This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 129.8.242.67This content was downloaded on 27/07/2015 at 16:25Please note that terms and conditions apply.The permalloy problem and anisotropy in nickeliron magnetic filmsView the table of contents for this issue, or go to the journal homepage for more1964 Br. J. Appl. Phys. 15 531(http://iopscience.iop.org/0508-3443/15/5/310)Home Search Collections Journals About Contact us My IOPscienceBRIT. J . APPL.PWS.,1964, VOL.15 Thepermalloyproblemandanisotropy innickel- iron magnetic films B.LEWIS The General Electric Company Ltd., Hirst Research Centre, Wembley, Middlesex MS.received 4thDecember1963, in revised form 4th February 1964 Abstract.Flux-field methods have been developed for the separate measurement of the uniaxial anisotropy field HTJand the mean domain anisotropy field HA of magnetic flms.Hc= HA represents the ideal uniaxial case but many i3t1-1~have HA > Hu. If Hc is experimentally reduced to zero the residual domain anisotropy is randomly oriented; its most probable origin iscrystalanisotropy.The occurrence ofgood uniaxial propertieswithalowvalueofHu,as foundnear83% nickel-iron,isa consequence ofa minimum value of HA at that composition.The composition de- pendence of HA and the IOWvalue of HA at 83 % nickel represent a new aspect of the permalloy problem.It is proposed that the magnetocrystalline anisotropy is reduced by crystal oriented magnetoelastic energy arising from isotropic stress and anisotropic magnetostriction.Near 83 % nickel these two anisotropy terms are mutually opposed and giveminimum net anisotropy at a composition which depends on the stress.The lower the crystal energy the more perfect is the alignment of magnetization caused by magnetic interactions.This alignment further reduces the effective local anisotropy until in the limiting case of perfect alignment N . = Hc.1.woduction Observations of angular and magnitude dispersion of anisotropy (Crowther 1963, West 1961), hi te hard axis coercivity (Prutton and Bradley 1960) and unidirectional rotational hysteresis (Doyle and Prutton 1963, Robinson 1962) have shown that all films possess some randomly oriented anisotropy, as well as uniaxial anisotropy.Oblique incidence anisotropy (Smith,Huber,Cohenand Weiss1960), localized stresses, impurities and imperfections (Pugh 1963) and crystal anisotropy (Robinson1962) have been suggested as origins of the randomly oriented anisotropy in magnetic films. For nickel-ironalloys near 80 % nickel, good uniaxial properties are associated with low anisotropy energy.In particular, films with the lowest values of uniaxial anisotropy have been found bySmith (1961) to have the lowest dispersion, and byPruttonand Bradley (1960) to have the closest approach to the rotational switching predicted for uniaxial films. The implication is that the randomly oriented anisotropy energy also has a minimum value near 80% nickel. The range of nickel-ironalloys near 75 % nickel, known as the permalloys, show unusual properties in bulk material as well as in films.The dependence of the properties of nickel- iron Nmson composition and substrate temperature can be considered as new aspects of the general problem of explaining the special properties of these alloys, the so-called perm- alloy problem (Bozorth 1953). It is usual to describe the properties of uniaxial magnetic films interms of an anisotropy energy coefficient K and the corresponding anisotropy field HK = 2K/Ms, where Ms is the saturation magnetization,However, it is not possible to represent all the anisotropy effects i n real films by a single parameter, and it is found that values of the anisotropy field obtained by diferentmethods do not agree (Doyle 1962, Feldtkeller 1963).Two anisotropy para- meters are used in the present work, and since both have previously been represented by HK it seems preferable to use two new symbols HV and HA. 2-Uniaxialand domain anisotropy fields The anisotropy energy of a magnetic domain is the energy associated with variation of 53 1 532B. Lewis the angle 8between the domain magnetization vector and a datum direction.The anise. tropy effects observed in magnetic films represent the sum of several components: [i) field aligned uniaxial anisotropy; (ii) stress aligned uniaxial anisotropy; (iii) uniaxial anisotropy originating from oblique incidence of the evaporant beam; (iv) randomly oriented magneto- crystalline anisotropy;(v)randomlyorientedstressanisotropy;(vi)randomlyoriented anisotropy associated with impurities and structural defects. 2.1.Theuniaxialanisotropyfield For an area of film a few millimetres across, the three uniaxial components (i), (ii) and (iii) can each beconsidered uniform in magnitude anddirection,and theirsum isalso uniaxial.The netuniaxialcomponentEc(8) of the anisotropy energy ofasmall area of film can, therefore, be expressed as ETJ= U sin2 0(where 8= 0 is the easy axis) and can be represented by a net uniaxial anisotropy field HE = 2U/Ms. Inorderto measure thenetuniaxial anisotropy fielditisnecessary toeliminate the effects ofsmall-scale randomly oriented anisotropy which may be of comparable or even of greater magnitude.If a unidirectional field large enough to pull the magnetization out ofthemoststablelocal easydirection isapplied theanisotropyenergy which favours alignment ofthe magnetization along the meaneasy axis can beobtained from torque measurements. For the present work a method using a loop tester wasdeveloped.If a large sweep field is applied to a film, close to the easy axis, the net direction ofrotatiofi of magnetiza- tion vectors at each reversal is controlled by the mean uniaxial component ofanisotropy energy.The symmetry position when (owing to angular dispersion of the easy axis) equal flux rotates clockwise and counter-clockwise, can be determined by observing the orienta- tion for zero transverse flux change with a pick-up coil whose axis is perpendicular to the sweep field.If the film is then rotated through a small angle athere is a net rotation of flux in one sense at each reversal.The observed transverse flux can be restored to zero (forreversal fromonesaturationdirection) byapplying abiasfield HB perpendicular to the sweep field, so that it acts against the restoring force of the uniaxial anisotropy. Equating magnetostatic and anisotropy torquesfor magnetization Ms,on the point of reversing, and for uniaxial anisotropy energy EV = U sin2 0, Hence HBMS = U sin 2a N 2Ua. HTJ= 2U/Ms = H B / ~ .The effect ofa high proportion of randomly oriented anisotropyon this measurement wasto reduce the transverse flux foragiven misorientation.Thesensitivity wasthen lower, but it remained possible to use the method to establish the easy axis and to measure HTJ,even for films with nearly isotropic hysteresis loops. 2.2.The randomlyorienteddomainanisotropy field The randomly oriented anisotropy components (iv), (v) and (vi), do not (by definition) affectthenetuniaxialanisotropy.Theydocontributetothetotaldomainanisotropy energy&(e),which controls the direction of magnetization of an individual domain and which represents the sum ofall the anisotropy components listed above, together with a termdue tointeraction with neighbouring domains.The response ofthe domain mag- netization to a weak field, i.e. the initial susceptibility K,,is determined by E-&(@. In zero field the domain vector is directed along a local easy axis for which EA(@ is a minimum and 0= Bo,say.The contribution of the domain to the susceptibility, by rota- tion in a small field, has been given by Lewis and Street (1958): Here ais the angle between the local easy axis and the axis along which K~is measured; EA" (e,)= d2A/d02with6= e,,and represents the rotational stability ofthe magnetka- tion vector. K , = dM/ dH = Ms2 sin2EA" (0,).(2) The permalloy problem and anisotropy in magnetic j l ms533 K~= MS sin2 ./HA (4) can be used to define an anisotropy field H-4which controls small rotations of the mag- netizationfromitsequilibriumorientation.Inadirectionperpendiculartotheaxis giving K ~ , the susceptibility K~ is Mscos2 a/&.Hence K~+ K~ = Ms/HA.Thesusceptibility for agivenareaoffilmisobtainedbysumming thecontributions from all domains within the area.Since for each domain K~+ K~is independentofa, for the whole area we have whereHA is now the mean domainanisotropyfield for the area. Whenthedistribution ofdomain vectororientationswithinthe area is isotropic then K~= K~= Ms / ~ HA. When the distribution is unaxial the susceptibilities K e and K halong and perpendicular to the alignment axis respectively, are given by equation (4)as K~= 0 In practice most so-called uniaxial films have local easy axes which are dispersed within a fewdegreesoneithersideofthemeaneasyaxis.However,since sin2 aN 0 and cos2 a - 1when ais small westill have K~N 0 and Kh- MsIH-4.Thus, for a low dis- persionfilm,theanisotropyfield obtained withalooptesterbythe familiar hardaxis susceptibility method is HA, asdehed more generally byequation (4). For an ideal uniaxial film the randomly oriented anisotropy components are zero and Ha = HTJ. For other films HA > Hr;. K O-kK~= Ms/ HA( 5 )and K h = Ms/HA. Figure1.Measurement ofthe mean domain anisotropy field HA.(a) Normalized flux change in a field 10.5 Oeas functions of the orientation of the applied field.The flux measurements were made after saturation by a unidirectional field along the axes marked 'saturate'.HTJ= 3 . 7Oe. (b) The same measurements repeated after the plate had been bent to reduce flu to 0.1 Oe. 3.Experimentalresults 3. 1. Domainanisotropy field f oranormal jilm Figurel ( a)gives resultsofsome measurements madeon a film deposited on an alu- miniumplate by evaporation from an ingot of composition 80:17:3 Ni:Fe:Co. Susceptibility is recorded in figure l ( a)as normalized flux change A+/ ds for a field change of10. 5 Oe.Before each measurement a d.c. saturation field was applied along the axis marked'saturate'and then reduced to zero. 534B. Lewis Saturation along theeasy axis, or at 45" to theeasy axis,ordemagnetization byan alternating field along any axis, gave initial susceptibility curves closely following a sin2 a variationwithorientation.This showed that thedomaindistribution wassubstantially uniaxial and confinned that only rotational processes were involved.(Boundary movement would give maximum susceptibility along the easy axis.)Using equation (5) and evaluating dH~&/ ( 44, + 4+h)gave H-4= 6 Oe as the value of anisotropy field controlling magnetka. tion vectors lying in local easy axes close to the mean easy axis. After saturation along the hard axis the initial susceptibility was small and nearly iso- tropic.Thissuggests thatthemajorityofmagnetization vectors remained close to the hard axis, in a locked configuration with high binding energy, and a minority relaxed to the easy axis controlled by a lower value of anisotropy field.Experimentally, the occur- rence oflowsusceptibility after hard axis saturation was always associated with a high value of remanence: indicating substantial alignment along the hard axis. 3. 2. Reductionofuniaxial anisotropy The uniaxial anisotropy of a film can be varied by the addition ofanother component of uniaxial anisotropy.This can conveniently be done by bending the substrate, and the observed change of uniaxial anisotropy can then be used to determine the isotropic magneto- strictive coefficient A,ofa film (Mitchell, Lykken and Babcock 1963). Whenthe plateusedforthemeasurements givenin figurel(a) was bentconcavely, perpendicular to the easy axis, to give a strain 75xat the centre, the uniaxial aniso- tropy HTJ was reduced from 3.7 to 0.1 Oe.The value of A,derived from this result is - 0.6xProperties of the film in this state are given in figure I@).Open hysteresis loops were observed at all orientations and the coercivity was nearly isotropic.The initial susceptibility was also nearly isotropic after the 6lrii had been demagnetized by reducing to zero an alternating field applied along any axis.The value of X.4was 7 Oe.After satura- tion byad.c. field along the O",45Oor 90" axis, thedomaindistribution, as tested by measuring initial susceptibility, was anisotropic, with a majority of magnetjzation vectors aligned close to the direction of the saturation field,This behaviour is known as 'rotatable initial susceptibility', and is generally found in films with abnormally high random aniso- tropy,as described by Flanders, Pruttonand Doyle (1963), for example, and by Lehrer (1963).Here it has been produced in a normal ~ by reducing the uniaxial anisotropy. Cohen (1963) has observed the development of rotatable initial susceptibility behaviour and highdispersion in magnetic films by an annealing treatmentwhich reduced the uniaxial anisotropyand increased the coercivity.The common feature in these experiments isa high ratio of HA to Hu. As the measurements recorded in figure l(b) show, the mean local anisotropy field was 7 Oe, regardless of the degree of alignment of the magnetization, or the axis of alignment. Thus,asexpected, in theabsence ofuniaxialanisotropythe filmpossessed randomly oriented domain anisotropy. Figure 2shows the variation withstrainofHE,HA andseveral other properties, for anotherl?lm evaporatedfroman8017:3Ni:Fe:Coingot.Hysteresisloopsforthis film, before bending, showed nearly perfect uniaxial anisotropy with a low value ofhard axis coercivity.HA was only slightly greater than Hr,and easy axis angular dispersion, measured by the Crowther (1963) method, to the quartiles of thedistribution curve was aq =3". The uniaxial anisotropy field was found to change linearly with strain, negative values corresponding to rotation of the easy axis through 90".To facilitate comparison with HA, these values ofHU arealso plottedas positive values in figure 2.As HU decreased aq increased and when HIT was zero the film was isotropic, with equal coercivities along the 0"and90" axes; HA was thenslightly less than the value for the unbentplate.High compressive or tensile strain gave easy axes along 0'and 90", respectively, with HE - HA and low values of aqand hard axis coercivity.The variationof uniaxial anisotropy with straincorrespondedtoamagnetostrictive coefficient As= 0.8 x10-6,showingthat, The permalloy problem and anisotropy in magnetic films535 &hough this film was nominally similar to the f3m whose properties are given in figure 1,be composition was in fact somewhat richer in iron. Other plates, of the same nominal composition, selected to cover a range of preparation variables and values of HA, HU and aP all showed qualitatively similar behaviour to that recorded i nfigure 2. men HV= 0 thedomainanisotropyfieldisrandomlyoriented and hasthevalue (H~),.Addition of a uniaxial component HL- to (HA), increases the observed value of HA. men HU is large enough to align the magnetization completely along the HU easy axis, the randomly oriented variations ofanisotropy average out to zero and the mean domain ajjisotropy field HA is then equal to Hu.The angular dispersion ofthe easy axis varies from isotropic when Hu Huand HA > Hu both indicate dispersion. The anisotropy fieldobtainedbythe hysteresisloop method is H-4.Valuesgiven by Smith (1959) and by Blades (1959) agree quite well with HA in figure 3 but do not go beyond 85 % nickel.Prutton and Bradley (1960), whose resultsare included in figure 3, found a minimum at 83 % nickel for a substrate temperature during deposition of 250"~. However, theirvaluesofanisotropyfield werehigherand rose moresteeply on either side of the minimum. I t hasalreadybeenestablished that only whenHc< HA do we findperfectuniaxial The permalloy problemand anisotropy in magnetic$lms537 anisotropy.The separate measurement of the variation of HU and H-4with composition has now shown that the dominant feature is a minimum in HA and HA)^.It is, therefore, of particular interest to understand the factors which determine (H&and to examine the lower limit of attainable values. 4.n e permalloy problem The minimumdomainanisotropy field (H&ofa magnetic lilm is controlled bythe magnetoelasticandmagnetocrystalline energiesofitscrystallites.Theremayalsobe anisotropy associated with impurities or structuraldefects, but unless these are ordered, their anisotropy will be on too small a scale to have any resultant effect on the crystallite anisotropy.Themagnetostrictivecoefficientsandthemagnetocrystallineanisotropy constant for nickel-ironalloys have been measured by Bozorth and Walker (1953) and are givenin figure 4(a).The magnetocrystalline anisotropycoefficientiszeroat75% nickel and is 5000 erg ~ m - ~ at 83% nickel. 65707580859095100Composition(% nickel) (0) ( b )Figure4.Crystal anisotropy (including magnetoelastic term) for nickel-iron &s.(a) Magneto- crystallheandmagnetostrictive coefficients.-K1;----- &II;- 0 - &oo- %III; - - (b)Calculated rotational stability Ec"(6,)derived from figure 6for zerostress and for isotropic planarstressup= 2xlo9 dyncm-?.Experimentaldatafrom figure 3areincludedforcomparison.- MSHA= EA"(e,,);- -MsHA(Prutton andBradley1960);- 0 - calculated Ec"(6,)forU= 0;- 2 - calculated E,''(6,)for up= 2xlo9 dyn cm-2. Bulk nickel-ironalloys in the same composition range also have high initial susceptibili- ties. The highest values are obtained with 78 % nickel-iron which has been given the perm- alloy heat treatment.The occurrence of maximum permeability at this composition and - J .lll)/Kl. 23 538B. Lewis the sensitivity to heat treatment constitute the permalloyproblem.It was this situation which stimulated the determination of the crystal anisotropy and magnetostrictive constants by Bozorth and Walker.Bozorth (1953) has suggested a qualitative explanation of the observed behaviour consistent with this data, based on boundary movement (for which no quantitative theory exists) as the principal magnetization process.It is argued elsewhere (Lewis 1964) that the rotational process is more likely. In the analogous problem with nickel-ironfilms the magnetization process is known to be rotation, equation (2) applies, and we haveto explain a minimum value ofEAft(e,). I t is proposed that, for bulk nickel-ironalloys and for films, the observed variation with composition of crystal anisotropy energy can be reconciled with the known composition dependence ofthe crystal constants byinclusion in the crystal energy of magnetoelastic energy terms arising from anisotropic magnetostriction in conjunction with macroscopicauy isotropic stress. 4.1.Crystalanisotropy including magnetoelastic terms elastic terms, is given by Lee (1955) as The orientation-dependent energy ofa cubic magnetic crystal, including the magneto- Ec = K,(u?at+ at a31 fa32u12)- 4 Aloou(u12y12+ a,2yz'- a32yy32) - 3 A~l l u(al az~G'z4- a2u3y2y3a3a1y3y1).(6) The U'Sare the direction cosines of the magnetization and the y'sare the direction cosines ofa linear stress U,referred tothe crystallographic axes; K, is the first-order magneto- crystalline anisotropy coefficient and A,,,and A,,,are the two principal magnetostrictive constants. Knorr and Hoffman (1 959) have used this equation to examine the uniaxial anisotropy produced by a fibre axis structure in the presence of a planar stress.The same problem with anisotropic stress has been considered by Pugh, Boyd and Freedman (1960) Freedman (1962) has used an equation with five magnetostrictive constants to calculate the anisotropy energy of(001) films ofnickel witha planarstress.The high symmetry made the rotational energy independent of Aloeand A,,,(or h,and h,)but left anisotropy terms in h,and h4.For polycrystalline films having randomly oriented crystallites the stress may be in the planeofthefilmormayberandomlyoriented.Planarstress varyingwithsubstrate temperature between f 10 and - 4 xlo9 dyn cm-2 has been observed by Prutton (1962) and by Weiss and Smith (1962).The case ofa 6.h with planarstress is analysed in the appendix.Foreach crystallite, whenKlisnegative,as it isbetween75%and100% nickel, and the magnetostriction is isotropic, the easy axis orientation in the plane of the filmisdetermined bythe firsttermofequation (A6)andisneara[ill] axis.When A,,,#XI,,thesecond termofequation (A6)gives minimum energy with magnetization close to a [I001 axis, and the two terms together give a rotational stability Ec" (6,)which may be less than the value for Kl alone at the same composition.The calculated variation with composition ofthemeanvalue ofEc"(e,)for apolycrystalline film withup= 0 and 2 xlo9 dyn cm-2 is shown in figure 4(b).For zero stress E,"(6,)is zeroat 75% nickel.For up = 1 and 5xlo9 dyn cm-2 the minimum value occurs at 82 % and 84 % nickel, respectively, and is slightly below the value for up = 0 at the same composition. Stress has very little effect in the range 84 % to100 % nickel because of the low value of The case of randomly oriented internal stress is treated elsewhere, with particular reference tobulknickel-ironalloys (Lewis1964), andshows a similar variationof Ec" (6,)with composition to that for planarstress. (A100 - h113Kl (see figure 4(4). 4.2.Magnetic interactions In the calculation of6, and Ec" (6,)theeffectofmagnetic interactionshas not been taken into account.As has been pointed out by Blades (1959), by Crowther (1963) and byThe permalloy problem and anisotropy in magnetic films539 Pug-,(1963), the effect ofalignment along a common axis is to weaken the influence of local variations ofanisotropy energy. The calculated value ofEc"(e,)for 83 % nickel is 6000 erg ~m- ~, which is about twice the observed value of EA"(e,).To estimate the effectiveness of magnetostatic and exchange energy in aligning the magnetization we can make use ofcalculations byRother (1962). Hefound that the ripple amplitude (i.e. the amplitude of deviations from the mean easy axis) is proportional to the crystal anisotropy coefficient C, and to the square of the crystal- lite diameter D. When C = 6000 erg ~ m - ~ and D = 500 Athe amplitude is 6". The easy axis angular dispersion uq is a parameter which is closely related to the ripple amplitude.Experimentalresultsforthevariationofuqwithcompositionwerevery similar to thoserecorded bySmith (1961).aq was&-lo near83% nickel, &4'at 75% and 87 % nickel and &8'at 90 % nickel. The rotational stability is greatest for the state of minimum crystal energy with magnetiza- tion along the crystallite easy axes.A ripple amplitude or easy axis dispersion ofonly a few degrees implies considerable deviation from the crystallite easy axes and a consequent reduction in the stability.Thus values ofC and D which give a small ripple amplitude also give small values of EA" (e,)and (a&. The substantial equality of HA and HO between 75 % and 85 % nickel shown in figure 3 suggests that over this range ofcompositionmagnetic interactions have so reduced the effective crystal anisotropyenergy that the domain anisotropy is dominated by the field- aligned uniaxial anisotropy.At each end of the HA composition curve there is an increase ofangular dispersion, and thestabilityincreases faster than the coefficientC.Prutton and Bradley'scurve may benarrowerbecauseoflarger crystallite size D. Clow (1961) hasfound experimentally that an increase of grain size at constant composition increases the easy axis angular dispersion. 4.3.Conclusions Summing up, we have a situation which is too complex for a complete quantitative treat- ment.However,ithasbeenshownthatthecombinationofmagnetocrystalline and magnetoelastic anisotropyenergy canaccountfor minimum anisotropyenergy at83 % nickel when the stress is 2xloDdyn cm-'.Lower stress gives a lower value of minimum anisotropy energy ata slightly lower percentage ofnickel.The zero value of isotropic magnetostriction A,near 82% nickel is not directly related to the low anisotropy energy, but has the practical advantage of reducing the sensitivity of the iTlmto applied stress.Mag- netic interactions further reduce the effective crystal anisotropy energy, and this effect is enhanced by small crystallite size. Acknowledgments This project wascarried out in co-operationwithSalford Electrical Instruments Ltd., andInternationalComputersandTabulators(Engineering) Ltd.I should like to thank my colleagues for constructive criticism and Dr.M. PruttonandDr. K. D. Leaverfor useful discussion.Mr. H. A.J onescarriedout themathematical analysis given in the appendix. Appendix Crystal anisotropyofa polycrystallinemagnetic film subjectedto planarstress Take axes Ox, Oy in the plane of the film, and Oz perpendicular to the film.Planar stress isrepresentedbystressesup inanytwoperpendiculardirections, sayalongOx and Oy. The orientationofa crystal with respect to the ( x,y ,z )axes can be specified bythe three angles shown in figure 5.9 is the angle between the[001] axis and the x yplane. Eis theangle between the projectionof[001] in the xy planeand Ox.+is the angle between[loo] and OP where OP is the line in which the plane through [OOl] perpendicular to the xy plane meets the (001) plane. 540B. Lewis z COOll Figure 5. Angles 4,ly and Erelating the axes of a randomly oriented crystallite to xand y axes in the plane of the film. The direction cosines ofOx with respect to the crystal axes are denoted by (yl,y 2,y3)where y2 = - sin 6 cos+f cos 8 sin 4 sin$ y1 = sin 6 sin+-+cos 6 sin 4 cos+y3= cos 6 cos 4 . ('41) I 1. The direction cosines of Oy with respect to the crystal axes are denoted by(y;,y2',y i )where ('42) y i = - cos 6 sin++ sin t sin 4 cos$ y i = cos 6 cos ++ sin 6 sin +sin $ y i = sin 8 cos 4 If the magnetization is in the xyplane and makes an angle 5 with Ox then its direction cosines with respect to the (x, y ,z) axes are (cos 5, sin 5,0),and with respect to the crystal axes are al= y1 cos 5 + y isin 5 = sin (6- 5)sin+-+cos (8 - 5)sin +cos$ a2 = y2 cos 5 + y i sin 1 = - sin (6- 1)cos+a3 = y 3cos5 cos (6- 1) sin +sin+y3/sin 5 = cos (6- 5)cos 6. a1 = - sin 0 sin$ f cos 8 sin 4 cos $1 a2 = sin 0 cos ++ cos 0 sin +sin$ a3 = cos 0 cos 4 Putting 5 - 6 = 0 (A3) 1- Now the crystal energy for a linear stress U,equation (6), can be written E,= Kl ( ut a2 + aZ2a2-+u3*a12)- Q (Al,,- Alll)o(a12y12T a22y22+ a32y2) - 4 Al l l 4al yl+ azyz4- Q3Y3I2644) alyla2y2f a3y3= COSi.(A5) and for stress along Ox Hence foraplanarstress withcomponentsoP alongOx = (yl,y 2,y 3)andoP along OY= hi , Y21, Y3') Ec= K1(al2a2 + a 2 u 2+ a2a12) - 3(~100- Al l ,)op(a,"(y,2+- Y 1'9+ a,2(y22 + yi 2)-I-a32(y32-I- Y i 2)) - Q Alllopcos2 5 - 3 Alllopsin2 5. Substituting from equations (AI)and (A2) Ec= K1(al2a2 4-a22a32+ a32a12)- $ (Aloo- Alll)oP[(sin2$ + sin24 cos2 $) a12 + (cos2 ++ sin2 4sin2 +) a22 -/-cos2 +a32]- 8 Alllop.(A6) The permalloy problem and anisotropy in magneticjlms541 The problem is then to find, for given values of K,,A,,,,A,,,,U, 4 and $ the value 0 = 8, for which Ec is a minimum. The equation obtained by putting6Ec/S0 = 0 is too compli- cated to be solved except by trial, and 8,is most easily found directly from equation (A6). Foreach set of parameters Ec has been computed for values of0from 0" to 180"in 15" intends;8, was thenfound byinterpolationandthesecond differential (S2Ec/60z)o=8, calculated.The process was repeated for values of 4 from 0" to go",and$ from 0" to 45",in 15" intervals.The mean value of ihe second differential Ec" (e,)was then evaluated for the chosen values of K,,A,,,and A,,,. - .-1g o- E I I I II/-- Normalizedmaqnetoelasticcoefficient(Aloo-hlll)uD/4 Figure6.Calculated values of E,"(e,)in terms of the magnetoelastic coefficient (Aloa- 1,,J op, averaged over a polycrystalline film with weak magnetic interactions between crystallites.Both functions are plotted on scales normalized with respect to Kl. U I -2 -4 -6(hioo-hlll)uppositive .-- z Since thereisonlyoneangle-dependent magnetoelastic term,equation(A6)canbe dividedthroughbyK,,andEc" (S,)/K,evaluatedintermsofasingleparameter (A,,,- A1,3up/K,.Results are given in figure 6, the four quadrants covering positive and negative values of(A,,,- A,,,)up and K,.Ec"(e,)is always positive.When K,=0, Ec"(8,)= 0.64 I (A,,,- A,,,)up I.When K,is positive (A,,,- Alll)/K,is always large for nickel-ironalloys(see figure 4),therefore Ec"(O,)/K, hasonly beencalculatedfor (Al oo- A,,,) up/K1 > 2,and for up = 0. References BLADES,J. D., 1959, J.Appl. Phys.,30, 260s. BOZORTH,R. M.,1953, Rev. Mod. Phys., 25,42. BOZORTH,R. M., and WALKER,J. G., 1953, Phys. Rev., 89,624. CLOW, H., 1961, Nature, Lond.,191,996. &HEN,M. S.,1963, J. Appl. Phys., 34,1841. C~OWTHER,T. S.,1963,J.Appl. Phys., 34,580. DOYLE, W. D., 1962, J.Appl. Phys.,33,1769. DOYLE, W. D., and PRUTTON, M., 1963, J.Appl. Phys., 34,1077. FELDTKELLER,E., 1963, Physics Letters, 7, 9. FLANDERS,P. 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