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Orientational states of magnetization in epitaxial (1 1 1)-oriented iron garnet films
Journal of Magnetism and Magnetic Materials 195 (1999) 575}582
Orientational states of magnetization in epitaxial (1 1 1)-oriented iron garnet "lms Sergii B. Ubizskii * R & D Institute for Materials, Scientixc Research Company &Carat', 202 Stryjska street, 290031, Lviv, Ukraine State University &Lvivska Politechnika', Lviv, Ukraine Received 19 May 1998; received in revised form 5 January 1999
Abstract A (1 1 1) oriented single-crystal iron garnet "lm having mixed cubic and uniaxial anisotropy was investigated. Taking into account the "rst two constants of cubic anisotropy an expression for anisotropic part of free energy was derived in the polar coordinate system connected with "lms crystallographic directions. Its minimization allowed to obtain possible orientational phases. While only the "rst cubic anisotropy constant is taken into account, there are a symmetric phase of &easy axis' anisotropy and two angle phases with &easy cone' anisotropy. By adding the second cubic anisotropy constant into consideration one more symmetric phase may be found that correspond to &easy plane' type of anisotropy. Orientation of easy magnetization directions are analyzed as well as their in#uence on domain structure con"gurations. 1999 Elsevier Science B.V. All rights reserved. Keywords: Iron garnet "lms; Magnetic anisotropy; Magnetic phases; Easy magnetization directions; Domain structure
1. Introduction Epitaxial "lms of rare earth iron garnets (REIG) have been under intensive investigation from 70}80s due to their prospects for application in the bubble domain memory devices. Films for those uses should have a su$ciently large induced uniaxial magnetic anisotropy in perpendicular to the "lm surface direction that is an appropriate condition for the stable bubble domains existence. As a promised fact it was considered the "lm application with this type of anisotropy in magnetooptical devices for signal processing as well. The "lms based on pure yttrium iron garnets (YIG) with low magnetic anisotropy were utilized only for microwave applications. These "lms have an anisotropy to be close to the &easy plane' type, i.e. the magnetization vector deviates from the "lm surface plane on a small angle. However, as their use is mainly based on gyromagnetic
* Corresponding author. Tel.: 380-322-632-219; fax: 380-322-632-228. E-mail address: [email protected] (S.B. Ubizskii) 0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 1 6 5 - 1
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properties in fully magnetized state, the peculiarities of anisotropy, domain structure and remagnetization were out of detailed studies. In recent years an interest has arisen to utilization of REIG "lms in non-traditional applications namely for sensor devices for visualization of the magnetic "eld irregularity distribution and the ultra-low "elds magnetometry [1,2]. For the "rst case the best active medium could be a REIG "lm that would have an anisotropy of the &easy plane' type and be in a single-domain state [3]. In the second case to get the best parameters of magnetometric sensor it is necessary to provide some "xed type of anisotropy [2]. To satisfy the above requirements it is necessary to elucidate conditions for existence of one or another equilibrium state of the magnetization orientation in the iron garnet "lm. This work presents a study of conditions for appearing of di!erent orientational phases in epitaxial (1 1 1)-oriented iron garnet "lms that have been derived on the base of the minimization procedure applied to anisotropic part of the magnetic "lm free energy. The attention will be given to in#uence of anisotropy constants, while in other materials, for example, in highly magnetostrictive Tb Dy Fe alloy, the external stress could determine the change of number and orientation of easy magnetization directions [4].
2. Theoretical background To analyze an equilibrium state we choose the polar coordinate system shown in Fig. 1. The magnetization direction is de"ned by a polar angle h counted from direction of the "lm normal (crystallographic direction [1 1 1]) and azimuthal angle u that is counted from [1 1 0] direction in the "lm plane. In the general case a stable equilibrium position of the magnetization vector is determined by conditions of minimum for the anisotropic part of the magnetic "lm free energy E that is a function of the magnetization vector orientation: *E "0, *u
(1)
*E "0, *h
(2)
*E *E *E ! '0, *u*h *u *h
(3a)
*E '0. *u
(3b)
Fig. 1. Polar coordinate system based on crystallographic axes of the epitaxial (1 1 1) oriented iron garnet "lm.
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The energy density E in our case is a sum of the following contributions: the cubic anisotropy energy E ; the induced uniaxial anisotropy energy E ; the magnetostatic energy that comprises of a magnetic's energy in external "eld EH and a demagnetization "eld energy E . The last contribution is su$ciently depended on the presence and type of domain structure in magnetic "lm. In the "rst stage of our study to de"ne a quantity and type of uniform orientational magnetic phases in epitaxial iron garnet "lm without external "eld it is su$cient to consider a single domain state of magnetic "lm. For this case the demagnetization "eld energy E is determined only by the shape of the magnetic sample, i.e. by "lm state of subject under examination. The magnetoelastic interaction energy will not be considered separately because its part resulting in spontaneous magnetostriction deformation and stresses arising during remagnetization is described phenomenologically as additional contribution to the cubic anisotropy energy constants K and K [5]. Another part of magnetoelastic energy connected with uniaxial strains due to the "lm-substrate lattice parameter mismatch contributes in the same manner to the uniaxial anisotropy constant K together with growth-induced uniaxial anisotropy [6]. Obtained general expressions in chosen coordinate system for the total energy density E of (1 1 1)-oriented epitaxial iron garnet "lm in zero "eld as well as expressions for derivatives used in Eqs. (1) and (2) (3) are given in the appendix, where magnetic "elds H , H and H correspond to external "elds in the directions [1 1 0], [1 1 2 ] and [1 1 1].
3. Results and discussion Handling of the equations system (1)}(2) provides equilibrium solutions which correspond to di!erent orientational states of magnetization, i.e. orientational phases. Equilibrium orientational phases that could arise in the absence of the external magnetic "eld are presented in Table 1 together with anisotropy types that they correspond to. For a stable equilibrium the presented solutions determine the directions of easy magnetization axes (EMA) of the iron garnet "lm. The number of spatially di!erent EMAs for obtained phases are also given in the Table 1. For angular phases B and C the value of polar angle h for the EMA equilibrium position could be found from equation: 3(16K #K )sin 2h#28(6K #K )sin 4h#23K sin 4h G2(2(3 ) (16K #K )cos 2h!8 ) (6K #K )cos 4h #5K cos 6h)#288 ) (4pM!2K )"0, where signs in &G' correspond to phases B and C, respectively.
(4)
Table 1 Equilibrium orientational states of magnetization in the epitaxial (1 1 1) oriented iron garnet "lms in zero external "eld Orientational phase
Anisotropy type
Polar angle h
Azimuthal angle u
Number of &easy axes'
A
&easy axis'
B
&easy cone'
C
&easy cone'
D
&easy plane'
0 p 0(h(p/2 p/2(h(p 0(h(p/2 p/2(h(p p/2
u u !p/6#(2p/3 ) ) n p/6#(2p/3 ) ) n p/6#(2p/3 ) ) n !p/6#(2p/3 ) ) n p/3 ) n
1 1 3 3 3 3 6
n is an integer.
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In classi"cation after [7] the A and D orientational phases should refer to symmetrical ones. It means that a change of any "lm parameters (saturation magnetization M and anisotropy constants K , K , K ) in the phase equilibrium area the orientation of magnetization vector is invariant. In contrary, the B and C phases are angular because an orientation of magnetization vector, particularly, a polar angle h, depends on "lm parameters over all area of the phase existence. The azimuthal angle of the magnetization vector is undetermined for the A phase. In this case the free energy density does not depend on angle u, and the condition for equilibration stability of the orientational state A is expressed by the inequality: *E '0 *h
(5)
or 12K #4K #9 ) (4pM!2K )(0. (6) For the rest phases the stability condition of the magnetization vector equilibrium position is determined by inequalities (3). The stability conditions for the B and C phases equilibration take the form (2K ) [10!15 cos 2h#6 cos 4h!cos 6h] $(2 ) [9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h] ) ; ([3 ) (16K #K ) cos 2h#56 ) (6K #K ) cos 4h#69K sin 6h] $(2 ) [6 ) (16K #K ) sin 2h!32 ) (6K #K ) sin 4h#30K sin 6h] #(4pM!2K ) cos 2h)'0, 2K ) [10!15 cos 2h#6 cos 4h!cos 6h] $(2 ) [9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h](0, where signs &$' correspond to phases B and C, respectively. For the D phase these conditions become:
(7a)
(7b)
(4pM!2K !3K )K !6K'0, (8a) K '0. (8b) It should be noted the orientational phase D could not be stable if the "rst cubic anisotropy constant K is taken into account only, i.e. K "0. An equilibrium at u"p/3 ) n and h"p/2 could be only unstable in this case. The phase boundary that devides phases B and C in this case is discribed by equation K "0. The negative and positive values of K correspond to phases B and C, respectivelly. In more general case the boundaries of phase stability dependent obviously upon all "lm material parameters M , K , K and K . In general case they are also dependent from applied magnetic "eld and temperature. So, the whole phase diagram of orientational states of magnetic "lms under consideration has at least four dimensions at "xed "eld and temperature. As an example, the sections of phase diagram for magnetization orientational states in zero "eld is shown in Fig. 2 in coordinates (K , K ) at di!erent K levels with "xed value of saturation magnetization M "135 G to be inherent to pure YIG magnetization at room temperature. As can see from Fig. 2 the &easy plane' -type anisotropy is possible at positive values of K . It has been shown in Ref. [8] the YIG second cubic anisotropy constant reverses its sign on positive one and becomes more than K in absolute magnitude when garnet is doped by iridium in concentration near 0.02 per formula unit. At that, if the uniaxial anisotropy will not be too much positive the conditions for arising of the &easy plane' anisotropy can be achieved (see Fig. 2).
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Fig. 2. An example of the orientational phases' diagram of iron garnet "lm in (K , K ) cross section in zero external "eld and M "135 G, K "!10 erg cm\ (a), K "0 (b), K "#10 erg cm\ (c). Level lines represent the constant polar angle h and its value is given in degrees.
As one can see from the Table 1 each of obtained phases A}D have energy equivalent but spatially di!erent EMA directions (the opposite EMA directions for the phases A}C are indicated in two rows in the Table 1). It is evident from this that there could concurrently exist several energetically equivalent domain areas with di!erent magnetization directions for each phase. The appearance of domains in iron garnet "lms may somewhat shift the boundaries of the stable orientational phases existence and the EMA positions in angular phases, because in this case one more addendum connected with domain wall (DW) energy will be added to the "lm free energy, and the demagnetization "eld energy will properly change. The magnetization vectors in particular domains and DWs themselves are oriented along EMAs. Hence, a domain structure that could arise in di!erent orientational phases will be di!erent in dependence of total quantity of spatially dissimilar EMAs. For the &easy axis' anisotropy type only 1803 DWs are possible because only two opposite EMA direction exist in this phase. It is apparent that such DWs may exist in all orientational phases. The presence in angular phases of three energy equivalent and spatially di!erent EMAs and two opposite sub-phases of the same type can
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Fig. 3. Typical domain structure of epitaxial yttrium iron garnet "lm having an &easy cone' anisotropy (a) and the scheme of domain boundaries con"guration that can be seen (b): solid arrows correspond to the B1 (C1) phase and dashed ones } to the B2 (C2) phase.
Fig. 4. Projections of the magnetization vector in the stable equilibrium state on the "lms plane (1 1 1) and &easy axes' orientation in the B1}B2 phases (a), C1}C2 phases (b) and D phase (c): directions deviated up from the (1 1 1) plane are shown by solid arrows; doted arrows show the direction deviated down from the "lm plane; dashed arrows represent directions that lie strictly in the "lm plane.
result in the existence of DWs with the magnetization vector's turn angle to be near 603 and 1203 in addition to 1803 ones. Those one could be seen in Fig. 3a, where microphoto of domain structure is shown consisting from domains of the B phase type of YIG "lm. Six spatially dissimilar EMAs in the D orientational phase strictly lie in the "lm plane with angles 603 between them. That is why DWs with the magnetization vector turned on 603 and 1203 accurately are possible, and an external view of domain structure in D phase should be the same as in angular phases. A discrepancy between domain structure appeared in angle phases and in the D phase is that the EMAs projections on the "lm plane have crystallographic orientations along 11 1 22 axes in the "rst case and are strictly directed along 11 1 02 axes for the second one as can be seen in Fig. 4.
4. Conclusions The work presents the results of an analysis for possible orientational magnetization states in the absence of external magnetic "eld made on the basis of a phenomenological description of the main contributions in the free energy anisotropic part for iron garnet "lm and minimization of total energy density. It was shown the stable equilibrium position is available for orientational phases with &easy plane' and &easy cone' anisotropy types if only the "rst cubic anisotropy constant is taken into account. In this case the
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magnetization orientation to be exactly in-plane is unstable. Considering the second cubic anisotropy constant leads to stable phases with &easy plane' anisotropy type that, as it was predicted, might be realized, for example, in YIG "lms doped with iridium at concentration near 0.02 atoms per formula unit. The determined directions for the easy magnetization axes explain the domain structure peculiarities, particularly a presence of 603 and 1203 domain boundaries, that arise in iron garnet "lms. It has been shown that the azimuthal orientation of stripe domains in angular phases coincides with the crystallographic axes 11 1 22 and with the 11 1 02 family in the &easy plane' phase. Appendix A K E" ) [21#4 cos 2h#7 cos 4h!4(2 sin 3u(2 sin 2h!sin 4h] 96 K # ) [44#24 cos 2h#36 cos 4h#24 cos 6h!(1015 cos 2h#6 cos 4h!cos 6h) 3456 ;cos 6u!2(2(9 sin 2h!12 sin 4h#5 sin 6h) sin 3u] #K sin h#2pM cos h!M ((H cos u#H sin u) sin h#H cos h), V W X
(A.1)
*E 1 " ) [K (10!15 cos 2h#6 cos 4h!cos 6h) sin 6u *u 576 !(2 [9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h] ) cos 3u] #M (H sin u!H cosu) sin h, V W
(A.2)
1 *E "! ) [8 ) [(6K #K ) sin 2h#3 ) (7K #K ) sin 4h#3K sin 6h] 576 *h #K (5 sin 2h!4 sin 4h#sin 6h) cos 6u #2(2 ) [3 ) (16K #K ) cos 2h!8 ) (6K #K ) cos 4h#5K cos 6h] ) sin 3u] #(K !2pM) sin 2h!M ((H cosu#H sinu)cos h!H sin h), Q V W X *E 1 " ) [2k (10!15 cos 2h#6 cos 4h!cos 6h) cos 6u *u 192
(A.3)
#(2 )[9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h] ) sin 3u] #(M (H cos u#H sinu) sinh), V W *E !1 " ) [8 ) [(6k #k ) cos 2h#6(7k #k ) cos 4h#9k cos 6h] *h 288 #k (5 cos 2h!8 cos 4h#3 cos 6h) cos 6u!(2 ) [6(16K #K ) sin 2h !32(6K #K ) sin 4h#30K sin 6h] ) sin 3u] #(2K !4pM) cos 2h#M ((H cos u#H sin u) sin h#H cos h), Q V W X
(A.4)
(A.5)
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1 *E " ) [k (5 sin 2h!4 sin 4h#sin 6h) sin 6u *u*h 96 !(2 ) [3 ) (16K #K ) cos 2h!8 ) (6K #K ) cos 4h#5K cos 6h] ) cos 3u] #M (H sin u!H cos u) cos h. V W
(A.6)
References [1] M.R. Koblischka, R.J. Wijngaarden, Supercond. Sci. Technol. 8 (1995) 199. [2] A.Ya. Perlov, A.I. Voronko, P.M. Vetoshko, V.B. Volkovoy, MMM-93 Conference Minneapolis, MN, 1993. Abstracts, p. 128. [3] L.A. Dorosinskii, M.V. Indenbom, V.I. Nikitenko, Yu.A. Ossip'yan, A.A. Polyanskii, V.K. Vlasko-Vlasov, Physica C. 203 (1992) 149. [4] D.C. Jiles, J.B. Thoelke, J. Magn. Magn. Mater. 134 (1994) 143. [5] S. Krupic\ ka, Physik der Ferrite und der verwandten magnetischen Oxide, Prague, 1973. [6] P. Hansen, Thin Solid Films. 114 (1984) 69. [7] K.P. Belov, A.K. Zvezdin, A.M. Kadomtseva, R.Z. Levitin, in: Orientational transitions in rare earth magnetics, Nauka, Moscow, 1979. [8] P. Hansen, R. Krishnan, J. Phys. 38 (1977) C1}147.
Orientational states of magnetization in epitaxial (1 1 1)-oriented iron garnet "lms Sergii B. Ubizskii * R & D Institute for Materials, Scientixc Research Company &Carat', 202 Stryjska street, 290031, Lviv, Ukraine State University &Lvivska Politechnika', Lviv, Ukraine Received 19 May 1998; received in revised form 5 January 1999
Abstract A (1 1 1) oriented single-crystal iron garnet "lm having mixed cubic and uniaxial anisotropy was investigated. Taking into account the "rst two constants of cubic anisotropy an expression for anisotropic part of free energy was derived in the polar coordinate system connected with "lms crystallographic directions. Its minimization allowed to obtain possible orientational phases. While only the "rst cubic anisotropy constant is taken into account, there are a symmetric phase of &easy axis' anisotropy and two angle phases with &easy cone' anisotropy. By adding the second cubic anisotropy constant into consideration one more symmetric phase may be found that correspond to &easy plane' type of anisotropy. Orientation of easy magnetization directions are analyzed as well as their in#uence on domain structure con"gurations. 1999 Elsevier Science B.V. All rights reserved. Keywords: Iron garnet "lms; Magnetic anisotropy; Magnetic phases; Easy magnetization directions; Domain structure
1. Introduction Epitaxial "lms of rare earth iron garnets (REIG) have been under intensive investigation from 70}80s due to their prospects for application in the bubble domain memory devices. Films for those uses should have a su$ciently large induced uniaxial magnetic anisotropy in perpendicular to the "lm surface direction that is an appropriate condition for the stable bubble domains existence. As a promised fact it was considered the "lm application with this type of anisotropy in magnetooptical devices for signal processing as well. The "lms based on pure yttrium iron garnets (YIG) with low magnetic anisotropy were utilized only for microwave applications. These "lms have an anisotropy to be close to the &easy plane' type, i.e. the magnetization vector deviates from the "lm surface plane on a small angle. However, as their use is mainly based on gyromagnetic
* Corresponding author. Tel.: 380-322-632-219; fax: 380-322-632-228. E-mail address: [email protected] (S.B. Ubizskii) 0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 1 6 5 - 1
576
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properties in fully magnetized state, the peculiarities of anisotropy, domain structure and remagnetization were out of detailed studies. In recent years an interest has arisen to utilization of REIG "lms in non-traditional applications namely for sensor devices for visualization of the magnetic "eld irregularity distribution and the ultra-low "elds magnetometry [1,2]. For the "rst case the best active medium could be a REIG "lm that would have an anisotropy of the &easy plane' type and be in a single-domain state [3]. In the second case to get the best parameters of magnetometric sensor it is necessary to provide some "xed type of anisotropy [2]. To satisfy the above requirements it is necessary to elucidate conditions for existence of one or another equilibrium state of the magnetization orientation in the iron garnet "lm. This work presents a study of conditions for appearing of di!erent orientational phases in epitaxial (1 1 1)-oriented iron garnet "lms that have been derived on the base of the minimization procedure applied to anisotropic part of the magnetic "lm free energy. The attention will be given to in#uence of anisotropy constants, while in other materials, for example, in highly magnetostrictive Tb Dy Fe alloy, the external stress could determine the change of number and orientation of easy magnetization directions [4].
2. Theoretical background To analyze an equilibrium state we choose the polar coordinate system shown in Fig. 1. The magnetization direction is de"ned by a polar angle h counted from direction of the "lm normal (crystallographic direction [1 1 1]) and azimuthal angle u that is counted from [1 1 0] direction in the "lm plane. In the general case a stable equilibrium position of the magnetization vector is determined by conditions of minimum for the anisotropic part of the magnetic "lm free energy E that is a function of the magnetization vector orientation: *E "0, *u
(1)
*E "0, *h
(2)
*E *E *E ! '0, *u*h *u *h
(3a)
*E '0. *u
(3b)
Fig. 1. Polar coordinate system based on crystallographic axes of the epitaxial (1 1 1) oriented iron garnet "lm.
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The energy density E in our case is a sum of the following contributions: the cubic anisotropy energy E ; the induced uniaxial anisotropy energy E ; the magnetostatic energy that comprises of a magnetic's energy in external "eld EH and a demagnetization "eld energy E . The last contribution is su$ciently depended on the presence and type of domain structure in magnetic "lm. In the "rst stage of our study to de"ne a quantity and type of uniform orientational magnetic phases in epitaxial iron garnet "lm without external "eld it is su$cient to consider a single domain state of magnetic "lm. For this case the demagnetization "eld energy E is determined only by the shape of the magnetic sample, i.e. by "lm state of subject under examination. The magnetoelastic interaction energy will not be considered separately because its part resulting in spontaneous magnetostriction deformation and stresses arising during remagnetization is described phenomenologically as additional contribution to the cubic anisotropy energy constants K and K [5]. Another part of magnetoelastic energy connected with uniaxial strains due to the "lm-substrate lattice parameter mismatch contributes in the same manner to the uniaxial anisotropy constant K together with growth-induced uniaxial anisotropy [6]. Obtained general expressions in chosen coordinate system for the total energy density E of (1 1 1)-oriented epitaxial iron garnet "lm in zero "eld as well as expressions for derivatives used in Eqs. (1) and (2) (3) are given in the appendix, where magnetic "elds H , H and H correspond to external "elds in the directions [1 1 0], [1 1 2 ] and [1 1 1].
3. Results and discussion Handling of the equations system (1)}(2) provides equilibrium solutions which correspond to di!erent orientational states of magnetization, i.e. orientational phases. Equilibrium orientational phases that could arise in the absence of the external magnetic "eld are presented in Table 1 together with anisotropy types that they correspond to. For a stable equilibrium the presented solutions determine the directions of easy magnetization axes (EMA) of the iron garnet "lm. The number of spatially di!erent EMAs for obtained phases are also given in the Table 1. For angular phases B and C the value of polar angle h for the EMA equilibrium position could be found from equation: 3(16K #K )sin 2h#28(6K #K )sin 4h#23K sin 4h G2(2(3 ) (16K #K )cos 2h!8 ) (6K #K )cos 4h #5K cos 6h)#288 ) (4pM!2K )"0, where signs in &G' correspond to phases B and C, respectively.
(4)
Table 1 Equilibrium orientational states of magnetization in the epitaxial (1 1 1) oriented iron garnet "lms in zero external "eld Orientational phase
Anisotropy type
Polar angle h
Azimuthal angle u
Number of &easy axes'
A
&easy axis'
B
&easy cone'
C
&easy cone'
D
&easy plane'
0 p 0(h(p/2 p/2(h(p 0(h(p/2 p/2(h(p p/2
u u !p/6#(2p/3 ) ) n p/6#(2p/3 ) ) n p/6#(2p/3 ) ) n !p/6#(2p/3 ) ) n p/3 ) n
1 1 3 3 3 3 6
n is an integer.
578
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In classi"cation after [7] the A and D orientational phases should refer to symmetrical ones. It means that a change of any "lm parameters (saturation magnetization M and anisotropy constants K , K , K ) in the phase equilibrium area the orientation of magnetization vector is invariant. In contrary, the B and C phases are angular because an orientation of magnetization vector, particularly, a polar angle h, depends on "lm parameters over all area of the phase existence. The azimuthal angle of the magnetization vector is undetermined for the A phase. In this case the free energy density does not depend on angle u, and the condition for equilibration stability of the orientational state A is expressed by the inequality: *E '0 *h
(5)
or 12K #4K #9 ) (4pM!2K )(0. (6) For the rest phases the stability condition of the magnetization vector equilibrium position is determined by inequalities (3). The stability conditions for the B and C phases equilibration take the form (2K ) [10!15 cos 2h#6 cos 4h!cos 6h] $(2 ) [9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h] ) ; ([3 ) (16K #K ) cos 2h#56 ) (6K #K ) cos 4h#69K sin 6h] $(2 ) [6 ) (16K #K ) sin 2h!32 ) (6K #K ) sin 4h#30K sin 6h] #(4pM!2K ) cos 2h)'0, 2K ) [10!15 cos 2h#6 cos 4h!cos 6h] $(2 ) [9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h](0, where signs &$' correspond to phases B and C, respectively. For the D phase these conditions become:
(7a)
(7b)
(4pM!2K !3K )K !6K'0, (8a) K '0. (8b) It should be noted the orientational phase D could not be stable if the "rst cubic anisotropy constant K is taken into account only, i.e. K "0. An equilibrium at u"p/3 ) n and h"p/2 could be only unstable in this case. The phase boundary that devides phases B and C in this case is discribed by equation K "0. The negative and positive values of K correspond to phases B and C, respectivelly. In more general case the boundaries of phase stability dependent obviously upon all "lm material parameters M , K , K and K . In general case they are also dependent from applied magnetic "eld and temperature. So, the whole phase diagram of orientational states of magnetic "lms under consideration has at least four dimensions at "xed "eld and temperature. As an example, the sections of phase diagram for magnetization orientational states in zero "eld is shown in Fig. 2 in coordinates (K , K ) at di!erent K levels with "xed value of saturation magnetization M "135 G to be inherent to pure YIG magnetization at room temperature. As can see from Fig. 2 the &easy plane' -type anisotropy is possible at positive values of K . It has been shown in Ref. [8] the YIG second cubic anisotropy constant reverses its sign on positive one and becomes more than K in absolute magnitude when garnet is doped by iridium in concentration near 0.02 per formula unit. At that, if the uniaxial anisotropy will not be too much positive the conditions for arising of the &easy plane' anisotropy can be achieved (see Fig. 2).
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Fig. 2. An example of the orientational phases' diagram of iron garnet "lm in (K , K ) cross section in zero external "eld and M "135 G, K "!10 erg cm\ (a), K "0 (b), K "#10 erg cm\ (c). Level lines represent the constant polar angle h and its value is given in degrees.
As one can see from the Table 1 each of obtained phases A}D have energy equivalent but spatially di!erent EMA directions (the opposite EMA directions for the phases A}C are indicated in two rows in the Table 1). It is evident from this that there could concurrently exist several energetically equivalent domain areas with di!erent magnetization directions for each phase. The appearance of domains in iron garnet "lms may somewhat shift the boundaries of the stable orientational phases existence and the EMA positions in angular phases, because in this case one more addendum connected with domain wall (DW) energy will be added to the "lm free energy, and the demagnetization "eld energy will properly change. The magnetization vectors in particular domains and DWs themselves are oriented along EMAs. Hence, a domain structure that could arise in di!erent orientational phases will be di!erent in dependence of total quantity of spatially dissimilar EMAs. For the &easy axis' anisotropy type only 1803 DWs are possible because only two opposite EMA direction exist in this phase. It is apparent that such DWs may exist in all orientational phases. The presence in angular phases of three energy equivalent and spatially di!erent EMAs and two opposite sub-phases of the same type can
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Fig. 3. Typical domain structure of epitaxial yttrium iron garnet "lm having an &easy cone' anisotropy (a) and the scheme of domain boundaries con"guration that can be seen (b): solid arrows correspond to the B1 (C1) phase and dashed ones } to the B2 (C2) phase.
Fig. 4. Projections of the magnetization vector in the stable equilibrium state on the "lms plane (1 1 1) and &easy axes' orientation in the B1}B2 phases (a), C1}C2 phases (b) and D phase (c): directions deviated up from the (1 1 1) plane are shown by solid arrows; doted arrows show the direction deviated down from the "lm plane; dashed arrows represent directions that lie strictly in the "lm plane.
result in the existence of DWs with the magnetization vector's turn angle to be near 603 and 1203 in addition to 1803 ones. Those one could be seen in Fig. 3a, where microphoto of domain structure is shown consisting from domains of the B phase type of YIG "lm. Six spatially dissimilar EMAs in the D orientational phase strictly lie in the "lm plane with angles 603 between them. That is why DWs with the magnetization vector turned on 603 and 1203 accurately are possible, and an external view of domain structure in D phase should be the same as in angular phases. A discrepancy between domain structure appeared in angle phases and in the D phase is that the EMAs projections on the "lm plane have crystallographic orientations along 11 1 22 axes in the "rst case and are strictly directed along 11 1 02 axes for the second one as can be seen in Fig. 4.
4. Conclusions The work presents the results of an analysis for possible orientational magnetization states in the absence of external magnetic "eld made on the basis of a phenomenological description of the main contributions in the free energy anisotropic part for iron garnet "lm and minimization of total energy density. It was shown the stable equilibrium position is available for orientational phases with &easy plane' and &easy cone' anisotropy types if only the "rst cubic anisotropy constant is taken into account. In this case the
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magnetization orientation to be exactly in-plane is unstable. Considering the second cubic anisotropy constant leads to stable phases with &easy plane' anisotropy type that, as it was predicted, might be realized, for example, in YIG "lms doped with iridium at concentration near 0.02 atoms per formula unit. The determined directions for the easy magnetization axes explain the domain structure peculiarities, particularly a presence of 603 and 1203 domain boundaries, that arise in iron garnet "lms. It has been shown that the azimuthal orientation of stripe domains in angular phases coincides with the crystallographic axes 11 1 22 and with the 11 1 02 family in the &easy plane' phase. Appendix A K E" ) [21#4 cos 2h#7 cos 4h!4(2 sin 3u(2 sin 2h!sin 4h] 96 K # ) [44#24 cos 2h#36 cos 4h#24 cos 6h!(1015 cos 2h#6 cos 4h!cos 6h) 3456 ;cos 6u!2(2(9 sin 2h!12 sin 4h#5 sin 6h) sin 3u] #K sin h#2pM cos h!M ((H cos u#H sin u) sin h#H cos h), V W X
(A.1)
*E 1 " ) [K (10!15 cos 2h#6 cos 4h!cos 6h) sin 6u *u 576 !(2 [9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h] ) cos 3u] #M (H sin u!H cosu) sin h, V W
(A.2)
1 *E "! ) [8 ) [(6K #K ) sin 2h#3 ) (7K #K ) sin 4h#3K sin 6h] 576 *h #K (5 sin 2h!4 sin 4h#sin 6h) cos 6u #2(2 ) [3 ) (16K #K ) cos 2h!8 ) (6K #K ) cos 4h#5K cos 6h] ) sin 3u] #(K !2pM) sin 2h!M ((H cosu#H sinu)cos h!H sin h), Q V W X *E 1 " ) [2k (10!15 cos 2h#6 cos 4h!cos 6h) cos 6u *u 192
(A.3)
#(2 )[9 ) (16K #K ) sin 2h!12 ) (6K #K ) sin 4h#5K sin 6h] ) sin 3u] #(M (H cos u#H sinu) sinh), V W *E !1 " ) [8 ) [(6k #k ) cos 2h#6(7k #k ) cos 4h#9k cos 6h] *h 288 #k (5 cos 2h!8 cos 4h#3 cos 6h) cos 6u!(2 ) [6(16K #K ) sin 2h !32(6K #K ) sin 4h#30K sin 6h] ) sin 3u] #(2K !4pM) cos 2h#M ((H cos u#H sin u) sin h#H cos h), Q V W X
(A.4)
(A.5)
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1 *E " ) [k (5 sin 2h!4 sin 4h#sin 6h) sin 6u *u*h 96 !(2 ) [3 ) (16K #K ) cos 2h!8 ) (6K #K ) cos 4h#5K cos 6h] ) cos 3u] #M (H sin u!H cos u) cos h. V W
(A.6)
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