RIPPLE THEORY AND THE TRANSVERSE SUSCEPTIBILITY OF (100) SINGLE CRYSTAL NICKEL FILMS

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RIPPLE THEORY AND THE TRANSVERSE SUSCEPTIBILITY OF (100) SINGLE CRYSTAL

NICKEL FILMS

D. Pomfret, M. Prutton

To cite this version:

D. Pomfret, M. Prutton. RIPPLE THEORY AND THE TRANSVERSE SUSCEPTIBILITY OF

(100) SINGLE CRYSTAL NICKEL FILMS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-

399-C1-401. �10.1051/jphyscol:19711139�. �jpa-00213957�

JOURNAL DE PHYSIQUE Colloque C 1, supplement au n° 2-3, Tome 32, Fevrier-Mars 1971, page C I - 399

RIPPLE THEORY AND THE TRANSVERSE SUSCEPTIBILITY OF (100) SINGLE CRYSTAL NICKEL FILMS

D. POMFRET

St. John's College, York, England and M. PRUTTON University of York, England

Résumé. — La théorie de Hoffmann des fluctuations de l'aimantation dans les couches minces polycristallines a 'été appliquée à des couches minces monocristallines de plan parallèle à (100) présentant un « skew » dû à des joints de grain à faible désorientation. La susceptibilité transversale de couches minces (100) de Ni sur support de LiF ou de Cu a été mesurée dans les directions (001) et (011) et comparée avec la théorie pour différentes dimensions des régions désorien- tées. Les couches minces déposées au-dessus de 250 °C sont le mieux décrites par un modèle de « skew » à grande longueur d'onde avec des grains de 0,15 [x environ et un écart-type de 6 x 10"4 rads pour le « skew ». Les résultats sont en accord avec des observations de microscopie électronique faites sur des couches minces de même type.

Abstract. — Hoffmann's ripple theory for random polycrystalline films has been applied to (100) single crystal films containing skew due to low angle grain boundaries. The transverse susceptibility of (100) Ni films on LiF and Cu substrates has been measured in (001) and (011) directions and compared with the theory for various scales of skewed region. Films deposited above 250 °C are found to be best described by a long wavelength skew model with grain sizes of about 0.15 t«n and r.m.s. skew of about 6 x 10~2 radians. These results are consistent with electron diffraction observations on similar films.

I. Introduction. — Ripple theories of the magne- tization distribution in uniaxial, polycrystalline films have been applied with considerable qualitative success to the description of some of their observed magnetic properties. After Feldtkeller's [1] original experiments several authors [2-5] have used Hoff- mann's theory [6] to describe the transverse suscep- tibility of polycrystalline permalloy films. The trans- verse susceptibility of (100) single crystal films of nickel shows similar deviations [7] from a single domain description to those found in uniaxial poly- crystalline films. The objects of this paper are to present the results of :

a) an application of Hoffmann's ripple theory to the case of (100) single crystal films in which the local anisotropy is caused by misoriented grains of size D ;

b) an examination of this theory for susceptibility data obtained from (100) nickel films.

II. Theory. — Hoffmann's [6] formulation of ripple theory and his notation are used throughout this paper. The case of a (100) single crystal film containing low angle grain boundaries is analogous to a poly- crystalline case with zero inhomogeneous local aniso- tropy Ks but with skew 9{x) which varies from grain to grain. The local anisotropy E^k is then given by

£*(r) = ^ sin² 2(<p⁰ - 0 ( r ) ) d ) where K^ is the fourth order magnetocrystalline

anisotropy constant. Following Hoffmann, equation (1) can be expanded to second order in the skew angle 6 in terms of components parallel to unit vectors m^t and m² which are respectively parallel and per- pendicular to the average magnetization M .

Terms for the local effective field he(( acting on the magnetization can be derived and the only two

which differ from Hoffmann are the effective single domain field h(a) which is given by

h{a) = h cos (a — <p0) — m1 cos 4 <p0 , (2) and the effective skew field /i^skew(a) which is given by

^skew(«) = 8 mi 6(f) s m 4 q>o + 30 m2 0(r) cos 4 <p0 . (3) The high coefficient of m2 in equation (3), is due to the strong angular dependence of the torque acting on M in a biaxial film and leads to large effects due to small skew 9.

The transverse susceptibility % is then given by

\ = M X < K(t{a) > =

= ah(a) + Bh-"(a) + Eh-"(u) . (4) In the manner discussed by Harte [8] and Hoffmann [6], the values of B, E, n, and m depend upon the size of a region of constant 9 (a grain) with respect to the coupling volume (Table I). The value of the constant a is determined by M2/2 Kt and the sensi- tivity of the apparatus used to measure x-

The expression for x is o n' y valid provided that the applied field h is greater than the blocking field h„

which is given by :

*.-©*"•

III. Experimental. — The experimental data for the variation of the transverse susceptibility with the field applied in the plane of the film has been obtained using a transverse Kerr effect magneto- meter [6] [7, 9]. The films were prepared as described in reference 10.

Graphs of susceptibility versus applied field were obtained in both easy [011], and hard [001], directions for r.m.s. values of the alternating (tickle) field

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711139

D. POMFRET AND M. PRUTTON

Theoretical expressions for the constants in equation (4) for dzyerent scales of grain size D Evaluated for jilms with grains extending throughout the thickness

Very short wavelength skew D

<

coupling lengths Short wavelength skew D

-

coupling lengths Long wavelength skew D

S

coupling lengths

between 1.5 and 6 Oe. The data were then corrected by at most 14

%

by extrapolation to zero tickle field for each value of the applied field H and thus plots of 1/x versus H or h could be obtained (Fig. 1). A

FIG. 1.

-

Experimental susceptibility data and fitted curves for a (100) Ni film 1,000 A thick deposited on to LiF at 350 *C.

0 Experimental points.

A

Fitting points.

- - -

Single domain susceptibility. - Fitted curve using equation (4) long wavelength skew. ^-.-m-- Fitted curve using short wavelength

skew.

value for the anisotropy field 2 K1/M was obtained by extrapolating the easy direction data to l / ~ = 0.

All applied fields could then be normalized and the experimental data replotted as a function of h(ol).

Theoretical curves defined by equation (4) and Table I were fitted at 3 points to the experimental data above

h,. The parameters a, B, and E determined by the fitting were used to estimate h, and the data taken as valid if all fitting had been carried out above h,,.

N. Results and Discussion.

-

It has been possible to fit equation (4) for both long and short wavelength skew to the experimental data for all films except the one deposited at 200 OC. In this case, only short or very short wavelength forms could be made to fit the data, this being consistent with the smaller grain size in a film deposited at a lower temperature.

All the other films studied could be best described by the long wavelength skew model, since fits to short wavelength skew led to larger values of the grain size D which appears to be an inconsistent conclusion.

The difference between the calculated single domain susceptibility and the observations was found to be predominantly an h-'l4(a) or h-3/4(a) form, indica- ting as expected that the transverse internal dema- gnetizing field was most important in determining the susceptibility. The data obtained for a number of the (100) Ni films measured are summarised in Table D where it has been assumed that

A = 1 X 1 0 - ~ erg/cm and M = 480 e.m.u.

For hen(a) = 1 and K, = 7.2 X 104 ergs/cc, which is consistent with the data of Table 11, the coupling lengths are 120 h; along the average magnetization and 3,500

b;

perpendicular to the average magneti- zation. The grain sizes obtained by curve fitting are indeed large compared to the former length but are comparable to the latter length.

The accuracy to which Hk can be determined limits the accuracy of the results obtained. The slow decay of the h-lt4(a) term required to fit the susceptibility

Summary of data obtained fromjitting l/x(h) to equation (4) for various scales of skew

Thickness Deposition Substrate HP H k ha (*)D (*)

<

82 > 1 / 2 (**) Skew

(A) Temperature (oersteds) (oersteds) (pm) (radians) X 102

("C)

- -

^{L L}

- - - - -

1,000 350 LiF 3 50 280 1.17 0.12 6.6 long (short)

1 ,000 300 Cu 325 320 1.25 0.15 5.7 long (short)

2 , m 250 LiF 350 280 1.26 0.16 5.5 long (short)

1 ,Ooo 200 LiF 320 270 1.05 0.09 short (very short)

(*) The errors on D and

<

82

>

¹¹² due to errors in the experimental data are of the order of f 20 % and f 50% respectively.

(**) The values for h,, D and

<

^{8 2}

>

¹¹² are obtained for the unbracketed skew model.

RIPPLE THEORY AND THE TRANSVERSE SUSCEPTIBILITY OF (LOO) SINGLE CRYSTAL C 1 -401

observations means that extrapolation of the linear part of easy direction 1 / ~ curve leads to overestimation of H, by a t most 10

%.

The use of a lower value of H, does not affect the use of an h-1/4(o() term in the susceptibility but does reduce slightly the values of D and ( 02

>

obtained. In principle it should also be possible to determine 2 H, from the separation of the hard and easy direction susceptibility curves.

However, in all single crystal Ni films examined so far there is a consistent gain change between the signals measured in the hard and easy directions. This diffe- rence amounts to between 13

%

and 30

%

and may be due to anisotropy in the transverse Kerr effect.

Such anisotropy has been discussed theoretically by Donovan [ I l l and detected in the Faraday effect in nickel by Heavens [12].

Transmission electron microscopy and diffraction was used on Ni films 1,000

A

thick grown at 300 OC in LiF and could be used t o place bounds on

< ^O2 >

and to estimate D. The spot size in transmission elec- tron diffraction was such that

<

⁰²

>

^{' l 2} < 4 ^X 10-2 radians. The boundaries of the extinction contours observed in transmission electron microscopy showed fluctuations in position on a scale of 2,000

a

^which

could be associated with fluctuations in the deviation from the Bragg condition due to the changing grain orientations. The low angle grain boundary model of an epitaxial film is supported by the X-ray rocking curve observations by Holloway [13].

It is concluded that this extension of Hoffmann's ripple theory can be used to describe the transverse susceptibility of epitaxial (200) Nicke! films.

References FELDTKELLER (E.), Zeits J Physik., 1965, 176, 510.

UCHIYAMA (S.), Jap. J. Appl. Phys., 1967, 6, 1.

KEMPTER (K.) and HOFFMANN (H,), Phys. Stat. Sol., 1969. 34. 237.

D ~ Y L E (W. D.) and RNNEGAN (T. F.), J. Appl Phys., 1968, 39, 3355.

LEAVER (K. D.). PRUTTON (M.) and WEST (F. G.). ,*

~ h y s : Stat. ' s o l , , 1966, 15, 267.

HOFFMANN (H.), Phys. Stat. Sol., 1969, 33, 175.

PRUTTON (M.), J. Appl. Phys., 1968, 39, 1153.

HARTE (K. J.), J. Appl. Phys., 1968, 39, 1503.

[g] CHAMBERS (A.), POMFRET (D.) and PRUTTON (M.) J. Sci. Inst., 1967, 44, 181.

[l01 CHAMBERS (A.) and PRUTTON (M.), Thin Solid Films, 1967. 1. 235.

[l11 DONOVAN (B.) and METCALF (T.), Proc. Phys Soc., 1965, 86, 1179.

[l21 HEAVENS ^(0. S.) and MILLER (R. F.), Proc. Roy. SOC., 1962, A266, 547.

[l31 HOLVWAY (B.), Thin Films in Physical Investiga- tions, Ed. J. C. Anderson, 1966, Wiley.