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Dipolar effects upon order in ferromagnets with random anisotropy
/l lm
Journal of Magnetism and Magnetic Materials 125 (1993) 310-314 North-Holland
Dipolar effects upon order in ferromagnets with random anisotropy J.C.S. L6vy a n d J.J.G. L6vy Laboratoire de Magn~tisme des Surfaces, Universitd Paris 7 Denis Diderot, 75251 Paris 05, France Received 25 November 1992
Dipolar effects upon ordering into magnetic zones of ferromagnets with randomly directed anisotropy are demonstrated to be efficient for thin films with different shapes for magnetic zones, from discs to columnar bubbles.
1. Introduction In recent years a considerable amount of work has been devoted to magnetic order in materials where strong ferromagnetic exchange competes with random anisotropy [1-3]. The magnetic structure that results from this competition is known to show a local ferromagnetic order, the size of which is characterized by a ferromagnetic correlation length (FCL), i.e. a typical radius for nearly spherical zones, which is large compared with the interatomic distance a. There is a large interest in the competition between order and statistics, since a similar competition occurs in spin glasses [4,5] with, basically, a similar analysis into zones characterized by a correlation length, while competition between deterministic exchange and anisotropy is quite different and leads to Bloch and N6el walls [6], i.e. well defined arrangements with finite extensions. Moreover, the magnetic structure that results from the competition of exchange and random anisotropy in discrete systems, i.e. lattices, is only slightly shifted by the introduction of an external magnetic field or of a constant anisotropy term that, however, breaks down the spherical symmetry [3], which Correspondence to: Dr J.J.G. L6vy, Laboratoire de Magn6tisme des Surfaces, Universit6 Paris 7 Denis Diderot, 75251 Paris 05, France.
implies a special critical behavior. Such a constant anisotropy term can be caused by a magneto-elastic effect applied to the internal stress which occurs during sample preparation. Among various possible realizations of realistic materials with these properties, soft amorphous ferromagnets (SAFs) ensure both a high value of exchange fields and a large distribution of sites with different 'crystalline' field values [7]. These SAF materials, as a matrix, can be doped with a rather large concentration range of rare earths (RE) of individual large anisotropy. Such doping can provide as large a density of local anisotropy energy as required. At each site occupied by a rare earth, the easy axis of anisotropy is determined by the local 'crystalline' field and in an amorphous structure, these axes are nearly random, with correlation lengths that is typically the nearest-neighbor distance b between atomic sites occupied by rare earths. Thus RE-doped SAFs enable us to follow the competition between exchange and random anisotropy when the density of the anisotropy energy, i.e. of the RE, varies over a wide range [8], since magnetically correlated clusters can be neglected. In doped SAFs, electron conductivity is usually so low that dipolar effects remain completely unscreened. Then, since dipolar effects are not easily compatible with ferromagnetic order or with the order which is induced by a random distribution of anisotropy
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J. C. S. Ldvy, J.J.G. Ldvy / Dipolar effects in f erromagnets
directions, the competition between the three terms (exchange, random anisotropy and dipolar effects) is interesting. The first goal of this paper consists in this analysis. A special case of interest is that of weak random anisotropy where magnetic zones are still expected to occur when dipolar fields are neglected [1,3]. Experimentally, in the absence of anisotropy, dipolar effects are known to lead to magnetic structures with opposite domains of various shapes such as stripes, bubbles and mazes in thin films [9] and to other structures in large samples [10]. Realistically, a critical value for random anisotropy versus dipolar effects is expected to occur in order to stabilize the zone structure. Another interesting point concerns the application of such a theory to thin films and multilayers since the ferromagnetic correlation length depends strongly on dimensionality 2 or 3. Since magnetic structures with overlapping magnetic zones lead to giant magnetic susceptibilities [3], direct applications are also involved for thin films and multilayers. The existence of magnetic zones is essentially an effect of the discreteness of atom packing. An isotropic zone of N atoms in a space of dimension d has a typical correlation length (FCL) R - - N 1/d. Thus, the magnetization gradient in this zone is proportional to N - 1 / d and the density of exchange energy •e is proportional to N-2/d:
E"e
l
=
2
( 1 / 2 ) A (ViM~,) - - A N - 2 / a ,
(1)
where A is the exchange constant and M is the local magnetization which is the spin density. The discrete version of anisotropy energy, with a local uniaxial behavior, reads: H A = - (1/2)EI(M.
nr) 2,
(2)
where n r is the unit vector of the local easy axis of magnetization at site r, and I is the local uniaxial anisotropy constant, which is assumed to be everywhere the same in this simple approach. The direction n R is assumed to be randomly distributed, i.e. with a correlation length b which is negligible compared with R, the size of the zone, and is a function of the R E density. Up to
311
an additive constant and a multiplicative factor, the discrete anisotropy energy of the zone is the sum of N cosines of random angles 20r, with 0 r = ( M , n r ) , exactly as a component of the result of N random flights as introduced by Pearson [11] and Lord Rayleigh [12] in relation to different topics and reported in the classical paper by Chandrasekhar [13]. The law of large numbers [14] enables us to estimate the anisotropy energy in the zones as: EA--- - k N 1/2,
(3)
with k as the anisotropy factor; the density of anisotropy energy thus reduces to: • A -------- KN - 1/2
(4)
with K as the anisotropy parameter. Neglecting dipolar effects, the total energy density • reads: e = A ( N ) -2/d _ K ( N ) -1/2
(5)
The minimization of this energy density with respect to N provides the optimal value of N : N o for the zone with: N O = ( 4 A / K ) (2d)/(4-d).
(6)
When the dimension d = 4, the exchange and anisotropy share the same variation with N, the minimum energy density is reached for N infinite [1], there is only one single zone. If the dimension d is larger than the marginal dimension 4, at N O the energy density is maximal, and the concept of a zone is no longer valid. For a 3D space, a minimum energy density is reached at No,3 = ( 4 A / K ) 6, with R0, 3 = ( 4 A / K ) 2, while for a 2 D space (i.e. a surface) the minimum energy density is reached for No,2 = ( 4 A / K ) 2 and R0, 2 = 4 A / K . The difference between these two values of the FCL length Ro, i is quite large and can become physically meaningful when the thickness t of thin films or the typical thickness t' of the films which form a multilayer, is varied. If these thicknesses are large compared with Ro,3, i.e. in a real 3D case, spherical zones with radius Ro,3 must be observed. If the thicknesses t lies in the intermediate range I, with I = [ R o , 2 , Ro,3] , cylindrical
312
J.C.S. L~vy, J.J.G. L~vy / Dipolar effects in ferromagnets
zones (i.e. bubbles) are expected to occur, with a typical radius p = ( 2 A / K ) t 1/2, which belongs to the interval I ' , with I ' = [(2A/K) 3/2, R0,3]. If the thicknesses t are lower than Ro, 2, cylindrical zones (i.e. discs) are expected to occur, with the same expression as before for the radius p, which now lies in the interval I", with I " = [Ro,z,(2A/K)3/2]. Thus large variations in the sizes and shapes of these zones must be observed either when thicknesses are varied over a short range, or when R E concentrations are slightly varied, since the effective value K of anisotropy depends upon the rare earth composition. For thicknesses t of a few R0,3, surface effects must still lead to bubble-shaped zones with axis normal to the film or multilayer planes. The precise purpose of this p a p e r is to introduce dipolar contributions when dealing with a rather thick film, i.e. with t >> R. This also means that the surface pinning of spins, which is a strong factor for bubble-shaped or disc-shaped zones, can be neglected. In section 2 generalities on the Hamiltonian and on the assumed shapes of magnetic zones are given, while the variational process is applied in section 3 and the nature of optimal zones is derived and discussed in section 4.
2. Generalities In this problem, energy must be counted at three levels: at a local level, with a spin Hamiltonian H ; at the level of a zone, with a zone energy E; and at the global level with an energy density E. The spin S per unit volume defines the local magnetization M, which is assumed to take everywhere the same magnitude in this low-temperature problem, but a nearly r a n d o m orientation. Because of this, the average magnetization m of a zone has a lower magnitude than M. The exchange Hamiltonian H e reads: H e = -(1/2)J•S
i'Sj,
(7)
i,j
which leads to the density of exchange energy expressed by eq. (1). The anisotropy Hamiltonian
was given in eq. (2). The dipolar field h a at site i reads, as usual: h d = ~_~[(3rijSj.rij-Sjri2j)/r~],
(8)
J
where rii is the vector which links sites i and j. Finally, the Z e e m a n Hamiltonian H z for a magnetic field H reads: n Z : --].b E S i " i
H,
(9)
with a magnetic constant /z; quite obviously a constant anisotropy contribution, i.e. with a given direction, can be integrated in this term as well as the dipolar field contribution. A zone with an average magnetization m induces inside this zone a dipolar field h a of which, because of symmetry arguments to be developed later, the effective part is collinear with m. Since this effective dipolar field depends on the shape of this magnetic zone, such a shape effect must be introduced when taking into account the symmetry axis defined by the magnetization axis m. With this symmetry, the zone is expected to be nearly a rectangular parallelepiped with typical sides R, according to directions perpendicular to m, and R ' , according to the m direction. O f course, it also means an ellipsoidal shape of the zone. Thus the volume of a zone is proportional to RZR ', with a form factor and the number N of atoms in a zone is up to a correcting density factor: N = R Z R ' / a 3, where a is the nearestneighbor distance between magnetic sites, i.e. a is lower than b. These relations are easily translated for a space of dimension d, with:
N = Rd-lR'a-d.
(10)
Then, the square of the magnetization gradient has for average magnitude ( ~ M ~ ) 2 = [(2/R 2) q-(1//g'2)], when d = 3, because of symmetry. For arbitrary d, it leads to:
which enables the exchange energy within the zone to be estimated. The estimation of the dipolar energy of the zone can be deduced from that of the dipolar
J.C.S. L3vy, Z Z G. Ldvy / Dipolareffects in ferromagnets field h d. Outside the zone, the average magnetization is zero and the weight of the integrating factor is strongly decreasing, and thus the resulting dipolar field due to the long-range part can be neglected. Within the zone, each local magnetization can be analyzed into two parts: namely, the average magnetization m and the rest, which is strongly fluctuating and has a null average, because of the distribution of anisotropy sites (RE), i.e. with a typical correlation length b. Because of such strong fluctuations, the highly fluctuating part gives no resulting contribution and the effective dipolar field is due only to the average magnetization applied to the whole zone. A constant magnetization is known to lead to a dipolar field which is practically constant within the zone with the exception of a sharp zone boundary where the estimated relative difference between surface and bulk dipolar fields is only a few per cent [15]. With the assumed symmetries of the zone, this effective dipolar field h d is parallel to the magnetization m, with a ratio of proportionality A, and reads:
h d = Am,
313
ergy must be calculated in a single zone. First, the exchange energy in a zone reads: E e = (1/2)aM2[(d-
1)Rd-3R' + R d - I R ' - ' ] . (13)
T h e dipolar energy E d for a single zone is due to the Z e e m a n effect created by the effective dipolar field, it reads:
Ed = - Al~m2R d- 1R',
(14)
and the anisotropy energy E A is a sum of N ' =
Rd-IR'b -d terms due to each RE: E A = - ( 1 / 2 ) IM 2 ~ '
cos20.
This sum can be analyzed into two terms, a m e a n t e r m which is proportional to N', and a fluctuation term which is proportional t o N '1/2 and depends upon the orientation of the m e a n magnetization m. Thus the optimization of this fluctuation term determines the m e a n magnetization orientation, and E A reads: E A =
- (1/4)
(12)
IMER d- 1R'b-d
-- ( 1 / 4 ) IM2R(d-1)/ER'1/Eb -d/2. where the ratio A is a demagnetizing factor associated with t h e zone geometry. T h e n it can be assumed that the optimization process to be developed later leads to a unique optimal zone geometry, for a given value of A; thus, the optimal zone geometry and A will be determined self-consistently at the end of this calculation. Practically, this assumption is not very constraining since the only opposition to a single-zone geometry comes from the requirement of packing the sample space with such zones and, as noticed from the very definition of zones, covering (i.e. overlapping) is not at all forbidden. Geometrical constraints with covering zones are weak [16], and a nearly unique zone profile and p a r a m e t e r A is expected to occur.
3. The variational process
(15a)
(15b)
Finally, the m e a n magnetization m is also due to this fluctuating term and takes the form: m E=
N-
(16)
1 M 2.
The calculation of the total energy per zone E t enables us to define a m e a n total energy per spin: e t = E t / N and an energy density • =et/(ad), which reads:
E I M Z = ( a l 2 ) [ ( d - 1 ) R -2 + R ' - 2 ] - A ~ R - ( d - 1)R,- 1 _ i / ( 4 b a)
-IR-(d-1)/2R'-l/2/(4bd/2).
(17)
The optimization of the energy density • versus p a r a m e t e r s R and R ' gives, after some manipulation, the set of equations:
R=R'=x
2,
x4-d + 2 A x 4 - Z d - - 4 A = O ,
(18a)
A = 4Al.~adbd/2/I,
(18b)
with Since there is no effective energy coming from the interaction of different zones, the total en-
A = 2abd/2/I
and
314
J.C.S. L~vy, J.J.G. L~vy / Dipolar effects in ferromagnets
where the p a r a m e t e r s A and A are related to a unit anisotropy.
4. The optimal zones
The conclusion of eq. (18a) is that optimal zones are spherical. In dimension 2, eq. (18b) shows that dipolar effects act as an extra anisotropy. In dimension 3, equation (18b) reads as a cubic equation: x 3 - 4Ax 2 + 2A = 0,
(19)
which, in the case of a weak anisotropy, leads to an optimal value of R = A/2A, i.e. the radius of a magnetic bubble and in the opposite case of a weak dipolar effect leads to the classical value x = 4A [3]. T h e resolution of this cubic equation is quite classical [17], with the criterion for the existence of three real roots: 128A3/A + 27 < 0, which means a negative value of A. The self-consistent treatment of the zone behavior leads to the conclusion that, if the zone can be spherical, i.e. R = R ' , it minimizes the energy, there is no dipolar asymmetry, thus A = 0, and therefore there is no dipolar effect. This assumption is satisfied either when the film thickness t is just one layer, when it defines a real 2D problem; or when the film thickness t is larger than R0,3, when it gives rise to a classical 3D problem with no effective dipolar effect upon zone shape or size. However, when the film thickness lies in the range [a,R0,3], realistic optimal
zones are ellipsoids because of geometric constraints, and the demagnetizing factor A can take positive or negative values. Thus dipolar effects on magnetic zones are restricted to this range of film thicknesses which can be observed in thin films and multilayers where magnetic zones are forced to be non-spherical.
References [1] Y. Imry and S.K. Ma, Phys. Rev. Lett. 35 (1975) 1399. [2] J. Villain and B. Semeria, J. Physique 44 (1982) L889. [3] E.M. Chudnovsky, W.M. Saslow and R.A. Serota, Phys. Rev. B (1986) 251; W. Saslow, Phys. Rev. B (1987) 3454. [4] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. [5] M. Gabay and G. Toulouse, Phys. Rev. Lett. 47 (1981) 201. [6] C. Kittel and J. Gait, in: Solid State Physics, vol. 3, eds. F. Seitz and D. Turnbull. (Academic Press, New York, 1956) p. 437. [7] G. Suran, M. Rivoire and J.C.S. L6vy, J. Appl. Phys. 67 (1990) 5649; J. Magn. Magn. Mater. 123 (1993) 52. [8] G. Suran and M. Rivoire, Preprints [9] A.H. Bobeck and E. Della Torre, in: Magnetic Bubbles, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1975). [10] I. Privorotskii, in Thermodynamic Theory of Domain Structures (Keter, Jerusalem, 1976). [11] K. Pearson, Nature 77 (1905) 294. [12] Lord Rayleigh, Phil. Mag. 10 (1980) 73. [13] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [14] W. Feller, in: An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971). [15] H.J.G. Draaisma and W.J.M. de Jonge, J. Appl. Phys. 64 (1988) 3610. [16] C.A. Rogers, Proc. London Math. Soc. 8 (1958) 609. [17] Handbook of Mathematical Functions, eds. M. Abramowitz and I.A. Stegun (Washington, 1964).
Journal of Magnetism and Magnetic Materials 125 (1993) 310-314 North-Holland
Dipolar effects upon order in ferromagnets with random anisotropy J.C.S. L6vy a n d J.J.G. L6vy Laboratoire de Magn~tisme des Surfaces, Universitd Paris 7 Denis Diderot, 75251 Paris 05, France Received 25 November 1992
Dipolar effects upon ordering into magnetic zones of ferromagnets with randomly directed anisotropy are demonstrated to be efficient for thin films with different shapes for magnetic zones, from discs to columnar bubbles.
1. Introduction In recent years a considerable amount of work has been devoted to magnetic order in materials where strong ferromagnetic exchange competes with random anisotropy [1-3]. The magnetic structure that results from this competition is known to show a local ferromagnetic order, the size of which is characterized by a ferromagnetic correlation length (FCL), i.e. a typical radius for nearly spherical zones, which is large compared with the interatomic distance a. There is a large interest in the competition between order and statistics, since a similar competition occurs in spin glasses [4,5] with, basically, a similar analysis into zones characterized by a correlation length, while competition between deterministic exchange and anisotropy is quite different and leads to Bloch and N6el walls [6], i.e. well defined arrangements with finite extensions. Moreover, the magnetic structure that results from the competition of exchange and random anisotropy in discrete systems, i.e. lattices, is only slightly shifted by the introduction of an external magnetic field or of a constant anisotropy term that, however, breaks down the spherical symmetry [3], which Correspondence to: Dr J.J.G. L6vy, Laboratoire de Magn6tisme des Surfaces, Universit6 Paris 7 Denis Diderot, 75251 Paris 05, France.
implies a special critical behavior. Such a constant anisotropy term can be caused by a magneto-elastic effect applied to the internal stress which occurs during sample preparation. Among various possible realizations of realistic materials with these properties, soft amorphous ferromagnets (SAFs) ensure both a high value of exchange fields and a large distribution of sites with different 'crystalline' field values [7]. These SAF materials, as a matrix, can be doped with a rather large concentration range of rare earths (RE) of individual large anisotropy. Such doping can provide as large a density of local anisotropy energy as required. At each site occupied by a rare earth, the easy axis of anisotropy is determined by the local 'crystalline' field and in an amorphous structure, these axes are nearly random, with correlation lengths that is typically the nearest-neighbor distance b between atomic sites occupied by rare earths. Thus RE-doped SAFs enable us to follow the competition between exchange and random anisotropy when the density of the anisotropy energy, i.e. of the RE, varies over a wide range [8], since magnetically correlated clusters can be neglected. In doped SAFs, electron conductivity is usually so low that dipolar effects remain completely unscreened. Then, since dipolar effects are not easily compatible with ferromagnetic order or with the order which is induced by a random distribution of anisotropy
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J. C. S. Ldvy, J.J.G. Ldvy / Dipolar effects in f erromagnets
directions, the competition between the three terms (exchange, random anisotropy and dipolar effects) is interesting. The first goal of this paper consists in this analysis. A special case of interest is that of weak random anisotropy where magnetic zones are still expected to occur when dipolar fields are neglected [1,3]. Experimentally, in the absence of anisotropy, dipolar effects are known to lead to magnetic structures with opposite domains of various shapes such as stripes, bubbles and mazes in thin films [9] and to other structures in large samples [10]. Realistically, a critical value for random anisotropy versus dipolar effects is expected to occur in order to stabilize the zone structure. Another interesting point concerns the application of such a theory to thin films and multilayers since the ferromagnetic correlation length depends strongly on dimensionality 2 or 3. Since magnetic structures with overlapping magnetic zones lead to giant magnetic susceptibilities [3], direct applications are also involved for thin films and multilayers. The existence of magnetic zones is essentially an effect of the discreteness of atom packing. An isotropic zone of N atoms in a space of dimension d has a typical correlation length (FCL) R - - N 1/d. Thus, the magnetization gradient in this zone is proportional to N - 1 / d and the density of exchange energy •e is proportional to N-2/d:
E"e
l
=
2
( 1 / 2 ) A (ViM~,) - - A N - 2 / a ,
(1)
where A is the exchange constant and M is the local magnetization which is the spin density. The discrete version of anisotropy energy, with a local uniaxial behavior, reads: H A = - (1/2)EI(M.
nr) 2,
(2)
where n r is the unit vector of the local easy axis of magnetization at site r, and I is the local uniaxial anisotropy constant, which is assumed to be everywhere the same in this simple approach. The direction n R is assumed to be randomly distributed, i.e. with a correlation length b which is negligible compared with R, the size of the zone, and is a function of the R E density. Up to
311
an additive constant and a multiplicative factor, the discrete anisotropy energy of the zone is the sum of N cosines of random angles 20r, with 0 r = ( M , n r ) , exactly as a component of the result of N random flights as introduced by Pearson [11] and Lord Rayleigh [12] in relation to different topics and reported in the classical paper by Chandrasekhar [13]. The law of large numbers [14] enables us to estimate the anisotropy energy in the zones as: EA--- - k N 1/2,
(3)
with k as the anisotropy factor; the density of anisotropy energy thus reduces to: • A -------- KN - 1/2
(4)
with K as the anisotropy parameter. Neglecting dipolar effects, the total energy density • reads: e = A ( N ) -2/d _ K ( N ) -1/2
(5)
The minimization of this energy density with respect to N provides the optimal value of N : N o for the zone with: N O = ( 4 A / K ) (2d)/(4-d).
(6)
When the dimension d = 4, the exchange and anisotropy share the same variation with N, the minimum energy density is reached for N infinite [1], there is only one single zone. If the dimension d is larger than the marginal dimension 4, at N O the energy density is maximal, and the concept of a zone is no longer valid. For a 3D space, a minimum energy density is reached at No,3 = ( 4 A / K ) 6, with R0, 3 = ( 4 A / K ) 2, while for a 2 D space (i.e. a surface) the minimum energy density is reached for No,2 = ( 4 A / K ) 2 and R0, 2 = 4 A / K . The difference between these two values of the FCL length Ro, i is quite large and can become physically meaningful when the thickness t of thin films or the typical thickness t' of the films which form a multilayer, is varied. If these thicknesses are large compared with Ro,3, i.e. in a real 3D case, spherical zones with radius Ro,3 must be observed. If the thicknesses t lies in the intermediate range I, with I = [ R o , 2 , Ro,3] , cylindrical
312
J.C.S. L~vy, J.J.G. L~vy / Dipolar effects in ferromagnets
zones (i.e. bubbles) are expected to occur, with a typical radius p = ( 2 A / K ) t 1/2, which belongs to the interval I ' , with I ' = [(2A/K) 3/2, R0,3]. If the thicknesses t are lower than Ro, 2, cylindrical zones (i.e. discs) are expected to occur, with the same expression as before for the radius p, which now lies in the interval I", with I " = [Ro,z,(2A/K)3/2]. Thus large variations in the sizes and shapes of these zones must be observed either when thicknesses are varied over a short range, or when R E concentrations are slightly varied, since the effective value K of anisotropy depends upon the rare earth composition. For thicknesses t of a few R0,3, surface effects must still lead to bubble-shaped zones with axis normal to the film or multilayer planes. The precise purpose of this p a p e r is to introduce dipolar contributions when dealing with a rather thick film, i.e. with t >> R. This also means that the surface pinning of spins, which is a strong factor for bubble-shaped or disc-shaped zones, can be neglected. In section 2 generalities on the Hamiltonian and on the assumed shapes of magnetic zones are given, while the variational process is applied in section 3 and the nature of optimal zones is derived and discussed in section 4.
2. Generalities In this problem, energy must be counted at three levels: at a local level, with a spin Hamiltonian H ; at the level of a zone, with a zone energy E; and at the global level with an energy density E. The spin S per unit volume defines the local magnetization M, which is assumed to take everywhere the same magnitude in this low-temperature problem, but a nearly r a n d o m orientation. Because of this, the average magnetization m of a zone has a lower magnitude than M. The exchange Hamiltonian H e reads: H e = -(1/2)J•S
i'Sj,
(7)
i,j
which leads to the density of exchange energy expressed by eq. (1). The anisotropy Hamiltonian
was given in eq. (2). The dipolar field h a at site i reads, as usual: h d = ~_~[(3rijSj.rij-Sjri2j)/r~],
(8)
J
where rii is the vector which links sites i and j. Finally, the Z e e m a n Hamiltonian H z for a magnetic field H reads: n Z : --].b E S i " i
H,
(9)
with a magnetic constant /z; quite obviously a constant anisotropy contribution, i.e. with a given direction, can be integrated in this term as well as the dipolar field contribution. A zone with an average magnetization m induces inside this zone a dipolar field h a of which, because of symmetry arguments to be developed later, the effective part is collinear with m. Since this effective dipolar field depends on the shape of this magnetic zone, such a shape effect must be introduced when taking into account the symmetry axis defined by the magnetization axis m. With this symmetry, the zone is expected to be nearly a rectangular parallelepiped with typical sides R, according to directions perpendicular to m, and R ' , according to the m direction. O f course, it also means an ellipsoidal shape of the zone. Thus the volume of a zone is proportional to RZR ', with a form factor and the number N of atoms in a zone is up to a correcting density factor: N = R Z R ' / a 3, where a is the nearestneighbor distance between magnetic sites, i.e. a is lower than b. These relations are easily translated for a space of dimension d, with:
N = Rd-lR'a-d.
(10)
Then, the square of the magnetization gradient has for average magnitude ( ~ M ~ ) 2 = [(2/R 2) q-(1//g'2)], when d = 3, because of symmetry. For arbitrary d, it leads to:
which enables the exchange energy within the zone to be estimated. The estimation of the dipolar energy of the zone can be deduced from that of the dipolar
J.C.S. L3vy, Z Z G. Ldvy / Dipolareffects in ferromagnets field h d. Outside the zone, the average magnetization is zero and the weight of the integrating factor is strongly decreasing, and thus the resulting dipolar field due to the long-range part can be neglected. Within the zone, each local magnetization can be analyzed into two parts: namely, the average magnetization m and the rest, which is strongly fluctuating and has a null average, because of the distribution of anisotropy sites (RE), i.e. with a typical correlation length b. Because of such strong fluctuations, the highly fluctuating part gives no resulting contribution and the effective dipolar field is due only to the average magnetization applied to the whole zone. A constant magnetization is known to lead to a dipolar field which is practically constant within the zone with the exception of a sharp zone boundary where the estimated relative difference between surface and bulk dipolar fields is only a few per cent [15]. With the assumed symmetries of the zone, this effective dipolar field h d is parallel to the magnetization m, with a ratio of proportionality A, and reads:
h d = Am,
313
ergy must be calculated in a single zone. First, the exchange energy in a zone reads: E e = (1/2)aM2[(d-
1)Rd-3R' + R d - I R ' - ' ] . (13)
T h e dipolar energy E d for a single zone is due to the Z e e m a n effect created by the effective dipolar field, it reads:
Ed = - Al~m2R d- 1R',
(14)
and the anisotropy energy E A is a sum of N ' =
Rd-IR'b -d terms due to each RE: E A = - ( 1 / 2 ) IM 2 ~ '
cos20.
This sum can be analyzed into two terms, a m e a n t e r m which is proportional to N', and a fluctuation term which is proportional t o N '1/2 and depends upon the orientation of the m e a n magnetization m. Thus the optimization of this fluctuation term determines the m e a n magnetization orientation, and E A reads: E A =
- (1/4)
(12)
IMER d- 1R'b-d
-- ( 1 / 4 ) IM2R(d-1)/ER'1/Eb -d/2. where the ratio A is a demagnetizing factor associated with t h e zone geometry. T h e n it can be assumed that the optimization process to be developed later leads to a unique optimal zone geometry, for a given value of A; thus, the optimal zone geometry and A will be determined self-consistently at the end of this calculation. Practically, this assumption is not very constraining since the only opposition to a single-zone geometry comes from the requirement of packing the sample space with such zones and, as noticed from the very definition of zones, covering (i.e. overlapping) is not at all forbidden. Geometrical constraints with covering zones are weak [16], and a nearly unique zone profile and p a r a m e t e r A is expected to occur.
3. The variational process
(15a)
(15b)
Finally, the m e a n magnetization m is also due to this fluctuating term and takes the form: m E=
N-
(16)
1 M 2.
The calculation of the total energy per zone E t enables us to define a m e a n total energy per spin: e t = E t / N and an energy density • =et/(ad), which reads:
E I M Z = ( a l 2 ) [ ( d - 1 ) R -2 + R ' - 2 ] - A ~ R - ( d - 1)R,- 1 _ i / ( 4 b a)
-IR-(d-1)/2R'-l/2/(4bd/2).
(17)
The optimization of the energy density • versus p a r a m e t e r s R and R ' gives, after some manipulation, the set of equations:
R=R'=x
2,
x4-d + 2 A x 4 - Z d - - 4 A = O ,
(18a)
A = 4Al.~adbd/2/I,
(18b)
with Since there is no effective energy coming from the interaction of different zones, the total en-
A = 2abd/2/I
and
314
J.C.S. L~vy, J.J.G. L~vy / Dipolar effects in ferromagnets
where the p a r a m e t e r s A and A are related to a unit anisotropy.
4. The optimal zones
The conclusion of eq. (18a) is that optimal zones are spherical. In dimension 2, eq. (18b) shows that dipolar effects act as an extra anisotropy. In dimension 3, equation (18b) reads as a cubic equation: x 3 - 4Ax 2 + 2A = 0,
(19)
which, in the case of a weak anisotropy, leads to an optimal value of R = A/2A, i.e. the radius of a magnetic bubble and in the opposite case of a weak dipolar effect leads to the classical value x = 4A [3]. T h e resolution of this cubic equation is quite classical [17], with the criterion for the existence of three real roots: 128A3/A + 27 < 0, which means a negative value of A. The self-consistent treatment of the zone behavior leads to the conclusion that, if the zone can be spherical, i.e. R = R ' , it minimizes the energy, there is no dipolar asymmetry, thus A = 0, and therefore there is no dipolar effect. This assumption is satisfied either when the film thickness t is just one layer, when it defines a real 2D problem; or when the film thickness t is larger than R0,3, when it gives rise to a classical 3D problem with no effective dipolar effect upon zone shape or size. However, when the film thickness lies in the range [a,R0,3], realistic optimal
zones are ellipsoids because of geometric constraints, and the demagnetizing factor A can take positive or negative values. Thus dipolar effects on magnetic zones are restricted to this range of film thicknesses which can be observed in thin films and multilayers where magnetic zones are forced to be non-spherical.
References [1] Y. Imry and S.K. Ma, Phys. Rev. Lett. 35 (1975) 1399. [2] J. Villain and B. Semeria, J. Physique 44 (1982) L889. [3] E.M. Chudnovsky, W.M. Saslow and R.A. Serota, Phys. Rev. B (1986) 251; W. Saslow, Phys. Rev. B (1987) 3454. [4] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. [5] M. Gabay and G. Toulouse, Phys. Rev. Lett. 47 (1981) 201. [6] C. Kittel and J. Gait, in: Solid State Physics, vol. 3, eds. F. Seitz and D. Turnbull. (Academic Press, New York, 1956) p. 437. [7] G. Suran, M. Rivoire and J.C.S. L6vy, J. Appl. Phys. 67 (1990) 5649; J. Magn. Magn. Mater. 123 (1993) 52. [8] G. Suran and M. Rivoire, Preprints [9] A.H. Bobeck and E. Della Torre, in: Magnetic Bubbles, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1975). [10] I. Privorotskii, in Thermodynamic Theory of Domain Structures (Keter, Jerusalem, 1976). [11] K. Pearson, Nature 77 (1905) 294. [12] Lord Rayleigh, Phil. Mag. 10 (1980) 73. [13] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [14] W. Feller, in: An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971). [15] H.J.G. Draaisma and W.J.M. de Jonge, J. Appl. Phys. 64 (1988) 3610. [16] C.A. Rogers, Proc. London Math. Soc. 8 (1958) 609. [17] Handbook of Mathematical Functions, eds. M. Abramowitz and I.A. Stegun (Washington, 1964).