Ferromagnetic resonance in nickel around the curie temperature

0022-3697/81/08068I-06S02.W/0 Pergamon Press Ltd.

1. Phys. Chm. Solids Vol. 42, pp. 681-686. 1981 Printed in Great Britain.

FERROMAGNETIC RESONANCE IN NICKEL AROUND THE CURIE TEMPERATURE STIC HARALDSON and LARS PETTERSSON Department

of Solid State Physics, Institute of Technology, University of Uppsala, Uppsala, Sweden

(Received 21 July 1980;accepted 23 December 1980) Abstract-The ferromagnetic resonance behaviour of nickel up to and above Tc is investigated. It is shown that up to the Curie point, which is 357.8”C,the resonance can be described by the Gilbert equation with the parameters g = 2.2OkO.02 and A =(4.0*0.5)x l@s-‘. Above r,, however, it is an open question if the Gilbert equation describes the resonance adequately. No sign of a decreasing linewidth above 365°C is seen. On the contrary, the resonance broadens rapidly and fades away. The exoeriment is oerformed in a new wav with the cavity kept at room temperature and the temperature of the sample is monitored during the run of the sp&trum.

The

ferromagnetic

INTRODUCTION resonance behaviour

of nickel

has

been rather extensively studied in the past, and all authors seem to agree on the results for temperatures below 300°C. There are, however, very different opinions about what happens with the resonance when the Curie temperature is approached. Rodbell[l] states that the resonance can be adequately described by the LandauLifshitz equation of motion and a constant g- and Avalue up to the Curie temperature. Actually, he seems to use the Kittel resonance condition all the way up to T,, which he sets equal to 360°C. He says, however, nothing explicitly about the high temperature measurements. Bhagat et al. [2,3] on the other hand, claim that none of the known equations of motion, Gilbert, LandauLifshitz or Wangsness, can describe the resonance behaviour of Ni in the vicinity of the Curie point. Spb;rel and Biller[4] found that in a region above T,, 365°C < T< 385”C, the linewidth rapidly falls off with increasing temperature, with maximum in the linewidths at 365°C. Below this temperature, there seemed to be a 10°C discrepancy between their measurements and those of Bhagat et al. These very different results compelled us to try to perform high temperature resonance experiments in a slightly different way, which above all, gave us a possibility to monitor the temperature of the sample while the spectra were taken. APPARATUSAND SAMPLES

The spectrometer is an ordinary Varian V45 X-band spectrometer with a standard rectangular TE,,, cavity and 100kHz field modulation. Instead of heating the whole cavity in an oven we heat the sample inside the cavity. The sample is mounted on the disk shaped end of a thin copper rod (see Fig. I), which is thermally insulated from a copper bottom plate through a stainless steel cylinder. To obtain strain free mounting only one end of the sample is glued to the copper disk. A heating coil is wound around the lower part of the copper rod. The winding of the coil should be bifilar not to create an unwanted magnetic field. For that reason, we made the

heater from a stainless steel shielded chrome-constantan thermocouple wire. The temperature of the sample is directly measured with a thermocouple that comes up through the stainless steel cylinder and an axial hole in the copper rod. The whole is surrounded by a single-walled quartz tube which can be connected to a vacuum pump. We normally work with a vacuum better than 10e5Torr. As only the thin copper rod has to be heated the power needed is rather low, some few watts are sufficient to reach about SOO”C,which is the highest temperature we have used. As the cavity stays at room temperature, its properties are unaffected by the heating and we can use the sensitive, high-Q Varian cavity without loss of sensitivity. The resonant frequency of the empty cavity is 9.3 GHz, with the whole set up inside it the frequency rises to 9.5 GHz. The Q-factor decreases slightly but the sensitivity and the signal line shape are not visibly affected. We have found that the temperature of the sample rises about 2°C when a microwave power of about I5 mW is applied to the cavity, due to the eddy current loss in the copper rod and the sample. Thus, if in an experiment the temperature is measured somewhere else and these readings are checked against the temperature of the sample without applied microwave power, the measurements may be in error. Extreme cleanliness is also essential. A grain of dust, which gives no EPR-signal at room temperature, may, when heated, develop a disturbing signal. The bulk Ni single crystals used in these studies were spark-cut from a piece (nominal purity 99.997%) supplied by Materials Research Corporation. One sample was grown in our laboratory using the floating zone method. All samples were cylinders, with the cylinder axis coinciding with a [lOO]-direction. The length and the diameter of the cylinders were typically 5 and 1 mm respectively. After the samples had been spark-cut they were first etched and then electropolished in a 60% sulphuric acid solution. Finally they were electroplated with a thin layer of copper, leaving only the central part of the cylinder exposed to the microwaves.

681

S. HARALDSON and

682 To vacuum

L. PETTERSSON

pump

MAGNETIZATION

To analyse resonance data, one needs to know the magnetization. The field and temperature dependence of the magnetization in nickel was investigated in the 1920’s by Weiss and Forrer[S], and their data have since been used. We considered it worth while to measure the magnetization as a function of field and temperature directly on one of our samples. We found, that for our sample, we could use the equation of state given by Arrott and Noakes [6],

with the following parameter values: y = 1.31, p = 0.386, M, = 40.0. 103A/m, T, =0.308”C and r, = 357.8”C. A comparison with Arrott and Noakes’ paper shows, that the only parameter which has a value different from that needed to fit Weiss and Ferrer’s data is Tc. Figure 2 shows measured and calculated M vs H curves in the temperature interval of interest, 350-36X. RESULTS AND DISCUSSION

In Fig. 3 the resonance field and the line width are plotted vs the temperature. As can be seen, the resonance field and the line width increase very rapidly around 7’,. Above 365°C the line has become so broad that it is impossible to determine the resonance field and the line width with any reasonable accuracy. It is, however, possible to follow the line higher up in temperature and see how it broadens and fades away. We have not been able to see any narrowing of the line, as reported by Spore1 and Biller[4], but up to 365°C our data on resonance field and line width agree with their results. Hence, we have the same discrepancy, about 10°C with the results reported by Bhagat and RothsteinB].

Fig. 1. (1) Copper plate, (2) stainless steel tube, (3) copper rod, (4) heater, (5) copper leads to the heaters, (6) thermocouple attached to the sample, (7) sample, (8) quartz tube, (9) O-ring groove.

Spectra were taken both during heating up and cooling down. No significant differences were observed. In addition, in order to minimize any temperature drift of the sample, data were not taken until the temperature had been stable for about 30min. The magnetic field was applied along the cylinder axis and measured using a NMR-probe.

I 5x IO5

E

Crx-

t

x-x-x

Px-x-x-

-x-‘x

-x-x x-

)(-x-X

>

X-

,X-

350 352

--x356

-x-x

-x-x

-x ,-x

X-X

-x

-x358

,-x362

7

x/x.XHx-

6 Fig. 2. Calculated (-_)

A/m

and experimental

(x) M-H curves.

“C

Ferromagnetic

resonance in nickel around the Curie temperature

683

)x

105 E

h . 8 5

I

I 300

310

320

330

L

I

340

350

360

370

“C Fig. 3. Experimentalresonance fields (x ) and line widths (0). -

We have also introduced the following parameters

For the analysis we used the Gilbert equation diU -=-/kJ(~x&$MxvQ!f dt

a=” ILOY%

L

where M and fl are the sums of the static and time dependent magnetizations and fields, M, is the static magnetization parallel to the applied field, y is the gyromagnetic ratio (=g&h), A is the exchange stiffness constant and A is the relaxation parameter. As we are interested only in the temperature region around T, the anisotropy can be disregarded. Following the standard procedure, first given by Ament and Rado 171,we obtain (TJt jL!,)m, -j&n,

are calculated with g = 2.19, A = 4 x 109s-l.

- k, = - l 2K2mx

(3a)

jnm, t (1 t 77t j&!,)m, = - e2K2m,

(3b)

2je2m, + 2jc2kx = - l 2K2k,

(34

m, t k, = 0.

(4)

Equations (3a) and (b) come from the equation of motion, while eqns (3~) and (4) follow from Maxwell’s equations. Here, lower case letters denote dynamic quantities and we have assumed y = m(O)ei(m’-ky) b = h(O)ei(ot-kY) _e= _e(O) ei(o’-ky).

L,=h POYK l/2 so=

2

( POwa

>

K=keSo &2A CLOM,~~O~ and have neglected the displacement current dQ/dt in the Maxwell’s equation. The solution to the eigenvalue eqn (3) gives 3 damped waves travelling into the metal. In the case of Ni at temperatures above 300°C one of the three waves will be dominating meaning that the exchange term will have negligible effect on the lineshape. In order to simplify the equations for this situation we put E = 0. Thus we get

j$=

jk2 --

poocr

_

i12-(1t~tjs1L)2 a2 - (7 t j&5)(1 t 71t j&L.)

(‘)

If fU is small (= 0), the damping will be high if 0’ = V(Qt 1). This is the ordinary Kittel resonance condition, (o/~or)~ = H(H + M). The damping will be small if fi = 1 t 7 (antiresonance) or (o/pay) = H t M. Antiresonance will appear only when 0 > 1, i.e. M < (dpoy). In the case of Ni (g = 2.2)

684

S. HARALDSON and L. PETTERSSON

at 9.5 GHz, this means that M is less than 2.45 x 10’ A/m, i.e. the temperature is higher than 320°C. The microwave power absorbed in a metal is proportional to the real part of the surface impedance, which can be shown to be Z = (j/c/a). In an ESR-FMR-experiment, however, the signal, which is recorded, is proportional to Q-‘, where Q is the quality factor of the cavity. Haraldson and Smith[8] have shown that the resonance signal will be correctly described by the surface impedance, but the antiresonance signal not. Using the perturbation method outlined by Lax and Button[9] they obtained A(l/Q) = C x Re((jW2) - l/K), where C is a constant.

According to a remark by Lax and Button and Haraldson and Smith the numerical results of perturbation-theory should be taken with some caution. However, the quantitative results or the functional form of the solution should be valid. Therefore we introduce a complex parameter a.

(6) (Yis determined from the best fit to the observed lineshape. It is important to note that LYhas a pronounced effect on the antiresonance, where k becomes small,

2 xY’Os, 6 A/m

x/

x’

Fig. 4. Experimental

Fig. 5. Experimental

(----_) and calculated ( X ) resonance

(--_)

and calculated

curve at 322.3”C. g = 2.19; A = 3.7 x I@ SC’, m = 0.3 t 0.2 j.

( :: ) resonance curve at 340°C. g = 2.19, A = 3.7 x 109 s-l, (Y= 0.3 + 0.2 j

Ferromagnetic

resonance in nickel around the Curie temperature

685

&

4x xx

x’

i’ X

I

X

,I,

I x

1I, X

z’ V

Fig. 6. Experimental

(-_)

Fig. 7. Experimental (-

and calculated

5

yh”,; x 4

X X

( x ) resonance curve at 350°C. g = 2.19, A = 3.9 x 109 s-l, a = 0.3

t 0.2j.

) and calculated (x) resonance curve at 357.8”C (T,). g = 219, A =4x l@s(I = 0.3 +0.2 j.

Fig. 8. Experimental (-)

and calculated resonance curve at 3656°C. 0, g = 2.40,A = 4.5 X I@ SK’,a = 0.3 X0.2 j. x,g=2.19,A=4.2~1~s-‘,a=1.0tOj.

686

S. HARALDSONand L. PETTERSSON

while its effect on the resonance is negligible. When, however, the magnetization becomes small and hence the antiresonance starts to merge with the resonance, the parameter (Y is very important for the calculated total lineshape, which is an antiresonance line superposed on a resonance. To obtain the g-factor and A-value we calculated the resonance curve for various temperatures from eqns (l), (5) and (6) for different values of the parameters g, A and (Yand compared with the experimental spectra. Figures 4-8 show some curves from 322 to 365°C. It should be noted that in the SI system of units A is equal to Acts times 4~. In Figs. 1 and 2, the full lines show resonance field and line width calculated with g = 2.19 and A = 4x 109s-‘. It must be pointed out, that it is not possible to obtain the g-factor and A-value only from the line position and width. CONCLUSION

We can not see any sign of a decreasing line width. Up to about T, the resonance behaviour can be described A= with a constant g = 2.20 * 0.02 and (4.0 + 0.5) x lo9 SK’. Above T,, the situation is more uncertain. It is not possible to reproduce the experimental spectra with the same parameter values as before. It

should be noted, that making A variable does not help. Either the g-factor has to be increased, or the parameter (Y must vary. In Fig. 8 we compare two calculated curves, one with constant g and one with constant (Y, with the experimental spectrum at 365.6”C. Here we do not find it possible to state, that one calculated curve is better than the other. At higher temperatures we have not found it meaningful to try to fit any calculated curve. What happens is that the high field peak of the resonance almost totally disappears, and at the same time the signal strength decreases rapidly. so any base-line drift will severely distort the line. Hence, we feel inclined to leave the question open, if the Gilbert equation correctly describes the resonant behaviour above T,. REFERENCES

D. S., Physics 1, 279(1965). 2. BhagatS. M. and ChicklisE.P., Phys. Reo. 178,828 (1969). I. Rodbell

3. Bhagat S. M. and Rothstein M. S., L de Physique 32, Cl-777 (1971). 4. Spore1F. and Biller E., Solid State Commun. 17, 833 (1975). 5. Weiss P. and Forrer R., Ann. de Phys. 5, 153(1926). 6. Arrott A. and Noakes J. E., Phys. Reu. Lett. 19, 286 (1%7). 7. Ament W. S. and Rado G. T., Phys. Rev. 97, 1386(1954). 8. Haraldson S. and Smith LJ., J. Phys. Chem. So/ids 35, 1237 (1974). 9. Lax B. and Button K. J., Microwave Fe&es and Ferrimagnefics, Chap. 8. McGraw-Hill, New York (1962).