Ferromagnetic domain size of Tb

Journal of Magnetism and Magnetic Materials 125 (1993) 319-322 North-Holland

Ferromagnetic domain size of Tb T.J. M c K e n n a , D.J. I s b i s t e r , S.J. C a m p b e l l a n d D . H . C h a p l i n Department of Physics, University College, The University of New South Wales, Australian Defence Force Academy, Canberra ACT 2600, Australia

Received 5 January 1993

The temperature dependence of ferromagnetic domain size for Tb is calculated over the range 0 K to the ferromagnetic ordering temperature Tc ~ 221 K and compared with earlier calculations and experimental data. The present results show a decrease in domain size with increasing temperature in broad agreement with the earlier calculations. Discrepancies between sets of calculations and experiments at high temperatures (T > 180 K) may be due to domain nucleation and/or thermal history effects.

1. Introduction

Study of the esoteric magnetism of the rare earth elements continues to attract the attention of theoreticians and experimentalists, particularly, in the latter case, in the investigation of high-purity single crystals. As examples, McEwan [1] has outlined recent developments in the understanding of the helifan structure (intermediate between the well known helical and fan structures) of the heavy rare earth metal H o and noted its potential applicability to the magnetism of light rare earths such as Nd. Similarly, the complex magnetic structure of Sm continues to attract attention with interest centred on its twostage antiferromagnetic ordering process and the nature of these transitions [2-4]. In the case of the heavy rare earth metal Tb of well known and relatively straightforward magnetic structure (e.g. ref. [5]), the behaviour of chirality d o m a i n s - - l i n k e d with the narrow region of helical spin antiferromagnetism which separates the paramagnetic ( T N ~ 229 K) and ferroCorrespondence to: Prof. S.J. Campbell, Department of

Physics, University College, The University of New South Wales, Australian Defence Force Academy, Canberra ACT 2600, Australia.

magnetic (T¢ ~ 221 K) r e g i o n s - - h a s been investigated using neutron topography (e.g. ref. [6]). The t e m p e r a t u r e dependence for such antiferromagnetic domains has been examined in detail by us recently using t e m p e r a t u r e modulation of the ac magnetic susceptibility for a high-purity single crystal of Tb [7]. T h e t e m p e r a t u r e hysteresis observed for the magnetic susceptibility could be well accounted for by a simple model of spiral spin structures with a decrease in the thickness of antiferromagnetic domain walls as the temperature increases. An extension of this work to examination of temperature- and field-induced relaxation effects in the ferromagnetic region suggested changes in domain sizes with t e m p e r a t u r e and the related movement of the domain wall pinning centres [8,9]. Evidence of a decrease in domain size with increasing t e m p e r a t u r e for ferromagnetic Tb was obtained by L6ffer and Rauch [10] by neutron depolarisation and small-angle scattering experiments. Their results for the m e a n domain size were obtained for polycrystalline samples cooled in zero field and then field cycled at fixed temperatures, leading to the m e a n values of ~ 2 . 3 Ixm at 114 K a n d ~ 1 . 9 Ixm at 1 9 2 K . More direct evidence for a change in the domain width d with t e m p e r a t u r e was obtained by Corner et al. [11] in their observation of the domain

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T.J. McKenna et aL / Ferromagnetic domain size of Tb

patterns by evaporation of iron on the polished surface of high-purity single crystal Tb. The domain width was found to decrease from d ~ 14 izm at 95 K to d ~ 9 ixm at 180 K whereas an increase to d ~ 13 i~m was observed at ~ 215 K as the temperature increased towards the ferromagnetic ordering point ( Tc ~ 221 K). Corner et al. also presented a theoretical calculation for d based on the 180 ° Bloch wall slab domain model, with d proportional to (K6)1/4/o " (where K 6 is the basal plane anisotropy and o~ is the reduced spontaneous magnetisation). Using experimental data for K 6 [12] and tr [13], the calculated values of d showed a monotonic decrease as the temperature increased, failing to replicate the observed increase in d towards Tc. In commenting on this discrepancy, Corner et al. noted that the experimental ~r [13] used in the calculation did not go to zero until 243 K, some 14 K above T N ~ 229 K. As mentioned above, our measurements of the ac magnetic susceptibility and transient enhancement of susceptibility of Tb were consistent with changes in domain size as the temperature was varied [9]. Given that our measurements covered the region (180-221 K) where C o m e r et al. [11] observed a difference between theory and experiment, we were interested to re-examine this discrepancy. H e r e we present a further calculation of the variation in domain size with temperature for ferromagnetic Tb and compare the results with experimental observations [11].

in agreement with the model of Corner et al. [11]. Birss et al. [12] suggested that the single-ion contribution derived by Callen and Callen [16] gives an acceptable fit to their experimental data, with = [13/2 [ ,,.~- 1( or ( T ) ) ] .

K6(T)/K6(O)

[13/2(x) is the reduced hyperbolic Bessel function and . ~ - 1 is the inverse Langevin function. Using theoretical tr(T) values derived from Darby [17] for J = 6 (the appropriate angular momentum for Tb), the basic quantity involved in the calculation is in(X) of Callen-Callen's analysis, where [,+1/2(x) =

in(x) io( X-~--

io(X )

sinh(x)

- - ,

(4)

X

and [ 10395 4725 210 1) i6(x) = ~ X7 + - 7 + --X3 + sinh x (10395

For uniaxial magnetic materials such as Tb, the slab domain width d is proportional to y l / 2 / t r , where y is the domain wall energy [14]. For such domain walls y is, in turn, proportional [15]. Thus in considering the domain to wall width relative to its value at T = 0 we set

d(T)-- d(T) d(O)

[K6(T)/K66(O)] 1/4 or(T)

,

(1)

(3)

H e r e the in(x) are the Bessel functions at half-integer order n + 11~2evaluated at imaginary argument, in(x) = ~/'rr/2xJn+l/2(ix) [18]. For the particular case followed by Callen-Callen, n = 6 and the corresponding functions are explicitly given below

-

2. Numerical analysis

(2)

1260

21) coshx,

(5)

with _~ 3 / 2 ( x ) = i 6 ( x ) / i o ( X ) . These results were generated by a simple recurrence relationship, given as

ik(X )

= -(2k -

1)ik_l(X)/X+ik_2(X ).

(6)

The actual arguments x of the [13/2(x) were determined as the numerical solutions to the nonlinear Langevin equation at a series of temperatures Ts selected throughout the ferromagnetic region where X =..~a - 1(O's) ,

(7)

T.J. McKenna et al. / Ferromagnetic domain size of Tb

and the well-known Langevin function S a ( x ) is defined as 1 - ~ ( x ) = coth x - - ,

(8)

X

with trs being the known theoretical variable. The numerical procedure involved a N e w t o n - R a p h son iterative search for the solutions x of eq. (7) for each magnetisation value ors. It was necessary to evaluate the modified Bessel functions i,(x) in double precision arithmetic with the rest of the program in single precision arithmetic. T h e quantities [f13/2(..~--1(O's))]l/4/Ors were evaluated for use in the following graphical presentation of our results.

3. Results and discussion T h e calculated values for the relative domain width d(T) up to T = 218 K are shown in fig. 1 along with a cubic fit to these calculated values.

|.0'

0.8'

© 0.6'

0.4

0.2

0.0 0

* 100

i 200

T(K) Fig. 1. Calculated values of the relative domain width d(T) (eq. (1)) as a function of temperature. T h e squares represent the actual values calculated with the trend of the data shown by the cubic fit to these values (lower line). T h e u p p e r line represents previous calculated values [11] normalised to the present results at T = 0 K. T h e circles represent the normalised experimental values of Corner et al. [11].

321

Also shown for comparison are the experimental and theoretical values of Corner et al. [11], normalised to our value at T - 0 K. The present results show the same trend as the earlier theoretical values [11]. In neither case, however, do the calculations show an increase in domain width for T > 180 K as observed experimentally [11]. As already mentioned, Corner et al. [11] used experimental values for K 6 and tr in their domain width calculations. They suggested that the discrepancy between theory and experiment could be due to their use of experimental tr data which did not go to zero until 243 K, some 14 K above TN. They postulated that the observed increase in domain width at higher temperatures might represent K 6 remaining non-zero while tr approached zero at the critical t e m p e r a t u r e (which they described as T¢ rather than the correct TN). However, using theoretical values of o-(T) and K6(T) (where K6(T) itself is a function of tr(T)), we have shown that K6(T) continues towards zero at T i at a sufficient rate not to dominate the reduction in cr(T) in eq. (1). O u r previous m e a s u r e m e n t s of ferromagnetic susceptibility are consistent with changes in domain width as the t e m p e r a t u r e varied [9], but did not give any indication of the polarity of such changes. Our observation of t e m p e r a t u r e hysteresis in the ferromagnetic susceptibility suggests a behaviour of the domain structure consistent with a thermal history depending on whether the sample was warmed through the ferromagnetic region or cooled from above Tc. Such hysteresis was most marked above T ~ 180 K where Corner et al. [11] observed the increase in domain widths. Since tr(T) is non-zero at Tc the process of ferromagnetic domain nucleation and denucleation below T~ could occur over an extended t e m p e r a t u r e range (e.g. ref. [19]). Furthermore, since Tb possesses an antiferromagnetic domain structure above T¢ this may influence its ferromagnetic domain structure on cooling, just as the antiferromagnetic domain structure obtained on warming is influenced by its ferromagnetic history [7]. The increase in ferromagnetic domain size observed above T ~ 180 K [11] is considered to be linked with such ferromagnetic domain nucleation a n d / o r thermal history effects.

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T.J. McKenna et al. / Ferromagnetic domain size of Tb

4. Conclusions C a l c u l a t i o n of the f e r r o m a g n e t i c d o m a i n w i d t h for T b b a s e d o n t h e o r e t i c a l values for o-(T) a n d K 6 ( T ) shows similar t r e n d s to the results obt a i n e d by C o r n e r et al. [11] u s i n g e x p e r i m e n t a l values for tr a n d K 6. N e i t h e r c a l c u l a t i o n leads to a n i n c r e a s e in d o m a i n w i d t h as o b s e r v e d experim e n t a l l y for T > 180 K. A s t h e r e is n o intrinsic m e c h a n i s m for such a d i s c r e p a n c y it is suggested that d o m a i n n u c l e a t i o n or t h e r m a l history effects could a c c o u n t for this difference in b e h a v i o u r .

Acknowledgements This work was s u p p o r t e d in p a r t by a g r a n t from the A u s t r a l i a n R e s e a r c h G r a n t s Scheme. T.J. M c K e n n a a c k n o w l e d g e s the s u p p o r t of the A u s t r a l i a n D e f e n c e Force.

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