Interfacial magnetic anisotropy in nanoscale magnetic multilayers

Interfacial magnetic anisotropy in nanoscale magnetic multilayers

Materlal~ Actenc e and Engineering, B6 (1990) 137-145 137 Interfacial Magnetic Anisotropy in Nanoscale Magnetic Multilayers D J SELLMYER, Z S SHAN a...

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Materlal~ Actenc e and Engineering, B6 (1990) 137-145


Interfacial Magnetic Anisotropy in Nanoscale Magnetic Multilayers D J SELLMYER, Z S SHAN and S S JASWAL Behlen Laborator~ of Physics and Center for Materials Regearc h and Analvszs, Unl~erslt~ oj Nebrast~a, Lmcoln, NE 68588-0111 (UAA) (Recewed February l, 1990)

Abstract Expertmental and theoretical research on magnetic antsotropy of nanoscale multllayers is revtewed The Importance of mterfaces and the occurrence of perpendicular magnettc amsotropy ts emphaszzed Transttton metal multdayers wtth non-magnetlc and rare earth metals are discussed, and slmtlartttes m properttes are poznted out A detatled model of perpendtcular antsotropy due to smgle-ton znteracttons ts outhned 1. Introduction The subject of lnterfacml magnetism and especially of magnetic anisotropy at Interfaces m magnetic multflayers is a relatively new area of condensed matter physics Control of the local atomic environments m artificially-structured muhilayers pernuts the tailoring of magnetic properties of thin films In ways not tmagmed one or two decades ago In additmn to questions revolving magnetism and phase transmons in restricted dimensions, these materials have possible Important technological lmphcatlons, parUcularly in erasable magneto-optic and perpendicular recording materials Both these applications rely on perpendicular magnetic anlsotropy (PMA), and It is on this phenomenon, and the degree to which ~t is understood m magnetzc multdayers, that we shall focus our attention m this paper The understanding of magnetic amsotropy m three-dimensional crystalline sohds ~s still at a fairly rudimentary level Thus it is not surprising that our fundamental knowledge of surface and Interface anlsotroples, at least for quantum mechanical quantitative predictions, is in its infancy In the course of this paper we will refer often to various types of magnetic multilayers It will be helpful to define the structural aspects of some 0921-5107/90/$3 50

ldeahzed, hmmng cases with the help of Figs 1 and 2 Figure l(a) exhibits a coherent layered structure or, equwalently, a single-crystal superlattice A composltlonally-modulated alloy (CMA) with disorder at the interfaces is shown schematically in Fig l(b) Increasingly complex models of crystalline, amorphous and mixed nonmagnetic and magnetic multilayers are shown m Figs l(c)-(f) Particularly by employing either a hght or a heavy rare earth (RE) m combmatmn with a transition metal (TM), it is possible to construct ferromagnetic or ferrlmagnetlc structures as shown Even more complexity is introduced when one allows for single-ion random anasotropy on RE Ions possessing orbital angular 0 0 0 0 0 0 • • •

0 0 0 0 0 0

0 0 0 0 • @ •

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Fig 1 Schemauc examples of vanous types of (a)-(d) nonmagnetic, (el, (f) magnetic multilayers © Elsevier Sequola/Pnnted m The Netherlands




RE (A)





disorder at the interface 4 brier review ~s pre sented of theoretical concepts that need to be developed for tbas class of materials No attempt ts made to discuss all magnenc multIlayers comprehensively, in terms ol etther their magnetic structures or magnettc amsotropy Recentl~y, Jm and Ketterson have summarized the magnenc, transport and superconducting properties ot a large number of artificial metalhc superlattlces Ill That revle~ and several other treatises ma) be consulted for references to a large number of papers on magnetic multtlayers i 2-6 ]


2. Magnetic anisotropy: experiment ----





-"- RE TM




(C) Fig 2 Schemanc diagrams of (a) ldeahzed RE-TM with perfectly sharp interfaces, (b) sine-wave modulated mululayer with CRE(Z)and CTM(Z) representmg the atomac fraenon of the RE and TM along the growth &recUon of the mulnlayer, (c) possible RE-TM amorphous alloy magnetic structures for hght and heavy REs with random amsotropy, and for gadohmum with no random anlsotropy

momenta, and for the possibility of amorphous CMAs, as shown schematically m Fig 2 Some examples of the types of magnetic structures possible In such CMAs are shown in Fig 2(c) GdFe and GdCo are simple ferrlmagnetlc structures because gadohmum is an S-state ion The "fanned" R E structures shown for the neodyrmum and dysprosmm structures result from random magnetic amsotropy m the assumed amorphous structure of the C M A Tl~s amcle reviews our understanding of mterfacial magneUc anlsotropy in magnetic multilayers A number of T M - N M (where NM represents a non-magneuc metal) and R E - T M examples will be discussed The SLmllanties m the amsotropy data for these two classes of multilayers wdl be emphasized, along wath the effects of

In this section we discuss the experimental determination of interface anisotropy and show some typical results Figures 3-8 show some typical data for the product of the bllayer penod 2 and the measured umaxlal amsotropy K u' as a ttmcnon of magnenc layer tbackness t for multilayers of cobalt with gold, copper and platinum, and also several R E - T M muttflayers The measured uniaxial amsotropy, or effective amsotropy, is determined m a standard way by measuring the area between the parallel and perpendicular magnetization curves That ~s, the measured amsotropy K.' ~s gwen by


Ku'=(! Had


i , - ( ! Ha a M L


where H a IS the applied field When the magnetization is approximately constant, the internal field H i s gwen by H = H a - NdM


where N d is the demagueUzanon factor It then follows that the intrinsic anlsotropy ~s K u = Ku'+ ~(Nd ± - Nd")M, a



K u = K j + 27rM, z


for a thin film with a perpendicular demagnetizatton factor of 47r From the energy viewpoint, the anlsotropy energy per umt area, for a T M - N M multdayer, can be written at 2 K j = 2 K , + 2 K L n d + ( K v + K , t - Kde)t


where K,, K,nd, K~, K~t and Kde represent rater-


.... \

I .... ~..




I .... .^.

I ....









" ~ . 2 n 300°C

Ku tco 0 (mJ/m 2)


~o ¢0

nL 250~C


I-0 4

::~ ,<


-0 8 0





Co tl~ckness (~.) -12

Fig 5 Plot of Z K j ~s too tor A u - C o and Cu-Co multdavers


-o •

10 15 too(A)


25 04

Fig 3 Plot of K . ' t o o v s too for A u - C o multllayers, before and after anneahng at 250 and 300°C (1 InJ In - z = 1 erg cIn- 2)[7]





+ XAFe/7,~Nd o XAFe/14ANd • XAFe/28ANd


o l




tpt= 17 5A





5 10 15 20 THICKNESSOF Fe (A)

Fig 6 p lot of 2K. vs 4e tor vanous Nd-Fe mululayers [10]


\ \


4% i






2 O0




"~ "4










10 20


N'~x ~\x

30 4 0


80 70

LAYER THICKNESSOF Fe(.&) Fig ? Plot of 2 K u' ~s t w ior vanous Dy-Fe multllayers [] 1]


tco(~.1 Fig 4 Plot of Ku~too v s too for P t - C o multdayers [8]

face, induced interface, mtnnmc volume, stress, and demagnetization energaes respectively Tins equation is often slmphfled to 2Ku ' = 2K, + (Kv - 2:rM,2) t


and it serves as a basis for discussing data such as those of Figs 3 - 8 A n u m b e r of pomts should be made about these data (1) Often, for t>~ 10 A, the data fall on a straight line with a slope and mtercept presum-

ably related to the second and first terms of eqn (6) respectwely (2) For t~< 10 A, there is a tendency for a maximum to be seen in 2Ku'(t) and an approach to zero as t--* 0 Tins is the region where the structure begins to exinbit sigmficant disorder and, at least for several R E - T M multllayers, where the structure becomes composmonally-modulated amorphous In dlscussmg a structurally sensitive property such as PMA, g will be useful to keep m mind the two hmltmg regions of the data, t e the "large" t region where the individual layers are crystalhne and where distinct interfaces exist, and

140 2 I








prosmm thickness but the intercept (2K,) does not [11] Addmonal data on D y - C o show that both the intercept and slope vary as a function of RE thickness These data can be reconcded w~th small-angle X-ray diffraction data which mdlcate a trend from more sharp to more diffuse Interfaces m the series Nd-Fe, Dy-Fe. D y - C o [13] Table 1 contains interface amsotropy values for most of the magnetic multilayers on which systematic data have permttted analysts via eqn (6) The references must be consulted for details on crystalline texture and interface structure vartatlons which can lead to varying values of K, for multllayers of two elements prepared by different methods In the following section, we discuss some theoretical concepts relevant to understandmg magnetic amsotropy in general We then outhne models for understanding multllayers of the two general classes mentioned above



c) cY~-I

/<-2 -3




Layer-thickness of



Co (~,)

Fig 8 Plot of ZKu' vs tcotor 8 A Dy-t A Co [12]

the "small" t data where disorder ts significant because the layer thicknesses correspond to several monolayers or fewer (3) In Fig 3, for Au-Co, a slgmficant shift towards posltwe 2Ku' (PMA) is seen as a result of annealmg This ~s interpreted as a sharpening of diffuse interfaces, by heat treating the ~mnusc~ble metals at a relatwely low temperature [7] These sharpened interfaces give rise to an enhanced interface amsotropy K, (4) For R E - T M multilayers of Nd-Fe (Fig 7), K, seems to be approximately independent of neodymium thickness [10]. However, for Dy-Fe (Ftg 8), the slope of the lines vanes with dysTABLE 1

3. Magnetic anisotropy: theory The magneUc anlsotropy arises trom dipole-dipole and spin-orbit mteracUons which are briefly reviewed below

3 1 Dipole-dipole mteractlom The long-range interaction energy between the dipoles m, and mj separated by a distance r,~ is gwen by m,

u,j -

m j - 3 ( m , ry)(mj r,j)/r~j2 3 r,j


Interface amsotropy values in selected multflayers

System Mo-NI[111] Cu-NI[111] Cu-NI[100] P d - C o [ l l 1] Pd-Co[111] P d - C o [ l 11] Au-Co[001] Au-Co[001] Pt-Co[001] Cu-Co[f c c ] Nd-Fe Dy-Fe Dy-Co Tb-Co Er-Fe Gd-Fe Gd-Co

Preparation method a S S S S V V S MBE V MBE S S S S S S S

Temperature h


T (K)

terg cm "~/

30 42 42 300 300 300 300 300 300 300 300 300 300 300 300 300 300

- 0 54 - 0 12 - 0 23 + 0 16 + 0 26 + 0 55 + 0 1-0 5 + 13 + 0 42 + 0 55 + 0 18 +25 + 0 4-0 8 >0 - 0 -0 - 0

aS, sputtered, V, vapor deposmon, MBE, molecular beam epttaxy bMeasurement temperature

Reference 14, 16 16 17 17 17 7 9 8 9 10 1l 12, 18 19 12 12,


13 13 13


T h e total energy from this term can be written as [20]



2Era,_ H,


where H, is the local field at the site t which does not include any contribution from m, T h e dipolar contribution to the amsotropy energy is determined from eqn (8) as the dipoles are aligned in various directions m the sample T h e local field is calculated by Lorentz's method This is done by introducing a spherical cavity around the site t of a size larger than the lnterdlpole distance but smaller than the macroscoplc distances H, is related to the macroscopic field H b y H'= H - H¢ + ~ -mj+3(m,~ ru)r,,/r,, e /#,



where H c is the macroscopic field due to the material inside the cavity and the summation in the last term is over the dipoles in the cavity Assuming magnetization M to be constant over the volume of the cavity


4:~M 3


This term does not contribute to the anlsotropy energy T h e last term in eqn (9) IS zero for cubic and higher symmetries Mlzoguchi and Cargill have estimated the contribution of this term for G d - C o amorphous films [21] T h e macroscopic field H, as determined by Maxwelrs equations, is due to the volume charge ( - V 3/) and the surface charge (M h where h is a unit vector normal to the surface) For an infimte sample with constant M, H is zero Mlzoguchl and Cargill [21] found a contribution from this term to the anisotropy energy due to fluctuations In M For a fimte sample, surface charges give rise to the depolarization field determmed by the shape of the sample ( - 4 ~ M for M normal to the surface of an infinitely large thin slab) This term favors the In-plane over the perpendicular orientation of M m the slab by the energy 2 ~ M 2 T h e procedure described above for the dipole-dipole energy breaks down for very thin films because the macroscopic theory falls [22] In such a s~tuatlon, one must perform the discrete sums in eqn (8)ff possible

3 2 Spin-orbit mteracttons T h e spin-orbit interaction is given by the oneparticle operator H,o = lct2 a{F(r)xp}


where a, a, F and p represent the fine structure constant, electron spin, electric field and momentum operators respectively Th~s interaction is much more difficult to handle theoretically Before the advent of computers, Van Vleck approximated the spin-orbit energy by a shortrange pseudodlpolar term of the form of eqn (7) where 1/% 3 is replaced by some function f(ru) [23] Ndel extended this procedure to study the surface and interface contributions to the anlsotropy energy [24] In recent years, attempts have been made to evaluate this contribution in simple systems within the framework of band theory [25, 26] T h e normal procedure is to calculate the self-consistent spin-polarized band structure without spin-orbit interactions T h e self-consistent potential and wavefunctlons are then used to calculate the matrix elements of the total Hami1tonlan including H,o T h e Hamlltonlan matrix is dlagonahzed at a very large number of points in the Brlllouln zone to find the total energy of the system T h e anlsotropy energy is obtained by repeating this calculation as a function of the spin orientation It is instructive and useful to look at this contribution from the point of view of perturbation theory, as outlined below 3 2 1 Transztton metals For the partially localized d states responsible for magnetism m these systems, the orbital angular m o m e n t u m is quenched, t e ( L , ) = ( L ~ ) = (Lz) = 0 In such a case, the second-order perturbation due to the spin-orbit Hamfltonlan 2 L S gives the anlsotropy energy of an ion [27] E, = D{3Sz 2 - S(S + 1)}


where D is the anisotropy constant Thus, the transition metal amsotropy can be analyzed In terms of the single-Ion model with the help of eqn (12) and a procedure developed in the following sections 3 2 2 Rare earths As the f electrons in the rare earths are fairly locahzed, their orbital m o m e n t u m is not quenched in a solid Their electronic states are the same as in the atom, t e [LSJJz) as determined by the spin-orbit coupling For the anlsotropy


energy, one is interested in the energy ot the ground state wath Jz = [J[ m the crystal potentml, as determined by the first-order perturbation theory for different &rections of z This is known as the single-ion amsotropy Keeping only the first term m the tesseral harmomc expansmn of the crystal potential energy, the energy of the tth rare earth mn can be written as [28] E, = aj(r2)A2"{3J 2 - J ( J = 2ctj(r-)A 2 J-

+ 1)} (J>l)


where % is the Stevens factor, (r 2) IS the quantum mechamcal average of the square of the radius of the 4f orbit, and A2 ° IS a crystal-field term A2° is given by

A2°°c ~ ql !

3 cos 2 0j - 1 ~

t 14)


where qj is the charge on the lth mn at a distance rj and 0j is the angle between rj and the z direction In general, especially for a complex structure involving ~tmerant electrons, knowmg the values of qj is difficult and m some cases even the s~gns of qj are controversml Also, eqn (14) does not contain the effects of crystal field on the charge distribution around the ion t itself This again illustrates the &fficulty of calculating magnetic amsotropies from first principles

3 3 Thmfilms and multtlayers For thin ferromagnetic films and slabs, magnenc amsotropy data such as those rewewed by Gradmann [29] have been analyzed m terms of an equatmn similar to eqn (6), except that K, is replaced by K,, the surface amsotropy It should be pointed out, however, that ff the thickness of the film corresponds to only a few monolayers or one monolayer, the concept of a volume amsotropy K, and the demagnetization energy 1 "~ (:NdM~") become unclear or undefined Under these circumstances, the magnenc amsotropy must agmn be deternuned as a difference between the total energaes, calculated w~th the magnetlzanon in the two relevant directions Dramsma and de Jonge [22] calculated the dipole-dipole amsotropy energy numencally for several ideal ferromagnetic structures m the form of ultrathln films, and found that the amsotropy energy deviates from the macroscopic value of 2zrM 2 for thicknesses less than about 12 A Thus the macroscopm theory breaks down for thick-

nesses less than about 12 A tor cubic and hexagonal lattices The researchers also show that the contributions from the last term m eqn (9) can be quite slgmficant tor tetragonal lattices with

~/a¢ l Suna considered the effect ol domains in calculating the occurrence o~ a perpendmular magnetic ground state of a multflayer film [30] He evaluated the dipolar energy for ferromagnetm-non-magnetlc layers, for M perpendicular to the plane of the film, and contmning alternately polarized stnp domains Under certain condv tions, the film behaves as a uniform medium with amsotropy arising from the layering Suna discussed experimental results on Pd-Co in relation to his theory The demagnetizing field and magnetostatlc energy of a thin film with surface roughness were considered by Bruno [31] The surface roughness was shown to give nse to an effective perpendicular anisotropy which can be related to parameters characterizing the roughness Bruno discussed the results and their relevance to experiment and to magnetocrystallme surface amsotropy Gay and lhchter have recently computed the spm anisotropy of ideal ferromagnetm multflayers of ~ron, nickel and vanadmm Self-consistent band theoretical methods were used as outlined above [26] The researchers lound that the easy &rectlon of magnetization is perpendicular to the plane of the monolayer for iron and vanadium but m the plane for nickel The results for iron lead to an anlsotropy energy of about 0 4 meV(atom)- 1, which is somewhat larger than the measured values of K, However, these should not necessanly be directly comparable Gay and Rmhter found that the dipole-&pole anisotropy energy for a monolayer of iron IS an order of magmtude smaller than the spin-orbit contribution In general, ttus seems to be the case tor transition metal-rare earth systems To extend the fundamental, band-theory approach to magnetic multIlayers, the following procedure would be reqmred The multfiayer, w~th an assumed single-crystal superlattice structure, would have to be subjected to band and total energy calculations, as a funcuon of magnetic metal layer thickness, for m-plane and perpendicular magnetization If this were done for several t values, a plot of 2K u' could be made and the intercept would presumably gave a quantity 2K, which would be comparable with that measured experimentally wa eqn (6) This


agenda appears very formidable, although with increasing supercomputer power, it may be feasible before too long

structure, the sum o v e r / i n eqn (14) is replaced with an integral weighted with an amsotroplc pair dlstnbutlon function [21 ] P,, = r/j(z)R,,(r)(1 +fl, cos 0j+

4. Amorphous RE-TM multilayers A major goal of our recent work [10-13] has been to understand the PMA of amorphous R E - T M multllayers in terms of the amsotropic pair correlations Figure 9 extubIts the intrinsic amsotropy K u as a function of TM layer ttuckness for several R E - T M series One of the main problems in developing a model is to assess the importance of dipole-dipole compared with single-ion contributions to K u We see from Fig 9 that the amsotropy Ku for Dy-Fe, Dy-Co and Tb-Fe is roughly an order of magmtude larger than that of Gd-Fe and G d - C o As gadohmum has no spin-orbit Interactions, it is reasonable to attribute the main origin of the perpendicular anisotropy of the dysprosium and terbium multilayers to single-ion amsotropy of the rare earth 1on Thus a model incorporating single-ion rateractions (eqn (13))and a slnusoidal compositional modulation (relevant to the multllayers consldered here) has been developed [12] The compositional modulation is ~]TM (~,) = ~TM 0q-




where #']TM1Sthe atomic fraction of the TM and A is the peak-to-peak amplitude of the modulation For a composltlonally-modulated amorphous ]






4 5ATb/xJ,Fe 60ABy/x~Co 50~Dy/xAFe






/ 4/ / ~ ~













~2 of





F1g 9 Intrinsic amsotropy characterlsUcs for vanou~ R E - T M multflayers The amsotropy data for several Gd-Fe samples are m the shaded area Data for several G d - C o samples are neghglbly small or negative [ 12]



where r/j(z) is the atomic fraction of the/th neighboring atom, R,j(r) is the lsotroplc part of the distribution and flj is the lowest-order anlsotroplc contribution The amsotropy in the pair distribution function can arise from the composition modulation of multdayers and strains at the various interfaces

The stress in a magnetic material gives rise to magnetoelastlc amsotropy energy due to inverse magnetostrlctlon (magnetostrlctlon is the strain 2 = Al/l m the direction of magnetization) This contribution is given by [32] Em~ = 2~,O s i n e 0


where 2, is the saturation magnetostrlctlon, o is the stress and 0 is the angle between M, and o (o is negative for compressive stress) Knowing 2, and o, one can find the contribution of this term to the anlsotropy energy This is not easy to do in complex systems such as multllayers However, this effect is contained m single-ion anlsotropy (eqn (13)) through the pair distribution functions and therefore the results of eqn (13) can be compared directly with the experimental data We have found for Dy-Fe multllayers that the anisotropy is the same for mica, tantalum, copper and mylar substrates, ruling out any slgmficant magnetostrlCtlve effect at the sample-substrate interface This effect is also not expected to be significant at the Dy-Co Interfaces because of their diffuse nature Thus the compositional modulation appears to be the m a n source of the anlsotroplc pair distributions and magnetic anlsotropy The lowest-order contribution to the crystal field term A2° comes from the first-order term in the expansion of r/from eqn (15) about the rare earth ion, it is proportional to A / 2 Also, the firstorder term in q has the maximum value at the interface and the charges q, are expected to be most significant in the interface region Thus, as expected, most of the anlsotropy comes from the interface region Combining the results for A2 ° with E, in eqn (13), the amsotropy energy K u for a CMA can be written as A 9 Ku = ~ - (MRE- )





I ÷ -


exper' l|erlt; t~eor'y

u f__ 0a

In general, the study of magnetic multllayers, where the interfaces play a dominant role, holds high interest because of the new phystcs and apphcattons that may follow from such materials


Acknowledgments O






4 8 Layer-thickness



12 of Co (~)

Fig 10 A comparison between the calculated and expenmental anlsotropy for a series of Dy-Co multdayers with 6 A layers of dysprosium [12]

We are grateful for financml support from the National Science Foundation under grants DMR 8605367 and INT 8715441 We thank Y J Wang and J X Shen for asslstaDce and helpful dtscusszon Thanks arc also due to various colleagues for sharing thetr data vta prcprlnts before pubhcatlon References

where ~ is a constant and (MRE2) lS the average of the square of the saturation magnenzatlon of the rare earth subnetwork in the easy direction [12] With ~ as an adlustable parameter, eqn (18) is fitted to the K~ data for 6 A D y - X A Co samples and the results are shown m Fig 10 Because of the single parameter, the agreement between the experunental data and calculated results ~s remarkable The fitted value of ~ = 5 26 x 10 -~ cm leads to an average value of the smgle-~on anlsotropy parameter D = 2 x 10-17 erg, which ~s reasonable m terms of the typical value of the single-ion random-amsotropy parameter in amorphous RE-TM alloys This model gives similar fits for other RE-TM CMA which wdl be pubhshed elsewhere [13] 5. Discussion The begmmngs of some order have been achieved m the expertrnental study of mterfacml magnetic anlsotropy for magnetic multdayers Although a number of K~ values have been determined, there have been no first-principle attempts to calculate them When the thickness of the magnetic layer IS less than several atomic dmmeters, slgmficant dzsorder sets in for both the TM-NM and RE-TM multflayers The composmonal modulation is lost m the limit t-~0, where a homogeneous, lsotroplc film results We have developed a model relevant to RE-TM multfiayers with slgmficant single-ion amsotropy From the similarity of the data of these multdayers and Co-Au and Co-Pt multflayers, this model is hkely to be vahd for such disordered multllayers as well

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