A classical theory of superparamagnetic relaxation

Journal of Magnetism and Magnetic Materials 3 (1976) 219-233 © North-Holland Publishing Company

A CLASSICAL THEORY OF SUPERPARAMAGNETIC RELAXATION D.A. SMITH and F.A. De ROZARIO Physics Depart,neat, Monash University, Clayton, Victoria 3168, Australia Received 19 March 1976; in revised form 7 May 1976 A Fokker-Planck equation for the relaxation of a classical ferromagnetic particle coupled to a classical heat bath is derived from the Nakajima-Zwanzig equation. The equation of motion for the mean magnetization of an ensemble of pattides is found to be closed only under special circumstances. In the strong motional narrowing limit the equation of motion reduces to the Bloch equations in the limit MH .~ kBT, i.e. for small particles, and to the Landau-Lifshitz equation in the opposite limit. For the motional narrowing region in tote the particular case of uniaxial anisotropy is analysed, giving an equation of motion which for large particles reduces to a modified Landau-Lifshitz equation with g-shift and a reduced damping constant. This equation cannot be meaningfully identified with the Gilbert equation. Approximate expressions for superparamagnetic relaxation rates by Kramers' method are obtained for the case of (i) triaxial (i.e. orthorhombic) and (ii) cubic (K +ve and -ve) anisotropy, assuming large energy barriers. The results supplement Brown's expression for uniaxial anisotropy and show a more complicated dependence on the Landau-Lifshitz parameter }, than the linear dependence found by Brown. For small h the rates tend to constant values compatible with the transition.

1. Introduction and discussion This paper is primarily an attempt to resolve two separate controversies, one being well known and the ather largely unnoticed, in the area of ferromagnetic, or superparamagnetic, relaxation. The first controversy is of long standing and conceres the status of two different phenomenological equations of motion for the magnetic moment M of a ferromagnetic body *. These are the Landau-Lifshitz [1 ] and the Gilbert [2] equations, given respectively by 21;/= 7M X n

M ~ M X (MX H)

absent as the magnitude of the moment is assumed fixed. This implies that the Landau-Lifshitz equation i s not compatible with damping mechanisms which change the magnitude of M, e.g. all mechanisms invoking spin waves. Magnon damping processes have been investigated by Haas and Callen [4], whose re. suits confirm this conclusion. The Gilbert equation is reputedly derived from the existence of a Rayleigh dissipation function. However, the two equations can be written in the same mathematical form. By using the Landau -Lifshitz decomposition, (2) can be expressed in the form of (1) with coefficients

(11

-

"[G

v

1 + (nl'r 34) 2 '

N; =

n

(3)

1 + (1"117114)2

and fir= _ _

,MXH . . . . n .... M X M .

, ........

The controversy then turns on whether the damping coefficients are disposable parameters. Most authors have preferred to use the Gilbert equation on the grounds that the energy dissipation [5], and tl,e inverse of the time required for magnetization revers~t [6], remain finite as the Gilbert damping parameter rl tends to infinity. This behaviour is in fact due entirety to the presence of r/in the denominator of the above expression for the effective Landau-IJfshitz damping pararaeter ~ , thus reducing the argument to a semantic-

(2)

,),.M2

Other cqaations have also been proposed [3] but are not widely used. The Landau-Lifshitz equation is simply an expansion of2l;/in terms of the orthogonal basis set M, M X H and M X (M × fir), the term in M being * In this paper the symbol M will be u3ed for magnetic moment and not, as is customary, for magnetization. 219

220

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

al one over which damping constant is the more "physicar'. Experimentally, one finds from resonance methods that the dimensionless damping parameter a = rl['~M is always smaller than unity so that 7G -~ 7, ~ ~ h and (1)and (2)are practically equivalent. The highest known value of a appears to be 0.16 for Ni at 0 K f~om the work of Bhagat and Lubitz [7], and values of ~ 1 0 - 2 are more typical. Pulse switching measurements in the high-field, uniform rotation region [5,8], give values of a several times larger than the corresponding resonance values but still less than unity. In the next section a somewhat idealised model of magnetic relaxation is developed which is sufficiently general to define the limitations of the Landau-Lift shitz equation and to give a unified description of paramagnetic and ferromagnetic relaxation. The model is restricted in that it does not include mechanisms which change the magnitude of M and that both the magnetic body itself and its environment, i.e. the source of relaxation, are assumed to obey the laws of classical mechanics. Damping is assumed to be due to a random magnetic field h(:) whose statistical properties are lmown, i.e. the standard deviation A/7 and correlation time r c. We consider an ensemble of magnets and derive a Fokker-Planck equation for the distributioa P(M, t) of magnetic moment at time t, i.e. 0P

-~i + V. J = O,

(4)

where J(M, t) is the corresponding flux and V = ~/aM. The results differ according to the degree of motional narrowing [of. eq. (31)]. In the strong motional narrowing limit (24) the flux has the form

J = ("rM X H - M2 X--~-M X (M X 1t) ) P - DVtP '

(5)

viz. a drift term of the form J~/P where Jl~is given by the Landau-Lifshitz equation (1), and an isotropic diffusion term. Vt is the tangential gradient def'med in (25). A result of thi~ fo..rm ha~ haan ahtnin~d hy Rraxxtn [Q] by a rather different argument. Brown started from a phenomenological equation ..~ motion (in fact the Gilbert equation) with a white noise field added. From this Langevin equation he was able to deduce the corresponding Fokker-Planck equation, thus justifying the existence of the diffusion term. However, this procedure does not justify a particular form of damping

term since this must be inserted ab initio in the equation of motion. The equation of motion for the mean moment (M) at time t is readily obtained from the Fokker-Planck equation, and is not in general dosed. We shall refer to a closed equation of this type as a Bloch equation, although the term is generally used in a more restrictive sense. Bloch equations result in two limiting cases. For paramagnetic particles, such that MH ~ k B 7", the offginal Bloch equation is recovered with isotropic relaxation towards the equilibrium field. The other simple case is that of a single ferromagnetic particle, where the Landau-Lifshitz equation is obtained with the same coefficient that occurs in (5). When the strong motional narrowing condition is relaxed, the Fokker-Hanck equation becomes considerably more complicated and can be obtained in simple form only for a uniaxial anisotropy field. The current density is given by (33) and displays anise. tropic diffusion, a shift in precession frequency and a mixed term of the". form M X VtP. More importantly, the coefficients are functions of the precession frequency w z (also a function Of Mz) in the anisotropy field. Again, Bloch equations for the mean magnetic moment occur only in the limiting cases of low and high fields. In the paramagnetic case the original Bloch equations appear with different longitudinal and transverse relaxation times, given by (45). For a single ferromagnetic particle, an equation of" l_andau-Lifshitz type appears whose coefficients are functions of w z. In particular, ~,(¢Oz) is proportional to the power spectrum of the random field at the same frequency, and will drop off at frequencies above r e 1. Considered as a function of re, X(COz) is linear for small re, rises to a maximum near ~zrc --- 1 and is expected to fall for larger values. There is an amusing similarity with the damping constant ~ (3) of the Gilbert equation, which also goes through a maximum as 7/increases. However, it is not possible to force the r-dependence of h(COz) into the Gilbert form, or anything remotely S~m~l~r by ° o,,~or, l° , h , ~ o ,,¢ *. To . . . . ~,,ao ,~.o.o is no theoretical foundation for either the Landau*

For the special ease of markovian modulation, it is found that MWz) = flM2A~rc/[ 1 + (tOzrc)2 ] which appears to be of the ,Gilbert form r/l[1 + (n/~M)2 ] if n ~ r e. This is not so; the choice r~= #3/2A2r e equates the numerators but then the maxima occ'ar at r e = to~-Iand ~,/ttMA2 respectively. The second value is usually the smaller.

D.A. Smith, F.A. de Rozatio / Theory of superparamagnetic relaxation Lifshitz or the Gilbert equations as such, although there are certainly limits to the amount of damping that can be achieved by varying the correlation time r c. The second topic of this paper is currently not controversial, though perhaps R should be. There is an expression, due-to Brown [9], for the relaxation time r of a superparamagnetic particle in a uniaxial anisotropy field V (M) = -KM~z/M2

(K > 0)

(6)

which is generally regarded as a satisfactory quantification of Neel's theory of superparamagnetism. Brown used the Gilbert equation in his derivation, but if the Landau-Lifshitz equation is substituted then this result is r

721'1~,nkB 1"]

7H c e-~l¢ (H c = 2K/M)

(7)

assuming no external field, and//K >> 1. Note that the right.hand side is linear in the damping constant ~,. This rather simple result can be compared with Kramers' results for the expected jump rate of a brownian particle over a potential hill [ 10], particularly as the two calculations involve the same assumption of a sta. tionary current flow. Kramers found that when the particle motion near the top of the barrier is overdamped the jump rate is inversely proportional to particle viscosity, which is the direct analogue of ~. For underdamping the jump rate becomes constant at a,.?proximately the value suggested by the transition state theory of chemical reactions, which assumes that particles at the top of the well are in thermal equilibrium*. There is no obvious correspondence in the dependence of 1/r on damping constant for these two problems; in particular the right-hand side of (7) does not go to zero as ~ -." 00. Brown's original expression using the Gilbert equation, which is (7) with ~, replaced by ~<;, certainly goes to zero as r / ~ o.. However, it appears that the physical damping parameter should be 7~since it arises as a direct consequence of the Fokker-Planck equation. To gain further understanding of the magnetic version of Kramers' problem we have obtained approximate solutions for the relaxation rate for anisotropy * Kramers also predicted that the jump rate would fall linearly to zero at much smallervalues of the viscosity, thus contradicting the previous resalt. The reason for this is not clear.

22 ~

potentials other than the simple uniaxial case considered by Brown. The resulting dependence on X is not linear and reflects the more complicated depen. denee on viscosity found in the original Kramers' problem. This behaviour can be understood by considering the flow pattern of the points of the en~mbk: in the appropriate phase space, which for the magnetic problem is the surface of the sphere l/h'] = M. The problem is to find a stationatT solution of the Fokker-Planck equation (4), (5) which transports magnetic moment between those points on the surface where V is a minimum, so that sources or sinks must be located at the minima. For uniaxi~il anisotropy as given by (6) the minima are at the poles and any path comlecting them must rise to the maximum potential V :: 0 at the equator. The resulting stationary distribution is constant around each potential contour, which is a circle of latitude. The corresponding current densit,.,, is the sum of a polar current uniformly distributed ,Lround each contour and an independent axial prece., sion current, which in no way affects the strength or ~istribution of the polar current. Now consider the triaxial anisotropy pote~tial I~ (KxM2x + K y M : + KzM2z) V(M) = - M---

(8)

with, say, K x < Ky < K z. The contours are shown in fig. 3a. The energy barrier between the poles is now a function of the chosen path between the poh:,~; the critical paths, for which the barrier is least, pass through the saddle points on the equator. In this case the stationary distribution varies around the ,.~quator and will be concentrated near the saddles. It follows that the form of this distribution, and hence the flow pattern, is a function of the precession current, which will attempt to sweep magnetic moment away from the saddle points along the contours. Consequently, the relaxation rate (65) is a function of the dimensionless damping parameter c~= h/,/M and the equatorial variation of the potential; a similar situation holds in the ori~n~A K~amers problem in p, q space. (65) i~ ~k,~ smaller than (7) by a factor (k 8 T/K) 112 because the flow of magnetic moment is channelled through the saddle points instead of being spread evenly around the equator. Similar conclusions occur when the anisotropy is cubic. In both these cases it is found that the relaxation rate is an increasing function of a, being linear in a for

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

222

c~~, 1 and tending to a constant value as ~ -+ 0. Fig. 1 illustrates this for the cubic case. There is a plateau region for small c~where file rate is approximately independent of damping and the "transition,state" concept is applicable. However, the correspondence with Kr",aners' problem is not complete, since for large a the transition rate increases to become linear in a, whereas the transition rate between wells for a Brownian partide decreases as the viscosity increases in the overdamped case. The difference arises because in Kramers' problem we are looking at the relaxation of the position variable which is conjugate to the momentum variable that experiences viscous damping, whereas in the magnetic problem there is only one variable, i.e. magnetic moment, which plays both roles. it is now apparent that the transition-state theory can be applied to superparamagnetic particles when < 1, since the rate at which magnetic moment arrives at the saddle points is then determined by the precession current alone. However, the case of uniaxial anisotropy is now an exception to the rule since no saddle points exist; then the precession current confines the magnetic moment to one hemisphere or the other and so the relaxation rate is zero when ~ -, 0, as eq, (7) shows. The transition-state theory must fail anyway for very small values of c~, since a minimal amount of damping is necessary to establish thermal equilibrium between the different contours of the potential. At lower values of a, the relaxation rate is expected to fall to zero at a = O; the reader is referred to Kramers' paper for a discussion of this delicate point.

2. Fo~er-Planck equation for a superparamagnetic part~icle The model that will be used for the derivation of a Fokker-Planck equation is that of a classical magnetic moment M, of fixed magnitude, which sees a random magnetic field h. The moment will be referred to as the '~system" S, and the random field is due to a "bath" B containing a large number of magnetic sources. The bath is assumed to be in thermal equilibrium at temperature T and to obey classical statistics. The hamiltonian of the model is therefore +

+

(9)

where = v(M),

=

~:IC1 = - M . h(Qi, P i )

(10)

.

The magnetic potential for the system S is V(M), where H = -1V V is the corresponding magnetic field. The bath is assumed to be specified by a complete set of canon:ical coordinates (Qi, Pi) and to obey Hamilton's equations of motion, from which the random field h acquires its time dependence. The classical dis. tribution fimction p for the combined system satisfies the Liou~ille equation

,6-- -~p - - ( ~ , p)

(11)

which defines the liouviUian .0 in terms of a Poisson bracket. For example, if X and Y are two functions of the ba':h coordinates only,

(X, r ) :

.

(aX ar

aX ar) •

(12)

From the classical description of a magnetic moment, it follows that the system S also obeys hamiltonian dynamics, and therefore the Poisson bracket notation applies universally. The following identity is useful in this respect; if F and G are arbitrary functions of the magnetic moment M then *

(F(M), I

0

1

~ _ ~ . .

=

(VF x vc). M,

(13.)

C~

2

Fig. 1.

* (13) follows from the basic Poisson bracket (M~,Mfl)=~,e~y

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

where 7 is the magnetogyric ratio of the moment. Hence the liouvillian _nS, which corresponds to the system hamiltonian 9dS, is 2S

=

- 7 ( M XV V)'V

(14)

as expected from the equation of motion for M. We require an equation of motion for the reduced probability distribution P(M, t) for the system S at time t, to be obtained from (11) by projecting out the bath coc,rdinates. Techniques for obtaining such master equations are by now very well developed, and we shall use an approximate version of the generalised master equation of Nakajima and Zwanzig, viz.

223

exp(-~ a ) B =ff~Q~p

exp(_/~B )

is the canonical distribution for the bath and ~ = 1[kl3T, k B being Boltzmann's constant. A detailed discussion and derivation will be found in the review by Haake [ 11 ]. (15) is an approximation of low-order in 9t 1 to the Nakajima-Zwanzig equations which is expected to be valid when

,x

I o(M)l,

(16)

where Ica(M)l is the precession frequency, in general dependent on M. In the strong motional narrowing regime the time dependence of S can be neglected in the damping term and (15) reduces to the Pauli master OP(M, t)/i)t = - . e s P equation with Born approximations to the scattering rates. As it stan~, (15) is non-markovian but it will be instructive to make the Markov approximat,;on at a :(-es+'eB ," later stage. 0 05) The processing of the terms in (15) is easily carried out with the aid of (13) and the usual decomposition where ~ Q, ~ P are differentials of all the bath corules for the Poisson brackets of products. The first ordinates term of the right-hand side is easily put into the form -V. J where J is the precession current -7(M X V V)P. For the second term, we start with the identity clo

+f d,ffwoWe

,ne(m,t-O,

(17)

.eI (Be) = - ( ~ I , BP) = Bh . (M, P) + (h, B) . M e .

The exponential factor in (15) that acts on the above has the effect of moving ail varaibles back in time by an amount 1", the time dependence of system variables being calculated on the basis o f ~ S alone and of bath varizbl,:s on ~ B alone. Hence let M ( - r ) = e x p ( - 2 st)M,

h(-r) = exp(- Z?Br ) h ,

(18)

and ~o exp [-('/~S + 2B) r] "~l BP(M, t - ~')=Bh(-r)" (/l~(-r),P(M(-r), t - r ) ) + ( h ( - r ) , B ) ' M ( - r ) P ( M ( - r ) , t - ~ ) . (19) The evaluation of the integrand may be completed by a further application of the identities (17) to (19). The average over bath coordinates thus involves the autocorrelation function of the random field, which is ti~e tens v,r quantity

(20)

J(t) = (h(t)h)= f f ~Q°OP Bh(t)h .

Note, incidentally, that sinc~ the bath is in equilibrium the random process {h(t)} is stationary and so Jc~(-t) = J~(t). The other averages required are d

D.A. Smith, F.A. de Rozario / Theory of superparamagnetie relaxation

224

and, finally, a rather curious identity which is the classical analogue of the (quantum.mechanical) one that the trace of the commutator of two operators is zero, i.e.

f f cDQOe

}3=0,

if either X or Y ~ 0 as IQii, IPil ~ **. Hence the master equation is obtained in the form of a continuity equation •e --+v. i)t

o

(4)

where

J(M, t) = - 7 ( M X V V) P(M, 0 + 7 2 f

M X l(r)"

OM(-r)

-r) X

dr

0

+

MX

d

r)]dr

(21)

0 on integrating the last term by parts over r. The Markov approximation P(M(-r), t - r) ~ P(M, O

(22)

can now be made in (21), giving the basic Fokker-Planck equation where the second and third terms of the righthand side describe diffusion and damping respectively. (22) would be exact if these terms were absent because the system S would then evolve causally under the potential ~: In general, P(M(-r), t - r) differs appreciably from P(M, t) for times r greater than the Bloch relaxation times T 1 and T 2. Hence (22) is valid if TI, T2 >> r c

(23)

where r c is the correlation time of the modulation. F~om now on it will be assumed that the modulation field h(t) is isotrt, pie, i.e. that the tensor J is a multiple J of the unit tensor. It is quite probable that in actual situations this will be the case even when the modulation is generated within the particle, e.g. by spin-lattice interactions.

Case A: Strong motional narrowing. This limiting case occurs when the modulation is fast compared with the precession period, i.e., Ito(M)l% < 1, wh 're ~(M) = gradient

....~'~r~i . . . . .

(24) +. . . . . . . . .

Vt =-(l//~2)M" X (Jlif XV)

;~'" ¢. . . . . . . . . . old rH(jux =

v

lip' .

JLL

ll~

li,,~'l,,.lililYl~llllll~,oJlilll,

ii.lij

,l,lli,,lllllli~

I.lllll.l~

II.clllllii~)li.,llllilll,l

(25)

which excludes the radial derivative. The basic parameters of the modulation are the rms modulation amplitude A in frequency units, where

A2 = 72 J(t = O)

(26)

D.A. Smith, F.A. de Rozario [ Theory of superparamagnetic relaxation

225

and the correlation time re =(/g(t)dt)/J

(t = O).

(27)

In an expansion in powers of I¢o(M)lrc, the leading term in (21) is

J = - 7 ( M X V V ) P - k(Vt V)P - DVtP

(28)

which is just eq. (6), and D =M2A2rc ,

k = lifO.

(29,30)

(30) also happens to be the condition for the canonical distribution P0 c: exp [-/3V(M)] at the bath temperature T to be a stationary solution of the Fokker-Planck equation, as is easily verified. An equivalent statement is the fluctuation-dissipation theorem, which is the relation between k arm modulation parameters obtained by eliminating D from (29) and (30). Case B: Motional narrowing in general. The fundamental condition of motional narrowing is that A r c ,~ 1

(31)

which, by (16), includes (24) as a special case. The diffusion and damping terms now depend on the detailed dynamics of free precession in the potential V(M), since if (24) is not satisfied (weak motional narrowing) the effect of the non-secular components of the modulation is partially washed out by the free spin motion. A quantitative discussion is possible only for the simplest kind of potential, viz. a uniaxial potential V = V(Mz). The precession frequency co(Mz) will be abbreviated to ~Oz,and so the free spin dynamics are specified by the function of (18), viz. (32)

M ( - r ) = Mzg - 2 X (g X 214)cos(cOzr) + t X M sin(coz r).

The other vector that appears in (21) is aP(M, t)/aM(-r) which transforms with r in exactly the same way. Some routine manipulation of vectors enables the current density to be written in the form

J = [~ + a ( % ) M z ]M X [H(Mz)~IP- E(~z)MzM X V tP

-D(~z)VtP- [ D ( O )

-

D(

z)IMX

t(MX e). vtP

- -

M X [M X H(Mz)£]P ,

(33)

where

7H(Mz) = w z , and

O( coz ) = n,y2M2 J( t~z ) ,

E(~oz) = 7r72r(coz)

X(,,,z) =

5(tOz) = BE(coz)

(34)

in terms of the power spectrum of the modulation

J(w) 1 f J(t) cos cot dt =

0

(35)

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

226

and its Hilbert transform

1

doy

(36)

where P denotes Cauchy's principal value. The result is slightly simpler in spherical polar coordinates; (33) then reads that

aP Jo = -k(C°z) H(Mz) sin 0 P - D(toz)M_ 1 -~ffaP+E(coz) M COS0 a~ 1 a/' Jo = - [ 3 ' + a(c%)M cos 0] H(Mz)sin 0 P - E(o~z)M cos 0 ~aP - [D(O)sin20 + D(coz) cos20 ] Msin 0 a~ (37) The diffusion term is anisotropic. The z-component of the random field (i.e., the secular part) causes diffusion in the ~ coordinate with diffusion constant D(0) but the x- and y-components are associated with the diffusion constant D(~z). The diffusion tensor does in fact satisfy Onsager's symmetry relations [12] provided that one remembers to reverse the magnetic field H(Mz); this follows from the fact that (35) and (36) are even and odd functions o t w respectively. The damping term is entirely due to the nonsecular part of the modulation; this can be understood because it is linear in the field H(Mz) and so is associated with inelastic transitions. There is also a shift in precession frequency linear in l"(COz),which would in quantum-mechanical language be called a self-energy. Note also that in the strong motional narrowing limit J and r can be replaced by their zero-frequency values and the earlier result (28) is recovered. It is possible to generalise this derivation to the quantum case, bat the price to be paid is that the classical probability distribution P(M, t) must be replaced by a quasiprobability distribution of the Wigner type [11 ], which is nut necessarily a real number in the range (0,1). This entails a large increase in the complexity of the calculation, particularly as we do not yet have representation-independent definitions of such distributions. A partial generalization would be to a classical system with a quantal bath, but even this requires a careful analysis of the passage to the classical limit and so one might as well consider the completely quantal case from the outset.

3. The Bloch equations The mean magnetic moment of an ensemble of moments at time t is given by

~(t)) =f M P(M, t) dM.

(38)

Its equation of motion follows straight-forwardly from the Fokker-Planck equation (4) since

d(M(t))/dt =f M(aP/at) dM = -J'Mv. J dM =f J dM.

(39)

in general this is not a ciose~i equation for the mean moment. We shall term any such closed equation of motion a Bloch equation, in the spirit if not the letter of the original usage. The main purpose of this section is to isolate the conditions under which Bloch equations exist in the framework of the basic model, and to compare them with existing phenomenological equations. For Case (A), i.e. strong motional narrowing, eq. (5) is appropriate. The diffusion term can be transformed with the help of the identity

MX (MX re)-- V. (MMe)- V(M2e)- 2Me.

(40)

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

227

It is essentiE1 to treat the terms of the right-hand side in the same way despite the fact that IMI is, for all ensembles of interest, constant; this feature will be handed automatically by the distribution itself. Hence

O~(Oat = ~ , f ( ~ x v v ) , e d i - ~ ( O > - .

(41)

X(MXVV)edm

which is not generally reducible to a Bloch equation. However, the most important feature is now at hand, that there are two different forms of damping term on the right-hand side whereas in expression (5) for the FokkerPlanck current density there is only one. A crude estimate ['or the ratio of the third to the second term is I~fH/D = MH/k B T so that by varying this ratio a single description of paramagnetie and ferromagnetic relaxation is obtained. In the paramagnetic (zero-field) limit the damping term is of the original Bloch form and the longitudinal and transverse relaxation times T1, T 2 are given by

l I T 1 = l/r' i

=

2D/M2-

2A2rc

(42)

.

In the opposite limit the Landau-Lifshitz damping term predominates. The first term on the fight-hand side of (41), i.e. the precession term, does not reduce to a Bloch equation except when V is due to an external magnetic field (VV constant). The last.term, which is the Landau-Lifshitz dam. ping term, is even less obliging, but if we are only interested in a single ferromagnetic particle then there is no need to consider an ensemble any more and the Landau-Lifshitz equation (1) is justified in its entirety. For Case (B), which is motional narrowing generally with a~uniaxial potential,.the current is given by (33). Identities similar to (40) can be used to transform the diffusion terms. The resulting equation of motion is

dOi/(t)) dt

-

f ( ~ + ~(,,.,z)m,

,

d

)

+ ~z din= Nztr(,.,,z)l M X Z(Mz)ee dM

dM-----~(M22

-

MzM)P dM

x tMx n(m )el e dM. where M l = bl - Mz2. The preceding remarks concerning reducibility to a Bloeh equation apply equally here. Bloch equations result, firstly, for paramagnetic particles in an external magnetic field, in which ease ~z can be taken outside the integrals. ~ then recover the well-known Bloch equations [13].

~> = -~z>/r!,

r-[

2D(~z)

M2

1

'

t

"r -'r(1 + e(coz)/COz). The structure of the expressions (45) is also standard [13 ]. For a single ferromagnetic particle, (46) reduces to a modified form of the Landau-lifshitz equation, viz.

r2

--X("°z) . . .X. . .[,M . . . .X/-/), . m2 M

(46)

(44)

where m

and I/T'2 " DfO)/M 2 is the secular contribution. Here

= "r(mz)M x H

= "r' X He -
1

(43)

N

--

1

2rt

÷

1

rI

,

(45)

where 7(Mz) = 3, + MzS(Mz), which differs from (1)as the coefficients are functions of Mz. As discussed in the introduction, the Gilbert equation can also be put into this form but the coefficients cannot meaningfully be equated with those of (46).

228

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

The conditions of validity for these Bloch equations follow from the basic conditions ( 1 6 ) a n d (23). They are displayed graphically in fig. 2, using the Markov form of the correlation function J(r). In the paramagnetic limit, use of (45) and (29) shows that (23) is equi~alent to the condition (3 I) for motional narrowing. If (23) and (16) are combined one finds that Io~ziT2 ~, 1, i.e. resonance linewidth ,~ resonance field. These are the conditions derived by Wangness and Bloch [ 14] and Redfield [15] for the quantum case. In the ferromagnetic limit, inspection of (46) shows that the equivalent relaxation times are given by

1 _ I _ ' % k(~z)H MzJ(¢%) MH A2re r l=r 2 M M - M J(O) k By

;Io

o Z O.

~z

(47) "1~¢

a

which are functions of Mz. Hence condition (23) becomes

MH J ( ~ z )

kBT J(0) (Arc)2 < 1

(48)

~z with the aid of (34) and (29) and neglecting the M z. dependence. Condition (16), when combined with (23), gives a linewidth condition

I~-lr2 >> [% rJ(0)/Mtt:(~z)] 1/2

.=!

MH

(49)

These results differ from the paramagnetic ones mainly in the presence of the factor MH/k B T, which by definitioo is larger than unity for ferromagnetic particles. This raises the question of whether the validity conditions can be satsified for all particle volumes, since M is proportional to volume. It is useful to distinguish between extrinsic and intrinsic modulation mechanisms for the modulating field, since their dependence on particle volume may be different. The following crude model is useful in this context. Let the particle moment M be composed of N atomic moments Mi, i = (1, N), each of magnitude M/N and seeing a field h i (t). Thus N h(t) = ! ~-, Ni=l as each field is shared over all N elementary moments. Modulatin~ fields at different atoms are assumed to be uncorrelated. Thus an intrinsic mechanism may be simulated by fields hi(t ) of equal variance at every atom, giwn " g A = N -"~/2 . For extrinsic mechanisms

,.T(,,.,z )

\ \

WEAK ~.=! MOTIONAL

\MH /

-|

~Oz

"~c

b

Fig. 2. The domain of validity of the Bloch equation of motion for (a) para_m..agneticand (b) feT~omagnetieparticles, shown as the shaded region in the space of the variables A and r c. In (b), the region a > 1 where the Gilbert equation differs from the Landau-Lifshitz equation lies above the dotted line. The derived equation (46) deviates from the Landau-Lifshitz form when weak motional narrowing applies, i.e. 7c > ItOzI-1 . The Markov form for (20) was used to construct the boundary curves; this corresponds to J(to) = J(0)/(1 + to:~r2e)and

229

D.A. Smith, EA. de Rozario / Tehory of superparamagnetic relaxation

various N-dependences are possible, depending on the spatial distribution of the modulating field. A common example is exchange anisotropy [16], for which the modulation is confined to the surface of the particles; for a spherical panicle this gives A 0cN - 2/3. In the limit of large N, intrinsic mechanisms will dominate, In that case, the N-dependence drops out of (47) and (48), and condition (16), or equivalently (49), is guaranteed. Finally. it must be said that the approxima,e forms of the Bloch equation derived in this section do not always predict relaxation towards the equilibrium average moment. The obvious offender is the first equation of (44), which predicts relaxation to zero (Mz) ratb~r than to the equilibrium value. Now the exact equations of motion, i.e. (41) and (43), are correct in this regard; the equilibrium average moment is always a solution of these equations because the Boltzmann P0 ~ exp [-~V(M)] is in fact a stationary solution of the corresponding Fokker-Planck equations. This may be seen directly from (21); i f P = P0 then the damping and diffusion currents cancel and the precession current becomes divergenceless. This being so, P can be replaced by P - P0 in the Fokker-Planck equation and (M(t)) by (M(t)) - ~/)0 in the Bloch equations, where 04)0 is the equilibrium average. The error is due to the way in which the paramagnetic limit was taken; the so-called "destination vector" (M>0 is of first order at least in H~ whereas the relaxation times (42) were evaluated to zeroth order only. The LandauLifshitz equation for a ferromagnetic particle does not require correction, since the damping term is proportional to Vt V which is zero when M assumes a stable equilibrium value. The derivation of the Bloeh equations has been confined to static magnetic fields. From Redfield's work [15] we knew that the same equations will apply in the presence of small r.f. fields and that the average moment at time t will relax to its equilibrium value in the instantaneous field at that time, provided that condition (23) is fail'died.

obtained an approximate analytic solution for the case of a lhigh potential barrier, i.e., K ~, k a T, by a method due to Kramers [10]. In this section the anal. ogous solutions are obtained for triaxial and cubic anisotropies. 4.1. Triaxlal anisotropy

The anisotropy potential has the form

where it is assumed that K x < Ky < K z. The contours of constant V are the free precession trajectories and are shown in fig. 3a. The z-directions are the minima of V, the y-directions are saddle points and the x-directions maxima. There are two energy barriers

and we assume that Ayz >>kB r ,

Brown [9] has discussed the relaxation of a superparamagnetic particle in a uniaxial anisotropy field in terms of the Fokker-Planck equation, and inter alia,

(51)

Axy ~, kB T .

The first inequality implies that most of the representative points of the ensemble in phase space are confined near the minima A and B, and that the ther.. mally activated flow of points between A and B will be slow by comparison with the precession period. Hence the distribution function P(M, t),can be regarded as time-independent, i.e., a stationary solution V.S=O

(52)

is required, where J is the Fokker-Planck current. It is assumed that J is given by the Landau-Lifshitz form (28). (52) determines the form of the distribution function, from which the populations n A and n B at the minima and the total current I from A to B can all be calculated. It is found that I = W(n A - riB). At . . . . . .

4. Relaxation times for superparamagnetic particles

(8)

v(M)- M2

(53)

--,-,,~¢han n f p a r t i c ' l a ~ a t t h e

minima means

that t] A = - t~B = I. Hence the net magnetic moment of the ensemble, which is prcportional to n A - nB, relaxes according to d(nA - n B ) dt

=

nA - n B r

,

1 r

-=2W.

(54)

D.A. Smith, F.A. de Rozario / Theory of superparamagnetie relaxation

230

r is the relaxation time for the net moment along the z-axis. The second inequality of (51) implies that the points of the ensemble at the equator are confined near the saddle points S and T. I f ~ e distribution function is written as /'(/tO = ~'(M) exp[-flV(M)]

(55)

the corresponding equation for ~', i.e. x

OVt2~"+ ('yMX V V A B Z

.A

•VIO" Vt~"= 0 ,

(56)

need be solved only in the vicinity of the saddle points as the entire flow pattern is an analytic continuation of the flows at the saddles. Consider the saddle point S, i.e. 0 = ~rr and $ = ~r in spherical polar coordinates. In terms of local coordinates x = cos ~, z = cos 0 (56) is approximated near S by the equation

D(~xx + ~ z z ) - 2(XAxyX + 7MAyzZ)~x F

- 2(TMAxyX - XAyzZ)~;z = O.

x

(57)

With Kramers we suppose that ~ is a function only of u = ax - z, where a is some constant. This leads to the differential equation

Y

B +

8

Au '(u) = 0 ,

(58)

where Z

A =

2(a +a)

Ayz (59)

~(a 2 + 1) k B T ' L

,

a

G

= ½ [-a(i + ,) + /a2 0 + 8)2 +

(60)

and

8 = Axy/Ay z . Y

C

~ = VTM

(61)

also h = 1319(30). Choosing the +re si,gn in (60) ensures that A is positive and that a = 0 (i.," ,, u is a function of 0 only) when there is no equatorial energy barrier (6 = 0). The particular--'--'" solu ti0n c,)

Fig. 3. Contours of constant anisotropy potentials for (a) the triaxial potential (8) and (b), (c) the cubic potential (66) with K +ve and - r e respectively. The minima axe labelled A, B, C, 13.... and S is a saddle-point. The dashed lines are critical paths b ~.tween minima.

~(u) = c(A/2rr)l/2 f

exp(-{Au2)dt, ,

t62)

u

where c is determined by normaliza':ion, has the limiting values of zero at B (where u = 1) and c at A (where

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

u = - 1 ) , since A >> 1. This solution is appropriate when ~teady all particles are in well A. Rather than construct the complete solution we will simply use eqs. (53) and (54) with n B = 0. ~ e number of particles at A is then given by in. tegrating the distribution with ~"- c over the region 0 < 0 < 01 where 01 is small but sufficiently large to exhaust the population in well A. Thus I

= cM 2

2~r

sin 0 dO 0

.f 0

exp[-~V(O,¢9] d~

(63)

To calculate the total current from A to B we integrate the polar current Jo around any circle of latitude. In particular we may choose the equator 0 = ½rt and furthor simplify the problem by integrating through the saddle points since almost all the current is concentrated there. Because of the symmetry of the potential we integrate around one saddle point and double the result. Now 7

OV

Jo = sinO

D O~')e_t3v MOO

so at 0 = ½rr ~2

I = 2M

Jo d~,

f

(~ about ~ ~)

''

where q~l, q~2 are azimuthal angles on either side of ~lr sufficiently large for the rise in poteatial to exceed k BT. Again the exact values are not important. Thus a+Ot

I = 2cD a--~,¢xp(13Ky).

(64)

Hence the relaxation time, from (53) a ad (54), is given by l_4t_, --

-

-- ~u

. , [ 1 +8~ 1/2TAyz -I- t x )

-

,~¢x

[ AYZ~ •

ensemble points are channelled through the saddle p,,ints rather than uniformly distributed over the equator. Furthermore, the pro-exponential factor varies nonlinearly with the damping parameter a and this dependence is itself a function of the equato/ial barrier as measured by 8. For a ,~ 1, the relaxation rate is independent of a over a reasonably wide range of values of 8 On fact, for a 2 ,~ 8 ,~ a - 2 ) and for a >> 1 the rate is linear in ct. 4.1. Cubic anisotropy

~rcM2 k B T

= Ayz(l + 8)1/2 exp(0d~z).

231

I v-~!

On comparing with the corresponding expression (7) for uniaxial anisotropy, the relaxation rate here is seen to be considerably smaller since the square-root factor, which is here of order unity, is of order (K/ kBT)I/2 ~" 1 in the uniaxial case. This is because the

For this case the simplest anisotropy potential is the quartic polynomial K (M~xM2 + M2M~2 + M2zM2x)

(66)

which has two different types of potential surfaces for K positive and negative. A sextic term will not be ineluded. K > 0: as indicated in fig. 3b. the minima (V = 0) lie along the coordinate axes with the saddle points (V = IK) situated midway between two minima. The maxima (V = •-K) lie along the various (111) dire.::tions. 3 As in the triaxial case we assume that all barriers are much larger than thermal values, This means that the current is quasi-stationary and is confined to regions around the saddle points. Denoting the minima at the two poles by A and B and those around the equator by C, D, E, F, we have rate equations for the populations of the six minima h A = - 4 W n A + W(n C + n D + n E + n F) , h B = - 4 W n B + W(n c + n D + n E + n F ) ,

(67)

with similar equations for n C to n F . Here, W is the transition rate between adjacent wells, and is assumed to be the same for all such pairs because of the symmetry of V. The z-component of magnetic moment is proportional to n A - n a and hence satisfies (54) with a re!~_xatio_n time l/r =4W. (68) r can be determined by imposing the initial concaition that only n A is non-zero. If the total c'~rrent flowing from A is denoted by I then we have I -- - h A and hence from (67) 1/ r = l/n A" The equations defining the distribution function

232

D.A. Smith, F.A. de Rozario / Theory o.f superparamagnetic relaxation

are given by (55) and (56) and need be solved only in the region of the saddle point S(0 = ~ lr, $ = 0) with (66) as the potential. Changing to local variables z = 2 (sin 0 - I t v ~ ) and y - sin ¢, (56) is approximated around S by

and are labelled in fig. 3c. With this l~belling, the rate equations for the populations of the minima are h A = --3WnA+W(nc+nF+nG), hi3 = --3WnB + W(nD + nE +

D(~zz + ~yy) - K[(-~ 7My - kz)~ z + (TMz + l k y ) ~ y ] =0.

(69) As before we assume that ~"is a fanction of u - - a y - z only, and obtain the particular solution

~u) = c

exp(-~ Au 2) du

f

V~ff u

(70)

h c = - 3 W n c + W(n A + n E + n i l ) , h D = - 3 W n D + W(n B + n F +

nG),

h E = - 3 W n E + W(n B + n c + riG), nF = - 3 W n F + W(nA + niD + n i l ) ,

with a+t~ K A = ot(l+a2_) kB T, a=~

nil),

[~+8

--

(71)

3al ,

(72)

and c is a normalization constant. At point A on the sphere ~ ~ c. In this case the number of particles in the well at A is (73)

n A = cM2~rk B T / K .

The net current flowing from A is obtained by calculating the current passing through the saddle point S and multiplying the result by four. This gives I = 4M

f 2 Jo sin ~ rr d~b

h G = - 3 WnG + W(n A + n D + niE), ti H = - - 3 W n H + W(n B + n C + nF).

(76)

The rate equations for the net magnetic moments along the various (111) directions (AB, CD, etc.) are coupled since these directions are not mutually orthogonal. In fact the coordinate axes ( x y z ) are eigenaxes of this relaxation matrix. Thus the corresponding components m x=(l/~f3)(n A-n B +n C-n D

- n E + nF - nG + n i l ) , my

=O/x/Y) (hA

- nB - nC + nO

-hE + nF +/~G - nil),

(cp about O)

m s = (I/v5) (hA - nB + nc - nD

~ _ 4 ~ cD a a+ ¢ exp(qt~x) ,

(74)

where 0 .-1~ rr and the integral is evaluated for K >>kBT. Hence the relaxation time is 1 _ 4V'2 7K (a + o0 exp(-K/4k s T)

7"

71" /~f

+ nE - nF + n G - nil).

of the net moment satisfyseparaterate equations of the form (54) with (78)

!/r-2W. (75)

K < 0: whex~K is negative the maxima of the po. tential (V = O) ire along the coordinate axes, with saddle points (V = - i iKI) in the directions between. The minima (V = - ~ IKI) are along the (111) directions,

(77)

If we again impose the ix~itial condition that only n A is non-z, ~ then the total current I = - t i A . Using (70) we get l / r = 21]3nA. Eq. (56) is approximated near S by the equation D(~zz + ~yy) -

IKI (27My - hz) ~z

- IKI (~Mz + 2Xy) ry = 0.

(79)

D.A. Smith, F.A. de Rozario / Theory of superparamagnetic relaxation

where z = cos 0 and y = cos(~ + ~) are local variables about the saddle point S at (110), i.e. 0 = ~ 2 ' = ~e. The solution ~'(u), where u = ay - z, that limits to c at the point A, is given by (62) with

=

f exp(-[A.2) du

with a + c~

A - a ( 1 +a2)

IKI

kBT;

a = ½{x/qa2 + 8 - 38}.

(80) (81)

The calculation of n A is eased by using axes (x'y'z') with the z' axis along A. Thus

31r cM2 kB T nA = 2 [KI exp(/31KI/3)

(82)

and

I = 3M

Jo d~ (~ about I n)

fined to the saddles, which leads to a single exponen. tial factor. A flow which is spread out over the equator can be si~aulated by an additional term in I/T of the form (7) appropriate for uniaxial anisotropy. Thus for K > 0 there would be an additional contribution of the form (TK/M) ( ~ O 1/2 exp(~K), whereas (75) goes as (TK/M) exp(-~ ILK); the uniaxial flow is thus frozen out for ~K sufficiently large. The above is at variance with the conclusions drawn by Aharoni and Eisenstein [18] from a numerical solution of the Fokker-Planck equation, but more recent work by the same authors [19] appears to be in complete agreement with our results.

Acknowledgements One of us (F.A. de Rozario) acknowledges financial support from a Commonwealth Postgraduate Scholarship. We are much indebted to Dr. I. Eisenstein for making available his recent work and for a detailed comparison of analytic results.

References

¢t where 0 = ~ n. The relaxation rate is

l _ 2V~ 7[K! T 3~ M ( a + a ) e x p ( - I K l / 1 2 k n T ) "

233

(83)

For both signs of K, the relaxation rates are again smaller than in the uniaxial case because of the absence of a factor (K/k B T) 1/2. Also, the pre-exponential factor is again constant for (~ ,~ 1 and linear in ~ for ~ ~, 1. It appears that formulae (75) and (83) are in better agreement with experimentally obtained relaxation rates in ferromagnets with cubic anisotropy than Brown's formula, which should be reserved for uniaxial materials. Thus Krop e~ al. [17] find that the pre-exponential factor f is ~ 108 sec- 1 for i3-Co particles over a wide range of values of K/kBT. Using the quoted values o f K a n d M and noting that K < 0 here, formula (83) in the low-aamping limit (~ = 0 , a = V~) gives f = 5 X 109 sec- i at 293 K. Assuming a ~ i, the observed constancy o f f is predicted by our formulae, in which the temperature dependence comes only from the facto~ K/M, which varies only weakly with temperature, and not from the damping constant which in some cases may be strongly temperature-dependent [7]. The derivation of (75) and (83) assumes that the flow of representative points between the poles is con-

[1] L.D. Landau and E.M. Lifshitz, Phys. Z. Sowjetuniop 8 (1935) 153. [2] T.L. Gilbert, Phys. Rev. 100 (1955) 1243. [ 3] For a recent review of phenomenological equations and ferromagnetic resonance in general see S.M. Bhagat, "Techniques of Metals Research", Ed. E. Passaglia (Wiley, New York, 1973) Vol. VI, Part 2, p. 79. [4]C.W. Haas and ll.B. Callcn, Magnelism, Ed. G.T. Rado and H. Suhl, (Academic Press, Nev, York, 1963) Vol. I, p. 449. [5] E.M. Gyorgy, Magnetism,Ed. G.T. Ratio and H. Suhl (Academic Press, Hew York, 1963) Vol. lIl, p. 525. [6l R. Kikuchi, J. Applied Phys. 27 (1956) 1352. [7] S.M. Bhagat and P. Lubitz, Phys. Rev. B10 (1974) 179. [8] D.O. Smith, Magnetism, Ed. G.T. Rado and H. Suhl (Academic Press, New York, 1963) Vol. lit, p. 465. [91 W.F. Brown, Jnr., Phys. Rev. 130 (1963) 1677. [1o1 H.A. Kramers, Physica 7 (1940) 284. [Zll F. Haake, Quantum Statistics in Optics and Solid-State Physics, (Springer-Verlag, Berlin, 1973) Vol. 66, p. 98. t121 L. Onsager, Phys. Rev. 37 (1931) 405 and 38 (1931) 2265. l a , ~ i C.P. Slich~,:r, Princip!es of Magnetic Resonance. (Harper and Row, New York, 1963). [141 R.K. Wangness and ~:. Bioch, Phys. Rev. 89 (1953) 728. ltsi A.G. Redfield, IBM J. Res. 1 (1957) l q. [16} I.S. Jacobs and C.P. Bean, Magnetism, L6. G.T. Rado and H. Suhl (Academic Press, New York, 1963) Vo!. Ill, p. 271. [171 K. Krop, J. Korecki, J Zukrowski and W. Karas, Int. J. Magnetism 6 (1974) 19. [18] A. Aharoni and 1. Eisenstein, Phys. Rev. B 11 (1975) 514. [19] I. Eisenstein and A. Aharoni (to be published).