Compositional and Angular Dependence of the Magnetostriction of Thin IronNickel FilmsE. N. Mitchell, G. I. Lykken, and G. D. Babcock Citation: Journal of Applied Physics 34, 715 (1963); doi: 10.1063/1.1729522 View online: http://dx.doi.org/10.1063/1.1729522 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/34/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetoimpedance of thin film meander with composite coating layer containing Ni nanoparticles J. Appl. Phys. 115, 17A323 (2014); 10.1063/1.4865319 Kinetic model for dependence of thin film stress on growth rate, temperature, and microstructure J. Appl. Phys. 111, 083520 (2012); 10.1063/1.4704683 Magnetization depth dependence in exchange biased thin films Appl. Phys. Lett. 89, 072504 (2006); 10.1063/1.2336742 Composition and irradiationtemperature dependence of the uniaxial anisotropy energy of largegrainironnickel alloy thin films J. Appl. Phys. 44, 5575 (1973); 10.1063/1.1662199 Compositional and Thickness Dependence of the Ferromagnetic Anisotropy in Resistance of IronNickel Films J. Appl. Phys. 35, 2604 (1964); 10.1063/1.1713808

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JOURNAL OF APPLIED PHYSICS VOLUME 34, NUMBER 4 (PART 1) APRIL 1963

Compositional and Angular Dependence of the Magnetostriction of Thin Iron-Nickel Films*

E. N. MITCHELL,t G. I. LYKKEN,t AND G. D. BABCOCK

University oj North Dakota, Grand Forks, North Dakota (Received 27 August 1962)

Theo:~tical consid;rati?n is given to the b~havior of the anisot;opy field ?f iron-nickel films ranging in CO~POSltIO~ fro~ 65 10. NI-~5%. Fe to 90% NI-lO% Fe as a functIon of applIed external stresses and mag­netIc fields III vanous directIOns III the plane of the film. It is postulated that internal stresses in the substrate film ~ystem may account for observed ?is~repancies. An expressi?n is derived which relates the magneto­elastic parameter (,.,) to the magnetostnctron (A,) of the film. ThiS parameter is measured as a function of comp~sition and the predicted value of A, agrees within the limits of precision to that observed for bulk matenal. As~uming t~e films to be ~sotropic in the plane of the substrate, a relation between applied stress and change III t.he amsot.ropy field IS developed and within limits experimentally verified. It is postulated that observed discrepancies are due to nonlinear anisotropic internal stresses and dispersion in the direction of the easy axis of the film.

INTRODUCTION

I T is the purpose of this paper to report in detail theoretical considerations and experimental verifi­

cation of the magnetoelastic behavior of thin films of nickel and iron ranging in composition from 65% Ni-35% Fe to 90% Ni-lO% Fe. A classical energy equation is developed based on phenomenological parameters. The equilibrium conditions for different states of applied stress and external field are derived . ' the parameters m the theory are related to parameters which can be measured in the laboratory, and the theory is compared with experiment. It is found (1) that the value of the saturation magnetostriction parameter (A8) is in substantial agreement with the values found for bulk material in the composition range studied, (2) that A. can be represented as isotropic with respect to direction in the plane of the film, (3) that the mag­netoelastic parameter (Tj) is substantially independent of the magnitude of the strain applied externally during measurement, (4) that within the limits of precision of the experiment the magnetoelastic parameter (Tj) is not a function of the substrate temperature during deposi­tion, and (5) that the model postulated is in general verified though discrepancies are observed which may be due to residual strains in the substrate and dispersion in the direction of the easy axis.

THEORETICAL CONSIDERATIONS

It has been assumed and to a considerable extent verifiedl- 3 that the energy due to field-induced anisot­ropy in a thin film can be represented as K sin2A, where K is the phenomenological anisotropy constant and A is the angle between the observed original easy axis and the direction of the saturation magnetization (M.). It

* This research was supported in part by a grant from the National Science Foundation.

t Present address: Physics Department, University of North Carolina, Chapel Hill, North Carolina.

1 D. O. Smith, J. App!. Phys. 29, 264 (1958). 2 C. D. Olson and A. V. Pohm, J. App!. Phys. 29, 274 (1958). 3 M. Takahashi, J. App!. Phys. 33, 1101 (1962).

has also been assumed and indirectly verified4 that the energy density in bulk material due to a uniaxial stress (0') which is applied at an angle (B) with respect to the direct.ion of the saturation magnetization in a single domam can be represented as 1A.O'sin2B, where As is the saturation magnetostriction constant for the mate­rial under consideration. In what follows it is assumed that the field-induced anisotropy is a consequence of the application of the field (in the absence of texture axis effects5) and that in the absence of perturbing in­fluences the observed original easy axis should be aligned with the direction of the applied field during deposition. For nonzero magnetostrictive materials the observed original easy axis might not lie along the depo­sition field direction due to residual anisotropic stress in the material in the plane of the film. It is postulated in what follows that a set of residual stresses (Ti) exist at angles (1';) with respect to the observed original easy axis [see Fig. 1 (a)]. In this study an external stress (0') can be applied at an angle (f3) with respect to the ob­served original easy axis, an external field (H) can be applied at an angle (a) with respect to the observed original easy axis, and the observed original easy axis is oriented at an angle (if;) with respect to the field that was applied during the deposition of a film. As a con­sequence of these various perturbing influences the saturation dipole moment per unit volume (M.) is oriented at an angle (0) with respect to the observed original easy axis. It is further assumed that A8 is iso­tropic, i.e., has the same value in all directions in the plane of the film). As a consequence the energy density may be represented in the following manner:

E=K sin2(0+if;)+!AsO' sin2(fi-0) +!A.Li T; sin2('Y;-0)-MsH cos(a-O). (1)

It is explicitly assumed that the energy due to the field­induced anisotropy is referenced with respect to the

4 R. M. Bozorth, Ferromagnetism (D. Van Nostrand Company Inc., Princeton, New Jersey, 1951), p. 612. '

5 D. O. Smith, J. App!. Phys. 30, 264S (1959).

715

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716 MITCHELL, LYKKEN, AND BABCOCK

B.

H

ORIGINAL

EASY

AXIS DEPOSITION

fIELD

yf3 '0

(a)

V r--------r--------~

o .".

f3 (b)

FIG. 1. (a) Co­ordinate system for proposed model; (b) allowed regions in which equilibrium may be accomplished by application of stress at specified angles ({1).

deposition field direction. Equilibrium is determined for a given set of external conditions when JE/O(} = ° and J2E/a02>0. If one applies the above conditions to the case in which no external field or stress is applied (H=O, 0-=0) then one finds that

(2)

If one considers the case in which an external field is ap­plied at right angles to the observed original easy axis but no external stress is applied (H¥-O, a=7r/Z, 0-=0) one finds for the equilibrium condition upon substitu­tion of (2) that

ZK cos2.r+3}..8Li T; cos2'Yi-M.EI/sinO=0. (3)

The hysteresis loop of the film when driven at right angles to the observed original easy axis is reversible and such that M is proportional to H for small drive. In this region the switching is presumably by rotation and the component of the saturation dipole moment along the drive direction is M =Ms sinO. If this process were extrapolated until the dipole moment was just aligned with the applied field then it would be necessary to apply a field (Hk) such that M/M 8 =H/Hk=sinO, and substituting in (3) above one obtains

Next consider the case in which the film is extended (or compressed) along the difficult axis and driven along the difficult axis (Hrf'O, a=7r/Z, q¥-O, (3=7r/Z). From the equilibrium condition one finds that

H k' = (2K cosZ.r)/ M.+ (3}..sLi Ti COS2'Yi)/Ms - (3}..sq)/M., (5)

where H k' is the new anisotropy field in the case when an external stress is applied.

Defining AHk=Hk-Hk ' and observing that if the stress is the result of an elongation (s) then 0-= Ys, where Y is the Young's modulus of the material, one finds from (4) and (5)

AHk =3}..sq/Ms=3}".Ys/M •.

Defining 7]= AH k/ s then

7]=3}..8Y/M •.

(6)

(7)

Consider now the case in which no external field is applied but an external stress is applied at an angle i3 (q¥-O, H = 0). The equilibrium condition reduces to the following upon substitution of (2):

sin20o[K cosZ.r+!}..sLi Ti COS2'YiJ -!}..8q sin2({3-(}o) =0, (8)

where (}o is the value of (} when the stress 0- is applied at an angle B with H=O. Substituting q= Ys as above, combining (4), (7), and (8), and simplifying, one finds that

(sin20o)/[sin2(iJ-Oo)J= 3Aso-/ (MsHk) = 7]s/Hk. (9)

Finally consider the case in which an external field is applied at right angles to the new easy axis which is de­termined by the above stress (qrf'O, H¥-O, a=7r/2+0o). Again the hysteresis loop will be reversible and such that M is proportional to H for small drive as in the case when an external field is applied perpendicular to the observed original easy axis in the absence of an applied stress. Defining (}=Oo+</I (see Fig. 2) one sees that </I represents the angle between the new easy axis and the direction of M. in the presence of a field H. Hence M/M.=H/Hko=sin</l, where H kO is the anisot­ropy field measured about the new easy axis. Writing (1) for the present case and substituting 0=00+</1, one can find the equilibrium condition by determining aE/ao= aE/J</I= 0. Using (2) and (4) in the equilibrium equation the internal strain variables can be repre­sented in terms of Hk. Finally using (9) to represent q

in terms of H k one obtains

HkO=[Hk sin2(3J/[sin2«(3-0o)]' (10)

Combining (9) and (10) one obtains

7]= (H kO sinZOo)/(s sin2(3). (11)

In order that the above equations may be valid it is necessary to determine the conditions under which

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MAGNETOSTRICTION OF THIN IRON-NICKEL FILMS 717

NEW EASY AXIS WITH STRAIN S

Hand ____________ I

DEPOSITION FIELO

~~~UP ~KNIFE EDGES

FIG. 2. Schematic diagram of strain and loop display apparatus as set up for measurement of H kO • In this case (3-()o=7r/4 and a-8o=7r/2.

equilibrium can be established by the application of an external stress (0') at an appropriate angle ({3). This requires that in addition to aE/ae=O one must have a2E/afJ2>0. The results of this determination as a function of (3 and eo are shown in Fig. 1 (b) for the case of a positive magnetostriction material (X.>O). The unshaded regions are those in which equilibrium can be established and the sign of the stress is that required for a positive magnetostriction material. If the material has negative magnetostriction the sign of 0' changes in Fig. 1 (b) in all allowed regions.

If one examines the energy equation (1) (excepting the term in H) one sees that this function has a period 7r and is of even order. Hence examination of one region of tension and one of compression in Fig. 1 (b) is suffi­cient to examine the model.

In the absence of stresses applied at angles other than {3=7r/2 one determines the orientation of the easy axis by observing a characteristic symmetry of the hysteresis loop of the film when driven along the diffi­cult axis to saturation. This criterion fails when large external'Stresses are applied at angles other than {3=7r/2. In order to examine the model in detail it was necessary to determine the direction of the new equi­librium axis (i.e., determine eo) experimentally when a stress was applied at some angle other than {3=7r/2.

If one examines the variation of H kO at equilibrium as a function of a variation of the magnitude of 0' when the stress is applied at a specified angle {3 one finds utilizing (9) and (10) that

aH kO/aO' = (aH kO/ae) (ae/aO') =nXs cos2({3-eo)]/M.. (12)

From this one sees that if {3-eo=7r/4 then H kO will be at an extremum. This criterion was used to determine

eo in all the experimental work in which O'~O and {3~7r/2.

APPARATUS AND EXPERIMENTAL PROCEDURE

In the experiments which were performed in the laboratory 1/1, a, {3, eo, s, H k, AH k, and H kO were measured in various combinations and compared with the theo­retical relations in the last section.

The basic test apparatus for these experiments was a transistorized 60-cps hysteresis loop display appa­ratus using a pick-up coil composed of two pile wound coils, the axes of which were parallel to each other and mounted perpendicular to the surface of the film under study and so oriented as to pick up the net flux change along the direction of alternating drive. The signal so detected was integrated and displayed as a hysteresis loop on an oscilloscope. For purposes of determining H it in the various parts of the experiment photographs were made of the loop for small drives and for drives sufficient to saturate the film along the difficult axis.

Stresses were applied to the films by bending the substrate which was supported between two parallel knife edges. The bending was accomplished by deflect­ing the center of the substrate by means of a third movable knife edge parallel to the first two. A lever coupled to the center of the substrate rotated a mirror which was part of an optical lever system. The gain of the over-all level system was 330. With this apparatus the deflection of the center of the slide could be meas­ured with a precision of 0.003 mm. The entire straining apparatus could be oriented at any arbitrary angle with respect to the drive field in the loop display apparatus.

Figure 2 shows a schematic diagram of the apparatus as set up to determine H kO when a strain s is applied at an angle {3~7r/2 such that {3-0o=7r/4. To arrive at this configuration one first determines the observed original easy axis by determining the original difficult axis which is perpendicular to it. From knowledge of the field deposition direction one can determine 1/1. A strain s is applied at some angle {3 while the film is driven mag­netically along a direction a. The slope of the hysteresis loop is observed with the pick-up and s is varied until this slope is an extremum which according to (12) assures that a-Oo=7r/2.

One group of films was deposited on 25-mmX 75-mm microscope slides and a second group of films was de­posited on 1-mm-thick circular glass plates which were 75 mm in diameter. In the case of the former sub­strates, strain could be applied only along the major dimension of the substrate. In the case of the circular slides the substrates could be rotated with respect to the straining apparatus so that the strain could be applied along any axis in the plane of the film.

The strain and resultant stress that was imposed on the film by such a deformation of the substrate could only be inferred and the over-all straining effects depend on the assumed deformations that take place in the

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718 MITCHELL, LYKKEN, A:>ID BABCOCK

substrate. It was assumed that the substrate could be treated as a homogeneous beam and that the median plane parallel to the film surface was bent but un­changed in dimensions. If the film was on the concave side of the substrate in the deflected position then the film was assumed to be compressed along an axis in the plane of the film and perpendicular to the mechanical supports; if the film was on the convex side it would be extended along this axis, and in either case from a knowledge of the thickness of the substrate, the de­flection at the center of the substrate, and the separa­tion of the mechanical supports it was possible to cal­culate the strain imposed on the film along the above specified axis. In the case of the rectangular substrate the strain imposed at the center of the film was

(13)

where y was the deflection at the center of the slide, t the thickness of the substrate, and C the distance between the fulcrums. Averaging the strain over the dimensions of the film (8 mm in diameter in this case) it was found that

(14)

where all dimensions were measured in millimeters. In the case of a circular substrate (see Fig. 2) of

radius (r) supported symmetrically on fulcrums which were separated by a distance C the strain at the center of the film was found to be

s= ~tyer~3[ - 2C/r+(r~2(rLtf2)!

+2 sin~1(e/2r)], (15)

and setting r=IC the average strain was found to be

s= 5.21ty/ (2. (16)

Evaluation of the effect of the applied strain on the film was complicated by the fact that when the film was elongated in one direction the film was shortened at right angles to this direction. A similar effect must have occurred in the various parts of the substrate. In a cylindrical rod this effect is reflected by the magnitude of the Poisson ratio of the material which is about 0.23 for glass and 0.31 for nickel. Assuming these values for the relative change in length in a direction perpendicular to the direction of the initial strain it was found that a compression along the initial direction gave rise to a compression at right angles to the initial strain direc­tion. As an approximation the strain of the film in a direction parallel to the surface of the film and at an angle (~) with respect to the original direction was assumed to be

sW=s(0.2+0.8 COS2~). (17)

The contribution due to such a complex of strains was

calculated to be

(18)

where E(IT,{3) was the energy contribution due to a stress (IT) in the film at an angle {3. Assuming that IT= Ys, the important conclusion reached by these last arguments was that the energy contribution due to an externally imposed strain was proportional to the de­flection measured at the center of the substrate. Evalu­ation of the constants in the theory depends on the validity of some assumptions just made.

The films used in this experiment ranged in composi­tion from 65% Ni~35% Fe to 90% Ni~10% Fe. Com­position was determined by winding appropriate amounts of iron and nickel wire each 0.009 inches in diameter onto a O.060-in.-diam tungsten filament in a total amount sufficient to make a film nominally 3000 A thick. As much of the metal as possible was removed by evaporation and the end composition of the film was assumed to be that of the original charge. The evapora­tion time was of the order of 60 sec.

Evapora tion was performed in an all metal dry vacuum system6 in which the pressure was about 5 X 10-6 mm Hg during deposition in a magnetic field of 20 Oe. The substrates were radiantly heated in a massive copper holder in which the substrate tempera­ture was monitored by a thermocouple cemented with Sauereisen cement to a dummy substrate.

RESULTS

Assuming As, Y, and M. to be constant for a given film, (6) predicts that Hk' should be directly propor­tional to s. From Fig. 3 it can be seen that this was the case for 5 of the 6 samples represented. It should be noted that 159-1 and 159-2 refer to two different meas­urements of the same film. In 5 cases of 33 a measurable nonlinearity was observed. In all of these measurements the film was compressed or extended along a direction defined by /3= n/2. If the magnetostriction was positive the film was compressed and if the magnetostriction was negative the film was extended.

Assuming As, Y, and Ms to be a function only of the composition (7) predicts that 1] should be a single­valued function of the composition. By measurement of the slopes of curves such as those shown in Fig. 3 the results tabulated in Table I and plotted in Fig. 4 were obtained. The division of the data into 3 groups was based on the assumption that those films for which H k

and/ or y; was large might have contained high internal anisotropic strains. It is believed that such strains might

6 E. N. Mitchell, Proc. N. Dakota Acad. Sci. 14, 70 (1960).

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MAGNETOSTRICTION OF THIN IRON-NICKEL FILMS 719

TABLE 1. Summary of data for dependence of 'I on composition.

COI%osition Film No." (0 Ni) "v (degrees) Hk(Oe) 'Ib

179 65 -8 8.4 69X 1()3 118c 68 13 5.0 1180e 68 11 4.8 51 120c 68 4 5.1 56 1200c 68 7 6.9 115e 68 6 5.9 50 1150e 68 2 5.9 53 1770e 69 8 3.6 53 172 70 3 8.7 63 176 71 -5 7 49 110e 72 -9 3.2 46 1100e 72 -7 7.9 53 178e 73 -11 6.0 42 1780e 73 12 4.0 46 1070e 75 -9 ;'.1 41 111c 77 2 3.2 32 1110e 77 0 4.3 36 1090e 78 -3 3.7 34 106c 79.5 -4 2.6 23 1060e 79.5 -4 3.3 23 108e 81 6 4.8 1080e 81 1 4.9 10 98e 82 -1 2.0 1 960c 84 1 1.6 -12

154c 85 -3 2.6 1590c 85 -12 3.4 -19 99c 86 9 2.4 -25

102c 87 -9 3.1 -39 1020e 87 9 2.1 100c 88 -1 2.2 -36 lOOc 88 4 2.2 -44

• The notation c or oc following a given film number specified whether the film examined was positioned over the center of the evaporation filament or displaced from the center.

b No ~ measurement was recorded for those films in which the data did not fall on a straight line when plotted as in Fig. 3.

have masked the anticipated results for reasons which will be explained later.

The values reported here disagree quantitatively with earlier values reported by Mitchell and Lykken.7 On comparison of the two experiments these values are more reliable because of determinant errors in the earlier experiment. The compositional dependence, however, agrees with this and other earlier work. 5 •7

Using polycrystalline data for M. and A8 for bulk materialS of the same composition as that of the melt (Fig. 5) shows 1/ for the films plotted as a function of A./ M. for bulk material. Assuming Y to be at most a slowly varying function of the composition in this region and considering the correction introduced by (18) one would expect from (7) to find a straight line in the above plot. The slope of the above line would be 0.6'11' Y if the films have the same magnetostriction as the bulk material. The straight line through the origin in Fig. 5 has a slope such that Y=2.1Xl012 dyn/cm2

,

which is the Young's modulus of bulk 78.5% Ni-21.5% Fe material. 9

7 E. N. Mitchell and G. 1. Lykken, J. Appl. Phys. 33, 1170 (1962).

8 R. M. Bozorth, Ferromagnetism (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1951), pp. 109,667.

9 O. Engler, Ann. Physik 31, 145 (1938).

20

• NO. 119 165-35) 18 A NO. flOC 172-28)

• NO. 1~-2 (15-25) 16 • NO. IOBC (81-19) •

o NO. 99C (86-14)

14 o NO. IOOC (88-12)

" NO. 159-1(75-25 12

;;:; 9 10 .>< X

O·~L-*2~~4~~6~~~8-L~IO~~~~~14~~16 lSI XIO-~

FIG. 3. Dependence of H k' on stress applied perpendicular to the observed original easy axis for selected films.

A better fit with the data could be obtained by as­suming a different value for Y (1.9X 1012 or 2.0X 1012

dyn/cm2). Hence the results reported here are in sub­

stantial agreement with bulk data for polycrystalline material. At the lower end of the nickel composition range studied, and to a certain extent at the upper end, the agreement is not too good. Here the magnetostric­tion is much higher and discrepancies with the theory may be expected to appear first. The experimental

70 + 0

60 X HK ~5; 1"'1~5' L .+ o HK >5; 1"'1~5"

50 + 0++ 11/11>5" + +

x x 3 x

20 x CiI 0 .... 10 ~ X

.... 0 0 x

-g -10

~ )(

-20 + x x

-30

x x x

-50 0

-60

-7~5 70 75 so 85 90

MELT COMPOSITION "N i

FIG. 4. Experimental dependence of 'I on composition. Data are divided into three groups as indicated using anisotropy field (H k) and rotation of easy axis ("v) as criteria for grouping.

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720 MITCHELL, LYKKEN, AND BABCOCK

xHKS5;11/I1s.5°

• HK>5; Il/Ils5°

+ 11/11> 50

-20 -15 -10 X 10-9

•

7)

70 X 103

60

50

40

30

20

10 86"- 84"-

I I

-5

x -20

-30

-40

-50

-60

-70 X IO~

fi"

+ f ++ +

+ +

""THEORETICAL x SLOPE

+

20MS Xl0-9

FIG. 5. Measured value of '1 for films plotted as a function of As/Ms for bulk material of the same composition. The line indi­cates the predicted relation using Young's modulus for bulk Permalloy.

error in 1/ was at most 5% for any given measurement. The composition of the source was measured pre­

cisely, but alloying of the nickel or iron preferentially to the filament may have altered the end composition of the film, and in the case of the film of composition nearest zero magnetostriction (98C) an error in com­position of at most t% nickel (which was possible in this experiment) would make the measured values for this film agree better with bulk values. Films of nomi­nally 70% Ni for which the data do not agree with the expected curve would have to experience a composition drift of as much as 4%. This could have happened in instances but not as repeatedly as was here observed. Part of the difficulty in this region is presumed to be experimental since in some of these films it was not possible to strain along the original difficult axis «(3=11/2) since the substrates were rectangular and the easy axis was not properly aligned with respect to the edges of the slide.

A series of 15 films of nominally 75% Ni-25% Fe material were deposited at deposition substrate tem­peratures which were in the interval from 50° to 400°C. Within the limits of the precision of this experiment 1/ was found to be independent of substrate temperature in this interval, with a mean value of 39.3X 103 Oe and an average mean deviation of 2.1X108 Oe.

In order to study the angular dependence of As in these films and to further investigate this theory a series of films were deposited on circular slides and were strained at various angles «(3) with respect to the easy axis. The position of the new easy axis (00) was de­termined by requiring (3-00=11/4 [see Eq. (12)J and finding the extremum in H kO when cr was varied con­tinuously. Since sin2 (8-00) = 1 a plot of sin200 as a

TABLE II. Summary of data for '1/Hk for selected films.

Composition Measured Film No. (%Ni) ~(Oe) Hk(Oe) ~/Hk slope

144 75 44 X 10' 5.6 7.9 X10' 7.2 X 10' 152 75 48 4.7 10.4 9.1 169 85 18 2.3 8 9.6 159-1" 75 70 to 43 b 6.2 11.3 to 6.9' 8.5 159-2" 75 77 to 31 b 5.6 13.7 to 5.5' 7.1 164 75 44 6.5 7 6.5

It The number after the dash represents a specific measurement on the film.

h The range of 71 allowable due to a nonlinear relation between Hk' and s. c The range of allowable l1/Hk as a consequence of a range in 71.

function of s should result in a straight line for which the slope is 1// H k [see Eq. (9)].

Such data are plotted in Fig. 6 for several films and the results are tabulated in Table II.

In these results 00 has been limited such that o ::::00 -:::: ·1l/4. Those films which exhibited positive mag­netostriction were extended and those which exhibited negative magnetostriction were compressed. One con­cludes from these data that for films 144, 152, and 169 the theory is satisfactory within the limits of the re­producibility of the experiment and that the value of As is the same in all directions examined.

In an effort to comprehensively evaluate the theory, film No. 159 was subjected to a remeasurement of the type just described for the verification of (9). Simul­taneously with the latter measurement H kO was meas­ured. From these data by the use of (10) and (11) 1/ and H k were calculated as a function of (3. These results are shown in Fig. 7 and the ratio 1// H k is shown in Fig. 6.

Figure 6 shows that while the slope (1// H k) is a constant for a given set of measurements this ratio is changing with time (see 159-1 and 159-2). If this model

1.0

.9

.8

.7

.6

crl' '" z .5 iii

.4

.3

.2

2 4 6 8

lsi 10

.. NO. 152

C NO. 159-1

x NO. 144

• NO. 159-2

• NO. 164

12 14 16 X 10-5

FIG. 6. Experimental relation between angular rotation of the easy axis and applied stress at specified angles (1l" /4 <fJ < 1l" /2).

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MAGNETOSTRICTION OF THIN IRON-NICKEL FILMS 721

14 a 13

12

II

0 9 ... ~ 8 a ...J :>

7 ~ .. 6 a a a !,! a a a

" :z: 5

4

3

2

~5' 50" 55' 60' 65' f3 70" 75' 80" 85' go'

a

5' 50' 55 60 85 90"

FIG. 7. Top-values of H k predicted theoretically from observed modified anisotropy fields (H kO ), applied stress, and position of the new easy axis; bottom-values of "1 predicted from the same data.

is correct 1/ and H k should be constants independent of {3. Except for the results of measurement at {3 = SOo and

{3 = S5 ° the values of 1/ and H k are found to be constants within the limits of precision of this experiment. It is believed that the difficulty with the measurements at {3=SO° and S5° may be attributable to dispersion in the direction of the easy axis. In as much as dispersion has been observed by other experimentersio this difficulty is to be expected since {3 is no longer uniquely defined and the difficulty will be compounded when f3 is not equ'al to 90° but is not too different from 90°. .

Afterward, the measurements for determining 1/ in which the film was strained along the difficult axis (f3=7r/2) were repeated and this result is plotted in Fig. 3. Repeated direct measurement of H k after various straining operations of this film showed it to be ranging from 5.5 to 7.5 Oe which is more than is usually observed. Figure 3 shows that H k' was not a linear function of s. The nonlinearity seems to have been in­creasing as the film aged and/or was mechanically worked as is shown in Fig. 3 (see No. 159-1 and No. 159-2).

The deviation from linearity could be explained in terms of a change in H k • Examination of (1) and (4)

10 D. O. Smith, J. App!. Phys. 32, 70S (1961).

shows that this model assumes that H k is constant as here defined [see Eq. (4)]. If k, 'Yi, and Ti were varying with time or mechanical working, the model might still be valid but the derivation oversimplified. The energy due to (J was calculated for all directions in the plane of the film (lS) assuming linear changes in stress along a direction such as one determined by 'Yi. Hence a linear change in Ti was considered but nonlinear terms were neglected. If the film was highly stressed (as it seemed to be) then the measured variation of Hk and the ob­served nonlinearity in H k' as a function of s could be explained in terms of nonlinear changes in internal stress. The variation in H k as measured would be an example of a nonlinear irreversible internal stress. The nonlinearity of H k' as a function of s would be an example of a nonlinear reversible change in the internal stress.

The effects observed could be explained in terms of changes in K but if this is the case the frequency of the effect would be expected to be independent of the mag­netostriction of the material. Examination of Fig. 5 indicates that deviations from the predicted theory are more common for the more magnetostrictive materials than for those of near the composition for zero magnetostriction.

This discussion explains why films with abnormally high H k and/or if; were separated from the rest of the sample in Fig. 4 and 5. If the model is correct and H k

and/or if; were high because the anisotropic internal strain was abnormally high, then the probability of nonlinear reversible changes in the internal strains would be greater giving rise to erroneous values for 1/.

The value of 1/ and H k as measured directly should be equal to the values calculated from {3, 0o, and H kO. This is found to be true to the precision with which 1/ and H k can be determined directly.

A set of measurements on fIlm No. 159 were made in which 0<00 <7r/4, and 37r/4<f3 <7r, and a plot such as those of Fig. 6 was made. The data fell very well on a straight line of slope equal to S.2X 103 indicating that to the extent that the model was valid in the first quad­rant, it was also valid in the second.

A more detailed examination of all the data taken on film No. 159 indicated discrepancies which can be ex­plained assuming that nonlinear stress-strain effects do occur in the film which increase in frequency with the amount of working of the film. Examination of all the films deposited on circular substrates also indicates the presence of such effects. This may be due to the manner in which the circular substrates are made and handled.

CONCLUSIONS

One concludes that the classical theory as here ex­tended can be used to describe the magnetostrictive behavior of these thin ferromagnetic films and that their behavior is much like that of bulk polycrystalline Permalloy. Specifically the following is concluded:

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722 MITCHELL, LYKKEN, AND BABCOCK

(1) that the value of As is in substantial agreement with the values for bulk polycrystalline material of similar composition, (2) that A8 is isotropic in the plane of the film in agreement with the findings of direct structure studies,!l (3) that within limits '1/ is independent of the external strain during measurement, (4) that '1/ is in­dependent of the substrate temperature during deposi­tion, (5) that if strain is applied repeatedly to the film the magnetic anisotropy is modified irreversibly and probably also in a reversible fashion, and (6) that this modification of the magnetic anisotropy is probably due to a nonlinear change in the internal stresses and con­sequent change III the magnetostrictive energy anisotropy.

11 R. F. Adamsky, J. App!. Phys. 31, 289S (1960).

One sees that '1/ is probably a good parameter for de­termining the composition of a film provided that nonlinear strain effects do not enter into the measurement.

From an applications point of view one can see that the average magnetostriction must be low for, if it is not, it will be impossible to make films of the same anisotropy on a commercial basis due to anisotropic stresses introduced in fabrication. In addition, aniso­tropic internal stresses in such films could lead to easy axis rotation, and dispersion in both magnitude and direction of the preferred magnetic easy axis. If the average magneostriction is low but the deviation from that average is large in different parts of a given film then the difficulties just mentioned still exist.

JOURNAL OF APPLIED PHYSICS VOLUME 34. NUMBER 4 (PART 1) APRIL 1963

Application of the Simplified Cylindrical Distribution Function: Orientation in Linear Polyethylene

M. E. MILBERG

Scientific Laboratary, Pard M otar Company, Dearborn, Michigan

(Received 15 June 1962)

The simplified cylindrical distribution function has been applied to the determination of the degree of orientation in a moderately oriented linear polyethylene fiber. The function has been calculated using the first two and ten lines of the scattering pattern and, though poorly resolved, yields useful orientation infor­mation. It is concluded that the greater the knowledge of the molecular structure of the material under investigation the less the amount of intensity data required to produce a useable function. Although the technique is applicable to both crystalline and noncrystalline materials, the standard methods, applicable only to crystalline materials, are simpler. Therefore, the method can be considered to apply mainly to moderately oriented noncrystalline fibers.

INTRODUCTION function3 can be written

m 00 RATHER simple and straightforward methods1.2 exist for obtaining structural information from

the x-ray scattering patterns of oriented crystalline high-polymer fibers. However, aside from trial and error methods, cylindrical distribution analysis provides the only sound method of obtaining such information when the fibers are noncrystalline. In a recent article,3 a simplified cylindrical distribution function was de­veloped. This function gives a considerable amount of structural information about moderately oriented, cylindrically symmetric systems and can be obtained with substantially less labor than can the corresponding complete function.

D'(r,a)=L L D2n'/(r)P2n(Cosa), (1)

For a moderately oriented material having a scatter­ing pattern containing anisotropic lines or halos and a very nearly isotropic "background" the simplified

1 J. J. Hermans, P. H. Hermans, D. Vermaas, and A. Weidinger, Rec. Trav. Chim. 65, 427 (1946).

2 R. A. Sack, J. Polymer Sci. 54, 543 (1961). 3 M. E. Milberg, J. App!. Phys. 33, 1766 (1962).

i=l n=l

where

D2n,/(r) = (-1)n47rG2n'ifooo S2Hi(S)hn(27rSr)dS, (2)

and

G2n ,i=H4n+1) 10" Gi(rp)P2n(cosrp) sinrpdrp. (3)

The variables Gi(rp) and Hi(S) are defined by

1/(S,rp) = Gi(rp)Hi(S), (4) where

1/ (S,tp) = Ii(S,tp) - Io,i(S), (5) and

Io,i(S)=! 10'11' I i (S,rp) sinrpdtp. (6)

Ii (S,tp ) is the corrected intensity distribution in the ith

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