The Magnon Excitations and Magnetic Properties of Quasi-One-Dimensional Organic Polymer Ferromagnets
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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.
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 , DMeFc TCNE , and a three-dimensional molecular ferromagnet named PiroPAN . 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 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  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  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  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  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 , 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 , 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
@ - 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 
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)12 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:
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 aabb in the product of SzS: . Introducing the standard Bogoljubov transformation into (4), the Hamiltonian becomes
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
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 , 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
5 , = - ( A , - B + [(B + A,), - 4B211/2}3
A, - 2C, + A2 + XI]] 3 8 + A1 A1 - B
1 [(B + A,) - 4B2I1I2
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
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 ) ] } .
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 Greens functions 
Gkk(w) = ((ak; 7 F k k ( w ) = (
474 GANG SU. DE-ssm Xue. FA-SI~EN LI, and MO-I.IN GE
In tcrms of the equation of motion of Greens 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. tempx...