Field-induced anisotropy in the NiMn system with Pt impurities

Journal of Magnetism and Magnetic Materials 109 (1992) 323-331 North-Holland

Field-induced anisotropy in the NiMn system with Pt impurities Y. i~ner, B. Aktas, F. A p a y d i n a n d C. K a p t a n o g l u Hacettepe Unit,ersity, Department of Physics, Beytepe, Ankara, Turkey Received 2 August 1991; in revised form 1 October 199.:

We have carried out ferromagnetic resonance (FMR) and magnetization measurements on Ni76_xMn24Pt.,. (x = 0, 1, 1.5, 2, 3) alloys. After cooling the samples down to 4.2 K in a magnetic field of 10 kG, the FMR and dc magnetization measurements have been done in the applied field either parallel (n-FC case) or antiparailel (r-FC case) to the cooling field direction for parallel geometry (static magnetization parallel to the surfa z of samples). It has been observed from the FMR and de magnetization data that both the unidirectional and uniaxiai anisotropies are significantly induced with increasing platinum impurity atoms. The ratio of the intensity of the resonance line for the r-FC case to that for the n-FC case increases with platinum as well. Furthermore, the line shapes are highly correlated with platinum. This asymmetric behaviour relative to the applied field direction has been discussed by taking into account the surface induced mode and introducing a directional character to the surface anisotropy in the Rado-Weertman (R-W) general surface boundary conditions.

1. Introduction The disordered NiMn system is one of the most investigated alloys which exhibits spin glass (SG) properties. T h e s e binary alloys show either SG or reentrant ( R E ) character for Mn content above or below 25 a t % respectively. T h e latter gives a p a r a m a g n e t i c ( P M ) - f e r r o m a g n e t i c (FM) transition at a t e m p e r a t u r e Tc followed .by a second transition to a reentrant phase at a lower temperature Tf. T h e R E state is also referred to as a mixed states of FM and SG. T h e hysteresis loop of R E alloy shifts towards a negative field after cooling the sample down to a sufficiently low t e m p e r a t u r e in a magnetic field (FC case). The amount of this shift is assigned as an induced unidirectional anisotropy ( H A). D C magnetiza[ i o n LIJ, r~l ..u . .b . .~ . . ~ [ ) t+:k:l:+~, ol c=oR c r,~ .l~,SzStl'~tt.V - a ; +; . ; h , ac ~ . l t ) l l l t . v rL~,j, t ~ " r JA, I [5,6] and magnetoresistivity [7,8] show that NiMn system consists of a large number of small and similar magnetic domains, each of which has an unidirectional anisotropy along its saturation magnetization direction. Recent E S R [4,9], magnetization and ac susceptibility [2,10,11] also pro-

posed some evidence indicating that the unidirectional anisotropy is not rigidly linked to the lattice along its initial direction, but it can be elastically rotated by the applied field. Prejean, Joliclerc and Monod [12] have shown that the addition of a few hundred ppm of Au and Pt nonmagnetic impurities in CuMn alloys considerably widens the hysteresis loop in the low-temperature SG state. Subsequently, Levy and Fert [13] have suggested a theory based on s p i n - o r b i t coupling including DzyaloshinskyMoriya ( D - M ) type interaction and they have shown that this anisotropy would be strongly enhanced due to strong spin-orbit coupling. O u r recent magnetization data on NiMnPt ternary alloy [11] have also displayed that both the unindirectional and uniaxial anisotropies are sig•. . , ~ ¢ ~ r , , - ~ - l ~ , IIIII~,~IILI~"

;~I~.-,~1

111~,,~ l ~ t ~ ' ~ ' u

l~,~r ,at-lrt~n¢~ O~ ~,,Ji,~.,&~,.4,11l~

D t A. ~.

;& m n l l r ~ tI ;[ol ~ ~ &li,~.,~i t"

ti ~a

NiMn alloy. In th~s study, we have used ESR and dc magnetization techniques together to bring out the role of Pt impurities on the anisotropies for several Pt concentrations (1, 1.5, 2, 3 at% Pt). We o b s e ~ e d that Pt impurities induce not only the anisetropies, but also change the resonance line

0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

324

E Oner et al. / FieM-induced anisotropy in

shapes. We have proposed that the resonance lines consist of usual F M R bulk mode and a surface induced mode. Thus we are able to interpret the peculiar resonance line shapes by taking into account the Rado-Weertman ( R - W ) F M R model [18], in which we have introduced a directional character to the, surface induced anisotropy.

120

Pt-impurity NiMn

(ai

FC

I

~

=4.2,K

NI76Mn2~.

~

I

I

I

-12-

~

{b)

o

2. Experimental techniques and results

I

I

Ni75Mn2/.ptl

r I

I

i

I

I

I

I

3

The samples were prepared by melting together high-purity constituent in an rf furnace. Dislike specimens (3 mm diameter, 0.2 mm thick) were cut from the ingot. The details of sample preparation were given in our previous study [6]. The magnetization was measured with a horhemade vibrating-sample magnetometer, the sensitivity of which was about 10 -5 emu. The frequency of vibration was equal to 88 Hz and was provided by the internal oscillator of a PAR 124 phase sensitive detector which constituted the central part of the whole set-up. The FMR studies were performed with a Varian Associate reflection spectrometer at fixed frequency (9.25 GHz). The temperature was controlled by a helium-flow cry:~tal from 4.2 to 300 K (Oxford Helium Flow Cryo.,'.tat), which was used in both magnetization and FMR measurements on the same samples. Fig. 1 shows the familiar FC hysteresis loops, plotted just after cooling the samples in He-- 10 kG down to 4.2 K for Ni76_xMn24Pt., (x = 0, 1, 2, 3). As can be seen, all of the hysteresis loops are asymmetric with respect to the applied field. This asymmetry of these loops was attributed to the

v

,o,

~: ~2 0

Ir :

:

i

NtTt,Mn2t,Pt2

!

I

I

I

I

"*

12-I~1 oI

~

NiT3Mn2tP~ ,3

:

I

I

-~21 1

i -B

-6

-z.

-2

0

i

2

!

a

I

{5

l

g

.*--

H(kG)

Fig. 1. Hysteresis loops measured after cooling the samples from 300 to 4.2 K in H c= 10 kG for Ni7~,_xMn24Ptx, (a) x = 0; (b) x = 1; (c) x = 2; (d) x = 3. Note that the coercivity gradually increases with Pt impurity atoms while the unidirectional anisotropy, which is defined as the shift of the center of the loops, has its maximum value H A = 1000 kG for 2 at% Pt. The anisotropy parameters deduced from these hysteresis loops where the center of the loop (Hay = HA) is calculated to be K ~ / M s while the hysteresis half-width ( A H = HK) is calculated to be 2K 2 / M s.

elastic anisotropy rotation [11]. As evident from fig. 1, while the coercive field gradually increases with Pt content up to 3 at% Pt (which is our

Table 1 Parameters used to calculate the magnetic field dependence of the absorption of rf radiation by Niv6_xMnzaPt x (x = 0, 1, 2, 3) Parameter 4"rrM,, [G] at H = 10 kG Exchange stiffness A [erg/cm] Uniaxial anisotrc,py K 2 [erg/cm 3] Unidirectional anisotropy K t [erg/cm 3] Surface uniaxial anisotropy k ~ [erg/cm ~] Surface unidirectional anisotropy k t [erg/cm 2] Damping parameter T~-! [Hz] Resktivity p [o,f~ cm]

Niv6Mn24

NiTsMn.~4Pt i

Ni-,,Mn2aPt2

Niv3~:|n 2,,Pt 3

2300 5 × 10- 7 1.4 X 104 5.6X 104 0.06 0.22 5 X 109 55

2350 8× 10 -7 2.4 X 104 1.1X lO s 0.065 - 0.25 5.5 X 109 68

2400 2× 10 -6 2.8 X 104 1.9X lO s 0.10 - 0.52 7.5 X 109 72

2500 8 x 10 -6 4 X 104 1.Tx lO s 0.21 - 0.93 8X 109 74

Y. Oner et aL / FieM-induced anisotropy in Pt-irnpurity NiMn

n- FC~ . ~ ~

n- FC ~

_

r-FC

.-

(~)

/ r-FC

~

N%Mn2t'Ptl

(b)

..~'_

~.. 3

.~ 6.

0

n - FC

r- FC

. /• /, ~

[

n. FC ~

N~Tt'~n2t'Pt2 ....

Icl

lines. These anomalous behaviours occurring in the spectra of NiMnPt ternary alloys become more pronounced with increasing Pt concentration compared with those of binary NiMn alloys. In addition, the resonance lines for r-FC cases are in some way different (line shapes, line widths, resonance fields etc.). It is clear that separation between the resonance fields of the lines for n-FC an r-FC cases is due to the unidirectional anisotropy ( ~ 2HA). These values are in good agreement with those deduced from magnetization data. On the other hand, both the resonance curves for n-FC and r-FC cases are shifted as a whole to the lower field side owing to the large coercivity with Pt. Similar behaviour was seen on AuFe [14,15]. In order to see the correlations

r-FC T=42K .~ ,"/ \

,,,

Ni73Mn2~Pt 3

.¢ o

;

325

~

~

i

~-FC ~ "

"A=Z'dOSe~J/cm

---f' ~' R=0..~ ~0.06

s"

.V

~. ~=us~o~,m~:r~

/

X

~

H(kG)

Fig. 2. Experimental (full line) and calculated (squares) spectra for parallel geometry in FC cases for Ni7~,_xMn24Pt x with (a) x = 0, (b) x = l, (c) x ---.2, and (d) x = 3. The experimental spectra were taken for increasing and subsequently decreasing field for both positive and negative direction. The theoretical curves were fitted to the experimental spectra fcr only increasing fields (for both positive and negative field directions). Note that the spectra for the r-FC case exhibit hysteresis, especially for Ni74Mn24Pt 2 and Ni73Mn24Pt 3, in consistence with increasing hysteresis in the magnetization as shown in fig. 1.

~

~ •~ , =

= I

/

/

/

~

_ A~ d ~ r ~ "

~

~

~/~

~

~ =038e~cm~

k~=O.O~e~c~

Y ~r~-~,~0~/~

.... ~ I

~

~ I c I ~ ~

~

~ : 5 ~ ° e~,~

~ ~:~?e~/~ro.o?~/~

~

~ ~=~s~m/~ ~a~¢~

" ll0K

highest Pt concentration used in this study), the unidirectional anisotropy first increases and then reaches a maximum value ( ~ 1000 G) at 2 at% Pt. The concentration dependence of the unidirectional anisotropy and coercive field are given in table !. We present the experimental (full corve) and calculated FMR curves for Niv6_~.Mn24Ptx (x = 0, 1, 2, 3) alloys in fig. 2. All of the resonance spectra were taken under the same conditions as those for the magnetization. As shown in the fig. 2, the resonance lines are too asymmetric to be represented by either Lorentzian or Gaussian

~

A=a~'~0~e~ ~=0 k2=a07e~c~

I /= I e

~

J

,-7-1

~=4~K1=0 K2=Sxl03erg/cr 'l~t 1/T2=9x109s'1

0

2

H(kOe)

3

Fig. 3. Experimental (full line) and calculated (squares) spectra of Ni74.sMn24Ptl,5 for some selecicd temperatures after cooling the sample down to 4.2 K in the external field of 10 kG being parallel to the surface of disk-like specimen. The fitting parameters (KI, K z, k~, k:, A, 1 / T 2) are indicated in the figure for both the n-FC and r-FC cases. Note that the directional component of the surface anisotropy fields k 1 decreases rapidly and then becomes zero where the directional bulk anisotropy K~ vanishes. The signs of all k~'s are negative.

Y. (Jner et al. / Field-induced anisotropy in Pt-impurity NiMn

326

between the line shapes and the field induced anisotropies, we have also studied the temperature dependence of the FMR spectra for Ni74.sMn24Pt~.5 alloy. Fig. 3 shows both experimental and theoretical FMR spectra for some selected temperatures. The temperature dependence of line shapes are exhibited in fig. 4 by giving peak-to-peak linewidth, ZlHpp, and the ratio A/B of upper-lobe amplitude to that of the lower lobe. In addition, the bulk anisotropy values obtained from fitting the calculated spectra to the corresponding experimental ones are given in fig. 5. From these figures, it can be seen that the line shapes for the n-FC and r-FC cases are strongly correlated with the unidirectional anisotropy. Many authors have shown that the absorption line of the FM metallic samples ineludes two different modes: one at the lower field side is usual an FMR bulk mode, the other at higher field side the surface induced mode. Recently Teale et al. [16] have discussed the dependence of the surface mode on both the exchange and surface anisotropy energy in details. They have concluded that the resonance line shapes are considerably modified by the surface anisotropies. Since the unidirectional anisotropy in the bulk causes only the shift of the

resonance line to the negative field as a whole, the different amount of distortions between n-FC and r-FC cases lead us to consider the fact that the unidirectional anisotropy on the surfaces could differ from that in the bulk. Taking into account this fact, we used an R - W based model in order to account for the behaviour of the resonance lines, adding directional character to the surface anisotropy (see appendix).

3. Discussion

The calculation of d(Re(Z'))/d(/-/..) as a function of H~ was performed on a prime computer. We wrote a program by which the computed and measured curves can be displayed on the screen for comparison. The program can be used interractively when the user adjusts the parameters to obtain by eye a fit between theory and experiment. The values for M~ and M ( H ) in eq. 1 (see appendix), have been taken from the static magnetization data. The remaining seven parameters, H A, H~:, ~/, T2, k~, k 2 and A have to be determined by fitting the calculated curves to the corresponding experimental ones. These curves together are shown in figs. 2 and 3. The fitting

80O

3

600

~2 JNi74. Mn24Ptl5 (

0

ta.

40OI-J~ ~001

-

r

Fig. 4. 'Fhe temperature dependence of line width (A.Ho~,) and , 4 / B ratio for Ni74.sMn24Pt ~5. Note that both curves split into two branches (for the n-FC and r-FC case) below T = 30 K where the unidirectional anisotropy vanishes.

Y. ~ner et al. / FieM-induced anisotropy in Pt-impttrity NiMn .

.

.

.

.

.

I

•

.

~

.

.

.

.

, ....

I{

M= 3 0 K

H (kOe) -12

.m

0

I Ni. 74.5 Mn24 Ptl.5 ]

Tr-_ 115 K • L

i

0

50

327

.

i

T(K)

I0O

~50

Fig. 5. The temperature dependence of the anisotropy field obtained from fitting for Ni74.sMn24Ptl. 5 alloy. Note that the lower branch below T = 30 K corresponds to the uniaxial anisotropy field H R. Inset shows M vs H hysteresis loop recorded under the same conditions as those for FMR spectra for this alloy. The anisotropies fields obtained from both magnetization and F M R data seem to be in good agreement.

parameters for these curves at 4.2 K are given in table 1. In order to bring out the effect of the rather large hysteresis of the magnetization, we also recorded the spectrum while reducing the applied field after i~nmediately completing the spectrum for increasing field for both the n-FC and r-FC cases. While no hysteresis is observed for the n-FC case, the F M R spectrum for the r-FC case shows significantly large hysteresis increasing with Pt impurities, in accordance with the static magnetization data. It is also noted that the intensity of the resonance lines observed for the r-FC case is bigger than that for the n-FC case in NiMnPt ternary alloys in contrary to those in NiMn binary

alloys. In other words, the resonance lines of NiMnPt ternary alloys for n-FC cases are broader than for r-FC cases, but the contrary situation was exhibited in binary NiMn alloys. These differences of intensity a n d / o r line width are enhanced with increasing Pt concentration. In other words, the line shapes are closely correlated with the field induced anisotropy. In order to clarify this effect, we have analyzed the FMR spectra for Ni24.sMn24Ptl..s as a function of temperature. As shown in fig. 3, the relative intensity of the spectra for r-FC and n-FC cases decreases with decreasing of the unidirectional anisotropy and the goes to one at T = 30 K where the directional component of the anisotropy vanishes (see fig. 5). In addition, both the line width and the A/B ratio curves are splitted into two branches at 30 K; one of which eerresponds to the n-FC case, the other to the r-FC case as indicated in fig. 5. It should be noted that the directional anisotropy not only shifts the resonance lines for the n-FC and r-FC cases, but also reflects itself on the line shapes in different ways for each case. This leads us to suggest that each resonance line consists of two different modes, namely usual FMR bulk mode and surface induced mode, and the overlapping of these two modes are different for n-FC and r-FC cases due to the surface induced directional anisotropy. In order to account for this behaviour, we have used Rado and Weertman theory by introducing a surface induced unidirectional anisotropy term. This theory gives results in almost perfect agreement with experimental data even in the low field side of the resonance lines where some anomalous behaviours occur. For NiTaMn24Pt2 and Ni73Mn24Pt3, the resonance for the n-FC case occurs at about zero field and therefore half of the resonance lines v~hmu remains ~'"- ~ iiegativc field side, a part which corresponds to the lower field side of the spectrum for the r-FC case. That is, the well appearing at about zero field for the r-FE case is the remaining part of the absorption curve for the n-FC case. We stress that such asymmetric lines obsen, ed for these alloys consist of two unresolved overlapping ~bsorption lines; one of which corresponds to the usual FMR bulk mode, the other appears

328

Y. ~ner et al. / Field-induced anisotropy in Pt-impurity NiMn

at higher field to the surface induced mode. The model used here involves both resonance lines mentioned above. Some significant general results of the calculation are as follows. While the parameters H A, H K, H a and ~/produce a shift of only the resonance field positions of the whole line, the remaining parameters k t, k 2 and A affect the shape of the absorption curves. Of these, k I and k2 change the position of the surface induced mode relative to the main usual FMR mode. In fact, the role of k 2 on the separation of these two modes are the same for both the n-FC and r-FC cases, but k~ has a directional effect on their relative position. In other words, the more negative k l is, the more separated these two modes are for the n-FC case (or closer for the r-FC case giving a more intense line relative to the n-FC case). Thus the relative intensity of the n-FC and r-FC cases changes with the sign of k~. On the other hand, the ratio of the intensity of the surface mode to he bulk mode gradually increases with increasing A. As for M, it changes not only the positions of the resonant curves, but also modifies the shape of the resonance line. In other words, the shoulder appearing at the lower field side of the main FMR mode comes closer, more and more, to zero field while flatting with increasing M as shown in figs. 2a and b. As can be seen in table 1, the values of k~ change ~ign and become more and more negative by adding Pt impurities to Ni76Mn24 binary alloys. This can be attributed to the unidirectional anisotropy differences in the bulk and near the surface. The value of the uniaxial component k 2 of the surface anisotropy increases together with the coercivity increment proportional to the Pt concentration as expected. The unidirectional anisotropy H A in the bulk first increases and acifieves its maximum value ( ~ 1000 G) at 2 at% Pt. As for A, its effective value increases from 5 × i0 -7 e r g / c m for Ni76MN24 to 8 × 10 -~' e r g / c m for Ni73Mn24Pt 3. This rise of the effective exchange stiffness is due to strong coupling between neighbouring spins (e.g., by Heisenberg type interaction) associated with total magnetic anisotropy. The spectroscopic splitting factor g remains practically constant at 2.16 for all speci-

mens. The magnetization at H = 10 kG slightly increases from M = 23 e m u / g for Ni76Mn24 t o M = 25 e m u / g for Ni73MN24Pt 3. It is also important to note that the total anisotropy decreases with increasing temperature while the uniaxial bulk anisotropy H~: first increases untill joining the total anisotropy curve and then decreases rapidly. Similar behaviour has been observed in NiMn alloys [4,21]. The line widths for the n-FC and roFC cases meet each other at T = 30 K and increase both gradually up to T = 50 K and then abruptly rise up to T = 70 K, accompanied by a rapid decrease of anisotropy in this temperature region. This behavior is presumably due to an inhomogeneous broadening contribution to be associated to exchange narrowing of the local D - M anisotropy fields [15]. In zero-field cooling (ZFC) cases, no FMR spectra were observed because of very large line widths arising from a randomly distributed highly enhanced unidirectional anisotropy with Pt impurities at temperatures up to 30 K. As a result, FMR lines have been interpreted by introducing directional character of the surface anisotropy to the R - W general surface boundary condition. The agreement between theory and experiment is almost as good as it could be desired. We have also found that Pt impurities lead to a large broadening of the hysteresis cycle (and a strong increase of the associated irreversibilities) and an enhancement of the unidirectional anisotropies. Furthermore, the effective exchange stiffness constant is found to be correlated with the field induced anisotropy. The magnetization data are quite consistent with FMR data as well.

Acknowledgements This work was supported by the Scientific and Technical Resear,~h c',~,,.,.il ,~f Turkey (TBAG819 and TBAG-851).

Appendix We will analyze the problem of absorption (or surface impedance problem) in ferromagnetic

E (~ner et al. / FieM-induced anisotropy in Pt-impurit3, NiMn

metallic alloys for the parallel geometry: the xz plane lies in the surface of the FM specimen with a constant magnetic field H directed along the z axis. According to the method, first introduced by Ament and Rado [17], and Rado and Weertman [18], and subsequently applied to particular cases by many authors [16,19-21], we begin the analysis by considerations of the microwave propagation properties in an FM medium of thick disk-like specimen. The direction of the propagation of the electromagnetic field is inward normal to the surface along the y axis. Magnetic phenomena in such a medium may be described by the well known phenomenological equation 1 dMx. ,,

3'

/

T2

(!)

for Bloch (B type) damping used in our study, where O

0

3

V,,=i~+iv~+i. 00~1

;

- 00~2

(2)

where C~ = 1 + 2 r / + i2~ 2

i 2 = - 1,

C 2 = r/( r / + 1) - ~ 2 + i4~2( 1 + r/), C3

- i2~2((1 +r/

=

and

g2 = ( to - i/T2) / ( 4"n'TM: ),

H A = -T-K~/M~, H K = 2K2/M.~,

Mx. ),

+ "~2s2 V 2 M

Q~' - C I Q 4 + C z Q 2 - C 3 = 0,

•/ = H,/(4'n'M:); H t = H + H A + H K ;

-~i " = M × , H + H~ - --~ V"E"" i }

secular equation for the propagation constant q of the microwave field:

Q2 = q2A /(2arM? ),

1

2A

329

" ~]O~ 3

ce~, a2, a 3 are the cosines of the angles between M and i,., i:., i. which are unit vectors along the x, y, z direction respectively. Also, y = ge/(2mc) is the gyromagnetic ratio of the electrons which provides the magnetization M and A is the exchange stiffness constant. The second term in the bracket of eq. (1) is the demagnetizing field which is negligible for parallel geometry. The third term describes an effective field arising from the volume anisotropy energy density Eani and the forth term denotes an effective field due to the exchange interaction. Here, we adopt the expression for the bulk anisotropy energy density employed for a spin glass system by

and 1/o-=19 is the electrical resistivity in ~12 cm. It should be noted that M~ is the magnetization of the specimen where the resonance occurs, M~ is the technical saturation magnetization. In our experiment the FMR absorption is proportional to the change of the real part of the surface impedance, which is defined as the ratio of the tangential components of electric and magnetic microwave fields (der.3ted by e[~ and h.~, respectively) at the surface ol the material (y = 0) and is given by

Z'=(2"rrA)l/Z(pM)-12cx

10 -3 ~

y=0

Solution of eq. (2) yields three positive solutions Q. (n = 1, 2, 3) for the propagation constants and three corresponding waves propagating in the metal. The boundary conditions at the surface (the tangential components of e and h must be continuous across the surface) are given as 3

Ean i = - K l a 3 - K 2 c e

~.

In addition, we apply an oscillating magnetic field h of w being linearly polarized along the x axis. Using the Maxwell equations and the above Landau-Lifshitz ( L - L ) equation, eq. (1) for the motion of the magnetization vector M, we obtain a

E h .~,.,, ! = h .'~, n=

( 3 a)

1

3

E O,,h°~.,, = Z'h°~,

(3b)

n=l

where h°~ is the microwave magnetic field outside

330

Y. O n e r

et al. / F i e M - i n d u c e d

the surface and h °X d / are three fields just inside the surface. We can obtain two remaining boundary conditions needed to determine the ratios of amplitudes of the three waves using eq. (1) and Maxwell's equations as follows: Q , ] - 2i~ 2 m° •,',,, =

8i,rre2

(4a)

h.°~,,,'

iO(Q'2'- 2ie2) h ° m°-~'' = 8irr,z(1 + r / - Q , ] ) "'""

(4b)

For the exchange boundary condition we used the equation 2,4

a n i s o t r o p y in P t - i m p u r i t y N i M n

where H,, = k~/M S and H k = 2kz/M.~. Here, n(r) labels correspond to the case in which the applied field is directed along the cooling field direction (when the field is reversed). The lower (upper) sign belongs to the n-FC (r-FC) case. Now, we are in the position to calculate the normalized surface impedance Z ' in terms of parameters Kt, K z, k~, kz and A. Combining eq. (7) with (4), we can get two independent equations for h °X , t l 's and we also have two independent equations from eq. (3). Altogether, therefore, there are four independent equations. In order to obtain a nonvanishing solution, the coefficients determinant must vanish. This requirement leads us to the surface impedance

~M

~ M X --Oy + T~urr= 0; T, = M X neff,

(5)

which was proposed along with some special forms by Rado and Weertman. This equation implies that Tsua includes all surface torque densities arising from forces other than FM exchange. We first propose the surface e n e r ~ density as the sum of directional and axial form, i.e.,

Z'=

{QI(B2A3

/{(BzA3

-

B3A2) + (B3AI

+(B~A2-B2At) },

-

B1A3)

(8)

+ ¢:: ) ( 8 '~'ie ) 2

ig2(Q,,2 - 2ie2)(Q,, + ~:~..")

k~ .

Bit

M~,y + ~ , = .

BIA3)

where

which leads to the induced surface field 2k2a 2 .

-

+Q3(B, A2-B2A,)}

A,, = (O~2 - 2ie 2)

E~ = - k 2 a ~ - k l a 3,

H~,.r=

B3A2) + Q2(B3A!

-

~

8wie2(1 + ~/_ Q,2,)

(6)

By putting eq. (6) into (5), separating into x and y components and combining these components with eq. (4), one can easily obtain the b o u n d a ~ condition in component form as

This is the expression which was employed to compute the real part of Z ' , which is proportional to the power absorbed in the sample, as a function of a steady field H.

3

W n=

(Qit+s.,-

¢,,r~

o

0,

(7a)

References

1

3 0 ~ (Q,, + ~.n , r x) ?7,~..it =0,

(7b)

~ I

where m °X,II and m~,, are the oscillatoff compo.', nents of the magnetization at the specimen surface, ~it,~ = T- (8,rrA)- ~/~" H ~ . I , ~X ~'

=

-

~

,

[1] J.S. Kouvel, W. AbduI-Razzaq and Kh.H. Ziq, Phys. Rev. B 35 (1987) 1768, and references therein. r.,lt~jT. Sato, Phys. Rev. B 41 (i990) 2 5 5 0 . [3] H. Hurdequint, J.S. Kouvei and P. Monod, J. Magn. Magn. Mater. 31-34 (1983) 1429. [4] B. Aktas, Y. Oner and E.A. Harris, Phys. Rev. B 39 (1989) 528, and references therein. [5] S. Senoussi and Y. 0net, Phys. Rev. B 28 (1983) 455. [6] Z.H. Durusoy and Y. 0net, Phys. Rex'. B 42 (1990) 6831. [7] S. Senoussi and Y. (~ner, J. Magn. Magn. Mater. 40 (1983) 12. [8] S. Senoussi and Y. Oner, J. Appl. Phys. 55 (1984) 1472.

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