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Anisotropy, magnetostriction and micromagnetism
1445 ANISOTROPY, M A G N E T O S T R I C T I O N AND M I C R O M A G N E T I S M M.W. M U L L E R * Max-Planck-Institut fiir Metallforschung, Stuttgart, W. Germany Experiments can be designed in which domain nucleation in magnetic films is dominated by intrinsic properties rather than by irregularities and imperfections. The results of such experiments are interpreted by micromagnetic theory to yield measurements of the magnetocrystalline anisotropy and of the magnetoelastic constants.
The c o m m o n magnetomechanical methods for the measurement of the magnetocrystalline anisotropy and of the magnetoelastic coefficients are of little use for determining these properties when the magnetic material is in the form of a thin film, especially when the film is attached to a much thicker substrate. In many such films the domain structure can be readily observed b y means of the Faraday or magnetooptic Kerr effects. We show here how the behaviour of the domain structure in properly applied magnetic fields can be interpreted to yield detailed and accurate information about the anisotropic and magnetoelastic properties of some films. An extremely simple-minded example of this idea is the extinction of d o m a i n s - s a t u r a t i o n in an ideal uniaxial crystal when the internal hard-axis field reaches the value 2 K d M . Much more detailed information can be derived from domain nucleation data with the help of micromagnetic nucleation theory. The principles of such a theory have been k n o w n for a long time, but it remained largely without application because experimentally observed domain nucleation conditions differed widely from its predictions, a discrepancy that b e c a m e k n o w n as " B r o w n ' s p a r a d o x " [1]. The premature domain nucleation that constitutes Brown's paradox occurs because imperfections and irregularities destabilize the metastable ("supercooled") saturated state of the specimen long before the theory of the perfectly regular specimen predicts instability. Attempts of a theory of domain nucleation at imperfections have been only partially successful, and they do not yield any information about the intrinsic properties of the materials [2]. Thus * Supported by a U.S. Senior Scientist Award from the Alexander von Humboldt Foundation; permanent address: Dept. of Electrical Engineering, Washington University, St. Louis, Missouri. Physica 86-88B (1977) 1445-1446 © North-Holland
the key to successful applications of nucleation theory to the analysis of experimental results has been the recognition that premature domain nucleation cannot occur if the saturated state of the specimen does not b e c o m e metastable; that is to say, if the applied field is adjusted so that nucleation becomes a second order rather than a first order phase transition [3, 4]. The procedure may be outlined as follows: The (Gibbs) free energy density in the magnetic material at a temperature well below the Curie (or N6el) temperature, where ]M[ can be assumed constant, is a functional of the orientation of M, as given b y the position-dependent direction cosines ai(x), and of the strains eli(x), and it depends on the uniform applied field H. w = w(t~i, Ooti/axj, eli; Hi).
(i)
To minimize the energy we use the variational principle &o
f w dv = 0.
(2)
The condition for nucleation is determined by solving the Euler equations of (2) with the largest applied field Hn that yields a non-trivial nucleation mode Otin(X), eijn(X). In keeping with the usual experimental practice, this ordinarily means that the direction h = H / H remains fixed as the magnitude of H is reduced until domain nucleation occurs. To discuss the nature of the nucleating phase transition, assume that the Euler equation has been solved for a nucleation mode Mn, • n which deviates slightly from the saturated state Mn(x) = Mo(H,) + Amn(X),
(3a)
where A is an amplitude. en(x) - Eo(H.) +
AEn(x),
(3b)
1446 Then the energy can be written as a power series tO(a)
= ~ w dv = tOo+ tOlA + tO2A2 + tO3A3 + 0)4 A 4 + . . . .
(4)
It is convenient (and always possible) to formulate the calculation so that to~ ~ 0 . The condition for domain nucleation then is to2 = 0; and nucleation is a second order phase transition if to3 = 0 and to4 > 0. The expressions for to1, to2,.., of course contain the properties of the magnetic material such as its saturation magnetization, exchange, anisotropy, and magnetoelastic constants, its geometrical demagnetizing factors, as well as the three components Hi of the applied field. Thus for a given material and geometry the two equations
nx, n y , nz) = 0, to3(Kmat, shape; nx, ny, nz) = O. tol(Kmat, shape;
(5)
define a one-dimensional locus (space curve) for the tip of the H-vector along which secondorder nucleation is expected to o c c u r . t Conversely, measurement of this locus provides a set of data from which the material constants can be inferred. The locus is measured by placing the specimen on a rotating stage within two orthogonal coils. Second order nucleation is characterized by the gradual appearance, as the field is reduced, of a very faint domain pattern of the form (one hopes) predicted by the theory. If the orientation of the field is " w r o n g " , so that the nucleation is first order, the domain pattern appears more suddenly, with greater contrast. Also, since under these conditions the initially nucleating pattern is unstable and matures rapidly, the observed pattern may differ from the predictions of the linearized nucleation theory. For example, in some useful groups of measurements on bubble garnet films [4, 5], the observed stable second order pattern consists of
t It may happen that to3 = 0 in a volume rather than on a surface in H-space, in which case eqs. (5) define a range of second order nucleation.
stripe domains as predicted by the theory, but a very small deviation of the field orientation from the stability locus in either direction results in the appearance of a bubble lattice of the corresponding polarity. In these measurements the locus is very sensitive to cubic and stray anisotropies of the film but quite insensitive to its magnetoelastic properties. Because of this, a simpler nucleation theory which neglects not only magnetostriction but also the exchange and part of the dipolar energy, in an approximation appropriate for very thick films, has been used by Hubert et al. to determine anisotropy constants [4]. The micromagnetic calculation [5] predicts not only a small correction to the second-order nucleation locus, but also the geometry of the incipient domains; in the case of the bubble films, stripe domains of a width and orientation that depends sensitively on the film's magnetoelastic properties. We have compared the most detailed measurements carried out to date, on a mixed G d T m Y garnet film epitaxially grown on gadolinium gallium garnet, with the predictions of the micromagnetic nucleation theory. We conclude from this comparison that the magnetoelastic coefficients deduced from the observations are largely, but not entirely, accounted for by the single-ion contributions to the magnetostriction. The results strongly suggest that the film's magnetoelastic tensor is not cubic, but has a c o m p o n e n t b44 -~ 2 x 106 erg cm -3. It seems quite plausible that this non-cubic term may arise from the same site-ordering that gives rise to the uniaxial anisotropy of the film. Work is under way to test this conjecture. References [1] W.F. Brown, Jr., Micromagnetics (Interscience, New York, 1963). [2] A. Aharoni, Rev. Mod. Phys. 34 (1962) 227. [3] M.H. Yang and M.W. Muller, Trans. IEEE MAG-7, (1971) 705. [4] A. Hubert, A.P. Malozemoff and J.C. DeLuca, J. Appl. Phys. 45 (1974) 3562. [5] M.H. Yang and M.W. Muller, J. Magnetism Mag. Mat'ls 1 (1976) 251.
The c o m m o n magnetomechanical methods for the measurement of the magnetocrystalline anisotropy and of the magnetoelastic coefficients are of little use for determining these properties when the magnetic material is in the form of a thin film, especially when the film is attached to a much thicker substrate. In many such films the domain structure can be readily observed b y means of the Faraday or magnetooptic Kerr effects. We show here how the behaviour of the domain structure in properly applied magnetic fields can be interpreted to yield detailed and accurate information about the anisotropic and magnetoelastic properties of some films. An extremely simple-minded example of this idea is the extinction of d o m a i n s - s a t u r a t i o n in an ideal uniaxial crystal when the internal hard-axis field reaches the value 2 K d M . Much more detailed information can be derived from domain nucleation data with the help of micromagnetic nucleation theory. The principles of such a theory have been k n o w n for a long time, but it remained largely without application because experimentally observed domain nucleation conditions differed widely from its predictions, a discrepancy that b e c a m e k n o w n as " B r o w n ' s p a r a d o x " [1]. The premature domain nucleation that constitutes Brown's paradox occurs because imperfections and irregularities destabilize the metastable ("supercooled") saturated state of the specimen long before the theory of the perfectly regular specimen predicts instability. Attempts of a theory of domain nucleation at imperfections have been only partially successful, and they do not yield any information about the intrinsic properties of the materials [2]. Thus * Supported by a U.S. Senior Scientist Award from the Alexander von Humboldt Foundation; permanent address: Dept. of Electrical Engineering, Washington University, St. Louis, Missouri. Physica 86-88B (1977) 1445-1446 © North-Holland
the key to successful applications of nucleation theory to the analysis of experimental results has been the recognition that premature domain nucleation cannot occur if the saturated state of the specimen does not b e c o m e metastable; that is to say, if the applied field is adjusted so that nucleation becomes a second order rather than a first order phase transition [3, 4]. The procedure may be outlined as follows: The (Gibbs) free energy density in the magnetic material at a temperature well below the Curie (or N6el) temperature, where ]M[ can be assumed constant, is a functional of the orientation of M, as given b y the position-dependent direction cosines ai(x), and of the strains eli(x), and it depends on the uniform applied field H. w = w(t~i, Ooti/axj, eli; Hi).
(i)
To minimize the energy we use the variational principle &o
f w dv = 0.
(2)
The condition for nucleation is determined by solving the Euler equations of (2) with the largest applied field Hn that yields a non-trivial nucleation mode Otin(X), eijn(X). In keeping with the usual experimental practice, this ordinarily means that the direction h = H / H remains fixed as the magnitude of H is reduced until domain nucleation occurs. To discuss the nature of the nucleating phase transition, assume that the Euler equation has been solved for a nucleation mode Mn, • n which deviates slightly from the saturated state Mn(x) = Mo(H,) + Amn(X),
(3a)
where A is an amplitude. en(x) - Eo(H.) +
AEn(x),
(3b)
1446 Then the energy can be written as a power series tO(a)
= ~ w dv = tOo+ tOlA + tO2A2 + tO3A3 + 0)4 A 4 + . . . .
(4)
It is convenient (and always possible) to formulate the calculation so that to~ ~ 0 . The condition for domain nucleation then is to2 = 0; and nucleation is a second order phase transition if to3 = 0 and to4 > 0. The expressions for to1, to2,.., of course contain the properties of the magnetic material such as its saturation magnetization, exchange, anisotropy, and magnetoelastic constants, its geometrical demagnetizing factors, as well as the three components Hi of the applied field. Thus for a given material and geometry the two equations
nx, n y , nz) = 0, to3(Kmat, shape; nx, ny, nz) = O. tol(Kmat, shape;
(5)
define a one-dimensional locus (space curve) for the tip of the H-vector along which secondorder nucleation is expected to o c c u r . t Conversely, measurement of this locus provides a set of data from which the material constants can be inferred. The locus is measured by placing the specimen on a rotating stage within two orthogonal coils. Second order nucleation is characterized by the gradual appearance, as the field is reduced, of a very faint domain pattern of the form (one hopes) predicted by the theory. If the orientation of the field is " w r o n g " , so that the nucleation is first order, the domain pattern appears more suddenly, with greater contrast. Also, since under these conditions the initially nucleating pattern is unstable and matures rapidly, the observed pattern may differ from the predictions of the linearized nucleation theory. For example, in some useful groups of measurements on bubble garnet films [4, 5], the observed stable second order pattern consists of
t It may happen that to3 = 0 in a volume rather than on a surface in H-space, in which case eqs. (5) define a range of second order nucleation.
stripe domains as predicted by the theory, but a very small deviation of the field orientation from the stability locus in either direction results in the appearance of a bubble lattice of the corresponding polarity. In these measurements the locus is very sensitive to cubic and stray anisotropies of the film but quite insensitive to its magnetoelastic properties. Because of this, a simpler nucleation theory which neglects not only magnetostriction but also the exchange and part of the dipolar energy, in an approximation appropriate for very thick films, has been used by Hubert et al. to determine anisotropy constants [4]. The micromagnetic calculation [5] predicts not only a small correction to the second-order nucleation locus, but also the geometry of the incipient domains; in the case of the bubble films, stripe domains of a width and orientation that depends sensitively on the film's magnetoelastic properties. We have compared the most detailed measurements carried out to date, on a mixed G d T m Y garnet film epitaxially grown on gadolinium gallium garnet, with the predictions of the micromagnetic nucleation theory. We conclude from this comparison that the magnetoelastic coefficients deduced from the observations are largely, but not entirely, accounted for by the single-ion contributions to the magnetostriction. The results strongly suggest that the film's magnetoelastic tensor is not cubic, but has a c o m p o n e n t b44 -~ 2 x 106 erg cm -3. It seems quite plausible that this non-cubic term may arise from the same site-ordering that gives rise to the uniaxial anisotropy of the film. Work is under way to test this conjecture. References [1] W.F. Brown, Jr., Micromagnetics (Interscience, New York, 1963). [2] A. Aharoni, Rev. Mod. Phys. 34 (1962) 227. [3] M.H. Yang and M.W. Muller, Trans. IEEE MAG-7, (1971) 705. [4] A. Hubert, A.P. Malozemoff and J.C. DeLuca, J. Appl. Phys. 45 (1974) 3562. [5] M.H. Yang and M.W. Muller, J. Magnetism Mag. Mat'ls 1 (1976) 251.