GANG Su et al.: Magnon Excitations and Magnetic Properties of Ferromagnets 467

phys. stat. sol. (b) 166, 467 (1991)

Subject classification: 75.10; S12.1

Department of Physics, Lanzhou University ') ( a ) , and Theoretical Physics Division, Nankai Institute of Muthematics, Nankai University2) ( b )

The Magnon Excitations and Magnetic Properties of Quasi-One-Dimensional Organic Polymer Ferromagnets

BY GANG Su (a), DE-SHENG XUE (a), FA-SHEN LI (a), and Mo-LIN GE (b)

The magnon excitations and magnetic properties of a theoretical model proposed for quasi one dimensional organic polymer ferromagnets are investigated. It is found that two non-degenerate magnons exist in the system, and their excitations are analog with those ones in ferrites. The specific heat, magnetization, and zero-field susceptibility of the system are also obtained, and found that the model shows some exotic behaviors.

Es werden Magnonenanregungen und die magnetischen Eigenschaften eines fur quasi-eindimensionale organische Polymer-Ferromagnete vorgeschlagenen Modells untersucht. Es wird gefunden, dal3 zwei nichtentartete Magnonen im System existieren und daB ihre Anregungen sich in Analogie mit denen in Ferriten befinden. Die spezifische Wiirme, Magnetisierung und Nullfeld-Suszeptibilitiiten des Systems werden ebenfalls erhalten, und es wird gefunden, dab das Modell einige ungewohnliche Verhaltensweisen aufweist.

1. Introduction

In recent years, several research groups in the world have successfully synthesized several organic ferromagnets from organic molecules and hydrocarbons, such as oligomers [l], polyBIPO [2], DMeFc TCNE [3], and a three-dimensional molecular ferromagnet named PiroPAN [4]. This seems to be peculiar in the range of our common knowledge. As is well known, the magnetic order of transition metal compounds and rare earth-based metal compounds mainly originates from the interactions between the itinerant or localized 3d electrons and the itinerant s electrons and between the localized 4f electrons and the itinerant 5 s electrons, respectively. However, in organic polymer molecular ferromagnets there are few magnetic impurities, why can in these organic substances appear magnetic order? The search for the origin will therefore be a considerably interesting subject presently.

Many years ago, McConnell[5] suggested that either the Heitler-London spin exchange between the positive spin density on one radical and the negative spin density on another or the admixing of a virtual triplet excited state with the ground state for a chain of alternating radical cation donors and radical anion acceptors can stabilize the ferromagnetic coupling in molecular solids. Lately, Ovchinnikov [6] theoretically proposed that very high spin molecules with magnetic domains can exhibit ferromagnetic order, and he has thus given several organic molecule designs. Miller and Epstein [7] recently developed the McConnell model and believed that ferromagnetic exchange in molecular systems can be

') Lanzhou 730000, People's Republic of China. *) Tianjin 300071, People's Republic of China.

468 GANG Su, DE-SHEN XUE, FA-SHEN LI, and Mo-LIN GE

stabilized via admixing of the ground state with a virtual charge-transfer excited state. They stated that the extended McConnell model and its mathematical realization in the generalized Hubbard model can provide a convenient means to explore ferro-, antiferro-, and ferrimagnetic phenomena in organic molecular solids. Besides, Nasu [8] investigated a quasi-1D organic ferromagnet called m-polydiphenylcarbene (m-PDPC) by using a periodic Kondo-Hubbard model, and assumed that the correlation between n electrons in a benzene ring of the material as well as their itineracy were taken into account by the Hubbard model, and the ferromagnetic correlations produced by the unpaired n electrons and the nonbonding localized electrons at bridge sites were described by a periodic Kondo model. All these models give a rough description of the origin of magnetic order in organic materials. More recently, Ovchinnikov and Spector [4] also developed a theoretical generalization of ferromagnetic ordering in organic materials, and assumed that when the radicals of an organic molecule are arranged in a definite way, the system of radicals will show ferromagnetic order, while exchange interaction between two neighboring organic radicals is antiferromagnetic. They stated that the model is suitable for the ferromagnetic quasi-1D molecular systems mentioned above, but they did not give a further analysis and calculation of the magnetic excitations and magnetic properties of such a theoretical model. Although up till now it is difficult to completely account for the magnetic order in organic molecules, we believe that this theoretical design may include some essential, reasonable core. Meanwhile, the model is also of considerable interest from the viewpoint of purely theoretical physics, although it seems to be crude from its appearance. Therefore, it is necessary to investigate some of its intrinsic characteristics. In this paper, starting from the theoretical pattern designed by Ovchinnikov and Spector [4], we will study the magnon excitations and magnetic properties of the theoretical model proposed for a quasi 1D-organic polymer ferromagnet. In Section 2, we write down the model Hamiltonian in which we assume that the antiferromagnetic exchange is isotropic, but the ferromagnetic one is anisotropic. Besides, the magnon excitations and magnetic properties of the theoretical molecular system, and corresponding numerical results, are also given in this section. The final Section 3 gives brief discussions and conclusions.

2. The Model Hamiltonian and Formalism for a Linear Spin Wave Theory

As shown in [4], the theoretical model for a quasi-1D organic ferromagnet (for instance, polyBIPO) can be approximately constructed in the way shown in Fig. 1. It is assumed that a radical of the material has only one residual spin S, and any detailed information on such a radical is neglected, and that exchange interaction between two neighboring organic radicals is antiferromagnetic. According to this theoretical picture, the upper chain (called 1 chain) can be treated as an antiferromagnetic spin chain, and the lower one (called m chain) is naturally taken as a ferromagnetic spin chain, i.e., this model can be equivalently

0 ; Jz

Fig. 1. The arrangement of radicals in organic magnet sfor the theoretical con- sideration

I

@ - 3 ! ! L - * L - a

Magnon Excitations and Magnetic Properties of Organic Polymer Ferromagnets 469

considered as an antiferromagnetic spin chain coupled antiferromagnetically with a ferromagnetic chain. This equivalent model is also of interest in theoretical physics. According to this theoretical model, the first organic polymer ferromagnet polyBIPO has been synthesized, whose molecular structure can be presented as [4]

As can be seen, the above structure can be indeed simplified in the way of Fig. 1. We assume that every radical in Fig. 1 has a non-compensated spin S, the 1 chain with

exchange integral J , > 0 is isotropic, the m chain with interaction J 3 ,!, J 3 , > 0 is anisotropic, and the coupling J 2 > 0 between 1 and m chains is isotropic. Since there is no purely one-dimensional ferromagnet, we may conceive that there perhaps exists a weaker 3D interchain interaction, which may be ignored at low temperatures. Therefore, the model Hamiltonian reads as

2 = 2 1 + P2 + 2 3 ,

= J 1 c sl ’ sL+A 9

I , A

2 2 = ~2 C S l + a . S m + a h i m j 1,m.a

where a = 2 4 , A and a are the lattice spacings of 1 and m chains, respectively, and

[S;, Sq] = iE,,,SThkj, S; = S ( S + 1) ; k, j E 1, m chains. (2)

We can further assume that the theoretical lattice is bipartitive and can be divided into A and B sublattices. On the A(B) sublattice, the vacuum state is the s” = S(-S) state, i.e., the A sublattice describes the up-spins and the B sublattice the down-spins.

2.1. The magnon excitations

In order to investigate the magnon excitations of the system, we have to introduce the usual Holstein-Primakoff (H-P) transformation. At low temperatures, we only consider the low-lying excitation states. So we can approximately expand the Bose operators in H-P transformation as follows:

S: 2 (2S)’12 a, , S, = (2s)’” u+ , S:, = S - .+ai for a E A , (3a)

SA L (2S)’” bf , S , L (2S)1’2 b j , S i j = bf b j - S for b E B . (3b)

The operators ai and bj satisfy the usual commutation relations. Inserting (3) into (l), the Hamiltonian for the linear magnons is then written in momentum space as follows:

470

with

GANG Su, DE-SHEN XUE, FA-SHEN Lr, and Mo-LIN GE

where y k = cos k (the lattice spacing A can be taken as unity), and N is the total site number of the m chain. Above we have neglected the nonlinear terms a’ab’b in the product of SzS: . Introducing the standard Bogoljubov transformation into (4), the Hamiltonian becomes

with

B - A , 1 2 2

E2(k) = ~ + - [ ( B + A,)2 - 4C,2]”2,

where cl(k) and c2(k) are two branches of the low-lying spin wave excitation spectrum of the system. They are nondegenerate and are different from the usual ones corresponding to the antiferromagnetic case, which may illustrate from another aspect that the magnetic and thermodynamic properties of the system are similar to some kind of ferromagnetic and magnetic order should appear in our system. In general, one can expect that there exist three magnon branches from three spins per unit cell. However, we should point out that we believe that the two magnon excitations from the up-spins of the 1 chain and the up-spins of the m chain are physically identical and can merge into one when diagonalizing the Hamiltonian (l), and therefore the information of in-phase and out-of-phase collective magnetic excitations is just the combined result of the actual three magnons from three spins, which does not change the final physical substance. On the other hand, we find that the magnons of the system have zero point energies as in the antiferromagnetic case. The ground state energy of such magnons is

where

Magnon Excitations and Magnetic Properties of Organic Polymer Ferromagnets 47 1

k - - - - - c

Fig. 2. Two nondegenerate magnon excitation spectra 8 , ( k ) and RZ(k) vs. wave vector (the lattice spacing A = 1). where S = lj2. (a) J , = 0. 1 eV, J z = 0.05 eV, JSL = 0.08 eV, J , , , = 0.09 eV; (b) 0.1, 0.1, 0.05, 0.05; (c) 0.1, 0.1, 0.1, 0.05

Particularly, when k = 0 and J , , , = J31, then J. = 1. Equation (9) shows that there exists an orientation deviation of spins on each sublattice in the ground state. For S = 1/2, J , = 0.1 eV, J , = 0.05 eV, J , = 0.08 eV, and J, , , = 0.09 eV, we have J. = 0.085, then Eground = -0.1989N. From the two aspects of the system having zero point energy and nondegenerate two-branch magnon excitation spectrum, we find that the behavior of the system is analogous to a ferrimagnet, and therefore some magnetic features may be similar to those of ferrites. For some sets of parameter values, the schematic plots of excitation spectra versus wave vector ( A = 1) are shown in Fig. 2. We see that neither of the two magnon excitation spectra shows a sound-wave-like dispersion in the limit of long wavelengths, which means that the magnon excitations of the system are not antiferromagnetic ones. Instead, ( k ) and c 2 ( k ) possess some characteristics of ferromagnetic magnons.

472 GANG Su, DE-SHEN XUE, FA-SHEN LI, and Mo-LIN GE

T ( K ) - Fig. 3. The specific heat vs. temperature ( k , = l), where S = 112. (a) J , = 0.1 eV, J , = 0.1 eV, J31 = 0.05 eV. J , , , = 0.05 eV; (b) 0.1, 0.05, 0.08, 0.09; (c) 0.1, 0.1, 0.01, 0.05

2.2 The thermodynamic properties of the magnons

At temperature T, the internal energy of the magnons is

E ( T ) = C ( d g , > r ~ ~ ( k ) + C ( P : B A - E A ~ ) , k

where ( O ) , = Tr [ O exp (-&’/T)]/Tr Iexp (-%/T)], and the Boltzmann constant k , is taken as unity. Through straightforward calculation we get

Consequently, the specific heat of the magnons per site at low temperatures can be obtained from

T 2 l 1 N k i [exp ( q ( k ) / T ) - 11’ [exp (&*(k)/T) - 11’

For given parameter values, the numerical results of C ( T ) are shown in Fig. 3. We see that the specific heat of the system approaches zero at very low temperatures. Particularly, a specific heat jump occurs in the case of isotropic ferromagnetic coupling (5, ,, = J,,), which illustrates that a phase transition may appear. However, it is experimentally shown that a quasi-one-dimensional organic ferromagnet is highly anisotropic [2], and thus this property of C ( T ) cannot occur in such organic materials, In the long-wavelength limit, assuming cl, E~ 9 T , (13) can be straightforwardly calculated by changing the summation into an integral,

(13) 1. G ( k ) exp (El(k)/T) + E m exp (EZ(k)/T) C ( T ) = - -c

blagnon Excitations and Magnetic Properties of Organic Polymer Ferromagnets

where

1 2

5 , = - ( A , - B + [(B + A,), - 4B211/2}3

A, - 2C, + A2 + XI]] 3 8 + A1 A1 - B

1 [(B + A,)’ - 4B2I1I2

473

1 [(B + A,), - 4B2]’/2 [A2 - 2C + A, + 2C1]} 3B + A , A1 - B

We should point out that (14) is only valid for the case of anisotropic ferromagnetic coupling, i.e., J , , , =+ J3L. The reason is that when J 3 , , = J, , the integral has a singularity at k = 0, and the result of (14) is naturally invalid for such a case. From the above, we observe that the behavior of the specific heat for our system at low temperatures is very different from the usual, purely antiferromagnetic or ferromagnetic, spin wave theory.

2.3 The magnetization at low temperatures

In the usual spin wave theory, the magnetization of sublattice A (all up-spins) is given by

(15)

and the magnetization of sublattice B has the following form:

1 MB(T) = NgpB (GT ( b : b k ) T - s, 9

M ( T ) = N g f i B - -1 ( ( a : a k > T - < b L b k ) T ) ] } .

(16)

where g is the Lande factor and pB the Bohr magneton. From the theoretical design of the system, therefore, the total magnetization of the system can be obtained as

(17) i [ i k

Now, if we know the thermodynamic expectation values (bk+bk)T and ( a t a , ) , , then M ( T ) can be easily calculated. So we define the double-time Green’s functions [9]

Gkk’(w) = ((ak; 7 F k k ’ ( w ) = (<bk; bL))o. (18)

474 GANG SU. DE-ssm Xue. FA-SI~EN LI, and MO-I.IN GE

In tcrms of the equation of motion of Green’s function, it is not hard to derive

Obviously, Eii’ and 4:’ are alternative rnagnon excitation spectra of the system. From (19) and (20), we get the thermodynamic averages

By means of (171, (21). and (221, we can therefore determine the total magnetization of thc system. For somc given parametcr values, the reduced magnetization m(T) =i M(?’)INgp,, versus temperature is shown in Fig. 4. where the summation over k is limited lo the first

Fig. 4. The reduwd magnetization m(T) = M(T)/Ngp, vs. tempxaturc, where S = 1/2. (a) J , = 0.01 eV. J t = 0.005 cY. J,, = 0.008cV. .I,, Z= 0 . M eV; (b)fi.O\, U.O1.0.~1.O.OOS~ (c)O.OI, 0.01. 0.005, 0.005

Mognon Excitations and Magnetic Properties of Organic Polymer Ferromagnets 415

Brillouin zone of the lattice. We note that the magnetization of the system varies strongly with temperature. The Curie temperatures for cases a, b, and c (see Fig. 4 ) are 291, 445, and 512K, respectively. From the parameter values used for the above estimation, we find that the integrals affect the Curie temperature drastically, which is also consistent with physical considerations. Besides, the temperature dependence of magnetization is qualitativ- ely in agreement with the experimental results for polyBIPO given by Cao et al. [2], if one assumes that the magnetic order of the molecule comes from two magnetic phases in the whole temperature range. Hence, it may be concluded that there indeed exists spontaneous magnetization in the theoretical quasi-1D ferromagnet, which differs completely from the usual, pure 1D ferromagnet. The reason for this point will be discussed in Section 3.

2.4 The zero-field susceptibility

When the magnetic field H (along the z-axis) is applied to the system, the field dependent Hamiltonian, after performing the H-P transformation, will be

Ck = C k .

The corresponding field dependent magnetization reads

M ( H ) = NgpB{S - - n b ( H ) l )

with

l 7 A ~ ’ ( H ) A L ~ ) ’ ( H ) + exp [E l i ) ’ (H) /T] - 1 exp [El:)’(H)/T] - 1

(241

B - A ; 1

2 - 2 pi)’. ( 2 ) ‘ - + - [(B’ + A;)2 - 4C,2]”2.

476 GANG Su, DE-SHEN XUE, FA-SHEN LI, and Mo-LIN GE

Fig. 5. The zero-field susceptibility ~ ( 0 , T ) vs. temperature (kB = I), where S = 1/2. (a) J , = 0.1 eV, J , = 0.05eV, J, , = 0.08eV,J3,, = 0.09eV; (b)0.1,0.1,0.05,0.05; (c)O.l,O.l, 0.1,0.05

From (24) to (26), we can obtain numerically the field dependence of the magnetization at finite temperatures. In addition, the field dependence of susceptibility x ( H ) can be derived from

From (27), we can also easily obtain the susceptibility of the system versus the applied field through a numerical computation. However, these properties will be presented elsewhere.

In the case of zero field, (27) is reduced to

~ ( 0 , T ) is just the zero-field susceptibility, which versus temperature is shown in Fig. 5, where the same parameter values are taken as for the specific heat. We notice that the behavior of ~ ( 0 , 7') is very peculiar. At lower temperatures, ~ ( 0 , T ) varies slowly with temperature, and nearly approaches a constant. On the contrary, at high temperatures, ~ ( 0 , T ) increases with temperature. In the isotropic case (curve b), ~ ( 0 , T ) depends on temperature more strongly than in the anisotropic cases (curves a and c). Whether this kind of strange behavior of the zero-field susceptibility is suitable for a quasi-1D ferromagnetic molecular system, is perhaps to be adjudicated by experimental results.

3. Discussions and Conclusions

As is well known, there is no pure 1D ferromagnet due to thermal fluctuations of magnons destroying ferromagnetic order. Mathematically, if one calculates the integral (17) to obtain the spontaneous magnetization in the pure one-dimensional case, the integral will diverge. However, in our case of antiferromagnetic coupling between a ferromagnetic spin chain and an antiferromagnetic one, it is possible that a residual moment remains after destroying of thermal excitation of magnons at low temperatures. In addition, the integral (17) is not

Magnon Excitations and Magnetic Properties of Organic Polymer Ferromagnets 477

divergent due to the appearance of the coefficients Bj'), Biz), Ail), and Ai2). This point differs fully from the pure 1D case, which can also be intuitively examined using (21) and (22). Meanwhile, quasi-1D organic ferromagnets [l, 2,4] have indeed been obtained in exper- iments. Therefore, it is reasonable to expect the spontaneous magnetization in the quasi-1D ferromagnets, although we have neglected the weaker interchain interaction.

Above we have investigated the magnon excitations and magnetic properties of a theoretical model proposed for quasi-one-dimensional organic polymer ferromagnets. We believe that the theoretical design of [4] is available, and it is meaningful to investigate some of its characteristics. As a consequence, we find that the system possesses two non-degenerate magnons whose excitation behavior is similar to ferromagnetic ones but not the same. Besides, the magnons have zero-point energies, as in the antiferromagnetic case. Thus the magnetic excitations of the system approach ferrimagnetic ones, i.e., the physical system constructed in the way shown in Fig. 1 will exhibit some properties of ferrites. This point is verified by calculating the magnetic and thermodynamic properties of the model. Specific heat, magnetization, and zero-field susceptibility behave in some unusual way. For instance, a finite jump of the specific heat at low temperatures occurs in the case of isotropic ferromagnetic interaction ( J , ,, = J J (Fig. 3), the Curie temperature in the isotropic case is larger than in the anisotropic case, the magnetization varies almost linearly with temperature (see Fig. 4), the zero-field susceptibility varies with temperature very slowly at low temperatures (Fig. 5), etc. Whether all of these characteristics of the theoretical system are available for quasi-1D organic polymer ferromagnets, will be examined by future experiments. Although we have investigated the linear spin-wave excitations of the model, it is possible that the nonlinear effects of spin wave excitations will affect the magnetic behavior of the system. This is left for future study. We should point out that due to the lack of enough experimental data on the parameters of organic magnets, we are forced to select freely and conveniently three sets of values of parameters for calculating the curves a, b, and c in Fig. 2 to 5, and these values are not found at present to have any relation to experimental data. Therefore, the theoretical results obtained above cannot be quantitatively compared with the corresponding experiments in order to check whether the theoretical model is reasonable. Up to now the above consequences can be used to interpret qualitatively some experimental phenomena in polymer ferromagnets.

Acknowledgements

We would like to thank Magnetism Laboratory, Chinese Academy of Sciences for partially financial support. We also thank the referee for the careful reading the manuscript and helpful comments.

References

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478 GANG Su et al.: Magnon Excitations and Magnetic Properties of Ferromagnets

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(Received August 17, 1990; in revisedforrn April 11, 1991)