Magnetization, AC susceptibility and microwave absorption in Tb and Gd70Tb30

Journal of Magnetism and Magnetic Materials 20 (1980) 56-66 0 North-Holland Publishing Company

MAGNETIZATION,

AC SUSCEPTIBILITY

D.M.S. BAGGULEY, J.P. PARTINGTON, Clarendon Laboratory,

AND MICROWAVE

ABSORPTION

IN Tb AND Cd,oTb30

J.A. ROBERTSON * and R.C. WOODS

Oxford, OX1 SPlJ, UK

Received 17 August 1979

Measurements of magnetization, ac susceptibility and microwave absorption at 9.5 and 35 GHz, have been carried out with single crystals of Tb and Gd70Tb30. The critical temperatures for Tb were in agreement with previous published results. The Curie temperature for Gd7OTb30 was measured to be (275 f 2) K. Anisotropy constants have been derived from the microwave measurements. The value of K2 at 0 K is estimated to be 8.8 X lo* erg cme3 (Tb) and 1.3 X 10’ erg cme3 (Gd70Tb30). At 77 K, the value of K6 is estimated to be 6 X lo4 erg crnM3 (Cd7gTb30).

Microwave absorption in Tb might also be expected to arise from a reasonably straightforward resonance process since the magnetic phase structure is comparaiively simple. Below 221 K Tb is a ferromagnet, with the spins aligned in the (0001) plane, between 221 and 230 K the zero field structure is a basal plane spiral which can be quenched by applying a small magnetic field (less than 0.8 kc), whilst above 230 K , Tb is paramagnetic obeying the Curie-Weiss law with a Curie constant equal to that for the free tri-positive ion. However, the anisotropy and magnetoelastic energy terms are much larger than is the case for Gd. Data from inelastic neutron scattering [lo] and magnetostatic anisotropy measurements [ 11,121,

1. Introduction Microwave absorption in the hexagonal crystals of the heavy rare earth elements has been investigated by a number of workers [I-S]. Only in the case of Gd, however, is a straightforward microwave resonance observed which can be interpreted directly using the classical electromagnetic theory for ferromagnetic resonance in an anisotropic crystal. A variety of absorption phenomena have been observed in the other rare elements Tb, Dy, Ho and Er, but rather limited progress has been made in explaining these results using the spin wave theory of refs. [681. In order to provide additional experimental evidence for the microwave absorption in the rare earth metals which exhibit a helical antiferromagnetic phase, Bagguley, Liesegang and Robinson [9] investigated magnetization and microwave absorption in GdY alloys. With suitable choice of composition, these alloys provided examples of the most straightforward features of the spiral spin and basal plane ferromagnetic configurations found in the elements, at the same time having relatively small crystal anisotropy and magnetoelastic interactions. The results could generally be described in terms of classical ferromagnetic resonance theory.

indicate that the spin wave gap arising from these effects is too large below 200 K for ferromagnetic resonance to be observed at 9.5 or 35 GHz. Extensive microwave absorption is in fact observed at both of these frequencies. The addition of Gd to Tb quenches the spiral phase and reduces the anisotropy and mag netoelastic terms. The GdTb alloy series therefore provides the opportunity to investigate a basal plane ferromagnet having reasonably strong interactions but without the complication of a helical phase. Magnetization and microwave absorption data have been previously reported for Gd2sTb7s [ 131. The present paper gives results for the magnetization, ac susceptibility and microwave absorption obtained from a re-examination of Tb and a new series of measurements on Gd70Tb30 (70 at% Gd, 30 at% Tb).

* On study leave from: Macquarie University, New South Wales, Australia, 2 113. 56

D.M.S. Bagguley et al. /Magnetization.

2. Materials preparation The single crystals used for the measurements reported in this paper were grown by Hukin at the Clarendon Laboratory using a zone refining technique with a water cooled horizontal crucible in very high vacuum [ 141. An rf induction coil, mounted outside the vacuum chamber, was moved slowly along the length ot’an ingot and crystal grains of order 1 cm3 were grown at the trailing edge of the molten zone. Single crystal disks were cut from one of these grains and prepared as described in ref. [9]. Offcuts from the disks were used for X-ray fluorescence analysis and the final composition of the alloy crystal was found to be in agreement with that calculated from the starting materials. Metallographic and X-ray examination of the finished disks indicated that the crystals were generally of high quality with few oxide inclusions. The purity of the material (Rare Earth Products (UK) Ltd. sublimed grade) is expected to be better than 99.9%.

3. Equipment The static magnetization was measured by the Curie method using a chemical balance to weigh the sample in an inhomogeneous magnetic field. A servomechanism was incorporated in the balance to ensure that the sample was always weighed at the same position in the field gradient. The sample holder was mad,e from high purity copper and the suspension was twisted glass fibre coated with PTFE (“Tygaflor” grade T3, manufactured by Fothergill and Harvey, Ltd.). Since the sample was free to rotate about the axis of suspension the magnetization was always measured along an easy direction in the plane of the crystal disk. Standard samples of Ni and Fe were used to calibrate the balance. The saturation magnetization at 293 K was taken’to be 55.1 emu g-’ for Ni and 217.6 emu g-’ for Fe [ 151. The apparatus operated over the range 10 to 350 K, the temperature being determined from the boiling point of standard refrigerants and by means of an FeAu vs. chrome1 thermocouple. The ac susceptibility was investigated using two small pickup coils wound in opposition; one of these coils contained the sample disk, the other was air

microwave absorption in Tb. Gd,oTbJO

51

cored. The coils were placed in a uniform oscillating magnetic field on which a steady bias field could be superimposed. The out of balance signal from the coils was proportional to the ac susceptibility of the sample, it was synchronously detected and the resulting dc signal recorded as a function of temperature or bias field. This apparatus operated over the range 20 to 300 K and the temperature was determined as for the magnetization balance described in the previous paragraph. Microwave measurements were carried out at 9.5 and 35 GHz using conventional low frequency field modulation spectrometers. The 9.5 GHz spectrometer had a rectangular cavity operating in the TEol 1 mode with field modulation at 61 Hz. The 35 GHz spectrometer had a cylindrical TEolI mode cavity and field modulation at 139 Hz. The differential power absorption, @P/W), was derived using phase sensitive detection techniques and recorded as a function of the applied magnetic field, H,at particular temperatures. A good signal to noise ratio was generally achieved for half-linewidths less than 2 kG. The operating range for the spectrometers was 10 to 350 K and again the temperature was determined as described above.

4. Experimental

results

In this section we summarize the results of measurements of(i) static magnetization, (ii) ac susceptibility, (iii) microwave absorption. Both Tb and GdTOTbJO have the hexagonal hcp crystal structure. Disks corresponding to the (0001) and (1 TOO) crystallographic planes were cut for each material, the finished disks having nominal dimensions of 0.2 mm thickness and 5 mm diameter. Measurements have therefore been carried out with the magnetic field applied in planes containing (a, b) and (a, c) axes of the hexagonal lattice, the in-plane demagnetizing factor (the term N in eq. (1)) being approximately 0.4 in each case. For the ferromagnetic phase and the (0001) disks, the easy axis in the plane was (1 iO0) for Tb at all temperatures whilst, for Gd7,,Tb30, it was (1 iO0) below 160 K and changed smoothly to (1 1200, for temperatures above 240 K where the hexagonal anisotropy was, of course, very small. The corresponding easy axis for Cd is also (1 1200, but this

58

D.M.S. Bagguley et al. /Magnetization.

may not necessarily provide the underlying reason for the change observed in Gd,eTbso. Oxide impurities can occur in the sample as platelets lying in { lOi2) planes with a hexagonal star configuration (H&in, private communication). Such impurities will distort the local field and may give rise to a weak basal plane anisotropy having (1120) an easy axis. For the (IiOO) disks the in-plane easy axis was [ 1 1201 at all temperatures for both materials. The experiments have been carried out in the temperature range 20 to 300 K with extension to 10 and 350 K in particular cases. The static magnetization and ac susceptibility measurements were taken either at fixed temperatures whilst increasing the applied magnetic field or, alternatively, as the sample warmed slowly towards room temperature in a constant external field. A few measurements of the ac susceptibility were carried out as the sample cooled down to 100 K. When measuring the static magnetization the sample was cooled in zero field and cycled up to 9 kG at the lowest temperature before starting a variable temperature run. The microwave data were taken in the conventional manner by sweeping the magnetic field (maximum value 11 kG at 9.5 GHz and 20 kG at 35 GHz) as the sample warmed slowly towards room temperature with a few additional measurements as the sample cooled down. The temperature drift over an absorption line was usually not more than 1 K and this could be reduced to less than 0.5 K when required for a particular measurement. 4. I. Static magnetization: Tb The results for Tb are in good agreement with those published previously [16-l 81. The specific magnetization at constant external field is plotted as a function of temperature in fig. 1 and the ferromagnet to spiral phase transition can be seen on the curves taken in 180 and 480 G. The constant mag netization observed below the Curie temperature, T,, in small fields, indicates that the magnetic configuration is a domain structure for which H-NCM)=O.

(1)

Here His the external magnetic field, (M) is the macroscopic average magnetization of the sample and N is the demagnetizing factor in the plane of the disk.

microwave

absorption

in Tb. Gd,,,Tb3,-,

100

300

200 Temperature

(K)

Fig. 1. Magnetization curves for Tb: . . . . . . , Gd70Tb30: -. The curves refer to (1 l?O) in (liO0) disk where explicitly labelled; unlabelled curves refer to the easy direction in (0001) disk.

For Tb, T, was (221 f 2) K and the Ne’el temperature, TN, was (230 f 1) K. 4.2. Static magnetkation: Cd 7&‘3~,, The results for Gd70Tb30 are given in fig. 1. The low temperature saturation magnetization for the alloy was measured to be 294 f 15 emu g-’ in good agreement with the values 282 and 286 emu g-’ calculated from the moments per atom for ferromagnetic Cd and Tb given in refs. [ 17,191. The magnetization curves showed no indication of a spiral phase and T,, determined from the sharp corner in the low field magnetization curves (20,211, was (275 f 2) K. These results are consistent with the measurements for the ahoy Cd2sTb7s reported in ref. 1131 4.3. The ac susceptibility: Tb The ac susceptibility of Tb was investigated in order to confirm the magnetization data for the temperature variation of the critical field for the spiral phase, to determine the field values at which domains were eliminated from the sample in the ferromagnetic phase, and to indicate whether the domain wall motion was subject to significant viscous damping. For these experiments the oscillating field was aligned with any additional bias field; it was either a sine wave at 61 Hz or a square wave at 300 Hz and the

D.M.S.Bagguley et al. /Magnetization, nlicroware absorption in Tb. Gd,oTb 30 !

1

,

I

a

1 100

150 Temperature

200

250

(K)

02 t

L--_l

kk---4vm Temperature

230 (K)

Fig. 2. (a) Recorded curves for ac susceptibility of Tb, (0001) disk, taken at constant bias fields. The curve for 0.4 kG has been displaced vertically for clarity of presentation. The dashed curve refers to a sample cycled to 11 kG at 77 K before starting the measurements; (b) transition fields for Tb, (0001) disk. l domain wall transition; n, v boundary of spiral phase.

peak amplitude was less than 2 G. Typical recorded curves for the ac susceptibility taken at constant bias field as the sample warmed slowly towards room temperature are given in fig. 2a. The two sharp features near 220 and 230 K indicate the boundaries of the spiral phase. The dashed line curve refers to measurements carried out after the sample had been cooled to 77 K in zero field and then cycled to 11 kG at that temperature before applying the bias and starting the experimental run. The continuous curves show a broad low temperature feature near 150 K which was only observed when the sample had not been cycled initially at 77 K and when the oscillating field amplitude was less than 10 G peak. A similar feature has been reported

59

previously for zero bias field in ref. [22]. Our experiments indicate that this effect is not relevant to the present microwave measurements where the sample was cycled repeatedly to a high magnetic field. The experiments using square wave modulation were entirely consistent with those using sine wave excitation. The rise time of the magnetization was less than 4 /JS at all temperatures. However, the amplitude of the induced magnetization observed both with sine wave and square wave excitation in zero bias field was much smaller than the value (M/N) which would be expected from eq. (1). The short rise time indicates that the reduction in amplitude is unlikely to arise from viscous damping of the wall motion and is more probably due to domain wall pinning. The results obtained from experiments in which the bias field was cycled to 11 kG at fixed temperatures are given in fig. 2b. The points in the range 0.7 to 0.9 kG below 220 K refer to the field at which the domains were eliminated as indicated by the sharp decrease in the ac susceptibility when the sample became single domain. Detailed measurements were carried out between 200 and 220 K where a stepfunction feature was observed in the microwave absorption (see section 4.5). Between 225 and 230 K the ac susceptibility had a sharp maximum at the bias field value corresponding to the phase boundary of the spiral spin structure. The low temperature boundary between 219 and 224 K was not detected in these measurements, presumably because a mixed phase was present. The points indicating the low temperature boundary have therefore been taken from curves of the type shown in fig. 2a. The (0001) plane disk was found to be isotropic for all the measurements described in this section. For the (1 TOO) disk measurements were carried out along [ 1 1201; the results were in agreement with those for the (0001) disk. A few experiments were carried out as the sample cooled down from room temperature to 100 K. All the features observed in the warming up experiments were reproduced without temperature shift to the accuracy of our measurements (?2 K). 4.4. The ac susceptibility: Cd ,oTbjo Measurements of ac susceptibility similar to those described in section 4.3 have been carried out using

60

D.M.S. Bagguley et al. /Magnetization.

61 Hz sine wave excitation. The data taken at constant bias field as the sample warmed slowly towards room temperature showed no evidence of a spiral phase in agreement with the static magnetization experiments reported in section 4.2. Typical recorded curves are given in fig. 3; the curves taken in zero bias field clearly demonstrate that there is no spiral phase. The broad peak near 240 K for the bias field H = 0.4 kG arises from the multidomain to single domain transition which occurs when NM(T) = H; there is an additional sharp peak near T,, on this curve, where the magnetization is strongly field dependent. The broad low temperature feature observed in Tb was again recorded near 140 K. The intensity of this feature was considerably reduced but (in contrast with Tb) was not completely eliminated by cycling the sample to 11 kG at 77 K before starting a warm up experiment. The field values at which the sample became essentially single domain were determined from experiments in which the bias field was cycled repeatedly to 11 kG as the sample slowly warmed towards room temperature. These results are plotted together with the microwave data in fig. 8 and it can be seen that they correspond with the low field ‘step’ transition observed in the microwave absorption (section 4.6). It can also be seen that the domain transition field has fallen to 0.4 kG near 240 K, confirming the

2kG

I 100

microwave absorption in Tb, Gd7gTbS0

description of the variable temperature experiment given in the previous paragraph. The (0001) plane disk was found to be isotropic in these measurements. For the (1 TOO) disk measurements were carried out along [ 1 1201 and the results were in agreement with those for the (0001) disk. 4.5. Microwave absorption: Tb The microwave absorption measurements are summarised in fig. 4. They are in general agreement with those reported previously [4,23,24]. However, the improved quality of the present crystal has enabled measurements at 35 GHz to be extended to lower temperatures than in the earlier work. The present results show essentially no anisotropy in the (0001) plane except at 195 K and 9.5 GHz, where an angular dependence with hexagonal symmetry was found, but the peak to peak amplitude was not more than 200 G. This differs from the original work in ref. [4], where microwave absorption was only observed along (1 l?O), but is in agreement with the measurements taken at 98 GHz [S]. We suggest that the earlier observation of strong anisotropy arose from the poorer quality of the crystal. The feature which was assigned to the spiral to ferromagnet phase transition in ref. [4] has now been observed at temperatures well below T,. It appears to arise from the transition multidomain to single domain ferromagnetic configuration in the sample. We still confirm,

(liO0)

I

I

I

200 Temperature

-

I 300

(K)

Fig. 3. Recorded curves for ac susceptibility of Gd7OTb30 taken at constant bias fields along (1120). The curves refer to (0001) disk except for the one labelled (1TbO). The dashed curve was recorded after the sample had been cycled to 11 kG at 77 K. Curves have been displaced vertically for clarity of presentation.

Temperature

(K)

Fig. 4. Microwave absorption data for Tb (0001) disk. q(liOO), n cll?O)at 35 GHz;o CliOO), l(ll~O) at 9.5 GHz; . - 98 GHz from ref. [S]; . . . . transition fields from fig. 2b.

D.M.S. Bagguley et al. /Magnetization,

however, that the microwave absorption in the ferromagnetic regime occurs at roughly the same fields for the two widely differing microwave frequencies 35 and 9.5 GHz, and that this absorption moves into higher magnetic fields as the temperature decreases. In the (1 TOO) disk, no field dependent absorption was observed for the [OOOl] direction but along [ 1 1201 the absorption corresponded to the high field tail of a resonance line. The peak absorption along [ 1120] would have occurred in very low magnetic fields but was not in fact observed because the sample had a multidomain configuration. Above TN, well defined resonance lines were observed for all directions in the (0001) disk and along [ 1 1 201 in the (1 TOO) disk. The lines moved rapidly into higher fields as the temperature increased. A field dependent absorption was also observed at 9.5 GHz, in the temperature range 77 K < T < 220 K, when the microwave magnetic field was parallel to the applied field. This ‘parallel’ field effect could be observed separately at 9.5 GHz where the rectangular cavity mode was linearly polarized but was always mixed into the ‘perpendicular’ absorption at 35 GHz where the cavity mode had circular symmetry. The ‘parallel’ configuration does not give rise to absorp tion in a normal ferromagnetic resonance experiment when the material is saturated, it must arise from a component of the magnetization which is not aligned with the applied field. In our experiments the ‘parallel’ field absorption occurred in low fields, it was isotropic in the (0001) disk and presumably arose from the multidomain configuration in the sample. The half-linewidth, taken as half the difference in field between maximum and minimum points on the differential absorption curve, was a minimum at TN. In the ferromagnetic phase the linewidth was frequency dependent, being narrower at 9.5 GHz and scaling roughly as the square root of the frequency. These results are presented in fig. 7.

microwave absorption in Tb. Gd,oTb

30

I a

Fig. 5. Typical differential absorption curves for GdTOTbJO, (0001) disk at 100 K and 9.9 GHz; (a) [ 1 ITO], (b) (liOO]. Magnetic field increases to the left.

4.6. Microwave absorption: Gd 70Tbjo Typical absorption curves for Gd70Tb30 at 100 K in figs. 5 and 6. The results are summarised in fig. 8 and show an overall temperature dependence similar to the corresponding results for Tb given in fig. 4. There are, however, significant differences in detail. At low temperatures anisotropy with hexa-

61

are shown

Fig. 6. Typical differential absorption curves for Gd70Tb30, (0OOl)diskat lOOKand35GHz;(a) [ll?O],(b) [lTbO]. Magnetic field increases to the left.

D.M.S.

Bagguley et al. /Magnetization,

tnicro,clave absorption

it, Tb, G’d70Th30

4-

3-

2-

lJ I

I

L

100 Temperature

I

I

200 (K)

I

I

300

Irig. 7. Linewidths for microwave absorption. Tb: L. 35 GHz; o 9.5 GHz. Gd,,-,Tb30: e 35 GHz; . 9.5 CHz.

gonal symmetry was observed in the (0001) disk. This anisotropy was observable down to 20 K at 35 GHz but was more clearly resolved at 77 K. Results for this temperature are plotted in fig. 9, where it can be seen that the easy direction is (1 iO0); note that the position of the absorption line is frequency dependent. At temperatures below 77 K the resonance line was well resolved along (1 TOO) but broadened and became distorted along (11 TO), suggesting that the magnetization was not fully aligned along the hard direction at these temperatures. Between 150 and 275 K the line observed at 9.5 GHz continued to

100 Temperature

200

300

(K)

Fig. 8. Microwave absorption data for Gd7uTbJo, (0001) disk: 0 ciioo), 9 (1150) at 35 GHz; 0 ciioo), 0 (IlTo) at 9.5 GfIz. Corresponding data for [ 11201 in (1iOO) disk, A 35 GHz, -9.5 CHz. o transition fields derived from ac susceptibility. The arrow at the top of the diagram indicates Tc.

I

I

[ioicj

I

1

[oiiciJ

Field Dir!zn Fig. 9. Angular dependence of the microwave absorption in Gd7oTb3o, (0001) disk at 77 K, showing hexagonal anisotropy. n 35 GHz; l 9.5 GHz.

move into lower fields as the temperature increased whereas at 35 GHz the line moved into higher fields and continued relatively smoothly into the paramagnetic regime. A similar effect has been reported previously for Gd2sTb7s [ 131. In the paramagnetic phase the resonance line moved raprdly into high fields at both frequencies and broadened considerably; it was reasonably intense up to 350 K which is the highest temperature at which our equipment will operate. Between 170 and 275 K a low field ‘step’ transition was clearly observed at both 35 and 9.5 GHz. This transition occurred at the field for which the sample became essentially single domain as determined from the ac susceptibility measurements described in section 4.4; it was not frequency dependent. A differential absorption peak was observed at 9.5 GHz in this temperature range, at these same field values, when the microwave magnetic field was parallel to the applied field. It was isotropic in the (0001) plane and, as for the case of Tb reported in section 4.5, presumably arises from the change in domain configuration. In the (1 TOO) disk, microwave absorption was observed along [ 11?0]. At low temperatures, the absorption was similar to that described previously for Tb and appeared to be the high field tail of a resonance line. The extent of the absorption decreased considerably as the temperature increased and near 250 K it developed into a sharp low field resonance line with a broad high field shoulder. The sharp low field line was anisotropic in the (1 iO0)

D.&Y.

Baggule_v et al. /Magnetization,

plane, occurring at a constant component of the applied field along [ 1 1201. The resonance line observed in the paramagnetic phase developed from this sharp low field line whilst the broad shoulder became weaker and moved into high fields outside the range of the magnets. Considerable anisotropy remained at room temperature and at 35 GHz the resonance field changed from about 10 kG along [ 1120] to a value in excess of 20 kG along [OOOl ] . The temperature dependence of the linewidth is given in fig. 7. Here the points refer to (liO0) in the (0001) disk, for which directions good line shapes were observed over the whole temperature range. The linewidth has a minimum at T, and, although the metallurgical quality of the alloy crystal was not as good as that for Tb, the width for the alloy was always smaller than that for the element. 5. Analysis and discussion of the microwave absorption The analysis of ferromagnetic resonance experiments is usually carried out by solving the coupled set of equations: V,e + (jw/c)(h + 47rM) = 0, V,h

t &J/C)@ t 4rro/jo)e

dM/df = y(MhHr,t)

+ damping term.

63

line. The lines observed in the present series of experiments were broad, and a problem arises in choosing the best form of the damping term in eq. (4). When the Bloch-Bloembergen expression is used [27] this term may be written -[(iM, t jMy)/Tz t k(M, Mo)/T, ] whereas the corresponding expression for Gilbert-Landau-Lifshitz damping [28] is -(a/lAQo(,,k). If the linewidths are assigned wholly to such terms the position of maximum power absorption will be shifted several hundred gauss from the undamped position. The two expressions, however, give shifts in opposite field directions and it is not clear which form is to be preferred. Our estimates of anisotropy constants are therefore limited in accuracy by this uncertainty of interpretation. It should also be remarked that it is necessary to work with the full relation given in eq. (5) since there is an implicit shift, additional to that arising from damping, due to the particular combination of p I and ~(2. This causes the maximum absorption to occur at a higher field value than would be determined by ~2 alone. 5.1. Tb:220K
(2) =0 ,

microwave absorption in Tb, Gd,oTbJo

(3) (4)

Eqs. (2) and (3) are Maxwell’s electromagnetic field equations with e, h. M, the usual field vectors, and eq. (4) describes the motion of the magnetization with Hint the effective field acting on the spin system including anisotropy, magnetoelastic and exchange terms. Examples of this type of analysis may be found in refs. [25,26,9]. The electrical conductivity, u, is assumed to be a constant and the microwave power absorption, P, is described in terms of an effective field dependent complex permeability @r - ipa). The microwave experiments described in the present paper measured aP/iJH, where

This method has been generally adequate for Cd [ 1] and for the GdY alloys 191. We shall use it to interpret the present data when the microwave absorp tion had the appearance of a ferromagnetic resonance

Over the temperature range above TN, 230 K < T < 250 K, where reasonably good signal intensity was achieved, the microwave absorption had the form of a straightforward resonance line and the position of the line was clearly frequency dependent. It is unlikely, however, that the complete linewidth is due to the relaxation processes discussed in the previous section since, in the paramagnetic phase, the singleion crystal field splitting of the J = 6 ground state spreads the absorption over a range of magnetic fields. In these circumstances it is not possible to estimate the shift in position of the line center caused by the relaxation terms but it may be noted that the position of maximum absorption scales quite closely with frequency, as would be the case for a paramagnetic spin system when linewidth contributions are neg lected. Immediately below TN the microwave absorption occurs in relatively small magnetic fields, but at 35 GHz the peak absordtion occurs at a field high enough to transform the sample into a single domain. The axial anisotropy is large but the hexagonal anisotropy and magnetoelastic terms may be ignored. The crystal

D.hf.3. Bagguky et af. /Magnetization, microwave absorption in Tb, Gi,0Tb30

64

anisotropy

energy, FA, may therefore be written:

5.2. 7%: T < 220 K

FA = -K2 sin2tI, where 6 is the respect to the implicit in eq. for the (0001)

(6)

polar angle of the magnetization with [OOOl] axis. Neglecting the line shift (5), the resonance frequency relation plane is:

a2

2K2 0r = [Htc4n-n?MtM I t-

[H

NM]

1 y2T;

(7)

’

In deriving this equation a Bloch-Bloembergen damping term has been used in eq. (4). Evidently the peak absorption occurs for H = NM when K2 is large. We have developed a computer program to fit the differential absorption curves observed at 35 GHz to the more accurate relation for (BP/aH) given in eq. (5). This analysis determined the best value of K2 at 220 K to be tl .l X lOa erg cmm3 which is in good agreement with estimates given by previous workers (see table 1). It should be observed, however, that ref. [5] does not include a correction for the lineshift in the analysis of the results. We also give an estimate for K2 at 0 K in table 1, assuming the M3 law to be valid [29,30]. This is an approximation both on account of the restricted range of validity of the M3 law and also because the axial anisotropy constant, Ka, includes contributions from higher order spherical harmonics. However, the static magnetization measurements in ref. [ 181 show that, in practice, the cube law is obeyed between 2 and 210 K.

Table 1 Summary of anisotropy constants

C&o-ho

Tb

(4

@I

(d

(cl

6.1

3.1

8.8

1.3

K2WO)

1.0

0.4

1.1

0.34

K6(77)

10-Z

-

-

6 x lO-4

Kz(O)

Units of K2, Kg: erg crnm3 X lO*;(a)ref. (c) present work.

[18],(b)ref.

[S],

The microwave absorption experiments show no evidence of the spiral phase in Tb; the sample is effectimely ferromagnetic at all temperatures below TN. In small magnetic fields the sample has a multidomain structure and the internal field remains zero for H < NM. A transition in the microwave absorption may therefore be expected to occur when the applied mag netic field penetrates the sample giving rise to the step feature which occurs near 1 kG for 150 K < T < 230 K. In the low field multidomain region additional power loss can arise from the component of microwave magnetic field parallel to the external field; this loss will decrease to zero for H > NM as the sample becomes a single saturated domain along H. The demagnetizing terms due to the low field domain wall distribution will give rise to a complex resonance response for the spin system [31], but it seems reasonable to expect the absorption due to the component of microwave magnetic field perpendicular to the external field to increase when the sample becomes single domain and the normal ferromagnetic resonance mode can be excited. The extensive microwave absorption observed at low temperatures for H > NM occurs with the sample, in principle, a single domain. However, the absorption does not appear to be influenced by the hexagonal anisotropy acting on the spin system. Thus, taking the hexagonal anisotropy constant K6 = 0.17 X 10’ erg cmm3 andM= 2214 emu cme3 at 140 K [ 181, the microwave absorption should exhibit anisotropy with a peak to peak amplitude of (72K6/M) = 5.5 kG. Except for the small anisotropy observed at 195 K and 9.5 GHz, having a peak to peak amplitude of 200 G, the absorption was isotropic in the (0001) plane. This behaviour contrasts with that of Gd70Tba0 shown in fig. 9 where the peak to peak amplitude is 2.5 kG. It is probably significant that the absorption, for a given frequency and temperature, occurs in lower fields for [ 11 TOO]in the (1 TOO) disk than is the case for the corresponding direction in the (0001) disk. This is characteristic of the uniform mode in normal ferromagnetic resonance experiments with a crystal having [OOOl] the hard direction of magnetization. For this case, K2 is positive and the resonance frequency for [ 11201 in the (1 TOO) disk is given by:

D.M.S. Bagguley et al. /Magnetization. microwave absorption in Tb, GdlOTb30

0r = [H-NM+%1 slz

x [Ht(4n-N)w

t-

Table 2 Magnetization and axial anisotropy constant for Gd7oTb3o

1 r2T;

65

’

The peak absorption moves to lower fields because the demagnetizing term (411- N) has changed brackets as compared with eq. (7). The observed change in position of the microwave absorption suggests that it is derived from the spin system and can sense the response of the uniform mode. The low temperature absorption is isotropic in the (0001) plane and so it cannot be due to the closing of the spin wave gap by a magnetic field along a hard basal direction (1 1700, [6,4,8]. Moreover, inelastic neutron scattering experiments [lo] show that the lattice strain arising from the magnetoelastic coupling remains ‘frozen’ at its equilibrium value and does not follow the dynamical motion of the spin system at zero wave vector. In these circumstances a field applied along (1lzO) does not close the spin wave gap. Chow and Keffer [32] have proposed that the low temperature absorption arises from a magnon-phonon coupled mode with wave vector equal to the reciprocal skin depth for the microwave field. The strength of the magnon-phonon coupling depends in first order upon the ratio (ok/&) where wk is the unperturbed phonon frequency and Rk the corresponding magnon frequency. The power loss is therefore field dependent through & and might be isotropic for the (0001) disk, in practice, because of the variety of different phonon polarizations which can in principle be excited. There is some doubt as to whether the strength of the coupling is adequate to account for the observed intensity [33], but it may be that this can be enhanced by the presence of impurities in the crystals (oxide inclusions are frequently present in the rare earth metals) and by the modulation of the internal field at the surface because of the oscillatory spatial variation of the exchange interaction. We have examined the consequences of wander in the [OOOl] axis which might occur in an actual single crystal. For our present samples, direct X-ray measurements indicate that the deviation of this axis from the mean position is less than $” and, although Kz = lo8 erg cmw3, this angular spread appears to be too small to give rise to any significant modification of the spin wave mode.

T W

M (emu cm -3)

K2U’l (erg cm -3 x 108)

K#Mf3 (cgs units x 10-q

180 200 220 240 260

1720 1600 1480 1280 1080

0.53 0.43 0.34 0.24 0.14

1.04 1.05 1.05 1.1 1.1

5.3. Cd,oTb30:

175 K < T<275

K

In this temperature range the microwave absorption observed at 35 GHz in the (0001) disk moves to higher magnetic fields as the temperature increases. The absorption line is isotropic and so we interpret the data in terms of the decrease of Ka, neglecting hexagonal anisotropy and magnetoelastic terms. The computer program was used, as for the case of Tb in section 5.1, to fit the differential absorption curves. The values of K, derived from this analysis over the temperature range 180 K < T < 260 K are given in table 2. Since K2 varies closely as M3 over this temperature range, we have estimated the value for 0 K assuming the relation to hold over the complete temperature range. This gives a value for K2(0) = 1.3 X 10’ erg crne3 which is of the expected order of magnitude and somewhat less than that for Tb given in table 1. The position of peak absorption at 9.5 CHz, computed using the values of K2 (7) given in table 2 and eq. (5), was found to occur in fields of order 600 G throughout the range 180 K < T < 275 K. The marked difference in the temperature dependence of the absorption at 35 and 9.5 GHz can therefore be explained. 5.4. Gd,0Tb30:20K
(9)

66

D.M.S. Bagguley et al. /Magnetization.

with r#~the polar angle of the magnetization measured with respect to [ 11 ZO]. The peak to peak amplitude for the angular dependence of the resonance field is approximately (72Ke/M), providing a direct measure of Kg. We estimate K6 = 6 X lo4 erg cme3 at 77 K and that K6 varies as h122 above this temperature in good agreement with the I(1 + 1)/2 power law [29,30]. Below 77 K the absorption line for (1 iOO> remains reasonably narrow, but along a hard (1lzO) direction the line broadens considerably indicating that the magnetization is not fully aligned with the applied field. Moreover, the low temperature observations are not entirely consistent with a description in terms of hexagonal anisotropy alone since the absorption along (ll?OO, starts at a lower field than is the case for ( 1TOO). This may indicate that the anisotropy constant K6 has differing values for measurements along different crystal axes [34-361 or that magnetoelastic terms are significant.

Acknowledgements The authors are grateful to Dr. D.A. Hukin for providing the single crystal materials used in these experiments. Mrs. M. Hoggins, Mr. D. Morris, Mr. C. Read and Mr. F. Wondre contributed to the materials preparation and instrumentation. J.P.P. and R.C.W. wish to acknowledge the award of S.R.C. post-graduate studentships and J.A.R. wishes to thank Professor B. Bleaney for extending to him the facilities of the Clarendon Laboratory whilst on study leave.

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microwave absorption in Tb, GdyOTbSO

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