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The equations of motion of micromagnetics in Hamiltonian form
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Journalof magnetism and magnetic materials
Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
The equations of motion of micromagnetics in Hamiltonian form M.A. Wongsam *, R.W. Chantrell Department of Physics and Electronics, Keele University, Keele, Staffs ST5 5BG, UK
Received 29 June 1995
Abstract
The equations of motion describing the response of a ferromagnetic system in the micromagnetic approximation are developed by applying Hamilton's principle to an action functional constructed from a Lagrangian density composed of the usual free energy expression, together with a suitably chosen expression for the kinetic energy density so as to produce the characteristic Larmor precession associated with magnetic spin systems under the influence of an external field. A suitable choice of the form of the generalised momenta and the standard form of the Rayleigh dissipation energy density functional, leads to equations of motion which exhibit similar behaviour to the Landau-Lifshitz-Gilbert form. It is shown that with our Hamiltonian formulation, micromagnetic simulations of systems with very many degrees of freedom can be performed with a substantial increase in the upper limit on the timestep size.
1. Introduction
Much attention has recently focused on the micromagnetic description of systems with many degrees of freedom, with the intention of acquiring a better understanding of macroscopic processes, without appealing to a treatment at the atomic level. In order to achieve this, it is necessary to account for a variety of forces, all of which contribute significantly, and hence, irreducibly, to the response of ferromagnetic systems. These forces include the exchange, magnetocrystalline, and magnetostatic forces. In addition, most experimental situations require the application of an extemally applied magnetic field. In general, all of these forces need to be accounted for from point to point throughout the sample. Many of the models in use utilise energy-minimising algorithms for the purposes of simulating the stationary states of ferromagnetic systems. However,
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many of the energy minimisation techniques are idiosyncratic, and cannot be applied generally. Moreover, properties such as the ferromagnetic resonance and the reversible transverse susceptibility require a description of the transient response of the systems. In addition, magnetisation reversal is an intrinsically dynamic phenomenon and energy minimisation cannot be guaranteed to find the correct magnetic state after irreversible transitions. Consequently the dynamic approach to micromagnetics is often used. For this purpose, the standard approach has been to utilise the equations of motion in the Landau-Lifshitz form. There are two forms of the equation of motion corresponding to two different phenomenological models of the dissipation. A good discussion comparing the two formulations has been given by Mallinson [1]. The older form due to Landau and Lifshitz is usually written dM a dt = T ( M X H ) - ~ - ~ M × ( M × H ) , (l)
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where Y is the Landau-Lifshitz gyromagnetic constant, and ot is the Landau-Lifshitz damping constant, and where M ( x , t ) is the magnetisation. The more recent Gilbert form is dt = Y G ( M X H ) -
MX
at
'
(2)
where Tc is the Gilbert gyromagnetic constant and a G is the Gilbert damping constant. Clearly, when there is negligible damping the two forms are virtually equivalent. In this case, the magnetization precesses eternally, at a frequency of 2.8 MHz Oe -~. When large damping is considered, profound differences arise between the two forms. Mallinson concluded that the Gilbert form is to be preferred on purely physical grounds. However, in order to drive the system to equilibrium workers have used either formulation, using the damping constant as a means of optimising the rate of descent to a minimum of the energy surface. One of the difficulties inherent in performing micromagnetic simulations involving both local and non-local interactions lies in the instabilities arising out of the stiffness of the differential systems, and not as a result of the nonlinearity of the Landau-Lifshitz equation as mentioned in Ref. [2]. This imposes severe restrictions on the choice of timestep in addition to those resulting from the form of the differential equations. In this paper we are concerned with the form of the differential equations and the effect this has on the choice of timestep. In solving dynamical equations with damping when many degrees of freedom are involved, two considerations affect the choice of timestep. Consider a ferromagnetic sample ~ with surface ~. The dynamical state at some time t is specified if M ( x , t ) is known at all x E {/2 U ,~}. In what follows, quantities with lowered indices refer to those quantities evaluated after a number of time increments indicated by the index. Raised indices refer to components of vectors or tensors. First, there is the requirement of convergence; that is to say, the solution trajectory should converge onto a limiting trajectory in phase space as $ t ~ 0 for all t. There will therefore be a requirement that the residuals at time t i, which may be denoted by Oi(x), satisfy
Io;;,(x)l <
a,
(3)
235
where E represents some tolerance. This represents a limiting condition on the choice of ~ t. Additionally, for problems in which relaxation to a stationary state occurs, there is also the question of the termination of the algorithm. If M = M~v, where Ms is the saturation magnetisation, and v is the reduced magnetisation, the stability of monotonous absolute convergence to a stationary state requires that
I~vil 2 _< I~vi_~l 2, Vx e~1"2,
(4)
where ~ v k is the change in v k produced when t k --* tk+ ~ = t i -I- 8tk, SO that one has (5)
Vk+ ,( X,tk+ ,) = Vk+ ,( X , t , + ~ t , )
(6)
-: Vk( X , t k ) + BVk( X , t k ) .
Such a requirement may impose an upper limit on the timesteps 8t~ independently of the requirement of convergence of the solution trajectory mentioned above. This is a strong stability condition and would not be imposed during an irreversible transition of the magnetisation state. The Landau-Lifshitz equation can be written (7)
b=TvXH-avXvXH.
Therefore, in a small time increment ~t, 8v = yv X H~t-
ctv X v X H ~ t .
(8)
At time t~, suppose there is an increment 8 t~ resulting in an increment in the components of reduced magnetisation 8v i such that v~+ 1 = v i + ~v~. Then the Landau-Lifshitz equation can be written
~l]i~- ~l(vi_ , 4-~Vi_l) XH~t-ol(vi_
X(vi_, + ~v,_,) = yvi_l XHBt+
XH$t
1 X
~/2i_
(9)
ySvi_ 1XHSt
-- OlVi_ 1 X Vi_ 1 X H S t -- Of/)i_
I 4-~b*i_ l)
(10)
I XH~t
- a S v i _ 1 X vi_ 1 X H S t --Ol~Vi_ 1 X ~lfli_ 1 X H ~ t
(11)
= ~)Vi- 1 + "Y~3Vi- 1 X H S t --
O~Idi_ I X ~/)i_ l X H ~ t
(12)
- ctSv~_ 1 X vi_ l X H ~ t --Ol.~l.)i_ l X ~Vi_ 1 X H ~ t ,
(13)
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where in the final step, Eq. (8) evaluated at time t i_ 1 has been used to replace the first and third terms of the penultimate step. Therefore, since Eq. (8) must be evaluated subject to the constraint v k • v k = 1, and for any vectors a, b and c, there is an identity a × b × c- b(a. c)- c(a.b), ~l)i = ~ V i - 1 "]- " Y ~ V i - 1 × -- O l V i - l ( ~ V i
H S t - c~Svi(
V i_ 1 " H ) ~ t
(14)
- 1" U ) S t .
For relaxation processes, there is therefore a trade off between timestep and relaxation rate - increasing the dissipation constant leads to a shortening of the maximum timestep. In this paper we present a reformulation of the dynamic problem using the Hamiltonian formalism which enables the integration timestep to be increased without loss of accuracy leading to a significant increase in computational speed.
Taking the scalar product of both sides of this equation,
I~vil 2 = [~vi-112 +
"y2~t2l?3Vi
- 1 ×
2. Reformulation of the dynamical equations in Hamiltonian form
HI2
+ ot28t2( vi_ , • H)21~v,_ ~12
(15)
+ O~2~t2(SVi - 1 "H)21vi-112 - - 2 0 t (~Vi_ 1 " n ) l ~ V i_ tlZSt -- 20~'y~/2(~/,'i_ 1 × H ) "
(16)
l?i_ l(~g?i_ 1" H ) , (17)
where the constraint condition and the vector identity given above have both been applied again. This relates the increment in the reduced magnetisation at time t~ to the increment at time t i_ 1" The criterion that absolute monotonous convergence takes place can be stated as a requirement that 1~3vil2 - [~v i_ j[2 < 0, that is,
~t < B / A ,
(18)
One can reformulate the equations of motion according to a variational principle [3], by constructing a suitable kinetic energy density 3-. Let b i denote the partial derivative of vi with respect to time, which can be called the ith component of the magnetisation rate vector, and U, etc., denote the higher derivatives. Also, let ~t "i~= ~v~/Ox j denote the components of the 'magnetisation gradient tensor'. Also, let Ev(V i, ~t'iJ; t) and Es(v i) be the volume and surface free energy densities, respectively, where E v may or may not depend explicitly on the time according to whether the external field is considered to be time dependent. The total free energy will then be
E[ v i, .4~'iJ; t] = foEv( v i, ..~,ij; t) d O
where A = ] / 2 [ ~ v i _ 1 x n l 2 + ~2(vi_ 1 . H )
2
+ f Es( v i) d ~ .
I~vi_ll 2 (19)
+0~2(~Vi_I "H)2)[Iyi_I ]2 --20f')/(~Vi_ 1 " n ) " V i _ l ( ~ V i _
B = 2 a ( v~. H)l~v,_ 112.
1XH),
(20)
(21)
There is therefore an inverse quadratic dependence on c~, with ~t having a clearly defined maximum. Convergence of the solution, that is to say, relaxation, will not take place unless cz > 0, since a --) 0, implies ~ t ~ 0. At the other extreme, as ot ~ ~, ~ t is proportional to 1 / a , and hence, also tends to zero.
(22)
In particular, for continuous magnetisation distributions, since one has V . v =.,g#, where in what follows, the summation convention due to Einstein has been adopted, the exchange and magnetostatic contributions to the internal energy, respectively, can be written as
Eexch = C.#"~.~ "'~, Ms2 r Emagn = - - v . j~
2
(23) x - x' , 3-.KJJ(x') d,~,
Ix-x
(24)
where Eexch a n d Emagn are the exchange and magne-
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M.A. Wongsam, R.W. Chantrell/Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
tostatic energy densities, respectively. Then, form the Lagrangian density . ~ ( v i, b i, j~,ij; t) by the prescription . ~ = J - - E v. The system Lagrangian is then defined as
L= fo.~.~'(vi,[ji,~iJ;t)d12- fzEsd~,
(25)
237
density, Y . This may be written as a functional in the hi:
o~r[ bil = ½f rlbi bi dt,
(29)
where r/ is a material constant with the dimensions of energy density times time. The generalised dissipative forces are then
and the action integral is then
~F ~/)i "
Q~iss -
(30)
qJ[ v~, b~, JWJ; t, , t2 ] = ft2 f _oc~d12 dt t~
1"1
With these additional forces taken into account, the evolution of the system is governed by
- f £ 2 f Es d,~ dr.
(26)
Minimisation of gt then gives the dynamical response of the system to the forces derivable from the potential E. Following Sudarshan and Mukunda [4], the equations determining the necessary and sufficient conditions for the action functional to be stationary are found to be OV i
31 ab i
..~J &j¢,,l
ax j a..~fi j
. =0, av'
=
Vx~Z,
O, Vx E 12,
aZ
~ aS"
OV i
at a[; i
a_~
a
aEs
.. ~ J - - -
av i
a~',J
=
aS"
a9r
a x j a./~¢ij
a[3 i
0.
O,
(31) (32)
(27)
These are the equations of motion in Lagrangian form. Again following Sudarshan and Mukunda [4], the Hamiltonian form of the equations of motion can be derived by considering a Legendre transformation:
(28)
pi =
where the first of these represents the bulk conditions, and the second represents the boundary conditions. These conditions are valid provided that variations in v ~ are zero at t I and t 2. These have to be solved for each value of the time t subject to the prescribed initial conditions, which for most applications can be given as b ~= 0 throughout 12 U ~. In addition, for problems in micromagnetics there are also constraint conditions to be satisfied at every x ~ {12 U X}, and an auxiliary problem of the magnetic scalar potential, q), which must be solved simultaneously with the equations of motion. The constraint conditions are incorporated using the method of Lagrange, and the magnetic scalar potential is determined as the solution of Poisson's equation V2q~ = 4"rrV • M in 12. These conditions give the response of the system to the forces which are derivable from the potential E. However, the relaxation of the system is determined by non-conservative dissipative forces, which can be derived from a Rayleigh dissipation energy
(33)
a/)i '
a ~ ==-p i D i - , . ~ ,
where pi is the canonical generalised momentum distribution conjugate to v i, and ,Tg' is the Hamiltonian density conjugate to t. With these definitions, the evolution equations become aX
p i + -av i
a
&,~'
ax i &zt"ij
a,Tg"
ab'
(34)
(35)
ap i &,~ aE s . ~J+ . =0,
a.t[ u
a,9r + ---= = o,
av '
(36)
where pi is the partial derivative of the ith generalised momentum, and with initial conditions chosen to suit the problem. As an example, consider the case of a single dipole in an external field (HX,0,0) without damping. Then the total potential is E = - M s V . H + A(v 1), where A is an as yet undetermined multi•
v
-
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M.A. Wongsam, R.W. Chantrell / Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
plier expressing forces of constraint, according to the method of Lagrange. Hence, in this case one has a ~ i J = 0,
(37)
age" a" i
(38)
MsHi + 2Av i.
Following Brown [3], consider now a form of the kinetic energy density linear in bi, given by ~ - = A i ( v k ) b g,
pressing the magnetisation rates is tautologous, and hence, instead, one is generally required to solve the set consisting of Eqs. (34) and (36), subject to initial conditions and constraints. A solution exists to the equations of motion if dissipative forces are taken into account. The first two equations to be solved are
0 = - * l b I + M s H 1 - 2Av 1, 1
~,(.,)2
(39)
where Ag(v k) are functions to be specified. Then, applying Eq. (33) produces p i = A i ( v k ) . Suppose that, in the example above, the p~ are chosen to have the form
(47)
U3/)I] = __*l/32 __ 2Av 2.
[UI/~3
(48)
Seeking a solution representing damped motion and consistent with the equations of constraint, vz = a(t)cos O(t),
(49)
pX= 1/y,
(40)
v3 = a(/)sin O(t),
(50)
pY = --vz//Tvx ,
(41)
v 1 = 1/1 - a 2 ,
(51)
pZ = v y / / y v x.
(42)
Then, transposing the equation in p~ from Eq. (34) for A, one finds that
MsH x A= - -
Substituting this into the remaining equations for pY and pZ from Eq. (34), along with the expressions above, leads one to
1 [ av z ~v x } vy _ v~ -v z = -MsHXT, y(v~) 2 at at 1 -
b 2 = d cos 0 - at0 sin 0,
(52)
b 3 = d sin 0 + a/J cos 0,
(53)
b' = - ad/1/1 - a 2 ,
(54)
(43)
2v x
+
which have rates given by
[uxaU y
-
T( v x)= [
aU x ] -
.~
at
at
(44)
where a < 1. From Eq. (47),
Ms H1 A= 2 ~
*lad + 211-a21
"
(55)
Substituting Eqs. (49)-(54) into Eq. (48),
Uz
] = -Msl-l~. y(1
1
a 2)
(~/1 - a 2 d s i n 0
(45) a 2 fi sin 0 +dl
Now, seeking a solution (v ~, v ~, v z) -- (t~ cos tot, a sin tot, b) representing precessional motion, one finds that at to cos to t
M s H xot cos to t
yb
b
'
a 2
a0 cos 0 +
= -- *ld cos 0 + *la0 sin 0
(46)
or t o = T M s H x. In other words, a system with a kinetic energy density given by Eq. (39) responds to an applied field by precessing with an angular velocity that is dependent only on the applied field and material constants. For such a system, Eq. (35) e x -
--
M s H 1 a cos 0 -
l/i" -
a 2
*la 2 fi cos 0 1 -- a 2
(56)
Equating the coefficients of sin 0 leads to /} =
d -
y*la[1 - a 2 ]3/2 '
(57)
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of Magnetism and Magnetic Materials I52 (1996) 234-242
plier expressing forces of constraint, according to the method of Lagrange. Hence, in this case one has ax 7
a”4f’J
=o,
(37)
az = -M,H’ aui
+ 2A”‘.
(38)
Following Brown [31, consider now a form of the kinetic energy density linear in C’, given by 9-=AA’(vk)fii,
(39)
where Ai are functions to be specified. Then, applying Eq. (33) produces pi = Ai( Suppose that, in the example above, the pi are chosen to have the form
pressing the magnetisation rates is tautologous, and hence, instead, one is generally required to solve the set consisting of Eqs. (34) and (36), subject to initial conditions and constraints. A solution exists to the equations of motion if dissipative forces are taken into account. The first two equations to be solved are 0=
-qfi’+M,H’-2hv’,
(47)
1 1 1.3
--vu
1 - 7)2- 2hv2.
_“3fi’
=
Y(“‘)2
(48)
Seeking a solution representing damped motion and consistent with the equations of constraint, v2 = u( t)cos O(t),
(49)
pX= l/Y,
(40)
“‘=a(t)sin8(t),
(50)
py=
(41)
vl=K-T,
(51)
(42)
which have rates given by
-vz/yvx,
pz = vY/yv”. Then, transposing the equation for A, one finds that A=-
in px from E@. (34)
M,H” 2””
fi2=d!cos8--ffbsin8,
(52)
fi3=Cisin8+aBcos8,
(53)
fi’ = -aci/&-z,
(54)
(43)
*
Substituting this into the remaining equations for py and pz from Eq. (341, along with the expressions above, leads one to
where a I 1. From Eq. (471, M,H’
lpi
A= 2\/1-;;i-
--
Substituting
Eqs. (49)-(54)
Y(1 -a’>
(45)
CYWcos WI yb
=
MsHxacos b
(uX, uy, vL) = ((Y cos wf, precessional motion, one
(55)
2[1-a’]’
1
-
Now, seeking a solution (Ysin wf, b) representing finds that
+
into E!q. (481,
h sin e
CT i
+i‘iTdc0s
u’cisin
e+
6-7
8 1
= -7+cos8+&sin8 wt ’
(W
or o = yM, H”. In other words, a system with a kinetic energy density given by Eq. (39) responds to an applied field by precessing with an angular velocity that is dependent only on the applied field and material constants. For such a system, IQ. (35) ex-
M,H’ UCOS 6
-
4s
Equating 8= -
~&~COS -
the coefficients
l-u2
8
’
(56)
of sin 8 leads to
b
yqu[1 -
u2 ]3’2 ’
(57)
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M.A. Wongsam, R.W. Chantrell / Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
Defining
pation ('0 --> 0) results in no convergence (~t --->0) as in the Landau-Lifshitz case. As "0 ~ ~, since
2K, y 0 [ 1 - y ~ ] F(y) =
(76)
K 2 y02 + 1
H l _ 'oh 1 =
the solution can be written correct to the second order in t, as 4Kl Yo [
Y=Yo l+Ft+K-2y--~o+l
1-2y¢~
K2y~(1-y2) ] F ~ ) +O(t3). K2YZo+I
-
(77)
2v ~
,
(85)
in order that 8t remain positive, it is necessary that '0 be bounded above by H~/b ~. It can be seen that as "0 approaches Hl/b 1 from below, the upper bound on 8 t diverges. There is therefore a clear computational advantage over the Landau-Lifshitz form of the equations of motion. It must be noted, that care still has to taken over the choice of 8t so as to ensure convergence of the trajectory in phase space, and methods appropriate to stiff differential systems are to be preferred.
3. Comparison with the Landau-Lifshitz equation It is interesting to compare the application of the dynamics formulated above with the Landau-Lifshitz dynamics now in current use. The Hamiltonian form of the equations of motion can be written, "00 = H - 2Xv - p .
(78)
Proceeding as with the treatment of the LandauLifshitz equation, '08v = ( H -
2Av) 8 t - ~p;
(79)
therefore,
2ASvi_ 18t,
4A28 t 2
[~vi- 112 _ _ '20
(81)
4A~ t ]~Vi_1[2 _ __lSvi_'0 112
(82) < 0
(83)
from which results ~t < "0/A.
~E
vi~V i,
(86)
(80)
on using Eq. (79) in the last step. The criterion for absolute convergence is i~v/i2 _
For more general problems involving energy densities and many degrees of freedom, one has an effective field composed of for instance, exchange, bulk and surface magnetocrystalline, magnetostrictive and magnetostatic components, as well as the external fields. These can be formulated as generalised forces Q, S conjugate to the generalised coordinates v, according to
faQi~vid~+ f Si~vidZ=
"0~[yi~-(H-eA[Vi_l--[-~Vi_l])~t-~pi_ I = "0~vi- 1 -
4. Simulations
(84)
Clearly, from this expression, the case of zero dissi-
which expresses the principle of virtual work, where the Qi are the components of any generalised internal and body forces, S ~ are the components of any generalised surface forces which will appear as contributions to the boundary conditions, and E is a total potential energy functional. For such problems, the solution represented by Eqs. (52)-(54) with a ( t ) and O(t) given as the solutions to Eqs. (60) and (58) has to be time-stepped, and the forces Qi, S ~ updated at every iteration. In order to use this solution, a local coordinate system must be defined with the effective field parallel to the x-axis. Comparisons have been made between calculations performed using the Landau-Lifshitz form of
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241
2.5"10 = -
2.0"10= -
1.5"10=o
•
•
1.0"10 =-
5 . 0 " 1 0 =~-
m •
! 0 . 0 " 1 0° 0 . 0 ° 1 O*
5.0" 1 0 ~
1.0" 1 0 H
1.5 ° 1 0 ~
Fig. 1. Comparison of elapsed times for some stationary states calculated using the Landau-Lifshitz TLt" and Hamiltonian TH dynamics.
the equations of motion and those using the Hamiltonian form with the above solution. The specimens were small rectangular platelets of polycrystalline
Cobalt with random in-plane uniaxial magnetocrystalline anisotropy. An effective field formalism was used, incorporating exchange, anisotropy, Zeeman
Fig. 2. Single-domain stationary state at 1500 Oe of a 780 × 380 × 40 A Co platelet, 4 × 4 × 1 square grains. The view represents the (x,y) plane at z = 20 A, where x is measured along the long axis in the plane of the platelet, y along the short axis in the plane, and z through the platelet thickness.
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M.A. Wongsam, R.W. Chantrell / Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
and magnetostatic interactions, the exchange and magnetostatic interactions being modelled according to a finite-difference scheme. The calculation of the magnetostatic interaction integrals was accelerated using a two-level hierarchical approach, the details of which will be reported separately. In these tests, the Landau-Lifshitz dissipation c o n s t a n t , aL L , was chosen to have a very large value, giving a fast relaxation [1]. The timestep in each case was chosen adaptively to give a stable evolution to the stationary state independent of the size of the timestep. Fig. 1 shows a comparison of elapsed times multiplied by a / M 2 for various values of the external field. A largely linear relation between the elapsed times TEL for the LandauLifshitz, and TH for the Hamiltonian cases is observed, with a ratio of approximately T H / T L L = 0.08147. The calculated stationary states were qualitatively and quantitatively similar for both approaches. Fig. 2 shows the single-domain state at 1500 Oe. for a 780 × 380 X 40 A polycrystalline Co platelet, showing the characteristic ripple predicted by theory [5,6] and experiment [7]. Here, the magnetisation distribution is resolved onto a 40 X 20 X 1 computational lattice and the grains are represented by 4 × 4 X 1 computational sublattices sharing common in-plane uniaxial anisotropy directions. A more detailed study of these systems, comparing a range of values of the intergranular exchange coupling, will be reported separately.
Comparisons have been carried out with the application of the Landau-Lifshitz form of the equations of motion. These comparisons indicate that a substantial improvement in computational performance can be achieved since the timestep required is generally much less than in an optimally damped system. However, more extensive tests will have to be carfled out, on much larger systems in terms of the number of degrees of freedom, so as to compare the dynamics of the two models when many more stable stationary states are possible. The development of the equations of motion in Hamiltonian form, opens vast possibilities for research in micromagnetics, since the classical methods of transformation theory and Hamilton-Jacobi theory become available and applicable. There is therefore a potential for the development of theoretical micromagnetics additional to the implications for computational micromagnetics described here.
Acknowledgements The work reported here was supported by grants from the SERC, and carried out within the framework of CAMST-2. RWC is grateful to Dr R.J. Veitch for handling the editorial duties for this paper, including the anonymous review.
References 5. Conclusions A model of the transient response of ferromagnetic systems in the micromagnetic approximation using an effective field formalism has been developed based on the Hamiltonian formulation of the equations of motion. The formulation takes explicit account of the constraints, and is hence valid in any coordinate system.
[1] J.C. Mallinson, IEEE Trans. Magn. 23 (1987) 2003. [2] Y. Nakatani, Y. Uesaka and N. Hayashi,Jpn. J. Appl. Phys. 28 (1989) 2485. [3] W.F. Brown, Micromagnetics(Interscience,New York, 1963). [4] E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A Modem Perspective(Wiley, Chichester, 1974). [5] H. Hoffman,J. Appl. Phys. 35 (1964) 1790. [6] H. Hoffman,J. Appl. Phys. 39 (1968) 873. [7] H.Y. Wang, J.N. Chapman and S. McVitie, J. Magn. Magn. Mater. 104-107 (1992) 329.
ja
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Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
The equations of motion of micromagnetics in Hamiltonian form M.A. Wongsam *, R.W. Chantrell Department of Physics and Electronics, Keele University, Keele, Staffs ST5 5BG, UK
Received 29 June 1995
Abstract
The equations of motion describing the response of a ferromagnetic system in the micromagnetic approximation are developed by applying Hamilton's principle to an action functional constructed from a Lagrangian density composed of the usual free energy expression, together with a suitably chosen expression for the kinetic energy density so as to produce the characteristic Larmor precession associated with magnetic spin systems under the influence of an external field. A suitable choice of the form of the generalised momenta and the standard form of the Rayleigh dissipation energy density functional, leads to equations of motion which exhibit similar behaviour to the Landau-Lifshitz-Gilbert form. It is shown that with our Hamiltonian formulation, micromagnetic simulations of systems with very many degrees of freedom can be performed with a substantial increase in the upper limit on the timestep size.
1. Introduction
Much attention has recently focused on the micromagnetic description of systems with many degrees of freedom, with the intention of acquiring a better understanding of macroscopic processes, without appealing to a treatment at the atomic level. In order to achieve this, it is necessary to account for a variety of forces, all of which contribute significantly, and hence, irreducibly, to the response of ferromagnetic systems. These forces include the exchange, magnetocrystalline, and magnetostatic forces. In addition, most experimental situations require the application of an extemally applied magnetic field. In general, all of these forces need to be accounted for from point to point throughout the sample. Many of the models in use utilise energy-minimising algorithms for the purposes of simulating the stationary states of ferromagnetic systems. However,
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many of the energy minimisation techniques are idiosyncratic, and cannot be applied generally. Moreover, properties such as the ferromagnetic resonance and the reversible transverse susceptibility require a description of the transient response of the systems. In addition, magnetisation reversal is an intrinsically dynamic phenomenon and energy minimisation cannot be guaranteed to find the correct magnetic state after irreversible transitions. Consequently the dynamic approach to micromagnetics is often used. For this purpose, the standard approach has been to utilise the equations of motion in the Landau-Lifshitz form. There are two forms of the equation of motion corresponding to two different phenomenological models of the dissipation. A good discussion comparing the two formulations has been given by Mallinson [1]. The older form due to Landau and Lifshitz is usually written dM a dt = T ( M X H ) - ~ - ~ M × ( M × H ) , (l)
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where Y is the Landau-Lifshitz gyromagnetic constant, and ot is the Landau-Lifshitz damping constant, and where M ( x , t ) is the magnetisation. The more recent Gilbert form is dt = Y G ( M X H ) -
MX
at
'
(2)
where Tc is the Gilbert gyromagnetic constant and a G is the Gilbert damping constant. Clearly, when there is negligible damping the two forms are virtually equivalent. In this case, the magnetization precesses eternally, at a frequency of 2.8 MHz Oe -~. When large damping is considered, profound differences arise between the two forms. Mallinson concluded that the Gilbert form is to be preferred on purely physical grounds. However, in order to drive the system to equilibrium workers have used either formulation, using the damping constant as a means of optimising the rate of descent to a minimum of the energy surface. One of the difficulties inherent in performing micromagnetic simulations involving both local and non-local interactions lies in the instabilities arising out of the stiffness of the differential systems, and not as a result of the nonlinearity of the Landau-Lifshitz equation as mentioned in Ref. [2]. This imposes severe restrictions on the choice of timestep in addition to those resulting from the form of the differential equations. In this paper we are concerned with the form of the differential equations and the effect this has on the choice of timestep. In solving dynamical equations with damping when many degrees of freedom are involved, two considerations affect the choice of timestep. Consider a ferromagnetic sample ~ with surface ~. The dynamical state at some time t is specified if M ( x , t ) is known at all x E {/2 U ,~}. In what follows, quantities with lowered indices refer to those quantities evaluated after a number of time increments indicated by the index. Raised indices refer to components of vectors or tensors. First, there is the requirement of convergence; that is to say, the solution trajectory should converge onto a limiting trajectory in phase space as $ t ~ 0 for all t. There will therefore be a requirement that the residuals at time t i, which may be denoted by Oi(x), satisfy
Io;;,(x)l <
a,
(3)
235
where E represents some tolerance. This represents a limiting condition on the choice of ~ t. Additionally, for problems in which relaxation to a stationary state occurs, there is also the question of the termination of the algorithm. If M = M~v, where Ms is the saturation magnetisation, and v is the reduced magnetisation, the stability of monotonous absolute convergence to a stationary state requires that
I~vil 2 _< I~vi_~l 2, Vx e~1"2,
(4)
where ~ v k is the change in v k produced when t k --* tk+ ~ = t i -I- 8tk, SO that one has (5)
Vk+ ,( X,tk+ ,) = Vk+ ,( X , t , + ~ t , )
(6)
-: Vk( X , t k ) + BVk( X , t k ) .
Such a requirement may impose an upper limit on the timesteps 8t~ independently of the requirement of convergence of the solution trajectory mentioned above. This is a strong stability condition and would not be imposed during an irreversible transition of the magnetisation state. The Landau-Lifshitz equation can be written (7)
b=TvXH-avXvXH.
Therefore, in a small time increment ~t, 8v = yv X H~t-
ctv X v X H ~ t .
(8)
At time t~, suppose there is an increment 8 t~ resulting in an increment in the components of reduced magnetisation 8v i such that v~+ 1 = v i + ~v~. Then the Landau-Lifshitz equation can be written
~l]i~- ~l(vi_ , 4-~Vi_l) XH~t-ol(vi_
X(vi_, + ~v,_,) = yvi_l XHBt+
XH$t
1 X
~/2i_
(9)
ySvi_ 1XHSt
-- OlVi_ 1 X Vi_ 1 X H S t -- Of/)i_
I 4-~b*i_ l)
(10)
I XH~t
- a S v i _ 1 X vi_ 1 X H S t --Ol~Vi_ 1 X ~lfli_ 1 X H ~ t
(11)
= ~)Vi- 1 + "Y~3Vi- 1 X H S t --
O~Idi_ I X ~/)i_ l X H ~ t
(12)
- ctSv~_ 1 X vi_ l X H ~ t --Ol.~l.)i_ l X ~Vi_ 1 X H ~ t ,
(13)
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where in the final step, Eq. (8) evaluated at time t i_ 1 has been used to replace the first and third terms of the penultimate step. Therefore, since Eq. (8) must be evaluated subject to the constraint v k • v k = 1, and for any vectors a, b and c, there is an identity a × b × c- b(a. c)- c(a.b), ~l)i = ~ V i - 1 "]- " Y ~ V i - 1 × -- O l V i - l ( ~ V i
H S t - c~Svi(
V i_ 1 " H ) ~ t
(14)
- 1" U ) S t .
For relaxation processes, there is therefore a trade off between timestep and relaxation rate - increasing the dissipation constant leads to a shortening of the maximum timestep. In this paper we present a reformulation of the dynamic problem using the Hamiltonian formalism which enables the integration timestep to be increased without loss of accuracy leading to a significant increase in computational speed.
Taking the scalar product of both sides of this equation,
I~vil 2 = [~vi-112 +
"y2~t2l?3Vi
- 1 ×
2. Reformulation of the dynamical equations in Hamiltonian form
HI2
+ ot28t2( vi_ , • H)21~v,_ ~12
(15)
+ O~2~t2(SVi - 1 "H)21vi-112 - - 2 0 t (~Vi_ 1 " n ) l ~ V i_ tlZSt -- 20~'y~/2(~/,'i_ 1 × H ) "
(16)
l?i_ l(~g?i_ 1" H ) , (17)
where the constraint condition and the vector identity given above have both been applied again. This relates the increment in the reduced magnetisation at time t~ to the increment at time t i_ 1" The criterion that absolute monotonous convergence takes place can be stated as a requirement that 1~3vil2 - [~v i_ j[2 < 0, that is,
~t < B / A ,
(18)
One can reformulate the equations of motion according to a variational principle [3], by constructing a suitable kinetic energy density 3-. Let b i denote the partial derivative of vi with respect to time, which can be called the ith component of the magnetisation rate vector, and U, etc., denote the higher derivatives. Also, let ~t "i~= ~v~/Ox j denote the components of the 'magnetisation gradient tensor'. Also, let Ev(V i, ~t'iJ; t) and Es(v i) be the volume and surface free energy densities, respectively, where E v may or may not depend explicitly on the time according to whether the external field is considered to be time dependent. The total free energy will then be
E[ v i, .4~'iJ; t] = foEv( v i, ..~,ij; t) d O
where A = ] / 2 [ ~ v i _ 1 x n l 2 + ~2(vi_ 1 . H )
2
+ f Es( v i) d ~ .
I~vi_ll 2 (19)
+0~2(~Vi_I "H)2)[Iyi_I ]2 --20f')/(~Vi_ 1 " n ) " V i _ l ( ~ V i _
B = 2 a ( v~. H)l~v,_ 112.
1XH),
(20)
(21)
There is therefore an inverse quadratic dependence on c~, with ~t having a clearly defined maximum. Convergence of the solution, that is to say, relaxation, will not take place unless cz > 0, since a --) 0, implies ~ t ~ 0. At the other extreme, as ot ~ ~, ~ t is proportional to 1 / a , and hence, also tends to zero.
(22)
In particular, for continuous magnetisation distributions, since one has V . v =.,g#, where in what follows, the summation convention due to Einstein has been adopted, the exchange and magnetostatic contributions to the internal energy, respectively, can be written as
Eexch = C.#"~.~ "'~, Ms2 r Emagn = - - v . j~
2
(23) x - x' , 3-.KJJ(x') d,~,
Ix-x
(24)
where Eexch a n d Emagn are the exchange and magne-
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tostatic energy densities, respectively. Then, form the Lagrangian density . ~ ( v i, b i, j~,ij; t) by the prescription . ~ = J - - E v. The system Lagrangian is then defined as
L= fo.~.~'(vi,[ji,~iJ;t)d12- fzEsd~,
(25)
237
density, Y . This may be written as a functional in the hi:
o~r[ bil = ½f rlbi bi dt,
(29)
where r/ is a material constant with the dimensions of energy density times time. The generalised dissipative forces are then
and the action integral is then
~F ~/)i "
Q~iss -
(30)
qJ[ v~, b~, JWJ; t, , t2 ] = ft2 f _oc~d12 dt t~
1"1
With these additional forces taken into account, the evolution of the system is governed by
- f £ 2 f Es d,~ dr.
(26)
Minimisation of gt then gives the dynamical response of the system to the forces derivable from the potential E. Following Sudarshan and Mukunda [4], the equations determining the necessary and sufficient conditions for the action functional to be stationary are found to be OV i
31 ab i
..~J &j¢,,l
ax j a..~fi j
. =0, av'
=
Vx~Z,
O, Vx E 12,
aZ
~ aS"
OV i
at a[; i
a_~
a
aEs
.. ~ J - - -
av i
a~',J
=
aS"
a9r
a x j a./~¢ij
a[3 i
0.
O,
(31) (32)
(27)
These are the equations of motion in Lagrangian form. Again following Sudarshan and Mukunda [4], the Hamiltonian form of the equations of motion can be derived by considering a Legendre transformation:
(28)
pi =
where the first of these represents the bulk conditions, and the second represents the boundary conditions. These conditions are valid provided that variations in v ~ are zero at t I and t 2. These have to be solved for each value of the time t subject to the prescribed initial conditions, which for most applications can be given as b ~= 0 throughout 12 U ~. In addition, for problems in micromagnetics there are also constraint conditions to be satisfied at every x ~ {12 U X}, and an auxiliary problem of the magnetic scalar potential, q), which must be solved simultaneously with the equations of motion. The constraint conditions are incorporated using the method of Lagrange, and the magnetic scalar potential is determined as the solution of Poisson's equation V2q~ = 4"rrV • M in 12. These conditions give the response of the system to the forces which are derivable from the potential E. However, the relaxation of the system is determined by non-conservative dissipative forces, which can be derived from a Rayleigh dissipation energy
(33)
a/)i '
a ~ ==-p i D i - , . ~ ,
where pi is the canonical generalised momentum distribution conjugate to v i, and ,Tg' is the Hamiltonian density conjugate to t. With these definitions, the evolution equations become aX
p i + -av i
a
&,~'
ax i &zt"ij
a,Tg"
ab'
(34)
(35)
ap i &,~ aE s . ~J+ . =0,
a.t[ u
a,9r + ---= = o,
av '
(36)
where pi is the partial derivative of the ith generalised momentum, and with initial conditions chosen to suit the problem. As an example, consider the case of a single dipole in an external field (HX,0,0) without damping. Then the total potential is E = - M s V . H + A(v 1), where A is an as yet undetermined multi•
v
-
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M.A. Wongsam, R.W. Chantrell / Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
plier expressing forces of constraint, according to the method of Lagrange. Hence, in this case one has a ~ i J = 0,
(37)
age" a" i
(38)
MsHi + 2Av i.
Following Brown [3], consider now a form of the kinetic energy density linear in bi, given by ~ - = A i ( v k ) b g,
pressing the magnetisation rates is tautologous, and hence, instead, one is generally required to solve the set consisting of Eqs. (34) and (36), subject to initial conditions and constraints. A solution exists to the equations of motion if dissipative forces are taken into account. The first two equations to be solved are
0 = - * l b I + M s H 1 - 2Av 1, 1
~,(.,)2
(39)
where Ag(v k) are functions to be specified. Then, applying Eq. (33) produces p i = A i ( v k ) . Suppose that, in the example above, the p~ are chosen to have the form
(47)
U3/)I] = __*l/32 __ 2Av 2.
[UI/~3
(48)
Seeking a solution representing damped motion and consistent with the equations of constraint, vz = a(t)cos O(t),
(49)
pX= 1/y,
(40)
v3 = a(/)sin O(t),
(50)
pY = --vz//Tvx ,
(41)
v 1 = 1/1 - a 2 ,
(51)
pZ = v y / / y v x.
(42)
Then, transposing the equation in p~ from Eq. (34) for A, one finds that
MsH x A= - -
Substituting this into the remaining equations for pY and pZ from Eq. (34), along with the expressions above, leads one to
1 [ av z ~v x } vy _ v~ -v z = -MsHXT, y(v~) 2 at at 1 -
b 2 = d cos 0 - at0 sin 0,
(52)
b 3 = d sin 0 + a/J cos 0,
(53)
b' = - ad/1/1 - a 2 ,
(54)
(43)
2v x
+
which have rates given by
[uxaU y
-
T( v x)= [
aU x ] -
.~
at
at
(44)
where a < 1. From Eq. (47),
Ms H1 A= 2 ~
*lad + 211-a21
"
(55)
Substituting Eqs. (49)-(54) into Eq. (48),
Uz
] = -Msl-l~. y(1
1
a 2)
(~/1 - a 2 d s i n 0
(45) a 2 fi sin 0 +dl
Now, seeking a solution (v ~, v ~, v z) -- (t~ cos tot, a sin tot, b) representing precessional motion, one finds that at to cos to t
M s H xot cos to t
yb
b
'
a 2
a0 cos 0 +
= -- *ld cos 0 + *la0 sin 0
(46)
or t o = T M s H x. In other words, a system with a kinetic energy density given by Eq. (39) responds to an applied field by precessing with an angular velocity that is dependent only on the applied field and material constants. For such a system, Eq. (35) e x -
--
M s H 1 a cos 0 -
l/i" -
a 2
*la 2 fi cos 0 1 -- a 2
(56)
Equating the coefficients of sin 0 leads to /} =
d -
y*la[1 - a 2 ]3/2 '
(57)
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of Magnetism and Magnetic Materials I52 (1996) 234-242
plier expressing forces of constraint, according to the method of Lagrange. Hence, in this case one has ax 7
a”4f’J
=o,
(37)
az = -M,H’ aui
+ 2A”‘.
(38)
Following Brown [31, consider now a form of the kinetic energy density linear in C’, given by 9-=AA’(vk)fii,
(39)
where Ai are functions to be specified. Then, applying Eq. (33) produces pi = Ai( Suppose that, in the example above, the pi are chosen to have the form
pressing the magnetisation rates is tautologous, and hence, instead, one is generally required to solve the set consisting of Eqs. (34) and (36), subject to initial conditions and constraints. A solution exists to the equations of motion if dissipative forces are taken into account. The first two equations to be solved are 0=
-qfi’+M,H’-2hv’,
(47)
1 1 1.3
--vu
1 - 7)2- 2hv2.
_“3fi’
=
Y(“‘)2
(48)
Seeking a solution representing damped motion and consistent with the equations of constraint, v2 = u( t)cos O(t),
(49)
pX= l/Y,
(40)
“‘=a(t)sin8(t),
(50)
py=
(41)
vl=K-T,
(51)
(42)
which have rates given by
-vz/yvx,
pz = vY/yv”. Then, transposing the equation for A, one finds that A=-
in px from E@. (34)
M,H” 2””
fi2=d!cos8--ffbsin8,
(52)
fi3=Cisin8+aBcos8,
(53)
fi’ = -aci/&-z,
(54)
(43)
*
Substituting this into the remaining equations for py and pz from Eq. (341, along with the expressions above, leads one to
where a I 1. From Eq. (471, M,H’
lpi
A= 2\/1-;;i-
--
Substituting
Eqs. (49)-(54)
Y(1 -a’>
(45)
CYWcos WI yb
=
MsHxacos b
(uX, uy, vL) = ((Y cos wf, precessional motion, one
(55)
2[1-a’]’
1
-
Now, seeking a solution (Ysin wf, b) representing finds that
+
into E!q. (481,
h sin e
CT i
+i‘iTdc0s
u’cisin
e+
6-7
8 1
= -7+cos8+&sin8 wt ’
(W
or o = yM, H”. In other words, a system with a kinetic energy density given by Eq. (39) responds to an applied field by precessing with an angular velocity that is dependent only on the applied field and material constants. For such a system, IQ. (35) ex-
M,H’ UCOS 6
-
4s
Equating 8= -
~&~COS -
the coefficients
l-u2
8
’
(56)
of sin 8 leads to
b
yqu[1 -
u2 ]3’2 ’
(57)
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Defining
pation ('0 --> 0) results in no convergence (~t --->0) as in the Landau-Lifshitz case. As "0 ~ ~, since
2K, y 0 [ 1 - y ~ ] F(y) =
(76)
K 2 y02 + 1
H l _ 'oh 1 =
the solution can be written correct to the second order in t, as 4Kl Yo [
Y=Yo l+Ft+K-2y--~o+l
1-2y¢~
K2y~(1-y2) ] F ~ ) +O(t3). K2YZo+I
-
(77)
2v ~
,
(85)
in order that 8t remain positive, it is necessary that '0 be bounded above by H~/b ~. It can be seen that as "0 approaches Hl/b 1 from below, the upper bound on 8 t diverges. There is therefore a clear computational advantage over the Landau-Lifshitz form of the equations of motion. It must be noted, that care still has to taken over the choice of 8t so as to ensure convergence of the trajectory in phase space, and methods appropriate to stiff differential systems are to be preferred.
3. Comparison with the Landau-Lifshitz equation It is interesting to compare the application of the dynamics formulated above with the Landau-Lifshitz dynamics now in current use. The Hamiltonian form of the equations of motion can be written, "00 = H - 2Xv - p .
(78)
Proceeding as with the treatment of the LandauLifshitz equation, '08v = ( H -
2Av) 8 t - ~p;
(79)
therefore,
2ASvi_ 18t,
4A28 t 2
[~vi- 112 _ _ '20
(81)
4A~ t ]~Vi_1[2 _ __lSvi_'0 112
(82) < 0
(83)
from which results ~t < "0/A.
~E
vi~V i,
(86)
(80)
on using Eq. (79) in the last step. The criterion for absolute convergence is i~v/i2 _
For more general problems involving energy densities and many degrees of freedom, one has an effective field composed of for instance, exchange, bulk and surface magnetocrystalline, magnetostrictive and magnetostatic components, as well as the external fields. These can be formulated as generalised forces Q, S conjugate to the generalised coordinates v, according to
faQi~vid~+ f Si~vidZ=
"0~[yi~-(H-eA[Vi_l--[-~Vi_l])~t-~pi_ I = "0~vi- 1 -
4. Simulations
(84)
Clearly, from this expression, the case of zero dissi-
which expresses the principle of virtual work, where the Qi are the components of any generalised internal and body forces, S ~ are the components of any generalised surface forces which will appear as contributions to the boundary conditions, and E is a total potential energy functional. For such problems, the solution represented by Eqs. (52)-(54) with a ( t ) and O(t) given as the solutions to Eqs. (60) and (58) has to be time-stepped, and the forces Qi, S ~ updated at every iteration. In order to use this solution, a local coordinate system must be defined with the effective field parallel to the x-axis. Comparisons have been made between calculations performed using the Landau-Lifshitz form of
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241
2.5"10 = -
2.0"10= -
1.5"10=o
•
•
1.0"10 =-
5 . 0 " 1 0 =~-
m •
! 0 . 0 " 1 0° 0 . 0 ° 1 O*
5.0" 1 0 ~
1.0" 1 0 H
1.5 ° 1 0 ~
Fig. 1. Comparison of elapsed times for some stationary states calculated using the Landau-Lifshitz TLt" and Hamiltonian TH dynamics.
the equations of motion and those using the Hamiltonian form with the above solution. The specimens were small rectangular platelets of polycrystalline
Cobalt with random in-plane uniaxial magnetocrystalline anisotropy. An effective field formalism was used, incorporating exchange, anisotropy, Zeeman
Fig. 2. Single-domain stationary state at 1500 Oe of a 780 × 380 × 40 A Co platelet, 4 × 4 × 1 square grains. The view represents the (x,y) plane at z = 20 A, where x is measured along the long axis in the plane of the platelet, y along the short axis in the plane, and z through the platelet thickness.
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M.A. Wongsam, R.W. Chantrell / Journal of Magnetism and Magnetic Materials 152 (1996) 234-242
and magnetostatic interactions, the exchange and magnetostatic interactions being modelled according to a finite-difference scheme. The calculation of the magnetostatic interaction integrals was accelerated using a two-level hierarchical approach, the details of which will be reported separately. In these tests, the Landau-Lifshitz dissipation c o n s t a n t , aL L , was chosen to have a very large value, giving a fast relaxation [1]. The timestep in each case was chosen adaptively to give a stable evolution to the stationary state independent of the size of the timestep. Fig. 1 shows a comparison of elapsed times multiplied by a / M 2 for various values of the external field. A largely linear relation between the elapsed times TEL for the LandauLifshitz, and TH for the Hamiltonian cases is observed, with a ratio of approximately T H / T L L = 0.08147. The calculated stationary states were qualitatively and quantitatively similar for both approaches. Fig. 2 shows the single-domain state at 1500 Oe. for a 780 × 380 X 40 A polycrystalline Co platelet, showing the characteristic ripple predicted by theory [5,6] and experiment [7]. Here, the magnetisation distribution is resolved onto a 40 X 20 X 1 computational lattice and the grains are represented by 4 × 4 X 1 computational sublattices sharing common in-plane uniaxial anisotropy directions. A more detailed study of these systems, comparing a range of values of the intergranular exchange coupling, will be reported separately.
Comparisons have been carried out with the application of the Landau-Lifshitz form of the equations of motion. These comparisons indicate that a substantial improvement in computational performance can be achieved since the timestep required is generally much less than in an optimally damped system. However, more extensive tests will have to be carfled out, on much larger systems in terms of the number of degrees of freedom, so as to compare the dynamics of the two models when many more stable stationary states are possible. The development of the equations of motion in Hamiltonian form, opens vast possibilities for research in micromagnetics, since the classical methods of transformation theory and Hamilton-Jacobi theory become available and applicable. There is therefore a potential for the development of theoretical micromagnetics additional to the implications for computational micromagnetics described here.
Acknowledgements The work reported here was supported by grants from the SERC, and carried out within the framework of CAMST-2. RWC is grateful to Dr R.J. Veitch for handling the editorial duties for this paper, including the anonymous review.
References 5. Conclusions A model of the transient response of ferromagnetic systems in the micromagnetic approximation using an effective field formalism has been developed based on the Hamiltonian formulation of the equations of motion. The formulation takes explicit account of the constraints, and is hence valid in any coordinate system.
[1] J.C. Mallinson, IEEE Trans. Magn. 23 (1987) 2003. [2] Y. Nakatani, Y. Uesaka and N. Hayashi,Jpn. J. Appl. Phys. 28 (1989) 2485. [3] W.F. Brown, Micromagnetics(Interscience,New York, 1963). [4] E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A Modem Perspective(Wiley, Chichester, 1974). [5] H. Hoffman,J. Appl. Phys. 35 (1964) 1790. [6] H. Hoffman,J. Appl. Phys. 39 (1968) 873. [7] H.Y. Wang, J.N. Chapman and S. McVitie, J. Magn. Magn. Mater. 104-107 (1992) 329.