NGYUEN CHI THANH and R. KRISHNAN: Anisotropy of Magnetostriction in Col-,Zr, 313

phys. stat. sol. (a) 90, 313 (1985)

Subject classification: 18.2.1; 2; 21.1

Institute of Physics, Polish Academy of Sciences, Warszawal) (a) and C N R S Laboratoire de Magnetisme, Jleudon- Bellsvuez) ( b )

Anisotropy of Magnetostriction in Coevaporated Col- ,Zr, Amorphous Thin Films

BY NGUYEN CHI THANH (a) and R. KRISHNAN (b)

Magnetostriction of amorphous Co,,Zr,, thin films is measured by means of strain modulation ferromagnetic resonance a t room temperature. It is shown that magnetostriction of investigated films has anisotropic character and has to be described by four independent magnetostriction constants. The results support applicability of the columnar structure model of magnetostriction to the investigated system.

Es wird die Magnetostriktion von amorphen diinnen Co,,Zr,,-Schichten mittels spannungsmodu- lierter ferromagnetischer Resonanz bei Zimmertemperatur gemessen. Es wird gezeigt, daB die Magnetostriktion der untersuchten Schichten anisotrop ist und durch vier unabhangige Magneto- striktionskonstanten beschrieben werden muB. Die Ergebnisse bestiitigen die Anwendbarkeit des Spaltenstrukturmodells der Magnetostriktion auf das untersuchte System.

1. Introduction

It has recently been proposed that three mechanisms are responsible for the magneto- striction of amorphous thin films [l]. These are: one-ion mechanism [2 to 41, two-ion mechanism [ 5 ] , and mechanism connected with anisotropic microstructure of the film (columnar structure of the film) [6]. All these models lead to anisotropy of magneto- striction. The anisotropy of magnetostriction is particularly strong in the case of films in which columnar structure exists. It seems that one of the best systems to study the anisotropy of magnetostriction are the amorphous Col--zZrz thin films. It was shown by Krishnan et al. [7 ] that the coevaporated amorphous Col-sZr, (0.074 5 x 2 0.27) thin films showed the presence of columnar microstructure.

The main purpose of this paper is to determine all magnetostriction constants and all components of the magnetoelastic tensor for the amorphous Co,,Zr,, thin films in order to check the character of magnetostriction in this system.

2. Magnetoelastic Energy and Magnetostriction in Amorphous Thin Films

Consider an amorphous ferromagnetic thin film, let the film lie parallel to the XOY plane, the 2-direction be the film normal (as shown in Fig. 1). In order to give directly the measured saturation magnetostriction constants a t a constant stress and at a given temperature, it is better to use the stress and the direction cosines of magnetization as the fundamental variables. Thus the magnetoelastic energy can be written in the form

1) Al. Lotnikow 32/46, 02-668 Warszawa, Poland. z, 92190 Meudon-Bellevue, France.

314

I ' NGYUEN CHI THANH and R. KRISHNAN

Fig. 1. The film with coordinate axes

In this equation, ukl are the stresses expressed as a second-rank tensor with k inter- changeable with 1,a6 are the three direction cosines of magnetization, Aijkl are the first- order magnetostriction terms expressed as a fourth-rank tensor (the Aijkl tensor is similar to the photoelastic I T i j k l tensor).

For amorphous thin films the magnetoelastic tensor Aijal has the following form [2 to 61 (written in Voigt notation):

A =

where

-All - All

A31 4 3

0 0 O A 4 , 0 0 0 0 0 O A , , O 0 0 0 0 0

The magnetoelastic energy can be written then in the form

The form of the saturation magnetostriction can be obtained from the equation for the magnetostrictive strain in any direction which is given by the formula

where the energy E is usually expressed as a change from the magnetoelastic energy of the demagnetized state, pl, pz, and p3 are the direction cosines of the direction for magnetostrictive strain with respect to the X , Y , and Z axes defined above, and 1 is the saturation magnetostriction. It is clear that the form of magnetostriction depends on the magnetization distribution in the demagnetized state of the investigated film.

The expression for saturation magnetostriction in an amorphous thin film with perpendicular anisotropy ( K , > 0) will be given by

Anisotropy of Magnetostriction in Coevaporated Col -%Zr, Amorphous Films 315

where AA, AB, ic, and A D are the magnetostriction constants which are connected with the components of the magnetoelastic tensor Aijkl by the following equations:

The saturation magnetostriction i n an amorphous thin film with easy plane type anisotropy ( K , < 0) will be given by the following expression:

A = AT[2(%/% -k a2pd2 - (1 - a:) (1 - p i ) - (a181 -k %$z) -k + A:[&1 - $1 - -k azpz) f- f A8L.gpg - + a & ~ ) a&] + + 4AD*(nlB1 + a z B z ) a3P3 (8)

where Ax, A:, A:, and A,* are the magnetostriction constants which are connected with the components of the magnetoelastic tensor Aijkl by the following equations :

1: = {- (A11 - A12) ) 1: = (A31 - A12) - (A11 - A12) , 1: = (A33 - A13) Y

1 = A*[(az, - f) p1" + (a; - f) pi + 2a1a2p1p2 - (a23 - f) (1 - p31 -

2.; = (AT +A: +A: + 2A4,). (9) The expression for saturation magnetostriction in an amorphous isotropic thin film

will be given by 0

- A%d - f) pf + (4 - $1 p2" + 2061azj31/9* f- 2(a; - f) (1 - 823) -

+ ~ A D ( w % + a3B3 , (10) - (alp1 + azpz) W%1 - - f) pg - (a181 -k %/%) -k

where the magnetostriction constants &,A!, A!, and A! are connected with the com- ponents of the magnetoelastic tensor Aijkl by the following equations :

0

0

0

AA = 3 ('11 - '12) - f ('31 - ' 412) 9

I C = - f (A33 - A13) >

A! = - % [(A11 - '12) f- - A1211 9

(11) AD = - f (A,i - A12) + ('433 - f 6&I Now turning to (6) to ( l l ) , it is evident that the magnetostriction in amorphous thin films will be isotropic and described by only one magnetostriction constant 2, (= All) if the following relations are fulfilled :

(12) A - A - A - - L A - - L A - - L A - - . L A 1 2 - 13- 31- 2 11- 2 33- 3 4 4 - 3 6 6 '

In the opposite case, the magnetostriction has an anisotropic character and has to be described by four independent magnetostriction constant,s connected with the com- ponents of the magnetoelastic tensor Aijk l by ( 7 ) ) (9), and (11).

3. Measurement of Magnetostriction

3.1 Theoretical analysis

There are several methods which are most commonly used to measure magnetostric- tion of amorphous magnetic materials, namely : a) strain gauge extensometry [8], b) the so-called three-terminals capacitance dilatometer [9 to 111, c) the cantilever- capacitance method [12, 131, d) the strain modulation ferromagnetic resonance (SMFMR) method [14, 151, e) the so-called small-angle magnetization rotation method

316 NGYUEN CEI THANH and R. KRISHNAN

[16,17]. The choice of a particular method depends mainly on sample geometry, the sensitivity required, and the temperature range of the measurement to be performed. To the best of our knowledge, the only method which enables one to determine all components of the magnetoelastic tensor as well as all four magnetostriction constants in amorphous materials is the SMFMR method introduced independently by Henning and den Boef [14] and by Szymczak et al. [15] in 1978. The most important advantage of this method is its extraordinarily high sensitivity making it possible to detect R values as small as 1141.

Below formulae will be given which relate the shift of the ferromagnetic resonance line to the applied uniaxial stress on amorphous thin films.

In the presence of a magnetic field H sufficient to achieve technical saturation and an uniaxial stress 0 the free energy of the film can be written as follows:

E, = -HM,[sin 8 sin 8 H cos (pl - p l H ) + cos 8 cos 8,] + 2 n ~ f cos2 8 + + K, sin2 8 - a(All - A12) sin2 8 sin2 8, cos2 (q - p,) - - a(A3, - A12) cos2 8 sin2 8, - o(A,, - A13) c0s2 8 cos2 8, -

- f aA,, sin 28 sin 28, cos (pl - 9,) , (13)

where 8 and pl, 8 H and plE, 8, and pa are polar and azimuthal angles of magnetization M , static magnetic field H , and uniaxial stress a, respectively, the first term is the Zeeman energy, the term with coefficient K, represents the magnetic anisotropy energy of the film, 2zM: cos2 8 is the demagnetization energy, and the last four terms represent the magnetoelastic energy, tensile stress is taken positive.

The ferromagnetic resonance condition is given by [18]

where y is the gyromagnetic ratio and the derivatives of E, are taken a t the equili- brium orientation of M which is found from aE,/a8 = 0 and aE,/apl = 0. In a fixed- frequency FMR experiment the stress-induced shift of the resonance field AH = = H, - Ho is obtained by evaluating (14) for a = 0 and a + 0 and subtracting the results.

There are several orientations of the static magnetic field H a n d the uniaxial stress 0, for which the shift of the FMR line can be easily calculated. Below the analysis of these particular cases will be performed.

3.1.1 The uniaxial stress 0 lies in the f i lm plane and i s applied parallel lo the radio frequency field h. The static magnetic field €I can be rotated in the plane which is perpendicular to the film plane and perpendicular with the uniaxial stress a. (For example, H can be rotated in the XOZ plane, a is applied parallel with the Y axis, and h is applied parallel to the Y axis, see Fig. 2)

In this case, the total free energy of the system is given by the following expression :

E, = -HM,[sin 8 sin 8, cos (q - p l ~ ) + cos 8 cos O H ] + 2nMt COS$ 8 + + K, sin2 8 - a(All - A12) sin2 8 sin2 pl - o(A,, - A12) cos2 8 , (15)

where 8 and pl a t equilibrium must satisfy both aE,li38 = 0 and aE,/apl = 0. Equa- tion i3Et/i3pl = 0 always holds when pl = c p H . This means that the magnetization M

Anisotropy of Magnetostriction in Coevaporated Col -%Zr, Amorphous Films 317

Fig. 2. H can be rotated in the XOZ plane, u 1 1 h and is applied parallel to the Y axis

ii

always lies in t'he XOZ plane, in which H is applied. i3Et/a8 = 0 is given by

20 -H sin (8, - 8) + 4nMeff + - (& - A,,)] sin 28 = 0 ,

Ms

where 4nMeff = (2Ku/Ms) - 4 z M s . The second derivatives i32E,/i302, i32Et/i3p2, and iYEt/i38 8q.1 are substituted into (14). The resonance condition has the form

In (17), H and 8 must satisfy (16). When 0 = 0 the resonance condition of the system is given by

In (18) H, is the field value at resonance and Bo the polar angle of magnetization a t equilibrium, they must satisfy the following equation :

-Ha sin ( 8 , - 8,) + 2nMeff sin 28, = 0 . (19)

Putting the expressions H = H, + AH and 8 = 8, + A8 into (16) and (17) and neglecting the terms of second degree of A H , A8, and A , and their products, the shift of the ferromagnetic resonance line has the form

20 AH = - JfS

- [H, cos (e, - 8,) + 4 n ~ ~ ~ ~ cos 2e,] x G-1 +

1 20 sin 8, cos 28, - - 1 L sin 28, x G-l ) (20) 2 + - (A12 - '31)

MS where

H, sin ( 8 , - 8,) - SnMeff sin 28, L = - ctg 8,. H, ~ O S (e, - 8,) + 4 n ~ , f f cos 28,

318 NCYUEN CHI THANH and R. KFLISHNAN

3.1.2 The uniaxial stress is applied perpendicular to the static magnetic field and parallel with the radio frequency field. They lie in the plane which is per- pendicular to the f i lm plane. (For example, H 10, ci ( ( h, and H , t ~ , and h lie in XOZ plane, as shown in Fig. 3)

The total free energy of the system, in tjhis case, is given by (13). Because of H 1 0,

hence cos ( y H - q~,) = -1. Thus the equation 8Ekl8p = 0 always holds when q~ = 91,. This means that the magnetization lies in the XOZ plane, in which H is applied. i3Et/i30 = 0 becomes

x [(A,, - A31) sin2 0, + (Al3 - AS3) cos2 0,] sin 20 (21)

The resonance condition is given by d A,, sin 20, s} x rf)z = { H S + 20 (A,, - sin2 e, - -

sin 0 M s MS sin 0 20 Jfs

x H cos (0, - 0) + 4nMeff cos 20 - - [ (A, , - &) sin2 0, +

+ (Al3 - A33) cos2 0,] cos 28 - - A,, sin 20, sin 28 . (22) I 20 1

MS The shift of the ferromagnetic resonance line is expressed as follows:

20 AH = - - (All - A12) sin2 0, [H, cos (6, - 0,) +4nMeff cos 20,] x G-l +

Ms

20 sin OH MS sin 0, + - [(A,, - A31) sin2 0, + (A13 - cos2 e,i H, -- x

1 1 x cos 20, - - L sin 20, G-1 + i 2

0 + - A,, sin 20, ctg 0, [H, cos (0, - 0,) + 4zMeff cos ZOO] + MS

(2 sin 20, + L cos 20,) sin 0,

t' a

Fig. 3. Zi -1- u, o 1 1 h, and H, a, h lie in the XOZ plane

Anisotropy of Magnetostriction in Coevaporated Col -zZr, Amorphous Films 319

3.1.3 The uniaxial stress i s applied parallel with the radio frequency field and lies in the plane which is perpendicular to the f i lm plane. The static magnetic field lies in the f i lm plane and is perpendicular to the uniaxial stress. (For example, 0 I I h and lies in XOZ plane, H i s applied along the Y axis, see Pig. 4)

The total free energy of the system, in this case, is given by (13). The equilibrium of magnetization is found from the following equations:

a

MS H sin 8 sin O H sin (p - VIf) + - (All - A12) sin2 8, sin2 8 sin (p - p,) 4

a 1 + - A,, MS -H[sin 8, COB 8 cos (p - pa) - sin 8 cos O H ] + 2nMeff sin 28 -

a a -_ (All - A12) sin2 8, sin 28 cos2 (p - qo) - - [ (A, , - A31) sin2 8, + MS MS

+ (Al3 - .433) cos2 O,] sin 28 + - A,, sin 28, cos 26 cos (p - p,) = 0 . (25)

(24) sin 28, sin 28 sin (p - p,,) = 0 ,

d

MS

If a = 0, (24) and (25) have the following solutions: if 4nNeff > 0:

po = pH = 90" and 8, = O H = 90" for W, 2 4nMeff,

if 4nMeff < 0: po = pH = 90" and 8, = 8H = 90" for an arbitrary value of H, .

It has to be stressed that the obtained solutions are true only for such values of H,, which is large enough to saturate the magnetization. (It means for the case when domain structure does not exist in the sample.)

Putting the expressions H = Ho + AH, 8 = 8, + A8 and p = po + Ap into the equations (24) and (25) and neglecting the terms of second degree of AH, A8, Ap, A , and their products, the shift of ferromagnetic resonance line has the form

2a AH = - (All - A12) sin2 O0 [H, sin 8, + 4nMeff cos ZOO] x G,-l + Ms

320 NGYUEN CHI THANH and R. KRISHNAN

where 1

G -- [2H0 sin 8, + 4nMeff cos 28, + HoL cos 13,] . O - sin 8,

In the case when either 0 < 4nMeff 5 Ho or 4nMeff < 0 the shift of the ferromagnetic resonance line (26) reduces to

* (27) 20 HO - - [(A,, - A,l) sin2 8, + (A13 - 4,) cos2 O,] MS 2Ho - 4nMeff

In the case when 4nMeff > Ho the shift of the ferromagnetic resonance line (26) reduces to

2a MS 8~M~ff(l6n'N;ff - 2Ht) Ms

H0(16n2M& - Hi) .. .+-x 20 AH = - (All - A12) sin2 8 ,

H: x [ (A, , - A,,) sin2 8, + (Al3 - A,,) C O S ~ O,] - - ~ _ _ _ _ _ ~ ~ ~ ~ - - - - . 4 ~ ~ M ~ f f ( 1 6 ~ ~ M , & - 2H3

(28) 3.2 Measurement configurations

In order to determine all components of magnetoelastic tensor A i J k l as well as four magnetostriction constants, strain modulation ferromagnetic resonance (SMFMR) and ferromagnetic resonance (FMR) spectra are taken simultaneously in the two measurement configurations described below. The first configuration with the uni- axial stress applied parallel to the film plane and parallel with the radio frequency field h and when the static magnetic field H can be rotated in the plane perpendicular to the film plane is shown in Fig. 5 . I n this configuration, the FMR field orientation dependence of the shift of the FMR line, AH versus Or,, is given by (20). (All - A12) and (Al2 - A,,) are calculated by making use of a computer program based on the least square method. The components Ail, A1,, A,,, and A,, as well as the magneto- striction constants AA, An (or A;, A: or A:, A:) are determined from the relations (3)

M

Fig. 5. The first configuration for measuring components A,,, A,,, A,,, and A,, of magnetoelastic tensor i ~ i j k l as well as magnetostriction constants LA, LB (or A:, A; or a;, A:)

Ailisotropy of Magnetostriction in Coevaporated Col -zrz Amorphous Films 32 1

..

Fig. 6. The second configuration for measuring components A,,, A,,, and A,, as well as magneto- striction constants Ac, AD (or A:, 15 or A$, A",. a) u I I h and u is applied a t 45" to the film normal, H 1 u, and H is applied parallel to the film plane. b) u 1 1 h and u is applied a t 45" to the film normal, H 1 a, and H is applied a t 45" t o the film normal

and (7) (or (9) or (11)). The second configuration with the uniaxial stress a applied parallel to the radio frequency field h and a t 45" to the film normal and when the static magnetic field N can be rotated in the plane perpendicular to the uniaxial stress a is shown in Fig. 6. When the static magnetic field H is parallel to the film plane, the shift of FMR line is given by (27) if 4nM,ff < 0 or 0 < 4nM,ff 5 Ha and by (28) if 4zMeff > H a for 8, = 45".

Substituting into (27) or (28) the values of A,,, A,,, A,, resulted from the measure- ments in the first configuration and using the relation (3) and (7) (or (9), or (11)) the components A,, and A,, as well as the magnetostriction constant Lc (or 2;) or 2:) are determined.

Rotating the static magnetic field H in the plane perpendicular to the uniaxial stress 0 on 90" from the position, a t which the static magnetic field H is parallel to the film plane, it can be obtained the case, in which the static magnetic field H lies in the same plane as the uniaxial stress a and the film normal. In this case, the shift of PMR line is given by (23). Substituting into (23) the determined values of A,,, A1,, A,,, A,,, and A,, and using the relations (3) and (7) (or (9)) or (11)) the last com- ponent A,, of the magnetoelastic tensor A z j k l and the magnetostriction constant A D (or A;, or 2;) are determined.

4. Experiment and Results

The amorphous Co,,Zr,, thin films were prepared by coevaporation from two electron beam sources onto glass substrates held a t 30 "C. The amorphous nature of these films was confirmed by X-ray diffraction. The microstructure of the films was studied by a scanning electron microscope. The film thickness and the saturation magnetiza- tion 4nMs is equal to 250 nm and 0.23 T, respectively. The informations about prepa- ration and properties of these films were reported in detail in 171 by Krishnan et al.

The ferromagnetic resonance spectra of the coevaporated amorphous Co,,Zr,, thin films have been measured a t a frequency of 9.35 GHz in room temperature with the 21 physica (a) 90/1

322 NGYUEN CHI THANH and R. KRISHNAN

static magnetic field H applied a t various angles to the film normal. A typical spec- trum exhibits two or more peaks. The relative position of these peaks changes as the direction of H is changed in such a way that a t some field angles, the peaks may be widely separated, while a t others they strongly overlap. To analyze these results, a computer program was developed to find the best fit to these spectra by adjusting the effective anisotropy field H : = H , - 4nH, and gyromagnetic ratio y (= gese/2mc). From ths best fitting material parameter of HZ, it is possible to calculate the per- pendicular anisotropy K,. For the amorphous Co,,Zr,, thin films we find K,= = 0.46 x lo4 J/m3.

Typical FMR and SMFMR spectra measured in the first configuration are shown in Fig. 7 , In this measurement configuration, the experimental and calculated FMR field orientation dependences of the shift of the FMR line, AH versus O H , for the amorphous Co,,Zr,, thin films are shown in Fig. 8.

For the amorphous Co,,Zr,, thin films we find

A<j( x 106) =

L

I I I I I

1 5500 7500 9. H 17916 Aim) -

Fig. 7

1 ' 1.9 -0.3 -1.4 0 0 0 -0.3 1.9 --1.4 0 0 0 -1.5 -1.5 2.8 0 0 0

0 0 0 8.1 0 0 0 0 0 0 8.1 0

0 0 0 0 0 2.2

oh' - Fig.8

, 1 7"

Fig. 7. SMFMR and FMR spectrum of the amorphous Co,,Zr,, thin films measured in the first configuration described in Fig. 5 when OH = 5"

Fig. 8. The FMR field orientation dependence of the FMR line shift at frequency 9.35 GHz of the amorphous Co,,Zr,, thin films in the first configuration described in Fig. 5 (0 = 5 x lo5 N/m2). The points denote the measured values of AH. The solid curve 1 is a theoretical one calculated from (20) using AA =: (3.4 The solid curve 2 is a theoretical one calculated from (31) using I , = 1.5 x lo+

0.3) x low6 and ,IB = (1.2 0.3) x

Anisotropy of Magnetostriction in Coevaporated Col -*Zr2 Amorphous Films

and

323

L A = (3.4 f 0.3) x 1B = (1.2 0.3) X Ac = (-4.2 f 0.3) x 1, = (3.9 f 0.3) x lo-'. (30)

The results (29) and (30) show that the magnetostriction in the coevaporated amor- phous Co,,Zr,, thin films is strongly anisotropic, since for isotropic magnetostriction the relations (12) must be fulfilled. Furthermore, as it is seen in Fig. 8, the values of AH calculated from (20) using the following parameters:

(curve 1) are in good agreement with those of AH measured experimentally, while the values of A H , calculated from the equation

AA = (3.4 f 0.3) x and AB = (1.2 f 0.3) x

(31) 30 M ,

AH = -A,[H,, ~ O S (e, - e,,) + 4 n ~ ~ ~ ~ cos ze,] x ~ - 1 ,

and 1, = All, obtained by assuming that the magnetostriction of the amorphous Co,,Zr,, thin films is isotropic (curve 2 , Aq = 1.5 x are significantly different from the measured values of A H .

5. Conclusions From the results reported above the following conclusions can be drawn:

(i) The magnetostriction in the amorphous Co,,Zr,, thin films in which columnar structure exists [7] is anisotropic and has to be described by four independent magneto- striction constants. The results support applicability of the columnar structure model of magnetostriction [6] to the investigated system.

(ii) Up to now, the strain modulation ferromagnetic resonance method (SMFMR) seems to be the most suitable one for studying the anisotropy of magnetostriction in amorphous thin films.

Acknou9ledgement The authors are greatly grateful to Prof. H. Szymczak for suggesting this problem and for fruitful discussions.

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[12] E. KLOKHOLM, IEEE Trans. Magnetics 12, 819 (1976). [13] R. GONTARZ, H. RATAJCZAK, and P. SUDA, phys. stat. sol. 6, 909 (1964). [14] J. C. M. HENNINC and J. H. DEN BOEF, 5. appl. Phys. 16, 353 (1978). [15] H. SZYMCZAK, J. WOSIK, W. ZBIERANOWSKI, and A. D~BKOWSKI, in: Proc. Fourth Internat.

[16] S. KONISHI, S. SUGATANI, and Y. SAKURAI, IEEE Trans. Magnetics 5, 14 (1964). [17] K. NARITA, J. YAMASAKI, and H. FUKUNAGA, IEEE Trans. Magnetics 16,435 (1980). [18] S. SMITH and H. G. BELIERS, Phillips. Res. Rep. 10, 113 (1955).

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Conf. Microwave Ferrites, Jablonna (Poland) 1978 (p. 247).

(Received March 15, 1985) 21.