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Magnetization, AC susceptibility and microwave absorption in Dy and TbY alloys
Journal of Magnetism and Magnetic Materials 22 (1981) 291-305 © North-Holland Publishing Company
MAGNETIZATION, AC SUSCEPTIBILITY AND MICROWAVE ABSORPTION IN Dy AND TbY ALLOYS D.M.S. BAGGULEY, J.P. PARTINGTON, J.A. ROBERTSON * and R.C. WOODS Clarendon Laboratory, Oxford, OX1-3PU, UK Received 3 October 1980 Measurements of magnetization, ac susceptibility and microwave absorption at 9.5 and 35 GHz, have been carried out with single crystals ofDy, Tb74Y26 , Tb79Y21, Tbs IYI9- These crystals exhibit a stable helix phase at high temperatures; Tb74Y26, Tb79Y21, also exhibit a low temperature 'frozen-in' antiferromagnet state. The critical temperatures Te, TN, and the maximum critical field for the stable helix-ferromagnet transition are, Dy: 80 K, 179 K, 11 kG; Th74Y26:25 K, 198 K, 9.6 kG; Tb79Y 21 : 93 K, 202 K, 7.4 kG; TbslYI9:116 K, 205 K, 5.5 kG. Values for the exchange parameter (~(Q) - ~(0)) estimated from the data are about 0.2 X 10-a eV.
results provide evidence for a fan phase. There is additional, but not conclusive, evidence for such a phase from magnetization, magnetoresistance, and ultrasonic measurements [8,11,15,16]. We have, therefore, investigated again the ac susceptibility, magnetization and microwave absorption for a single crystal of Dy and in addition three alloys of TbY chosen to have de Gennes factors, (G), (and magnetic structures) similar to Dy. The actual values of (G) = CGTb, where c is the atomic concentration for the alloy, are Tb74Y26: 7.77, Tb.79Y21: 8.30, TbslY19: 8.51. These values are slightly greater than 7.08 which is characteristic of Dy, but were so chosen because the early neutron data from ref. [17] indicated that the Curie temperature, To, had fallen to 0 K for ~G) < 7.88. The present series of measurements generally conf'nan the previous microwave results for Dy [18,19] and are in accord with predictions from the values of (G) given above. Moreover, the features reported in refs. [11,16] have also been observed in the microwave absorption.
1. Introduction The magnetic phase diagram of dysprosium has been investigated in considerable detail using neutron diffraction [1,2], magnetization [2-8] and ultrasonic techniques [9-11]. Above 178 K, Dy is paramagnetic, obeying a Curie-Weiss law for the susceptibility with a Curie constant equal to that calculated for the free tri-positive ion; between 178 and 86 K the stable configuration in zero field is a straightforward basal plane spiral, the turn angle per layer changing smoothly from 43.2 ° to 26.5 ° as the temperature decreases; below 86 K the ground state is a basal plane ferromagnet with easy axis of magnetization along 1120). The spiral spin configuration can be converted into a ferromagnet by applying a magnetic field in the basal plane, the magnitude of the field required for this transformation increasing from zero at the ferromagnetic Curie temperature to about 11 kG near 178 K. The discovery of changes in the direction of easy magnetization at temperatures above 130 K [6,12,13] has stimulated further interest in the details of the magnetic structures present at the critical field transition, and these authors have suggested that their
2. Materials preparation The single crystals used for the present series of experiments were grown by Dr. D.A. Hukin at the Clarendon Laboratory. Crystal grains about 1 cm 3 in
* On study leave from: Macquarie University, New South Wales 2113, Australia. 291
292
D.M.S. Bagguley et al. /Magnetization and microwave absorption in Dy and Tb Y alloys
size were obtained by the method of zone passing along a water-cooled horizontal boat in very high vacuum [20]. The overall length of a roughly cylindrical multicrystal ingot was about 10 cm and the diameter 0.75 cm. Single crystal disks were cut from individual grains and prepared for measurement as described in ref. [21]. The orientation of each disk was checked by X-ray diffraction and the compositions for the alloys confirmed by X-ray fluorescence analysis. The quoted compositions are expected to
Fig. 1. Photomicrographs of prepared single crystal surfaces: (a) Tb?gY 21 (0001) disk; note the presence of platelet impurities in this sample, (b) Dy (0001) disk.
be correct to + 1 at%. Metallographic and X-ray examination of the finished disks indicated that the crystals were of reasonable quality; figs. 1a and b indicate the standard achieved. The purity against other metals is expected to be better than 99.9% (Rare Earth Products (UK) Ltd., sublimed grade).
3.
Equipment
The specific magnetization was measured by the Curie method using a standard chemical balance to weigh the sample in an inhomogeneous magnetic field. A servo-mechanism was incorporated in the balance to ensure that the sample was always weighed at the same position in the field gradient. For these measurements the sample disk was mounted in a holder suspended by a twisted glass fibre thread, the plane of the disk being horizontal; the sample could, therefore, freely rotate about the vertical axis and the results refer to the inplane easy direction of magnetization. Disks of Ni and Fe were used to calibrate the apparatus; the values for the saturation magnetization of these elements were taken to be 55.1 emu g-i and 217.6 emu g-l at 293 K [22]. The apparatus operated over the temperature range 10 to 300 K, the temperature being determined from the boiling points of standard refrigerants and by means of an FeAu vs. chromel thermocouple. One additional experiment was carried out at 4 K. We are indebted to Mr. M.R. Wells for allowing us to use his equipment for this measurement. The ac susceptibility was investigated using a double pick-up coil arrangement with the two coils wound in opposition; one coil contained the sample, the other was air cored. The coils were connected in series and placed together in a uniform oscillatory magnetic field on which a steady bias field could be superimposed. Both field directions and the winding axes of the coils were aligned. For this configuration, the out of balance signal from the pick-up 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 described above for the magnetization balance. The sample disk was fixed with adhesive tape inside the
293
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
pick-up coil and selected crystal orientations could be investigated. A direct indication of the easy axis of magnetization was achieved by fixing the sample to the end of a tufnol rod with contact adhesive and suspending the rod in a uniform magnetic field. The axis of the rod was vertical and its weight sufficient to ensure that the pla~e of the sample disk lay in the horizontal plane, which also contained the direction of the applied field. A pick-up loop was attached to the tufnol rod and a small ac modulation of the steady field gave rise to an output signal which provided a direct indication of the orientation of the sample. Rotation of the tufnol rod by hand enabled the 6-fold or 2-fold symmetry to be unambiguously verified. This ap,paratus operated from 20 to 300 K and the temperature was determined using an FeAu vs. chromel th~rmocouple. The microwave measurements were carried out using Conventional low frequency field modulation spectrqmeters operating at 35 and 9.5 GHz. Both spectrometers had TEot ~ mode resonant cavities but at 35 (~Hz the cavity was cylindrical whereas at 9.5 GHz a rectangular cavity was used. The field modulation frequencies were 139 and 61 Hz, respectively. Phase ~ensitive detection was used to derive recorded data pioportional to the differential power absorption. These ~pectrometers operated over the range 10 to 350 K and the temperature was determined as descril~ed above. The sample was attached lightly by adhesige to the end wall of the resonant cavity and a comphtte angular investigation of the absorption could 6e undertaken by rotating the applied magnetic field. The specific magnetization and ac susceptibility measurements were carried out either at fixed temperatures Whilst increasing the applied field or, alternatively, as the sample warmed slowly towards room temperature in a constant field. A few additional experiments were undertaken with decreasing magnetic fields and with decreasing temperatures to identify hysteresis effects. The microwave data were recorded in the usual manner, with the field being increased from zero to maximum (12 kG for 9.5 GHz and 20 kG for 35 GHz) at regular intervals as the sample warmed slowly towards room temperature. The temperature drift during a complete field sweep was usually not more than 2 K, it was probably not
more than 1 K over the field range covering an absorption feature and this could be reduced to 0.5 K when required for a particular measurement.
4. Underlying theoretical relations Detailed discussions of the spinwave excitations and magnetic phase transformations for the heavy rare earth metals have been given by a number of authors (see for example the reviews in refs. [23-26]). Many different notations and approximations have been used in deriving relations for the interpretation of the experimental data. We shall therefore set out in this section the particular formulae which we have made use of when deriving the fundamental physical parameters from our results. The underlying theoretical description is based on a Hamiltonian of the form: = -
-
]Y
Jm
n,m n>rn
n
n
+
+
+
+ n
+
_
.y)61 +
+ n
J ( J - ½)
n
].
- J'y)
O)
In this equation the Cartesian axes x, y and z are taken along ( I 120), ( 1] 00) and [0001 ], respectively, gj is the Land~ factor, J is the total angular momentum of an atom, ~(Rnm ) is the exchange coupling parameter for atoms at lattice sites R n and R m corresponding to an exchange interaction energy - ( g j - 1)2 ~(Rnm) X Jn " Jm for each atom pair, H is an applied magnetic field, the terms V2 ... V~6 describe the single ion crystal field anisotropy energies, and terms containing C "r, (e~), (e~), B ~'2 represent the elastic and first order magnetoelastic contributions. The theory applies strictly to a rare earth element and treats the hexagonal close packed crystal struc-
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
294
ture as a Bravais lattice. For the TbY alloys the summations in eq. (1) are to be carried out over the magnetic atoms only and the magnetoelastic terms may need to be modified; we shall make the assumption that ~(Rnm) is unaffected by alloying at the concentrations used in the present series of experiments. No significant modifications to the formulae presented here seem to arise in a more refined analysis which takes account of the two interpenetrating hexagonal sublattices. Mathematical relations, useful to the experimentalist, are not usually derived from the full Hamiltonian given above. For example the crystal anisotropy and magnetoelastic terms are included, or neglected, according to their estimated relevance for a particular set of measurements. In addition we discard the terms in I,'4 and II6, and make use of the following relations derived from eq. (1) following the discussions in refs. [23-27].
Here M is the volume magnetization,la n the atomic moment.
m -- M(T)/M(O) = ~n (T)/Isn(O),
(4c)
(Jn)r = mJn.
(4d)
(iv) The paramagnetic Curie-Weiss temperatures, Ol, 0// ko. = ~gs - 1 y s ( j + 1)c ~p(o) + }v2(J-
(Sa)
~ ) q + }),
kO//= }(gs - 1)2J(J. + 1)c ~p(0) - ~ v2 (s - })(J + }),
(Sb)
~ p ( 0 ) - [9(q)]q=o in the paramagnetic phase, and
v2 < < (gs - 1) 2 9p(0). (v) The Ndel temperature, TN, for (0001) plane spiral kTN ={(gj - 1)2J(J + 1)cgs(Q)
q) The Fourier transforms for the exchange interactions ~(q) = ~
~(Rnm) eiq'Rnm,
+ ] V2 (J - ½)(J + ~),
(6)
~s(Q) -= [~(q)]q=O for the spiral phase and I12 < <
(2a)
( g s - 1)2 ~s(O).
~(R.,.) = ~1 ~q 9(q) e-iq'Rnm
(2b)
(vO The initial susceptibility, ×i, for (0001) plane spiral
~(Rnn) = 0,
(2c)
Xi,± =
m
~
9(q) = 0.
q -
(ii) The molecular field, Hmf
i=x,y,z,
+ 2V2 },
H~f -
m
T=0,
(gJ - 1)~
gsl3 i = x, y, z,
Oa)
m
(3b)
where c is the atomic concentration of Tb atoms in an alloy TbY; m is def'med by eq. (4c): 1 >~ m / > 0.
(7a)
T = 0.
(7b)
Here, NTb is number of Tb atoms per gram, Xi,± refers to measurements in (0001) plane, Xi,//refers to measurements along [0001 ] axis.
(rio The spinwave dispersion relation, (0001) plane spiral h 2 cos2 (q) = ~rJ 2 {(gj - 1)2c [2 ~s (Q) -
(ii0 The reduced magnetization, m
Jn,
~ s ( Q + q ) - ~ s ( Q - q)]}
X { ( g d - 1)2C[~s(Q) - ~s(q)] +2V2},
T = O,
(4a)
T = O.
(4b)
n
ltn(O) = -gd~Jn,
T=0
cm ~ ~(Rnm)(Ji),
T :/= 0,
111(0) = -gjfl ~
~s(2Q)],
Xi,//= NTbg~#2/( (gJ -- 1) 2 C[~s (O) - ~s(O)]
Hmf = _ (gJ - I)2 ~3 9(R.m),
gs~
NTb~O 2/(gs -- 1) 2 c [2 9s(Q) - ~s(0)
T = 0. j2 h2 t.Os2(q) = 2m-~ {(gJ - 1)2m2c
(8a)
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys x [2~s(Q) -
~s(O +q) - ~s(Q - q ) ] )
295
and near TN
X { ( g j - 1 ) 2 m 2 e [ ~ s ( Q ) - ~s(q)]
gs~nl = (gj
-
1 ) 2 m J c [ ~ f ( O ) - ~f(0)].
(10b)
A
+ 2V2Is/2 } ,
T=/: O.
(Sb)
Here q is along [0001] ;1(1+1/2) is the ratio of hyperbolic Bessel function of order (l + ½) to that of order ½, with the argument of the Bessel function the inverse Langev~ function of the reduced magnetization. At low T, I(1+1/2 ) = ml(l+l)12.
Here, the - / + signs refer to easy/hard direction for HI (0001) plane; Ht is usually interpreted as defining the high field boundary of the fan phase. (x) The critical field, Hc; (0001)plane spiral to ferromagnet
+ 2V2J1 +- 61V~61Js + ½B~'2X~J -1
The (0001) plane spiral transforms, in principle, to the (0001) plane ferromagnet when the free energy of the ferromagnet is less than that for the spiral. It is usually accepted that H c can be evaluated to sufficient accuracy with neglect of entropy terms. In the molecular field approximation:
+ gj~H + (Nz - Nx)M6o,q )
gH3mJHc = ½(gj - 1)2J(J + 1)m2 C[~s(Q) - ~f(0)]
(viii) The spinwave dispersion relation, (0001) plane f erromagnet
~/2~(q) = ~(gj_ 1)2Jc[~f(0)_ ~f(q)]
X ((gj - 1)2je[~f(0) - ~f(q)] _+361V~61Js
-IV~6[J6113/2 -~B3"z~'s(/~s/2)2 + ~XJ_/-/~c, ( l l a )
+ B%2)(rJ -1 + gj~tt + (N x - Ny)MSo, q },
T = 0.
and near TN (9a)
h2co~(q) = ( ( g j - 1)2mJc[~f(0)- ~f(q)]
g2{3mJnc = 2x(g2 - 1)2J(J + 1) X m 2 c [ ~ s ( Q ) - ~f(0)].
+ 2V2J]s/2m -1 + 61V~61JSf13/2m -1
(1 lb)
Here H c refers to easy direction, (0001) plane. If He < HI given in (ix) above, the transition from the spiral is first into the fan phase at H e and to the ferromagnetic phase at H1, with increasing field.
+ ½B%2)~TJ -1 m - l ( i s / 2 ) 2 + g j ~ t t + (N z - Nx)Mm6o, q ) X ( ( g s - 1 ) 2 m J c [ ~ f ( O ) - ~f(q)]
+- 361V~6[JSil 3/2m -1 + B 7'2 )O'J -1 m -1 (Is/2) 2 + gj~-I + (N~ - N y ) M m ~ o,q ) ,
T :/: 0.
5. Experimental results
(9b)
Here q is along [000 l]; Jl = ( J - ½), Js = ( J - ~) X (J - 1) ... (J - ½); -+ signs refer io easy/hard directions for H, (0001) plane; ;~'r is the magnetostriction coefficient; B 7,z ~,7 appropriate to alloy when necessary; N x , iVy, N z are the macroscopic demagnetizing factors; for l(t~l/Z) see (vii). Eq. (9a) is based on ref. [~7]. Eq. (9b) is based on ref. [24]. (ix) The critical field, H1: instability o f the (0001) plane ferromagnet
The (0001) plane ferromagnet is unstable unless the spinwave frequencies are real. From eq. (9b), the spinwave frequency will be imaginary for H < H1, where gJ~l-I1 = { ( g J - 1)2mJe[~f(Q)- ~f(0)]
(10a) 361 ~ IJS/:13/2m -1 - B %2~ J - lm- 1(/~Sl2)2),
In this section we summarize the results of detailed measurements of static magnetization, ac susceptibility and microwave absorption. In addition we report preliminary experiments indicating the temperature and field dependence for the direction of easy magnetization in the (0001) plane. Dy and the TbY alloys have the hexagonal crystal structure A3, P63/mmc. Disks about 5 mm diameter and 0.25 mm thickness were prepared from single crystal grains taken from the ingots described in section 2. For Tb74Y26 it was possible to cut (0001) and (1100) disks from the same grain; in the case of Tb79Y21, TbslY19, two separate grains had to be used and this accounts for the difference in composition. For Dy, only one disk, (0001), was available. The finished disk Tb74Y26, (1100) plane could not be prepared to a sufficiently high surface quality
296
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys
and, although this sample appeared to be adequate for the magnetization measurements, the microwave results were only of qualitative value. The same crystal disks have been used for all the measurements reported in this paper; the different sets of data may therefore be compared directly. The in-plane demagnetizing factor for a disk, 4rre in eq. (12), was approximately 0.4 when calculated using the relations given in ref. [28].
'
'
'
The results for the high temperature paramagnetic phase of the TbY alloys are given in fig. 2. It can be seen that the susceptibilities accurately obey the Curie-Weiss law. The curves refer to 1 g of material and are described by the relations:
' ,'/7,t,
/ ', ,o o1L I
.zx~;o//o/
:6 ,..%o
Vn o / o
#4 o o
7 Z,o A/
o
[]
//7/°"
///
200
0.068 Xs(74, 26) = T - 188
(0001) and ( l i 0 0 ) disks,
0.070 Y~L(79,2 1 ) - - T - 196
(0001) disk,
)0_(81, 1 9 ) -
(li00) disk.
0.076 T - 200
'
,~
=
5,1. Static magnetization measurements: paramagnetic Tb Y alloys
'
Here the Curie constants are in cgs units g-1, the Curie temperatures in K and )(1 represents the basal plane susceptibility. The accuracy of the Curie constant measurement is better than +5% and the Curie temperatures,better than +-2 K. Calculated values for the Curie constants are C(74, 26) = 0.062 cgs g-i ; C(79, 21) = 0.065 cgs g-X ; C(81, 19)'-- 0.066 cgs g-~. Evidently the only significant discrepancy, between the measured and calculated values, arises for the alloy Tbs lY19. However, it would be unwise to attach too great a weight to this difference on the basis of one series of measurements. On the other hand it will be observed that there is a similar discrepancy in the measured and calculated values for the saturation magnetization of this alloy (see section 5.2). 5.2. Static magnetization: magnetization curves Typical magnetization curves taken at fixed temperatures are presented in fig. 3a and b. In our equipment the sample was cooled to a low temperature
220 240 260 Temperature (K) Fig. 2. Inverse susceptibility for TbY alloys (units refer,to 1 g). Tbq4Y26, v(O001), n (1]00); Tb79Y 21, o (0001); Tb 81YI9,
o (li00). (20 or 77 K) relatively quickly, the refrigerant being applied directly to the outside of the sample chamber and the crystal cooling by thermal contact via helium exchange gas. For these conditions a 'frozen in' antiferromagnetic state was observed with Tb79Y21, Tb74Y26, which could be transformed into the ferromagnetic state by applying a magnetic field. In the case of Tb79Y2 l, at 77 K and below, the sample, remained ferromagnetic when the applied field was reduced to zero and did not subsequently return:to the antiferromagnetic state. For Tb74Y26 the sample remained ferromagnetic after cycling to a high field at 20 K but at 51 K or above the sample returned to the antiferromagnetic phase in a zero field. The 'frozen-in' state was not observed with TbslY~9 or with Dy, these samples became straightforwardly ferromagnetic on cooling below TeThe magnetization curves given in fig. 3a and bare plotted as a function of the internal magnetic field Hi, defined by the relation Hi = H - 4rre(M),
(12)
where H is the applied field, 4rre the in-plane demagnetizing factor and (3t) the macroscopic volume mag-
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys I
I
I
I
o~--o.~-o--~--o ,.,-o
,1= 250
I
~,--
'
~
200i
e-.
observed width was further increased if the measurements were not carried out sufficiently slowly; equilibrium times were frequently of order 15 min for field changes of a few gauss below 100 K. Values for the specific magnetization at 0 K, in emu g-i, are 00(74, 26) = 273;
~ 1001
297
00(79, 21) = 278;
oo(81, 19) = 267; ~
50
i ~//~,
,,___.=~.-.~
2
4
6
a
8
10
and the corresponding values calculated from the atomic concentration, taking the moment of Tb to be 9.28fl per atom, are
Field (kG)
o~(74, 26) = 272.5; i
.
i
i
i
i
i
i
0~(81, 19) = 288.2.
_o t. o ~ _ f . o F . . . . . .
.--.,, 250 9 f / f ~
%
t
._~
r,I,
5.3. Static magnetization: the magnetic phase boundaries
I
I
~IsolI
o~(79, 21) = 283.8;
Magnetic phase diagrams for the TbY alloys and for Dy are given in figs. 4 to 7. These diagrams include results from fixed temperature experiments
.--"
:[ lO0
14 50 II
I
e~ i " e - " ' e ~
I~ II
I
I
I
1
-~ I
2
h
I
3
field increasing
field decreasing
• • •
o
zx u
o
~ D
B
10
Fig. 3. Magnetization curves for (0001) disk, (1 i00) direction, (a) Tb74Y26 , (b) Tbq9Y21. Subsequent curves field increasing
[]
!
Field (kG)
Initial magnetization curve
/
12
I
Temperature (K)
I0 20 77
-m 8 iT
O
[] 121
D/
?
6
,n
~
o//
o~ i
C
.,,/ /, 50
netization. The magnetic transitions appear to be extremely sharp therefore, whereas, in terms of the applied field, the transition width was approximately 41toMs (Ms being the saturation volume magnetization). Moreover, in the cases of Tb79Y21 and Tb~4Y=6, the
[] / D
I d
A
,
,
100 150 Temperature (K)
I, 200
Fig. 4. Magnetic phase diagram for Tb74Y26 derived from measurements with (0001) and ( l i 0 0 ) disks. A: stable helix; B: stable ferromagnet; C: 'frozen-in' antiferromagnetic phase. Data points: % magnetization; o, a¢ susceptibility; o, 9.5 GHz microwave absorption; D, 35 GHz microwave absorption.
298
D.M.S. Bagguley et al. I Magnetization and microwave absorption in Dy and Tb Y alloys
i
14
[]
/
14
12
[]
12
10
/ []
?'
~ 8 k~
10 w
O
o' /O
-o 8 oj
iE_
//
6
lOG
4
O
6
i
/
7°
0
o
50
,
100 150 Temperature (K)
;_
12 10
[]/
(.9 8 0.) It.
6
8
, 50
[]oj
~ j
io,'7 ) ?1 0,,7"<>/ ~ " $--J /1, -d 100 150 Temperature ('K)
I
50
14
200
Fig. 6. Magnetic phase diagram for Tbs 1YI9 derived from measurements with (1 i00) disk. Notation as in fig. 4.
O
I
o
2O0
Fig. 5. Magnetic phase diagram for Tb79Y 21 derived from measurements with (0001) disk. Notation as in fig. 4.
I
A
/°'
2
)%o_eo/,
o
°'/ -
I
0
I
I
100 150 Temperature (K)
200
Fig. 7. Magnetic phase diagram for Dy derived from measurements with (0001) disk. Notation as in fig. 4.
(see section 4.2) and supplementary data derived f r o m experiments in which the sample warmed slowly towards room temperature in a constant magnetic field. When Tb74Y26 was cycled into the ferromagnetic state at 20 K and then warmed slowly in a magnetic field, the ferromagnet.stable helix phase boundary together with the h e l i x - p a r a m a g n e t i c transitions were observed. Alternatively, if the low temperature 'frozen-in' antiferromagnetic state was retained, warming in a field greater than 3.5 kG provided evidence for the additional antiferromagnet-ferromagnet boundary. Similar results were obtained with Tb79Y2 l, for which typical magnetization curves are given in fig. 8. The Curie temperatures, Tc, N~el temperatures, TN, and maximum critical fields,/-/~eax, were found to be:
Tb74Y26 Tb79Y21 Tbs1Y19 Dy
Tc (K)
TN (K)
H max (kG)
25 93 116 81
198 202 205 179
9.6 7.4 5.5 11.0
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys I
I
I
I~
100
150
299
I
~ " 300 ___
E
0 "---~ 0
g
c-
J
N 100 0
50 :
200
250
T e m p e r a t u r e (K)
Fig. 8. Magnetization curves for Tb79Y21 (0001) disk after rapid cooling to 20 K. v, 0.17 kG; % 1.34 kG; u, 3.60 kG; o, 5.8 kG; o, 7.25 kG;<, 7.75 kG. v, A, sample not cycled at 20 K, note three phase boundaries; u, o, % ,~, sample cycled at 20 K, note absence of 'frozen-in' phase. Dashed lines indicate magnetization observed when the 'frozen-in' phase has been eliminated. Arrow at top of diagram indicates TN.
5.4. The ac susceptibility
The ac susceptibility measurements complement those for the static magnetization and, in principle, provide data which identify the magnetic phase boundaries, indicate the degree of domain wall motion, and give some evidence for wall damping or pinning processes. Data points, derived from experiments using 61 Hz sine wave excitation with the bias field swept regularly from zero to 12 kG as the sample warmed slowly towards room temperature, have been included in figs. 4 to 7. The phase boundaries determined from these measurements are in good agreement with the results from static magnetization. Above 120 K the ac susceptibility showed a clear maximum at the D helix-ferromagnet phase transitions, becoming very intense near TN, at which temperature the signal disappeared. The magnetic phase boundaries were also indicated by changes in the ac susceptibility as the sample warmed slowly towards room temperature in a constant magnetic field. Sine wave excitation at 61 Hz was also used in these experiments and again the static magnetization results were confirmed. Typical recorded curves showing the changes at the phase boundaries are given in fig. 9. With this apparatus it was also possible to carry
out measurements as the sample cooled down slowly from room temperature. It was found that Tb79Y21 converted to the ferromagnetic phase at 72 K in zero applied field. This experiment confirms the 'frozen-in' nature o f the low temperature antiferromagnetic phase and also indicates a temperature hysteresis of about 20 K in Tc. i
i
I
I
N
2.5kG
<
--1 I
I
I
I
50
100
150
200
Temperature (K)
Fig. 9. Recorded curves for ac susceptibility (arbitrary units), Tb79Y21 taken at constant bias fields. The curves have been displaced vertically for clarity of presentation. The curves refer to. (0001) excepting that labelled (1 i 00), where the crystal direction was (11~0).
300
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
Preliminary experiments, using square wave modulation at 100 Hz, have been carried out to investigate the domain wall response time. These measurements indicate that domain wall pinning may be more significant than viscous damping in giving rise to the time dependent magnetization effects reported in section 5.2. 5.5. Magnetic anisotropy
The easy axis of magnetization was observed directly by hanging an (0001) disk in a uniform magnetic field. Changes in crystallographic orientation of the easy axis for Dy have been reported in refs. [6,12, 13]. Our experiments confirm the results given in ref. [13]. Below 130 K, the easy axis in the ferromagnetic phase was (1120); above 130 K, (1100) was the easy direction for fields just greater than the first critical field but this changed to (1120)at higher fields. We have carried out similar experiments with the TbY alloys. For our samples the easy axis was ~1100) in the ferromagnetic phase throughout the field and temperature range covered with our equipment (0 kG < H < 12 kG; 77 K < T < 200 K). We have observed no changes in orientation such as occur in Dy. It will be noted that (1100) is also the easy axis of magnetization for Tb. 5. 6. The microwave absorption
Measurements of microwave absorption in single crystals of Dy have been reported previously in refs. [18,19]. These authors observed field dependent absorption in the temperature range 90 to 180 K. The signal from the Dy crystal used for the present experiments was too weak for quantitative measurement at temperatures below 110 K. The microwave absorption features occur in the vicinity of the stable helix-ferromagnet transformation. The results from the present series of experiments are included in figs. 4 to 7. At low temperatures the microwave absorption corresponded to a broad decrease in power loss with increasing magnetic field. As the temperature increased this feature became narrower and at about 130 K a sharp low field edge developed, indicating an abrupt increase in power loss. This feature continued to increase in intensity up to TN, when the absorption assumed the
general shape of a paramagnetic resonance line, broadened rapidly, and became too weak for measurement a few degrees above this temperature. The absorption was of the same general form at both measuring frequencies for Dy and all the TbY alloys _ when the magnetic field was along (1100) or (1120). This is also in agreement with the previous microwave results for Dy [18,19]. No field dependent absorption was observed for the field along [0001]. In the present series of experiments an additional high field shoulder to the microwave absorption was observed at 35 GHz. This feature was clearly identified at temperatures above 130 K and presumably corresponds to the magnetization and resistivity singularities reported for Dy [8,16]. This additional absorption feature was not resolved in the present experiments at 9.5 GHz, probably because it occurred at, or above, the high field limit of the magnet used in that spectrometer. Microwave absorption was observed at 9.5 GHz, in the range 77 K < T < 200 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 polarised, but was always mixed into the '13erpendicular ' absorption at 35 GHz where the cavity mode had circular symmetry. The 'parallel' field configuration does not give rise to absorption in a normal ferromagnetic resonance experiment when the sample is saturated in the field direction and the presence of such an absorption indicates that the magnetization is not fully aligned by the external field. In the present series of experiments the 'parallel' field absorption at 9.5 GHz exhibited features similar to those found with the perpendicular configuration, suggesting that the microwave absorption phenomena occur when the magnetization is influenced by the domain structure at the phase boundary. 6. Discussion Values for the exchange parameters [~(Q) 9(0)] may be estimated from the experimental data presented in section 5 and the theoretical relations in section 4. From section 4(iv) and (v): k(TN - Oj.)
= ~(gj - 1)2J(J + 1)C[gs(Q) - ~p(O)] •
(13)
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys
301
Table 1 Summary of magnetic data (G)
Tc (K)
TN (K)
0t (K)
o0 (emu g-l)
Xi,_t_ X 10-3 (emu g-l)
H max (kG)
179 179 198
169
351
5.2
11.0
0.17
188
273
4.0
9.6
0.16
[gs(Q)
Dy *
7.08
Tbq4Y26
7.77
85 81 25
Tbq9Y21
8.30
93
202
196
278
3.4
7.4
0.10
TbalYl9
8.51
116
205
200
267
3.9
5.5
0.08
221
229
239
328
-
0.25
0.24
Tb*
10.5
-
9p(O)]
(x 10-s eV) (1)
(1) From TN - 01. (2) From Xi,± assuming 1~(2Q) - 9(0)1 ~- 41~(Q) - 9(0)1; assuming ~(2Q) = 9(0). (3) From H max. (4) From inelastic neutron scattering.
From section 4(vi): Xi,± = g1~oo/(gs - 1)2 Jc [2 ~s (Q)
- 9s(O) - 9s(2Q)] •
(14)
Values for [~s(Q) - ~s(0)] may be derived from this relation using the empirical observation that reasonable extreme limits are given by I~s(2Q) - ~s(0)l ~ 41~s(Q) - ~s(0)l or alternatively ~s(2Q) ~ ~s(0) depending on which experimental curves are selected from inelastic neutron scattering data. From section
4(x): gs[3mJI-pcncax = ~ ( g , / - 1)2j(j + 1)m2c
X [~s(Q] - ~f(0)].
(15)
Eq. (13) assumes that V2J 2 < < (ga" - 1)2j(j + 1) X c ffp (0), whilst the low temperature relation for ×i,1 has been quoted in eq. (14). This is reasonably satisfactory since the experimental results for Xi,± are not strongly temperature dependent. Eq. (15) is a high temperature approximation, hexagonal anisotropy and magnetoelastic terms have therefore been neglected. A contribution to the free energy which arises from the distortion of the crystal lattice at the phase transition (an elastic energy term) has also been discarded.
Values for the exchange parameters derived using eqs. ( 1 3 ) - ( 1 5 ) are given in table 1. Ignoring the refine ment of the subscripts which identify parameters referring strictly to the ferromagnetic, spiral and paramagnetic phases, it is evident that there is general agreement among the different sets of experimental data and that [~(Q) - 9(0)] ~ 0.2 × 10 -3 eV. W h e n H t from eq. (10a) is smaller than Hc from eq. (1 la), the (0001) plane ferromagnet is stable for H > He and the transition from the spiral phase to the ferromagnetic phase is direct. The highest temperature at which this direct transition may be expected to occur is determined by the condition H~ = He; for our samples this is about 130 K. At this temperature, therefore,
a*
½(gj-
1)2jmc[~(Q)-
9(0)] = E
(16)
(361V~61Jsm-xita/2 - B%2 ?~~J - l m - l (Is12) 2 } . The spinwave gap, A, at q = 0 at this temperature may be written, using eq. (9b), A = hoof(O) .~ [ 2 V 2 J i s / 2 m - l E ] '/2 .
(17)
This equation may be rewritten in terms of the macroscopic crystal anisotropy energy parameter, K2. If the crystal anisotropy energy is FA = - K 2 sin20, and a
302
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
[~s(Q) - ffs(O)]
[~s(O) - ~ f ( 0 ) ]
[~s(Q) - ~s(O)l
A (130 K)
A (130 K)
(X 10-3 eV) (2)
(X 10-3 eV) (3)
(X 10-3 eV) (4)
(X 10-3 eV) (5)
(X 10-3 eV) (6)
0.11 0.32
0.36
0.3
0.43
0.4
0.09 0.27
0.33
0.62
0.1 0.3
0.24
0.62
0.08 0.25 0.11 0.32
0.19
0.62
0.41
(5) From H 1 = H c. (6) From neutron data. * Values from ref. [33].
single ion model is valid, then [V2J]Sl2 m-1 ] = gs~K2 [M(T)] -1 so that A ~ [gjflK2E/M(T)] 1/2.
(18)
easy axis of magnetization for Dy is thought to arise from a decrease in the angular spread of the spin system in a fan phase [13,14]. Consider a simple linear triangular waveform representation of a fan, measuring the_spin angle q~in the (0001) plane with respect to (1120) and taking the z-axis along [0001 ]. Choose the origin for z at a plane where q~is an extremum; the fan may then be described by
The values of A(130 K) given in table 1 have been calculated from eq. (18) using values for K2, M(T), from ref. [15] and taking [~(Q) - 9(0)] from our own data. It can b e seen that there is quite reasonable agreement with the estimate of A from inelastic neutron scattering. Our experiments indicate that ×i,± is greater than ×i, // throughout the stable helix phase. This implies (compare eqs. (7a) and (7b)),
and the anisotropy energy per plane described by
v2 > I 0 ~ - 1)~C[~s(O) - ~ s ( 0 ) l ,
Wp = -IK661 cos 6~.
i.e. 1:2 > 0.03 X 10 -3 eV and correspondingly K2 > 1 X 108 erg cm -3 . This latter inequality is consistent with known values for K2. Evidently K 2 J z ~ 1.26 × 10 -3 eV, whereas kOt ~ 17 X 10 -3 eV and so the inequality assumed for eq. (13) can also be satisfied. It should be noted that, for the 'frozen-in' antiferromagnet phase of Tb79Y21 at 20 K, Xi,± ~ 48 X 10 -3 emu g-l. This is much larger than expected for a simple spiral configuration and may imply the presence of some ferromagnetic domains. The change in crystallographic orientation of the
The averaged anisotropy energy, WA, for an incom. mensurate fan may be obtained by integrating eq. (20) over one complete period, 0 ~< z ~< (2-1 ; 27r~ = Q.
= q~o(40z - I),
0 < z < 1[2C),
= q~o(3 - 40z),
1/2Q < z < 1/0,
WA = ( - I K ~ I/6¢o) sin 6¢o.
(19)
(20)
(21)
Evidently WA is a minimum for a fan centred on (1120) if¢o < 30 °, but for 30 ° < ¢o < 60 ° the minim u m energy corresponds to a fan about (1100). If, therefore, the fan amplitude immediately above the phase transition for Dy is greater than 30 ° (23 ° for the model of refs. [13,14]) the observed change in
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys
orientation of the easy direction of magnetization can be explained since increasing the field will reduce ¢o- No change in orientation of the easy axis of magnetization has been observed for the TbY alloys, but the microwave experiments suggest the possible existence of a fan phase; these observations can be reconciled if the amplitude of the fan is always less than 30 ° (23 °) and the sign before IK~I is taken positive, so making (1100) the easy axis as for Tb. A general theory of magnetic resonance in the heavy rare earth metals has been given in refs. [29, 30]. These authors assumed the microwave excitation to be essentially uniform throughout the sample; the microwave magnetic field couples to spinwave modes having wavevectors q = 0, q = Q, in the basal-plane spiral phase and to modes with q = 0 in the ferromagnetic phase. For an incommensurate spiral the frequency of the q = 0 mode is zero [31], whilst from eq. (8a) Cos(Q) ~ [ J ( g j - 1)/h] X {[2~(Q) - ~(2Q) - ~(0)1V2} 1/2; vs(Q) ~ 100 GHz, at 0 K, which is significantly larger than the microwave frequencies, 35 and 9.5 GHz, used in the present experiments. This estimate for us(Q) is probably adequate for T < 0.9 TN and so no marked field dependence of the microwave absorption is to be expected in the spiral phase. In the ferromagnetic phase at high temperatures, where the spiral phase is stable in zero applied field, eqs. (9a) and (9b) indicate that Ggf(0)• (1/h)[2gj[JH c V2JI] */2 , probably for T < 0.9 TN; again the excitation frequency is vf(0) 100 GHz. At low temperatures the spinwave gap arising from magnetoelastic and anisotropy terms determines the minimum excitation frequency. From eq. (9a) ogf(0)~ (1/h){2V2J~ [361K~ IJs + fir,z ;k-rj-i ] } 1/2, vf(0) I> 500 GHz. It seems unlikely, therefore, that a straightforward ferromagnetic resonance will be observed in Dy and the alloys Tb74Y26, Tb79Y21, TbalY~9, at 35 or 9.5 GHz except possibly at low temperatures for very high magnetic fields applied along a hard axis of magnetization in the (0001) plane (for this particular orientation the spinwave gap can be reduced to zero when magnetoelastic terms are negligible). Previous microwave experiments, u sing Dy, con firm this conclusion [ 18,19 ].
303
For the temperature range covering the stable spiral phase the microwave power loss in the sample may be expected to respond to the changes in magnetic structure at the critical fields. In the vicinity of He the change in spinwave frequencies, discussed in the previous paragraph, may give rise to a transition in the high frequency permeabihty, p, whilst the disappearance of superzone boundaries may cause a change in resistivity, p [ 16,32]. The microwave power loss in the sample depends essentially upon ~ p ) l n and may, therefore, change at the phase boundaries; such changes in loss at H c were reported in refs. [ 18, 19 ]. The present series of experiments suggest that further consideration of the detailed magnetic structures which occur at the phase transitions may be useful. Below 130 K the change in structure is expected to be direct from the spiral to the ferromagnetic phase. The width of this transition as measured by the external magnetic field will be of order 4rreM. The microwave measurements indicate a decrease in power loss when the structure changes from spiral to ferromagnet but although the onset of the change in loss is reasonably sharp, the microwave transition is not fully completed until the external field increment is three or four times 4rre.M. For T > 130 K, a high field shoulder in the microwave power loss presumably indicates the second phase boundary at H1. Consider the possibility of microwave resonant absorption in an extensive fan phase. For T > 130 K the fan is expected to form about the direction of the applied field, H [23,24,30]. A mode with q = 0 may be excited by the component of microwave magnetic field perpendicular to H. The frequency of this mode is of order (1/27rh)[2gj~HV2J1 ] 1/2, which is close to that for the ferromagnet q = 0 mode discussed previously and greater than 35 and 9.5 GHz, used in the present experiments. A component of the microwave magnetic field parallel to H in the (0001) plane, may excite the antisymmetric spinwave mode at q = +Q, for which UA(Q) ~ (1/27rh)[4V2Jlgj[3 X (H1 -/_/)]1/2 becoming zero at Hi. This mode could be excited at 35 GHz when (HI - H) ~ 1 kG and at 9.5 GHz when (H1 - H) ~ 0.1 kG. The intensity goes to zero atH~ in proportion to (HI - / / ) 1 / 2 and so the microwave absorption should be stronger at 35 GHz. The high field shoulder, mentioned above, could therefore originate from an excitation of the anti-
304
DalI.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
symmetric mode, -+Q, in an extensive fan phase. It seems unlikely, however, that an extensive homogeneous fan phase is actually present and it is more probable that a mixed phase having domains with fan and spiral configurations occurs. The presence of domain walls can have a marked effect on straightforward ferromagnetic resonance absorption since additional demagnetizing fields of order 47rM arise [34,35]. However, in the case of Dy and the TbY alloys, the walls are expected to be perpendicular to [0001 ], the additional demagnetizing terms have to be compared with the axial anisotropy field (V2J1/gj~3) > > 41rM, and so it is improbable that the walls modify significantly the estimate for the excitation frequency given in the previous paragraph. On the other hand the observed additional feature in the microwave absorption could simply reflect nonresonant magnetoresistance changes of the type reported in ref. [16], and will not indicate the excitation of a particular spinwave mode but arise from a further change in the quantity ~ p ) l / 2 mentioned earlier in this section.
7. Conclusions Measurements of magnetization, ac susceptibility, and microwave absorption indicate that TbY alloys having de Gennes factors near 8.0 exhibit magnetic phase diagrams similar to Dy. The microwave experiments suggest that a high temperature, high field, 'fan' phase is also present in the alloys but, in contrast with Dy, no change in crystallographic direction for the easy axis o f magnetization has been found. At low temperatures a 'frozen in' antiferromagnetic phase was observed in Tb74Y26, Tb79Y21. This phase appeared to be metastable up to a well defined, but temperature dependent, critical field. The transition into the ferromagnetic state was very slow at the phase boundary, where time constants of order 15 min were observed, suggesting that a (sample dependent) crystal lattice relaxation may be involved.
Acknowledgements The authors are grateful to Dr. D.A. Hukin for providing the single-crystal materials used in these experi-
ments. Mrs. M. Hoggins, Mr. D. Morris, Mr. G. Read and Mr. F. Wondre contributed to the materials preparation and instrumentation. Professor R.J. Elliott has made a considerable contribution throughout this work in discussions o f the underlying theory. J.P.P. and R.C.W. wish to acknowledge the award of SRC 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.
References [ 1[ M.K. Wilkinson, W.C. Koehler, E.O. Wollan and J.W. Cable, J. Appl. Phys. 32 (1961) 48S. [2] W.C. Koehler, J. Appl. Phys. 36 (1965) 1078. [3] D.R. Behrendt, S. Legvold and F.H. Spedding, Phys. Rev. 109 (1958) 1544. [4] R.G. Jordan and E.W. Lee, Proc. Phys. Soc. 92 (1967) 1074. [5] J.-L. Feron, G. Hug and R. Pauthenet, Z. Angew. Phys. 30 (1970) 61. [6] P.B. Fynbo, J. Phys. F7 (1977) 1349. [7] A. del Moral and E.W. Lee, J. Phys. F4 (1974) 280. [8] R. Herz and H. Kronmfiller, J. Magn. Magn. Mat. 9 (1978) 273. [9] M. Rosen and H. Klinker, Phys. Rev. B1 (1970) 3748. [10] S.B. Palmer, J. Phys. F5 (1975) 2370. [11] R.A. Treder and M. Levy, J. Magn. Magn. Mat. 5 (1977) 9. [12] P.H. Bly, W.D. Corner and K.N.R. Taylor, J. Appl. Phys. 40 (1969) 4787. [13] W.D. Corner and J. Welford, J. Magn. Magn. Mat. 3 (1976) 212. [14] P.B. Fynbo, J. Phys. F7 (1977) L199, L279. [15] J.-L. Feron, Thesis, Grenoble, France (1969). [16] A. Akhavan, H.A. Blackstead and P.L. Donoho, Phys. Rev. B8 (1973) 4258. [ 17] H.R. Child, W.C. Koehler, E.O. Wollan and J.W. Cable, Phys. Rev. 138 (1965) A1655. [18] F.C. Rossol and R.V. Jones, J. Appl. Phys. 37 (1966) 1227 [19] D.M.S. Bagguley and J. Liesegang, Proc. Roy. Soc. A300 (1967) 497. [20] D.A. Hukin, Clarendon Laboratory Rept., Oxford (1979). [21] D.M.S. Bagguley, J. Liesegang and K. Robinson, J. Phys. F4 (1974) 594. [22] J. Crangle and G.M. Goodman, Proc. Roy. Soc. A321 (1971)477. [23] T. Nagamiya, in: Solid State Phys., Vol. 20, eds. F. Seitz, D. Turnbuil and H. Ehrenreich (Academic Press, New York, 1967) p. 305. [24] B.R. Cooper, in: Solid State Phys., Vol. 21, eds. F.
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys Seitz, D. Turnbull and H. i~hrenreich (Academic Press, New York, 1968) p. 393. [25 ] R.J. Elliott, ed., Magnetic Properties of Rare Earth Metals (Plenum Press, London, 1972). [26] B. Coqblin, The Electronic Structure of Rare Earth Metals and Alloys (Academic Press, New York, London, 1977). [27] A.R. Mackintosh and H.B. M¢ller, in: ref. [25]. [28] J.A. Osborn, Phys. Rev. 67 (1945) 351. [29] B.R. Cooper, R.J. Elliott, S.J. Nettel and H. Suhl, Phys. Rev. 127 (1962) 57.
305
[30] B.R. Cooper and R.J. EUiott, Phys. Rev. 131 (1963) 1043. [31] R.J. EUiott and R.V. Lange, Phys. Rev. 152 (1966) 235. [32] R.J. Elliott and F.A. Wedgwood, Proc. Phys. Soc. 84 (1964) 63. [33] J.J. Rhyne, in: ref. [25]. [34] J. Smit and H.G. Beljers, Philips Res. Rep. 10 (1955) 113. [35] J.O. Artman, Phys. Rev. 105 (1957) 74.
MAGNETIZATION, AC SUSCEPTIBILITY AND MICROWAVE ABSORPTION IN Dy AND TbY ALLOYS D.M.S. BAGGULEY, J.P. PARTINGTON, J.A. ROBERTSON * and R.C. WOODS Clarendon Laboratory, Oxford, OX1-3PU, UK Received 3 October 1980 Measurements of magnetization, ac susceptibility and microwave absorption at 9.5 and 35 GHz, have been carried out with single crystals ofDy, Tb74Y26 , Tb79Y21, Tbs IYI9- These crystals exhibit a stable helix phase at high temperatures; Tb74Y26, Tb79Y21, also exhibit a low temperature 'frozen-in' antiferromagnet state. The critical temperatures Te, TN, and the maximum critical field for the stable helix-ferromagnet transition are, Dy: 80 K, 179 K, 11 kG; Th74Y26:25 K, 198 K, 9.6 kG; Tb79Y 21 : 93 K, 202 K, 7.4 kG; TbslYI9:116 K, 205 K, 5.5 kG. Values for the exchange parameter (~(Q) - ~(0)) estimated from the data are about 0.2 X 10-a eV.
results provide evidence for a fan phase. There is additional, but not conclusive, evidence for such a phase from magnetization, magnetoresistance, and ultrasonic measurements [8,11,15,16]. We have, therefore, investigated again the ac susceptibility, magnetization and microwave absorption for a single crystal of Dy and in addition three alloys of TbY chosen to have de Gennes factors, (G), (and magnetic structures) similar to Dy. The actual values of (G) = CGTb, where c is the atomic concentration for the alloy, are Tb74Y26: 7.77, Tb.79Y21: 8.30, TbslY19: 8.51. These values are slightly greater than 7.08 which is characteristic of Dy, but were so chosen because the early neutron data from ref. [17] indicated that the Curie temperature, To, had fallen to 0 K for ~G) < 7.88. The present series of measurements generally conf'nan the previous microwave results for Dy [18,19] and are in accord with predictions from the values of (G) given above. Moreover, the features reported in refs. [11,16] have also been observed in the microwave absorption.
1. Introduction The magnetic phase diagram of dysprosium has been investigated in considerable detail using neutron diffraction [1,2], magnetization [2-8] and ultrasonic techniques [9-11]. Above 178 K, Dy is paramagnetic, obeying a Curie-Weiss law for the susceptibility with a Curie constant equal to that calculated for the free tri-positive ion; between 178 and 86 K the stable configuration in zero field is a straightforward basal plane spiral, the turn angle per layer changing smoothly from 43.2 ° to 26.5 ° as the temperature decreases; below 86 K the ground state is a basal plane ferromagnet with easy axis of magnetization along 1120). The spiral spin configuration can be converted into a ferromagnet by applying a magnetic field in the basal plane, the magnitude of the field required for this transformation increasing from zero at the ferromagnetic Curie temperature to about 11 kG near 178 K. The discovery of changes in the direction of easy magnetization at temperatures above 130 K [6,12,13] has stimulated further interest in the details of the magnetic structures present at the critical field transition, and these authors have suggested that their
2. Materials preparation The single crystals used for the present series of experiments were grown by Dr. D.A. Hukin at the Clarendon Laboratory. Crystal grains about 1 cm 3 in
* On study leave from: Macquarie University, New South Wales 2113, Australia. 291
292
D.M.S. Bagguley et al. /Magnetization and microwave absorption in Dy and Tb Y alloys
size were obtained by the method of zone passing along a water-cooled horizontal boat in very high vacuum [20]. The overall length of a roughly cylindrical multicrystal ingot was about 10 cm and the diameter 0.75 cm. Single crystal disks were cut from individual grains and prepared for measurement as described in ref. [21]. The orientation of each disk was checked by X-ray diffraction and the compositions for the alloys confirmed by X-ray fluorescence analysis. The quoted compositions are expected to
Fig. 1. Photomicrographs of prepared single crystal surfaces: (a) Tb?gY 21 (0001) disk; note the presence of platelet impurities in this sample, (b) Dy (0001) disk.
be correct to + 1 at%. Metallographic and X-ray examination of the finished disks indicated that the crystals were of reasonable quality; figs. 1a and b indicate the standard achieved. The purity against other metals is expected to be better than 99.9% (Rare Earth Products (UK) Ltd., sublimed grade).
3.
Equipment
The specific magnetization was measured by the Curie method using a standard chemical balance to weigh the sample in an inhomogeneous magnetic field. A servo-mechanism was incorporated in the balance to ensure that the sample was always weighed at the same position in the field gradient. For these measurements the sample disk was mounted in a holder suspended by a twisted glass fibre thread, the plane of the disk being horizontal; the sample could, therefore, freely rotate about the vertical axis and the results refer to the inplane easy direction of magnetization. Disks of Ni and Fe were used to calibrate the apparatus; the values for the saturation magnetization of these elements were taken to be 55.1 emu g-i and 217.6 emu g-l at 293 K [22]. The apparatus operated over the temperature range 10 to 300 K, the temperature being determined from the boiling points of standard refrigerants and by means of an FeAu vs. chromel thermocouple. One additional experiment was carried out at 4 K. We are indebted to Mr. M.R. Wells for allowing us to use his equipment for this measurement. The ac susceptibility was investigated using a double pick-up coil arrangement with the two coils wound in opposition; one coil contained the sample, the other was air cored. The coils were connected in series and placed together in a uniform oscillatory magnetic field on which a steady bias field could be superimposed. Both field directions and the winding axes of the coils were aligned. For this configuration, the out of balance signal from the pick-up 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 described above for the magnetization balance. The sample disk was fixed with adhesive tape inside the
293
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
pick-up coil and selected crystal orientations could be investigated. A direct indication of the easy axis of magnetization was achieved by fixing the sample to the end of a tufnol rod with contact adhesive and suspending the rod in a uniform magnetic field. The axis of the rod was vertical and its weight sufficient to ensure that the pla~e of the sample disk lay in the horizontal plane, which also contained the direction of the applied field. A pick-up loop was attached to the tufnol rod and a small ac modulation of the steady field gave rise to an output signal which provided a direct indication of the orientation of the sample. Rotation of the tufnol rod by hand enabled the 6-fold or 2-fold symmetry to be unambiguously verified. This ap,paratus operated from 20 to 300 K and the temperature was determined using an FeAu vs. chromel th~rmocouple. The microwave measurements were carried out using Conventional low frequency field modulation spectrqmeters operating at 35 and 9.5 GHz. Both spectrometers had TEot ~ mode resonant cavities but at 35 (~Hz the cavity was cylindrical whereas at 9.5 GHz a rectangular cavity was used. The field modulation frequencies were 139 and 61 Hz, respectively. Phase ~ensitive detection was used to derive recorded data pioportional to the differential power absorption. These ~pectrometers operated over the range 10 to 350 K and the temperature was determined as descril~ed above. The sample was attached lightly by adhesige to the end wall of the resonant cavity and a comphtte angular investigation of the absorption could 6e undertaken by rotating the applied magnetic field. The specific magnetization and ac susceptibility measurements were carried out either at fixed temperatures Whilst increasing the applied field or, alternatively, as the sample warmed slowly towards room temperature in a constant field. A few additional experiments were undertaken with decreasing magnetic fields and with decreasing temperatures to identify hysteresis effects. The microwave data were recorded in the usual manner, with the field being increased from zero to maximum (12 kG for 9.5 GHz and 20 kG for 35 GHz) at regular intervals as the sample warmed slowly towards room temperature. The temperature drift during a complete field sweep was usually not more than 2 K, it was probably not
more than 1 K over the field range covering an absorption feature and this could be reduced to 0.5 K when required for a particular measurement.
4. Underlying theoretical relations Detailed discussions of the spinwave excitations and magnetic phase transformations for the heavy rare earth metals have been given by a number of authors (see for example the reviews in refs. [23-26]). Many different notations and approximations have been used in deriving relations for the interpretation of the experimental data. We shall therefore set out in this section the particular formulae which we have made use of when deriving the fundamental physical parameters from our results. The underlying theoretical description is based on a Hamiltonian of the form: = -
-
]Y
Jm
n,m n>rn
n
n
+
+
+
+ n
+
_
.y)61 +
+ n
J ( J - ½)
n
].
- J'y)
O)
In this equation the Cartesian axes x, y and z are taken along ( I 120), ( 1] 00) and [0001 ], respectively, gj is the Land~ factor, J is the total angular momentum of an atom, ~(Rnm ) is the exchange coupling parameter for atoms at lattice sites R n and R m corresponding to an exchange interaction energy - ( g j - 1)2 ~(Rnm) X Jn " Jm for each atom pair, H is an applied magnetic field, the terms V2 ... V~6 describe the single ion crystal field anisotropy energies, and terms containing C "r, (e~), (e~), B ~'2 represent the elastic and first order magnetoelastic contributions. The theory applies strictly to a rare earth element and treats the hexagonal close packed crystal struc-
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
294
ture as a Bravais lattice. For the TbY alloys the summations in eq. (1) are to be carried out over the magnetic atoms only and the magnetoelastic terms may need to be modified; we shall make the assumption that ~(Rnm) is unaffected by alloying at the concentrations used in the present series of experiments. No significant modifications to the formulae presented here seem to arise in a more refined analysis which takes account of the two interpenetrating hexagonal sublattices. Mathematical relations, useful to the experimentalist, are not usually derived from the full Hamiltonian given above. For example the crystal anisotropy and magnetoelastic terms are included, or neglected, according to their estimated relevance for a particular set of measurements. In addition we discard the terms in I,'4 and II6, and make use of the following relations derived from eq. (1) following the discussions in refs. [23-27].
Here M is the volume magnetization,la n the atomic moment.
m -- M(T)/M(O) = ~n (T)/Isn(O),
(4c)
(Jn)r = mJn.
(4d)
(iv) The paramagnetic Curie-Weiss temperatures, Ol, 0// ko. = ~gs - 1 y s ( j + 1)c ~p(o) + }v2(J-
(Sa)
~ ) q + }),
kO//= }(gs - 1)2J(J. + 1)c ~p(0) - ~ v2 (s - })(J + }),
(Sb)
~ p ( 0 ) - [9(q)]q=o in the paramagnetic phase, and
v2 < < (gs - 1) 2 9p(0). (v) The Ndel temperature, TN, for (0001) plane spiral kTN ={(gj - 1)2J(J + 1)cgs(Q)
q) The Fourier transforms for the exchange interactions ~(q) = ~
~(Rnm) eiq'Rnm,
+ ] V2 (J - ½)(J + ~),
(6)
~s(Q) -= [~(q)]q=O for the spiral phase and I12 < <
(2a)
( g s - 1)2 ~s(O).
~(R.,.) = ~1 ~q 9(q) e-iq'Rnm
(2b)
(vO The initial susceptibility, ×i, for (0001) plane spiral
~(Rnn) = 0,
(2c)
Xi,± =
m
~
9(q) = 0.
q -
(ii) The molecular field, Hmf
i=x,y,z,
+ 2V2 },
H~f -
m
T=0,
(gJ - 1)~
gsl3 i = x, y, z,
Oa)
m
(3b)
where c is the atomic concentration of Tb atoms in an alloy TbY; m is def'med by eq. (4c): 1 >~ m / > 0.
(7a)
T = 0.
(7b)
Here, NTb is number of Tb atoms per gram, Xi,± refers to measurements in (0001) plane, Xi,//refers to measurements along [0001 ] axis.
(rio The spinwave dispersion relation, (0001) plane spiral h 2 cos2 (q) = ~rJ 2 {(gj - 1)2c [2 ~s (Q) -
(ii0 The reduced magnetization, m
Jn,
~ s ( Q + q ) - ~ s ( Q - q)]}
X { ( g d - 1)2C[~s(Q) - ~s(q)] +2V2},
T = O,
(4a)
T = O.
(4b)
n
ltn(O) = -gd~Jn,
T=0
cm ~ ~(Rnm)(Ji),
T :/= 0,
111(0) = -gjfl ~
~s(2Q)],
Xi,//= NTbg~#2/( (gJ -- 1) 2 C[~s (O) - ~s(O)]
Hmf = _ (gJ - I)2 ~3 9(R.m)
gs~
NTb~O 2/(gs -- 1) 2 c [2 9s(Q) - ~s(0)
T = 0. j2 h2 t.Os2(q) = 2m-~ {(gJ - 1)2m2c
(8a)
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys x [2~s(Q) -
~s(O +q) - ~s(Q - q ) ] )
295
and near TN
X { ( g j - 1 ) 2 m 2 e [ ~ s ( Q ) - ~s(q)]
gs~nl = (gj
-
1 ) 2 m J c [ ~ f ( O ) - ~f(0)].
(10b)
A
+ 2V2Is/2 } ,
T=/: O.
(Sb)
Here q is along [0001] ;1(1+1/2) is the ratio of hyperbolic Bessel function of order (l + ½) to that of order ½, with the argument of the Bessel function the inverse Langev~ function of the reduced magnetization. At low T, I(1+1/2 ) = ml(l+l)12.
Here, the - / + signs refer to easy/hard direction for HI (0001) plane; Ht is usually interpreted as defining the high field boundary of the fan phase. (x) The critical field, Hc; (0001)plane spiral to ferromagnet
+ 2V2J1 +- 61V~61Js + ½B~'2X~J -1
The (0001) plane spiral transforms, in principle, to the (0001) plane ferromagnet when the free energy of the ferromagnet is less than that for the spiral. It is usually accepted that H c can be evaluated to sufficient accuracy with neglect of entropy terms. In the molecular field approximation:
+ gj~H + (Nz - Nx)M6o,q )
gH3mJHc = ½(gj - 1)2J(J + 1)m2 C[~s(Q) - ~f(0)]
(viii) The spinwave dispersion relation, (0001) plane f erromagnet
~/2~(q) = ~(gj_ 1)2Jc[~f(0)_ ~f(q)]
X ((gj - 1)2je[~f(0) - ~f(q)] _+361V~61Js
-IV~6[J6113/2 -~B3"z~'s(/~s/2)2 + ~XJ_/-/~c, ( l l a )
+ B%2)(rJ -1 + gj~tt + (N x - Ny)MSo, q },
T = 0.
and near TN (9a)
h2co~(q) = ( ( g j - 1)2mJc[~f(0)- ~f(q)]
g2{3mJnc = 2x(g2 - 1)2J(J + 1) X m 2 c [ ~ s ( Q ) - ~f(0)].
+ 2V2J]s/2m -1 + 61V~61JSf13/2m -1
(1 lb)
Here H c refers to easy direction, (0001) plane. If He < HI given in (ix) above, the transition from the spiral is first into the fan phase at H e and to the ferromagnetic phase at H1, with increasing field.
+ ½B%2)~TJ -1 m - l ( i s / 2 ) 2 + g j ~ t t + (N z - Nx)Mm6o, q ) X ( ( g s - 1 ) 2 m J c [ ~ f ( O ) - ~f(q)]
+- 361V~6[JSil 3/2m -1 + B 7'2 )O'J -1 m -1 (Is/2) 2 + gj~-I + (N~ - N y ) M m ~ o,q ) ,
T :/: 0.
5. Experimental results
(9b)
Here q is along [000 l]; Jl = ( J - ½), Js = ( J - ~) X (J - 1) ... (J - ½); -+ signs refer io easy/hard directions for H, (0001) plane; ;~'r is the magnetostriction coefficient; B 7,z ~,7 appropriate to alloy when necessary; N x , iVy, N z are the macroscopic demagnetizing factors; for l(t~l/Z) see (vii). Eq. (9a) is based on ref. [~7]. Eq. (9b) is based on ref. [24]. (ix) The critical field, H1: instability o f the (0001) plane ferromagnet
The (0001) plane ferromagnet is unstable unless the spinwave frequencies are real. From eq. (9b), the spinwave frequency will be imaginary for H < H1, where gJ~l-I1 = { ( g J - 1)2mJe[~f(Q)- ~f(0)]
(10a) 361 ~ IJS/:13/2m -1 - B %2~ J - lm- 1(/~Sl2)2),
In this section we summarize the results of detailed measurements of static magnetization, ac susceptibility and microwave absorption. In addition we report preliminary experiments indicating the temperature and field dependence for the direction of easy magnetization in the (0001) plane. Dy and the TbY alloys have the hexagonal crystal structure A3, P63/mmc. Disks about 5 mm diameter and 0.25 mm thickness were prepared from single crystal grains taken from the ingots described in section 2. For Tb74Y26 it was possible to cut (0001) and (1100) disks from the same grain; in the case of Tb79Y21, TbslY19, two separate grains had to be used and this accounts for the difference in composition. For Dy, only one disk, (0001), was available. The finished disk Tb74Y26, (1100) plane could not be prepared to a sufficiently high surface quality
296
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys
and, although this sample appeared to be adequate for the magnetization measurements, the microwave results were only of qualitative value. The same crystal disks have been used for all the measurements reported in this paper; the different sets of data may therefore be compared directly. The in-plane demagnetizing factor for a disk, 4rre in eq. (12), was approximately 0.4 when calculated using the relations given in ref. [28].
'
'
'
The results for the high temperature paramagnetic phase of the TbY alloys are given in fig. 2. It can be seen that the susceptibilities accurately obey the Curie-Weiss law. The curves refer to 1 g of material and are described by the relations:
' ,'/7,t,
/ ', ,o o1L I
.zx~;o//o/
:6 ,..%o
Vn o / o
#4 o o
7 Z,o A/
o
[]
//7/°"
///
200
0.068 Xs(74, 26) = T - 188
(0001) and ( l i 0 0 ) disks,
0.070 Y~L(79,2 1 ) - - T - 196
(0001) disk,
)0_(81, 1 9 ) -
(li00) disk.
0.076 T - 200
'
,~
=
5,1. Static magnetization measurements: paramagnetic Tb Y alloys
'
Here the Curie constants are in cgs units g-1, the Curie temperatures in K and )(1 represents the basal plane susceptibility. The accuracy of the Curie constant measurement is better than +5% and the Curie temperatures,better than +-2 K. Calculated values for the Curie constants are C(74, 26) = 0.062 cgs g-i ; C(79, 21) = 0.065 cgs g-X ; C(81, 19)'-- 0.066 cgs g-~. Evidently the only significant discrepancy, between the measured and calculated values, arises for the alloy Tbs lY19. However, it would be unwise to attach too great a weight to this difference on the basis of one series of measurements. On the other hand it will be observed that there is a similar discrepancy in the measured and calculated values for the saturation magnetization of this alloy (see section 5.2). 5.2. Static magnetization: magnetization curves Typical magnetization curves taken at fixed temperatures are presented in fig. 3a and b. In our equipment the sample was cooled to a low temperature
220 240 260 Temperature (K) Fig. 2. Inverse susceptibility for TbY alloys (units refer,to 1 g). Tbq4Y26, v(O001), n (1]00); Tb79Y 21, o (0001); Tb 81YI9,
o (li00). (20 or 77 K) relatively quickly, the refrigerant being applied directly to the outside of the sample chamber and the crystal cooling by thermal contact via helium exchange gas. For these conditions a 'frozen in' antiferromagnetic state was observed with Tb79Y21, Tb74Y26, which could be transformed into the ferromagnetic state by applying a magnetic field. In the case of Tb79Y2 l, at 77 K and below, the sample, remained ferromagnetic when the applied field was reduced to zero and did not subsequently return:to the antiferromagnetic state. For Tb74Y26 the sample remained ferromagnetic after cycling to a high field at 20 K but at 51 K or above the sample returned to the antiferromagnetic phase in a zero field. The 'frozen-in' state was not observed with TbslY~9 or with Dy, these samples became straightforwardly ferromagnetic on cooling below TeThe magnetization curves given in fig. 3a and bare plotted as a function of the internal magnetic field Hi, defined by the relation Hi = H - 4rre(M),
(12)
where H is the applied field, 4rre the in-plane demagnetizing factor and (3t) the macroscopic volume mag-
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys I
I
I
I
o~--o.~-o--~--o ,.,-o
,1= 250
I
~,--
'
~
200i
e-.
observed width was further increased if the measurements were not carried out sufficiently slowly; equilibrium times were frequently of order 15 min for field changes of a few gauss below 100 K. Values for the specific magnetization at 0 K, in emu g-i, are 00(74, 26) = 273;
~ 1001
297
00(79, 21) = 278;
oo(81, 19) = 267; ~
50
i ~//~,
,,___.=~.-.~
2
4
6
a
8
10
and the corresponding values calculated from the atomic concentration, taking the moment of Tb to be 9.28fl per atom, are
Field (kG)
o~(74, 26) = 272.5; i
.
i
i
i
i
i
i
0~(81, 19) = 288.2.
_o t. o ~ _ f . o F . . . . . .
.--.,, 250 9 f / f ~
%
t
._~
r,I,
5.3. Static magnetization: the magnetic phase boundaries
I
I
~IsolI
o~(79, 21) = 283.8;
Magnetic phase diagrams for the TbY alloys and for Dy are given in figs. 4 to 7. These diagrams include results from fixed temperature experiments
.--"
:[ lO0
14 50 II
I
e~ i " e - " ' e ~
I~ II
I
I
I
1
-~ I
2
h
I
3
field increasing
field decreasing
• • •
o
zx u
o
~ D
B
10
Fig. 3. Magnetization curves for (0001) disk, (1 i00) direction, (a) Tb74Y26 , (b) Tbq9Y21. Subsequent curves field increasing
[]
!
Field (kG)
Initial magnetization curve
/
12
I
Temperature (K)
I0 20 77
-m 8 iT
O
[] 121
D/
?
6
,n
~
o//
o~ i
C
.,,/ /, 50
netization. The magnetic transitions appear to be extremely sharp therefore, whereas, in terms of the applied field, the transition width was approximately 41toMs (Ms being the saturation volume magnetization). Moreover, in the cases of Tb79Y21 and Tb~4Y=6, the
[] / D
I d
A
,
,
100 150 Temperature (K)
I, 200
Fig. 4. Magnetic phase diagram for Tb74Y26 derived from measurements with (0001) and ( l i 0 0 ) disks. A: stable helix; B: stable ferromagnet; C: 'frozen-in' antiferromagnetic phase. Data points: % magnetization; o, a¢ susceptibility; o, 9.5 GHz microwave absorption; D, 35 GHz microwave absorption.
298
D.M.S. Bagguley et al. I Magnetization and microwave absorption in Dy and Tb Y alloys
i
14
[]
/
14
12
[]
12
10
/ []
?'
~ 8 k~
10 w
O
o' /O
-o 8 oj
iE_
//
6
lOG
4
O
6
i
/
7°
0
o
50
,
100 150 Temperature (K)
;_
12 10
[]/
(.9 8 0.) It.
6
8
, 50
[]oj
~ j
io,'7 ) ?1 0,,7"<>/ ~ " $--J /1, -d 100 150 Temperature ('K)
I
50
14
200
Fig. 6. Magnetic phase diagram for Tbs 1YI9 derived from measurements with (1 i00) disk. Notation as in fig. 4.
O
I
o
2O0
Fig. 5. Magnetic phase diagram for Tb79Y 21 derived from measurements with (0001) disk. Notation as in fig. 4.
I
A
/°'
2
)%o_eo/,
o
°'/ -
I
0
I
I
100 150 Temperature (K)
200
Fig. 7. Magnetic phase diagram for Dy derived from measurements with (0001) disk. Notation as in fig. 4.
(see section 4.2) and supplementary data derived f r o m experiments in which the sample warmed slowly towards room temperature in a constant magnetic field. When Tb74Y26 was cycled into the ferromagnetic state at 20 K and then warmed slowly in a magnetic field, the ferromagnet.stable helix phase boundary together with the h e l i x - p a r a m a g n e t i c transitions were observed. Alternatively, if the low temperature 'frozen-in' antiferromagnetic state was retained, warming in a field greater than 3.5 kG provided evidence for the additional antiferromagnet-ferromagnet boundary. Similar results were obtained with Tb79Y2 l, for which typical magnetization curves are given in fig. 8. The Curie temperatures, Tc, N~el temperatures, TN, and maximum critical fields,/-/~eax, were found to be:
Tb74Y26 Tb79Y21 Tbs1Y19 Dy
Tc (K)
TN (K)
H max (kG)
25 93 116 81
198 202 205 179
9.6 7.4 5.5 11.0
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys I
I
I
I~
100
150
299
I
~ " 300 ___
E
0 "---~ 0
g
c-
J
N 100 0
50 :
200
250
T e m p e r a t u r e (K)
Fig. 8. Magnetization curves for Tb79Y21 (0001) disk after rapid cooling to 20 K. v, 0.17 kG; % 1.34 kG; u, 3.60 kG; o, 5.8 kG; o, 7.25 kG;<, 7.75 kG. v, A, sample not cycled at 20 K, note three phase boundaries; u, o, % ,~, sample cycled at 20 K, note absence of 'frozen-in' phase. Dashed lines indicate magnetization observed when the 'frozen-in' phase has been eliminated. Arrow at top of diagram indicates TN.
5.4. The ac susceptibility
The ac susceptibility measurements complement those for the static magnetization and, in principle, provide data which identify the magnetic phase boundaries, indicate the degree of domain wall motion, and give some evidence for wall damping or pinning processes. Data points, derived from experiments using 61 Hz sine wave excitation with the bias field swept regularly from zero to 12 kG as the sample warmed slowly towards room temperature, have been included in figs. 4 to 7. The phase boundaries determined from these measurements are in good agreement with the results from static magnetization. Above 120 K the ac susceptibility showed a clear maximum at the D helix-ferromagnet phase transitions, becoming very intense near TN, at which temperature the signal disappeared. The magnetic phase boundaries were also indicated by changes in the ac susceptibility as the sample warmed slowly towards room temperature in a constant magnetic field. Sine wave excitation at 61 Hz was also used in these experiments and again the static magnetization results were confirmed. Typical recorded curves showing the changes at the phase boundaries are given in fig. 9. With this apparatus it was also possible to carry
out measurements as the sample cooled down slowly from room temperature. It was found that Tb79Y21 converted to the ferromagnetic phase at 72 K in zero applied field. This experiment confirms the 'frozen-in' nature o f the low temperature antiferromagnetic phase and also indicates a temperature hysteresis of about 20 K in Tc. i
i
I
I
N
2.5kG
<
--1 I
I
I
I
50
100
150
200
Temperature (K)
Fig. 9. Recorded curves for ac susceptibility (arbitrary units), Tb79Y21 taken at constant bias fields. The curves have been displaced vertically for clarity of presentation. The curves refer to. (0001) excepting that labelled (1 i 00), where the crystal direction was (11~0).
300
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
Preliminary experiments, using square wave modulation at 100 Hz, have been carried out to investigate the domain wall response time. These measurements indicate that domain wall pinning may be more significant than viscous damping in giving rise to the time dependent magnetization effects reported in section 5.2. 5.5. Magnetic anisotropy
The easy axis of magnetization was observed directly by hanging an (0001) disk in a uniform magnetic field. Changes in crystallographic orientation of the easy axis for Dy have been reported in refs. [6,12, 13]. Our experiments confirm the results given in ref. [13]. Below 130 K, the easy axis in the ferromagnetic phase was (1120); above 130 K, (1100) was the easy direction for fields just greater than the first critical field but this changed to (1120)at higher fields. We have carried out similar experiments with the TbY alloys. For our samples the easy axis was ~1100) in the ferromagnetic phase throughout the field and temperature range covered with our equipment (0 kG < H < 12 kG; 77 K < T < 200 K). We have observed no changes in orientation such as occur in Dy. It will be noted that (1100) is also the easy axis of magnetization for Tb. 5. 6. The microwave absorption
Measurements of microwave absorption in single crystals of Dy have been reported previously in refs. [18,19]. These authors observed field dependent absorption in the temperature range 90 to 180 K. The signal from the Dy crystal used for the present experiments was too weak for quantitative measurement at temperatures below 110 K. The microwave absorption features occur in the vicinity of the stable helix-ferromagnet transformation. The results from the present series of experiments are included in figs. 4 to 7. At low temperatures the microwave absorption corresponded to a broad decrease in power loss with increasing magnetic field. As the temperature increased this feature became narrower and at about 130 K a sharp low field edge developed, indicating an abrupt increase in power loss. This feature continued to increase in intensity up to TN, when the absorption assumed the
general shape of a paramagnetic resonance line, broadened rapidly, and became too weak for measurement a few degrees above this temperature. The absorption was of the same general form at both measuring frequencies for Dy and all the TbY alloys _ when the magnetic field was along (1100) or (1120). This is also in agreement with the previous microwave results for Dy [18,19]. No field dependent absorption was observed for the field along [0001]. In the present series of experiments an additional high field shoulder to the microwave absorption was observed at 35 GHz. This feature was clearly identified at temperatures above 130 K and presumably corresponds to the magnetization and resistivity singularities reported for Dy [8,16]. This additional absorption feature was not resolved in the present experiments at 9.5 GHz, probably because it occurred at, or above, the high field limit of the magnet used in that spectrometer. Microwave absorption was observed at 9.5 GHz, in the range 77 K < T < 200 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 polarised, but was always mixed into the '13erpendicular ' absorption at 35 GHz where the cavity mode had circular symmetry. The 'parallel' field configuration does not give rise to absorption in a normal ferromagnetic resonance experiment when the sample is saturated in the field direction and the presence of such an absorption indicates that the magnetization is not fully aligned by the external field. In the present series of experiments the 'parallel' field absorption at 9.5 GHz exhibited features similar to those found with the perpendicular configuration, suggesting that the microwave absorption phenomena occur when the magnetization is influenced by the domain structure at the phase boundary. 6. Discussion Values for the exchange parameters [~(Q) 9(0)] may be estimated from the experimental data presented in section 5 and the theoretical relations in section 4. From section 4(iv) and (v): k(TN - Oj.)
= ~(gj - 1)2J(J + 1)C[gs(Q) - ~p(O)] •
(13)
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys
301
Table 1 Summary of magnetic data (G)
Tc (K)
TN (K)
0t (K)
o0 (emu g-l)
Xi,_t_ X 10-3 (emu g-l)
H max (kG)
179 179 198
169
351
5.2
11.0
0.17
188
273
4.0
9.6
0.16
[gs(Q)
Dy *
7.08
Tbq4Y26
7.77
85 81 25
Tbq9Y21
8.30
93
202
196
278
3.4
7.4
0.10
TbalYl9
8.51
116
205
200
267
3.9
5.5
0.08
221
229
239
328
-
0.25
0.24
Tb*
10.5
-
9p(O)]
(x 10-s eV) (1)
(1) From TN - 01. (2) From Xi,± assuming 1~(2Q) - 9(0)1 ~- 41~(Q) - 9(0)1; assuming ~(2Q) = 9(0). (3) From H max. (4) From inelastic neutron scattering.
From section 4(vi): Xi,± = g1~oo/(gs - 1)2 Jc [2 ~s (Q)
- 9s(O) - 9s(2Q)] •
(14)
Values for [~s(Q) - ~s(0)] may be derived from this relation using the empirical observation that reasonable extreme limits are given by I~s(2Q) - ~s(0)l ~ 41~s(Q) - ~s(0)l or alternatively ~s(2Q) ~ ~s(0) depending on which experimental curves are selected from inelastic neutron scattering data. From section
4(x): gs[3mJI-pcncax = ~ ( g , / - 1)2j(j + 1)m2c
X [~s(Q] - ~f(0)].
(15)
Eq. (13) assumes that V2J 2 < < (ga" - 1)2j(j + 1) X c ffp (0), whilst the low temperature relation for ×i,1 has been quoted in eq. (14). This is reasonably satisfactory since the experimental results for Xi,± are not strongly temperature dependent. Eq. (15) is a high temperature approximation, hexagonal anisotropy and magnetoelastic terms have therefore been neglected. A contribution to the free energy which arises from the distortion of the crystal lattice at the phase transition (an elastic energy term) has also been discarded.
Values for the exchange parameters derived using eqs. ( 1 3 ) - ( 1 5 ) are given in table 1. Ignoring the refine ment of the subscripts which identify parameters referring strictly to the ferromagnetic, spiral and paramagnetic phases, it is evident that there is general agreement among the different sets of experimental data and that [~(Q) - 9(0)] ~ 0.2 × 10 -3 eV. W h e n H t from eq. (10a) is smaller than Hc from eq. (1 la), the (0001) plane ferromagnet is stable for H > He and the transition from the spiral phase to the ferromagnetic phase is direct. The highest temperature at which this direct transition may be expected to occur is determined by the condition H~ = He; for our samples this is about 130 K. At this temperature, therefore,
a*
½(gj-
1)2jmc[~(Q)-
9(0)] = E
(16)
(361V~61Jsm-xita/2 - B%2 ?~~J - l m - l (Is12) 2 } . The spinwave gap, A, at q = 0 at this temperature may be written, using eq. (9b), A = hoof(O) .~ [ 2 V 2 J i s / 2 m - l E ] '/2 .
(17)
This equation may be rewritten in terms of the macroscopic crystal anisotropy energy parameter, K2. If the crystal anisotropy energy is FA = - K 2 sin20, and a
302
D.M.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
[~s(Q) - ffs(O)]
[~s(O) - ~ f ( 0 ) ]
[~s(Q) - ~s(O)l
A (130 K)
A (130 K)
(X 10-3 eV) (2)
(X 10-3 eV) (3)
(X 10-3 eV) (4)
(X 10-3 eV) (5)
(X 10-3 eV) (6)
0.11 0.32
0.36
0.3
0.43
0.4
0.09 0.27
0.33
0.62
0.1 0.3
0.24
0.62
0.08 0.25 0.11 0.32
0.19
0.62
0.41
(5) From H 1 = H c. (6) From neutron data. * Values from ref. [33].
single ion model is valid, then [V2J]Sl2 m-1 ] = gs~K2 [M(T)] -1 so that A ~ [gjflK2E/M(T)] 1/2.
(18)
easy axis of magnetization for Dy is thought to arise from a decrease in the angular spread of the spin system in a fan phase [13,14]. Consider a simple linear triangular waveform representation of a fan, measuring the_spin angle q~in the (0001) plane with respect to (1120) and taking the z-axis along [0001 ]. Choose the origin for z at a plane where q~is an extremum; the fan may then be described by
The values of A(130 K) given in table 1 have been calculated from eq. (18) using values for K2, M(T), from ref. [15] and taking [~(Q) - 9(0)] from our own data. It can b e seen that there is quite reasonable agreement with the estimate of A from inelastic neutron scattering. Our experiments indicate that ×i,± is greater than ×i, // throughout the stable helix phase. This implies (compare eqs. (7a) and (7b)),
and the anisotropy energy per plane described by
v2 > I 0 ~ - 1)~C[~s(O) - ~ s ( 0 ) l ,
Wp = -IK661 cos 6~.
i.e. 1:2 > 0.03 X 10 -3 eV and correspondingly K2 > 1 X 108 erg cm -3 . This latter inequality is consistent with known values for K2. Evidently K 2 J z ~ 1.26 × 10 -3 eV, whereas kOt ~ 17 X 10 -3 eV and so the inequality assumed for eq. (13) can also be satisfied. It should be noted that, for the 'frozen-in' antiferromagnet phase of Tb79Y21 at 20 K, Xi,± ~ 48 X 10 -3 emu g-l. This is much larger than expected for a simple spiral configuration and may imply the presence of some ferromagnetic domains. The change in crystallographic orientation of the
The averaged anisotropy energy, WA, for an incom. mensurate fan may be obtained by integrating eq. (20) over one complete period, 0 ~< z ~< (2-1 ; 27r~ = Q.
= q~o(40z - I),
0 < z < 1[2C),
= q~o(3 - 40z),
1/2Q < z < 1/0,
WA = ( - I K ~ I/6¢o) sin 6¢o.
(19)
(20)
(21)
Evidently WA is a minimum for a fan centred on (1120) if¢o < 30 °, but for 30 ° < ¢o < 60 ° the minim u m energy corresponds to a fan about (1100). If, therefore, the fan amplitude immediately above the phase transition for Dy is greater than 30 ° (23 ° for the model of refs. [13,14]) the observed change in
D.M.S. Bagguley et aL / Magnetization and microwave absorption in Dy and Tb Y alloys
orientation of the easy direction of magnetization can be explained since increasing the field will reduce ¢o- No change in orientation of the easy axis of magnetization has been observed for the TbY alloys, but the microwave experiments suggest the possible existence of a fan phase; these observations can be reconciled if the amplitude of the fan is always less than 30 ° (23 °) and the sign before IK~I is taken positive, so making (1100) the easy axis as for Tb. A general theory of magnetic resonance in the heavy rare earth metals has been given in refs. [29, 30]. These authors assumed the microwave excitation to be essentially uniform throughout the sample; the microwave magnetic field couples to spinwave modes having wavevectors q = 0, q = Q, in the basal-plane spiral phase and to modes with q = 0 in the ferromagnetic phase. For an incommensurate spiral the frequency of the q = 0 mode is zero [31], whilst from eq. (8a) Cos(Q) ~ [ J ( g j - 1)/h] X {[2~(Q) - ~(2Q) - ~(0)1V2} 1/2; vs(Q) ~ 100 GHz, at 0 K, which is significantly larger than the microwave frequencies, 35 and 9.5 GHz, used in the present experiments. This estimate for us(Q) is probably adequate for T < 0.9 TN and so no marked field dependence of the microwave absorption is to be expected in the spiral phase. In the ferromagnetic phase at high temperatures, where the spiral phase is stable in zero applied field, eqs. (9a) and (9b) indicate that Ggf(0)• (1/h)[2gj[JH c V2JI] */2 , probably for T < 0.9 TN; again the excitation frequency is vf(0) 100 GHz. At low temperatures the spinwave gap arising from magnetoelastic and anisotropy terms determines the minimum excitation frequency. From eq. (9a) ogf(0)~ (1/h){2V2J~ [361K~ IJs + fir,z ;k-rj-i ] } 1/2, vf(0) I> 500 GHz. It seems unlikely, therefore, that a straightforward ferromagnetic resonance will be observed in Dy and the alloys Tb74Y26, Tb79Y21, TbalY~9, at 35 or 9.5 GHz except possibly at low temperatures for very high magnetic fields applied along a hard axis of magnetization in the (0001) plane (for this particular orientation the spinwave gap can be reduced to zero when magnetoelastic terms are negligible). Previous microwave experiments, u sing Dy, con firm this conclusion [ 18,19 ].
303
For the temperature range covering the stable spiral phase the microwave power loss in the sample may be expected to respond to the changes in magnetic structure at the critical fields. In the vicinity of He the change in spinwave frequencies, discussed in the previous paragraph, may give rise to a transition in the high frequency permeabihty, p, whilst the disappearance of superzone boundaries may cause a change in resistivity, p [ 16,32]. The microwave power loss in the sample depends essentially upon ~ p ) l n and may, therefore, change at the phase boundaries; such changes in loss at H c were reported in refs. [ 18, 19 ]. The present series of experiments suggest that further consideration of the detailed magnetic structures which occur at the phase transitions may be useful. Below 130 K the change in structure is expected to be direct from the spiral to the ferromagnetic phase. The width of this transition as measured by the external magnetic field will be of order 4rreM. The microwave measurements indicate a decrease in power loss when the structure changes from spiral to ferromagnet but although the onset of the change in loss is reasonably sharp, the microwave transition is not fully completed until the external field increment is three or four times 4rre.M. For T > 130 K, a high field shoulder in the microwave power loss presumably indicates the second phase boundary at H1. Consider the possibility of microwave resonant absorption in an extensive fan phase. For T > 130 K the fan is expected to form about the direction of the applied field, H [23,24,30]. A mode with q = 0 may be excited by the component of microwave magnetic field perpendicular to H. The frequency of this mode is of order (1/27rh)[2gj~HV2J1 ] 1/2, which is close to that for the ferromagnet q = 0 mode discussed previously and greater than 35 and 9.5 GHz, used in the present experiments. A component of the microwave magnetic field parallel to H in the (0001) plane, may excite the antisymmetric spinwave mode at q = +Q, for which UA(Q) ~ (1/27rh)[4V2Jlgj[3 X (H1 -/_/)]1/2 becoming zero at Hi. This mode could be excited at 35 GHz when (HI - H) ~ 1 kG and at 9.5 GHz when (H1 - H) ~ 0.1 kG. The intensity goes to zero atH~ in proportion to (HI - / / ) 1 / 2 and so the microwave absorption should be stronger at 35 GHz. The high field shoulder, mentioned above, could therefore originate from an excitation of the anti-
304
DalI.S. Bagguley et al. / Magnetization and microwave absorption in Dy and Tb Y alloys
symmetric mode, -+Q, in an extensive fan phase. It seems unlikely, however, that an extensive homogeneous fan phase is actually present and it is more probable that a mixed phase having domains with fan and spiral configurations occurs. The presence of domain walls can have a marked effect on straightforward ferromagnetic resonance absorption since additional demagnetizing fields of order 47rM arise [34,35]. However, in the case of Dy and the TbY alloys, the walls are expected to be perpendicular to [0001 ], the additional demagnetizing terms have to be compared with the axial anisotropy field (V2J1/gj~3) > > 41rM, and so it is improbable that the walls modify significantly the estimate for the excitation frequency given in the previous paragraph. On the other hand the observed additional feature in the microwave absorption could simply reflect nonresonant magnetoresistance changes of the type reported in ref. [16], and will not indicate the excitation of a particular spinwave mode but arise from a further change in the quantity ~ p ) l / 2 mentioned earlier in this section.
7. Conclusions Measurements of magnetization, ac susceptibility, and microwave absorption indicate that TbY alloys having de Gennes factors near 8.0 exhibit magnetic phase diagrams similar to Dy. The microwave experiments suggest that a high temperature, high field, 'fan' phase is also present in the alloys but, in contrast with Dy, no change in crystallographic direction for the easy axis o f magnetization has been found. At low temperatures a 'frozen in' antiferromagnetic phase was observed in Tb74Y26, Tb79Y21. This phase appeared to be metastable up to a well defined, but temperature dependent, critical field. The transition into the ferromagnetic state was very slow at the phase boundary, where time constants of order 15 min were observed, suggesting that a (sample dependent) crystal lattice relaxation may be involved.
Acknowledgements The authors are grateful to Dr. D.A. Hukin for providing the single-crystal materials used in these experi-
ments. Mrs. M. Hoggins, Mr. D. Morris, Mr. G. Read and Mr. F. Wondre contributed to the materials preparation and instrumentation. Professor R.J. Elliott has made a considerable contribution throughout this work in discussions o f the underlying theory. J.P.P. and R.C.W. wish to acknowledge the award of SRC 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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