ~ 1 - x ) / 3 ] O 4 - 2 for partially reduced -,/-Fe20 3, where ( ) indicates A- and [ ] indicates B-sites, and where [] represents B-site vacancies. This peak is reminiscent of that of cubic magnetocrystalline anisotropy K 1 versus concentration of Fe 2÷ in the mixed ferrite 3+ ( M n l2_+ x F e x3 + )[Fe2_xFe ~2 + ]O 42 - . The positive contribution of Fe 2+ to K 1 was attributed to the same effective Hamiltonian (2), though with • t/hc = 1300 cm -1 > kT, I AS I [8]. Experimentally, however, Fe 2+ begins to contribute negatively for x > 0.2. The reason suggested for the elimination of the positive contribution was that, when x is sufficiently great, electrons resonate quantum-mechanically from one B-site to another at high frequency, creating conventional energy bands. Formation of bands further quenches L, invalidating eq. (2) and diminishing anisotropy from the prediction based on eq. (2). One may cite the d-band width of about 2 × 104hc [cm -1] in Fe304 which one can infer from results of an ab initio band calculation [17]. The effectiveness

of this quenching mechanism is indicated by comparing this band width with the maximum spinorbit splitting 2S I A(Fe2+)l/hc = 320 cm-1. Some very recent work with smaller particle size [18] yields results similar to fig. 6f for the effect of polyphosphate treatment. However, the preliminary chemical reduction of ~-Fe20 3 has a much smaller effect on H e than observed in the works cited above [4-7]. This result can be reconciled with the present theory by taking Nun= Nz_, meaning that the areas of the two kinds of growth terraces were nearly equal. Thus it appears that the ratio of these areas may vary to some extent from one experiment to another. Any absolute prediction of coercivity increase A H c will be uncertain, first because the influence of surface anisotropy on coercivity is not understood, and second because the particles are very irregular and variable in size [15]. A simple, generous way of estimating an upper bound of A H c is to assume that during remagnetization an amount of energy equal to the barrier energy U is dissipated. Thus we balance the increase of irreversible switching work of a complete hysteresis loop in one particle (two full reversals) J = 2 × 2UA, where A is the area of one (ll0)-facet, against the area increase of a rectangular hysteresis loop K = 4MsAHcAW, where W is the particle width and M S ( = 440 e m u / c m 3) is the saturation magnetization per unit magnetic volume. We may estimate the maximum anisotropy barrier (stage D in fig. 5) with U =/3 x 23/2a-2/2, where the ~1 factor assumes sharp levels (o-= 0 ) a n d Nil = N~_. Thus we have from J > K the bound AH c < U/MsW= 21/2/3/MsWa 2. In this way we find from eq. (5) and/3113oo/hc = 19 cm -1 for the doublet ground-state case q~t < - k T , - bASI the result AHc < 170 Oe. This bound is low compared to the maximum phosphate-induced increases found by Itoh et al. to range between 180 and 1000 Oe, as recorded in figs. 6 a - e [4-6]. However it is in better accord with the 340 Oe increase (over the stoichiometric-~ case) observed by Spada et al., which is seen in fig. 6f [7]. If we took the singlet ground-state case q~t > kT, IAS[ in order to be consistent with table 1, along with the assumption Eli > EL_, then we could just barely match the experimental A H c = 1000

J.C. Slonczewski / Surface anisotropy and coercivity in ferrimagnetic ~-Fe 2O3

Oe with a value of ~t//hc = 80 cm-1 which would not satisfy the condition 4~t > kT = 210 hc [cm-l]. A more accurate formula than our/3 z_ = A2S2/3~t would be needed. Even though this possibility requires tuning of (~t to a small value, it cannot presently be ruled out. The question of the sign of (J~t might be settled by directly measuring the surface anisotropy (e.g. using torque curves in (110) single-crystal films) versus T and comparing the result with alternative accurate analytical formulae roughly expressed by eqs. (3) and (4). F o r the doublet case with optimum phosphate treatment we predict U = 0.7 e r g / c m 3 at T = 300 K, and an order of magnitude greater at T = 0. Any of our theoretical estimates of A H c is an upper bound because nothing guarantees that all of the barrier energy is dissipated during reversal. Indeed micromagnetic modeling (without surface anisotropy)'predicts that inhomogeneous demagnetization causes the reversal to progress gradually from both ends of the particle toward the middle [19]. This gradual progression would imply a kind of wall motion giving rise to less coercivity change than estimated by our procedure. Thus it is not easy to reconcile the foregoing theory with the larger experimental Hc-peaks reported. One possible way is suggested by the fact that the chemically measured surface area of 3,-Fe20 3 powder is generally much greater than expected from typical particle dimensions in powders observed microscopically [18]. Suppose, then, that the number of smaller-than-typical particles is very large, or if each particle has cavities or crevasses which the phosphate can enter. Then the value of W appropriate to our equations is really smaller than that observed in a microscope, giving us a proportionally larger prediction of A H c. In particular, let us suppose that the quantum efficiency of phosphate molecules per subsurface B-site is unity. Then our work inequality J >__K becomes / 3 N m _> 4 # s A H c where N m is the number of polyphosphate molecules deposited on one gram of particle, and /z s is its magnetization per gram. Then the assumption /3/hc = 19 cm -1 predicts A H c < 7000 Oe, which is consistent with experiment even if a large fraction of the molecules are wasted by binding to (001)-, (111)-,

377

or other less effective facets, or by not reaching the magnetic particle at all. We imagine here that at the peak composition one end of each negatively charged linear polyphosphate complex ion adhering to a (ll0)facet binds orthogo~ally to the particle surface, as indicated in fig. 3, in such a way as to drive a single electron out of one /__-type subsurface site. These estimates support the proposal, based on chemical and spectroscopic investigation, that one iron ion is bonded to the oxygen ion at the main chain terminal of (NaPO3) n [6]. This would explain why sodium tripolyphosphate, whose ring structure has no ends, produces the weaker effect shown by the lower curve in fig. 6b [6]. The moderate effectiveness of monomeric P205 is attributed to the subsequent formation of chain-like (HPO3) n [6].

Acknowledgements It is a pleasure to acknowledge the stimulus of a visit to the Center for Magnetic Recording Research at the University of California in San Diego during February of 1989, when this research was begun. The author is very grateful to A.E. Berkowitz for drawing his attention to the problem addressed in this article. He also heartily thanks A.E. Berkowitz, H.N. Bertram, F.E. Spada, F.T. Parker, M. Carey and C. Sommers for helpful discussions and access to research results prior to publication.

Appendix. An alternative model We remark on the alternative, at first sight more natural, possibility that the Fe 2+ ions bearing the surface anisotropy are subject to a noncubic crystalline-electric field perturbation due to proximity of the crystallite surface instead of the trigonal field inherent to the spinel structure. This condition could exist for Fe 2+ located within our postulated surface-reconstructed FezO region lying between the spinel and the particle surface which is indicated in fig. 3.

378

J.C. Slonczewski / Surface anisotropy and coercicity in ferrimagnetic y-Fe 20 ~

In such a crystalline-electric field of general symmetry (Hamiltonian ~e~gS) all orbital degeneracy is removed. The three crystal-field de-eigenstates on an octahedrally coordinated site satisfy ~fg.~I k ) = VkLk) and their wave functions can generally be written

I k) = (Ckxyz + Ckyxz + Ckzxy)f(r ),

(A.I)

where the subscripts x, y, z are 'indices' which identify the basic d~ orbitals and k takes on 'index values' x', y', z'. The real transformation coefficient matrix Ckj effectively constitutes a real orthogonal coordinate-axis transformation between the original cubic axes xyz and the new x'y'z'. We choose to order the energy eigenvalues according to Vx, >_ Vy, > Vz,. In order to treat the spin-orbit perturbation ASa.L we calculate the matrix elements of L = - ir × V. In the new frame, the only non-vanishing ones, apart from adjoints, w o r k o u t to ( x ' r L y , 1 2 " ) = (y'[Lz, J x ' ) = ( z ' J L x , l y ' ) = - i . To secondorder perturbation theory, one finds for the d ~ ground-state wave function I z ' ) (with S = 2) the triaxial one-ion anisotropy: Wgs=-l~2S2[(Vy,-

Vz,) -1

2

-1

2

(A.2) We see that the new coordinate axes k = Ckxx + Ckyy + C~zz, with k = x ' , y', z' are the principal axes of anisotropy, with x' the easy axis and z ' the hard axis. Note the important counter-intuitive property, due to quantization as in the case of eq. (3), that decreasing the degree of crystal-field symmetry by increasing the energy denominators, V~,y,- Vz, decreases the magnitude of one-ion anisotropy coefficients. It could happen that the easy axis x ' of WgS is along the particle length parallel to spinel axis (110), while the barrier (saddle) axis y ' is (001), so that the anisotropy-barrier height would be ~ r e c . = Wgs(~y' = 1) -- Wgs(~l' x, = 1)

(A.3) For example, an epitaxially formed surface layer

of FeO whose rocksalt structure is strained by lattice mismatch would have the correct symmetry for this description. Since this formula is similar to our/3 L = A2S2/3~t in section 4, this model would not be more difficult to reconcile with the large magnitude of the phosphate effect on H c. However, no natural explanation of the very existence of a maximum would be forthcoming. For this reason we prefer the foregoing model of subsurface spinel B-sites whose variously directed local trigonal axes make possible level shifts which correlate with signs of one-ion anisotropy and thus make possible the maximum.

References [1] See review by A.E. Berkowitz, IEEE Trans. Magn. MAG-22 (1986) 466. [2] A.R. Corradi et al., in: Ferrites: Proc. of the Int. Conf. (September-October 1980) Japan, p. 526. [3] A.E. Berkowitz, F.E. Parker, E.L. Hall and G. Podolsky, IEEE Trans. Magn. MAG-24 (1988) 2871. [4] F. Itoh and M. Satou, Jpn. J. Appl. Phys. 14 (1975) 2091. [5] F. Itoh, M. Satou and Y. Yamazaki, IEEE Trans. Magn. MAG-13 (1977) 1385. [6] F. ltoh and M. Satou, Nippon Kagaku Kaishi 8 (1982) 1281 (in Japanese). [7] F.E. Spada, A.E. Berkowitz and N.T. Prokey, J. Appl. Phys. 69 (1991) 4475. [8] J.C. Slonczewski, J. Appl. Phys. 32 (1961) 253S. [9] D. Khalafalla and A.H. Morrish, J. Appl. Phys. 43 (1972) 624, [10] W.M. Mularie and M.P. Sharrock, J. Appl. Phys. 69 (1991) 4938. [11] William Low, Paramagnetic Resonance is Solids (Academic Press, New York and London, 1960) chap. 13. [12] J.C. Slonczewski, Phys. Rev. 110 (1958) 1341. [13] J. Smit and H.P.J. Wijn, Ferrites (John Wiley, New York and Eindhoven, 1959) p. 137. [14] Y. Fukumoto, K. Matsumoto and Y. Matsui, J. Appl. Phys. 69 (1991) 4469. [15] E.L Hall and A.E. Berkowitz, J. Mater. Res. 1 (1986) 836. [16] R. Wiesendanger, I.V. Shvets, D. Biirgler, G. Tarrach, H.-J. Giintherodt and J.M.D. Coey, Z. Phys. B 86 (1992) 1. [17] M. Penicaud, B. Siberchicot, C.B. Sommers and J. Kiibier, J. Magn. Magn. Mater. 103 (1992) 212. [18] F.E. Spada and A.E. Berkowitz, private communication. [19] M.E. Schabes and H.N. Bertram, J. Appl. Phys. 64 (1988) 5832.