Interpolation formulae between Kramers escape rates for axially symmetric and non-axially symmetric potentials for single-domain ferromagnetic particles

Journal of Molecular Liquids 114 (2004) 105–111

Interpolation formulae between Kramers escape rates for axially symmetric and non-axially symmetric potentials for single-domain ferromagnetic particles David J. McCarthy School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Abstract In this paper the connection between the work of Kramers on the escape of particles over potential barriers and that of Brown on magnetic stability is briefly outlined. The case of magnetic stability in the context of an axially symmetric potential is also briefly outlined. The need for interpolation formulae between various regimes of approximation is discussed, and those discovered to date are given. 䊚 2004 Elsevier B.V. All rights reserved. Keywords: Magnetic field; Kramers; Brown

1. Introduction 1.1. The escape of mechanical particles from a potential well In 1940, Kramers published a paper w1x in which he dealt with a collection (approximately 1023 in number) of small particles. He considered them trapped in a region of space by a conservative force. For simplicity he took the one-dimensional case, but extension to many dimensions is straightforward. This force being conservative meant that it could be described by a potential U and the force could be derived from this potential dU function by the formula Fsy . This potential funcdx tion may be looked on as a ‘potential well’ trapping the particles (see Fig. 1). While these particles receive no energy from outside, they will remain trapped in the well ad infinitum. However, Kramers now assumed that these particles were in constant contact with a heat source (e.g. the surroundings which could be at room temperature), which supplied energy on an ongoing basis to the particles so that a small number of them (say approx. 1000ys) could attain sufficient energy to escape from the well. (In order to give an idea of the scale of the problem, it is worth pointing out that the particles would E-mail address: [email protected] (D.J. McCarthy).

take approximately 1020 s for all of them to quit the well. The earth is currently approximately 1022 s old!). Kramers then calculated the escape rate for the particles under these conditions. 1.2. Magnetic stability A large unmagnetised block of magnetic material consists of several domains as shown in Fig. 2 in which the directions of the magnetisation are such as to cancel each other out. The domains are separated by so-called ‘Bloch walls’ some 100 atoms thick in which the orientation of the magnetisation vectors change gradually from the orientation in one domain to that in the other. When a magnetic field H is applied, the magnetisations tend to reorientate where necessary in order to align with the field. As we increase the field, more and more domains align with the field. After a certain stage the magnetisation of all the domains are aligned with the field and no matter by how much we increase the field, the overall magnetisation of the block of material will not increase. We now consider a particle so small that it consists of a single domain. This single-domain particle will have an anisotropy. This will be due to many causes (shape, structure, imperfections, etc.) but for simplicity we will take it to be due to its shape only. The shape of the particle we will take to be an ellipsoid (see Fig.

0167-7322/04/$ - see front matter 䊚 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2004.02.041

106

D.J. McCarthy / Journal of Molecular Liquids 114 (2004) 105–111

Fig. 1. The potential function. Particles are trapped at A. A small number obtain sufficient energy to escape over the potential barrier at C. These escape over the peak to B.

3). The particle will be easier to magnetise in the directions AB and BA rather than in a direction perpendicular to them. These directions AB and BA are called the easy axes of magnetisation. If a magnetic field aligns the magnetisation of a number of such particles, grouped in close proximity, in the direction AB, say; when the field is switched off, the tendency will be for the magnetisations to reorientate in order for the overall magnetisation to return to zero. However, in order for an individual magnetisation to reorientate, it will have to ‘flip’ through 1808 (to align with the other easy axis, BA). Furthermore, before it reaches this state, it will have to pass through the 908 point. This will require extra energy. This will come from the heat of the surroundings. This shows that the situation for single-domain ferromagnetic particles is very similar to the situation for mechanical particles dealt with by Kramers.

Fig. 3. Diagram of a single domain ferromagnetic particle showing the easy axis of magnetisation AB. B

expCy D

4. 5.

6. 2. Details of Kramers’ work 7. Kramers assumed 1. The particles are initially trapped at A. 2. The barrier height is very large compared with KT. 3. In the well, the number of particles with energy between U and UqdU is proportional to

Fig. 2. Magnetisation vectors in a block of unmagnetised material.

UE FdU. kT G

That is a Maxwell–Boltzmann distribution is attained extremely rapidly in the well. Quantum effects are negligible. The escape of particles over the barrier is extremely slow, so that the disturbance to the Maxwell–Boltzmann distribution is negligible at all times. That is the minimum is a metastable state. Once a particle escapes over the barrier, it never returns. A typical particle of the reacting system may be modelled by the theory of Brownian motion taking account of the inertia of the particles. He then derived the equation

≠r p ≠r ≠B ≠r E syKry qh CrpqmkT F ≠t m ≠x ≠p D ≠p G

(2.1)

where p is the momentum of the particle, x is the position of the particle, m is the mass of a particle, t is the time, r is the density of the particles, k is Boltzmann’s constant, T is the absolute temperature, h is a ≠U phenomenological damping constant, Ksy is the ≠x conservative force, where U is the potential of the force K. This is the one dimensional Klein–Kramers equation. Kramers assumed further that

D.J. McCarthy / Journal of Molecular Liquids 114 (2004) 105–111

i. near the minimum, the function U may be written 1 UfUAq mvA2 ŽxyxA.2 2 ii. near the max 1 UfUCy mvC2 ŽxyxC.2 2

GsA9

wherevA, vC are constants, in fact v2As

d2U dx2

)

d 2U dx2

)

v2Cs

(2.4) xsxA

(2,5)

flux across barrier number of particles in the well

Now for IHD we must have A9sy

y

z B h2 DU E C F | q1 exp y D 4v2C kT G ~

y

(2.6)

(2.7)

A9s

(3.3)

vA expŽybDU. 2p

A™y

h q 2vC

y

I vA F expCy D kT 2p kT G EsEC

the motion, EC is the energy at the saddle point. Kramers, at this stage confessed that he could not obtain an expression for the range of damping between the VLD and IHD domains.

(3.5)

as h™`

(3.6)

A™bhI as h™0. They came up with the solution for A

(2.8)

where I' ¢ p dx is the action over one cycle of

h2 q1 4vC2

and

y

Gsh

(3.4)

where

w

DU E

(3.2)

hI kT

AŽD.sxy

and B

h2 q1 4vC2

While for VLD, we need

GsA

for the escape rate G, where DU is the difference in the values of U at the maximum and at the minimum. 2. The case of very low damping (VLD). Where, on changing from position and momentum variables to angle and action variables he obtained the expressions B DU DU E F expCy D kT kT G

h q 2vC

They then assumed that G could be written

he obtained the expression vA w xy h q 2p y 2vC

(3.1)

1 kT

xsxC

He then solved this equation in two limiting cases w2x 1. The case of intermediate to high damping (IHD).Where, on considering

Gsh

vA expŽybDU. 2p

where b'

xA is the x coordinate of the minimum, xC is the x coordinate of the maximum.

Gs

They noted that the escape rates above may be written in the form:

(2.2)

(2.3)

107

h q 2vC

y

w

x

=exp

y

1 2p

|

`

y`

z h2 | q1 4vC2 ~

w z S B 1 EW T T 2 FX lnx1yexpU TyDCl q T| D V 4 GY~ y

1 lq 4 2

z

|

dl

(3.7)

~

where (3.8)

3. The paper of Melnikov and Meschkov

D'bhI

In 1986, two Russian scientists, Melnikov and Meschkov, published a paper w3x to remedy this deficiency.

This function has the desired limits as h™0 and h™ `.

D.J. McCarthy / Journal of Molecular Liquids 114 (2004) 105–111

108

be written as

4. The work of Brown In the 1960s and 1970s Brown published a number of papers w4,5x on the problem of magnetic relaxation. He used as his starting point the Gilbert equation for a particle of magnetisation M in a magnetic field H: w dM dM z sgM=xHqh | dt dt ~ y

(4.1)

where g is the gyromagnetic ratio s magnetic moment , h is a phenomenological damping angular momentum constant. With some (relatively) easy manipulation, this equation reduces to dM sbMsŽM=H.qaŽM=H.=M dt

(4.2)

where g a' ; Ž1qa2.Ms

b'aa;

a'hgMs;

Ms')M)

Letting M'y=V, we find, after a lot of manipulation that this reduces to the Fokker–Planck equation for the density of magnetic moment orientations in spherical polar coordinates (1, u, w):

(4.6)

ns0

The eigenvalues pn and the corresponding eigenfunctions Fn(u, w) and amplitudes An are determined by the requirements of single-valuedness and finiteness, and initial conditions. The lowest eigenvalue p0s0. As in the case of mechanical particles, we are looking ≠W f0. for a quasi stationary solution, i.e. the case ≠t Now we cannot, in general, solve by separating the variables as F(u,w)sQ(u)F(w)

(4.7)

since VsV(u,w) depends on both u and w and because ≠V ≠V both and both appear. ≠u ≠w Two things are, however, helpful. 1. Under steady state conditions, which we have except in the very early stages, we find that p2,p3,......4p1 and A2,A3,.....
≠W 1 s ≠t sinu

=

`

Ws 8 AnFnŽu,w.expŽypnt.

WfW0qA1F1(u,w)exp(yp1t)

BS ≠V ≠w a ≠V W ≠W Ez 1 T T Ub X Fq sinuCT y TWqk9 ≠u y sinu ≠w Y ≠u G~ sinu DV ≠u

x

|

≠ wB ≠V b ≠V E k9 ≠W z FWq = xCa q | ≠w yD ≠u sinu ≠w G sinu ≠w ~

(4.3)

where, on insisting that in the equilibrium situation, we have the Maxwell–Boltzmann distribution of velocities, we find that

(4.8)

and so ≠W fyp1A1F1Žu,w.expŽyp1t. ≠t

(4.9)

and ≠ ŽWyW0.fyp1A1F1Žu,w.expŽyp1t. ≠t

(4.10)

So k9s

hkT

b s . b 2

1 qh2Ms g2

(4.4)

≠ ŽWyW0. ≠t fyp1sythe escape rate WyW0

(4.11)

If we now assume a solution of the form WŽu,w,t.'TŽt.FŽu,w.

(4.5)

we see that the solution, W, to the above equation may

This last equation tells us that the rate of build up of particles in the metastable well is approximately yp1 ´ and so the escape rate is approximately qp1. The Neel relaxation time, which is the average time for a particle

D.J. McCarthy / Journal of Molecular Liquids 114 (2004) 105–111

to cross the maximum into the other region is the 1 inverse of the escape rate and so is . p1 It is worth mentioning here that we have two space variables viz u, w in the Fokker–Planck equation in the magnetic case, while in the case of mechanical particles we have one space variable and one momentum variable. In the case of mechanical particles we were dealing with a simple one-dimensional maximum for the barrier over which the particles have to escape. In the case of magnetic particles, the barrier will not be a maximum, but rather a saddle point. The problem of solving the Fokker–Planck equation or of finding the smallest non-vanishing eigenvalue is intractible except in the simplest of cases. To overcome this, Brown established a method, which, in the limit of high potential barriers DU4kT, gave very good approximations. It involved approximate calculations of the number of particles crossing the barrier under the assumption that the potential function could be approximated to a quadratic function of the coordinates, i.e. he expanded the function as a Taylor series and truncated it at the quadratic term. He divided this by the number of particles in the well under a similar assumption (See Kramers work on mechanical particles. He makes a very similar assumption about the potential function). This gives an asymptotic expression for the escape rate, usually called ‘the Kramers escape rate for magnetic spins’ due to its similarity to the Kramers calculation for mechanical particles. 5. Interpolating functions

Now the VLD (very low damping) formula for the escape rate is found from consideration of the MFPT (mean first passage time), i.e. the average time it takes a magnetic particle to first reach the energy needed to flip its magnetisation (see Ref. w6x). vA bDEeybŽE 2p

(5.1)

yEA.

C

w

≠H

1

≠H

z

xŽ1yp2. ≠p dwy 1yp2 ≠w dp| EsE Cy

~

(5.2)

1

w

S a2Žc2yc1.2 W T T2 b U 2 X GIHDs v q T C T 2pavC yV 4 Y

x

y

aŽc2qc1. 2

z

|yc

(1) (1) ybŽVCyVA.

1

(5.6)

c2 e

~

where c 1 Ž 1. '

≠2E evaluated at the saddle point ≠Ža1Ž1..2

c 2 Ž 1. '

≠2E evaluated at the saddle point ≠Ža2Ž1..2

a1Ž1.,a2Ž1. is a local co-ordinate system where the Hessian of the energy is diagonalised in the vicinity of the bottom of the well. This means essentially that the energy may be approximated to the quadratic approximation: b w Ž 1. Ž 1 . 2 c1 Ža1 . qc2Ž1.Ža2Ž1..2z~ 2y x

(5.7)

|

c1, c2, a1 and a2 are defined in a similar way for the saddle point: VsVCq

bw c1Ža1.2qc2Ža2.2z~; 2y x

b a' shgMs; a

|

a'

ag 'h9; Ž1qa2.Ms

b'

1 ; kT

g 'g9; Ž1qa2.Ms vA'yc1(1)c2(1);

vC'yyc1c2;

VA'value of V at the minimum; VC'value of V at the saddle point.

p'cosu g 0 vA' yEpp E0ww Ms

(5.5)

The IHD (intermediate to high damping) formula is obtained from the Brown method and is given, in the case of a single well, by Ref. w7x

b'

where DE'a ¢

≠ 2E evaluated at the minimum. ≠p2

VsVAq

5.1. Interpolating function connecting the VLD and IHD cases for spins

GVLDs

E0pp'

109

(5.8)

(5.3) (5.4)

Note that since we are dealing with a saddle point, c1 and c2 are of opposite sign and so

yyc1c2gR.

(5.9)

D.J. McCarthy / Journal of Molecular Liquids 114 (2004) 105–111

110

Now since the two formulae have been found by two essentially different methods there is no reason to believe that they are compatible in the region between their regions of validity, and so we look for an interpolation formula for the crossover region. This was found to be similar to the Melnikov and Meschkov formula. Letting w

≠E

≠E

1

z

xŽ1yp.2 ≠p dwy 1yp2 ≠w dp| EsE

s'b ¢

Cy

(5.10)

~

The potential at (5.15) is the potential function for uniaxial anisotropy with a field H applied at an angle c to the z axis (in the xz plane). The potential at (5.14) is that with the field applied along the z axis. This potential function is axially symmetric wVsV(u only); V/V(w)x. The potential at (5.13) is the limiting case of (5.14) as H™0. Now in the case of the potential at (5.13). We find that the asymptotic expression for the escape rate is G0s

and w

x|

AŽD.'exp

1 p

`

w S Wz B 1 ET T 2 FX lnx1yexpU TyDCl q T| D V 4 GY~ y

y`

y

1 lq 4 2

z

|

a

ag 2 3 s 2 Ž1yh2.Ž1qh.eysŽ1qh. 2 bMsŽ1qa . yp

(5.12)

G

(5.17)

(5.18)

limG1sG0.

FGIHD

2

we see immediately that h™0

B as E D

G1s

~

we find that w8x GsAC

(5.16)

For that at (5.14) we find

dl

(5.11)

ag 2 3 ys s2e 2 bMsŽ1qa . yp

However, using the potential at (5.15) we get an expression which, as h™0, yields an escape rate looking like w2x

fills the gap between VLD and IHD. B as E

as As a™0 AC F™ D a G a

G 2f

D

a

(5.19)

yh

and so G™GVLD and as a

for any angle c. wThe expression for the IHD and high barrier escape rate is:

B as E

™` AC

constant

F™1 and so G™GIHD G

5.2. Interpolating formula connecting IHD axially symmetric and non-axially symmetric escape rate formulae

G 2s

Let us now look at three types of possible potential functions which, while simple, are of practical interest.

1

wS

x

b

T

U Tyc1c2q 2payyc1c2 yV

y

aŽc2qc1. 2

a2Žc2yc1.2 W T2 X

4

T Y

z

|v e A

~

ybŽVCyVA.

(5.20)

VsyKcos2u

(5.13)

Vsycos2uyHKMs cosu

(5.14)

If we focus on the term yyc1c2 in the denominator, it can be shown w2x that yyc1c2sconstyh near hs0. With no cancelling factor in the numerator this means that as h™0

(5.15)

G;

VsyKcos2uyKHMs(cosu cosc qsinu cosw sinc)

where K is a constant called the anisotropy constant, H is the magnitude of the applied magnetic field, Ms is the magnetisation of a single particle, u,w are the usual spherical polar co-ordinates (angles).

1

yh

™`x

(5.21)

wIn fact c1;h and c2;const.x, i.e. G2 diverges while G1™G0 (finite) as h™0. Now if the Kramers escape model is to be a good model (at least for high barriers),

D.J. McCarthy / Journal of Molecular Liquids 114 (2004) 105–111

then G2 must™G0 as h™0 also, and so we need an interpolation formula connecting G2 and G1 for small values of h in the IHD case. In order to establish the interpolation formula, we write the escape rate as

y ps QfŽj.

(5.22)

where

where

w dw | y ´qjcos ´

Q' lim

´™yj

w

x|

DUsVCyVA

y

We find

`

y ps e

yj

I0Žj.

(5.23)

where I0 is the modified Bessel function of the first kind of zero order, j'2sh;

a'hgMs

I0(j) may be written

zy1

| yjqt

seyjŽerfcyj.

y1

(5.29)

~

|

e

jcosw

dw.

(5.24)

0

This has been worked out for a transverse field only. If we apply a weak transverse field to the easy axis of magnetisation, we find the Kramers escape rate. 8as v yh A eybDU 1qa2 p

(5.25)

while a 1qa2

Referring back to the last section, we can show that 23y2 f0.900. Therefor j'0, Qs1, while for j/0, Qs p fore, there is a jump in the value of Q, and hence another crossover formula is needed. It can be shown that this should come into play around aysfj2. This has not been worked out to date. Acknowledgments

2p

5.3. Interpolating formula in the VLD limit

G0s2p

0

eytdt

5.4. Crossover formula for VLD near aysfj2

As2pa

G 2s

(5.28)

0

fŽj.'yp

1 I0Žj.s 2p

(5.27)

As2pa

2p

vA GsA eybDU, 2p

s'bK;

111

y ps vp e

A ybDU

(5.26)

i.e. G2™0 as h™0 while this is not so for G0. Again, if the model is to be a good one G2 and G0 should approach the same limit in the limiting case of zero field. The interpolating formula in this case works out to B E vA be Cagain writing GsA eybDUF D G p

The subject matter of this talk involves work carried out in collaboration with Prof. W.T. Coffey (Dublin University), Prof. D.S.F. Crothers (Queens University ´ ´ Belfast), Dr P.M. Dejardin (Universite´ de Perpignan), Dr D.A. Garanin (Universite´ de Versailles). The author would also like to thank the Dublin Institute of Technology for their continued support for this research. References w1x H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (4) (1940) 284–304, Utrecht. w2x W.T. Coffey, D.A. Garanin, D.J. McCarthy, Crossover formulas in the Kramers theory of thermally activated escape rates— application to spin systems, Adv. Chem. Phys. 117 (2001) 483. w3x V.I. Mel’nikov, S.V. Meshkov, J. Chem. Phys. 85 (1986) 1018. w4x W.F. Brown Jr, Phys. Rev. 130 (1963) 1677. w5x W.F. Brown Jr, IEEE Trans. Mag. 15 (1979) 1197. w6x D.J. McCarthy, W.T. Coffey, J. Phys. Cond. Matter 11 (1999) 10 586. w7x L.J. Geoghegan, W.T. Coffey, B. Mulligan, Adv. Chem. Phys. 100 (1997) 475. w8x P.M. Dejardin, ´ D.S.F. Crothers W.T. Coffey, D.J. McCarthy, Phys. Rev. E (2001).