spherical spin waves

S P H E R I C A L S P I N W A V E S * )

J. Kolgt~ek

lnstitute o f Physics, CzechosL Acad. ScL, ]Va Slovance 2, 180 40, Praha 8, Czechoslovakia

The spin waves in a two-sublattice �9 sphere are derived from the equation of motion for the sublattice magnetization. The pinning conditions are discussed and the resuhs are compared with those obtained by numerical diagonalization of corresponding Heisenberg hamiltonian. For a sphere with the diameter of about 40 lattice constants the continuum approximation is fully applicable for estimation of some ("long wavelength") spin wave energies and angle of the core spins. The transition probabilities seem to be considerably overestimated by this method.

INTRODUCTION

In a small ferrimagnetic sphere with two sublattices, there exist more than two magnetic excitations which can be excited by uniform external magnetic field. An approximate formula for the sum of the transition probabilities induced by such a field is derived in [1]. These modes are explicitly calculated for appropriately chosen examples in [2] by numerical diagonalization of the hamiltonian.

Using a more powerful method of the decomposition of a given vector into the eigenvectors of a sparse matrix described in [3, 4] it was possible to extend the computations to larger particles with the diameter of about 40 lattice constants.

I t was then observed that besides some surface modes and the uniform (ferrimagnetic resonance) mode there are some spherically symmetrical modes which can be closely approximated by spherical Bessel functions. I t i s the aim of this paper to show how such modes can be derived classically f rom the equations of motion for the sublattice magnetizations and to make a simple comparison with quantum mechanical results.

SPIN WAVES IN A SPHERICAL FERRIMAGNETIC PARTICLE

Let us suppose a ferrimagnetic material containing two magnetic sublattices A, B with spin numbers SA > SB and gyromagnetic ratios VA, YB- T h e sublattice magnetizations M A, M B are used as the main quantities characterizing the magnetic state of the substance.

They can be divided into a static part describing the exactly antiparallel ordering of spins (the classical ground state) and a dynamical one describing the interaction"

(1) M A = MAS + m A

M B = MBS + m s ,

*) Dedicated to Dr. Svatopluk KrupiCka on the occasion of his 65th birthday.

8 Czech. J. Phys. B 37 [1987]

J. Kold&k: Spherical spin waves

where

MAS = MAi z ; MA = nA~ASAh

MBs = - Mai= ; M a = naysSBh

with hA, nB being the numbers of spins A, B in the unit volume.

It is convenient to decompose the dynamical part into the circulary polarised components

(2) m A = mA i+ ~(r) exp (--ioet)

m a = - m B i + ~b(r) exp ( - i c o 0 ,

where i -+ = l/x/(2 ) (1, ___i, (3); the suffixes + and - denote the right hand polarized (rhp) and the left hand polarized (lhp) modes, respectively. The spatial dependence of the magnetization ~k(r) is supposed to be the saine for the A- and B-sublattices.

The magnetization must obey the classical equations of motion

(3) 0tMA = --TAMA X Hy

c3,M B = --~BM D x Heff(B),

where the effective fields, sec e.g., [5], are in this model written as

(4a) Hy = VaA(1 + b– V a) MA + VAB(1 + b�94 R V E) MB

(4b) Heff(B) ~-- VAB(1 "q- b2A V2)MA "j- VBB(1 "q- b2B V2)MB,

The static external field, anisotropy field an the dipolar interaction are not considered here.

The solution of the equations of motion is possible under the assumption that the dynamical parts of the magnetizations are small with only mild spatial variations, so that the terms containing quantities m and V2m in the second power can be neglected. Substituting (1), (2), and (4) into (3) it is possible to obtain linearized equations of motion in the form:

(TAYAAb�94 -- ~AVABb�94 V2~/ +

+ (~]AVABMBmA -- ~AVABMAmB + oemA)~1 = 0

and an analogous form for (4b). Because these equations must be fulfilled in the entire volume of the particle, it is

clear that the solution requires V2O tobe proportional to ~k, e.g.,

(5) v2r + k2O = 0

so that the linearized equations of motion become

(6) (yAVA.Ma -- yAVAab– + r m a + 7AVA.MA(b2B k2 -- 1) m. = 0

~ava.Ma(b~Ak 2 - 1) m A + (~'BVABMA -- 7.v..b..k2 2 M s - co) m S = 0

Czech. J. Phys. B 37 [1987] 9

J. KoldEek: Spherical spin waves

for the rhp modes. The same equations are valid for the lhp modes with oe < 0. Solutions of the Helmholtz equation (5) are well known and in spherical coordinates

r, 0, ™ they can be written as

(7) ~/k = Zl(kl*) Y/m(0, (p),

where ~ are the usual spherical functions and Z l denotes spherical Bessel, Neumann or Hankel functions. Only the spherical Bessel functions with l = 0

(8) ~kk = sin (kr)/(kr)

will be taken into account, as only these are regular in the origin and can contribute to the absorption of r.f. magnetic field in the case of spherically symmetrical particles.

It is important to note that this approximation is valid only for

(9) k ~ = - ; R>> a a

when the assumption of mild space variation of the magnetization holds. The two functions sin (klr)/(klr) , sin (k2r)/(k2r) with nonzero Ikll * Ik21 are

orthogonal on the sphere of radius R, if

(10) k2 sin (k lR) cos (k2R) = kl cos (klR) sin (k2R)

and thus for an arbitrarily chosen pinning condition [6, 7]

OErn + fl d_m_m = 0 dn

(dm/dn is the normal derivative of the dynamic magnetization m) there exists an infinite set of mutually orthogonal solutions (8) with k determined by

(11) tan (kR) _ fl . kR fl - OER

For a = 0 (the normal derivative of m i s zero on the surface - unpinned modes) the value k = 0 is aUowed; the corresponding function O(r) = 1 is orthogonal to all the functions with k determined by tan (kR) = kR. For fl = 0 (the dynamic part of the magnetization is zero on the surface - pinned modes) the values of k are multiples of rc/R.

In agreement with quantum mechanical results (see the next paragraph) the rhp modes will be supposed to be unpinned (a = 0), in contradistinction to lhp modes which will be supposed pinned (fl = 0). To motivate this choice it is necessary to discuss the dynamics of the surface layer. Above, for the sake of simplicity, a very unrealistic supposition was tacitly ruade, that the exchange parameters v are constant throughout the entire volume of the particle and this seems to provide no reason for choosing any particular pinning condition.

10 Czech, J. Phys. B 37 [1987]

J. KoldZek: Spherical spin waves

It is clear that the parameters v should change near the surface due to "missing interactions" and this can be used as the basis to discuss the pinning.

If the spatial dependence of VAA, VAB is the same as it is for VAB , then the rhp uniform mode ff(r) --- 1 with to = 0 remains the solution of the equation of motion (6). This mode is clearly unpinned. The theory is applicable only for small k, what for rhp modesimplies that the corresponding ca is small (see, e.g.,(17) below) and so the equations (6) are in a good approximation fulfilled even in the case of nonconstant v's.

This reasoning does not hold fot the lhp modes, where the dynamic term is compar- able with other terms in the equation (6) even for very small k, and so its applicability requires mA, mB to be zero (or to be small) in the region where v does change (near the surface). In other words, the spins at the surface are "off resonance" and so the dynamical magnetization should be zero there.

To summarize, for the spherical spin waves the k-numbers are determined by

(12) tan (kR) = kR for the rhp modes

k = j r c / R , j = 1 ,2 , . . . f o r t h e l h p m o d e s .

To make the resulting formulas shorter, a structure will be supposed, where every A-spin has z nearest neighbours of the B-type and vice versa, so that

(13) n A = n B = n/2 ; bas = bBA -- b = a / x / z ,

where a is the lattice constant. In such a structure the A - B interaction commonly happens to dominate, so that the A - A and the B - B interactions can be neglected in the first approximation. Consequently, it will be supposed here that

2z (14) VAA= VBB=0; VAB----- V - - - - 2 J .

nTAYB h2

The microscopical parameters 2J and the lattice constant a are used here instead of their phenomenological analogues v and b to make easier the comparison with the quantum mechanical results.

Using (13) and (14), the equations of motion (6) are simplified to

(15) (zSB -- oe') ma + zSaTA (a2k2/z - 1) mB = 0 78

zSa 7a (a2k2/z _ 1) mA + (zSA + ~') mB = 0 7a

and solving the resulting secular determinant the dispersion law

(16) co' = - � 8 9 A - Sa) +- (�88 -- Sa) + 2˜171 1/2

can be obtained. Here, fo' denotes the energy ho) of the spin waves in the units of

Czech. J. Phys. B 37 [1987] 11

J. Koldœ Spherical spin waves

- 2 J . For small k the energy of the spin waves be approximated as

(17) oe '= 2SASB a2k 2 (SA -- SB)

2SASB a2k2 y - z ( s A - sa) - ( s ~ : ?.)

for the rhp modes

for the lhp modes.

COMPARISON WITH A QUANTUM MECHANICAL MODEL

Since no experimental data have been found in the literature to test the theory as yet, the applicability of it can be tested only by comparison with other theoretical results.

To make possible a direct comparison with quantum mechanical results of ref. [-2], the spin wave amplitude (which is not determined by equations (6)) must be suitably chosen.

By excitation of one magnon described by quantum mechanical function lU) a

the mean value of the operator mx, (taxi is the x-th component of the magnetic moment located on the i-th spin)increases by the value 2S~~ 2 h2]U,I 2. Here, lUi[ z is the probability that there is a spin deviation on the i-th spin; the i-th spin is of the type (oE = A, B) and its radius vector is r v So the amplitude of the spin wave corresponding to the presence of one magnon in the partMe is

(18) m~ ~,(r,) = U,ny ,/(2S~).

Hence the quantum mechanical normalization requirement [1] can be written as

fV __ tD 2 (19) - I "~1~ - Ira"le d Z r = + 1 , 2'/�94 2y�8

where the sign + , - refers to the rhp, lhp modes, respectively. The integration runs here over all the volume Vof the particle.

The absorption of the unit r.f. magnetic field (corresponding to the quantum mechanical probability of transition) will be the square of the quantity

(20) w = (mA - m . ) f t i r ) d3r.

The normalized values of m A, ma can then be readily obtained:

[ 2 (kR - sin (kR) cos (kR]] Z(SA -- S,) + 2oe' l - 1/2 (21) mA = ~Ah + œ " " ~ A ~ ; ~ + - i o 3 3

[ Z(SA- S,) + 2oe'] -I/2 2 (kR - sin (kR) cos (kR)) -~a(z~ �9 ~~ J ma = 7Bh +

12 Czech. J. Phys. B 37 [1987]

J. Kold&k: Spherical spin waves

where the sign +, - again refers to the rhp, lhp, respectively. Using it, the integral in (19) is readily evaluated yielding explicit expressions

(22) w = (m A - ma) ~}~:R 3

w = 0

w = ( - 1) j§ 4R---~3 (mA -- ma) nj2

for the uniform rhp mode,

for other rhp modes,

for the lhp modes.

The absorption w 2 for the lhp modes depends on the mode number j also through the values mA, ma which are given by (21) with k and co' taken from (12) and (16). In this way we have got two infinite sets of solution; of course, only those fulfilling the condition (9) should be taken into account.

According to i t , besides one ferrimagnetic resonance line in the finite sphere it is possible to expect a number of resonances excited by the left hand polarized uniform magnetic field with energy larger than the energy of the exchange resonance observed in the infinite crystal. The intensity of these lines decreases with increasing energy.

It is instructive to see what happens, if the diameter of the particle increases to infinity. For arbitrarily chosen j the corresponding k (given by kR = in) tends to zero and, consequently, the (positive) energy of the lhp modes decreases to

(23) co '= Z(SA -- Sa)

which is the exchange resonance energy in an infinite crystal. The proportion mA/m 8 goes to 7A, rB and the total absorption related to unit volume is

2

(24) ̀ ~ =(VA-- V,) 2h 2 , SAS , for R ~ oe ./= i S A - S � 8 7

in accordance with the well known formulas for the exchange resonance in an infinite crystal.

For comparison of this phenomenological theory with quantum mechanical results equations (8, 12, 16, 21, 22) are used. Figure 1 (see plate I, p. 128a) displays some spin waves in the particle with the diameter of 42 lattice constants obtained by the numerical diagonalization of hamiltonian (see [2]) along with the corresponding analytical functions. Here, the values 5/2, 1, 2, 2.3 and 6 were used for SA, Sa, #A, ™ and z (the same as in [2]; note that VA,a = gA,a//a/h) and as a unit on the y-axis a precession angle of spins is chose, when only one uniform (ferrimagnetic resonance) mode is excited in the particle.

Taking into account the quadratic scale on the x-axis it is clear that the continuum approximation (in the figure represented by lines mostly overlapped by the points) is fully applicable for the cote spins. The precession angle of spins in the surface layer (about 3 lattice constants only) seems to be rather random; it is obviously due to the fact that the particles do not have spherical symmetry and so there is a great

Czech. J. Phys. B 37 [1987] 13


Table 1

Radius of the particle (R): 21 lattice constants Number of spins: 38 911

Type

Numerical This theory diagonalization

~2 ~2 k-number Energy Integration Summation Energy ~2

lhp n/R 9"07 10-08 9.21 9"09 5"79 2 n/R 9"29 3-27 2"28 3-32 1-45 3 n/R 9"63 2"05 1"02 9"65 0"61

rhp 1"43 n/R 0"15 0 0"00 0"15 0"00 2"46 rc/R 0"43 0 0"00 0"42 0"00 3"47 rc/R 0"82 0 0"00 0"79 0"00

variety of possible surroundings for the spins near the surface. Despite it can be clearly seen that for the rhp modes the surface spins are unpinned, and for the lhp modes there is a thin layer in which most of the spins are pinned (they are "off resonance").

Also the energies of the spin waves are closely approximated by the continuum approximation, as it can be seen from table 1. On the other hand, the absorption w 2, calculated according to (22) with normalization to the volume of 100a 3 (listed in the 4th column of the table 1), gives considerably larger values than those of the transition probabilities obtained by the numerical diagonalization (last column). This discrepancy seems to be partly due to the fact that in the approximation above the summation is substituted by integration. Another way of calculating w 2 is to use the continuum approximation only for estimation of the eigenvectors and, instead of integration (leading to formula (22), to sure over the lattice points (the same formula as in [2]). These values (listed in column 5) are somewhat smaller but, there remains some discrepancy which is obviously due to the contribution of the outer spins. The reason is that this simple theory does hot take into account the surface modes which contribute to the absorption, as was shown in [2-1.

CONCLUSION

A single exchange resonance mode with energy E~x is present in bulk material. In a very small ferrimagnetic particle the exchange resonance should split into a number of modes with energy larger than Eex. The energy of these lines can be predicted by the above-mentioned theory. With increasing diameter of the particle

] 4 Czech. J. Phys. B 37 [1987~


ail these energies shift towards Eex so that the resonance lines merge and a single slightly broadened line should be observed. The intensity of the line will be lower in comparison with the bulk material value but, as the true value depends considerably on the behaviour of the surface spins which are hot described properly by the continuum approximation, the estimation based on the above formulas can be taken for qualitative only.

The author wishes to thank Dr. V. Kambersk~ for many valuable discussions which enabled him to improve the text.

Received 31. 7. 1986.

References

[1] KolhCek J.: Czech J. Phys. B 33 (1983) 92. [2] Kol~Eek J.: Czech J. Phys. B 33 (1983) 1024. [3] Lewis J. G.: Algorithms for Sparce Matrix Eigenvalue Problems. STAN-CS-77-595 (1977). [4] Kol~Cek J.: to be published. [5] Turov E. A.: in Ferromagnitnyj rezonans (ed. S. V. Vonsovskii). Moscow, 1961 (in Russian). [6] Sparks M.: Phys, Rev. 81 (1970) 3831, 3869. [7] Rado G. T., Weertman J. R.: J. Phys. & Chem. Solids 11 (1959) 315.

Czech. J, Phys. B 37 [1987] ] 5

Plate I: J. Kold(ek: Spherical ~7)in waves (p. 8)

Fig. 1. Some "long wavelength" spin waves in a spherical particle with a diameter of 42 lattice constants. Precession angle from this theory (lines) compared with the precession angle obtained by numerical diagonalization of hamiltonian (• and �9 are for A- and B-sublattices; D i s the

distance from the centre).

Czech, J. Phys, B 37 [19871 128a

spherical spin waves

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