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Determination of magnetic anisotropy constants for garnet films from angular dependence of FMR
Journal of Magnetism and Magnetic Materials 53 (1985) 115-120 North-Holland, Amsterdam
115
D E T E R M I N A T I O N OF MAGNETIC A N I S O T R O P Y C O N S T A N T S FOR GARNET FILMS FROM ANGULAR D E P E N D E N C E OF FMR Y.Q. HE and P.E. W I G E N Department of Physics, The Ohio State University, Columbus, OH 43210, USA Received 7 January 1985; in revised from 18 April 1985
More precise analytic expressions for ferromagnetic resonance (FMR) of a {111 } garnet film have been derived, including the contribution of the uniaxial anisotropy term and higher order cubic anisotropy terms. The deviation between the magnetization equilibrium direction and the applied field direction are taken into consideration by using a series expansion of the deviation angle in terms of the energy derivatives as a power series in 1/H. The approximation has been taken to second order in the anisotropy fields. The angular dependence of FMR for a CaGe : YIG film has been analyzed with the magnetic anisotropy energy constants determined by fitting the experimental curves.
1. Introduction Measurements of the magnetocrystalline anisotropy constants of the magnetic garnet films are usually obtained from ferromagnetic resonance (FMR) techniques [1]. In general only the first order anisotropy constant K 1 and the uniaxial anisotropy constant K u have been given consideration. However, in some cases the higher order cubic anisotropy terms appear, and at certain temperatures it may be greater than or comparable in magnitude with K 1 [2]. The fact that the torque and the ferromagnetic resonance measurements often yield divergent results for the anisotropy constants may be attributed to the neglect of the higher-order terms in some circumstances. A recent theoretical study indicates that the higher order constants K 3 and K 4 may also be significant in some doped Y I G [3]. Hence an experimental determination of the higher order cubic anisotropy constants over a wide temperature range may provide a test of the applicability of the theoretical predications. Analytical equations used to calculate the F M R field orientation dependence of the resonance for {111} bubble garnet films were reported by Cronemeyer et al. [4]. That derivation is used to determine the K 1 and K u values of garnet films.
However, Cronemeyer et al. assume the applied field and the magnetization directions are colinear. One expects that the values of Kg's derived from experimental data depend strongly on the approximation made in the theoretical treatment. Makino et al. [5] has suggested a numerical method for determination of the magnetocrystalline anisotropy constants which are not quite amenable to routine characterization procedures. Below a more precise analytic expressions for ferromagnetic resonance of a ( l l l } - o r i e n t e d garnet film has been developed which includes the higher-order anisotropy energy terms.
2. FMR relations Magnetic garnet films grown on {l ll}-oriented substrates of Gd3GasO12 by the LPE method have an induced uniaxial magnetic anisotropy energy which has an easy axis in the [111] direction normal to the film surface. The films also possess a magnetocrystalline cubic anisotropy energy. Consider a thin film of ferromagnetic single crystal in a {111} plane (fig. 1). The orientation of the magnetization M and of the static magnetic field H with reference to the coordinate system []10], [112], [111] will be described by the angles 0,
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Y.Q. He, P.E. Wigen / Magnetic anisotropy constantsfor garnet films
116
noting the angular dependence which have the form [7].
[III]
fl = (3 - 6 cos20 0
q-
7 cos40
+ 4V~- sin 30 cos 0 sin30)/12
~
(3)
and f2 = [¢2- sin30 sin 30 + cos 0 (3 - 5 cos20)]2/108.
(4)
=- ciiea
[ilo] Fig. 1. The coordinate system used for the evaluation of the ferromagnetic resonance condition.
and OH, @H, respectively. The film is assumed to be magnetized to a single domain state ( M constant over the film). The resonance frequency ~0 can be expressed in terms of the second order angular derivatives of the free energy per unit volume by the relation [6]
Since the magnetostatic energy and the uniaxial anisotropy energy have the same angular form, these terms cannot be distinguished in the FMR. Thus the magnetostatic energy term will be eliminated and K o will be understood to mean an effective uniaxial anisotropy constant, K 0 = K u 2 ¢rMs2 which has the angular form fo = sin20.
(5)
The equilibrium orientation of M is found by the relation, E 0 - M H [sin 0/4 cos 0 cos(0 - 0/4) - c o s 0 , sin 0] = 0,
1 ME sinZ0e (FooF,~,-F~o)o,~,,
(1)
where M~ is the saturation magnetization, y is the gyromagnetic constant, 8e and @e represent the equilibrium direction of the magnetization and Foo, F~,~, and Fo, are the corresponding partial derivative of the free energy. The free energy per unit volume of the media has the form
F=M.H+E = -gH[sin
K2U2 + g3f ? + g4flf2,
(6b)
There is no general analytic solution to these equation. In order to solve eqs. (6a) and (6b) an iteration method of successive approximation can be used [8]. The angles 0 and 0 are expressed in terms of the known angle (OH, 0/4) specifying the orientation of H and certain small deviation 80 and 8, characterize the noncollinearity of the vectors M and H : 0 = OH + 80
0/4 sin 0 cos(@ - 0/4)
(7a)
and
+ cos OH cos 0] + Kusin20- 21rMZsin20 +Klf 1+
E, + MH sin 0/4 sin 0 sin(0 - 0/4) = 0.
(6a)
(2)
where the first term corresponds to the Zeeman energy, the second to the uniaxial anisotropy with an easy axis perpendicular to the plane o f the film, the third term is the demagnetization energy and the last four terms are due to the cubic magnetocrystalline anisotropy energy with fl and f2 de-
0 = 0/4 + 80.
(7b)
Expanding eqs. (6a) and (6b) about the applied external field orinetation and retaining only terms up to quadractic in 80 and 80, eqs. (6a) and (6b) can be solved approximately for 80 and 80 using 1/H as an expansion parameter. To second order in l/H, the deviations 80 and ~0 are given by the
Y.Q. He, P.E. Wigen / Magnetic anisotropy constantsfor garnet films
117
formulas: I-It = - ½ E h,& + ¼
60 = - E o / M H + ( EoEoo - E~ cos 0 . / 2 sin30H + E, Eo,/sin2OH)/M 2H:
h,e,
/=0
(8a)
4
and
)
( -~. / 2]11/2
/,j=0
6ep = - E J M H -
sinZ0n -(EoE~, cos On/sin3OH
EoEaq,/sin2OH - E, E4,÷/sin4OH)/M2H 2. (8b)
Expanding the derivatives of the free energy in a series of 80 and 8q, about the applied field orientation and using the relation (8a) and (8b), the resonance field Hf is given by
(~)2=H~+(Eoo+
E**/sin20n
+ E o cot OH)n r / M + [ e~cotZ0n + ( eooE** - corot,,
Eq. (11) represents a general resonance relation which is correct to second order in the anisotropy fields. The terms which contain the first order anisotropy fields are expected to be the same as those obtained by Cronemeyer, et al. [4] and Marysko [7]. The new terms express the second order anisotropy field contributions to the resonance relation. On the basis of the formula above, the cubic and uniaxial anisotropy energy constants can be calculated from a fit to the measured Hres versus angle curve.
3. Experiment and discussion
- e~,Eoo,- E~,)/sin20n + 2(EoE** + E, Eoq,/2)cos 0./sin30n
- ( E~ + E,E**o)/sin'O H - EoEoao]/M z,
(9) where the partial derivatives of the energy Eij are computed at the orientation of the applied field rather than at the equilibrium position of 0e and q~e as in eq. (1). The resulting resonance relations will be correct up to second order in the angle deviations 80 and 8q~. In order to get a clear picture, eq. (9) can be rewritten in terms of the anisotropy field as
+
h,e, I-It + E h , j t ' , j /,j~0
(,!0)
'
= o, 0 o )
where Pt and Pij are polynomials of the trigonometric functions of the coordinate angles of the applied field; h t, h~j are the corresponding anisotropy fields h t = K t / M and hij = K~Ky/M z. The expressions for the angular derivatives of the anisotropy energy are presented in the Appendix. It is now possible to express the resonance field H r in terms of a quadratic equation formula
The angle dependence of the ferromagnetic resonance has been observed on low magnetization garnet films containing calcium and germanium, CaGe : YIG, as a function of temperature. A sample used in an earlier work [2] was employed in this study. In this experiment it is convenient to rotate the applied field in the (i10) plane and to measure the angle from the [111] direction. Typical 8.7
G Y
r-~ B. zL _.J W la. W ~_~ Z
g
8.1
7. B
W 7.5 0
68 12fl ANGLE (DEG)
18fl
Fig. 2. The variation of the resonance field as a function of the angle measured from the [111] direction in an (111) oriented epitaxial C a G e : Y I G garnet thin film. The temperature is 300 K.
Y.Q. He, P.E. Wigen / Magnetic anisotropy constants for garnet films
118
12.0
9.6
S
S2:C v
c7
nlO. 0
~8. s
_.1 W
IJ_
U_
IJJ (.D
W U Z
8.0
~6.0
123 (./3
CO W rt"
n,,
4.0
6.6 0
180
60 120 ANGLE (DEG)
0
Fig. 3. S a m e as fig. 2 for T = 220 K.
performed for many different anisotropy values keeping the g-factor as an adjustable parameter. The coincidence between calculated and experimental data is judged by the root mean square error. The derived best-fitting values for film W6 are listed in table 1. In figs. 2-5 the solid line represents the best theoretical curves which are plotted by a HP9825A calculator and plotter. The coincidence between theoretical and experimental results is considered to be satisfactory without including the higher order anisotropy energy constants K 3 and K 4 for this analysis. The temperature dependence of K i could be obtained by repeating this treatment over the wide temperature range. The analytical expression derived above for the resonance condition greatly increases the efficiency with which the appropriate material parameters can be determined.
S v
o9.0 ..J W
8.@
~7.0 m w
6.0
0
iBZ
Fig. 5. S a m e as fig. 2 for T = 150 K.
10.0
UJ U Z
60 120 ANGLE(DEG)
60 120 ANGLE(OEG)
180
Fig. 4. S a m e as fig. 3 for T = 200 K.
experimental results for sample W6 are shown for various temperatures in figs. 2-5. The microwave frequency employed in all experiments was 23.4 GHz. The magnetization was determined by means of a vibrating magnetometer. The determination of the gyromagnetic ratio and the anisotropy constants was done by best-fitting to the data. In routine computations the optimization is usually
Acknowledgements The authors would like to thank J.H. Huang for experimental assistance. This work supported in part by NSF Grant #DMR-8304250.
Table 1 M a t e r i a l p a r a m e t e r s (film w r )
T
g
(K) 300 220 200 150
2.06 2.09 2.12 2.15
4"nMs
Ku
(GS)
Ka
(The Ki-values are given in 103 e r g / c m 3)
215 225 220 215
0.39 0.63 0.65 2.40
- 3.95 - 13.60 - 16.80 - 29.20
K2 -
3.00 3.10 3.52 4.40
Y.Q. He, P.E. Wigen / Magnetic anisotropy constants for garnet films
Appendix The partial angular derivatives of the anisotropy energy E can be written as
+ ¢2- sin 3~ sin20(3 - 4 sinE0)]/3,
f2o = sin 0[¢~- sin30 sin 3~ + cos 0(3 - 5 cos20)] x (5 cos20 - 1 + ¢~- sin 0 cos 0 sin 3q~)/18.
Eo = Kofoo + KlflO + K2f2o + 2K3flflo
f l , = v~- cos 3~ cos 0 sinaO,
+ K4(flof2 + flf2o), Eoo = Kofoo o + K l f w o + K2f200
f2, = cos 3~ sin30 [2 sin30 sin 3~
+ 2K3(f?o + f , f , oo) + g 4 ( f l o o f 2 + 2flof2o + flf2oo),
E, = KI/~ + K:f2~o+ K32flf~ + K , ( f l j ~ + f~k~), Etk ~ = gafloq~ -.1-g2f2,q~ .-.I-2K3( f?rb + f l f l q ~ )
+r,(f,**k + 2 f , , k , + f~k**), Eo, = K~flo, + K~f~o, + 2K3( f j ~ o + kf~o,)
+K4(flo,k + f,d~, + k , k o +kko,), Eo,~, = K~f~o~,~ + K2f20** + 2K3(fl~,q, fl 0
+ 2k,f,o, +faf, o,~,)
foao = 2 cos 20, floo = [29 sin20 - 28 sin40 - 4 + v ~ sin 20 sin 3@(3 - 8 sin20)]/3,
f2oo = ([2v~- sin 20 sin20 sin 3~ + c o s 20(3 - 5 cos20) + 10 sin20 cos20] × (30 cos20 - 6 + 3v~- sin 20 sin 3~) + (6v~- cos 20 sin 3~ - 30 sin 20)
× [¢~- sin40 sin 3¢ + sin 20(3 - 5 cos20)/2] ) / 1 0 8 , fl,~, = -3v~- sin 3,~ sin30 cos O,
+f~**ko + 2fl,kO, +kko**),
f2** = ¢~- sin30 [¢~- sin30 cos 6ff - s i n 3q~ cos 0(3 - 5 cos20)]/6,
+2g3(3fj1** + flfl***)
fl0, = - ~
+ ~:~(f,***k + 3fl**k,
f2o~, = ¢~- sin20 cos 3q~(2~/2- sin 3~ sin30 cos 0
+ 3fl,k** +f,k***), Eoo, = KlflOoq , -t- K2f2004 ~
+ 2g3 (2f~oflo, + f~,f~oo,) + K4 (floo,f~ + fieok, + 2f~oko
+ 2flokO, + fl,kOo + f, koo,), Eoo o = Kofoooo + Klf~ooo + K2f2ooo
"}-2 K 3 (3fl ofloo + fl/lOOO ) + K4 (flooof~ + 3flooko + 3flof200 +flf20oO), where
f~o =
+¢~ cos 0(3 - 5 cos20 )] / 1 8 ,
+ K4(floe#q, fz + 2flo,f2, + f, of Eq,* E*** = Klf~*** + K2f2***
cos 30 sinZO(4 sin20 - 3),
+ 9 cos20 - 10 cos40 - 1 ) / 6 ,
flo,~, = 3¢~- sin 3~ sin20(4 sin20 - 3), fzaq, q, = sin20 [2 cos 6@ sin30 Cos 0 --¢2- sin 3~(9 c o s 2 0 - 10 c o s 4 0 - 1 ) / 2 ] , fl*** = --9v~- cos 3~ sin30 cos 0,
f2~** = - s i n 30 [2 sin30 sin 6~ +V~- cos 3@ cos 0(3 - 5 cos20)/2],
floo~, = v~- sin 20 cos 3~(3 - 8 sin20), f2oo, = v~- cos 3,# [sin 40(3 - 5 cos20)
[sin 0 cos 0(7 sin=0 - 4)
119
+4v~- sin 3~ sin40(6 cos20 - 1)
120
Y.Q. He, P.E. Wigen /Magnetic anisotropy constantsfor garnet films
+ 2 sin20 sin 2 0 ( 2 0 cos20 - 7 ) ] / 1 2 , - 4 sin 2 0, [sin 2 0 ( 2 9 - 56 sin20) + 2 ¢ ~ sin 3 ~ ( 3 cos 20 - 8 cos 20 sin20 - 4
/2oo
sin220)]/3,
= { (5 cos 20 + v~- sin 20 sin 3~)
x [ 2 ¢ 2 sin 3 ~ ( 2 cos 20 sin20 + sin220 -- 2 sin40) + sin 2 0 ( 5 0 cos20 - 27)] + (v/2 cos 20 sin 3q~- 5 sin 2 0 ) X [4 cos 20(3 -- 5 cos20) + 1 0 sin220 + 16v~ sin30 cos 0 sin 3q,] } / 3 6 .
References
[1] P.E. Wigen, Thin Solid Films 114 (1984) 135. [2] C. Borghese, P. DeGasperis and P.E. Wigen, IEEE Trans. on Magn. MAG-18 (1982) 1301. [3] C. Rudowicz, Z. Naturf. 38a (1983) 540. [4] D.C. Cronemeyer, T.S. Plaskett and E. Klokholm, AlP Conf. Proc. 24 (1975) 586. [5] H. Makino and Y. Hidaka, Mater. Res. Bull. 16 (1981) 957. [6] J. Smit, H.G. Beljers, Philips Research Repts. 10 (1955) 113. [7] M. Marysko, Czech. J. Phys. B30 (1980) 1269. [8] W.Y. Jia and P.X. Zhang, Acta Phys. Sinica 25 (1976) 264.
115
D E T E R M I N A T I O N OF MAGNETIC A N I S O T R O P Y C O N S T A N T S FOR GARNET FILMS FROM ANGULAR D E P E N D E N C E OF FMR Y.Q. HE and P.E. W I G E N Department of Physics, The Ohio State University, Columbus, OH 43210, USA Received 7 January 1985; in revised from 18 April 1985
More precise analytic expressions for ferromagnetic resonance (FMR) of a {111 } garnet film have been derived, including the contribution of the uniaxial anisotropy term and higher order cubic anisotropy terms. The deviation between the magnetization equilibrium direction and the applied field direction are taken into consideration by using a series expansion of the deviation angle in terms of the energy derivatives as a power series in 1/H. The approximation has been taken to second order in the anisotropy fields. The angular dependence of FMR for a CaGe : YIG film has been analyzed with the magnetic anisotropy energy constants determined by fitting the experimental curves.
1. Introduction Measurements of the magnetocrystalline anisotropy constants of the magnetic garnet films are usually obtained from ferromagnetic resonance (FMR) techniques [1]. In general only the first order anisotropy constant K 1 and the uniaxial anisotropy constant K u have been given consideration. However, in some cases the higher order cubic anisotropy terms appear, and at certain temperatures it may be greater than or comparable in magnitude with K 1 [2]. The fact that the torque and the ferromagnetic resonance measurements often yield divergent results for the anisotropy constants may be attributed to the neglect of the higher-order terms in some circumstances. A recent theoretical study indicates that the higher order constants K 3 and K 4 may also be significant in some doped Y I G [3]. Hence an experimental determination of the higher order cubic anisotropy constants over a wide temperature range may provide a test of the applicability of the theoretical predications. Analytical equations used to calculate the F M R field orientation dependence of the resonance for {111} bubble garnet films were reported by Cronemeyer et al. [4]. That derivation is used to determine the K 1 and K u values of garnet films.
However, Cronemeyer et al. assume the applied field and the magnetization directions are colinear. One expects that the values of Kg's derived from experimental data depend strongly on the approximation made in the theoretical treatment. Makino et al. [5] has suggested a numerical method for determination of the magnetocrystalline anisotropy constants which are not quite amenable to routine characterization procedures. Below a more precise analytic expressions for ferromagnetic resonance of a ( l l l } - o r i e n t e d garnet film has been developed which includes the higher-order anisotropy energy terms.
2. FMR relations Magnetic garnet films grown on {l ll}-oriented substrates of Gd3GasO12 by the LPE method have an induced uniaxial magnetic anisotropy energy which has an easy axis in the [111] direction normal to the film surface. The films also possess a magnetocrystalline cubic anisotropy energy. Consider a thin film of ferromagnetic single crystal in a {111} plane (fig. 1). The orientation of the magnetization M and of the static magnetic field H with reference to the coordinate system []10], [112], [111] will be described by the angles 0,
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Y.Q. He, P.E. Wigen / Magnetic anisotropy constantsfor garnet films
116
noting the angular dependence which have the form [7].
[III]
fl = (3 - 6 cos20 0
q-
7 cos40
+ 4V~- sin 30 cos 0 sin30)/12
~
(3)
and f2 = [¢2- sin30 sin 30 + cos 0 (3 - 5 cos20)]2/108.
(4)
=- ciiea
[ilo] Fig. 1. The coordinate system used for the evaluation of the ferromagnetic resonance condition.
and OH, @H, respectively. The film is assumed to be magnetized to a single domain state ( M constant over the film). The resonance frequency ~0 can be expressed in terms of the second order angular derivatives of the free energy per unit volume by the relation [6]
Since the magnetostatic energy and the uniaxial anisotropy energy have the same angular form, these terms cannot be distinguished in the FMR. Thus the magnetostatic energy term will be eliminated and K o will be understood to mean an effective uniaxial anisotropy constant, K 0 = K u 2 ¢rMs2 which has the angular form fo = sin20.
(5)
The equilibrium orientation of M is found by the relation, E 0 - M H [sin 0/4 cos 0 cos(0 - 0/4) - c o s 0 , sin 0] = 0,
1 ME sinZ0e (FooF,~,-F~o)o,~,,
(1)
where M~ is the saturation magnetization, y is the gyromagnetic constant, 8e and @e represent the equilibrium direction of the magnetization and Foo, F~,~, and Fo, are the corresponding partial derivative of the free energy. The free energy per unit volume of the media has the form
F=M.H+E = -gH[sin
K2U2 + g3f ? + g4flf2,
(6b)
There is no general analytic solution to these equation. In order to solve eqs. (6a) and (6b) an iteration method of successive approximation can be used [8]. The angles 0 and 0 are expressed in terms of the known angle (OH, 0/4) specifying the orientation of H and certain small deviation 80 and 8, characterize the noncollinearity of the vectors M and H : 0 = OH + 80
0/4 sin 0 cos(@ - 0/4)
(7a)
and
+ cos OH cos 0] + Kusin20- 21rMZsin20 +Klf 1+
E, + MH sin 0/4 sin 0 sin(0 - 0/4) = 0.
(6a)
(2)
where the first term corresponds to the Zeeman energy, the second to the uniaxial anisotropy with an easy axis perpendicular to the plane o f the film, the third term is the demagnetization energy and the last four terms are due to the cubic magnetocrystalline anisotropy energy with fl and f2 de-
0 = 0/4 + 80.
(7b)
Expanding eqs. (6a) and (6b) about the applied external field orinetation and retaining only terms up to quadractic in 80 and 80, eqs. (6a) and (6b) can be solved approximately for 80 and 80 using 1/H as an expansion parameter. To second order in l/H, the deviations 80 and ~0 are given by the
Y.Q. He, P.E. Wigen / Magnetic anisotropy constantsfor garnet films
117
formulas: I-It = - ½ E h,& + ¼
60 = - E o / M H + ( EoEoo - E~ cos 0 . / 2 sin30H + E, Eo,/sin2OH)/M 2H:
h,e,
/=0
(8a)
4
and
)
( -~. / 2]11/2
/,j=0
6ep = - E J M H -
sinZ0n -(EoE~, cos On/sin3OH
EoEaq,/sin2OH - E, E4,÷/sin4OH)/M2H 2. (8b)
Expanding the derivatives of the free energy in a series of 80 and 8q, about the applied field orientation and using the relation (8a) and (8b), the resonance field Hf is given by
(~)2=H~+(Eoo+
E**/sin20n
+ E o cot OH)n r / M + [ e~cotZ0n + ( eooE** - corot,,
Eq. (11) represents a general resonance relation which is correct to second order in the anisotropy fields. The terms which contain the first order anisotropy fields are expected to be the same as those obtained by Cronemeyer, et al. [4] and Marysko [7]. The new terms express the second order anisotropy field contributions to the resonance relation. On the basis of the formula above, the cubic and uniaxial anisotropy energy constants can be calculated from a fit to the measured Hres versus angle curve.
3. Experiment and discussion
- e~,Eoo,- E~,)/sin20n + 2(EoE** + E, Eoq,/2)cos 0./sin30n
- ( E~ + E,E**o)/sin'O H - EoEoao]/M z,
(9) where the partial derivatives of the energy Eij are computed at the orientation of the applied field rather than at the equilibrium position of 0e and q~e as in eq. (1). The resulting resonance relations will be correct up to second order in the angle deviations 80 and 8q~. In order to get a clear picture, eq. (9) can be rewritten in terms of the anisotropy field as
+
h,e, I-It + E h , j t ' , j /,j~0
(,!0)
'
= o, 0 o )
where Pt and Pij are polynomials of the trigonometric functions of the coordinate angles of the applied field; h t, h~j are the corresponding anisotropy fields h t = K t / M and hij = K~Ky/M z. The expressions for the angular derivatives of the anisotropy energy are presented in the Appendix. It is now possible to express the resonance field H r in terms of a quadratic equation formula
The angle dependence of the ferromagnetic resonance has been observed on low magnetization garnet films containing calcium and germanium, CaGe : YIG, as a function of temperature. A sample used in an earlier work [2] was employed in this study. In this experiment it is convenient to rotate the applied field in the (i10) plane and to measure the angle from the [111] direction. Typical 8.7
G Y
r-~ B. zL _.J W la. W ~_~ Z
g
8.1
7. B
W 7.5 0
68 12fl ANGLE (DEG)
18fl
Fig. 2. The variation of the resonance field as a function of the angle measured from the [111] direction in an (111) oriented epitaxial C a G e : Y I G garnet thin film. The temperature is 300 K.
Y.Q. He, P.E. Wigen / Magnetic anisotropy constants for garnet films
118
12.0
9.6
S
S2:C v
c7
nlO. 0
~8. s
_.1 W
IJ_
U_
IJJ (.D
W U Z
8.0
~6.0
123 (./3
CO W rt"
n,,
4.0
6.6 0
180
60 120 ANGLE (DEG)
0
Fig. 3. S a m e as fig. 2 for T = 220 K.
performed for many different anisotropy values keeping the g-factor as an adjustable parameter. The coincidence between calculated and experimental data is judged by the root mean square error. The derived best-fitting values for film W6 are listed in table 1. In figs. 2-5 the solid line represents the best theoretical curves which are plotted by a HP9825A calculator and plotter. The coincidence between theoretical and experimental results is considered to be satisfactory without including the higher order anisotropy energy constants K 3 and K 4 for this analysis. The temperature dependence of K i could be obtained by repeating this treatment over the wide temperature range. The analytical expression derived above for the resonance condition greatly increases the efficiency with which the appropriate material parameters can be determined.
S v
o9.0 ..J W
8.@
~7.0 m w
6.0
0
iBZ
Fig. 5. S a m e as fig. 2 for T = 150 K.
10.0
UJ U Z
60 120 ANGLE(DEG)
60 120 ANGLE(OEG)
180
Fig. 4. S a m e as fig. 3 for T = 200 K.
experimental results for sample W6 are shown for various temperatures in figs. 2-5. The microwave frequency employed in all experiments was 23.4 GHz. The magnetization was determined by means of a vibrating magnetometer. The determination of the gyromagnetic ratio and the anisotropy constants was done by best-fitting to the data. In routine computations the optimization is usually
Acknowledgements The authors would like to thank J.H. Huang for experimental assistance. This work supported in part by NSF Grant #DMR-8304250.
Table 1 M a t e r i a l p a r a m e t e r s (film w r )
T
g
(K) 300 220 200 150
2.06 2.09 2.12 2.15
4"nMs
Ku
(GS)
Ka
(The Ki-values are given in 103 e r g / c m 3)
215 225 220 215
0.39 0.63 0.65 2.40
- 3.95 - 13.60 - 16.80 - 29.20
K2 -
3.00 3.10 3.52 4.40
Y.Q. He, P.E. Wigen / Magnetic anisotropy constants for garnet films
Appendix The partial angular derivatives of the anisotropy energy E can be written as
+ ¢2- sin 3~ sin20(3 - 4 sinE0)]/3,
f2o = sin 0[¢~- sin30 sin 3~ + cos 0(3 - 5 cos20)] x (5 cos20 - 1 + ¢~- sin 0 cos 0 sin 3q~)/18.
Eo = Kofoo + KlflO + K2f2o + 2K3flflo
f l , = v~- cos 3~ cos 0 sinaO,
+ K4(flof2 + flf2o), Eoo = Kofoo o + K l f w o + K2f200
f2, = cos 3~ sin30 [2 sin30 sin 3~
+ 2K3(f?o + f , f , oo) + g 4 ( f l o o f 2 + 2flof2o + flf2oo),
E, = KI/~ + K:f2~o+ K32flf~ + K , ( f l j ~ + f~k~), Etk ~ = gafloq~ -.1-g2f2,q~ .-.I-2K3( f?rb + f l f l q ~ )
+r,(f,**k + 2 f , , k , + f~k**), Eo, = K~flo, + K~f~o, + 2K3( f j ~ o + kf~o,)
+K4(flo,k + f,d~, + k , k o +kko,), Eo,~, = K~f~o~,~ + K2f20** + 2K3(fl~,q, fl 0
+ 2k,f,o, +faf, o,~,)
foao = 2 cos 20, floo = [29 sin20 - 28 sin40 - 4 + v ~ sin 20 sin 3@(3 - 8 sin20)]/3,
f2oo = ([2v~- sin 20 sin20 sin 3~ + c o s 20(3 - 5 cos20) + 10 sin20 cos20] × (30 cos20 - 6 + 3v~- sin 20 sin 3~) + (6v~- cos 20 sin 3~ - 30 sin 20)
× [¢~- sin40 sin 3¢ + sin 20(3 - 5 cos20)/2] ) / 1 0 8 , fl,~, = -3v~- sin 3,~ sin30 cos O,
+f~**ko + 2fl,kO, +kko**),
f2** = ¢~- sin30 [¢~- sin30 cos 6ff - s i n 3q~ cos 0(3 - 5 cos20)]/6,
+2g3(3fj1** + flfl***)
fl0, = - ~
+ ~:~(f,***k + 3fl**k,
f2o~, = ¢~- sin20 cos 3q~(2~/2- sin 3~ sin30 cos 0
+ 3fl,k** +f,k***), Eoo, = KlflOoq , -t- K2f2004 ~
+ 2g3 (2f~oflo, + f~,f~oo,) + K4 (floo,f~ + fieok, + 2f~oko
+ 2flokO, + fl,kOo + f, koo,), Eoo o = Kofoooo + Klf~ooo + K2f2ooo
"}-2 K 3 (3fl ofloo + fl/lOOO ) + K4 (flooof~ + 3flooko + 3flof200 +flf20oO), where
f~o =
+¢~ cos 0(3 - 5 cos20 )] / 1 8 ,
+ K4(floe#q, fz + 2flo,f2, + f, of Eq,* E*** = Klf~*** + K2f2***
cos 30 sinZO(4 sin20 - 3),
+ 9 cos20 - 10 cos40 - 1 ) / 6 ,
flo,~, = 3¢~- sin 3~ sin20(4 sin20 - 3), fzaq, q, = sin20 [2 cos 6@ sin30 Cos 0 --¢2- sin 3~(9 c o s 2 0 - 10 c o s 4 0 - 1 ) / 2 ] , fl*** = --9v~- cos 3~ sin30 cos 0,
f2~** = - s i n 30 [2 sin30 sin 6~ +V~- cos 3@ cos 0(3 - 5 cos20)/2],
floo~, = v~- sin 20 cos 3~(3 - 8 sin20), f2oo, = v~- cos 3,# [sin 40(3 - 5 cos20)
[sin 0 cos 0(7 sin=0 - 4)
119
+4v~- sin 3~ sin40(6 cos20 - 1)
120
Y.Q. He, P.E. Wigen /Magnetic anisotropy constantsfor garnet films
+ 2 sin20 sin 2 0 ( 2 0 cos20 - 7 ) ] / 1 2 , - 4 sin 2 0, [sin 2 0 ( 2 9 - 56 sin20) + 2 ¢ ~ sin 3 ~ ( 3 cos 20 - 8 cos 20 sin20 - 4
/2oo
sin220)]/3,
= { (5 cos 20 + v~- sin 20 sin 3~)
x [ 2 ¢ 2 sin 3 ~ ( 2 cos 20 sin20 + sin220 -- 2 sin40) + sin 2 0 ( 5 0 cos20 - 27)] + (v/2 cos 20 sin 3q~- 5 sin 2 0 ) X [4 cos 20(3 -- 5 cos20) + 1 0 sin220 + 16v~ sin30 cos 0 sin 3q,] } / 3 6 .
References
[1] P.E. Wigen, Thin Solid Films 114 (1984) 135. [2] C. Borghese, P. DeGasperis and P.E. Wigen, IEEE Trans. on Magn. MAG-18 (1982) 1301. [3] C. Rudowicz, Z. Naturf. 38a (1983) 540. [4] D.C. Cronemeyer, T.S. Plaskett and E. Klokholm, AlP Conf. Proc. 24 (1975) 586. [5] H. Makino and Y. Hidaka, Mater. Res. Bull. 16 (1981) 957. [6] J. Smit, H.G. Beljers, Philips Research Repts. 10 (1955) 113. [7] M. Marysko, Czech. J. Phys. B30 (1980) 1269. [8] W.Y. Jia and P.X. Zhang, Acta Phys. Sinica 25 (1976) 264.