- Email: [email protected]
Dipole-exchange modes of a thin ferromagnetic film
Volume 30A. number
PHYSICS
1
LETTERS
For these stress data the field is 200 off from the [llO] direction. Note that the intensity of the indiiidual lines varies but the total intensity of all lines remains at nine units. We should also point out that the intensity change occurs in less than a fraction of a second. Observation of such a rapid reorientation time would seem to rule out the possibility that the distortions are associated with nearby defects in the crystal. Also random strains seem to be unimportant because the intensities change even with very small stresses. An additional resonance line at high fields with an almost isotropic g factor neat 2.09 and having three units of intensity is included in the total intensity of nine units. It could be speculated that this line comes from the triplet state that arises when Jahn-Teller distortions in (111) directions are dynamically coupled [6]. *****
DIPOLE-EXCHANGE
MODES
OF
T. WOLFRAM
A THIN
The spin-\vave both dipole
Oaks.
modes of a thin ferromagnetic and exchange interactions.
film
The magnetostatic modes of a ferromagnetic plate in the absence of exchange interactions have been characterized by Damon and Eshbach [ 11. We present here the spin-wave spectrum of a thin film in the transition region in which the exchange and dipole energies are comparable *. Consider a thin film of thickness S in the x direction and infinite in the y and z directions with magnetization &I, and internal field Hi parallel to the z axis. The spin wave modes are solutions of Maxwell’s equations (V x h = 0 and V . (h+4rm) = 0) and the equations of motion for the magnetization, -iwm
= yM,
X [h + DV2m - (Hi/Mo)m].
In eq. (1) the quantities
/I and m are
* A L~L~W tvl,c‘ ol’ surl’;lce state exisrs
the oscil-
in this transition region in the C:IW th:lt the magnetization and applied field arc pcrpc’ndicular to the film surface [2].
2
FERROMAGNETIC
FILM
and R. E. DE WAMES
Received
ding
1969
References 1. M.D.Sturge. Solid State Physics. vol. 20. F.Seitz. D. Turnbull and H. Ehrenreich, eds. (Academic Press. 1967) p. 91. Resonance. ed. 2. F. S. Ham, Electron Paramegnetic S. Geschwind (Plenum Press. 1969). 3. T. L. Estle. G. K. Walters and M.deWit. Paramagnetic Resonance. vol. I. ed. W. Lo\v (Academic Press. 1963)p. 144. J. Phys. Sot. Japan 19 (1964) 187. 4. K.Morigaki. 5. D. C. von Hoene and R. C. Fedder. Bull. Am. Phys. Sot. 14 (1969) 48. Phys. Rev. A138 (1965) 1727: 6. F.S.Ham. M.Caner and R.Englman. J. Chem. Phgs. 44 (1966) 4054.
ic2rw:c Center. North American Thousand
8 September
(1)
Rockwell
California
Corporation
91360. US-4
15 July 1969
magnetized
parallel
to the surface
are
obtained
inclu-
lating components of the magnetic field and magnetization, respectively. The magnetic field is derived from a scalar potential q (/I = - Vlc/). Inside the film the potential is a linear combination of six plane waves whose propagation vectors, kl (normal to the surface), are the six roots of the characteristic equation:
‘s-ii + ak slk
=
-
$)(k,2 + $) + (0; -
(1/4~)[Hl/~o)
Y
+D(kf
+ k:
fi2)k;=
+ki)I,
0
(2)
where D is the exchange constant. The potential outside decays exponentially towards x = +m. The spin-wave eigenvalues and eigenvectors are determined by the boundary conditions. At x = 0 andS the tangential component of hand the normal component of (h + 4sm) are continuous. The solutions obtained using boundary conditions derived from microscopic spin-wave theory are essentially the same as those obtained by requiring the normal derivatives of m to vanish
Volume 30A, number 1
PHYSICS
m,+l/rb
T t
Fig. 1. Spin-wave surfaces with (upper) and without (lower) exchange. D = 0/47ryMo and !& = Hi/4TMo. ( for the regime discussed here). Fig. 1. is a schematic illustration of the mode surfaces for a ferromagnetic film with and without exchange. The surface state solutions correspond to points
on the sheet which lies above (C$$ + St, )g . The bulk modes are condensed into a single line along the ky axis but are separated by the dipole interaction at finite k, [ 11. With exchange the bulk
8 September 1969
LETTERS
8164/r)
-
Fig. 2. Spin-wave spectrum as a function of the anglee.
modes are shifted up and separated. The surface state cannot cross a bulk level and the curves repel each other. The eigenvectors for these models are admixtures of the bulk and surface states. The spectrum for a thin yttrium iron garnet film is shown in fig. 2 for a fixed value of (k$ + kz) as a function of the angle 0 between the propagating vector and the ky axis. In this calculation the following numerical values were used: S = 0.61 X 10-&m, D = 2.6 X lo-12cm2, 47rMo = 1750 Oe,Hi/JrMo = 0.5 and 0 = w/4syMo. The dashed curve indicated the position of the surface state without exchange. A detailed discussion of the effect of exchange on the surface state along the ky axis is given elsewhere [3].
References 1. R.W.Damon and J.R.Esbbach. J. Phys. Chem. Solids 19 (1960) 308. 2. R. E. De Wames and T. Wolfram, to be published. 3. T. Wolfram and R. E. De Wames, to be published.
3
PHYSICS
1
LETTERS
For these stress data the field is 200 off from the [llO] direction. Note that the intensity of the indiiidual lines varies but the total intensity of all lines remains at nine units. We should also point out that the intensity change occurs in less than a fraction of a second. Observation of such a rapid reorientation time would seem to rule out the possibility that the distortions are associated with nearby defects in the crystal. Also random strains seem to be unimportant because the intensities change even with very small stresses. An additional resonance line at high fields with an almost isotropic g factor neat 2.09 and having three units of intensity is included in the total intensity of nine units. It could be speculated that this line comes from the triplet state that arises when Jahn-Teller distortions in (111) directions are dynamically coupled [6]. *****
DIPOLE-EXCHANGE
MODES
OF
T. WOLFRAM
A THIN
The spin-\vave both dipole
Oaks.
modes of a thin ferromagnetic and exchange interactions.
film
The magnetostatic modes of a ferromagnetic plate in the absence of exchange interactions have been characterized by Damon and Eshbach [ 11. We present here the spin-wave spectrum of a thin film in the transition region in which the exchange and dipole energies are comparable *. Consider a thin film of thickness S in the x direction and infinite in the y and z directions with magnetization &I, and internal field Hi parallel to the z axis. The spin wave modes are solutions of Maxwell’s equations (V x h = 0 and V . (h+4rm) = 0) and the equations of motion for the magnetization, -iwm
= yM,
X [h + DV2m - (Hi/Mo)m].
In eq. (1) the quantities
/I and m are
* A L~L~W tvl,c‘ ol’ surl’;lce state exisrs
the oscil-
in this transition region in the C:IW th:lt the magnetization and applied field arc pcrpc’ndicular to the film surface [2].
2
FERROMAGNETIC
FILM
and R. E. DE WAMES
Received
ding
1969
References 1. M.D.Sturge. Solid State Physics. vol. 20. F.Seitz. D. Turnbull and H. Ehrenreich, eds. (Academic Press. 1967) p. 91. Resonance. ed. 2. F. S. Ham, Electron Paramegnetic S. Geschwind (Plenum Press. 1969). 3. T. L. Estle. G. K. Walters and M.deWit. Paramagnetic Resonance. vol. I. ed. W. Lo\v (Academic Press. 1963)p. 144. J. Phys. Sot. Japan 19 (1964) 187. 4. K.Morigaki. 5. D. C. von Hoene and R. C. Fedder. Bull. Am. Phys. Sot. 14 (1969) 48. Phys. Rev. A138 (1965) 1727: 6. F.S.Ham. M.Caner and R.Englman. J. Chem. Phgs. 44 (1966) 4054.
ic2rw:c Center. North American Thousand
8 September
(1)
Rockwell
California
Corporation
91360. US-4
15 July 1969
magnetized
parallel
to the surface
are
obtained
inclu-
lating components of the magnetic field and magnetization, respectively. The magnetic field is derived from a scalar potential q (/I = - Vlc/). Inside the film the potential is a linear combination of six plane waves whose propagation vectors, kl (normal to the surface), are the six roots of the characteristic equation:
‘s-ii + ak slk
=
-
$)(k,2 + $) + (0; -
(1/4~)[Hl/~o)
Y
+D(kf
+ k:
fi2)k;=
+ki)I,
0
(2)
where D is the exchange constant. The potential outside decays exponentially towards x = +m. The spin-wave eigenvalues and eigenvectors are determined by the boundary conditions. At x = 0 andS the tangential component of hand the normal component of (h + 4sm) are continuous. The solutions obtained using boundary conditions derived from microscopic spin-wave theory are essentially the same as those obtained by requiring the normal derivatives of m to vanish
Volume 30A, number 1
PHYSICS
m,+l/rb
T t
Fig. 1. Spin-wave surfaces with (upper) and without (lower) exchange. D = 0/47ryMo and !& = Hi/4TMo. ( for the regime discussed here). Fig. 1. is a schematic illustration of the mode surfaces for a ferromagnetic film with and without exchange. The surface state solutions correspond to points
on the sheet which lies above (C$$ + St, )g . The bulk modes are condensed into a single line along the ky axis but are separated by the dipole interaction at finite k, [ 11. With exchange the bulk
8 September 1969
LETTERS
8164/r)
-
Fig. 2. Spin-wave spectrum as a function of the anglee.
modes are shifted up and separated. The surface state cannot cross a bulk level and the curves repel each other. The eigenvectors for these models are admixtures of the bulk and surface states. The spectrum for a thin yttrium iron garnet film is shown in fig. 2 for a fixed value of (k$ + kz) as a function of the angle 0 between the propagating vector and the ky axis. In this calculation the following numerical values were used: S = 0.61 X 10-&m, D = 2.6 X lo-12cm2, 47rMo = 1750 Oe,Hi/JrMo = 0.5 and 0 = w/4syMo. The dashed curve indicated the position of the surface state without exchange. A detailed discussion of the effect of exchange on the surface state along the ky axis is given elsewhere [3].
References 1. R.W.Damon and J.R.Esbbach. J. Phys. Chem. Solids 19 (1960) 308. 2. R. E. De Wames and T. Wolfram, to be published. 3. T. Wolfram and R. E. De Wames, to be published.
3