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Microwave properties of magnetic garnet thin films
Thin Solid Films, 114 (1984) 135-186
135
ELECTRONICS AND OPTICS
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS P. E. WlGEN
Department of Physics, Ohio State University, 174 West 18th Avenue, Columbus, 0H43210 (U.S.A.)
The theories describing the dynamical excitations of the spin system of the ferromagnetic garnets are reviewed in this paper. Initially the microscopic description using quantum spin operator notation, normally used in the theoretical studies on the subject, is presented. The role of surface anisotropy energies in determining the boundary conditions of standing spin wave modes in thin films is developed in some detail in this model. The continuum model, which is the usual language of the experimentalist for analyzing ferromagnetic resonance data, is subsequently introduced and shown to be equivalent to the microscopic model. Some applications of the continuum model include ferromagnetic resonance, spin wave resonance, magnetostatic wave effects and non-linear excitations of the spin system.
1. INTRODUCTION
Single-crystal films of yttrium iron garnet (YIG) were first grown in 19671. Ferromagnetic resonance (FMR) was one of the earliest techniques used to investigate the fundamental properties as well as some of the magnetic material parameters of the film 2. As the growth of the more complex magnetic garnet films has developed, FMR has continued to be one of the most active and valuable techniques used to investigate their properties. FMR can be useful in the determination of many of the magnetic material parameters of the magnetic garnet films. These parameters include internal fields having uniaxial symmetry, first- and second-order cubic crystalline anisotropy fields and energy constants, g values, exchange constants and spin wave dispersion constants, magnetic loss parameters, magnetostrictive coupling constants and the magnetostatic wave characteristics. In addition, FMR has proved to be a valuable technique in many investigations of fundamental interest. These investigations include spin wave resonance, effects of surface anisotropy energy, magnetoelastic wave excitations, magnetoexchange branc~ repulsion, magnetoacoustic coupling, photomagnetic effects, domain wall resoenance, relaxation phenomena and high power (parallel-pumping) non-linear effects. 0040-6090/84/$3.00
© ElsevierSequoia/Printedin The Netherlands
136
P.E. WIGEN
This article is designed to serve as an introduction and/or guide to the novice or for the occasional user of the F M R technique who either investigates thin magnetic garnet films or evaluates magnetic material parameters. Theorists who work in the field have a tendency to develop their theories in terms of a microscopic model using quantum mechanical operators, while experimentalists usually describe their results in terms of a continuum model. This article represents an attempt to show the equivalence of these two approaches. The fundamentals of magnetism and the microscopic theory of spin waves are reviewed in Section 2 while the diversity of the surface energies and their influence on the boundary conditions and the normal modes of the spin wave excitations are described in detail in Section 3. The equivalence between the microscopic approach and the continuum model is developed in Section 4 while the effects of a non-parallel ground state and its contribution to an effective surface condition are considered in Section 5. Section 6 is devoted to the effects of inhomogeneous internal fields in the film and how they can dominate the surface effects in certain cases. Magnetostatic effects and some examples of their applications to devices are discussed in Sections 7 and 8. Finally, the high power responses are briefly discussed in Section 9. It is intended that these sections should give the reader the basic understanding of the application of F M R to magnetic garnet films. The references are intended to be typical examples of the application of F M R to garnet films but are neither a complete set nor a thorough review of the literature. Some of the topics reviewed in this paper were discussed in more detail by the present author in a previous publication 3. F M R is indeed a technique that is quite versatile and has proven to be useful in the study of the magnetic garnet films. It is expected that its application will continue to be relevant in the future and it is the hope of the author that this article will assist m a n y investigators in their efforts to analyze their data and to obtain parameters and/or information that will be useful for their purpose. 2.
THE MICROSCOPIC MODEL OF THE SPIN SYSTEM IN GARNETS
2.1. H e i s e n b e r g e x c h a n g e hamiltonian
In a microscopic electron system, the Pauli exclusion principle demands a totally antisymmetric wave function. Neglecting spin-orbit coupling, an orbital wave function which is symmetric will be coupled with a spin function which is antisymmetric and vice versa. For two individual electron wave functions, ~bi and ~bj, located at sites i and j respectively in the crystal lattice, the symmetric and antisymmetric orbitals will have the forms ~ym =
~Pi(rl)q~j(r2) + ~)i(r2)q~j(rl)
and
(1) t/Jam = ~ i ( r l ) ~ b j ( r 2 ) -
~)i(r2)~pji(rl)
where r 1 and r 2 are the spatial coordinates of the two electrons. The energies of these states are 4 Esy m =
K + Jij
and
(2) Eant = K - Jij
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
137
where K contains the energy due to kinetic and potential contributions and (
Jij =
e2 ~i(rl)t~j(r2) ~
)) ~b~(r2)~bj(rl
(3)
the exchange integral, arises from the spatially different forms of the symmetric and antisymmetric orbital wave functions. This purely quantum mechanical term has its origin in the region where the wave functions overlap and it splits the energy levels of the electronic system s . As the individual electronic wave functions in insulators are generally localized to the atomic site, the exchange integral in a macroscopic system is expected to be limited to nearest or at most next-nearest neighbors in the spin system. In the magnetic oxides, the exchange integral may be mediated through a connecting oxygen atom via the superexchange interaction 6. If sl and s 2 are the spin angular momentum vectors of the two electrons, the total angular momentum S = sl 4-s2 will be restricted to the eigenvalues 0 and 1 in the antisymmetric and symmetric spin states respectively. Then S 2 = sl 2 + s2 2 + 2s 1-s 2
(4)
which has the eigenvalues
S(S + 1) = 2s(s + 1) + 2s I -s 2
(5)
For s = 1/2, the operator ½+ 2sl "s2
(6)
will have the eigenvalues - 1 and + 1 in the states ~V,ymand ~Pantrespectively. Therefore the hamiltonian for the system can be equivalently expressed in terms of the spin operators by the relation 3ffo = K - (½+ 2s 1.s2)J0
(7)
In this relation, the electrons behave as if there existed a very strong magnetic-like coupling of the form s~ .s 2 between the spin vectors. However, the origin of the interaction is not magnetic but is due to a quantum mechanical Coulomb repulsion interaction which has no classical analogue. In the presence of an applied magnetic field H o, that portion of the total hamiltonian that is dependent on the orientation of the spins can be expressed in the form = Yrz + ~ E
(8)
where 3fazis the Zeeman energy and ~E is the exchange energy. The Zeeman term is given by
~Cz = g#BHo ~ S, z
(9)
where Ho defines the z axis, g is the Land6 g factor and #a the Bohr magneton. The positive sign is chosen as the electron spin is antiparallel to the magnetic moment. The exchange hamiltonian has the following form:
~E = --2 ~, J,jS,.Sj i>j
(10)
138
P.E. WIGEN
where the sum on i is over all lattice sites and in the exchange term the double sum is normally limited to nearest neighbors unless otherwise indicated and restricted to i > j so that the same interaction is not counted twice. For the parallel spin alignment the exchange term Ji~ is positive while the antiparallel spin alignment arises from a negative value of the exchange term. In magnetic solids, additional interactions between atoms will influence the energy of the spin system. Examples of such interactions include dipole-dipole interactions leading to magnetostatic demagnetization effects and spin-orbit interactions which contribute to the anisotropy energies of the system. These additional terms can be included in the spin hamiltonian by allowing the applied field H o to be replaced by an effective field Heff~ that determines the additional spinorientation-dependent terms in the hamiltonian. The hamiltonian for an assembly of exchange-coupled localized spins in a magnetic field is = g#, ~ H e f f i ' S / - 2 ~ JijS,'Sj i
(11)
i>j
2.2. Spin waves In the first excited state, it might be assumed that one spin Si is flipped parallel to the magnetic field while the others remain antiparallel to the field. As the value of J is of the order of 10 -14 erg ~ 100kB erg, where kB is the Boltzmann constant, this spin configuration corresponds to a large energy state. In comparison, the Zeeman energy for a single flipped spin is about lkB erg in a magnetic field of 10 kOe. In addition, such an excited state lacks the translational invariance of the hamiltonian in eqn. (11). The large exchange energy can be reduced by sharing the spin deviation between a large number of spins. In this mode, the angle between adjacent spins is small and, since the exchange energy is proportional to the cosine of the angle between adjacent spins, the exchange energy in the spin system is reduced. At low temperatures, such excitations, known as spin waves or magnons, determine the thermodynamic behavior of ferromagnets. It was Bloch v who first found that the states near the ground state could be approximated by sinusoidal spin waves. In evaluating the excited states it will be useful to express the hamiltonian in terms of the raising and lowering operators (12)
S + i ~- SXi + iSY i
and of SZ~. Equation (11) then has the form : g/~aHeff E SZi -- 2J .~ (S~i S~j + ½S +iS- j + ½S -iS +j) i
(13)
L,3
This hamiltonian will be considered for the following examples. 2.2.1. Linear chain with periodic boundary conditions and S = 1/2 The ground state eigenfunction for a linear chain of N spins whose hamiltonian is given by eqn. (13) can be expressed in terms of the single-particle spin down (magnetic moment up) eigenstates ~i as I//0
~--- 0 ~ 1 ~ t 2 0 ~ 3
• • • 0~ N
(14)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
139
This is the state realized at 0 K and represents the maximum alignment of spins or complete magnetization. As the temperature increases, the system will be excited out of the ground state into an excited state that may be thought of as containing a single reversed spin: $i =
° q °t2 " " °q - t fli°q + 1 "
" %v
(15)
where fl~ is the single-particle spin up eigenstate at the site i. However, ~k~is not an eigenfunction of the hamiltonian of eqn. (13). An eigenfunction can be formed by taking a linear combination of the ~,~ such as
Itlk = E cki~]i
(16)
i
Slater s has shown that the proper combinations in eqn. (16) are equivalent to a wave-like disturbance of wavevector k with the values of k determined by the periodic boundary conditions. Each spin i interacts with its two nearest neighbors i + 1 and i - 1 through the exchange integral J. Using the relation (~q I~ ( 1 / 2 ) 17'k) = l(~bilgl Wk~
(17)
the following set of coupled equations is obtained: {E + {(N -- 2)g/t a n o + ~ N - 5) J} C', + J(C k, +1 + cki-1) = 0
(18)
The solutions have the wave-like form (19)
cki = N - 1/2 exp(ikia)
where a is the distance between spins. The periodic boundary conditions require Ck~ to equal Cki +N, which yields the following values of k: kaN - 0, 1 ,2 . . . . . N 2x
1
(20)
Substituting eqn. (19) into eqn. (18), the energy E is found to be related to the wavevector k by E + ~ N - 2)g/~aH eft + {(N-- 5) d + 2J cos(ka) = 0 or
(21) E - Eo
= g/,tll H eff +
2J{ 1 - cos(ka)) = hook
where Eo = --½NglzaHeff-½NJis the ground state energy and hCOk is the energy required to excite the spin wave of wavevector k. In the limit of small values o f k, i.e. ka ~ 1, eqn. (21) reduces to the familiar form hfo/~ ~ g ~ B H elf +
Dk 2
(22)
where D = Ja 2 for a linear chain of spin 1/2. This dispersion curve is shown in Fig. 1. 2.2.2. Linear chain with end spins "'pinned" and S = 1/2 In thin magnetic garnet films, a considerable effort has been extended to observe and interpret the excitation of spin wave modes. The excitation of these modes is determined by the presence of an effective "surface" anisotropy field H 'urf acting on the spins in the first layer and in the Nth layer. In addition, the surface
140
P.E. W|GEN
g,u. B Heft-
D k 2 -->
Fig. 1. Spin wave dispersion curve calculated for Zeeman and exchange energies only. spins are not exchange coupled to a full set of nearest neighbors. Pincus 9 was the first to consider this problem by introducing symmetric surface terms into the hamiltonian: o'~surf = gltBHsurf(sz I + SzN)
(23)
The energy condition given in eqn. (18) is then replaced by {E + ½(N -- 2)gtt n H off + ~{N -- 5) J }Cki + J(C k, + 1
cki - 1
0
(24)
{e + ½ t N - 2)ggnHaf + g#nHsurf + ~ ( N - 3)d}C k, + dCk2 = 0
(25)
+
) =
for/=# 1,N. F o r i = 1
and for i = N
{E+~(N--2)glJBnaf+gHBnsurf+~N--3)d}CkN+JCkN
t = 0
(26)
The solutions are of the form
cki+ 1 = ~ cos(kia) + fl sin(kia)
(27)
and the energy is given by E = -- ~(N -- 2)g/~BH eft
-
-
~(N - 5) J - 2J cos(ka) -- g#B Hsurf
(28)
The surface field has no effect on the form of the dispersion relation but modifies the allowed values of k. Using this energy condition, the ratio of the coefficients of eqn. (27) is ct
J sin(ka) = - J{ 1 - cos(ka)} + g/~BHS"rf
(29)
F r o m this equation, it is observed that the surface pinning or spin wave b o u n d a r y condition is not a constant term for all spin wave modes but changes with k. At low values of k the surface anisotropy energy m a y dominate the b o u n d a r y condition giving a nearly pinned spin wave. However, as the k value increases, the higher exchange energy in the spin wave will dominate the surface anisotropy energy when Dk 2 > gl~a SH s"rf and a nearly free b o u n d a r y condition m a y exist. Using this ratio and eqns. (25) and (26), the following relation is obtained to determine the allowed
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
141
values of k: 2(1-~r) _ -~ ( l2k a ) - ~ cot(Nka) = -tr(2_~t) tan
cot(ka)
(30)
with g # a H surf ~r -
-
-
(31)
J
For cr = 1, the allowed modes have nodes at i = 0 and i = N + 1. For arbitrary tr, the nodal points are functions of k. The allowed values of k will be discussed in detail in Section 3. 2.2.3. Three dimensions and arbitrary S In YIG the spin S of the iron ion is 5/2. MOiler t 0 has extended the analysis to the case of three dimensions and arbitrary S. The result gives a dispersion relation of the form fifO/, =
g#s H e f t + 2SzJ( 1 - Yk)
(32)
where Ykis the geometric structure factor given by )'k = Z- 1 ~ exp(--ik'ri)
(33)
i
with r i being the vectors to the z nearest-neighbor magnetic ions. In a lattice of simple cubic symmetry and small values of k, eqn. (32) reduces to the familiar form hfOk ~ g / h 3 H elf + D k 2
(34)
where in this case D = ~SJza 2 = 2SJa 2 is the spin wave dispersion constant and a is the cube length. For YIG, D ~ 10 -2a erg cm 2 or 5 x 10 - 9 0 e cm 2 (ref. 11). 2.2.4. Spin waves in the garnet multisublattice model In the magnetic garnets there are up to 32 spins in the unit magnetic cell. For this case it is convenient to locate lattice sites by the relation riu = r~ + pu where the pu represent the intracell vectors to each of the v distinct magnetic sites in the magnetic cell giving rise to v recognizable sublattices in the ground state with all spins pointing up or down along a common axis. Oscillations away from the ground state are expressed in terms of spin deviation operators 12 over the v magnetic sites in the magnetic cell. The deviation operators are expanded in spin wave operators which are generally coupled and it is necessary to solve a secular equation of order v to find the normal modes. The general procedure has been discussed by Saenz 1a. In the application of this process to the magnetic garnets, the sum must be carried out over the various sites having dodecahedral symmetry {c}, octahedral symmetry [a] and tetrahedral symmetry (d). For details on the crystal structure see ref. 14. In YIG the {c} site is occupied by the non-magnetic yttrium ion and the problem reduces to a two-sublattice spin system. The details of the derivation for YIG have been evaluated by Harris ~s who found hfOo °p ~ -- lOJad
(35)
for the optical branch and ~lO)k a¢: ~, .5r (8Jaa -- 5Jad "t- 3Jdd)a2k 2 1o
(36)
142
P.E. WIGEN
for the acoustic branch. The k dependence of the 19 optical modes is not observed to be very strong. For the more complicated magnetic garnets where the {c} sites are occupied by rare earth ions the following relation is obtained for the acoustic mode: h(Okac ~,
200daa -- 125Jad + 75Jdd -- 20ddc Sc + 50Jac Sc 32k2 16(-5+6S~)
(37)
Harris 15 has also evaluated the coefficient of the terms of order k 4. Because of the relatively weak exchange between the iron ions at sites [a] and (d) and the rare earth ions at the {c} sites, the optical spectra will consist of 19 slightly perturbed high energy YIG modes plus 12 additional low lying optical branches. Since the garnet system is limited to a single acoustic type of spin wave branch with higher order optical spin wave branches separated to a relatively high frequency in zero magnetic field, the system will behave as a ferromagnetic array having spin wave energy Dk: given by eqn. (36) for YIG or eqn. (37) for the rare earth garnets. For the substituted or mixed garnets, Harris and Kirkpatrick 16 have evaluated the dispersion constant using percolation techniques. The experimental results are in good agreement with their model I 7.18 In YIG the frequency of the optical mode normally falls in the IR or visible region of the electromagnetic spectrum. In the rare earth garnets a two-sublattice approximation can be used by assuming that the iron atoms, which are strongly coupled together, can be treated as a single sublattice coupled less strongly to the rare earth sublattice. At some temperature or rare earth ion concentration, the condition gFeMve--gREMRE = 0 may be satisfied for which the total angular momentum will vanish. Under these exceptional circumstances, the resonance frequency of the optical mode will move into the microwave region. The temperature at which the angular momentum vanishes is often near the temperature at which MFe = MRE where the magnetization is compensated. Wangsness ~9 has shown that the small difference between the compensation points of the angular momentum and the magnetization also contributes to the Zeeman energy term associated with eqn. (37). The Land6 g value in the two-sublattice model has the form Ma-Mb gcff= Ma/g _ Mb/g b
(38)
Near the compensation point for the angular momentum, geff will become very large and change sign ~7. 2.3. Additional spin interactions in the harniltonian In addition to the Zeeman and exchange energies that are included in the spin hamiltonian in eqn. (11), a variety of additional spin interaction terms can be included. Typical terms include the following. 2.3.1. Single-ion anisotropy The crystal electric field may introduce second-order effects in the spin hamiltonian via spin-orbit or spin-spin interactions. These effects are important in the resonance spectrum of dilute paramagnetic salts, and they have been reviewed in detail by Bleaney and Stevens z°. If the local crystal field has uniaxial symmetry, e.g. crystals having tetragonal or trigonal symmetry with the unique axis in the z
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
143
direction, the crystal field contribution is ~¢ = ~ ~ {(SZ,)2 - ~ S ( S + 1)}
(39)
i
where ~ is the measure of the interaction. This term vanishes for S = ½. For small values of ~, this term can be included in the Zeeman hamiltonian by introducing an anisotropic g: ~,~ffz=
-gll#nHZo~S~,-g±lzB(HXo~iSX,+H'o~S" t
(40)
In ordered magnetic materials, such as the garnets, this will be included as an anisotropy energy in the system.
2.3.2. Terms coupling spin pairs In general, the pair hamiltonian ~vair can be expanded in a power series of appropriate symmetry in the spin operators:
°3~pair =
E E jklijSkiSlj
i>j k.l
k,l = x,y,z
(41)
If the spin system has cylindrical symmetry, this term can be expressed in the form °~pair = '~e 3t"°~d
(42)
where -~e is given in eqn. (10) and ~ d = E dij*(Si'Sj-- 3ru- 2(~i'riJ)(Sj'riJ)) i>j
(43)
do* = dij + Pij
(44)
du = g2fla2ru- 3
(45)
with
and
The do coupling term is the familiar long range rij-3 contribution arising from magnetic dipolar interactions and Po is the phenomenological short range pseudodipolar exchange interaction first introduced by van Vleck 21. These terms are often referred to as anisotropic exchange terms.
2.3.3. Dipole-dipole interaction The inclusion of the long range r~j-3 spin interactions complicates the procedure of diagonalizing the hamiltonian to determine the spin wave dispersion relation. However, this dipole-dipole term is a particularly important term as the long range of this term is responsible for the spin wave band associated with the spin wave dispersion relation and contributes significantly to the understanding of the relaxation processes 22. In a classic paper, Holstein and Primakoff (HP) 12 developed a spin wave theory for a spin hamiltonian which consists of the Zeeman, exchange and dipolar and/or pseudodipolar interactions:
oW= ggeHo. ~ Si-2 ~, JuSi'Sj+i~>jdu{Si'Sj-3rij-2(Si'rij)(Sj'rij) } i
i>j
" "
(46)
144
P.E. WIGEN
In their procedures, the spins are treated as quantized particles subject to creation and annihilation operators by using spin raising and spin lowering operators 23. Spin deviation operators are next introduced by using Bose creation and annihilation operators. Spin waves are introduced by making use of the Fourier expressions of the localized Bose operators. Finally a fourth transformation is required to decouple the states o f k and - k . The result is a diagonalized hamiltonian having a dispersion relation of the form hCOk
{(Dk z + g p B H e f f ) ( D k 2 + g l ~ a Heff + g # n 4 x m s i n 2 0 k ) } 1/2
=
(47)
where M is the magnetization of the medium, Ok is the angle between the propagation constant k and the magnetization M and H eft is the internal field: H elf = H o - 47tNz M
(48)
where Ho is the applied field and N= is the demagnetization factor along the z axis of the sample which is defined here as the direction of H eft . The dispersion relation, shown in Fig. 2, indicates the spin wave manifold proposed in the continuum model which will be discussed further in Section 6. For many experimental conditions of the magnetic excitations in the garnet systems, 4riM may be smaller than the internal field H eft. Under this condition, eqn. (47) can be approximated to the simpler relation he) k ~ D k 2 + g / I n Heff + g l ~ n 2 n M sinE0k
(49)
This relation is shown in Fig. 2, broken line, for sinE0k = 1. This is the relation that would be obtained by neglecting the final H P transformation that decouples the k and - k states. This approximation to the energy will be valid except where k is small, sin20a is near unity and the internal field is small. In YIG, D ~ l 0 - 2 a ergcm 2 and, for k = 106 c m - 1, D k 2 , ~ 10 - 1 6 erg ~ lkn erg
(50)
At room temperature, M = 140 e.m.u, c m - a for which g l ~ a 4 x M is 0.2ka erg so the approximation is good for k >/ l 0 6 c m - 1. In spin wave resonance experiments, the spectrometer usually operates at a ~O~ k
s
,./
../-~s,n,e~-o_
k,~-~TL--ApproximateDispersion / r~ / Relation g/~B2TrM I / ~ ~--r" "Exact DispersionRelation g~BHeff l _ _
Dk2 Fig. 2. Spin wave band calculated for Zeeman, exchange and dipole-dipole energies: given by eqn. (47); - - -, circular precession approximation given by eqn. (49).
, exact solution
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
145
constant radial frequency coo. In this case the applied field is varied to satisfy the spin wave resonance condition or Ho --, H~ in eqns. (47) and (48). At sin Oh = 1, the error in the approximation in eqn. (49) for YIG is 20~o at 3 GHz, 3% at 10 GHz and 0.5~o at 25 GHz. Thus the final HP transformation is only necessary in those experiments where H "ff is only slightly larger than the value necessary to saturate the material. This occurs in experiments at radio frequencies. Another case where the final transformation, which involves the ellipticity of the precession, is necessary is for high power or parallel-pumping experiments 24. 3.
SURFACE INHOMOGENEITIES AND SURFACE SPIN WAVES
3.1. Surface anisotropy energies In Section 2.2 the concept of an effective surface anisotropy field was introduced to provide the boundary condition to determine the allowed spin wave modes observed in the ferromagnetic resonance spectra of thin magnetic garnet films 25-28. In the experiments, it is observed that the spin wave spectrum may have a complex dependence on the orientation of the magnetic field and on the temperature. An example of the influence of these parameters on the spin wave spectrum is shown for a YIG film in Figs. 3 and 4. The weaker signal at higher fields in the parallel orientation is a non-propagating surface mode. While the extremes in most films may not always be as pronounced as those observed in this example, this author is not aware of a report where the spin wave spectrum in a thin magnetic
-t-1
4900
I
5000
I
5100
2900
I
5000
40*
t I
3 'oo
34C0
, 2700
!/
I, 2800
5100
29100
,90 °
5O"
3000
V
!~20
i 2620
MAGNETIC F1ELD(Oe)
Fig. 3. Spin wave spectra showing the first two high field modes at six orientations observed at 9.2 GHz and at room temperature for a YIG film (formed by chemical vapor deposition (CVD)) having a single surface mode. (From ref. 28.)
146
P.E. WIGEN
ToZ92K ,i
T:r3eK ,'i! f - -
I i i'
2500 T=256K
Oe 2 6 0 0
2400
2500
Oe T=IC8K ", X]CO,
l i
:,
, t i ! 2500
T=231K
!/ / 26CC
24C0
'
;
2500
T=2OK
q 2400
I
~'
,'i
i!,1 2500
'
2600
2500
2400
I,~ L~ V
2500
Fig. 4. Spin wavespectra showing the first two high fieldmodes in the parallelorientation as a function of temperature for a CVD YIG filmhaving a single surfacemode at room temperature. (From ref. 28.) garnet film does not reduce to a single mode at some "critical" angle as a function of the orientation of the magnetic field between the normal to the film plane (perpendicular resonance) and in the film plane (parallel resonance). This angle dependence of the pinning condition can be predicted by a combination of an isotropic energy and a unixial surface anisotropy energy. Depending on the sign of the energy terms, the high field non-propagating surface mode will be observed at magnetic field orientations between the critical angle and parallel resonance or between the critical angle and perpendicular resonance. Both conditions have been observed experimentally. These non-propagating surface modes differ from the magnetostatic surface modes of D a m o n and Eshbach (DE) 29 in that the propagation constant in the plane of the film is zero, i.e. the propagation constant is purely imaginary. In addition, decay lengths as short as 5 0 n m have been observed. Consequently, the field position of these surface modes is determined by their exchange energies and not by the magnetostatic effects discussed later. Non-propagating surface modes were first proposed by Wolf 3° in 1963 when a uniaxial surface energy was assumed to be present on the film surface. Puszkarski al-33 was the first to discuss the angular dependence of these surface modes, and this was followed almost simultaneously by the work of Sokolov and Tavger34, 35. Additional contributions on the topic of surface interactions have been made by Mills a6, Spalek and coworkers 37'38, Wachniewski 39 and D u d a 4°. The behavior of the surface mode in Figs. 3 and 4 can be interpreted in terms of a combination of surface energies, each having a unique angle dependence. As an introduction to the full description of these spectra, a model assuming a unidirectional surface anisotropy field will be considered initially 32. The calculations are
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
147
carried out for a hamiltonian consisting of a Heisenberg localized spin model having nearest-neighbor exchange and a Zeeman energy term oW=--2
JgSo'Sl+o,j,+g~tBEHefft'Slj
~ |,J,l + o,J'
(51)
Ij
Here the sum is carried out over the j spins located within the I layers of the spin system of a thin film. Jg is the exchange integral between nearest neighbors situated in layers I and l_+ g and Heft!is the effective magnetic field assumed to be uniform and unidirectional within a given layer. Using this hamiltonian, only circular precession modes will be predicted. The influence of the elliptical precession will be included later. Nearest neighbors of a given spin belonging to the same layer are designated by g = 0 and the two closest layers by g = _+ 1. Jo is then the exchange constant between spins within the I plane and J1 is the exchange constant between spins of different layers. The presence of an effective surface field acting only at the spins in the I = 0 layer and I -- N - 1 layer is represented by Heffo and Heft s _ 1 respectively. All other Heft! a r e assumed to be identical. In this approximation the spins at the surface of the system are subject to a surface field that is assumed to be a unidirectional phenomenological quantity that accounts for the pinning of the surface spins, thus determining the boundary conditions for the k values of the allowed spin wave modes. The energy of the surface spins is given by (52)
Esurf = Ein t + 2 S 2 z l J 1 A
where Eint ---- -- 2S2zoJo _ 4S2z 1J 1 - g#n S~" H eft
(53)
is the energy of the interior spins in their ground state. A is the pinning parameter which determines the boundary conditions and is given by A = 1-- g#n
...... f
(54)
2 S z l J l ~" rl
where ~ is the unit vector in the direction of the spin moment and z I is the number of nearest neighbors in the adjacent layer. The surface field H surf will either increase the energy of the surface spin giving an unpinned boundary condition or decrease the energy of surface spins giving a pinned boundary condition. The surface energy density is given by Esurf
=
--g#n
S a - 2 3" H surf
= 2S2zljla-2(A-
1)
(55)
where a is the lattice constant. Several characteristic features of the calculated k values of the spin wave spectrum are presented in Fig. 5. First, as A approaches - oo, the lowest order mode has an "absolutely" pinned boundary condition; k, = n ~ / L where L = Na. As A increases to unity, the lowest energy mode kt becomes the uniform or kl = 0 mode. The first antisymmetric mode k2 crosses the k2 = 0 condition at A = (N + 1)/(N-- 1). For A greater than unity, kx becomes imaginary or an acoustic surface wave. k2 also becomes imaginary forming an antisymmetric surface mode for A = (N + 1)/(N-- 1).
148
P . E . WIGEN
A --r
~
A C O U S ] IC SURFACE MODES L+J
i,
II
I
II
r I ,
i
ii~
!
'i,
'i,
~
i$ q
Ii i
I
i i
IL
5
,
li L
~
;I
4
i
I
2 I
[I
IiI i I
'~|
% I
II ';11 I
h I
I i i
Ii i i I
,,,,
/i
I
I
i,:/
O
\
2_ L H
*:i /L
I
'
I
G(t)
I ,3
I I
~
,,:
i d
....
-4 n:l
9
3
OPTICA[ SUf?FACE MOI)ES
Fig. 5. Graphical solutions of the characteristic equations as a function of the surface parameter A : , antisymmetric states; - - , symmetric states. The number n labels the allowed states. This graph represents a symmetric film having 11 layers. (From ref. 32.)
Similar conditions exist for the optical surface spin wave in the region A ~< - 1 but, because the moment of this m o d e is so small, they have not been observed in any experiments. The spatial dependence of the transverse m o m e n t of the first few low energy spin wave modes for different values of the surface parameter is shown in Fig. 6. It is noted that as A varies from - oo to unity the precession amplitude of the surface spins increases until the slope of the transverse m o m e n t at the surface is zero when A = 1. As A increases to + oo, the amplitude of the surface spins again decreases and, at A = + oo, the spin wave spectrum has transformed to be identical with that observed when A = - oo as the two surface modes have zero m o m e n t and will not be excited. For A slightly larger than unity, the first two modes are surface modes whose position shifts to lower energies and whose transverse m o m e n t decreases rapidly. A =-cO
A=O
A=I
A:2
A = + '::o SURFACE MODES
n=l
SURFACE MODES
n=l I n~ 2
J
i
i
°:31
1
i,,,2 1
,,=21
I
n=41
=
i
.:sl
.=51
W i I
I I r
I
II
L-I
L-I
L-I
)
L-I
L-I
i
L I
L-I
L-I
L-I
L-I
Fig. 6. The transverse component of the magnetization for a few spin wave modes of low n for various values of the surface parameter A. (From ref. 32.)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
149
The resulting spin wave spectra can be described by the following conditions on A. (a) A = 1 or Es,rt = 0 gives the natural condition at the surface. (b) A > 1 or Es,rf > 0 gives unpinned surface spins (surface modes and body modes). (c) A < 1 or Es,rf < 0 gives pinned surface spins (body modes only). At A = 1, only the uniform mode (k = 0) will be excited by a uniform r.f. magnetic field. All higher order spin wave modes will have odd symmetry about the center of the film and will not be excited. The angle dependence of the surface parameter A is implicit in eqn. (54) and can be expressed explicitly as A= 1
g # ~ ~'H surf 2SzlJ1
= 1
g/~n H surf cos(0o- 0)
2SzlJt
(56)
where 0 o is the angle between the z axis of the film and the orientation of a constant unidirectional surface field and 0 is the angle between ~ and the z axis. For a constant unidirectional surface field, A will be less than unity for I0 o - 01 < n/2 and greater than unity for 10o-01 > ~/2. For the condition 10o-01 = n/2, a critical angle is observed where A will be unity and only the uniform mode will be excited. The unidirectional surface anisotropy field predicts an angle dependence that has a lower symmetry than that observed in the experiments shown in Figs. 3 and 4. Higher order symmetries can be obtained by introducing the appropriate surface energy terms with the original hamiltonian in eqn. (51) 27. In fact any number of these interactions may be present at either surface of the YIG film. Thus to evaluate the influence of these surface energies on the spin wave spectra completely it is necessary to analyze the data in terms of two independent surface parameters, i.e. A s at the substrate surface and Ae at the free surface of the film where As = 1 - ~ aspPp(cos 0) and
(57) Af = 1 - ~ afvPv(cos O)
The Legendre polynomials Pp(cos 0) will represent the angle dependence of the various surface terms present in the hamiltonian and ap indicates the strength of that interaction. In the above analysis, the spins are in a symmetrical environment and the precession will be isotropic. In thin garnet films, the demagnetization factor gives rise to an anisotropic resonance condition and therefore introduces elliptical effects in the spin precession. These effects can be included in the microscopic approach by adding a single-ion anisotropy term of the form of eqn. (39)'~°. However, the elliptical effects are more naturally included in the continuum model considered in Section 4 and therefore will not be considered further here. In comparing the predictions of this model with experimental results, the nonelliptical model will be a valid approximation only in the limit of high frequency (oJly >>4nM) where the degree of ellipticity in the precession is minimized. In order to probe the surfaces of a thin magnetic film by spin wave resonance, it
150
P.E. WIGEN
is desirable that the interior of the film be homogeneous since volume inhomogeneities result in spin wave spectra that are determined only partly or not at all by the surface conditions 4~. Extremely homogeneous films of YIG 42 of the order of 1 lam thick can be routinely produced by epitaxial growth on gadolinium gallium garnet (GGG) substrates. The low intensity of higher order spin wave modes 43 compared with the intensity of the principal mode in films grown by liquid phase epitaxy (LPE) is evidence that the pinning parameters As and Af of eqns. (57) are nearly unity. This indicates that negligibly small surface energies are present in as-grown LPE films. In CVD films or annealed L P E films significant surface energy contributions are present zS-zS. In a typical CVD YIG film, a non-propagating surface spin wave mode is observed, as shown in Figs. 3 and 4. It was found to be localized at the Y I G substrate interface as confirmed by etching experiments 26. Films that are etched in hot phosphoric acid have any free surface impurities removed and yield a natural condition (Af = 1) at that surface. In this manner the surface energies at the filmsubstrate interface can be investigated without interference from the free surface interactions. The angle dependence of the mode positions and relative intensities are shown in Fig. 3 where only the two high field modes observed in the spectra are presented. The weak high field absorption observed at parallel resonance is identified as a surface mode while the next and strongest absorption is the first of the body modes. As the applied magnetic field is rotated towards the perpendicular orientation, the high field surface mode is observed to increase while the body modes all decrease in intensity. At the critical angle 0c, which is 40 ° for this film, the surface mode is transformed into the uniform precession mode and becomes the only mode observed. As the magnetic field is rotated towards the perpendicular orientation, the surface mode transforms into the first of the body modes and the higher order body modes are again observed. The temperature dependence of the spin wave resonance spectrum of the same film has been investigated from 2 to 300 K. While no significant change has been observed for the perpendicular resonance spectrum as the sample temperature is changed, the parallel resonance spectrum changes dramatically as shown in Fig. 4. On annealing the CVD YIG film in the atmosphere at a temperature between 750 and 1300°C, a surface anisotropy energy develops at each surface z8. As the surface energies have different magnitudes at each surface, the modes of pseudoeven-symmetry (even n values) are not antisymmetric and can be excited in the experiments. Figure 7 indicates the results of a series of experiments in which the pinning parameter at each surface and therefore the surface energies were measured as a function of the orientation of M. The YIG film used in the measurements had a [100] orientation, a thickness of approximately 0.46 ~tm and was annealed at 1300 °C for 2 h. The spectra were measured at 23.3 GHz. The pseudosymmetric surface mode is localized at the substrate surface whereas the pseudoantisymmetric surface mode is principally localized at the free surface. The theory of asymmetric boundary conditions was used to find the values of As and Af consistent with these quantities at each angle. The curves shown in Fig. 7 are the least-squares fits to the experimental data.
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
151
0.04
t 0.03
0.02
///I/
0
-0.02
I
o 02
//
-0.04
/ " - ~ SUBSTRATE SURFACE
-006
-0.08
-0101
-012
- 0 14 i
l
l
30 l
I t
l l 50 60 8 ( M - F i L M NORMAL}
-0.01
l
70
90
Fig. 7. The surface pinning parameter A at the substrate interface and free surface of an annealed YIG film calculated from the spin wave spectra (YIG[100]/GGG; YIG thickness, 0.46 pm; annealed for 2 h at 1300 °C; 292 K; 23.3 GHz). The curves on the fight-hand side are the data from 50 ° to 90 ° shown on an expanded vertical scale. (From ref. 28.)
Terms with unidirectional symmetry (odd p) were eliminated from the eqns. (57) since the spin wave spectrum remained unchanged when the direction of M was reversed. It was also found that only two terms in eqns. (57), namely P0 and/)2, were necessary to obtain a good fit to the data. Thus the surface parameters can be written as A~ = 1--aSo-a~2
3 cos20-1 2
and
(58) Af =
1-ato-af23
cos20- 1 2
These coefficients and the corresponding surface energies are given in Table I. The surface energy density values shown in Table I are calculated using the relation bp =
ap2S2ZlJ l a2
(59)
The detailed analysis of the data in terms of the magnitude, angle dependence and temperature dependence of the surface anisotropy energy allows a unique approach to the evaluation of various microscopic interactions that exist at magnetic surfaces. Only when the details of the surface anisotropy are understood can the boundary conditions and the allowed k values of the normal modes be determined with any degree of confidence. Then, once the k values are determined, the bulk spin wave properties such as the spin wave dispersion coefficient can be evaluated.
152
P.E, WIGEN
TABLE I THE COEFFICIENTSat EVALUATEDFROMTHE DATA IN FIG. 7 l
a l ( × l O -2)
b t ( × l O - 2 e r g c m 2)
Substrate interface 0 2 4
0.67_+0.05 7.47+_0.15 (0.72+_0.75)
-4.31 +0.32 -48.00+_0.96
-0.14+_0.03 2.23 +_0.10 (0.14+_0.32)
0.90+_0.19 14.33 +_0.64
Free surface 0 2 4
4.
FERROMAGNETIC RESONANCE IN THE CONTINUUM MODEL
4.1. Equation o f motion
The analysis of F M R experiments is usually performed in terms of a continuum model as opposed to the quantum mechanical expressions developed in the earlier sections. The transition from the spin hamiltonian in eqn. (51) to a set of classical field equations is accomplished by treating the spin operators as classical vectors. The correspondence of the quantum mechanical operators and the classical spin vectors can best be demonstrated by considering the time variations in the expectation values of the transverse components of the magnetic moments when a transition occurs between two allowed states of the hamiltonian. In terms of the spin operators, the transverse magnetic moments are given by
and
(60) (try) = - - g H n ( S y )
From quantum mechanics, the time dependence of such an operator can be expressed in terms of its commutator with the hamiltonian. Considering the Zeeman term as an example, J~'~z =
g#B H" S
(61)
= gpaHoSz
The time dependence of the components of the spin operators is given by the following: d
i
~s~ = ~[~sx] = d
~ [ S r = 7HoSx
d ~Sz =0
-yHoS,, (62)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
153
These equations are the component form of the vector equation
d_s= dt - S x ~/-i~
(63)
or in terms of the expectation values of the magnetic moment operators d(#> dt = - x ~ H o
(64)
where yHo = ho~ is the energy splitting between the two allowed quantum states being considered in the spin transition.Thus in the quantum mechanical model the expectation value of the transverse magnetic m o m e n t in a transition between two states has the same frequency as that required of a small transverse r.f.magnetic field in F M R . The addition of the exchange term to the hamiltonian can be made by expanding the variation in the neighboring spin S t in a Taylor seriesabout Si at the position i:
S t = S,+{(rt-r,)'V}Si+½{(rt-ri)'V}ZS,+...
(65)
Assuming that the ith site is a symmetry center, the second term in the expansion will not contribute to the exchange energy since contributions from opposite neighbors will cancel. In the approximation of nearest-neighbor interactions, the exchange hamiltonian reduces to the form
~'~,~ = - 2zJ ~ Si 2 - J ~_, S,. {(rt - r,). V} 2Si + . . . i
(66)
it
where z is the number of nearest neighbors. The first term in eqn. (66) determines the ground state energy and will not contribute to the spin dynamic energy terms. The second term can be further simplified according to the crystal lattice type. For a cubic lattice with lattice constant a the hamiltonian becomes
= +g#B ~., H'rfi'Si-2Ja2z ~ SrVZSi i
(67)
l
The quantum mechanical equation of motion for a spin Si is given by
ihff-~S, = [Si, oaf']
(68)
Using the relationship S x S = ihS, the commutator yields the expression d h-~tSi = --g#BS x H ©ff+ z2Ja2Si x V2Si
(69)
Since Si represents any of the spins in a finite sample, eqn. (69) can be rewritten in terms of the magnetization vector M as l d--M=-MxH'ff+2-~T~2MxVZM 7 dt Mo
(70)
where M = g#BNS and N is the number of spins per unit volume. The exchange
154
P . E . WIGEN
constant d is related to the exchange integral by the relation d
= zJS2/a
(71)
The first term on the right-hand side ofeqn. (70) represents the torques acting on the magnetization M placed in a magnetic field H eft while the second term can be thought of as the torque due to an exchange field H cx = - ( 2 s 4 / M o 2 ) V 2 M . The torque due to H off can be treated in terms of the free energy density of the system by the relation 44 z = ~, x ( - VE)
(72)
In this relation the effects of the applied field, magnetostatic energy and the anisotropy energy can then be handled as scalar contributions to the energy density rather than as vector fields. In addition, the ellipticity of the normal modes can be readily evaluated. The equation of motion can be solved by using the coordinate system shown in Fig. 8 in which the equilibrium direction of the magnetization is the ~, direction and small deviations from equilibrium are in the e0 and ~¢ directions. In this coordinate system the magnetization has the form M = Mo~, + moeo + m4,e¢
(73)
where mo = M 80, m~b = M sin 0 8q5 and 60 and 6~b are small deviations from the equilibrium direction. The unit vectors i,, e0 and i¢ are the standard unit vectors of a
Fig. 8. The coordinate system used for the evaluation of the ferromagnetic resonance condition in the continuum model.
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
155
spherical coordinate system. On substituting eqn. (72) into eqn. (70), the equation of motion has the form
1 d M = M VE+2T_~dzMxV2M
(74) dt Moo× Mo A phenomenological damping term can be easily introduced on the right-hand side of eqn. (74) to account for the line shapes in the spectra. These may have one of the following forms4S: the Landau-Lifshitz form i Mo2MX (M x H "ff) (75a) the Gilbert form Gt
--M Mo
xM
(75b)
the Bloch-Bloembergen form
Mx,y Mz-Mo ~2
(75c)
"el
In the remainder of this treatise, the damping terms will not be considered further. In the equilibrium orientation, the magnetization will not have a variation in either time or space and eqn. (74) reduces to the form M - - x VE = E0/~-- E~ cosec 0 e0 = 0 (76) Mo
where the notation Eo = dE~dOand E¢ = cOE/dc~is used. Equation (76) is just a statement that zero net torque acts on the magnetization in the equilibrium orient-'ion. The equation of motion is linearized by expanding eqn. (74) about the equilil+ium orientation and retaining only terms to first order in the deviation from equilit _Jum:
y \dr ]
=
(--Moo
VE + ~2-~2 ~(M x
t MO
(77)
V2M)
where
M
(-~o X VE) = M-~o~Mx VE + ~-~ x ~(VE)
(78)
From eqn. (72), the first term on the right-hand side is zero and the second term takes the form M cosec 0 - - × fi(VE) -~o (Eoemo+E~¢cosec 0 m¢,)~.o+ Mo
1
+-~oo(Eoomo+ Eo#cosec 0 m~)~¢
(79)
The equation of motion in linearized form reduces to the set of coupled equations 1 d
2~ 2 cosec 0 (Eocmo+Ecc,cosecOmg~)___~ooV me
and
(80) Id
I
2~/
2
~m¢ = ---;T(Eoomo, v,o + Eo¢cosec 0 me) +Moo V mo
156
P . E . WIGEN
For a magnetic medium in the region y t> 0, the solutions to eqns. (80) will have the form mo = {ctl sin(k 1Y) + fll cos(k xY) + 71 exp( - kzy)} exp( - loot)
and
(81) ms = {ct2 sin(klY) + f12 cos(kly) + 72 e x p ( - kEy)} exp(--itot)
where k 2 is the exponentially decaying surface mode required to satisfy Maxwell's equations at the y = 0 surface. Substituting eqns. (81) into eqns. (80) and setting cosec 0 to unity, the following amplitude relations are determined:
52
fiE
cq
fll
io,)/~ + Eo4,/M o Ee~b/Mo - Dkl 2 Eoo/Mo + Dka 2 = iog/7_Eoe/Mo
(82a)
and Y2 Yl
ko/y + Eo4o/Mo E¢¢JMo--Dk2 2
=
Eoo/M o - D k 2 2 ko/7 - Eo~/M o
(82b)
The dispersion relation is given by ( ~ ) Z = ( E o o +Okl.2)(E~b ¢ ++_Dkl,22)- EO¢~2
(83)
When these terms are evaluated for Y I G in which D = 2 ~ / M o = 0.5 x l0 - s 2, the value o f k 2 is 1.8 X 10 6 c m - 1 at 25 G H z or 1.1 x l06 cm 1 at 10 GHz. This corresponds to decay lengths of 5.5 nm and 9.0 nm respectively. In garnet films of thickness of the order of 1 lam, the fraction of energy absorbed by the mode associated with k 2 is negligible and the use of Maxwell's equations to consider the continuity at the boundary is unnecessary. The allowed values of k~ will be determined by the film thickness and the boundary conditions and correspond to the surface modes or bulk modes evaluated in Section 3. For a thin film having a uniaxial anisotropy constant K u with an easy axis normal to the film plane and the magnetization oriented normally to the film plane where 0o = ~bo = re/2, the derivative terms have the form Oecm
Eoo = E~O = M ( H o + Hu)
(84)
where H o is the applied external field and H. = 2 K u / M o is the effective anisotropy field. The dispersion relation from eqn. (83) then yields (.o
-- = Ho + H u - Dkl 2
(85)
and the ellipticity from eqns. (82) gives = i giving a circular precession as expected.
(86)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
157
When the magnetization is oriented in the plane of the film where 0o = n/2 and 4~o = 0,
Eoo = HoMo
E ~ = Mo(no - n,)
(87)
In this geometry the dispersion relation yields
( ~ ) 2 = {(Ho + Dk,2)(Ho- Hu + Dk12)}l/2
(88)
and the ellipticity gives
~X2
f12
~)2
=
=
: ( H o - - H u ~ ll2 't, 7
(89)
4.2. Boundary conditions The allowed values of kl will be determined by the boundary conditions established by the surface anisotropy energy density Es(O,@)which can be expressed in terms of the coordinates of M. The surface torque T~ is then obtained by the relation M T' = - - - x VE~
(90)
Mo
This term is balanced by the exchange torque at the surface given by the second term in the Taylor expansion of eqn. (65): T~
2~¢
aM
(91)
Mo~MX O---j
=
The gradient in the surface energy in eqn. (90) is
VE" = E'o~o+ E~ cosec 0 ~
(92)
The surface spins are assumed to lie parallel to the bulk spins to give a parallel ground state. Consequently only the higher order terms, arising from torques acting on the dynamic terms of the magnetization, will contribute to the boundary conditions. The first of these terms has the form (93) After proper manipulation, this term can be expressed as 1 s 8T'=--~o(EO~COsecOmo-E'~cotOcosecOmo+E'
s
s
if4o(E oomo + E o~ cosec Om~)#
(94)
This surface torque will be balanced by the surface exchange torque in eqn. (91) which has the form =
Mo\ oy
aY 7
(95)
158
P . E . W1GEN
Equating the terms in eqns. (95) and (91), the boundary conditions at the film surface are expressed in terms of the matrix equation
E
cosecobI l
Om~[O = _ 0y
(96)
[_(gs0,~ cosec 0 - es~ cot 0 cosec 0) 8s,¢,,~cosec26
d
where gu = Eu/2Jz¢" Choosing the usual geometry with 0o = rt/2, eqn. (96) becomes :mY4~t
= --
so0 ~s00
(97)
The solutions to the equation of motion in eqns. (80) can also be expressed as m o = { G cos(kly + 6) + 71 e x p ( - k2y)} e x p ( - i ~ t )
(98)
and similarly for toO. Here G -- (ct12+ fl 2)1/2 and tan 6
-
c~
fl~
(99)
The angle ~ is the spin wave phase shift observed at the boundary. Using the boundary conditions ofeqn. (97), Stakelon 46 has shown that the relations which the spin wave number must satisfy are kl tan 6 = k2K+ + ~rs°°~s~4~
(100)
kE+K_
where K+ = ½{o~s00(1___0") + ~s4~4~(1 +0")} O" = (1 __~,~+~r~-)- 1/2
and [2 + =
ioJ/]; + Ebo4aM o (Ebo0 -- Ebo~)/2Mo
In the last expression, the E b terms are those obtained previously from the energy in the interior of the film. 4.2.1. Unidirectional surface anisotropy
Let us consider the case of an effective surface energy density of the form E s = - Mo"/-Pu'f = Mo Hsu'f sin 0 cos(4~-th)
(101)
where q~ is the direction of the unidirectional anisotropy energy. This is equivalent to the unidirectional field proposed by Puszkarski in the microscopic model (see eqn. (54)) From this expression f0-r the surface anisotropy ESoo = ESe~¢ = HsUrfMo sin 0 cos(q~s-- ~)
and
(102) E~o4~ = 0
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
159
In addition a = 0. This is due to the absence of higher order anisotropy energy contributions in the bulk field terms. Then K± - H~UaM° 2,~t sin 0 cos(~bs- ~b)
(103)
and the dispersion relation gives
H,UrfMo
kl tan 6 = ~
sin 0. cos(q~,- 4~s)
(104)
For this example, there is no coupling between kx and k2 and the results are in exact agreement with those of Puszkarski for a unidirectional field.
4.2.2. Uniaxial surface anisotropy The surface anisotropy energy having the next higher order term in the spherical harmonics is a uniaxial term of the form E ~ = K s sin20 sin2q~
(105)
At 0 = n/2, the derivative terms become
ESoo = _ 2 K s sin2~ ES4,~ = _ 2KS(sin2~b - cos24~) and
ESo~ = 0 From these terms, K± -
(3 cos2fl- 1 + ~ sin2fl)
(106)
where fl = ~ / 2 - 4~ is the angle between the magnetization and the film normal. In eqn. (100) the terms in k2 will dominate the right-hand side and the boundary conditions reduce to k~ tan 6 ~ K± K s
~ - 2 ~ ( 3 cos2fl - 1 + a sin2fl)
(107)
A surface anisotropy of uniaxial symmetry has been identified in YIG films 27'28. Using the procedures outlined above, the surface torques can be evaluated and the phase shift at the surface as well as the spin wave spectrum can be calculated. For small values of K s, the dimensionless quantity k I tan 6/(KS/2d) determines the standing spin wave spectrum. In Figs. 9, 10 and 11 this relation is shown as a function of the orientation of Mo at 3 GHz, 9 G H z and 23 G H z respectively. The bulk energy density for these figures is assumed to have the form E b = 2nMo 2 sin20 sin2~b -- Moll 0 sin 0 cos(~b - ~bn)
(108)
where Ho is the applied external field having the coordinate values On = n/2 and 4~nThe three curves correspond to the following conditions. Curves a are the angle dependence predicted in Section 3 which was considered in the circular precession approximation where a = 0. Curves b are the angle dependence predicted from eqn. (107) with the elliptical term included. Curves c include the effects of the non-parallel
160
P.E. WIGEN
-2 Fig. 9. The angle dependence of the surface parameter for YIG obtained for a uniaxial surface anisotropy energy (magnitude of the surface anisotropy, 0.03 erg cm- 2; frequency, 3 GHz): curve a, prediction in the circular precession approximation; curve b, prediction in the elliptical precession approximation; curve c, prediction in the non-parallel ground state configuration. (From ref. 46.)
-2 Fig. 10. Same as Fig. 9 except for a frequency of 9 GHz.
~ c I
I
I
Fig. 11. Same as Fig. 9 except for a frequency of 23 GHz.
i
161
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
ground state of the magnetization which will be discussed in Section 5. It should be noted that the latter terms have a significant contribution except at high frequencies where to/y >>4 x M . A further evaluation of the dispersion relation in eqn. (83) requires an explicit form for the energy density in the magnetic material. The energy density of a magnetic material consists of Zeeman, magnetostatic or demagnetization, uniaxial anisotropy and crystalline anisotropy energies. The Zeeman energy density is given by Ez = - M ' H o = M o l l o sin 0 cos(~b - ~bu)
(109)
where H0 is the magnetic field applied in the x y plane for the coordinate system illustrated in Fig. 8. The magnetostatic energy is given by E m = 2riM. N" M
(110)
where N is the demagnetization tensor. For thin film geometry the magnetostatic energy can be expressed as E m = 2nMyz = 2riM02 sinZ0 sin2~
(111)
The uniaxial anisotropy energy having an easy direction normal to the film will be written as: E u = -- K u sin20 sin2~
(112)
Since the magnetostatic energy and the uniaxial anisotropy energy have the same angular form, these terms cannot be distinguished in the FMR. Thus the uniaxial term will be eliminated and the magnetostatic energy 2gM 2 will be understood to mean an effective uniaxial energy 27tM 2 - K u. The crystalline anisotropy energy can be written for a cubic structure as Ec=
+
113)
where Kx and K2 are the first and second order anisotropy energy constants, and i,j and k refer to the principal crystal axis directions [100], [010] and [001] respectively. In a majority of the cases involving magnetic garnet films, the second-order anisotropy energy constant is negligible and will not be included here. Additional references treating the K2 term are available in the literature 47'.8. In terms of the coordinate system in Fig. 8, the expression for a (111)-oriented film becomes M \4 =
-3
Y-4"'"-2
~'"Y-
fsin40 cos4~b sin't0 sin4~ cos40 ,~ ~ t-~+ K~[ ~ 3 x//-2 san't0 sin t~ c o s 3 ~ b _ 4 +-3-
3
~
y x~- ~ y , }
sin2(20) cos2~b Jr 8
sin2(20) sin(2~)}
(114)
162
P.E. WIGEN
The total energy density is then the sum of the four terms contained in eqns. (109), (111), (112) and (114). The equilibrium orientation of M is found by setting both Eo and cosec 0 E 4, equal to zero. Setting Eo = 0 implies that the equilibrium value of 0 is n/2. Setting E~ = 0 with 0 = n/2 yields the relation MoHo sin(~b- q~n)+ 2nMo 2 sin(2~b) +
+ K 1[~2 sin(2q~) { 1 -- 7 cos(2~b)} +~ cos2(b {2 cos(2~b)- 1}] = 0
(115)
There is in general no analytic solution to this equation and it will be retained in this form together with the dispersion relation. Evaluation of the second derivatives of the energy allows the dispersion relation to be solved explicitly. For the energy terms considered, Eoe~ = 0. Making the transformations ct ~ n/2 - (9n and fl ~ n/2 - ok, ct and fl become the angles between H a n d M respectively and the film normal. For a film with the [111] crystallographic orientation normal to the film plane and M a n d Hchosen to lie in a (110) plane, the derivatives yield the dispersion relation ( ~ ) Z = {Ho cos(o~-fl)-4nMo cos(2fl)+gl(fl)+ DkZ} ×
× {Ho cos(or - f l ) - 4nM o COS2fl -]- g2(fl) + Dk2 }
(t 16)
In addition, the equilibrium orientation of M is given by H o sin(or- fl) + 2 n M o sin(2fl) + ga(fl) = 0
(117)
where g~(fl) = HK(~ sin 2(2fl)--~cos 2 4 f l _ 2xf2 _ sin(2fl)(1 + ~2 sin 2fl) } g2(fl) ----- - - H K [ l c o s ( 2 f l ) + 7 c o s ( 4 f l ) + ~ - { s i n ( 2 f l )
g3(fl) =
- 2 sin(4fl)}]
(118)
H K [ 1 sin(2fl){ 1 + 7 c o s ( 2 f l ) } - - ~ sinZfl{ 1 + 2 cos(2fl)}]
and HK = 2K 1/M is the effective cubic anisotropy field. Setting k = 0 in eqn. (116) determines the magnetic field value for which the uniform mode resonance will occur at the frequency ~o: 1
H, cos(~t - fl) = 2riMo{cos2fl + cos(2fl)} - ~ {gl(fl) + gz(fl)} + f w \ 2) 1 /2 1( 2 (119) + ~ ~[4nMo{cos f l - cos(2fl)} + g2(fl) - - g l ( f l ) ] 2 + 4 ~ -) ~
The position of the nth spin wave mode with k = k, is then found from eqn. (116): Dk, z H, = H. (120)
cos(~-/~)
If the magnetic field is applied along the film normal (~ = fl = O) the expression
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
163
simplifies to the form H~± = Hux - Dk~ 2 09 4 M 2 = 7 + It eff+-~HK-Dkn 2
(121)
The dispersion relation in eqn. (116) and the equilibrium condition in eqn. (113) can be used to evaluate the film parameters D, 4 ~ M eff, H K and ~. The surface anisotropy energy at each surface can be evaluated from the angle dependence of the spin wave spectra. 5. NON-PARALLEL GROUND STATE
To this point, the description of the spin wave excitations has assumed that the ground state of the system is an array of parallel spins whose direction is determined by the equilibrium of the "bulk" torques. Several researchers 49-51 have suggested that surface torques derived from the surface anisotropy energy will give rise to a non-parallel spin ground state which will influence the spin wave excitation. Mills 36 has predicted changes in the magnetic surface ordering due to instabilities resulting from different bulk and surface exchange interactions. In this section the work of Stakelon46 is reviewed by extending the continuum model of Section 4 to calculate the effects of the surface anisotropy on the reorientation of the magnetization near the film surface and thereby to determine the non-parallel ground state and its contribution to the "dynamic" boundary conditions. For conditions in which 09/4n~M is less than unity, the dynamic effects of the non-parallel interior spins will be as important as the surface anisotropy energies acting directly at the surface plane. Recently Harada et al.52 have calculated the ground state configuration in the presence of a uniaxial surface anisotropy energy using a microscopic theory. The results they obtained are also consistent with the results of the continuum model reviewed here. Using the coordinate system described in Fig. 8, the magnetization M at any point will be given by a time-independent component M 0 which describes the orientation of the spins in the ground state plus a smaller transverse component mt which is time dependent and has a spatial variation normal to the film surface. The time dependence of the magnetization can then be written in the form M(y, t) = Mo~,(y) + mo(y) exp( - itot)~0(y)+ m~(y) exp( - kot)~(y)
(122)
Here the unit vector i,(y) is changing as a function of y in such a fashion that it is always parallel to the equilibrium direction of the magnetization Mo; the unit vectors e,o(Y) and i~(y) are thus modified to maintain a mutually orthogonal coordinate system. The surface torques /'So are obtained from the surface anisotropy energy density Es(O, dp) as given by eqn. (90). The surface torque is balanced by the exchange torque given by eqn. (91). For the non-parallel ground state, the static term in the surface torque will produce a bending of the spins near the surface which will be balanced by the exchange torque. In this case, the surface torque will be expanded in terms of the torque /'So acting on the static term of the magnetization M o and the
164
P.E. WIGEN
higher order terms 6T ~arising from the torques acting on the dynamic terms of the magnetization: TS= T~o +~STs
(123)
The static boundary conditions are given by (6~-Y~)= = o~b°'
--°~s~ c°sec20°
and
(124) (8~Y)= 0 ° ' = - ~ s ° o
where ~s
= ES / 2 ~
The angles q~o and 0 o are the equilibrium positions of the spins in the bulk of the magnetic medium while ~bo' and 0 o' are the angle derivatives evaluated at the surface. The behavior of the spins in the bulk region will be determined by the balance of the bulk torque terms with the exchange torque. Evaluating the exchange torques and equating them to the bulk torques gives the equations W - s i n 0cos O(c~') 2 = o~b0
(125a)
~b"+ 2 cot 0 0'~b' = ~ b
(125b)
and
which can be solved in conjunction with the boundary conditions in eqns. (124) to evaluate the ground state configuration of the spins. To evaluate the ground state configuration, the equations are linearized by letting O(y) = 0 o + ~O(y)
(126a)
and ~b(y) = ~bo+ 6q~(y)
126b)
where 60(y) and 6~b(y) are the deviations of the spins from the bulk equilibrium orientations 00 and ~bo respectively. If E s is zero, the variations in the orientation of the magnetization would be zero and the equilibrium orientation of the magnetization (0 o and ~o) will be given by the equilibrium condition EbO0
=
Eb°~ = 0
(127)
For a non-zero surface energy density and considering the usual condition 0 o = rc/2, eqns. (125) have the form ~50" = 8b°oo ~50+ ~b°o4, ~5q5
(128a)
6q~" = d~b°o,~ 60 + 8b°4,~ 8~b
(128b)
and
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
165
The boundary conditions from eqns. (124) become 8~,'(0)
=
- a's~
(129a)
80'(0) = - - ~ 0
(129b)
and
To solve these equations, a surface energy is considered that does not produce a torque out of the yz plane, i.e. ESo = 0, and a bulk energy term that is uniaxial such that Eb°o~ = 0. Then eqns. (128) and (129) reduce to 80' = 0
(130a)
8~b" = 8b0~4, 8~b
(130b)
and
The solution to eqns. (130) is 80 = 0
(131a)
84) = A 4, exp(Ay)
(131b)
and
where A 2 = ~bO . From the boundary condition in eqn. (129b)
8dp'(O) = AA4, = -- gs 4,
(132a)
which yields gs~ -
At~ = - - A
gs~
(132b)
(~b°~b~b)l/2
For YIG at a frequency of 25 GHz, a value of E ~ ~ E ~ of about 2 x 10 - 2 erg cm -2 (ref. 28) gives A = 1.3 x 106 cm-1 or a decay length of 7.7 nm. The magnitude A~ of the maximum deviation of the magnetization at the surface is about 0.02 rad or approximately 1°. The bending of the ground state magnetization near the film surface will produce a perturbation on the energy density in the surface region causing an additional torque on the magnetization that is appreciably within a region of a few decay lengths A - 1 from the surface. Beyond this region, the spin wave will take the form A cos(kly + 6 + ~') similar to that in eqn. (98) where 6' is the additional phase shift introduced by the perturbation in the surface regions. For small values ofkx, mo and m~ are not changed significantly by the perturbation and the additional torques can be included as an effective surface torque by integrating the volume terms over the effective surface region. The normal modes in the non-parallel ground state system require a solution of the equation of motion by considering the higher order terms of the surface torques 8/~ acting on the dynamic terms of the magnetization. Due to bending of the ground state magnetization near the surface, the variation in E b ~ and Ebo0 of eqns. (80) will produce the additional terms to the equations of motion 8 T O = EbOq~oO
Mo
8tfp mo -t- ET~_O° 8 0 /14o
m~
(133a)
166
P.E. WIGEN
and
8T~ =
Ebo0~ boo0080 2d 2 Mo 8q5 mo+ E.Mo mO+~o(SC~) mo
_
(133b)
From eqns. (131), the relations 80 = 0 and 8~b = -ES~,(Eb~¢) 1/2 exp(Ay) can be substituted into eqns. (133) and the resulting relations integrated to evaluate the spin wave phase change from the bulk region to the surface. The resulting phase change can then be treated as an effective surface torque in the equations of motion to give
6TSo
Es
Eb~bq~tb"~ 6m6
-
Eb~o
(134a)
Mo
and
8TSc~ - Eb°°~ ":" ~ moqmo Ebe~,~ Mo 4.4AM o
(134b)
The combination of these terms with the surface torques due to the boundary conditions given in eqn. (96) determines the final form of the surface torques: 1 /
E bO
\
- ( E ~~¢ +~Eb°¢+ ¢ ~ E s~m TS° = -Mo( ~} ,
(135a)
and 1
= Moo
Es
-Eb°oo#'E
~
°O"r E - - ~
~Es4,)X2/ 2~2iI \1/2")
~
mo
The magnitude of ES~ is comparable with that of ES~ although it may differ in the functional dependence on ~b. Similarly Eb°~/Eb¢~¢, is nearly unity in many situations with the result that the additional terms may contribute significantly to the spin wave spectrum. For the conditions described in Section 4 and Figs. 9-11, curves c represent the angle dependence of the pinning parameter including the effects of the inhomogeneous ground state. 6. VOLUME INHOMOGENEITY MODEL
In a thin film, the exchange interaction attempts to make V,M = 0 at the surface. If the internal field is everywhere uniform, a spin at the surface and one directly in from the surface must have the same value ofm since all spins will precess at the same frequency in a normal mode. A non-zero slope ofm would give an extra torque on the surface spin because of the missing neighbors. In addition to surface anisotropy energy, a change in the surface magnetization or an inhomogeneity in the anisotropy energy in the surface region of the film will also produce modes which are not unpinned. In the volume inhomogeneity model, the surface region is assumed to have a magnetization which drops off from its bulk value to some lower value in a thickness e which is much smaller than the film thickness as shown in Fig. 12. This model has been considered in detail by Sparks 53. For e # 0, the spins in the surface region are "off resonance" when the spins in the bulk region are "on resonance". As the surface spins are exchange coupled to the
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
167
bulk spins a phase change 6 in the spin wave will occur in the surface region. If the surface region is approximated as dropping suddenly to Mo/2 at a distance e/2, a phase change of n/2 will occur in a critical thickness ecr of the order of (nsl/Mo24) l/2 where ~¢ is the exchange constant. For YIG, ec, is about 55 nm in this model. If the surface region is much less than ec~,the surface spins will remain unpinned. Ife <~ ecr is not satisfied, the surface exchange cannot hold V,M = 0 at the surface and the low order spin wave modes will be partially pinned. Surfoce Region
Bulk Region
Surfoce Region Approximolion for Surfoce Region
-½~
i Z~
-vS
J
o
L.
,~
-4"
Fig. 12. Variation in the saturation magnetization across the thickness of the film. (From ref. 53.)
The case of a surface layer of different M is useful in establishing general features of the spin wave spectrum. The equation of motion for the magnetization in the circular precession approximation is given as 2~¢m"
--+/¢2m
Mo
(136)
= 0
where m" = d2m/~2yand r is the root of the secular equation of eqns. (80) when a variation in the magnetization is included:
(/co'X2 ) I/2 t¢2 : ~ - ~ ) +(27zMosin20)2~ - H d f
2~¢My" Mo 2
2nMosin20
(137)
For perpendicular resonance sin20 = 0 and x 2 reduces to t ° - H=ff 2~¢Mi' = 7 Mo 2
(138)
For parallel resonance, sin20 = 1 and f/'~\2
) 1/2
2~¢Mr,,
(139)
2
The thinner the boundary region the larger will be the M," term in eqn. (137). For sufficiently thin surface layers, this term will dominate the other terms in eqn. (137) and eqn. (136) will reduce to
Mom"= raM,"
(140)
Integrating once with m = Mo = 0 at the film surface gives
Morn' = raM;
(141)
Since these equations hold everywhere in the surface region, this expression will also
168
P.E. WIGEN
be valid at the edge of the bulk region. For a homogeneous magnetization, m' = 0 at the boundary of the bulk region and the spin wave is unpinned. Equation (138), with co/7 = HeffBq-Dk, 2, where HeffB is the effective field in the bulk region of the film and Dk, 2 is the exchange contribution to the frequency for spin wave modes, becomes K ± 2 ~-
Dk,, 2 +
neff B -- Hefts
2~'M " -'-Y
(142)
Mo 2
where H~efs is the effective field in the surface region of the film. The maximum value of Heffa-Heffs in the surface region is 4nMo. For Dk,,Z~ 4rtM, the term 2 d M f / M o 2, of the order of 4Die 2 in the surface region, is larger than the other terms on the right-hand side of eqn. (142) when 2~4 e2 ,~ - Mo2rt
for Dk. 2 ~ 4nMo
(143)
For Dk~ 2 >~4~M, the corresponding result is 2, e2 , ~ 7I
for Dkn 2 ~>47tMo
(144)
where k, = 2rt/2,. An expression valid in both limits is
2 d (1 ~2 ,~ ~cr 2 - -
Dk"2 \
Mo2,/l~\ + 4 ~ o o )
1 (145)
Ramer and Wilts 54'55 have used this model to analyze spin wave mode positions and intensities in thin YIG films. Their analysis extends the model to an asymmetric film having two surface layers and considers the ellipticity, angle dependence, anisotropy fields and g values in the bulk region and surface region of the film. The surface layer at the substrate interface is assumed to be due to a diffusion zone between the G G G substrate and the YIG film which has a composition Y3-rGdrFe5 xGaxO12 where x and y depend on growth conditions and annealing treatment. In attempting to duplicate the experimental spectrum, the surface layer thickness, magnetization and, in the substrate interface layer, the Mcd to Mve ratios were varied to give the best fit to the mode locations at all angles. Typical results for room temperature data are shown in Fig. 13 and the parameters used are shown in Table II. Experiments of spin wave resonance in garnet films produced for bubble devices show large deviations from the quadratic law 56"57. Having the composition Y2.ssLao.15Fe3.asGal.25012 and grown from a non-stirred melt, these materials have a growth-induced anisotropy and a stress-induced anisotropy which are very sensitive to the composition and growth conditions. The spin wave resonance data were used to identify various strata produced in the film during the growth process. The volume inhomogeneities were identified by observing the mode intensities while the film was etched to smaller and smaller thicknesses. Another set of experiments involving the volume inhomogeneity model involve resonance in ion-implanted films. In this case e >> ecr in Fig. 12 and the spectra from these "layered" films will exhibit resonances from each of the layers independently 5a. The shifts in the resonance field and their angular dependences will allow the
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
169
200 o
a++ +
e# +
3 D D ~a+8 #
D+ a
MAGNET ANGLE (deg)
90
Fig. 13. The angle dependence of the magnetic field separation of the observed and calculated positions H. of the spin wave modes from the calculated position H, of the uniform mode: +, calculated; Fq, experimental points for a film having a single surface mode at the parallel orientation. (From ref. 54.) TABLE II ANALYSISOF ThE DATASHOWNIN FIG. 13 (FROMREF. 54) Orientation Total thickness (ttm) Annealing temperature (°C) Annealing time (h) Frequency (GHz) A (erg cm- ~) T (~) 4nMb (Oe) ~b t s - I O e -
1)
L, (A) Mb/Ms
[100] 0.47 1000 6 9.16 3.593 x 10- ~ 4200 1735 1.767 x 10- t o
7b/•eff
4~MF¢(Oe) 4'/tMGd(Oe)
470 1.95 0.9977 1400 510
T (~) (K,) 1(erg cm -2) (K,)2 (erg cm- 2)
4270 0.1085 0
L, (/~) Mb/M" 7b/7,
e v a l u a t i o n of the p a r a m e t e r s in each layer. T h u s F M R has b e c o m e a viable tool for the e v a l u a t i o n of the effects of ion i m p l a n t a t i o n o n m a g n e t i c g a r n e t films 59-62. A typical e x a m p l e of this a p p l i c a t i o n is s h o w n in Figs. 14 a n d 15 from A l g r a e t al. 62 In Fig. 14, a new set of resonances is o b s e r v e d in a Y2.ssLao.l 5Gal.15Fe3.ssO12 film i m p l a n t e d with 10 ~4 N e + ions c m - 2 of 300 k e V energy. By careful e v a l u a t i o n of the p a r a m e t e r s , A l g r a e t al. were able to d e t e r m i n e the profile of the internal field as well as the e v a l u a t i o n of the d i s p e r s i o n c o n s t a n t in b o t h the b u l k region a n d the ion i m p l a n t e d region. T h e spin wave d i s p e r s i o n c o n s t a n t 2 ~ / / M was e s t i m a t e d to
170
P.E. WIGEN 4800
~
x
0 - Q--O-O'Ol~l~
/
4600
~
4400 0 B
4200
!
4000 bulk
"~
IOx I I I I 3400 3600 5800 4000 4200 4400 4600
ion-implantedregion
3800
0
H (Oe)~
200 210 220 230 240 2.50 L (ffm} ---->
Fig. 14. Spin wave resonance spectra of a Y2 a~La0.,~Gal.,sFe3.esO12 film recorded with the applied magnetic field perpendicular to the film: spectrum A, before implantation; spectrum B, after implantation with 10 ~4 Ne ÷ ions cm - 2 at 300 keV. The breaks in the spectra indicate a change in the instrumental gain. (From ref. 62.) Fig. 15. The variation in H eu with the film thickness. (From ref. 62.)
change from 1.58 x 10- s Oe in the implanted layer.
cm 2
in the bulk region of the film to 0.45 x 10 8 0 e
cm 2
7. MAGNETOSTATIC WAVES The theory of magnetostatic modes was originally given by White and Solt 63 and Walker 64'65 for spheroidal specimens. Since then there has been a considerable amount of experimental work on the study of magnetostatic modes in ellipsoidal specimens. The formal theory that was given by Walker for the magnetostatic modes of spheres was adapted by D E 29 to magnetically ordered films of infinite extent in the directions normal to the thickness of the film. Sparks 53 carried out some calculations for thin films using an alternative to the procedure of DE. Experiments on microwave absorption in thin Y I G films were performed by Tittmann 66'67, Storey et al. 68 and Borghese et al. 69 and in layered compounds by Reimann e t al. 7° The original version of the D E theory treated the magnetic materials as isotropic, which is often not the case. The inclusion of anisotropy in DE theory is straightforward and was done by Akhiezer et al. 71 and Schneider 72. Magnetostatic modes are magnetic excitations in a magnetically ordered system. They hax, e wavelengths 2 that are of the same order of magnitude as the linear dimensions of the specimen under consideration. For garnet films having a thickness greater than 5 lam the effects due to the exchange interaction are negligible. The magnetostatic modes can be regarded as normal modes of "vibration" of the magnetization vector M(r, t), which is a function of position r within the material at time t and can be described by the magnetostatic Maxwell equations VxH = 0
(146)
V.H = -4rcV.M
(147)
and
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
171
plus the constitutive torque equation: 1 aM T dt : g × H e f f
= M x { V E - h exp(-iogt)}
(148)
where the exchange contribution in eqn. (69) is neglected. The effective field in eqn. (148) has the form H etf =
H o - H u - 4nN'Mo + h exp(-iogt)
(149)
where H o is the applied field, Hu is the uniaxial anisotropy field, N is the demagnetization tensor and h is the transverse r.f. field. For a thin film lying in the x z plane (see Fig. 8) the only non-negligible demagnetization term is Nyy = 1. Magnetic garnet films grown epitaxially on a G G G substrate have a stress-induced anisotropy and, in the case of rare-earth-doped garnets, a growth-induced anisotropy with a symmetry axis normal to the plane. Having an energy expression of the form given in eqn. (112), the torque due to the anisotropy energy, defined by eqn. (72), produces an effective field H , = - V E , / M o along the y direction. In the small signal approximation, the magnetization can be written as M = Moil + m
e x p ( - iogt)
where fi is a unit vector in the direction of the effective field Heft and m is the small r.f. component that lies in the plane perpendicular to ti. In this approximation, eqn. (148) relates m to h and defines the dynamic susceptibility tensor ~ such that m = ~.h
(150)
The dynamic permeability tensor IXis related to Z by IX = 1 + 4 ~
(151)
where Ix depends on the direction of the applied field H o. While the normal modes can in principle be evaluated at an arbitrary orientation, the solutions become very complex. As a consequence, the magnetostatic modes are normally evaluated at two geometries. Firstly, for the magnetization vector lying in the plane of the film (Mo//~) and, secondly, for the magnetization vector being normal to the plane (Moll5,). Using the usual procedures for determining the equation of motion by neglecting high order terms and the losses, the H o//~ or parallel case of eqn. (148) reduces to iO~mz = - ~{(Ho - H u)my- Mohy }
(152)
i~omy = y ( - Horn z + M o h z )
(153)
and From eqns. (152) and (153) the terms in the dynamic permeability tensor can be readily derived: #xx = 1
I~xy = #yx = #xz = #zx = 0
B
/~zz = 1 + 4 x ~ / a y y
A
= 1 +4n/iB_t22
f2 /~y = - #yz = i4n A B - - t22
(154)
172
P . E . WIGEN
where A=--
Ho
Ho-H . B - - Mo
Mo
f2-
~o 7Mo
Similarly, for the perpendicular case (H0 is perpendicular to the film surface, i.e.
HoflY) the dynamic permeability tensor has the following terms: ].,tyy --~-
1
#y~ =/a~y = #r:, = Pxy = 0
pxx= p= = 1 +4/tC2
C
~,~2
(155)
f2 /ax~ = -/a,x = i4rt C2 _ 02 where C= f2-
H o - 4 ~ M + Hu
Mo (D
~Mo
The dispersion relation can be derived from Maxwell's equations: Vxh = 0
(156)
V.(h+4nm) = V'la'h = 0
(157)
Equation (156) permits the introduction of a magnetic potential ~ such that h = V~ and eqn. (157) becomes V'wV~ = 0
(158)
For a rectangular film, the normal modes are described by the frequency 09, the inplane wavevector ki, = (kx,kz) and the out-of-plane wavenumber ky. The general solution for eqn. (158) then takes the form ~i,t = {a exp(ikyy) + b e x p ( - ikry)} exp{i(kxx + kzz)} exp( - kot)
(159)
inside the film. Substitution of eqn. (159) into eqn. (158) yields kx 2 + flyy kr 2 + llzz kz 2 = 0
(160)
for the parallel case and lazz(kx 2 + kz 2) + ky 2 = 0
(161)
for the perpendicular case. Outside the film, ~ext = C1,2 exp( + ~cry)exp{i(kxx + kzz)} e x p ( - iogt)
(162)
for y > d/2 or y < - d / 2 . Equation (158) then yields Kr 2 = kx 2 + k z 2
(163)
For a given in-plane wavevector, eqn. (160) determines ky which, depending on the permeability, can be real (volume modes) or imaginary (surface modes).
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
173
Furthermore, the boundary conditions must be satisfied. This requires that the normal components of b and the tangential components of h be continuous: for
y = + d/2 ~y~ext :
(#yx~x "+"]~yy~y +/2yzt~z)¢in,
(164)
and ~=xt = ~,n,
(165)
This systemof four linear homogeneousequationsyields the characteristicequation
{xy2 + (k,,~yx + k=~y=)2-(kr#ry) 2} sin(kyd)- 2Kykr#yy cos(kyd) = 0
(166)
The dispersion relation for the parallel case falls into two categories, volume modes and surface modes. The boundary between the two regions is determined by a critical value in the ratio k=/kx. For volume modes ky 2 > 0 and the dispersion relation is
tan(kxdct1/2 + mr) = 1 +
2btyY~t1/2(1 + 172)1/2 if2 _ I/~y, 12r/2-- ~t/~ry2
(167)
and for surface modes (ky2 < 0) tanh(kxdl ctl 1/2) =
2/~yyIotl 1/2(1 + r/2)1/2 1 -t- r/2 -- I/G,I 2~2 -t- I• I/tyy2
(168)
where r/= k=/kx, ~ =/t==//~yy(r/=2 - r/2), ~/¢2 = _/~=- 1 and kr - - kxot 1/2. Figure 16 gives the traditional magnetostatic mode spectrum of a ferromagnetic slab magnetized parallel to the surface and for which H, is set equal to zero. A series of planes, corresponding to the different n values in eqn. (167), is shown for the volume modes in the frequency range between ~Ho and ?{Ho(Ho +4rcM)} 1/2. One surface is observed to rise above the high frequency volume mode band at a critical angle determined by r/¢. The upper frequency limit of this surface is 09 = ~(Ho
+ 2nM). The inclusion of the uniaxial anisotropy field has some very unusual effects on the shape of the magnetostatic mode pattern as shown in Fig. 16. The bottom of the
oT Surface / modes 4.,_._.=o=/f ( Ho+ 2 TrMo) ~
""
o
H
,~k=
< ' ~ w :i[HO(H°+ 4 "M°)]''.
w=lfHo
J
Fig. 16. Magnetostatic mode spectrum of a ferromagnetic slab magnetized parallel to the surface. The separation between the surface and the volume modes originating at ~c should be noted.
174
P.E. WIGEN
volume mode band at the og,kx plane is shifted to the frequency to = y{Ho(Ho _ Hu)} 1/2, while the frequency at kx = ky = 0 is ~o = 7{Ho(Ho - H u + 47tin)} 1/2. The most significant effect has to do with the surface modes. Under appropriate conditions, the uniaxial anisotropy field will distort the volume band for k~ :~ 0 and some of the modes in the surface branch will become volume modes. An example of this case will be discussed later. For the perpendicular orientation, only volume modes are allowed and the dispersion relation is given by
tanI#xx'/2(kx2+ky2)l/2d+nn}-2(-Itxx)l/Zl + #xx
(169)
where n is an integer. For the major series, n = 0. In this orientation, the role of the uniaxial anisotropy field is to shift the frequency of the magnetostatic modes by the amount Am = 7H,. However, the relative positions of the mode spectra will remain unchanged. In a typical magnetostatic resonance experiment, the samples are restricted to finite dimensions lx and lz. Thus the allowed resonances can be characterized by the quantized components of kx and kz given by the relations
kx - nx~
kz - ngt It
Ix
(170)
where nx and nz are assumed to be positive integers. The intensity of the absorption due to the excitation of a mode (kx,kz) by the microwave field h will be proportional to the value of m.h integrated over the sample, where m is the magnetization associated with this mode. For the parallel case m can be written in the form {0,mr(r) exp(-i~ot), m~(r) e x p ( - iogt)} where
_ 1)o~(x,y,z)
o~(x,y,z)
and
(171)
mr = #yz
8~(x,y,z) +(l~yr_ 1)8~(x,y,z) ~z 8y
according to eqns. (152) and (153). In practice, the dimensions of a sample are rather small compared with the dimensions of the cavity used in microwave absorption measurements. Therefore h can be regarded as independent of the position r throughout the region occupied by the specimen; and it can therefore be written as (hx,hr,h,)exp(-iogt). The absorption of power by the mode (nx~/lx,n~/lz) is then given by m. h = ½ Re(re.h*)
(172)
Substituting for m and h* and recalling that
= X(x)Y(y)Z(z) = {a exp(ikyy) + b exp(ikyy)} exp{i(kxx + k:)} e x p ( - i~ot)
(173)
175
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
the integration can be performed over the volume of the sample:
=
fleA
d"2
~d12 j-a/2
Y(y)dy
stntT)
sinlt--~- )
(174)
where A and B are functions of co, Hi,M o and h. From the form of this equation some general conclusions can be drawn. (i) There is a selection rule that no absorption will occur for any mode for which either nx or n= is even. (ii) If nx and n= are both odd, the mode intensity is inversely proportional to nx (x being the direction of the applied field). A typical example of a magnetostatic mode spectrum in a 12 ttm YIG film having a negligible value o f H u is shown in Fig. 17. The series of modes (n, 1) are the
•• (7,1}
(5.11
2572 2583 (1.3) (3.5)
1
(3,1) ~
2522 2534 41.5) (3,71
Imam mode)
(3.~7) 2818
r3,19).,,etc 28/.6 2871
/
f/
/'
2"177~1 2807
(1,13)-
(115)
2572 2583 (1.3) (3.5)
2666 2680 2706 2720 2743 2757 2777 {1.7) (3,9J {1.9] (3.11) 11.11) {3.13] (1fl3)
i3.15) 2790
/'
l
,,
289/. 2917 2935 295A
/'
2835 2862 2 8 8 6 2908 2929 2949 2966 11,17} i1.19J.-.,etc
Fig. 17. The magnetostatic mode spectrum at parallel resonance for a 12.0Wn Y I G film (planar dimensions: I~ = 0.58 mm; 1~ = 0.68 mm): spectrum a, main mode with lines on the low field Side and the first few lines on the high field side; spectrum b, continuation of spectrum a at higher fields; spectrum c, continuation of spectrum b at still higher fields and increased instrumental gain. (From rd. 68.)
176
P.E. WIGEN
surface modes for which t/is greater than r/c while the modes (1, n) or (3, n) are the bulk modes for which ~/is less than r/c. The observed separation between the modes and the values calculated from the DE theory show agreement that is typically within 1% or 2% of each other. For thin YIG films (less than 1 lam) the exchange energies become comparable with or greater than the magnetostatic energies 73. A number of publications by Wolfram and DeWames 74 and by Sparks 53 have considered these magnetoexchange modes. A typical example of the magnetoexchange branch repulsion is shown in Figs. 18 and 19. It is worth pointing out that the consequence of the presence of an easy plane anisotropy energy (Ku negative) is the existence of volume mode excitation within
Ilo =._.~4 n I00 9O 80 7O
.5
1~--20
Oe
4O 50 2O I0 0
,I
t 5O8O
o
o
:!
o
o
o
o
o
o
o
o
o
o
o
o
o
o o o
I I I I I I_LI I I I I I I I 2 3 4 5 6 7 8 9 I011121514
Mugnetic Field (Oe)
n
F i g . 18. Magnetoexchange
spectrum in a 0.48 ~tm Y I G film: the high field modes represent the magnetostatic modes in the n = 1 exchange branch while the lower field mode represents the magnetostatic modes in the n = 3 exchange branch. (From ref. 73.) F i g . 19. A plot of the mode number vs. field position for the first two magnetoexchange branches shown in F i g . 18. The repulsion of the modes as the n = 1 branch approaches the n = 3 branch should be noted. (From ref. 73.) (3,1)
(5,1}
(I,7)
(7,1) {I,I) (9,1) I
(I,21)
(I,19) A
(,,z31 ~
I
I 2:320
(11,1}
,|
II
I
I
I
I 2:360
I
I
I
I 2400
I
I
I
I 2440
I
I
I
I
I
I
2480
Mognetic Field (Oe) Fig. 20. A portion of the magnetostatic mode spectrum measured in a reduced calcium Y I G film having an anisotropy field H. = - 250 Oe: the gap in the surface m o d e region with modes (1,9)-(1,17) missing is clearly evident. (From ref. 70.)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
177
the series of surface modes. If this bulk excitation is suppressed, perhaps by an inhomogeneous anisotropy profile, a field gap will appear in the surface mode spectrum. This feature is clearly illustrated in Fig. 20. Finally, it should be noted that the existence of these surface modes is a direct consequence of the DE theory and is predicted in homogeneous materials. Their nature is very different from that of the surface excitations reviewed in Sections 3 and 4 which is due to the presence of a surface anisotropy energy. 8. PROPAGATING MAGNETOSTATIC WAVE DEVICES
The use ofmagnetostatic waves in a variety of device applications has led to the development of a large technological effort in this field. Magnetostatic wave dispersion delay lines have important applications in compressive receivers and for variable delay lines for phased array beam steering. These applications require large bandwidths but each application has its unique requirement for differential delay ranges. Magnetic garnet films with their variety of material parameters and geometries offer a large potential for application to these devices. As a result, magnetostatic wave properties are at present an active area of investigation. This section is not intended to be an extensive review of the device studies but rather an introduction to the variety of techniques available to generate devices of desired properties as well as to refer the reader to some of the appropriate sources. Magnetostatic waves are intrinsically suited for use in microwave delay lines 75 because they are readily generated at high frequencies and their velocity is frequency dependent. However, as observed from eqns. (167)-(169), magnetostatic waves are intrinsically dispersive and the group velocity is a function of frequency. Therefore the rate of change of the delay with frequency can be either positive or negative depending on the nature of the magnetostatic wave being propagated. In most cases of interest the magnetic bias field H o is along a principal axis because the directions of less symmetry are characterized by large "walk-off" angles where k is not along the desired direction of the device. A typical structure is shown in Fig. 21. The geometries of the device then dictate the nature of the waves being l
MJ:~tr~
-
~
Sul3strme
Fig. 21. Magnetostatic wave device geometry showing microstrip transducers and magnetic bias field directions: x direction, surface magnetostatic wave; y direction, backward volume magnetostatic wave; z direction, forward volume magnetostatic wave.
178
P.E.
WIGEN
investigated. These include surface waves, with H 0 and M o parallel to ~ and k parallel to g; forward volume waves, with H o and M 0 parallel to ~ and k isotropic in the xz plane; and backward volume waves, with H o and M o parallel to .~ and k parallel to .~. In garnet films used for magnetostatic wave devices, the magnitude of the uniaxial anisotropy energy is suppressed. As a result, K , is assumed to be zero in this section. Then the frequency ranges of these waves are as follows: for surface waves,
(Ho2+Ho4xMo) 1/2 <
(2 < H o + 2 7 t M o
(175)
for volume waves,
Ho < f2 < (Ho2 + Ho47tMo) 1/2
(176)
Thus the volume waves lie within the spin wave band while surface waves lie above it. The dispersion relations for the travelling waves associated with the three cases discussed above are shown in Fig. 22 for a Y I G film with an applied field H 0 of 712 Oe. The group velocity Vg = t3o.)/t3kis shown in Fig. 23. The group delay is given by 1/vg and is inversely proportional to the film thickness through the relation kd which is a function of frequency. i
6 i
I
=N >,
.
E
I
" . loo - FVW ~-
/
J Volume ~. _ w _ ~ , _
f
-2000
[
>
1000 Wave Number, cm-1
BVW
o
I
-lo00
50
2000
3
4
Frequency, GHz
Fig. 22. The magnetostatic mode dispersion relation for waves travelling in the positive y direction as defined in Fig. 21 : the parameters are those for YIG and a bias field of 0.4 x 4riM. (From ref. 75.) Fig. 23. The dependence of the group velocity on the frequency: the conditions are the same as those in Fig. 22. (From ref. 75.)
As observed in Fig. 23, the group velocity is non-linear in frequency. For device application, this non-linearity must be modified if the device is to be a non-dispersive delay line. For a fixed distance between input and outputteads, the group delay can be modified by means of a ground plane 76-79, by multiple layers a°-a7 or by using reflective arrays to change the frequency dependence of the path lengthsS8 93. When a ground plane is present, the dispersion relation will be significantly modified as the surface waves are not reciprocal. In addition its utility is limited by the ohmic losses in the metal plane 76 which are also frequency dependent. In general, the attenuation due to the ground plane is greatest in the regions where the ground plane is most likely to be useful and so one of the major limitations is the t i m e bandwidth product. A typical example of a ground plane group delay line operating at X-band
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
179
frequencies using forward volume waves is shown in Fig. 24. A YIG film 20 pm thick, spaced 20 ~tm from a silver ground plane and operating in a bias field Ho of 4800 Oe shows a 190 ns differential delay over a 1.2 GHz bandwidth 75.
250
S* 200
~" o
150
o
tO0
50
t
0
;
8,6
,i,
,18
10.2.
(GHz)
Frequency
Fig. 24. Group delay time v s . frequency for forward volume waves in a YIG film having a thickness of 20 ~tm and spaced 20 ~tm from a silver ground plane (applied field, 4800 Oe). (From ref. 75.)
Multiple-layer devices interact with each other and behave in some ways similarly to a layer adjacent to a ground plane in that the effective thickness is a function of the wavenumber. By proper choice of thicknesses, magnetizations and spacings of the layers, a device can be "tailored" to give delay characteristics without incurring the losses due to a ground plane• The band edges occur at different frequencies if the magnetizations of the two layers differ• Such a discontinuity in the dispersion relation has been calculated 84 and observed experimentally aa. The best results for the bandwidth are obtained when the magnetization of each layer is kept the same. A comparison for a device having two YIG layers of thickness 20 ~tm and separated by a non-magnetic layer of 40 I~m thickness with the calculated delay operating in the 8.5-9.4 GHz region is shown in Fig. 25 75. 2~
I
i
t
i
i
i
1
l
I
~200
150 ¢
1oo @
a
IOO
~0
0
8~0
I
I
l
I
I
I
90MHz/Div
Frequency. GHz
I
I
,
I
94OO
4
I 4.2
,
I 4.4
Frequency
i
i 4.6
(GHz)
Fig. 25. Group delay v s . frequency of the forward volume waves for a double-layer YIG film (each layer was 20 pm thick and the layers were separated by a 40 ~tm insulating layer). (From ref. 75.) Fig. 26. Group delay v s . frequency for a dispersive array formed by ion implantations. (From ref. 93.)
180
v.E. WIGEN
Dispersive magnetostatic delay lines can be "actively" synthesized by using reflective arrays. Three types of reflective arrays have been investigated. These include chemically etched grooves on the surface of the garnet s8"89, metal strips which may be continuous or broken 9°'9~ and periodic strips in which ions are implanted 92,93. The forward waves are best suited for use in a reflective array because they are isotropic and the wavenumber is not altered by oblique reflection and can be detected by a second transducer. By contrast, surface waves and backward volume waves are anisotropic and must be reflected back to the generator. Reflective arrays tend to have a narrow.bandwidth in order to avoid interference with higher order Brags reflections. The largest bandwidth of 0.4 G H z has been achieved at the C-band frequencies using forward waves in a delay line constructed with an ionimplanted array 93. The results are shown in Fig. 26 93. 9. NON-LINEAR RESPONSES In F M R experiments, certain effects are observed at high r.f. power levels that are not evident at low powers. In particular, if the r.f. field is applied parallel to the direction of the internal effective field H eft, as opposed to the usual F M R in which the r.f. field is applied perpendicular to the direction of H elf, no measurable power will be absorbed by the sample until a certain threshold power is achieved. These high power non-linear responses are designated as parallel-pumping experiments and were proposed independently by Morganthaler 94 and Schl6mann et al. 95 In the decade of the 1960s, many papers reporting spin wave relaxation measurements by the parallel pumping technique were published. A few of the earlier papers include refs. 96-100. The parallel-pumping experiment is a sensitive and versatile method of measuring the relaxation frequencies of various spin waves with different k values. Both the static field H 0 and the frequency cok of any desired spin wave can be investigated without having to satisfy the usual F M R resonance conditions. For a given spin wave very little r.f. power absorption is observed below some critical value of the r.f. field. Above this critical field value, the energy going out of the modes by relaxation cannot keep up with the input energy so the number of spin waves grows rapidly, giving rise to large amplitudes of precession. The power absorption for parallel pumping is ~
dM~\
h J dr~-'-.tim ) ...... ge
(177)
In a spherical isotropic sample, d M J d t will be zero and no coupling will exist. However, by including volume demagnetization effects, the spins precess in elliptical orbits and the precessional frequency of M~ will oscillate at twice the drive frequency. The spatial integral in eqn. (177) will vanish for a plane wave; consequently, a standing spin wave of wavevectors + k and - k must be considered. Considering the equation of motion in eqn. (70), where H e r r = Ho~ and exchange effects are neglected, the z component of the magnetization is given by
dM, dt
= -7(MxHy-MyHx)
(178)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
181
The magnetization for a standing wave along the x direction is given by Mx = m sin(kx) sin(tot) M r = m sin(kx) cos(tot)
(179)
The demagnetization field of these spin waves is determined by the magnetostatic Maxwell equations (eqns. (146) and (147)). The magnetic potential ~ of eqn. (158) can be obtained and the resulting demagnetization field is given by HxD = - 4xm sin(tokt) sin(kx)
(180)
and use of the double-frequency pump field h0 sin(2tot) (eqn. (177)) yields Pr.f. = 2 ~Vra2ho
(181)
The transverse magnetization term m 2 can be related to the spin wave occupation number as follows: i
MV-
dr Mz = 2pnnk
(182)
Using the relation Mz 2 -- M 2 - M~2 - My2
(183)
and the condition Mx2 + My 2 ,~ M 2
(184)
the following relation is obtained: m 2 = 8~Mnk
(185)
V Thus, from eqn. (181), Pr.f. --- 4n~lhlMhonk
(186)
In the steady state condition this power is dissipated into mode k at the rate 1
hto~(nk-- (n~))
(187)
where Tk is the relaxation time of the kth spin wave and (nk) is the thermal equilibrium value of nk. Equations (186) and (187) give the critical condition on nk: (nk)
nk = l - h o / h c k
(188)
where hck is the critical r.f. field for the wavevector k and is given by hck =
2tok(1/r Tk) tom
(189)
where tom = 4rcyM and the relation 2#a = ?h is used. If k is allowed to propagate in an arbitrary direction with respect to the applied
182
P.E. WIGEN
field, the ellipticity of the precession will be influenced, giving a general result hck
co°(1/7T~) com sin20k
=
(190)
The factor 1/sin20k favors instability of the rt/2 spin waves as shown in Fig. 2 and, if the p u m p frequency cop is high enough for (Ok = COp/2spin waves to exist in the spin wave band, they will be the first to become unstable. Thus, at high frequencies or a low magnetic field H 0, the experimental results are expected to give a measure of the relaxation frequency for the ~/2 spin waves : 1
(191)
= hck corn
7Tk
cop
Typical experimental results showing a "butterfly curve"1 o 1 are given in Fig. 27. As the applied field increases, the value of k for 0 = n/2 decreases until, at the minimum of the c u r v e , cop/2 lies just at the top of the spin wave manifold. For higher magnetic fields, a 0 = rt/2 spin wave is not available and the relaxation frequency increases sharply.
T I.O
*x~ g
o.8 o
_~ 0.6
g x
°x'~ ~ l ~
0.4
x
0.2 0
I
1.0
t
t.2
t
i
1.4
t
I
1.6
I
I
I
1.8
Static Externol Field Ho(kOe)
Fig. 27. Comparison of the parallel pump data in YIG at 9.3 OHz with the theoretical predictions: t , experimental points; ×, theory. (From ref. 102.) The shape of the butterfly curve is given by --
1
7T~
= A + B k + C sinZ(20k)
(192)
For Y I G magnetized in the [001] direction and COp-- 9.3 G H z lol, A is determined from the minimum of the butterfly curve to be 0.132Oe, C = 0.163 Oe and B = 4.90 x 10- 7 0 e cm. The A and B k terms are related to three-magnon relaxation processes 9s while the sin220k term is found to give the best fit for field values larger than the minimum point in the curve. In the theory reviewed above, the effective field in eqn. (70) was simplified by neglecting the effects of anisotropy energy. In a recent series of papers, Patton 1o2 has included these terms in the spin wave instability theorem. The main influence of the cubic anisotropy energy, apart from a general frequency shift due to a modified effective field, is the introduction of a dependence o f COk o n the azimuthal spin wave angle q~k"This introduces a sin2~bk term in the spin wave dispersion curve derived from eqn. (83). The dependence of the spin wave manifold on ~bk and k 2 is shown schematically in Fig. 28. It is noted that the cubic anisotropy energy changes the
183
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
dependence of the n/2 branch but not the 0 = 0 branch. As shown in the example, the field can be adjusted to give two values of a butterfly curve. Such effects have been observed t°3 and the effects of the inclusion of the anisotropy energy in the theory have accounted for the second dip in the butterfly curve shown in Fig. 29. ~",~ ~" Ok=~r/2 [O k
/
Ok= 0
1~/ /
-f
i I
l
/
',,!
o.as[-
0.20 I
!
chk = "rr/2
0.45
•
DkZ-~--.~
1.5
.
H"
.
I ~'¢ I 1,55 1.6
"~.. 'l '¢ I 1.65
Ho(kOe)
•
Fig. 28. Schematic representation of the available spin wave states at tok = ,t/2 for a [110]- magnetized YIG sample at low field. Fig. 29. Comparison of the parallel pump data from a [ll0]-magnetized YIG sample at 9.522 GHz: I , experimental data (from ref. 104); , theory (from ref. 105).
These instability experiments provide a probe for analyzing the limitations of the power-handling capability of various devices. Thus the relaxation parameter is an important physical quantity in situations where power may be a significant consideration in the application of the garnet material to certain devices. Another technique recently reported for selectively investigating parallelpumped spin waves with specific wave vectors is that of Brillouin light scattering104, lo5. Thus it is possible to observe spin wave scattering continuously from the thermal level excitations into the parametric spin wave regions as well as to investigate the dependence of the angle and the wavenumber of the pumped spin waves directlyl°s. This allows a detailed study of the parametric processes at all levels of microwave power. This technique is particularly applicable to magnetic garnet films. ACKNOWLEDGMENTS
Much of the material that is reviewed in this paper was developed from a course on magnetism that was recently given at Ohio State University. The author expresses his appreciation to his students for their careful reading of the manuscript as well as a special acknowledgment to Dr. Jedryka for her critical reviews and constructive comments as the paper was being developed. The partial support of the National Science Foundation in the preparation phase of this manuscript through Grant DMR-8304250 is also acknowledged.
184
P.E. WIGEN
REFERENCES
1 J.E. Mee, lEEETrans. Magn.,3(1967) 190. 2 M. Sparks, B. R. Tittmann and C. Newkirk, Phys. Lett. A, 28 (1968) 131. 3 P.E. Wigen, in A. Paoletti (ed.), Physics o f Magnetic Garnets, North-Holland, Amsterdam, 1978, pp. 196-269. 4 L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955, p. 234. 5 J . H . van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, Oxford, 1932. 6 P.W. Anderson, Magnetism, 1 (1963) 25-83. 7 F. Bloch, Z. Phys., 61 (1930) 206. 8 J.C. Slater, Phys. Rev., 35 (1930) 509. 9 P. Pincus, Phys. Rev., 118 (1960) 658. 10 C. M¢ller, Z. Phys., 82 (1933) 559. 11 E.E. Anderson, Phys. Rev. A, 134 (1964) 1581. 12 T. Holstein and H. Primakoff, Phys. Rev., 58 (1940) 1098. 13 A.W. Saenz, Phys. Rev., 125 (1962) 1940. 14 S. Geller, in A. Paoletti (ed.), Physics of Magnetic Garnets, North-Holland, Amsterdam, 1978, pp. 1-55. 15 A.B. Harris, Phys. Rev., 132 (1963) 2398. 16 A.B. Harris and S. Kirkpatrick, AlP ConJl Proc., 24 (1974) 99. 17 A.C. Gerhardstein, P. E. Wigen and S. L. Blank, Phys. Rev. B, 18 (1978) 2218. 18 D.K. Krahn, P. E. Wigen and S. L. Blank, J. Appl. Phys., 50 (1979) 2189. 19 R.K. Wangsness, Am. J. Phys., 24 (1956) 60. 20 B. Bleaney and K. W. H. Stevens, Rep. Prog. Phys., 16 (1953) 108. 21 J.H. van Vleck, Phys. Rev., 52 (1937) 1178. 22 A.M. Clogston, H. Suhl, L. R, Walker amd P. W. Anderson, J. Phys. Chem. Solids, 1 (1956) 129. 23 S.V. Tyablikov, Methods in the Quantum Theory o f Magnetism, Plenum, New York, 1967, p. 49. 24 M. Sparks, Ferromagnetic Relaxation Theory, McGraw-Hill, New York, 1964. 25 S.D. Brown, R. Henry, P. E. Wigen and P. J. Besser, Solid State Commun., 11 (1972) 1179. 26 J.T. Yu, R. A. Turk and P. E. Wigen, Phys. Rev. B, 11 (1975) 420. 27 P.E. Wigen and H. Puszkarski, SolidState Commun., 18 (1976) 363. 28 P.E. Wigen, T. S. Stakelon, H. Puszkarski and P. Yen, AlP Conf Proc., 29 (1975) 670. 29 R.W. Damon and J. R. Eshbach, J. Phys. Chem. Solids, 19 (1960) 308. 30 P. Wolf, J. Appl. Phys., 34 (1963) 1139. 31 H. Puszkarski, Phys. Status Solidi, 22 (1967) 355. 32 H. Puszkarski, Acta Phys. Pol. A, 38 (1970) 217, 899. 33 H. Puszkarski, Phys. Status Solidi B, 50 (1972) 87. 34 V.M. Sokolov and B. A. Tavger, Fiz. Tverd. Tela (Leningrad), 10 (1968) 1793 (Soy. Phys. Solid State, 10 (1968) 1412). 35 V.M. Sokolov, Fiz. Tverd. Tela (Leningrad), 11 (1969) 753 (Soy. Phys. Solid State, 11 (1969) 602). 36 D.L. Mills, AlP Conf. Proc., 34 (1976) 93. 37 J. Spalek and A. Z. Maksymowicz, Solid State Commun., 15 (1974) 559. 38 J. Spalek and W. Schmidt, SolidState Commun., 16 (1975) 193. 39 A. Wachniewski, Phys. Status SolidiB, 77(1976) 319. 40 A. Duda, Phys. Status SolidiB, 107(1981) 779. 41 R.D. Henry, S.D. Brown, P. E WigenandP. J. Besser, Phys. Rev. Lett.,28(1972) 1272. 42 C. Vittoria and J. J. Krebs, AlP Conf. Proc., 24 (1974) 486. 43 C. Vittoria and H. Lessoff, Phys. Rev. Lett., 17 (1976) 53. 44 D.O. Smith, J. Appl. Phys., 29 (1958) 264. 45 R.W. Teale, in A. Paoletti (ed.), Physics of Magnetic Garnets, North-Holland, Amsterdam, 1978, pp. 270-319. 46 T. Stakelon, J. Appl. Phys., 49 (1978) 1592. 47 M. Marysko, Czech. J. Phys. B, 10 (1980) 1269. 48 C. Borghese, P. DeGasperis and P. E. Wigen, IEEE Trans. Magn., 18 (1982) 1301.
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
185
H. Puszkarski, Acta Phys. Pol. A, 39 (1971) 597. J.C. Levy, E. Ilisca and J. L. Motchane, Phys. Rev. B, 5 (1972) 187, 1099. Y.J. Liu, Ph.D. Thesis, Yale University, 1974. I. Harada, O. Nagai and T. Nagamiya, Phys. Rev. B, 16 (1977) 4882. M. Sparks, Phys. Rev. B, 1 (1970) 3831, 3856, 3869. O.G. Ramer and C. H. Wilts, Phys. Status Solidi B, 73 (1976) 443. O.G. Ramer and C. H. Wilts, Phys. Status Solidi B, 79 (1977) 313. B. Hockstra, R. P. vanStapele and J. M. Robertson, J. Appl. Phys., 48 (1977) 382. B. Hockstra, J. M. Robertson and G. Bartels, AIP Conf. Proc., 29 (1975) 111. J.P. Omaggio and P. E. Wigen, J. Appl. Phys., 50 (1979) 2264. C.H. Wilts and S. Prasad, IEEE Trans. Magn., 17 (1981) 2405. K. Komenou, J. Zebrowski and C. H. Wilts, J. Appl. Phys., 50 (1979) 5442. J. Mada and K. Asama, J. Appl. Phys., 50 (1979) 5914. H.A. Algra, W. de Roode, R. A. Henskens and J. M. Robertson, J. Appl. Phys., 52 (1981) 2358. R.L. White and I. H. Solt, Phys. Rev., 104 (1956) 56. L. Walker, Phys. Rev., 105(1957) 390. L. Walker, J. Appl. Phys., 29 (1958) 318. B. Tittmann, AlP Conf. Proc., 5 (2) (1971) 1128. B. Tittmann, Solid State Commun., 13 (1973) 463. B.E. Storey, A. O. Tooke, A. P. Cracknell and J. A. Przyslawa, J. Phys. C, 10 (1977) 875; 13 (1980) 2079. C. Borghese, P. DeGasperis and R. Tappa, SolidState Commun., 22 (1973) 463 ; 25 (1978) 21. H. Reimann, Ph.D. Thesis, University of Zurich, 1978. H. Reimann, L. C. Hsia and P. E. Wigen, J. Appl. Phys., 52 (1981) 2264. A. Akhiezer, V. Bar'yakhtar and S. Peletminskii, Spin Waves, North-Holland, Amsterdam, 1968. B. Schneider, Phys. Status Solidi B, 51 (1972) 325. R. Henry, S. Brown, P. Wigen and P. Besser, Phys. Rev. Left., 28 (1972) 1272. T. Wolfram and R. E. DeWames, Prog. Surf. Sci., 2 (4) (1972) 233. M.R. Daniel, J. D. Adam and P. R. Emtage, J. Circuits, Syst. Signal Process., to be published. R.E. DeWames and J. Wolfram, J. Appl. Phys., 41 (1970) 5243. W.L. Bongianni, J. Appl. Phys., 43 (1972)2541. N . D . J . Miller, Phys. Status Solidi A, 37 (1976) 83. M.R. Daniel, J. D. Adam and T. W. O'Keefe, IEEE Trans. Magn., 15 (1979) 1735. T. Wolfram, J. Appl. Phys., 41 (1970) 4748.
A.K. GangulyandC. Vittoria, J. AppI. Phys.,45(1974)4665. H. Sasaki and N. Mikoshiba, Electron. Lett., 15 (1979) 172; J. Appl. Phys., 52 (1982) 3546. L.R. Adkins and H. L. Glass, Electron. Lett., 16 (1980) 503; J. Appl. Phys., 53 (1982) 8928. M.R. Daniel and P. R. Emtage, J. Appl. Phys., 53 (1983) 3723. P. Grunberg and K. Mika, Phys. Rev. B, 27 (1983) 2955. R.E. Camley, T. S. Rahman and D. L. Mills, Phys. Rev. B, 27 (1983) 261. P. Emtage and M. Daniel, Phys. Rev. B, 29 (1984) 212. M. Tsutsumi, Y. Sagaguchi and N. Kumanai, Appl. Phys. Lett., 31 (1977) 779. J.P. Perekh and H. S. Tuan, Appl. Phys. Lett., 30 (1977) 667; 31 (1978) 709. J.M. Owens, J. H. Collins, C. V. Smith and J. M. Chiang, Appl. Phys. Lett., 31 (1977) 781. J.M. Owens, C.V. Smith, S.N. LeeandJ. H. Collins, IEEETrans. Magn.,14(1978)820. R.L. Carter, J.M. Owens, C.V. SmithandK. W. Reed, J. Appl. Phys.,53(1982)2655. G. Volluet and P. Hartemann, in B. R. McAroy (ed.), Proc. Ultrasonics Syrup., 1981, IEEE, New York, 1981, p. 394. F.R. Morganthaler, Dissertation, Massachusetts Institute of Technology, 1959; J. Appl. Phys., 31 (1960) 955. E. Schl6mann, J. J. Green and U. Milano, J. Appl. Phys., 31 (1960) 3863. J. J. Green and E. Schl6mann, J. Appl. Phys., 33 (1962) 1358. T. Kohane and E. Schl6mann, J. Appl. Phys., 34 (1963) 1544. R.M. White and E. SchlSmann, J. Appi. Phys., 33 (1962) 2437. T. Kasuya and R. C. LeCraw, Phys. Rev. Lett., 6 (1961) 223. E.G. Spencer, R. C. LeCraw and R. C. Linares, Phys. Rev., 123 (1961) 1937. P.E. Sciden, Phys. Rev. A, 133 (1964) 728.
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P.E. WIGEN
101
Y.M. Yakovlev, Y. N. Burdin, Y. R. Shilnikov and T. N. Bushueva, Fiz. Tverd., Tela, 12 (1970) 3059 (Soy. Phys.--SolidState, 12 (1971) 2475). C.E. Patton, Phys. Status Solidi B, 92 (1979) 211 ; 93 (1979) 63. J.C. Sethares, M. H. Seavey, T. G. Purnhagen and M. R. Stiglitz, J. Appl. Phys., 35 (1964) 898. V.N. Venitsky, V. V. Eremenko and E. V. Matyushkin, Pis'ma Zh. Eksp. Teor. Fiz., 27 (1978) 239 ( JETP Lett., 27 (1978) 222); Zh. Eksp. Teor. Fiz., 77 (1979) 1965 (Soy. Phys. JETP, 50 (1979) 934). W. Wettling, W. D. Wilber, P. Kabos and C. Patton, Phys. Rev. Lett., 51 (1983) 1680
102 103 104
105
135
ELECTRONICS AND OPTICS
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS P. E. WlGEN
Department of Physics, Ohio State University, 174 West 18th Avenue, Columbus, 0H43210 (U.S.A.)
The theories describing the dynamical excitations of the spin system of the ferromagnetic garnets are reviewed in this paper. Initially the microscopic description using quantum spin operator notation, normally used in the theoretical studies on the subject, is presented. The role of surface anisotropy energies in determining the boundary conditions of standing spin wave modes in thin films is developed in some detail in this model. The continuum model, which is the usual language of the experimentalist for analyzing ferromagnetic resonance data, is subsequently introduced and shown to be equivalent to the microscopic model. Some applications of the continuum model include ferromagnetic resonance, spin wave resonance, magnetostatic wave effects and non-linear excitations of the spin system.
1. INTRODUCTION
Single-crystal films of yttrium iron garnet (YIG) were first grown in 19671. Ferromagnetic resonance (FMR) was one of the earliest techniques used to investigate the fundamental properties as well as some of the magnetic material parameters of the film 2. As the growth of the more complex magnetic garnet films has developed, FMR has continued to be one of the most active and valuable techniques used to investigate their properties. FMR can be useful in the determination of many of the magnetic material parameters of the magnetic garnet films. These parameters include internal fields having uniaxial symmetry, first- and second-order cubic crystalline anisotropy fields and energy constants, g values, exchange constants and spin wave dispersion constants, magnetic loss parameters, magnetostrictive coupling constants and the magnetostatic wave characteristics. In addition, FMR has proved to be a valuable technique in many investigations of fundamental interest. These investigations include spin wave resonance, effects of surface anisotropy energy, magnetoelastic wave excitations, magnetoexchange branc~ repulsion, magnetoacoustic coupling, photomagnetic effects, domain wall resoenance, relaxation phenomena and high power (parallel-pumping) non-linear effects. 0040-6090/84/$3.00
© ElsevierSequoia/Printedin The Netherlands
136
P.E. WIGEN
This article is designed to serve as an introduction and/or guide to the novice or for the occasional user of the F M R technique who either investigates thin magnetic garnet films or evaluates magnetic material parameters. Theorists who work in the field have a tendency to develop their theories in terms of a microscopic model using quantum mechanical operators, while experimentalists usually describe their results in terms of a continuum model. This article represents an attempt to show the equivalence of these two approaches. The fundamentals of magnetism and the microscopic theory of spin waves are reviewed in Section 2 while the diversity of the surface energies and their influence on the boundary conditions and the normal modes of the spin wave excitations are described in detail in Section 3. The equivalence between the microscopic approach and the continuum model is developed in Section 4 while the effects of a non-parallel ground state and its contribution to an effective surface condition are considered in Section 5. Section 6 is devoted to the effects of inhomogeneous internal fields in the film and how they can dominate the surface effects in certain cases. Magnetostatic effects and some examples of their applications to devices are discussed in Sections 7 and 8. Finally, the high power responses are briefly discussed in Section 9. It is intended that these sections should give the reader the basic understanding of the application of F M R to magnetic garnet films. The references are intended to be typical examples of the application of F M R to garnet films but are neither a complete set nor a thorough review of the literature. Some of the topics reviewed in this paper were discussed in more detail by the present author in a previous publication 3. F M R is indeed a technique that is quite versatile and has proven to be useful in the study of the magnetic garnet films. It is expected that its application will continue to be relevant in the future and it is the hope of the author that this article will assist m a n y investigators in their efforts to analyze their data and to obtain parameters and/or information that will be useful for their purpose. 2.
THE MICROSCOPIC MODEL OF THE SPIN SYSTEM IN GARNETS
2.1. H e i s e n b e r g e x c h a n g e hamiltonian
In a microscopic electron system, the Pauli exclusion principle demands a totally antisymmetric wave function. Neglecting spin-orbit coupling, an orbital wave function which is symmetric will be coupled with a spin function which is antisymmetric and vice versa. For two individual electron wave functions, ~bi and ~bj, located at sites i and j respectively in the crystal lattice, the symmetric and antisymmetric orbitals will have the forms ~ym =
~Pi(rl)q~j(r2) + ~)i(r2)q~j(rl)
and
(1) t/Jam = ~ i ( r l ) ~ b j ( r 2 ) -
~)i(r2)~pji(rl)
where r 1 and r 2 are the spatial coordinates of the two electrons. The energies of these states are 4 Esy m =
K + Jij
and
(2) Eant = K - Jij
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
137
where K contains the energy due to kinetic and potential contributions and (
Jij =
e2 ~i(rl)t~j(r2) ~
)) ~b~(r2)~bj(rl
(3)
the exchange integral, arises from the spatially different forms of the symmetric and antisymmetric orbital wave functions. This purely quantum mechanical term has its origin in the region where the wave functions overlap and it splits the energy levels of the electronic system s . As the individual electronic wave functions in insulators are generally localized to the atomic site, the exchange integral in a macroscopic system is expected to be limited to nearest or at most next-nearest neighbors in the spin system. In the magnetic oxides, the exchange integral may be mediated through a connecting oxygen atom via the superexchange interaction 6. If sl and s 2 are the spin angular momentum vectors of the two electrons, the total angular momentum S = sl 4-s2 will be restricted to the eigenvalues 0 and 1 in the antisymmetric and symmetric spin states respectively. Then S 2 = sl 2 + s2 2 + 2s 1-s 2
(4)
which has the eigenvalues
S(S + 1) = 2s(s + 1) + 2s I -s 2
(5)
For s = 1/2, the operator ½+ 2sl "s2
(6)
will have the eigenvalues - 1 and + 1 in the states ~V,ymand ~Pantrespectively. Therefore the hamiltonian for the system can be equivalently expressed in terms of the spin operators by the relation 3ffo = K - (½+ 2s 1.s2)J0
(7)
In this relation, the electrons behave as if there existed a very strong magnetic-like coupling of the form s~ .s 2 between the spin vectors. However, the origin of the interaction is not magnetic but is due to a quantum mechanical Coulomb repulsion interaction which has no classical analogue. In the presence of an applied magnetic field H o, that portion of the total hamiltonian that is dependent on the orientation of the spins can be expressed in the form = Yrz + ~ E
(8)
where 3fazis the Zeeman energy and ~E is the exchange energy. The Zeeman term is given by
~Cz = g#BHo ~ S, z
(9)
where Ho defines the z axis, g is the Land6 g factor and #a the Bohr magneton. The positive sign is chosen as the electron spin is antiparallel to the magnetic moment. The exchange hamiltonian has the following form:
~E = --2 ~, J,jS,.Sj i>j
(10)
138
P.E. WIGEN
where the sum on i is over all lattice sites and in the exchange term the double sum is normally limited to nearest neighbors unless otherwise indicated and restricted to i > j so that the same interaction is not counted twice. For the parallel spin alignment the exchange term Ji~ is positive while the antiparallel spin alignment arises from a negative value of the exchange term. In magnetic solids, additional interactions between atoms will influence the energy of the spin system. Examples of such interactions include dipole-dipole interactions leading to magnetostatic demagnetization effects and spin-orbit interactions which contribute to the anisotropy energies of the system. These additional terms can be included in the spin hamiltonian by allowing the applied field H o to be replaced by an effective field Heff~ that determines the additional spinorientation-dependent terms in the hamiltonian. The hamiltonian for an assembly of exchange-coupled localized spins in a magnetic field is = g#, ~ H e f f i ' S / - 2 ~ JijS,'Sj i
(11)
i>j
2.2. Spin waves In the first excited state, it might be assumed that one spin Si is flipped parallel to the magnetic field while the others remain antiparallel to the field. As the value of J is of the order of 10 -14 erg ~ 100kB erg, where kB is the Boltzmann constant, this spin configuration corresponds to a large energy state. In comparison, the Zeeman energy for a single flipped spin is about lkB erg in a magnetic field of 10 kOe. In addition, such an excited state lacks the translational invariance of the hamiltonian in eqn. (11). The large exchange energy can be reduced by sharing the spin deviation between a large number of spins. In this mode, the angle between adjacent spins is small and, since the exchange energy is proportional to the cosine of the angle between adjacent spins, the exchange energy in the spin system is reduced. At low temperatures, such excitations, known as spin waves or magnons, determine the thermodynamic behavior of ferromagnets. It was Bloch v who first found that the states near the ground state could be approximated by sinusoidal spin waves. In evaluating the excited states it will be useful to express the hamiltonian in terms of the raising and lowering operators (12)
S + i ~- SXi + iSY i
and of SZ~. Equation (11) then has the form : g/~aHeff E SZi -- 2J .~ (S~i S~j + ½S +iS- j + ½S -iS +j) i
(13)
L,3
This hamiltonian will be considered for the following examples. 2.2.1. Linear chain with periodic boundary conditions and S = 1/2 The ground state eigenfunction for a linear chain of N spins whose hamiltonian is given by eqn. (13) can be expressed in terms of the single-particle spin down (magnetic moment up) eigenstates ~i as I//0
~--- 0 ~ 1 ~ t 2 0 ~ 3
• • • 0~ N
(14)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
139
This is the state realized at 0 K and represents the maximum alignment of spins or complete magnetization. As the temperature increases, the system will be excited out of the ground state into an excited state that may be thought of as containing a single reversed spin: $i =
° q °t2 " " °q - t fli°q + 1 "
" %v
(15)
where fl~ is the single-particle spin up eigenstate at the site i. However, ~k~is not an eigenfunction of the hamiltonian of eqn. (13). An eigenfunction can be formed by taking a linear combination of the ~,~ such as
Itlk = E cki~]i
(16)
i
Slater s has shown that the proper combinations in eqn. (16) are equivalent to a wave-like disturbance of wavevector k with the values of k determined by the periodic boundary conditions. Each spin i interacts with its two nearest neighbors i + 1 and i - 1 through the exchange integral J. Using the relation (~q I~ ( 1 / 2 ) 17'k) = l(~bilgl Wk~
(17)
the following set of coupled equations is obtained: {E + {(N -- 2)g/t a n o + ~ N - 5) J} C', + J(C k, +1 + cki-1) = 0
(18)
The solutions have the wave-like form (19)
cki = N - 1/2 exp(ikia)
where a is the distance between spins. The periodic boundary conditions require Ck~ to equal Cki +N, which yields the following values of k: kaN - 0, 1 ,2 . . . . . N 2x
1
(20)
Substituting eqn. (19) into eqn. (18), the energy E is found to be related to the wavevector k by E + ~ N - 2)g/~aH eft + {(N-- 5) d + 2J cos(ka) = 0 or
(21) E - Eo
= g/,tll H eff +
2J{ 1 - cos(ka)) = hook
where Eo = --½NglzaHeff-½NJis the ground state energy and hCOk is the energy required to excite the spin wave of wavevector k. In the limit of small values o f k, i.e. ka ~ 1, eqn. (21) reduces to the familiar form hfo/~ ~ g ~ B H elf +
Dk 2
(22)
where D = Ja 2 for a linear chain of spin 1/2. This dispersion curve is shown in Fig. 1. 2.2.2. Linear chain with end spins "'pinned" and S = 1/2 In thin magnetic garnet films, a considerable effort has been extended to observe and interpret the excitation of spin wave modes. The excitation of these modes is determined by the presence of an effective "surface" anisotropy field H 'urf acting on the spins in the first layer and in the Nth layer. In addition, the surface
140
P.E. W|GEN
g,u. B Heft-
D k 2 -->
Fig. 1. Spin wave dispersion curve calculated for Zeeman and exchange energies only. spins are not exchange coupled to a full set of nearest neighbors. Pincus 9 was the first to consider this problem by introducing symmetric surface terms into the hamiltonian: o'~surf = gltBHsurf(sz I + SzN)
(23)
The energy condition given in eqn. (18) is then replaced by {E + ½(N -- 2)gtt n H off + ~{N -- 5) J }Cki + J(C k, + 1
cki - 1
0
(24)
{e + ½ t N - 2)ggnHaf + g#nHsurf + ~ ( N - 3)d}C k, + dCk2 = 0
(25)
+
) =
for/=# 1,N. F o r i = 1
and for i = N
{E+~(N--2)glJBnaf+gHBnsurf+~N--3)d}CkN+JCkN
t = 0
(26)
The solutions are of the form
cki+ 1 = ~ cos(kia) + fl sin(kia)
(27)
and the energy is given by E = -- ~(N -- 2)g/~BH eft
-
-
~(N - 5) J - 2J cos(ka) -- g#B Hsurf
(28)
The surface field has no effect on the form of the dispersion relation but modifies the allowed values of k. Using this energy condition, the ratio of the coefficients of eqn. (27) is ct
J sin(ka) = - J{ 1 - cos(ka)} + g/~BHS"rf
(29)
F r o m this equation, it is observed that the surface pinning or spin wave b o u n d a r y condition is not a constant term for all spin wave modes but changes with k. At low values of k the surface anisotropy energy m a y dominate the b o u n d a r y condition giving a nearly pinned spin wave. However, as the k value increases, the higher exchange energy in the spin wave will dominate the surface anisotropy energy when Dk 2 > gl~a SH s"rf and a nearly free b o u n d a r y condition m a y exist. Using this ratio and eqns. (25) and (26), the following relation is obtained to determine the allowed
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
141
values of k: 2(1-~r) _ -~ ( l2k a ) - ~ cot(Nka) = -tr(2_~t) tan
cot(ka)
(30)
with g # a H surf ~r -
-
-
(31)
J
For cr = 1, the allowed modes have nodes at i = 0 and i = N + 1. For arbitrary tr, the nodal points are functions of k. The allowed values of k will be discussed in detail in Section 3. 2.2.3. Three dimensions and arbitrary S In YIG the spin S of the iron ion is 5/2. MOiler t 0 has extended the analysis to the case of three dimensions and arbitrary S. The result gives a dispersion relation of the form fifO/, =
g#s H e f t + 2SzJ( 1 - Yk)
(32)
where Ykis the geometric structure factor given by )'k = Z- 1 ~ exp(--ik'ri)
(33)
i
with r i being the vectors to the z nearest-neighbor magnetic ions. In a lattice of simple cubic symmetry and small values of k, eqn. (32) reduces to the familiar form hfOk ~ g / h 3 H elf + D k 2
(34)
where in this case D = ~SJza 2 = 2SJa 2 is the spin wave dispersion constant and a is the cube length. For YIG, D ~ 10 -2a erg cm 2 or 5 x 10 - 9 0 e cm 2 (ref. 11). 2.2.4. Spin waves in the garnet multisublattice model In the magnetic garnets there are up to 32 spins in the unit magnetic cell. For this case it is convenient to locate lattice sites by the relation riu = r~ + pu where the pu represent the intracell vectors to each of the v distinct magnetic sites in the magnetic cell giving rise to v recognizable sublattices in the ground state with all spins pointing up or down along a common axis. Oscillations away from the ground state are expressed in terms of spin deviation operators 12 over the v magnetic sites in the magnetic cell. The deviation operators are expanded in spin wave operators which are generally coupled and it is necessary to solve a secular equation of order v to find the normal modes. The general procedure has been discussed by Saenz 1a. In the application of this process to the magnetic garnets, the sum must be carried out over the various sites having dodecahedral symmetry {c}, octahedral symmetry [a] and tetrahedral symmetry (d). For details on the crystal structure see ref. 14. In YIG the {c} site is occupied by the non-magnetic yttrium ion and the problem reduces to a two-sublattice spin system. The details of the derivation for YIG have been evaluated by Harris ~s who found hfOo °p ~ -- lOJad
(35)
for the optical branch and ~lO)k a¢: ~, .5r (8Jaa -- 5Jad "t- 3Jdd)a2k 2 1o
(36)
142
P.E. WIGEN
for the acoustic branch. The k dependence of the 19 optical modes is not observed to be very strong. For the more complicated magnetic garnets where the {c} sites are occupied by rare earth ions the following relation is obtained for the acoustic mode: h(Okac ~,
200daa -- 125Jad + 75Jdd -- 20ddc Sc + 50Jac Sc 32k2 16(-5+6S~)
(37)
Harris 15 has also evaluated the coefficient of the terms of order k 4. Because of the relatively weak exchange between the iron ions at sites [a] and (d) and the rare earth ions at the {c} sites, the optical spectra will consist of 19 slightly perturbed high energy YIG modes plus 12 additional low lying optical branches. Since the garnet system is limited to a single acoustic type of spin wave branch with higher order optical spin wave branches separated to a relatively high frequency in zero magnetic field, the system will behave as a ferromagnetic array having spin wave energy Dk: given by eqn. (36) for YIG or eqn. (37) for the rare earth garnets. For the substituted or mixed garnets, Harris and Kirkpatrick 16 have evaluated the dispersion constant using percolation techniques. The experimental results are in good agreement with their model I 7.18 In YIG the frequency of the optical mode normally falls in the IR or visible region of the electromagnetic spectrum. In the rare earth garnets a two-sublattice approximation can be used by assuming that the iron atoms, which are strongly coupled together, can be treated as a single sublattice coupled less strongly to the rare earth sublattice. At some temperature or rare earth ion concentration, the condition gFeMve--gREMRE = 0 may be satisfied for which the total angular momentum will vanish. Under these exceptional circumstances, the resonance frequency of the optical mode will move into the microwave region. The temperature at which the angular momentum vanishes is often near the temperature at which MFe = MRE where the magnetization is compensated. Wangsness ~9 has shown that the small difference between the compensation points of the angular momentum and the magnetization also contributes to the Zeeman energy term associated with eqn. (37). The Land6 g value in the two-sublattice model has the form Ma-Mb gcff= Ma/g _ Mb/g b
(38)
Near the compensation point for the angular momentum, geff will become very large and change sign ~7. 2.3. Additional spin interactions in the harniltonian In addition to the Zeeman and exchange energies that are included in the spin hamiltonian in eqn. (11), a variety of additional spin interaction terms can be included. Typical terms include the following. 2.3.1. Single-ion anisotropy The crystal electric field may introduce second-order effects in the spin hamiltonian via spin-orbit or spin-spin interactions. These effects are important in the resonance spectrum of dilute paramagnetic salts, and they have been reviewed in detail by Bleaney and Stevens z°. If the local crystal field has uniaxial symmetry, e.g. crystals having tetragonal or trigonal symmetry with the unique axis in the z
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
143
direction, the crystal field contribution is ~¢ = ~ ~ {(SZ,)2 - ~ S ( S + 1)}
(39)
i
where ~ is the measure of the interaction. This term vanishes for S = ½. For small values of ~, this term can be included in the Zeeman hamiltonian by introducing an anisotropic g: ~,~ffz=
-gll#nHZo~S~,-g±lzB(HXo~iSX,+H'o~S" t
(40)
In ordered magnetic materials, such as the garnets, this will be included as an anisotropy energy in the system.
2.3.2. Terms coupling spin pairs In general, the pair hamiltonian ~vair can be expanded in a power series of appropriate symmetry in the spin operators:
°3~pair =
E E jklijSkiSlj
i>j k.l
k,l = x,y,z
(41)
If the spin system has cylindrical symmetry, this term can be expressed in the form °~pair = '~e 3t"°~d
(42)
where -~e is given in eqn. (10) and ~ d = E dij*(Si'Sj-- 3ru- 2(~i'riJ)(Sj'riJ)) i>j
(43)
do* = dij + Pij
(44)
du = g2fla2ru- 3
(45)
with
and
The do coupling term is the familiar long range rij-3 contribution arising from magnetic dipolar interactions and Po is the phenomenological short range pseudodipolar exchange interaction first introduced by van Vleck 21. These terms are often referred to as anisotropic exchange terms.
2.3.3. Dipole-dipole interaction The inclusion of the long range r~j-3 spin interactions complicates the procedure of diagonalizing the hamiltonian to determine the spin wave dispersion relation. However, this dipole-dipole term is a particularly important term as the long range of this term is responsible for the spin wave band associated with the spin wave dispersion relation and contributes significantly to the understanding of the relaxation processes 22. In a classic paper, Holstein and Primakoff (HP) 12 developed a spin wave theory for a spin hamiltonian which consists of the Zeeman, exchange and dipolar and/or pseudodipolar interactions:
oW= ggeHo. ~ Si-2 ~, JuSi'Sj+i~>jdu{Si'Sj-3rij-2(Si'rij)(Sj'rij) } i
i>j
" "
(46)
144
P.E. WIGEN
In their procedures, the spins are treated as quantized particles subject to creation and annihilation operators by using spin raising and spin lowering operators 23. Spin deviation operators are next introduced by using Bose creation and annihilation operators. Spin waves are introduced by making use of the Fourier expressions of the localized Bose operators. Finally a fourth transformation is required to decouple the states o f k and - k . The result is a diagonalized hamiltonian having a dispersion relation of the form hCOk
{(Dk z + g p B H e f f ) ( D k 2 + g l ~ a Heff + g # n 4 x m s i n 2 0 k ) } 1/2
=
(47)
where M is the magnetization of the medium, Ok is the angle between the propagation constant k and the magnetization M and H eft is the internal field: H elf = H o - 47tNz M
(48)
where Ho is the applied field and N= is the demagnetization factor along the z axis of the sample which is defined here as the direction of H eft . The dispersion relation, shown in Fig. 2, indicates the spin wave manifold proposed in the continuum model which will be discussed further in Section 6. For many experimental conditions of the magnetic excitations in the garnet systems, 4riM may be smaller than the internal field H eft. Under this condition, eqn. (47) can be approximated to the simpler relation he) k ~ D k 2 + g / I n Heff + g l ~ n 2 n M sinE0k
(49)
This relation is shown in Fig. 2, broken line, for sinE0k = 1. This is the relation that would be obtained by neglecting the final H P transformation that decouples the k and - k states. This approximation to the energy will be valid except where k is small, sin20a is near unity and the internal field is small. In YIG, D ~ l 0 - 2 a ergcm 2 and, for k = 106 c m - 1, D k 2 , ~ 10 - 1 6 erg ~ lkn erg
(50)
At room temperature, M = 140 e.m.u, c m - a for which g l ~ a 4 x M is 0.2ka erg so the approximation is good for k >/ l 0 6 c m - 1. In spin wave resonance experiments, the spectrometer usually operates at a ~O~ k
s
,./
../-~s,n,e~-o_
k,~-~TL--ApproximateDispersion / r~ / Relation g/~B2TrM I / ~ ~--r" "Exact DispersionRelation g~BHeff l _ _
Dk2 Fig. 2. Spin wave band calculated for Zeeman, exchange and dipole-dipole energies: given by eqn. (47); - - -, circular precession approximation given by eqn. (49).
, exact solution
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
145
constant radial frequency coo. In this case the applied field is varied to satisfy the spin wave resonance condition or Ho --, H~ in eqns. (47) and (48). At sin Oh = 1, the error in the approximation in eqn. (49) for YIG is 20~o at 3 GHz, 3% at 10 GHz and 0.5~o at 25 GHz. Thus the final HP transformation is only necessary in those experiments where H "ff is only slightly larger than the value necessary to saturate the material. This occurs in experiments at radio frequencies. Another case where the final transformation, which involves the ellipticity of the precession, is necessary is for high power or parallel-pumping experiments 24. 3.
SURFACE INHOMOGENEITIES AND SURFACE SPIN WAVES
3.1. Surface anisotropy energies In Section 2.2 the concept of an effective surface anisotropy field was introduced to provide the boundary condition to determine the allowed spin wave modes observed in the ferromagnetic resonance spectra of thin magnetic garnet films 25-28. In the experiments, it is observed that the spin wave spectrum may have a complex dependence on the orientation of the magnetic field and on the temperature. An example of the influence of these parameters on the spin wave spectrum is shown for a YIG film in Figs. 3 and 4. The weaker signal at higher fields in the parallel orientation is a non-propagating surface mode. While the extremes in most films may not always be as pronounced as those observed in this example, this author is not aware of a report where the spin wave spectrum in a thin magnetic
-t-1
4900
I
5000
I
5100
2900
I
5000
40*
t I
3 'oo
34C0
, 2700
!/
I, 2800
5100
29100
,90 °
5O"
3000
V
!~20
i 2620
MAGNETIC F1ELD(Oe)
Fig. 3. Spin wave spectra showing the first two high field modes at six orientations observed at 9.2 GHz and at room temperature for a YIG film (formed by chemical vapor deposition (CVD)) having a single surface mode. (From ref. 28.)
146
P.E. WIGEN
ToZ92K ,i
T:r3eK ,'i! f - -
I i i'
2500 T=256K
Oe 2 6 0 0
2400
2500
Oe T=IC8K ", X]CO,
l i
:,
, t i ! 2500
T=231K
!/ / 26CC
24C0
'
;
2500
T=2OK
q 2400
I
~'
,'i
i!,1 2500
'
2600
2500
2400
I,~ L~ V
2500
Fig. 4. Spin wavespectra showing the first two high fieldmodes in the parallelorientation as a function of temperature for a CVD YIG filmhaving a single surfacemode at room temperature. (From ref. 28.) garnet film does not reduce to a single mode at some "critical" angle as a function of the orientation of the magnetic field between the normal to the film plane (perpendicular resonance) and in the film plane (parallel resonance). This angle dependence of the pinning condition can be predicted by a combination of an isotropic energy and a unixial surface anisotropy energy. Depending on the sign of the energy terms, the high field non-propagating surface mode will be observed at magnetic field orientations between the critical angle and parallel resonance or between the critical angle and perpendicular resonance. Both conditions have been observed experimentally. These non-propagating surface modes differ from the magnetostatic surface modes of D a m o n and Eshbach (DE) 29 in that the propagation constant in the plane of the film is zero, i.e. the propagation constant is purely imaginary. In addition, decay lengths as short as 5 0 n m have been observed. Consequently, the field position of these surface modes is determined by their exchange energies and not by the magnetostatic effects discussed later. Non-propagating surface modes were first proposed by Wolf 3° in 1963 when a uniaxial surface energy was assumed to be present on the film surface. Puszkarski al-33 was the first to discuss the angular dependence of these surface modes, and this was followed almost simultaneously by the work of Sokolov and Tavger34, 35. Additional contributions on the topic of surface interactions have been made by Mills a6, Spalek and coworkers 37'38, Wachniewski 39 and D u d a 4°. The behavior of the surface mode in Figs. 3 and 4 can be interpreted in terms of a combination of surface energies, each having a unique angle dependence. As an introduction to the full description of these spectra, a model assuming a unidirectional surface anisotropy field will be considered initially 32. The calculations are
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
147
carried out for a hamiltonian consisting of a Heisenberg localized spin model having nearest-neighbor exchange and a Zeeman energy term oW=--2
JgSo'Sl+o,j,+g~tBEHefft'Slj
~ |,J,l + o,J'
(51)
Ij
Here the sum is carried out over the j spins located within the I layers of the spin system of a thin film. Jg is the exchange integral between nearest neighbors situated in layers I and l_+ g and Heft!is the effective magnetic field assumed to be uniform and unidirectional within a given layer. Using this hamiltonian, only circular precession modes will be predicted. The influence of the elliptical precession will be included later. Nearest neighbors of a given spin belonging to the same layer are designated by g = 0 and the two closest layers by g = _+ 1. Jo is then the exchange constant between spins within the I plane and J1 is the exchange constant between spins of different layers. The presence of an effective surface field acting only at the spins in the I = 0 layer and I -- N - 1 layer is represented by Heffo and Heft s _ 1 respectively. All other Heft! a r e assumed to be identical. In this approximation the spins at the surface of the system are subject to a surface field that is assumed to be a unidirectional phenomenological quantity that accounts for the pinning of the surface spins, thus determining the boundary conditions for the k values of the allowed spin wave modes. The energy of the surface spins is given by (52)
Esurf = Ein t + 2 S 2 z l J 1 A
where Eint ---- -- 2S2zoJo _ 4S2z 1J 1 - g#n S~" H eft
(53)
is the energy of the interior spins in their ground state. A is the pinning parameter which determines the boundary conditions and is given by A = 1-- g#n
...... f
(54)
2 S z l J l ~" rl
where ~ is the unit vector in the direction of the spin moment and z I is the number of nearest neighbors in the adjacent layer. The surface field H surf will either increase the energy of the surface spin giving an unpinned boundary condition or decrease the energy of surface spins giving a pinned boundary condition. The surface energy density is given by Esurf
=
--g#n
S a - 2 3" H surf
= 2S2zljla-2(A-
1)
(55)
where a is the lattice constant. Several characteristic features of the calculated k values of the spin wave spectrum are presented in Fig. 5. First, as A approaches - oo, the lowest order mode has an "absolutely" pinned boundary condition; k, = n ~ / L where L = Na. As A increases to unity, the lowest energy mode kt becomes the uniform or kl = 0 mode. The first antisymmetric mode k2 crosses the k2 = 0 condition at A = (N + 1)/(N-- 1). For A greater than unity, kx becomes imaginary or an acoustic surface wave. k2 also becomes imaginary forming an antisymmetric surface mode for A = (N + 1)/(N-- 1).
148
P . E . WIGEN
A --r
~
A C O U S ] IC SURFACE MODES L+J
i,
II
I
II
r I ,
i
ii~
!
'i,
'i,
~
i$ q
Ii i
I
i i
IL
5
,
li L
~
;I
4
i
I
2 I
[I
IiI i I
'~|
% I
II ';11 I
h I
I i i
Ii i i I
,,,,
/i
I
I
i,:/
O
\
2_ L H
*:i /L
I
'
I
G(t)
I ,3
I I
~
,,:
i d
....
-4 n:l
9
3
OPTICA[ SUf?FACE MOI)ES
Fig. 5. Graphical solutions of the characteristic equations as a function of the surface parameter A : , antisymmetric states; - - , symmetric states. The number n labels the allowed states. This graph represents a symmetric film having 11 layers. (From ref. 32.)
Similar conditions exist for the optical surface spin wave in the region A ~< - 1 but, because the moment of this m o d e is so small, they have not been observed in any experiments. The spatial dependence of the transverse m o m e n t of the first few low energy spin wave modes for different values of the surface parameter is shown in Fig. 6. It is noted that as A varies from - oo to unity the precession amplitude of the surface spins increases until the slope of the transverse m o m e n t at the surface is zero when A = 1. As A increases to + oo, the amplitude of the surface spins again decreases and, at A = + oo, the spin wave spectrum has transformed to be identical with that observed when A = - oo as the two surface modes have zero m o m e n t and will not be excited. For A slightly larger than unity, the first two modes are surface modes whose position shifts to lower energies and whose transverse m o m e n t decreases rapidly. A =-cO
A=O
A=I
A:2
A = + '::o SURFACE MODES
n=l
SURFACE MODES
n=l I n~ 2
J
i
i
°:31
1
i,,,2 1
,,=21
I
n=41
=
i
.:sl
.=51
W i I
I I r
I
II
L-I
L-I
L-I
)
L-I
L-I
i
L I
L-I
L-I
L-I
L-I
Fig. 6. The transverse component of the magnetization for a few spin wave modes of low n for various values of the surface parameter A. (From ref. 32.)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
149
The resulting spin wave spectra can be described by the following conditions on A. (a) A = 1 or Es,rt = 0 gives the natural condition at the surface. (b) A > 1 or Es,rf > 0 gives unpinned surface spins (surface modes and body modes). (c) A < 1 or Es,rf < 0 gives pinned surface spins (body modes only). At A = 1, only the uniform mode (k = 0) will be excited by a uniform r.f. magnetic field. All higher order spin wave modes will have odd symmetry about the center of the film and will not be excited. The angle dependence of the surface parameter A is implicit in eqn. (54) and can be expressed explicitly as A= 1
g # ~ ~'H surf 2SzlJ1
= 1
g/~n H surf cos(0o- 0)
2SzlJt
(56)
where 0 o is the angle between the z axis of the film and the orientation of a constant unidirectional surface field and 0 is the angle between ~ and the z axis. For a constant unidirectional surface field, A will be less than unity for I0 o - 01 < n/2 and greater than unity for 10o-01 > ~/2. For the condition 10o-01 = n/2, a critical angle is observed where A will be unity and only the uniform mode will be excited. The unidirectional surface anisotropy field predicts an angle dependence that has a lower symmetry than that observed in the experiments shown in Figs. 3 and 4. Higher order symmetries can be obtained by introducing the appropriate surface energy terms with the original hamiltonian in eqn. (51) 27. In fact any number of these interactions may be present at either surface of the YIG film. Thus to evaluate the influence of these surface energies on the spin wave spectra completely it is necessary to analyze the data in terms of two independent surface parameters, i.e. A s at the substrate surface and Ae at the free surface of the film where As = 1 - ~ aspPp(cos 0) and
(57) Af = 1 - ~ afvPv(cos O)
The Legendre polynomials Pp(cos 0) will represent the angle dependence of the various surface terms present in the hamiltonian and ap indicates the strength of that interaction. In the above analysis, the spins are in a symmetrical environment and the precession will be isotropic. In thin garnet films, the demagnetization factor gives rise to an anisotropic resonance condition and therefore introduces elliptical effects in the spin precession. These effects can be included in the microscopic approach by adding a single-ion anisotropy term of the form of eqn. (39)'~°. However, the elliptical effects are more naturally included in the continuum model considered in Section 4 and therefore will not be considered further here. In comparing the predictions of this model with experimental results, the nonelliptical model will be a valid approximation only in the limit of high frequency (oJly >>4nM) where the degree of ellipticity in the precession is minimized. In order to probe the surfaces of a thin magnetic film by spin wave resonance, it
150
P.E. WIGEN
is desirable that the interior of the film be homogeneous since volume inhomogeneities result in spin wave spectra that are determined only partly or not at all by the surface conditions 4~. Extremely homogeneous films of YIG 42 of the order of 1 lam thick can be routinely produced by epitaxial growth on gadolinium gallium garnet (GGG) substrates. The low intensity of higher order spin wave modes 43 compared with the intensity of the principal mode in films grown by liquid phase epitaxy (LPE) is evidence that the pinning parameters As and Af of eqns. (57) are nearly unity. This indicates that negligibly small surface energies are present in as-grown LPE films. In CVD films or annealed L P E films significant surface energy contributions are present zS-zS. In a typical CVD YIG film, a non-propagating surface spin wave mode is observed, as shown in Figs. 3 and 4. It was found to be localized at the Y I G substrate interface as confirmed by etching experiments 26. Films that are etched in hot phosphoric acid have any free surface impurities removed and yield a natural condition (Af = 1) at that surface. In this manner the surface energies at the filmsubstrate interface can be investigated without interference from the free surface interactions. The angle dependence of the mode positions and relative intensities are shown in Fig. 3 where only the two high field modes observed in the spectra are presented. The weak high field absorption observed at parallel resonance is identified as a surface mode while the next and strongest absorption is the first of the body modes. As the applied magnetic field is rotated towards the perpendicular orientation, the high field surface mode is observed to increase while the body modes all decrease in intensity. At the critical angle 0c, which is 40 ° for this film, the surface mode is transformed into the uniform precession mode and becomes the only mode observed. As the magnetic field is rotated towards the perpendicular orientation, the surface mode transforms into the first of the body modes and the higher order body modes are again observed. The temperature dependence of the spin wave resonance spectrum of the same film has been investigated from 2 to 300 K. While no significant change has been observed for the perpendicular resonance spectrum as the sample temperature is changed, the parallel resonance spectrum changes dramatically as shown in Fig. 4. On annealing the CVD YIG film in the atmosphere at a temperature between 750 and 1300°C, a surface anisotropy energy develops at each surface z8. As the surface energies have different magnitudes at each surface, the modes of pseudoeven-symmetry (even n values) are not antisymmetric and can be excited in the experiments. Figure 7 indicates the results of a series of experiments in which the pinning parameter at each surface and therefore the surface energies were measured as a function of the orientation of M. The YIG film used in the measurements had a [100] orientation, a thickness of approximately 0.46 ~tm and was annealed at 1300 °C for 2 h. The spectra were measured at 23.3 GHz. The pseudosymmetric surface mode is localized at the substrate surface whereas the pseudoantisymmetric surface mode is principally localized at the free surface. The theory of asymmetric boundary conditions was used to find the values of As and Af consistent with these quantities at each angle. The curves shown in Fig. 7 are the least-squares fits to the experimental data.
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
151
0.04
t 0.03
0.02
///I/
0
-0.02
I
o 02
//
-0.04
/ " - ~ SUBSTRATE SURFACE
-006
-0.08
-0101
-012
- 0 14 i
l
l
30 l
I t
l l 50 60 8 ( M - F i L M NORMAL}
-0.01
l
70
90
Fig. 7. The surface pinning parameter A at the substrate interface and free surface of an annealed YIG film calculated from the spin wave spectra (YIG[100]/GGG; YIG thickness, 0.46 pm; annealed for 2 h at 1300 °C; 292 K; 23.3 GHz). The curves on the fight-hand side are the data from 50 ° to 90 ° shown on an expanded vertical scale. (From ref. 28.)
Terms with unidirectional symmetry (odd p) were eliminated from the eqns. (57) since the spin wave spectrum remained unchanged when the direction of M was reversed. It was also found that only two terms in eqns. (57), namely P0 and/)2, were necessary to obtain a good fit to the data. Thus the surface parameters can be written as A~ = 1--aSo-a~2
3 cos20-1 2
and
(58) Af =
1-ato-af23
cos20- 1 2
These coefficients and the corresponding surface energies are given in Table I. The surface energy density values shown in Table I are calculated using the relation bp =
ap2S2ZlJ l a2
(59)
The detailed analysis of the data in terms of the magnitude, angle dependence and temperature dependence of the surface anisotropy energy allows a unique approach to the evaluation of various microscopic interactions that exist at magnetic surfaces. Only when the details of the surface anisotropy are understood can the boundary conditions and the allowed k values of the normal modes be determined with any degree of confidence. Then, once the k values are determined, the bulk spin wave properties such as the spin wave dispersion coefficient can be evaluated.
152
P.E, WIGEN
TABLE I THE COEFFICIENTSat EVALUATEDFROMTHE DATA IN FIG. 7 l
a l ( × l O -2)
b t ( × l O - 2 e r g c m 2)
Substrate interface 0 2 4
0.67_+0.05 7.47+_0.15 (0.72+_0.75)
-4.31 +0.32 -48.00+_0.96
-0.14+_0.03 2.23 +_0.10 (0.14+_0.32)
0.90+_0.19 14.33 +_0.64
Free surface 0 2 4
4.
FERROMAGNETIC RESONANCE IN THE CONTINUUM MODEL
4.1. Equation o f motion
The analysis of F M R experiments is usually performed in terms of a continuum model as opposed to the quantum mechanical expressions developed in the earlier sections. The transition from the spin hamiltonian in eqn. (51) to a set of classical field equations is accomplished by treating the spin operators as classical vectors. The correspondence of the quantum mechanical operators and the classical spin vectors can best be demonstrated by considering the time variations in the expectation values of the transverse components of the magnetic moments when a transition occurs between two allowed states of the hamiltonian. In terms of the spin operators, the transverse magnetic moments are given by
and
(60) (try) = - - g H n ( S y )
From quantum mechanics, the time dependence of such an operator can be expressed in terms of its commutator with the hamiltonian. Considering the Zeeman term as an example, J~'~z =
g#B H" S
(61)
= gpaHoSz
The time dependence of the components of the spin operators is given by the following: d
i
~s~ = ~[~sx] = d
~ [ S r = 7HoSx
d ~Sz =0
-yHoS,, (62)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
153
These equations are the component form of the vector equation
d_s= dt - S x ~/-i~
(63)
or in terms of the expectation values of the magnetic moment operators d(#> dt = - x ~ H o
(64)
where yHo = ho~ is the energy splitting between the two allowed quantum states being considered in the spin transition.Thus in the quantum mechanical model the expectation value of the transverse magnetic m o m e n t in a transition between two states has the same frequency as that required of a small transverse r.f.magnetic field in F M R . The addition of the exchange term to the hamiltonian can be made by expanding the variation in the neighboring spin S t in a Taylor seriesabout Si at the position i:
S t = S,+{(rt-r,)'V}Si+½{(rt-ri)'V}ZS,+...
(65)
Assuming that the ith site is a symmetry center, the second term in the expansion will not contribute to the exchange energy since contributions from opposite neighbors will cancel. In the approximation of nearest-neighbor interactions, the exchange hamiltonian reduces to the form
~'~,~ = - 2zJ ~ Si 2 - J ~_, S,. {(rt - r,). V} 2Si + . . . i
(66)
it
where z is the number of nearest neighbors. The first term in eqn. (66) determines the ground state energy and will not contribute to the spin dynamic energy terms. The second term can be further simplified according to the crystal lattice type. For a cubic lattice with lattice constant a the hamiltonian becomes
= +g#B ~., H'rfi'Si-2Ja2z ~ SrVZSi i
(67)
l
The quantum mechanical equation of motion for a spin Si is given by
ihff-~S, = [Si, oaf']
(68)
Using the relationship S x S = ihS, the commutator yields the expression d h-~tSi = --g#BS x H ©ff+ z2Ja2Si x V2Si
(69)
Since Si represents any of the spins in a finite sample, eqn. (69) can be rewritten in terms of the magnetization vector M as l d--M=-MxH'ff+2-~T~2MxVZM 7 dt Mo
(70)
where M = g#BNS and N is the number of spins per unit volume. The exchange
154
P . E . WIGEN
constant d is related to the exchange integral by the relation d
= zJS2/a
(71)
The first term on the right-hand side ofeqn. (70) represents the torques acting on the magnetization M placed in a magnetic field H eft while the second term can be thought of as the torque due to an exchange field H cx = - ( 2 s 4 / M o 2 ) V 2 M . The torque due to H off can be treated in terms of the free energy density of the system by the relation 44 z = ~, x ( - VE)
(72)
In this relation the effects of the applied field, magnetostatic energy and the anisotropy energy can then be handled as scalar contributions to the energy density rather than as vector fields. In addition, the ellipticity of the normal modes can be readily evaluated. The equation of motion can be solved by using the coordinate system shown in Fig. 8 in which the equilibrium direction of the magnetization is the ~, direction and small deviations from equilibrium are in the e0 and ~¢ directions. In this coordinate system the magnetization has the form M = Mo~, + moeo + m4,e¢
(73)
where mo = M 80, m~b = M sin 0 8q5 and 60 and 6~b are small deviations from the equilibrium direction. The unit vectors i,, e0 and i¢ are the standard unit vectors of a
Fig. 8. The coordinate system used for the evaluation of the ferromagnetic resonance condition in the continuum model.
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
155
spherical coordinate system. On substituting eqn. (72) into eqn. (70), the equation of motion has the form
1 d M = M VE+2T_~dzMxV2M
(74) dt Moo× Mo A phenomenological damping term can be easily introduced on the right-hand side of eqn. (74) to account for the line shapes in the spectra. These may have one of the following forms4S: the Landau-Lifshitz form i Mo2MX (M x H "ff) (75a) the Gilbert form Gt
--M Mo
xM
(75b)
the Bloch-Bloembergen form
Mx,y Mz-Mo ~2
(75c)
"el
In the remainder of this treatise, the damping terms will not be considered further. In the equilibrium orientation, the magnetization will not have a variation in either time or space and eqn. (74) reduces to the form M - - x VE = E0/~-- E~ cosec 0 e0 = 0 (76) Mo
where the notation Eo = dE~dOand E¢ = cOE/dc~is used. Equation (76) is just a statement that zero net torque acts on the magnetization in the equilibrium orient-'ion. The equation of motion is linearized by expanding eqn. (74) about the equilil+ium orientation and retaining only terms to first order in the deviation from equilit _Jum:
y \dr ]
=
(--Moo
VE + ~2-~2 ~(M x
t MO
(77)
V2M)
where
M
(-~o X VE) = M-~o~Mx VE + ~-~ x ~(VE)
(78)
From eqn. (72), the first term on the right-hand side is zero and the second term takes the form M cosec 0 - - × fi(VE) -~o (Eoemo+E~¢cosec 0 m¢,)~.o+ Mo
1
+-~oo(Eoomo+ Eo#cosec 0 m~)~¢
(79)
The equation of motion in linearized form reduces to the set of coupled equations 1 d
2~ 2 cosec 0 (Eocmo+Ecc,cosecOmg~)___~ooV me
and
(80) Id
I
2~/
2
~m¢ = ---;T(Eoomo, v,o + Eo¢cosec 0 me) +Moo V mo
156
P . E . WIGEN
For a magnetic medium in the region y t> 0, the solutions to eqns. (80) will have the form mo = {ctl sin(k 1Y) + fll cos(k xY) + 71 exp( - kzy)} exp( - loot)
and
(81) ms = {ct2 sin(klY) + f12 cos(kly) + 72 e x p ( - kEy)} exp(--itot)
where k 2 is the exponentially decaying surface mode required to satisfy Maxwell's equations at the y = 0 surface. Substituting eqns. (81) into eqns. (80) and setting cosec 0 to unity, the following amplitude relations are determined:
52
fiE
cq
fll
io,)/~ + Eo4,/M o Ee~b/Mo - Dkl 2 Eoo/Mo + Dka 2 = iog/7_Eoe/Mo
(82a)
and Y2 Yl
ko/y + Eo4o/Mo E¢¢JMo--Dk2 2
=
Eoo/M o - D k 2 2 ko/7 - Eo~/M o
(82b)
The dispersion relation is given by ( ~ ) Z = ( E o o +Okl.2)(E~b ¢ ++_Dkl,22)- EO¢~2
(83)
When these terms are evaluated for Y I G in which D = 2 ~ / M o = 0.5 x l0 - s 2, the value o f k 2 is 1.8 X 10 6 c m - 1 at 25 G H z or 1.1 x l06 cm 1 at 10 GHz. This corresponds to decay lengths of 5.5 nm and 9.0 nm respectively. In garnet films of thickness of the order of 1 lam, the fraction of energy absorbed by the mode associated with k 2 is negligible and the use of Maxwell's equations to consider the continuity at the boundary is unnecessary. The allowed values of k~ will be determined by the film thickness and the boundary conditions and correspond to the surface modes or bulk modes evaluated in Section 3. For a thin film having a uniaxial anisotropy constant K u with an easy axis normal to the film plane and the magnetization oriented normally to the film plane where 0o = ~bo = re/2, the derivative terms have the form Oecm
Eoo = E~O = M ( H o + Hu)
(84)
where H o is the applied external field and H. = 2 K u / M o is the effective anisotropy field. The dispersion relation from eqn. (83) then yields (.o
-- = Ho + H u - Dkl 2
(85)
and the ellipticity from eqns. (82) gives = i giving a circular precession as expected.
(86)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
157
When the magnetization is oriented in the plane of the film where 0o = n/2 and 4~o = 0,
Eoo = HoMo
E ~ = Mo(no - n,)
(87)
In this geometry the dispersion relation yields
( ~ ) 2 = {(Ho + Dk,2)(Ho- Hu + Dk12)}l/2
(88)
and the ellipticity gives
~X2
f12
~)2
=
=
: ( H o - - H u ~ ll2 't, 7
(89)
4.2. Boundary conditions The allowed values of kl will be determined by the boundary conditions established by the surface anisotropy energy density Es(O,@)which can be expressed in terms of the coordinates of M. The surface torque T~ is then obtained by the relation M T' = - - - x VE~
(90)
Mo
This term is balanced by the exchange torque at the surface given by the second term in the Taylor expansion of eqn. (65): T~
2~¢
aM
(91)
Mo~MX O---j
=
The gradient in the surface energy in eqn. (90) is
VE" = E'o~o+ E~ cosec 0 ~
(92)
The surface spins are assumed to lie parallel to the bulk spins to give a parallel ground state. Consequently only the higher order terms, arising from torques acting on the dynamic terms of the magnetization, will contribute to the boundary conditions. The first of these terms has the form (93) After proper manipulation, this term can be expressed as 1 s 8T'=--~o(EO~COsecOmo-E'~cotOcosecOmo+E'
s
s
if4o(E oomo + E o~ cosec Om~)#
(94)
This surface torque will be balanced by the surface exchange torque in eqn. (91) which has the form =
Mo\ oy
aY 7
(95)
158
P . E . W1GEN
Equating the terms in eqns. (95) and (91), the boundary conditions at the film surface are expressed in terms of the matrix equation
E
cosecobI l
Om~[O = _ 0y
(96)
[_(gs0,~ cosec 0 - es~ cot 0 cosec 0) 8s,¢,,~cosec26
d
where gu = Eu/2Jz¢" Choosing the usual geometry with 0o = rt/2, eqn. (96) becomes :mY4~t
= --
so0 ~s00
(97)
The solutions to the equation of motion in eqns. (80) can also be expressed as m o = { G cos(kly + 6) + 71 e x p ( - k2y)} e x p ( - i ~ t )
(98)
and similarly for toO. Here G -- (ct12+ fl 2)1/2 and tan 6
-
c~
fl~
(99)
The angle ~ is the spin wave phase shift observed at the boundary. Using the boundary conditions ofeqn. (97), Stakelon 46 has shown that the relations which the spin wave number must satisfy are kl tan 6 = k2K+ + ~rs°°~s~4~
(100)
kE+K_
where K+ = ½{o~s00(1___0") + ~s4~4~(1 +0")} O" = (1 __~,~+~r~-)- 1/2
and [2 + =
ioJ/]; + Ebo4aM o (Ebo0 -- Ebo~)/2Mo
In the last expression, the E b terms are those obtained previously from the energy in the interior of the film. 4.2.1. Unidirectional surface anisotropy
Let us consider the case of an effective surface energy density of the form E s = - Mo"/-Pu'f = Mo Hsu'f sin 0 cos(4~-th)
(101)
where q~ is the direction of the unidirectional anisotropy energy. This is equivalent to the unidirectional field proposed by Puszkarski in the microscopic model (see eqn. (54)) From this expression f0-r the surface anisotropy ESoo = ESe~¢ = HsUrfMo sin 0 cos(q~s-- ~)
and
(102) E~o4~ = 0
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
159
In addition a = 0. This is due to the absence of higher order anisotropy energy contributions in the bulk field terms. Then K± - H~UaM° 2,~t sin 0 cos(~bs- ~b)
(103)
and the dispersion relation gives
H,UrfMo
kl tan 6 = ~
sin 0. cos(q~,- 4~s)
(104)
For this example, there is no coupling between kx and k2 and the results are in exact agreement with those of Puszkarski for a unidirectional field.
4.2.2. Uniaxial surface anisotropy The surface anisotropy energy having the next higher order term in the spherical harmonics is a uniaxial term of the form E ~ = K s sin20 sin2q~
(105)
At 0 = n/2, the derivative terms become
ESoo = _ 2 K s sin2~ ES4,~ = _ 2KS(sin2~b - cos24~) and
ESo~ = 0 From these terms, K± -
(3 cos2fl- 1 + ~ sin2fl)
(106)
where fl = ~ / 2 - 4~ is the angle between the magnetization and the film normal. In eqn. (100) the terms in k2 will dominate the right-hand side and the boundary conditions reduce to k~ tan 6 ~ K± K s
~ - 2 ~ ( 3 cos2fl - 1 + a sin2fl)
(107)
A surface anisotropy of uniaxial symmetry has been identified in YIG films 27'28. Using the procedures outlined above, the surface torques can be evaluated and the phase shift at the surface as well as the spin wave spectrum can be calculated. For small values of K s, the dimensionless quantity k I tan 6/(KS/2d) determines the standing spin wave spectrum. In Figs. 9, 10 and 11 this relation is shown as a function of the orientation of Mo at 3 GHz, 9 G H z and 23 G H z respectively. The bulk energy density for these figures is assumed to have the form E b = 2nMo 2 sin20 sin2~b -- Moll 0 sin 0 cos(~b - ~bn)
(108)
where Ho is the applied external field having the coordinate values On = n/2 and 4~nThe three curves correspond to the following conditions. Curves a are the angle dependence predicted in Section 3 which was considered in the circular precession approximation where a = 0. Curves b are the angle dependence predicted from eqn. (107) with the elliptical term included. Curves c include the effects of the non-parallel
160
P.E. WIGEN
-2 Fig. 9. The angle dependence of the surface parameter for YIG obtained for a uniaxial surface anisotropy energy (magnitude of the surface anisotropy, 0.03 erg cm- 2; frequency, 3 GHz): curve a, prediction in the circular precession approximation; curve b, prediction in the elliptical precession approximation; curve c, prediction in the non-parallel ground state configuration. (From ref. 46.)
-2 Fig. 10. Same as Fig. 9 except for a frequency of 9 GHz.
~ c I
I
I
Fig. 11. Same as Fig. 9 except for a frequency of 23 GHz.
i
161
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
ground state of the magnetization which will be discussed in Section 5. It should be noted that the latter terms have a significant contribution except at high frequencies where to/y >>4 x M . A further evaluation of the dispersion relation in eqn. (83) requires an explicit form for the energy density in the magnetic material. The energy density of a magnetic material consists of Zeeman, magnetostatic or demagnetization, uniaxial anisotropy and crystalline anisotropy energies. The Zeeman energy density is given by Ez = - M ' H o = M o l l o sin 0 cos(~b - ~bu)
(109)
where H0 is the magnetic field applied in the x y plane for the coordinate system illustrated in Fig. 8. The magnetostatic energy is given by E m = 2riM. N" M
(110)
where N is the demagnetization tensor. For thin film geometry the magnetostatic energy can be expressed as E m = 2nMyz = 2riM02 sinZ0 sin2~
(111)
The uniaxial anisotropy energy having an easy direction normal to the film will be written as: E u = -- K u sin20 sin2~
(112)
Since the magnetostatic energy and the uniaxial anisotropy energy have the same angular form, these terms cannot be distinguished in the FMR. Thus the uniaxial term will be eliminated and the magnetostatic energy 2gM 2 will be understood to mean an effective uniaxial energy 27tM 2 - K u. The crystalline anisotropy energy can be written for a cubic structure as Ec=
+
113)
where Kx and K2 are the first and second order anisotropy energy constants, and i,j and k refer to the principal crystal axis directions [100], [010] and [001] respectively. In a majority of the cases involving magnetic garnet films, the second-order anisotropy energy constant is negligible and will not be included here. Additional references treating the K2 term are available in the literature 47'.8. In terms of the coordinate system in Fig. 8, the expression for a (111)-oriented film becomes M \4 =
-3
Y-4"'"-2
~'"Y-
fsin40 cos4~b sin't0 sin4~ cos40 ,~ ~ t-~+ K~[ ~ 3 x//-2 san't0 sin t~ c o s 3 ~ b _ 4 +-3-
3
~
y x~- ~ y , }
sin2(20) cos2~b Jr 8
sin2(20) sin(2~)}
(114)
162
P.E. WIGEN
The total energy density is then the sum of the four terms contained in eqns. (109), (111), (112) and (114). The equilibrium orientation of M is found by setting both Eo and cosec 0 E 4, equal to zero. Setting Eo = 0 implies that the equilibrium value of 0 is n/2. Setting E~ = 0 with 0 = n/2 yields the relation MoHo sin(~b- q~n)+ 2nMo 2 sin(2~b) +
+ K 1[~2 sin(2q~) { 1 -- 7 cos(2~b)} +~ cos2(b {2 cos(2~b)- 1}] = 0
(115)
There is in general no analytic solution to this equation and it will be retained in this form together with the dispersion relation. Evaluation of the second derivatives of the energy allows the dispersion relation to be solved explicitly. For the energy terms considered, Eoe~ = 0. Making the transformations ct ~ n/2 - (9n and fl ~ n/2 - ok, ct and fl become the angles between H a n d M respectively and the film normal. For a film with the [111] crystallographic orientation normal to the film plane and M a n d Hchosen to lie in a (110) plane, the derivatives yield the dispersion relation ( ~ ) Z = {Ho cos(o~-fl)-4nMo cos(2fl)+gl(fl)+ DkZ} ×
× {Ho cos(or - f l ) - 4nM o COS2fl -]- g2(fl) + Dk2 }
(t 16)
In addition, the equilibrium orientation of M is given by H o sin(or- fl) + 2 n M o sin(2fl) + ga(fl) = 0
(117)
where g~(fl) = HK(~ sin 2(2fl)--~cos 2 4 f l _ 2xf2 _ sin(2fl)(1 + ~2 sin 2fl) } g2(fl) ----- - - H K [ l c o s ( 2 f l ) + 7 c o s ( 4 f l ) + ~ - { s i n ( 2 f l )
g3(fl) =
- 2 sin(4fl)}]
(118)
H K [ 1 sin(2fl){ 1 + 7 c o s ( 2 f l ) } - - ~ sinZfl{ 1 + 2 cos(2fl)}]
and HK = 2K 1/M is the effective cubic anisotropy field. Setting k = 0 in eqn. (116) determines the magnetic field value for which the uniform mode resonance will occur at the frequency ~o: 1
H, cos(~t - fl) = 2riMo{cos2fl + cos(2fl)} - ~ {gl(fl) + gz(fl)} + f w \ 2) 1 /2 1( 2 (119) + ~ ~[4nMo{cos f l - cos(2fl)} + g2(fl) - - g l ( f l ) ] 2 + 4 ~ -) ~
The position of the nth spin wave mode with k = k, is then found from eqn. (116): Dk, z H, = H. (120)
cos(~-/~)
If the magnetic field is applied along the film normal (~ = fl = O) the expression
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
163
simplifies to the form H~± = Hux - Dk~ 2 09 4 M 2 = 7 + It eff+-~HK-Dkn 2
(121)
The dispersion relation in eqn. (116) and the equilibrium condition in eqn. (113) can be used to evaluate the film parameters D, 4 ~ M eff, H K and ~. The surface anisotropy energy at each surface can be evaluated from the angle dependence of the spin wave spectra. 5. NON-PARALLEL GROUND STATE
To this point, the description of the spin wave excitations has assumed that the ground state of the system is an array of parallel spins whose direction is determined by the equilibrium of the "bulk" torques. Several researchers 49-51 have suggested that surface torques derived from the surface anisotropy energy will give rise to a non-parallel spin ground state which will influence the spin wave excitation. Mills 36 has predicted changes in the magnetic surface ordering due to instabilities resulting from different bulk and surface exchange interactions. In this section the work of Stakelon46 is reviewed by extending the continuum model of Section 4 to calculate the effects of the surface anisotropy on the reorientation of the magnetization near the film surface and thereby to determine the non-parallel ground state and its contribution to the "dynamic" boundary conditions. For conditions in which 09/4n~M is less than unity, the dynamic effects of the non-parallel interior spins will be as important as the surface anisotropy energies acting directly at the surface plane. Recently Harada et al.52 have calculated the ground state configuration in the presence of a uniaxial surface anisotropy energy using a microscopic theory. The results they obtained are also consistent with the results of the continuum model reviewed here. Using the coordinate system described in Fig. 8, the magnetization M at any point will be given by a time-independent component M 0 which describes the orientation of the spins in the ground state plus a smaller transverse component mt which is time dependent and has a spatial variation normal to the film surface. The time dependence of the magnetization can then be written in the form M(y, t) = Mo~,(y) + mo(y) exp( - itot)~0(y)+ m~(y) exp( - kot)~(y)
(122)
Here the unit vector i,(y) is changing as a function of y in such a fashion that it is always parallel to the equilibrium direction of the magnetization Mo; the unit vectors e,o(Y) and i~(y) are thus modified to maintain a mutually orthogonal coordinate system. The surface torques /'So are obtained from the surface anisotropy energy density Es(O, dp) as given by eqn. (90). The surface torque is balanced by the exchange torque given by eqn. (91). For the non-parallel ground state, the static term in the surface torque will produce a bending of the spins near the surface which will be balanced by the exchange torque. In this case, the surface torque will be expanded in terms of the torque /'So acting on the static term of the magnetization M o and the
164
P.E. WIGEN
higher order terms 6T ~arising from the torques acting on the dynamic terms of the magnetization: TS= T~o +~STs
(123)
The static boundary conditions are given by (6~-Y~)= = o~b°'
--°~s~ c°sec20°
and
(124) (8~Y)= 0 ° ' = - ~ s ° o
where ~s
= ES / 2 ~
The angles q~o and 0 o are the equilibrium positions of the spins in the bulk of the magnetic medium while ~bo' and 0 o' are the angle derivatives evaluated at the surface. The behavior of the spins in the bulk region will be determined by the balance of the bulk torque terms with the exchange torque. Evaluating the exchange torques and equating them to the bulk torques gives the equations W - s i n 0cos O(c~') 2 = o~b0
(125a)
~b"+ 2 cot 0 0'~b' = ~ b
(125b)
and
which can be solved in conjunction with the boundary conditions in eqns. (124) to evaluate the ground state configuration of the spins. To evaluate the ground state configuration, the equations are linearized by letting O(y) = 0 o + ~O(y)
(126a)
and ~b(y) = ~bo+ 6q~(y)
126b)
where 60(y) and 6~b(y) are the deviations of the spins from the bulk equilibrium orientations 00 and ~bo respectively. If E s is zero, the variations in the orientation of the magnetization would be zero and the equilibrium orientation of the magnetization (0 o and ~o) will be given by the equilibrium condition EbO0
=
Eb°~ = 0
(127)
For a non-zero surface energy density and considering the usual condition 0 o = rc/2, eqns. (125) have the form ~50" = 8b°oo ~50+ ~b°o4, ~5q5
(128a)
6q~" = d~b°o,~ 60 + 8b°4,~ 8~b
(128b)
and
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
165
The boundary conditions from eqns. (124) become 8~,'(0)
=
- a's~
(129a)
80'(0) = - - ~ 0
(129b)
and
To solve these equations, a surface energy is considered that does not produce a torque out of the yz plane, i.e. ESo = 0, and a bulk energy term that is uniaxial such that Eb°o~ = 0. Then eqns. (128) and (129) reduce to 80' = 0
(130a)
8~b" = 8b0~4, 8~b
(130b)
and
The solution to eqns. (130) is 80 = 0
(131a)
84) = A 4, exp(Ay)
(131b)
and
where A 2 = ~bO . From the boundary condition in eqn. (129b)
8dp'(O) = AA4, = -- gs 4,
(132a)
which yields gs~ -
At~ = - - A
gs~
(132b)
(~b°~b~b)l/2
For YIG at a frequency of 25 GHz, a value of E ~ ~ E ~ of about 2 x 10 - 2 erg cm -2 (ref. 28) gives A = 1.3 x 106 cm-1 or a decay length of 7.7 nm. The magnitude A~ of the maximum deviation of the magnetization at the surface is about 0.02 rad or approximately 1°. The bending of the ground state magnetization near the film surface will produce a perturbation on the energy density in the surface region causing an additional torque on the magnetization that is appreciably within a region of a few decay lengths A - 1 from the surface. Beyond this region, the spin wave will take the form A cos(kly + 6 + ~') similar to that in eqn. (98) where 6' is the additional phase shift introduced by the perturbation in the surface regions. For small values ofkx, mo and m~ are not changed significantly by the perturbation and the additional torques can be included as an effective surface torque by integrating the volume terms over the effective surface region. The normal modes in the non-parallel ground state system require a solution of the equation of motion by considering the higher order terms of the surface torques 8/~ acting on the dynamic terms of the magnetization. Due to bending of the ground state magnetization near the surface, the variation in E b ~ and Ebo0 of eqns. (80) will produce the additional terms to the equations of motion 8 T O = EbOq~oO
Mo
8tfp mo -t- ET~_O° 8 0 /14o
m~
(133a)
166
P.E. WIGEN
and
8T~ =
Ebo0~ boo0080 2d 2 Mo 8q5 mo+ E.Mo mO+~o(SC~) mo
_
(133b)
From eqns. (131), the relations 80 = 0 and 8~b = -ES~,(Eb~¢) 1/2 exp(Ay) can be substituted into eqns. (133) and the resulting relations integrated to evaluate the spin wave phase change from the bulk region to the surface. The resulting phase change can then be treated as an effective surface torque in the equations of motion to give
6TSo
Es
Eb~bq~tb"~ 6m6
-
Eb~o
(134a)
Mo
and
8TSc~ - Eb°°~ ":" ~ moqmo Ebe~,~ Mo 4.4AM o
(134b)
The combination of these terms with the surface torques due to the boundary conditions given in eqn. (96) determines the final form of the surface torques: 1 /
E bO
\
- ( E ~~¢ +~Eb°¢+ ¢ ~ E s~m TS° = -Mo( ~} ,
(135a)
and 1
= Moo
Es
-Eb°oo#'E
~
°O"r E - - ~
~Es4,)X2/ 2~2iI \1/2")
~
mo
The magnitude of ES~ is comparable with that of ES~ although it may differ in the functional dependence on ~b. Similarly Eb°~/Eb¢~¢, is nearly unity in many situations with the result that the additional terms may contribute significantly to the spin wave spectrum. For the conditions described in Section 4 and Figs. 9-11, curves c represent the angle dependence of the pinning parameter including the effects of the inhomogeneous ground state. 6. VOLUME INHOMOGENEITY MODEL
In a thin film, the exchange interaction attempts to make V,M = 0 at the surface. If the internal field is everywhere uniform, a spin at the surface and one directly in from the surface must have the same value ofm since all spins will precess at the same frequency in a normal mode. A non-zero slope ofm would give an extra torque on the surface spin because of the missing neighbors. In addition to surface anisotropy energy, a change in the surface magnetization or an inhomogeneity in the anisotropy energy in the surface region of the film will also produce modes which are not unpinned. In the volume inhomogeneity model, the surface region is assumed to have a magnetization which drops off from its bulk value to some lower value in a thickness e which is much smaller than the film thickness as shown in Fig. 12. This model has been considered in detail by Sparks 53. For e # 0, the spins in the surface region are "off resonance" when the spins in the bulk region are "on resonance". As the surface spins are exchange coupled to the
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
167
bulk spins a phase change 6 in the spin wave will occur in the surface region. If the surface region is approximated as dropping suddenly to Mo/2 at a distance e/2, a phase change of n/2 will occur in a critical thickness ecr of the order of (nsl/Mo24) l/2 where ~¢ is the exchange constant. For YIG, ec, is about 55 nm in this model. If the surface region is much less than ec~,the surface spins will remain unpinned. Ife <~ ecr is not satisfied, the surface exchange cannot hold V,M = 0 at the surface and the low order spin wave modes will be partially pinned. Surfoce Region
Bulk Region
Surfoce Region Approximolion for Surfoce Region
-½~
i Z~
-vS
J
o
L.
,~
-4"
Fig. 12. Variation in the saturation magnetization across the thickness of the film. (From ref. 53.)
The case of a surface layer of different M is useful in establishing general features of the spin wave spectrum. The equation of motion for the magnetization in the circular precession approximation is given as 2~¢m"
--+/¢2m
Mo
(136)
= 0
where m" = d2m/~2yand r is the root of the secular equation of eqns. (80) when a variation in the magnetization is included:
(/co'X2 ) I/2 t¢2 : ~ - ~ ) +(27zMosin20)2~ - H d f
2~¢My" Mo 2
2nMosin20
(137)
For perpendicular resonance sin20 = 0 and x 2 reduces to t ° - H=ff 2~¢Mi' = 7 Mo 2
(138)
For parallel resonance, sin20 = 1 and f/'~\2
) 1/2
2~¢Mr,,
(139)
2
The thinner the boundary region the larger will be the M," term in eqn. (137). For sufficiently thin surface layers, this term will dominate the other terms in eqn. (137) and eqn. (136) will reduce to
Mom"= raM,"
(140)
Integrating once with m = Mo = 0 at the film surface gives
Morn' = raM;
(141)
Since these equations hold everywhere in the surface region, this expression will also
168
P.E. WIGEN
be valid at the edge of the bulk region. For a homogeneous magnetization, m' = 0 at the boundary of the bulk region and the spin wave is unpinned. Equation (138), with co/7 = HeffBq-Dk, 2, where HeffB is the effective field in the bulk region of the film and Dk, 2 is the exchange contribution to the frequency for spin wave modes, becomes K ± 2 ~-
Dk,, 2 +
neff B -- Hefts
2~'M " -'-Y
(142)
Mo 2
where H~efs is the effective field in the surface region of the film. The maximum value of Heffa-Heffs in the surface region is 4nMo. For Dk,,Z~ 4rtM, the term 2 d M f / M o 2, of the order of 4Die 2 in the surface region, is larger than the other terms on the right-hand side of eqn. (142) when 2~4 e2 ,~ - Mo2rt
for Dk. 2 ~ 4nMo
(143)
For Dk~ 2 >~4~M, the corresponding result is 2, e2 , ~ 7I
for Dkn 2 ~>47tMo
(144)
where k, = 2rt/2,. An expression valid in both limits is
2 d (1 ~2 ,~ ~cr 2 - -
Dk"2 \
Mo2,/l~\ + 4 ~ o o )
1 (145)
Ramer and Wilts 54'55 have used this model to analyze spin wave mode positions and intensities in thin YIG films. Their analysis extends the model to an asymmetric film having two surface layers and considers the ellipticity, angle dependence, anisotropy fields and g values in the bulk region and surface region of the film. The surface layer at the substrate interface is assumed to be due to a diffusion zone between the G G G substrate and the YIG film which has a composition Y3-rGdrFe5 xGaxO12 where x and y depend on growth conditions and annealing treatment. In attempting to duplicate the experimental spectrum, the surface layer thickness, magnetization and, in the substrate interface layer, the Mcd to Mve ratios were varied to give the best fit to the mode locations at all angles. Typical results for room temperature data are shown in Fig. 13 and the parameters used are shown in Table II. Experiments of spin wave resonance in garnet films produced for bubble devices show large deviations from the quadratic law 56"57. Having the composition Y2.ssLao.15Fe3.asGal.25012 and grown from a non-stirred melt, these materials have a growth-induced anisotropy and a stress-induced anisotropy which are very sensitive to the composition and growth conditions. The spin wave resonance data were used to identify various strata produced in the film during the growth process. The volume inhomogeneities were identified by observing the mode intensities while the film was etched to smaller and smaller thicknesses. Another set of experiments involving the volume inhomogeneity model involve resonance in ion-implanted films. In this case e >> ecr in Fig. 12 and the spectra from these "layered" films will exhibit resonances from each of the layers independently 5a. The shifts in the resonance field and their angular dependences will allow the
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
169
200 o
a++ +
e# +
3 D D ~a+8 #
D+ a
MAGNET ANGLE (deg)
90
Fig. 13. The angle dependence of the magnetic field separation of the observed and calculated positions H. of the spin wave modes from the calculated position H, of the uniform mode: +, calculated; Fq, experimental points for a film having a single surface mode at the parallel orientation. (From ref. 54.) TABLE II ANALYSISOF ThE DATASHOWNIN FIG. 13 (FROMREF. 54) Orientation Total thickness (ttm) Annealing temperature (°C) Annealing time (h) Frequency (GHz) A (erg cm- ~) T (~) 4nMb (Oe) ~b t s - I O e -
1)
L, (A) Mb/Ms
[100] 0.47 1000 6 9.16 3.593 x 10- ~ 4200 1735 1.767 x 10- t o
7b/•eff
4~MF¢(Oe) 4'/tMGd(Oe)
470 1.95 0.9977 1400 510
T (~) (K,) 1(erg cm -2) (K,)2 (erg cm- 2)
4270 0.1085 0
L, (/~) Mb/M" 7b/7,
e v a l u a t i o n of the p a r a m e t e r s in each layer. T h u s F M R has b e c o m e a viable tool for the e v a l u a t i o n of the effects of ion i m p l a n t a t i o n o n m a g n e t i c g a r n e t films 59-62. A typical e x a m p l e of this a p p l i c a t i o n is s h o w n in Figs. 14 a n d 15 from A l g r a e t al. 62 In Fig. 14, a new set of resonances is o b s e r v e d in a Y2.ssLao.l 5Gal.15Fe3.ssO12 film i m p l a n t e d with 10 ~4 N e + ions c m - 2 of 300 k e V energy. By careful e v a l u a t i o n of the p a r a m e t e r s , A l g r a e t al. were able to d e t e r m i n e the profile of the internal field as well as the e v a l u a t i o n of the d i s p e r s i o n c o n s t a n t in b o t h the b u l k region a n d the ion i m p l a n t e d region. T h e spin wave d i s p e r s i o n c o n s t a n t 2 ~ / / M was e s t i m a t e d to
170
P.E. WIGEN 4800
~
x
0 - Q--O-O'Ol~l~
/
4600
~
4400 0 B
4200
!
4000 bulk
"~
IOx I I I I 3400 3600 5800 4000 4200 4400 4600
ion-implantedregion
3800
0
H (Oe)~
200 210 220 230 240 2.50 L (ffm} ---->
Fig. 14. Spin wave resonance spectra of a Y2 a~La0.,~Gal.,sFe3.esO12 film recorded with the applied magnetic field perpendicular to the film: spectrum A, before implantation; spectrum B, after implantation with 10 ~4 Ne ÷ ions cm - 2 at 300 keV. The breaks in the spectra indicate a change in the instrumental gain. (From ref. 62.) Fig. 15. The variation in H eu with the film thickness. (From ref. 62.)
change from 1.58 x 10- s Oe in the implanted layer.
cm 2
in the bulk region of the film to 0.45 x 10 8 0 e
cm 2
7. MAGNETOSTATIC WAVES The theory of magnetostatic modes was originally given by White and Solt 63 and Walker 64'65 for spheroidal specimens. Since then there has been a considerable amount of experimental work on the study of magnetostatic modes in ellipsoidal specimens. The formal theory that was given by Walker for the magnetostatic modes of spheres was adapted by D E 29 to magnetically ordered films of infinite extent in the directions normal to the thickness of the film. Sparks 53 carried out some calculations for thin films using an alternative to the procedure of DE. Experiments on microwave absorption in thin Y I G films were performed by Tittmann 66'67, Storey et al. 68 and Borghese et al. 69 and in layered compounds by Reimann e t al. 7° The original version of the D E theory treated the magnetic materials as isotropic, which is often not the case. The inclusion of anisotropy in DE theory is straightforward and was done by Akhiezer et al. 71 and Schneider 72. Magnetostatic modes are magnetic excitations in a magnetically ordered system. They hax, e wavelengths 2 that are of the same order of magnitude as the linear dimensions of the specimen under consideration. For garnet films having a thickness greater than 5 lam the effects due to the exchange interaction are negligible. The magnetostatic modes can be regarded as normal modes of "vibration" of the magnetization vector M(r, t), which is a function of position r within the material at time t and can be described by the magnetostatic Maxwell equations VxH = 0
(146)
V.H = -4rcV.M
(147)
and
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
171
plus the constitutive torque equation: 1 aM T dt : g × H e f f
= M x { V E - h exp(-iogt)}
(148)
where the exchange contribution in eqn. (69) is neglected. The effective field in eqn. (148) has the form H etf =
H o - H u - 4nN'Mo + h exp(-iogt)
(149)
where H o is the applied field, Hu is the uniaxial anisotropy field, N is the demagnetization tensor and h is the transverse r.f. field. For a thin film lying in the x z plane (see Fig. 8) the only non-negligible demagnetization term is Nyy = 1. Magnetic garnet films grown epitaxially on a G G G substrate have a stress-induced anisotropy and, in the case of rare-earth-doped garnets, a growth-induced anisotropy with a symmetry axis normal to the plane. Having an energy expression of the form given in eqn. (112), the torque due to the anisotropy energy, defined by eqn. (72), produces an effective field H , = - V E , / M o along the y direction. In the small signal approximation, the magnetization can be written as M = Moil + m
e x p ( - iogt)
where fi is a unit vector in the direction of the effective field Heft and m is the small r.f. component that lies in the plane perpendicular to ti. In this approximation, eqn. (148) relates m to h and defines the dynamic susceptibility tensor ~ such that m = ~.h
(150)
The dynamic permeability tensor IXis related to Z by IX = 1 + 4 ~
(151)
where Ix depends on the direction of the applied field H o. While the normal modes can in principle be evaluated at an arbitrary orientation, the solutions become very complex. As a consequence, the magnetostatic modes are normally evaluated at two geometries. Firstly, for the magnetization vector lying in the plane of the film (Mo//~) and, secondly, for the magnetization vector being normal to the plane (Moll5,). Using the usual procedures for determining the equation of motion by neglecting high order terms and the losses, the H o//~ or parallel case of eqn. (148) reduces to iO~mz = - ~{(Ho - H u)my- Mohy }
(152)
i~omy = y ( - Horn z + M o h z )
(153)
and From eqns. (152) and (153) the terms in the dynamic permeability tensor can be readily derived: #xx = 1
I~xy = #yx = #xz = #zx = 0
B
/~zz = 1 + 4 x ~ / a y y
A
= 1 +4n/iB_t22
f2 /~y = - #yz = i4n A B - - t22
(154)
172
P . E . WIGEN
where A=--
Ho
Ho-H . B - - Mo
Mo
f2-
~o 7Mo
Similarly, for the perpendicular case (H0 is perpendicular to the film surface, i.e.
HoflY) the dynamic permeability tensor has the following terms: ].,tyy --~-
1
#y~ =/a~y = #r:, = Pxy = 0
pxx= p= = 1 +4/tC2
C
~,~2
(155)
f2 /ax~ = -/a,x = i4rt C2 _ 02 where C= f2-
H o - 4 ~ M + Hu
Mo (D
~Mo
The dispersion relation can be derived from Maxwell's equations: Vxh = 0
(156)
V.(h+4nm) = V'la'h = 0
(157)
Equation (156) permits the introduction of a magnetic potential ~ such that h = V~ and eqn. (157) becomes V'wV~ = 0
(158)
For a rectangular film, the normal modes are described by the frequency 09, the inplane wavevector ki, = (kx,kz) and the out-of-plane wavenumber ky. The general solution for eqn. (158) then takes the form ~i,t = {a exp(ikyy) + b e x p ( - ikry)} exp{i(kxx + kzz)} exp( - kot)
(159)
inside the film. Substitution of eqn. (159) into eqn. (158) yields kx 2 + flyy kr 2 + llzz kz 2 = 0
(160)
for the parallel case and lazz(kx 2 + kz 2) + ky 2 = 0
(161)
for the perpendicular case. Outside the film, ~ext = C1,2 exp( + ~cry)exp{i(kxx + kzz)} e x p ( - iogt)
(162)
for y > d/2 or y < - d / 2 . Equation (158) then yields Kr 2 = kx 2 + k z 2
(163)
For a given in-plane wavevector, eqn. (160) determines ky which, depending on the permeability, can be real (volume modes) or imaginary (surface modes).
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
173
Furthermore, the boundary conditions must be satisfied. This requires that the normal components of b and the tangential components of h be continuous: for
y = + d/2 ~y~ext :
(#yx~x "+"]~yy~y +/2yzt~z)¢in,
(164)
and ~=xt = ~,n,
(165)
This systemof four linear homogeneousequationsyields the characteristicequation
{xy2 + (k,,~yx + k=~y=)2-(kr#ry) 2} sin(kyd)- 2Kykr#yy cos(kyd) = 0
(166)
The dispersion relation for the parallel case falls into two categories, volume modes and surface modes. The boundary between the two regions is determined by a critical value in the ratio k=/kx. For volume modes ky 2 > 0 and the dispersion relation is
tan(kxdct1/2 + mr) = 1 +
2btyY~t1/2(1 + 172)1/2 if2 _ I/~y, 12r/2-- ~t/~ry2
(167)
and for surface modes (ky2 < 0) tanh(kxdl ctl 1/2) =
2/~yyIotl 1/2(1 + r/2)1/2 1 -t- r/2 -- I/G,I 2~2 -t- I• I/tyy2
(168)
where r/= k=/kx, ~ =/t==//~yy(r/=2 - r/2), ~/¢2 = _/~=- 1 and kr - - kxot 1/2. Figure 16 gives the traditional magnetostatic mode spectrum of a ferromagnetic slab magnetized parallel to the surface and for which H, is set equal to zero. A series of planes, corresponding to the different n values in eqn. (167), is shown for the volume modes in the frequency range between ~Ho and ?{Ho(Ho +4rcM)} 1/2. One surface is observed to rise above the high frequency volume mode band at a critical angle determined by r/¢. The upper frequency limit of this surface is 09 = ~(Ho
+ 2nM). The inclusion of the uniaxial anisotropy field has some very unusual effects on the shape of the magnetostatic mode pattern as shown in Fig. 16. The bottom of the
oT Surface / modes 4.,_._.=o=/f ( Ho+ 2 TrMo) ~
""
o
H
,~k=
< ' ~ w :i[HO(H°+ 4 "M°)]''.
w=lfHo
J
Fig. 16. Magnetostatic mode spectrum of a ferromagnetic slab magnetized parallel to the surface. The separation between the surface and the volume modes originating at ~c should be noted.
174
P.E. WIGEN
volume mode band at the og,kx plane is shifted to the frequency to = y{Ho(Ho _ Hu)} 1/2, while the frequency at kx = ky = 0 is ~o = 7{Ho(Ho - H u + 47tin)} 1/2. The most significant effect has to do with the surface modes. Under appropriate conditions, the uniaxial anisotropy field will distort the volume band for k~ :~ 0 and some of the modes in the surface branch will become volume modes. An example of this case will be discussed later. For the perpendicular orientation, only volume modes are allowed and the dispersion relation is given by
tanI#xx'/2(kx2+ky2)l/2d+nn}-2(-Itxx)l/Zl + #xx
(169)
where n is an integer. For the major series, n = 0. In this orientation, the role of the uniaxial anisotropy field is to shift the frequency of the magnetostatic modes by the amount Am = 7H,. However, the relative positions of the mode spectra will remain unchanged. In a typical magnetostatic resonance experiment, the samples are restricted to finite dimensions lx and lz. Thus the allowed resonances can be characterized by the quantized components of kx and kz given by the relations
kx - nx~
kz - ngt It
Ix
(170)
where nx and nz are assumed to be positive integers. The intensity of the absorption due to the excitation of a mode (kx,kz) by the microwave field h will be proportional to the value of m.h integrated over the sample, where m is the magnetization associated with this mode. For the parallel case m can be written in the form {0,mr(r) exp(-i~ot), m~(r) e x p ( - iogt)} where
_ 1)o~(x,y,z)
o~(x,y,z)
and
(171)
mr = #yz
8~(x,y,z) +(l~yr_ 1)8~(x,y,z) ~z 8y
according to eqns. (152) and (153). In practice, the dimensions of a sample are rather small compared with the dimensions of the cavity used in microwave absorption measurements. Therefore h can be regarded as independent of the position r throughout the region occupied by the specimen; and it can therefore be written as (hx,hr,h,)exp(-iogt). The absorption of power by the mode (nx~/lx,n~/lz) is then given by m. h = ½ Re(re.h*)
(172)
Substituting for m and h* and recalling that
= X(x)Y(y)Z(z) = {a exp(ikyy) + b exp(ikyy)} exp{i(kxx + k:)} e x p ( - i~ot)
(173)
175
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
the integration can be performed over the volume of the sample:
=
fleA
d"2
~d12 j-a/2
Y(y)dy
stntT)
sinlt--~- )
(174)
where A and B are functions of co, Hi,M o and h. From the form of this equation some general conclusions can be drawn. (i) There is a selection rule that no absorption will occur for any mode for which either nx or n= is even. (ii) If nx and n= are both odd, the mode intensity is inversely proportional to nx (x being the direction of the applied field). A typical example of a magnetostatic mode spectrum in a 12 ttm YIG film having a negligible value o f H u is shown in Fig. 17. The series of modes (n, 1) are the
•• (7,1}
(5.11
2572 2583 (1.3) (3.5)
1
(3,1) ~
2522 2534 41.5) (3,71
Imam mode)
(3.~7) 2818
r3,19).,,etc 28/.6 2871
/
f/
/'
2"177~1 2807
(1,13)-
(115)
2572 2583 (1.3) (3.5)
2666 2680 2706 2720 2743 2757 2777 {1.7) (3,9J {1.9] (3.11) 11.11) {3.13] (1fl3)
i3.15) 2790
/'
l
,,
289/. 2917 2935 295A
/'
2835 2862 2 8 8 6 2908 2929 2949 2966 11,17} i1.19J.-.,etc
Fig. 17. The magnetostatic mode spectrum at parallel resonance for a 12.0Wn Y I G film (planar dimensions: I~ = 0.58 mm; 1~ = 0.68 mm): spectrum a, main mode with lines on the low field Side and the first few lines on the high field side; spectrum b, continuation of spectrum a at higher fields; spectrum c, continuation of spectrum b at still higher fields and increased instrumental gain. (From rd. 68.)
176
P.E. WIGEN
surface modes for which t/is greater than r/c while the modes (1, n) or (3, n) are the bulk modes for which ~/is less than r/c. The observed separation between the modes and the values calculated from the DE theory show agreement that is typically within 1% or 2% of each other. For thin YIG films (less than 1 lam) the exchange energies become comparable with or greater than the magnetostatic energies 73. A number of publications by Wolfram and DeWames 74 and by Sparks 53 have considered these magnetoexchange modes. A typical example of the magnetoexchange branch repulsion is shown in Figs. 18 and 19. It is worth pointing out that the consequence of the presence of an easy plane anisotropy energy (Ku negative) is the existence of volume mode excitation within
Ilo =._.~4 n I00 9O 80 7O
.5
1~--20
Oe
4O 50 2O I0 0
,I
t 5O8O
o
o
:!
o
o
o
o
o
o
o
o
o
o
o
o
o
o o o
I I I I I I_LI I I I I I I I 2 3 4 5 6 7 8 9 I011121514
Mugnetic Field (Oe)
n
F i g . 18. Magnetoexchange
spectrum in a 0.48 ~tm Y I G film: the high field modes represent the magnetostatic modes in the n = 1 exchange branch while the lower field mode represents the magnetostatic modes in the n = 3 exchange branch. (From ref. 73.) F i g . 19. A plot of the mode number vs. field position for the first two magnetoexchange branches shown in F i g . 18. The repulsion of the modes as the n = 1 branch approaches the n = 3 branch should be noted. (From ref. 73.) (3,1)
(5,1}
(I,7)
(7,1) {I,I) (9,1) I
(I,21)
(I,19) A
(,,z31 ~
I
I 2:320
(11,1}
,|
II
I
I
I
I 2:360
I
I
I
I 2400
I
I
I
I 2440
I
I
I
I
I
I
2480
Mognetic Field (Oe) Fig. 20. A portion of the magnetostatic mode spectrum measured in a reduced calcium Y I G film having an anisotropy field H. = - 250 Oe: the gap in the surface m o d e region with modes (1,9)-(1,17) missing is clearly evident. (From ref. 70.)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
177
the series of surface modes. If this bulk excitation is suppressed, perhaps by an inhomogeneous anisotropy profile, a field gap will appear in the surface mode spectrum. This feature is clearly illustrated in Fig. 20. Finally, it should be noted that the existence of these surface modes is a direct consequence of the DE theory and is predicted in homogeneous materials. Their nature is very different from that of the surface excitations reviewed in Sections 3 and 4 which is due to the presence of a surface anisotropy energy. 8. PROPAGATING MAGNETOSTATIC WAVE DEVICES
The use ofmagnetostatic waves in a variety of device applications has led to the development of a large technological effort in this field. Magnetostatic wave dispersion delay lines have important applications in compressive receivers and for variable delay lines for phased array beam steering. These applications require large bandwidths but each application has its unique requirement for differential delay ranges. Magnetic garnet films with their variety of material parameters and geometries offer a large potential for application to these devices. As a result, magnetostatic wave properties are at present an active area of investigation. This section is not intended to be an extensive review of the device studies but rather an introduction to the variety of techniques available to generate devices of desired properties as well as to refer the reader to some of the appropriate sources. Magnetostatic waves are intrinsically suited for use in microwave delay lines 75 because they are readily generated at high frequencies and their velocity is frequency dependent. However, as observed from eqns. (167)-(169), magnetostatic waves are intrinsically dispersive and the group velocity is a function of frequency. Therefore the rate of change of the delay with frequency can be either positive or negative depending on the nature of the magnetostatic wave being propagated. In most cases of interest the magnetic bias field H o is along a principal axis because the directions of less symmetry are characterized by large "walk-off" angles where k is not along the desired direction of the device. A typical structure is shown in Fig. 21. The geometries of the device then dictate the nature of the waves being l
MJ:~tr~
-
~
Sul3strme
Fig. 21. Magnetostatic wave device geometry showing microstrip transducers and magnetic bias field directions: x direction, surface magnetostatic wave; y direction, backward volume magnetostatic wave; z direction, forward volume magnetostatic wave.
178
P.E.
WIGEN
investigated. These include surface waves, with H 0 and M o parallel to ~ and k parallel to g; forward volume waves, with H o and M 0 parallel to ~ and k isotropic in the xz plane; and backward volume waves, with H o and M o parallel to .~ and k parallel to .~. In garnet films used for magnetostatic wave devices, the magnitude of the uniaxial anisotropy energy is suppressed. As a result, K , is assumed to be zero in this section. Then the frequency ranges of these waves are as follows: for surface waves,
(Ho2+Ho4xMo) 1/2 <
(2 < H o + 2 7 t M o
(175)
for volume waves,
Ho < f2 < (Ho2 + Ho47tMo) 1/2
(176)
Thus the volume waves lie within the spin wave band while surface waves lie above it. The dispersion relations for the travelling waves associated with the three cases discussed above are shown in Fig. 22 for a Y I G film with an applied field H 0 of 712 Oe. The group velocity Vg = t3o.)/t3kis shown in Fig. 23. The group delay is given by 1/vg and is inversely proportional to the film thickness through the relation kd which is a function of frequency. i
6 i
I
=N >,
.
E
I
" . loo - FVW ~-
/
J Volume ~. _ w _ ~ , _
f
-2000
[
>
1000 Wave Number, cm-1
BVW
o
I
-lo00
50
2000
3
4
Frequency, GHz
Fig. 22. The magnetostatic mode dispersion relation for waves travelling in the positive y direction as defined in Fig. 21 : the parameters are those for YIG and a bias field of 0.4 x 4riM. (From ref. 75.) Fig. 23. The dependence of the group velocity on the frequency: the conditions are the same as those in Fig. 22. (From ref. 75.)
As observed in Fig. 23, the group velocity is non-linear in frequency. For device application, this non-linearity must be modified if the device is to be a non-dispersive delay line. For a fixed distance between input and outputteads, the group delay can be modified by means of a ground plane 76-79, by multiple layers a°-a7 or by using reflective arrays to change the frequency dependence of the path lengthsS8 93. When a ground plane is present, the dispersion relation will be significantly modified as the surface waves are not reciprocal. In addition its utility is limited by the ohmic losses in the metal plane 76 which are also frequency dependent. In general, the attenuation due to the ground plane is greatest in the regions where the ground plane is most likely to be useful and so one of the major limitations is the t i m e bandwidth product. A typical example of a ground plane group delay line operating at X-band
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
179
frequencies using forward volume waves is shown in Fig. 24. A YIG film 20 pm thick, spaced 20 ~tm from a silver ground plane and operating in a bias field Ho of 4800 Oe shows a 190 ns differential delay over a 1.2 GHz bandwidth 75.
250
S* 200
~" o
150
o
tO0
50
t
0
;
8,6
,i,
,18
10.2.
(GHz)
Frequency
Fig. 24. Group delay time v s . frequency for forward volume waves in a YIG film having a thickness of 20 ~tm and spaced 20 ~tm from a silver ground plane (applied field, 4800 Oe). (From ref. 75.)
Multiple-layer devices interact with each other and behave in some ways similarly to a layer adjacent to a ground plane in that the effective thickness is a function of the wavenumber. By proper choice of thicknesses, magnetizations and spacings of the layers, a device can be "tailored" to give delay characteristics without incurring the losses due to a ground plane• The band edges occur at different frequencies if the magnetizations of the two layers differ• Such a discontinuity in the dispersion relation has been calculated 84 and observed experimentally aa. The best results for the bandwidth are obtained when the magnetization of each layer is kept the same. A comparison for a device having two YIG layers of thickness 20 ~tm and separated by a non-magnetic layer of 40 I~m thickness with the calculated delay operating in the 8.5-9.4 GHz region is shown in Fig. 25 75. 2~
I
i
t
i
i
i
1
l
I
~200
150 ¢
1oo @
a
IOO
~0
0
8~0
I
I
l
I
I
I
90MHz/Div
Frequency. GHz
I
I
,
I
94OO
4
I 4.2
,
I 4.4
Frequency
i
i 4.6
(GHz)
Fig. 25. Group delay v s . frequency of the forward volume waves for a double-layer YIG film (each layer was 20 pm thick and the layers were separated by a 40 ~tm insulating layer). (From ref. 75.) Fig. 26. Group delay v s . frequency for a dispersive array formed by ion implantations. (From ref. 93.)
180
v.E. WIGEN
Dispersive magnetostatic delay lines can be "actively" synthesized by using reflective arrays. Three types of reflective arrays have been investigated. These include chemically etched grooves on the surface of the garnet s8"89, metal strips which may be continuous or broken 9°'9~ and periodic strips in which ions are implanted 92,93. The forward waves are best suited for use in a reflective array because they are isotropic and the wavenumber is not altered by oblique reflection and can be detected by a second transducer. By contrast, surface waves and backward volume waves are anisotropic and must be reflected back to the generator. Reflective arrays tend to have a narrow.bandwidth in order to avoid interference with higher order Brags reflections. The largest bandwidth of 0.4 G H z has been achieved at the C-band frequencies using forward waves in a delay line constructed with an ionimplanted array 93. The results are shown in Fig. 26 93. 9. NON-LINEAR RESPONSES In F M R experiments, certain effects are observed at high r.f. power levels that are not evident at low powers. In particular, if the r.f. field is applied parallel to the direction of the internal effective field H eft, as opposed to the usual F M R in which the r.f. field is applied perpendicular to the direction of H elf, no measurable power will be absorbed by the sample until a certain threshold power is achieved. These high power non-linear responses are designated as parallel-pumping experiments and were proposed independently by Morganthaler 94 and Schl6mann et al. 95 In the decade of the 1960s, many papers reporting spin wave relaxation measurements by the parallel pumping technique were published. A few of the earlier papers include refs. 96-100. The parallel-pumping experiment is a sensitive and versatile method of measuring the relaxation frequencies of various spin waves with different k values. Both the static field H 0 and the frequency cok of any desired spin wave can be investigated without having to satisfy the usual F M R resonance conditions. For a given spin wave very little r.f. power absorption is observed below some critical value of the r.f. field. Above this critical field value, the energy going out of the modes by relaxation cannot keep up with the input energy so the number of spin waves grows rapidly, giving rise to large amplitudes of precession. The power absorption for parallel pumping is ~
dM~\
h J dr~-'-.tim ) ...... ge
(177)
In a spherical isotropic sample, d M J d t will be zero and no coupling will exist. However, by including volume demagnetization effects, the spins precess in elliptical orbits and the precessional frequency of M~ will oscillate at twice the drive frequency. The spatial integral in eqn. (177) will vanish for a plane wave; consequently, a standing spin wave of wavevectors + k and - k must be considered. Considering the equation of motion in eqn. (70), where H e r r = Ho~ and exchange effects are neglected, the z component of the magnetization is given by
dM, dt
= -7(MxHy-MyHx)
(178)
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
181
The magnetization for a standing wave along the x direction is given by Mx = m sin(kx) sin(tot) M r = m sin(kx) cos(tot)
(179)
The demagnetization field of these spin waves is determined by the magnetostatic Maxwell equations (eqns. (146) and (147)). The magnetic potential ~ of eqn. (158) can be obtained and the resulting demagnetization field is given by HxD = - 4xm sin(tokt) sin(kx)
(180)
and use of the double-frequency pump field h0 sin(2tot) (eqn. (177)) yields Pr.f. = 2 ~Vra2ho
(181)
The transverse magnetization term m 2 can be related to the spin wave occupation number as follows: i
MV-
dr Mz = 2pnnk
(182)
Using the relation Mz 2 -- M 2 - M~2 - My2
(183)
and the condition Mx2 + My 2 ,~ M 2
(184)
the following relation is obtained: m 2 = 8~Mnk
(185)
V Thus, from eqn. (181), Pr.f. --- 4n~lhlMhonk
(186)
In the steady state condition this power is dissipated into mode k at the rate 1
hto~(nk-- (n~))
(187)
where Tk is the relaxation time of the kth spin wave and (nk) is the thermal equilibrium value of nk. Equations (186) and (187) give the critical condition on nk: (nk)
nk = l - h o / h c k
(188)
where hck is the critical r.f. field for the wavevector k and is given by hck =
2tok(1/r Tk) tom
(189)
where tom = 4rcyM and the relation 2#a = ?h is used. If k is allowed to propagate in an arbitrary direction with respect to the applied
182
P.E. WIGEN
field, the ellipticity of the precession will be influenced, giving a general result hck
co°(1/7T~) com sin20k
=
(190)
The factor 1/sin20k favors instability of the rt/2 spin waves as shown in Fig. 2 and, if the p u m p frequency cop is high enough for (Ok = COp/2spin waves to exist in the spin wave band, they will be the first to become unstable. Thus, at high frequencies or a low magnetic field H 0, the experimental results are expected to give a measure of the relaxation frequency for the ~/2 spin waves : 1
(191)
= hck corn
7Tk
cop
Typical experimental results showing a "butterfly curve"1 o 1 are given in Fig. 27. As the applied field increases, the value of k for 0 = n/2 decreases until, at the minimum of the c u r v e , cop/2 lies just at the top of the spin wave manifold. For higher magnetic fields, a 0 = rt/2 spin wave is not available and the relaxation frequency increases sharply.
T I.O
*x~ g
o.8 o
_~ 0.6
g x
°x'~ ~ l ~
0.4
x
0.2 0
I
1.0
t
t.2
t
i
1.4
t
I
1.6
I
I
I
1.8
Static Externol Field Ho(kOe)
Fig. 27. Comparison of the parallel pump data in YIG at 9.3 OHz with the theoretical predictions: t , experimental points; ×, theory. (From ref. 102.) The shape of the butterfly curve is given by --
1
7T~
= A + B k + C sinZ(20k)
(192)
For Y I G magnetized in the [001] direction and COp-- 9.3 G H z lol, A is determined from the minimum of the butterfly curve to be 0.132Oe, C = 0.163 Oe and B = 4.90 x 10- 7 0 e cm. The A and B k terms are related to three-magnon relaxation processes 9s while the sin220k term is found to give the best fit for field values larger than the minimum point in the curve. In the theory reviewed above, the effective field in eqn. (70) was simplified by neglecting the effects of anisotropy energy. In a recent series of papers, Patton 1o2 has included these terms in the spin wave instability theorem. The main influence of the cubic anisotropy energy, apart from a general frequency shift due to a modified effective field, is the introduction of a dependence o f COk o n the azimuthal spin wave angle q~k"This introduces a sin2~bk term in the spin wave dispersion curve derived from eqn. (83). The dependence of the spin wave manifold on ~bk and k 2 is shown schematically in Fig. 28. It is noted that the cubic anisotropy energy changes the
183
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS
dependence of the n/2 branch but not the 0 = 0 branch. As shown in the example, the field can be adjusted to give two values of a butterfly curve. Such effects have been observed t°3 and the effects of the inclusion of the anisotropy energy in the theory have accounted for the second dip in the butterfly curve shown in Fig. 29. ~",~ ~" Ok=~r/2 [O k
/
Ok= 0
1~/ /
-f
i I
l
/
',,!
o.as[-
0.20 I
!
chk = "rr/2
0.45
•
DkZ-~--.~
1.5
.
H"
.
I ~'¢ I 1,55 1.6
"~.. 'l '¢ I 1.65
Ho(kOe)
•
Fig. 28. Schematic representation of the available spin wave states at tok = ,t/2 for a [110]- magnetized YIG sample at low field. Fig. 29. Comparison of the parallel pump data from a [ll0]-magnetized YIG sample at 9.522 GHz: I , experimental data (from ref. 104); , theory (from ref. 105).
These instability experiments provide a probe for analyzing the limitations of the power-handling capability of various devices. Thus the relaxation parameter is an important physical quantity in situations where power may be a significant consideration in the application of the garnet material to certain devices. Another technique recently reported for selectively investigating parallelpumped spin waves with specific wave vectors is that of Brillouin light scattering104, lo5. Thus it is possible to observe spin wave scattering continuously from the thermal level excitations into the parametric spin wave regions as well as to investigate the dependence of the angle and the wavenumber of the pumped spin waves directlyl°s. This allows a detailed study of the parametric processes at all levels of microwave power. This technique is particularly applicable to magnetic garnet films. ACKNOWLEDGMENTS
Much of the material that is reviewed in this paper was developed from a course on magnetism that was recently given at Ohio State University. The author expresses his appreciation to his students for their careful reading of the manuscript as well as a special acknowledgment to Dr. Jedryka for her critical reviews and constructive comments as the paper was being developed. The partial support of the National Science Foundation in the preparation phase of this manuscript through Grant DMR-8304250 is also acknowledged.
184
P.E. WIGEN
REFERENCES
1 J.E. Mee, lEEETrans. Magn.,3(1967) 190. 2 M. Sparks, B. R. Tittmann and C. Newkirk, Phys. Lett. A, 28 (1968) 131. 3 P.E. Wigen, in A. Paoletti (ed.), Physics o f Magnetic Garnets, North-Holland, Amsterdam, 1978, pp. 196-269. 4 L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955, p. 234. 5 J . H . van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, Oxford, 1932. 6 P.W. Anderson, Magnetism, 1 (1963) 25-83. 7 F. Bloch, Z. Phys., 61 (1930) 206. 8 J.C. Slater, Phys. Rev., 35 (1930) 509. 9 P. Pincus, Phys. Rev., 118 (1960) 658. 10 C. M¢ller, Z. Phys., 82 (1933) 559. 11 E.E. Anderson, Phys. Rev. A, 134 (1964) 1581. 12 T. Holstein and H. Primakoff, Phys. Rev., 58 (1940) 1098. 13 A.W. Saenz, Phys. Rev., 125 (1962) 1940. 14 S. Geller, in A. Paoletti (ed.), Physics of Magnetic Garnets, North-Holland, Amsterdam, 1978, pp. 1-55. 15 A.B. Harris, Phys. Rev., 132 (1963) 2398. 16 A.B. Harris and S. Kirkpatrick, AlP ConJl Proc., 24 (1974) 99. 17 A.C. Gerhardstein, P. E. Wigen and S. L. Blank, Phys. Rev. B, 18 (1978) 2218. 18 D.K. Krahn, P. E. Wigen and S. L. Blank, J. Appl. Phys., 50 (1979) 2189. 19 R.K. Wangsness, Am. J. Phys., 24 (1956) 60. 20 B. Bleaney and K. W. H. Stevens, Rep. Prog. Phys., 16 (1953) 108. 21 J.H. van Vleck, Phys. Rev., 52 (1937) 1178. 22 A.M. Clogston, H. Suhl, L. R, Walker amd P. W. Anderson, J. Phys. Chem. Solids, 1 (1956) 129. 23 S.V. Tyablikov, Methods in the Quantum Theory o f Magnetism, Plenum, New York, 1967, p. 49. 24 M. Sparks, Ferromagnetic Relaxation Theory, McGraw-Hill, New York, 1964. 25 S.D. Brown, R. Henry, P. E. Wigen and P. J. Besser, Solid State Commun., 11 (1972) 1179. 26 J.T. Yu, R. A. Turk and P. E. Wigen, Phys. Rev. B, 11 (1975) 420. 27 P.E. Wigen and H. Puszkarski, SolidState Commun., 18 (1976) 363. 28 P.E. Wigen, T. S. Stakelon, H. Puszkarski and P. Yen, AlP Conf Proc., 29 (1975) 670. 29 R.W. Damon and J. R. Eshbach, J. Phys. Chem. Solids, 19 (1960) 308. 30 P. Wolf, J. Appl. Phys., 34 (1963) 1139. 31 H. Puszkarski, Phys. Status Solidi, 22 (1967) 355. 32 H. Puszkarski, Acta Phys. Pol. A, 38 (1970) 217, 899. 33 H. Puszkarski, Phys. Status Solidi B, 50 (1972) 87. 34 V.M. Sokolov and B. A. Tavger, Fiz. Tverd. Tela (Leningrad), 10 (1968) 1793 (Soy. Phys. Solid State, 10 (1968) 1412). 35 V.M. Sokolov, Fiz. Tverd. Tela (Leningrad), 11 (1969) 753 (Soy. Phys. Solid State, 11 (1969) 602). 36 D.L. Mills, AlP Conf. Proc., 34 (1976) 93. 37 J. Spalek and A. Z. Maksymowicz, Solid State Commun., 15 (1974) 559. 38 J. Spalek and W. Schmidt, SolidState Commun., 16 (1975) 193. 39 A. Wachniewski, Phys. Status SolidiB, 77(1976) 319. 40 A. Duda, Phys. Status SolidiB, 107(1981) 779. 41 R.D. Henry, S.D. Brown, P. E WigenandP. J. Besser, Phys. Rev. Lett.,28(1972) 1272. 42 C. Vittoria and J. J. Krebs, AlP Conf. Proc., 24 (1974) 486. 43 C. Vittoria and H. Lessoff, Phys. Rev. Lett., 17 (1976) 53. 44 D.O. Smith, J. Appl. Phys., 29 (1958) 264. 45 R.W. Teale, in A. Paoletti (ed.), Physics of Magnetic Garnets, North-Holland, Amsterdam, 1978, pp. 270-319. 46 T. Stakelon, J. Appl. Phys., 49 (1978) 1592. 47 M. Marysko, Czech. J. Phys. B, 10 (1980) 1269. 48 C. Borghese, P. DeGasperis and P. E. Wigen, IEEE Trans. Magn., 18 (1982) 1301.
MICROWAVE PROPERTIES OF MAGNETIC GARNET THIN FILMS 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
185
H. Puszkarski, Acta Phys. Pol. A, 39 (1971) 597. J.C. Levy, E. Ilisca and J. L. Motchane, Phys. Rev. B, 5 (1972) 187, 1099. Y.J. Liu, Ph.D. Thesis, Yale University, 1974. I. Harada, O. Nagai and T. Nagamiya, Phys. Rev. B, 16 (1977) 4882. M. Sparks, Phys. Rev. B, 1 (1970) 3831, 3856, 3869. O.G. Ramer and C. H. Wilts, Phys. Status Solidi B, 73 (1976) 443. O.G. Ramer and C. H. Wilts, Phys. Status Solidi B, 79 (1977) 313. B. Hockstra, R. P. vanStapele and J. M. Robertson, J. Appl. Phys., 48 (1977) 382. B. Hockstra, J. M. Robertson and G. Bartels, AIP Conf. Proc., 29 (1975) 111. J.P. Omaggio and P. E. Wigen, J. Appl. Phys., 50 (1979) 2264. C.H. Wilts and S. Prasad, IEEE Trans. Magn., 17 (1981) 2405. K. Komenou, J. Zebrowski and C. H. Wilts, J. Appl. Phys., 50 (1979) 5442. J. Mada and K. Asama, J. Appl. Phys., 50 (1979) 5914. H.A. Algra, W. de Roode, R. A. Henskens and J. M. Robertson, J. Appl. Phys., 52 (1981) 2358. R.L. White and I. H. Solt, Phys. Rev., 104 (1956) 56. L. Walker, Phys. Rev., 105(1957) 390. L. Walker, J. Appl. Phys., 29 (1958) 318. B. Tittmann, AlP Conf. Proc., 5 (2) (1971) 1128. B. Tittmann, Solid State Commun., 13 (1973) 463. B.E. Storey, A. O. Tooke, A. P. Cracknell and J. A. Przyslawa, J. Phys. C, 10 (1977) 875; 13 (1980) 2079. C. Borghese, P. DeGasperis and R. Tappa, SolidState Commun., 22 (1973) 463 ; 25 (1978) 21. H. Reimann, Ph.D. Thesis, University of Zurich, 1978. H. Reimann, L. C. Hsia and P. E. Wigen, J. Appl. Phys., 52 (1981) 2264. A. Akhiezer, V. Bar'yakhtar and S. Peletminskii, Spin Waves, North-Holland, Amsterdam, 1968. B. Schneider, Phys. Status Solidi B, 51 (1972) 325. R. Henry, S. Brown, P. Wigen and P. Besser, Phys. Rev. Left., 28 (1972) 1272. T. Wolfram and R. E. DeWames, Prog. Surf. Sci., 2 (4) (1972) 233. M.R. Daniel, J. D. Adam and P. R. Emtage, J. Circuits, Syst. Signal Process., to be published. R.E. DeWames and J. Wolfram, J. Appl. Phys., 41 (1970) 5243. W.L. Bongianni, J. Appl. Phys., 43 (1972)2541. N . D . J . Miller, Phys. Status Solidi A, 37 (1976) 83. M.R. Daniel, J. D. Adam and T. W. O'Keefe, IEEE Trans. Magn., 15 (1979) 1735. T. Wolfram, J. Appl. Phys., 41 (1970) 4748.
A.K. GangulyandC. Vittoria, J. AppI. Phys.,45(1974)4665. H. Sasaki and N. Mikoshiba, Electron. Lett., 15 (1979) 172; J. Appl. Phys., 52 (1982) 3546. L.R. Adkins and H. L. Glass, Electron. Lett., 16 (1980) 503; J. Appl. Phys., 53 (1982) 8928. M.R. Daniel and P. R. Emtage, J. Appl. Phys., 53 (1983) 3723. P. Grunberg and K. Mika, Phys. Rev. B, 27 (1983) 2955. R.E. Camley, T. S. Rahman and D. L. Mills, Phys. Rev. B, 27 (1983) 261. P. Emtage and M. Daniel, Phys. Rev. B, 29 (1984) 212. M. Tsutsumi, Y. Sagaguchi and N. Kumanai, Appl. Phys. Lett., 31 (1977) 779. J.P. Perekh and H. S. Tuan, Appl. Phys. Lett., 30 (1977) 667; 31 (1978) 709. J.M. Owens, J. H. Collins, C. V. Smith and J. M. Chiang, Appl. Phys. Lett., 31 (1977) 781. J.M. Owens, C.V. Smith, S.N. LeeandJ. H. Collins, IEEETrans. Magn.,14(1978)820. R.L. Carter, J.M. Owens, C.V. SmithandK. W. Reed, J. Appl. Phys.,53(1982)2655. G. Volluet and P. Hartemann, in B. R. McAroy (ed.), Proc. Ultrasonics Syrup., 1981, IEEE, New York, 1981, p. 394. F.R. Morganthaler, Dissertation, Massachusetts Institute of Technology, 1959; J. Appl. Phys., 31 (1960) 955. E. Schl6mann, J. J. Green and U. Milano, J. Appl. Phys., 31 (1960) 3863. J. J. Green and E. Schl6mann, J. Appl. Phys., 33 (1962) 1358. T. Kohane and E. Schl6mann, J. Appl. Phys., 34 (1963) 1544. R.M. White and E. SchlSmann, J. Appi. Phys., 33 (1962) 2437. T. Kasuya and R. C. LeCraw, Phys. Rev. Lett., 6 (1961) 223. E.G. Spencer, R. C. LeCraw and R. C. Linares, Phys. Rev., 123 (1961) 1937. P.E. Sciden, Phys. Rev. A, 133 (1964) 728.
186
P.E. WIGEN
101
Y.M. Yakovlev, Y. N. Burdin, Y. R. Shilnikov and T. N. Bushueva, Fiz. Tverd., Tela, 12 (1970) 3059 (Soy. Phys.--SolidState, 12 (1971) 2475). C.E. Patton, Phys. Status Solidi B, 92 (1979) 211 ; 93 (1979) 63. J.C. Sethares, M. H. Seavey, T. G. Purnhagen and M. R. Stiglitz, J. Appl. Phys., 35 (1964) 898. V.N. Venitsky, V. V. Eremenko and E. V. Matyushkin, Pis'ma Zh. Eksp. Teor. Fiz., 27 (1978) 239 ( JETP Lett., 27 (1978) 222); Zh. Eksp. Teor. Fiz., 77 (1979) 1965 (Soy. Phys. JETP, 50 (1979) 934). W. Wettling, W. D. Wilber, P. Kabos and C. Patton, Phys. Rev. Lett., 51 (1983) 1680
102 103 104
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