XIAO-FENG PANG: Collective Excitation in Quasi- 1 D Antiferromagnets 237

phys. stat. sol. (b) 180, 237 (1993)

Subject classification: 71.45 and 75.10

International Centre for Material Physics, Academia Sinica, Shenyang, Chinese Centre of Advanced Science and Technology (World Laboratory), Beijing, and Department of Physics, Southwest Institute for Nationalities, Chengdu ')

Properties of a Collective Excitation in Quasi-One-Dimensional Antiferromagnets

BY XIAO-FENG PANG

Contributions of magnon-phonon coupling and magnon-magnon interactions to the formation of localized solitons in common quasi-one-dimensional antiferromagnets are closely studied and discussed by using Makhankov's and Pang's method with double-sublattice model. The anomalous effect in the (CH,),NMnCl, system is also discussed by using these results.

1. Introduction

The collective excitation and the motion of solitons caused by electron-phonon interactions and some other types of interaction in quasi-one-dimensional Heisenberg ferromagnetic systems have been studied and some interesting results are obtained [l, 21. Little attention has been paid, however, to similar problems in antiferromagnetic systems due to their complexity. The characteristics of the collective excitation in such systems, which are quite different from those in ferromagnetic systems, should be studied. We present in this paper our study on the collective excitation in ordinary anisotropic Heisenberg antiferromagnets with magnon-phonon and magnon-magnon interactions by means of a new method and a new wave function of the collective excitation.

2. Collective Excitation with Magnon-Phonon Interaction

If the double-sublattice model is adopted, the Hamiltonian of the Heisenberg antiferromagnet studied here can be expressed as

where T and I, are the kinetic and potential energies of lattice oscillations, respectively,

I ) Chengdu 610041, People's Republic of China.

238 XIAO-FENG PANG

with m being a spin “mass”, a, the lattice constant, and c, the sound velocity in the crystal which we set equal to unity in later calculations. Sftj, (k = x, y , z ) is the spin component at the i ( j ) site in k-direction.

Making the transformation: SGj) = (ST(j) + we change (1) into

A A

H = T + V + f C C [Jii+aSTSf+s + f (ti if6 + qi i+J (S+S;+, + S;S+++) i 6

+ ( t i i + a - qi i+a) (s+s~++, + s;S;+a)I

We again use the Dyson-Maleev representation of the spin operators by means of Bose creation and annihilation operators [l, 3, 41

and

where a+, a and b+, b are the creation and annihilation operators of the Heisenberg magnon field on the two sublattices, respectively. Substituting (3) into (2) and taking into account the symmetry of sublattices A and B and the distribution character such that the B sublattice neighbours the A sublattice, etc., we can write the Hamiltonian (2) approximately as

x (a+aiaib:+, + uib:+abi+Sbi+6 + a+ai+,a+ai + b:+abi+a~:bi+s)

l A A 8 i ~

- -CC (tii.6 + v i i + J

x (a’aiaibi+, + aibi+6b++6bi+6 + a;bi+,a‘ai + aib:+6b:+6bi+6), (4)

where the last four terms are anomalous terms resulting from magnon-magnon interactions. They can be neglected when the interactions are weak. In this case the above Hamiltonian of the system becomes quite simple which will be considered first in the following.

Properties of a Collective Excitation in Quasi-1D Antiferrornagnets 239

We here can apply the methods of Makhankov and Fedyanin [5] and Pang [6 to 91 to study the properties of collective excitations of one-dimensional antiferromagnetic system. In Heisenberg representation, the equations of the operators af and b, of sublattices A and B can, after some proper transformations, be written as

ihkf = bf, HI = s c J f f + # f + t s c ( 5 f f + S - Y f f + a ) bf+6

+ t s c ( 4 f f + 6 + Yff+a) b;+a 2 ( 5 a) 6

ihdf = [b f , HI = s c J f f , 6bf + t s c ( S f f + b - Y f f + 6 ) af+a 6 6

Now we suppose that the trial wave function of the collective excitation state of the quasi-particles of the system have the form of our wave function [6 to 91, i.e.,

where 10) is the vacuum state (ground state), clai and clbj are expansion coefficients related with the characteristics of the quasi-particles which obviously are functions of time and space, and i is a normalization constant.

= abf/A2 are the Schrodinger probability amplitudes. Using (6) we have from (5) Notice that (y(l)l af Iv(t)> = c(af/i2, (w(t)l b f IY(t)> = clbf/A2, and (Paf = claf/i2, (Pbf

ih@bf(t) = c J f f + 6 ( P b f ( t ) f s c ( 5 f f + 6 - ' l f f + s ) ( P a f + a ( t ) 6

+ t s c ( t f f + a + Y f f + 6 ) (P,*r+a(t). (7 b) 6

If the oscillation amplitude of the lattice is small and the magnon-phonon interaction is also weak, we may proceed to the continuum limit for the coefficients and the probability amplitudes. Dropping the terms with higher derivatives, as a result, we get

240 XIAO-FENG PANG

where a, is the average distance between neighbouring sites. Using (8) and bearing in mind the symmetry of the sublattices, we can easily derived the approximate equations of motion for (Paf(t) and (Pbf(t) as

We define qs(t) = q a f ( t ) + qbf(t); then we can obtain the following equation:

Now we shall consider the part representing the lattice oscillations. We have from (4)

A A

f $ 1 1 (ti,+,, - V i i + d ) ( (Pai(Pt j+d + Pa*i(Pbj+a) ' (10) i 6

It should be noted that with the wave function in the form (6) the anomalous terms in

Here we use the classical Hamilton equation the original Hamiltonian have been removed.

where M is the mass of a lattice point (atom, for example), us its displacement, a classical quantity. Having in mind the symmetry of sublattices A and B and the fact

Properties of a Collective Excitation in Quasi- 1 D Antiferrornagnets 24 1

that the neighbours of sublattice A belong to sublattice B and vice versa, and using again

A A A c c J i i + a I q a i I 2 = ( 2 J 0 - J l ( u b i + 1 - u b i - l ) $ Iqai1' > i d i

B B B c c J j j + a I q b j I 2 = c {2J0 - J 1 ( u b j + 1 - u b j - 1)) 1qb j I2 7

j a j

we can obtain from (10) and (1 1)

2 -M'af = K [ 2 u a f - u b f + l - 'bf-11 + J1s [ lqb f+112 - I q b f - 1 1 1

+ 4 '(r1 - rl) [qa f (q : f+ l - q,*,-l) f q a * f ( ' P b f + l - (Pbf - l ) ] >

2 -Mcbf = K[2ubf - u a f + l - u a f - l l + JIS[lCpaf+lI - I'Paf-1121

+ f S ( t l - Y1)[(Pbf((Pi$+I - q , * f - l ) + ( P z f ( q a f + l - q a f - 1 ) 1 ,

where K is the force coefficient, K = rnci/2a0. Defining uf = uaf + ubf and taking into account that Ma = Mb = M, the symmetry of

the sublattices, etc., we have no difficulty in obtaining the following equation in continuum approximation:

Equations (9) and (12) form a complete set of equations for the collective excitations in a Heisenberg antiferromagnetic system with magnon-phonon interactions. Now we proceed to find out the solutions to the equations.

In the case of a quasi-steady state we can assume that

Substituting (13) into (12), we have

16 physica (b) 180/1

242

We can get by solving the above equation

XIAO-FENG PANG

where C is an integration constant to be determined from boundary conditions. Substituting (15) into (9) and solving it we can then immediately find out the characteristics of the collective excitation caused by magnon-phonon coupling in an antiferromagnet.

We can prove that the localized soliton cannot be formed in an isotropic antiferromagnet if only magnon-phonon interactions exist, as is in the case for a ferromagnet.

For simplicity, we consider here only the anisotropic antiferromagnet with 5 = q (the others can, of course, be discussed in the same way). In this case, the situations J > 5 and J < 5 correspond to the easy magnetic axis (Oz) and the easy magnetic plane (xOy) in the antiferromagnet, respectively. Equation (9) in this case also reduces to

Its conjugate equation reads

where we have dropped the subscript of cp and u. Now we make a further transformation.

cp+ = cp + cp* , cp- = cp - cp* .

Then we have from (16) and (17)

a 2 au ax ax a 2 au

ih@- = 2 ( J , + 50) Scp+ + sa;t, 7 cp+ - Wl + 5,) Sa, - cp+ 1

ih@+ = 2(5, - 5,) Scp- - sa;tO 7 cp- - 2 ( J , - 5,) Sa, - cp- ax ax

In the case of V 4 c, or V z < Ka;/M, we can derived the non-linear equations of motion for cp (x, t ) as follows:

a2 au ax ax

2 2 2 a2 au ax2 ax

2 2 2 - h 2 @ + = 4S2(J; - 5;) (P+ - 4t0S a, 7 cp 1 - 8a0S2(J1J, - 5150) - cp+ , (18)

-h2@- = 4S2(J; - 5;) cp- - 4<,S a, - cp- - S U , S ~ ( J , J , - tito) ~ cp- . (19)

Adding (18) to (19) and then substituting (15) into it, we get

Properties of a Collective Excitation in Quasi-1 D Antiferrornagnets 243

where

It can be seen from (21) that C, = 0 and B, = 0 because J , = 5, and J , = t1 in the isotropic antiferromagnet. Thus there exists no soliton solution to (19), thereby proving the above conclusion.

The situation becomes different in the case of an anisotropic antiferromagnet. For instance, we can prove that for the antiferromagnet magnetized along the z-direction, the magnon-phonon coupling in this direction plays an important part in forming localized solitons.

We proceed from (20) by assuming [lo, 111

(22) cp = f ( x - V t ) ei(K’x-m*)

Substituting this into (20), we have [lo, 111

d f d@’

K’A, = V W , (V’ - A, ) - + ( B , + A,K” - ~ ’ ) f - C0f3 = 0 .

It becomes after integration

where we have set the integration constant equal to zero because of the boundary conditions. Integrating once more we arrive at

- 112

~ v; - f 2 ) ] df + const, - v !)

where

In the case of - w 2 < (AoB,)/(Ao - V’)), there are non-topological soliton solutions to (22) if the requirements C, / (A , - V’) > 0 and f ; > 0 are satisfied, i.e., if J , > 5 , and at same time either a) J,J, > tot1, V < min [I/KIM a,, 2t0Sao/hl or b) J o J l < 5051, 25,Sa,/h < V < I/KIM a,. The normalized solitary wave is then

cp = l/& sech [L;’(x - Vt)] exp i{K’x - ot} ,

244 XIAO-FENG PANG

- (d2/16) A,C$(A, - V2)2 . A,& ( A , - V 2 )

where K’ = V/o/A,, L, = 4(A, - V2)/C,a,, o2 = ~

If V 2 - A , > 0, i.e., C , / ( A , - V 2 ) < 0, (23) has the topological soliton solution

cp = l,/& tanh [L;’(x - Vt)] exp [i(K’x - ot)] .

It can be seen from the above conditions that only for the usual easy magnetic axis antiferromagnet, the localized soliton can be excited by the magnon-phonon coupling. This is a new result: no parallel has been obtained in the ferromagnet. In our case, the coupling of the longitudinal oscillations of the lattice with the magnon results in a remarkable change of the value of the transverse exchange integral of the antiferromagnet. It is the non-linear interaction caused by the coupling that is vital to the formation of the soliton. In this case the velocity of the soliton excited satisfies V < min [[/KIM ao, 2[,Sa,/h].

3. Collective Excitation Caused by Magnon-Magnon Interaction

So far we considered only the collective excitation caused by magnon-phonon interaction. In fact, when magnon-magnon interactions in the system become too strong to be not neglected, a new non-linear interaction source results which will contribute to the collective excitation. Obviously, the formation process and the properties of the collective excitation will alter accordingly, which should also be studied.

The Hamiltonian of the system in the case can still be expressed by (4), but the interactions between the magnons in the formula now represent the direct interactions between neighbouring magnons and the effects of a magnon upon the transfer of other magnons and upon the magnon “resonance”, etc., which are all two-magnon effects.

We still assume (6) to be the trial wave function of the collective excitation of the quasi-particles of the system. Meanwhile we adopt the method of quasi-average field approximations [12] when considering the effects of the anomalous correlation terms upon the soliton formation and the quasi-particle energy (the validity has been proved already for ferromagnets), that is

a;aiaib,r+, = (a: Ui) sib;+, + (sib;+,) .;ai - (a+ai) (aib;&), u+uiu+u++6 = (U+Ui) a+aif+b,

(26)

. . . . . . . . . . . . . . . . . .

Then the Hamiltonian of the system (4) becomes

Properties of a Collective Excitation in Quasi-1D Antiferrornagnets 245

where

From the second-quantized Hamiltonian, (27), and in the same way as in the last section, with the aid of Makhankov [5] and Pang's [6 to 91 method we obtained the apparent equations of motion for the operators a, and b, in Heisenberg representation,

abf = [b,, HI . ih - aa, at at ih - = [a,, HI,

Then, using the Schrodinger probability amplitude defined by (6),

'Pa/(t) = (V(t)I a/ IV(t)> = @a,(t)/i2 9 (Pb/(l) = <V(t)I b/ IV(t)> = a b f ( t ) / i 2 *

We can finally derive the non-linear equations for cpa,-(c) and (Pb/(t)r

+ - S (5,,+6 + 9,.r+b) & + a 2 6

+ lqb/+61z) (Db/+d + b a f I z p:f+d + Iqaf12 q b , + 6 l

-xT ( t J / + 6 + q/f+6) (baf1' + Iqbf+61z) &+6- (29) 1

Interchanging the symbols a and b in (29), we can get the corresponding equation for qb,(t). We then proceed to the continuum limit as before, dropping the terms with higher than third derivatives and taking into account the symmetry of sublattices A and B and the distribution characteristics mentioned above to approximate cp,(t) = cpa,(t) + (Pb/(t) as

ih@ = 2J0Sq + ( t o + rto) Scp* + s(t0 - 90) cp

1 1 250 + - ( to - 90) I d 2 cp - V ( t 0 + 90) Id2 q* 9 -[ 8

246 XIAO-FENG PANG

where the subscript of the functions in (30) has been dropped and a parameter v has been introduced which equals & in the case studied. Equation (30) differs from (9) only by an additional non-linear term resulting from the interactions between magnons. If higher non-linear terms and dispersion effects are neglected, there will be no correlation between magnon-magnon interactions and magnon-phonon interactions and they become two independent non-linear interaction sources. Meanwhile, the equation of the lattice oscilla- tions is still expressed by (11). Therefore, we can solve problems of this type by combining (15) with (30).

It can be proved that there is still no soliton solution for isotropic antiferromagnets even if the interactions between magnons are taken into account. In the case of isotropy, (30) becomes

a2 au ax ax ik@ = 2J,S(cp + cp*) + J,Sai cp* - 2J,a,S ~ (cp + cp*)

- 25, Id2 (cp + "*).

Its conjugated equation reads

az au ax2 ax -ik@* = 2J0S(cp* + cp) + J,Sa; - cp - 2J1u0S - (cp* + cp)

- 2Jo Id2 (cp* + V c p )

Introducing cp* = cp f 'p* and in the same way as applied to (18) to (20) we obtain

Comparing this with (20), it can be seen that now B , = 0 and only non-linear terms exist, and therefore soliton solutions of type similar to (23) cannot be found for this equation [lo, 111, thereby completing the proof. It can then be concluded that in the case of isotropic Heisenberg antiferromagnetic chains, neither magnon-phonon interactions nor interactions between magnons can not result in a collective excitation of the type of non-topological solitons, which is somehow similar to the situation in isotropic ferromagnetic chains [l].

For anisotropic antiferromagnets we also limit our discussion to the cases of easy magnetic axis (Oz) and easy magnetic plane (xOy) where (30) reduces to

a2 ik@ = 2J0Scp + 25,Scp* + <oS&,cp* -

ax au ax 2J,Suo -- cp

Again introducing cp+ = cp f cp* and adopting the same method applied to (18) to (20), we arrive at the equation for cp(x, t )

- 8 ( J 2 - ~'5;) S IcpI2 cp . (33)

Properties of a Collective Excitation in Quasi-1D Antiferromagnets 247

Combining it with (15), we have an equation quite similar to (20), i.e.,

a 2

ax @ - A 7 ~p + B c ~ - g (c~ I ’ ~p = 0 ,

where

A = A , = 45,S2&h2,

B = B, = [4S2(Ji - 5;)/h2] + 8(J,Jl - 5051) Sa,C,,

g = c, + 8 s ( J ; - v2{;)/hZ.

(34)

(35)

Therefore, we can find soliton solutions of the type (23) and (25) in this case. Simply replacing C,, A,, and B , in (23) and (25) by g, A , and B, we have the corresponding solutions which indicates that taking into account magnon-magnon interactions only changes the amplitude and velocity of the soliton and does not alter the fundamental type of the collective excitation. The magnon-magnon interactions enhance the effects of the non-linear interac- tions, thereby being helpful in the formation of more stable solitons. The reason lies in that g is always greater than C, when the existence of solitons is permitted. If we further notice that J , 9 J , and 5, 9 we can see that when there are magnon-magnon interactions, primarily they are only the soliton solution of the type (23) or (25) in the first-type velocity range. In this case

A B d 2 A - [g + 8 S ( J i - &)I2 . _ _ _ _ _ _ _ _ _ _ ~ AB d2Ag2

w 2 = - - A - V 2 16(A - V2)’ A - V 2 16 ( A - V 2 )

It should be noted that though magnon-phonon coupling and magnon-magnon interactions have the same effects on the formation of solitons, as can be seen from the above discussion, there are differences between the two types of interactions. In the first place, the mechanism of non-linear localized collective excitation caused by the first type is the breakdown of the kinetic symmetry, that is, it is caused by the interactions between magnon and lattice oscillation systems when in the steady state the soliton and the localized deformation depending on lattice oscillations propagate together with the same speed along the antiferromagnetic chain. The mechanism of excitation caused by the second type is the spontaneous breakdown of the symmetry brought about by the magnon-magnon interac- tions in the single-axis anisotropic antiferromagnet. However, the collective excitations resulting from both mechanisms have the same characteristics, i.e., the structural anisotropy, as its preliminary and peculiar condition. As said above, in the isotropic antiferromagnetic chains the two mechanisms may cancel each other and no soliton can exist as in the case of ferromagnetic chains. It also indicates that the excitation of solitons in the system is determined by the anisotropy of the system. Once anisotropy is brought about due to one or another reason, the magnon-phonon coupling and the non-linear interactions between magnons will make the magnons “self-trapping” in a range of the dimension 2L, in the one-dimensional chains and a stable soliton is excited. When the anisotropy changes, so do the amplitude, the energy, the momentum, and the number of solitons.

It can also be concluded that magic and interesting physical phenomena will come into play in antiferromagnets because of the formation of solitons resulting from the interactions in the anisotropic antiferromagnetic chain. Peculiar anomalous phenomena have indeed

248 XIAO-FENG PANG: Collective Excitation in Quasi-1 D Antiferromagnets

been observed in experiments and attempts have been made to explain then using the magnetic soliton model, though no theoretical expression for the soliton as (23) and (25) has been obtained [13, 141. For example, though no theoretical expression for magnetic solitons was available, Boucher et al. [14] recently used the soliton model to explain the phenomenon of nuclear spin-lattice relaxation (NSLR) in antiferromagnetic chains (CH,),NMnCl,. They obtained through measurement that the ratio T,' of NSLR of N15 in the antiferromagnet is a function of external field H (20 kA m-' < H < 80 k Am-') and temperature T (2 K 5 T 5 4.2 K) and observed that T;' diverged exponentially with HIT at a certain given temperature. With the theoretical results for the soliton we can now fully explain the excitation and the behaviour of the soliton in this kind of system, which in turn verifies the correctness of our theory.

It should also be pointed out that here we have only considered the case with no external fields. If any such field is present and if it is in the direction along the easy magnetic axis, its effect, due to the opposite magnetization directions of the double sublattice, will be equivalent to a periodic external field of the period 2a, which will strengthen the discreteness of the lattice making the continuum approximation no longer valid. However, if the direction of the external field is perpendicular to the antiferromagnetic spin direction, the continuum approximation will remain valid. Because of this reason, earlier experimental and theoretical studies mainly were concentrated on transverse fields, and not on longitudinal fields.

Acknowledgement

This work was supported by ICMP No. 91007.

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(Received February 25, 1993; in revised form August 23, 1993)

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