Pergamon Solid State Communications, Vol. 95, No. 1. 57-61. pp. 1995

Elsevier Science Ltd Printed in Great Britain.

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STUDIES ON THE FERROMAGNETIC STATE IN QUASI-ONE-DIMENSIONAL ORGANIC POLYMER FERROMAGNETS WITH COMPETITION BETWEEN ELECTRON-ELECTRON

INTERACTION AND ELECTRON-PHONON COUPLING*

Z. Fang and Z.L. Liu

National Laboratory of Laser Technology and Department of Physics, Huazhong University of Science and Technology, Wuhan. 430074, P.R. China

K.L. Yao

International Center of Material Physics, Academy Sinica, Shenyang, 110015, P.R. China

and

Z.G. Li

National Laboratory of Laser Technology, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China

(Received 9 September 1994 by A.H. MacDonald)

The possible ferromagnetic ground state of a theoretical model proposed for quasi-one-dimensional organic polymer ferromagnets is investigated in detail. Within the self-consistent-field Hartree-Fock approximation, allowing for full lattice relaxation, the system is studied self-consistently. We find a critical interaction, which is determined by the competition between the electron-electron interaction and the electron-phonon coupling. When electron-electron interaction is larger than the critical value, the ground state of the system is a stable ferromagnetic state. It is also found that in this stable ferromag- netic state the r-electrons SDW will become very strong, and almost zero dimerization happens for the main chain.

1. INTRODUCTION

IN RECENT YEARS, scientists have begun to design and synthesize a new class of ferromagnetic materials based on molecular rather than metallic or ionic lattices [l, 21. Several research groups in the world have successfully synthesized several quasi-one- dimensional organic ferromagnetics, such as poly- BIPO [3], m-PDPC [4] and p-NPNN [5]. The search for organic ferromagnets has become a challenge that has attracted considerable attention.

McConnell [6] first proposed to produce inter- molecular ferromagnetic interactions in organic molecules in 1963. Mataga [7] and Ovchinnikov [8] reported another strategy to prepare organic ferro-

* This work is supported by the National Natural Science Foundation of China.

magnets based on intramolecular ferromagnetic inter- actions in very large molecules. And this kind of alternant hydrocarbon, named poly-BIPO, has been indeed synthesized by A.A. Ovchinnikov et al. [9] and Y. Cao et al. [lo]. They successfully attached a kind of free-radical, which contained unpaired electrons, as side groups to the carbon backbone, schematically shown in Fig. 1. They found that this kind of quasi- one-dimensional polymer shows ferromagnetic properties at room temperature. However, about this kind of quasi-1D organic polymer ferromagnet, many things are still unclear, and at the present stage of the synthesis, only a few species show ferromag- netic properties. Further studies are encouraged.

As is well known, quasi-one-dimensional systems, like conducting polymers, charge-transfer solids (e.g., TTF-TCNQ), and halogen-bridged mixed- valence, transition-metal linear-chain complexes

57

58 QUASI-ONE-DIMENSIONAL ORGANIC POLYMER FERROMAGNETS Vol. 95, No. 1

PqyP)A R R lL

Fig. 1. The simplified structure of poly-BIPO. R’ means a kind of side free-radical containing an unpaired electron.

Schrieffer-Heeger (SSH) Hamiltonian to describe the main chain and the e-ph coupling. The e-e interaction is introduced by the Hubbard model. Then the Hamiltonian that has been employed for modeling the polymer is given by

(HMMC or MX chain), can show a variety of symmetry-broken ground states like bond-ordering- wave (BOW), charge-density-wave (CDW), spin- density-wave (SDW), spin-Peierls state and even superconducting state on a much lower energy scale. Their competition and possible coexistence are deter- mined by the interplay between the electron-electron (e-e) and electron-phonon (e-ph) interactions. For quasi-one-dimensional organic polymer ferromag- nets, one important problem is that the strong e-e interaction and e-ph coupling might change the magnetic properties of the system dramatically. Allowing for full lattice relaxation and considering the antiferromagnetic correlation between itinerant 7r- electrons and the localized unpaired electrons at side radicals, we have proposed a theoretical model for the quasi-one-dimensional organic ferromagnet poly- BIPO [ll]. We found that the system can show a ferromagnetic ground state as the result of the elec- tron-electron interaction and topological structure. In this article, based on the theoretical model pro- posed by us, we will give a detailed study on the possible ferromagnetic ground state of quasi-one- dimensional organic polymer ferromagnets poly- BIPO with competition between e-e interaction and e-ph coupling. Within the self-consistent-field Hartree-Fock approximation, the strong e-e inter- actions and e-ph coupling in a one-dimensional system are taken into account self-consistently. The mean-field ordering of a one-dimensional polymer chain due entirely to the Hartree-Fock approxima- tion may perhaps, but not necessarily, be qualitatively more characteristic of the quasi-one-dimensional solid than would be the theory of a single chain where no mean-field approximation is used. In Section 2 we will give the theoretical model and computational method. Results and discussions will be given in Section 3.

2. THEORETICAL MODEL AND COMPUTATIONAL METHOD

Figure 1 is the system employed for our study. The main zigzag chain consists of a-conjugated carbon atoms, and R’ is a kind of side radical containing an unpaired electron. We use the generalized Su-

h = H/t* = - C[l + (-l)i~j](Ci+,i,,Ci,, + H-C) i, 0

where i labels the carbon atoms along the chain and 0 (= (Y, /I) labels the direction of spin where a: denotes up and /3 denotes down spin, yi is the dimerization order parameter which describes the displacement of the bond length between the ith and the jth lattice sites, to is the transfer or overlap integral when yi = 0 for all i, Cit(r and Ci,o are creation and annihilation operators for a 7r-electron with spin g on the ith site respectively, X is the e-ph coupling parameter. u (> 0) is the effective repulsive energy, in units of to, between two x-electrons when they are on the same carbon atom, and ni,c = Ci+,Ci ~.

Here we assume’that every side radical has a non- compensated spin SiR. Since there is no purely one- dimensional ferromagnet, we may conceive that there exists chain-radical interaction, and we assume the couplingjS > 0 between the spin Si of 7r-electrons and the residual spin SIR of R radical is isotropic, as in [ 111. The radical R connects with the odd carbon atom,

Si = 0 i is even,

1 i is odd.

We use mean-field approximation to deal with the last term of the Hamiltonian in equation (1) h becomes

’ = - CL1 + (-l)i_Yi](Ci++l,vCi,g + H-C) i, c7

Using the self-consistent iterative method and self- consistent-field Hartree-Fock approximation [ 111, we can obtain the eigenenergies ei, the expansion coeffi- cients 2: of the molecular orbitals and the ootimized

Vol. 95, No. 1 QUASI-ONE-DIMENSIONAL ORGANIC POLYMER FERROMAGNETS 59

geometry yi from the following self-consistent itera- tive equations:

- (1 + (-l)‘Ji]Z,“,i+I - [l + (-l)‘yi]Z~,j-1

(occi

- (l + (-l)'Yi]z~,i+, -. [I + (-l)iYi]zf,i-,

+ [

jf GG) ucz;r~iz;!,i -7

P

(OCC)

(6)

Here, periodic boundary conditions are used, and (occ) means those states occupied by electrons. New values of the dimerization order parameter yi are calculated by minimizing the total energy E( yi) of the system with respect to yi,

E(yi) = - C [I + (-l)‘_Vi] i. n (

(occ)

+ 1 c Jq[lz;,i,* - ].$J]. (7) i II

(xc)

And the distribution of spin density of n-electrons along the main chain can be obtained self-consistently as

f!+rj = (np - n$/2

The starting geometry in the iterative optimization process is usually the one with zero dimerization. The stability of the optimization geometry is always tested by using another starting configuration and perform- ing the optimization once again. For all systems

F (E-E,, l/N < 0.5

T

-“(: ‘1. -x=0.5

h=O.l

i\ t

h=0.3 ii

\\ ",

\ \ \ '\‘\

‘\' \:

z -1.5 1 I I I 1

0.0 1.0 2.0 3.0 4.0 5.0 e

e-e REPULSION/(TRiWSFER ENERGY)

U

Fig. 2. Ratio of the total energy per electron to the transfer energy referenced from the nonmagnetic state.

included in this study, the same optimal ground state was reached, independent of the starting config- uration. The criterion for terminating the optimiza- tion is that between two successive iterations, the difference is less than low5 A for the dimerization order parameter and less than 10e5 for the spin density.

3. RESULTS AND DISCUSSION

We consider a chain of 80 carbon atoms (N = 80) and 40 side-radicals, and the periodic boundary condition is used. In order to study the ground state, we always fill the 7r-electrons in the lowest possible levels in every iterative step and we must minimize the total energy with respect to (&). Owing to the last two terms of the Hamiltonian in equation (3), the degeneration of spin has been lifted in this kind of system, and we must solve the given equations with different spins respectively. As a result, all the 7r- electrons along the main chain will form an antiferro- magnetic SDW. And mediated by the SDW, a ferro- magnetic order of the unpaired electrons of side radicals can be obtained [l 11.

At first, we give a calculation of total energy with different e-e interaction and e-ph coupling. Figure 2 shows the total energy of the ferromagnetic state referenced from the energy ENM of the nonmagnetic state, which is the state when jr = 0. We can easily see that when electron-electron interaction u is larger than a certain critical value, the ferromagnetic state is more stable than the nonmagnetic state. And the

60 QUASI-ONE-DIMENSIONAL ORGANIC POLYMER FERROMAGNETS Vol. 95, No. 1

0.5

0.4

ti

5 0.3

z 2

; 0.2

4

01

0.0 1 0.0 I .b 2.0 3.0 4.0 5.0 f

e-e REPCL%ON/(TRANSFER ENERGY)

U i.0

oIj&Ll__bu 0.0 2.0 4.0

e-e REPULSION/(TRANSFER ENERGY)

Fig. 3. The amplitude of SDW as a function of Fig. 4. The dimerization of the main chain as a electron-electron interaction u with different electron- function of electron-electron interaction u with dif- phonon coupling X. ferent electron-phonon coupling X.

critical value will increase with increasing of e-ph coupling X. So in order to obtain stable quasi- one-dimensional polymer ferromagnets, it is essential that the e-ph coupling X should be as small as possible, and e-e interaction u should be as large as possible.

Figure 3 shows the amplitude of the SDW, which can be expressed as S = (6;). as a function of e-e repulsion u with different e-ph coupling X. It is illustrated that beyond a critical interaction the spin density S will increase rapidly with the increasing of the e-e interaction. Because the ferromagnetic coupling between the spin of unpaired electrons at side-radicals is mediated by this kind of spin density fluctuation, we are convinced that this ferromagnetic exchange interaction will become stronger with increasing of u, and then the ferromagnetic ground state of the system will be more stable.

state. And in this stable ferromagnetic state the r-electrons SDW will become very strong, and almost zero dimerization happens for the main chain. In order to obtain a stable quasi-one-dimensional organic polymer ferromagnet, we should have smaller e-ph coupling and larger e-e interaction. In fact, for the actual highly anisotropic solid, the energy scale or the temperature scale is roughly the square root of: the Hartree-Fock scale for instability on one chain times the chain-chain interaction. The qualitative effect of the chain-chain interaction is to replace the Hartree- Fock condensation energy and transition temperature with much smaller values, but to prevent the situation where the transition temperature of an infinite one- dimensional chain in isolation is necessarily zero.

Figure 4 shows the dimerization y = ( yi) of the main chain with different e-e interaction and e-ph coupling. We can see that when the electron-electron interaction u increases above the critical value, the dimerization of the main chain drops rapidly to zero as the result of the competition between the e-e interaction and e-ph coupling. So in the stable ferromagnetic ground state, almost there is no dimer- ization for the main chain.

Throughout this article, we are deeply convinced that, when the K-conjugated systems are set in a definite way, it can show ferromagnetic properties. Although, at the present stage of the synthesis, only a few species show ferromagnetic properties. But we believe that this kind of conjugated n-electron system is a good candidate for organic polymer ferromagnet.

REFERENCES

In summary, we have found a critical interaction, which is determined by the competition between the e-e interaction and the eeph coupling. When e-e interaction is larger than the critical value, the ground state of the system is a stable ferromagnetic

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Vol. 95, No. 1 QUASI-ONE-DIMENSIONAL ORGANIC POLYMER FERROMAGNETS 61

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