Z. Phys. B 99, 425 428 (1996) ZEITSCHRIFT FOR PHYSIK B �9 Springer-Verlag 1996

The CDW transition in quasi-one-dimensional organic polymer ferromagnets: effects of next-nearest neighbor hopping interactions Z. Fang 1, K.L. Yao 2, Z.L. Liu 3

1 Center of Theoretical Physics, CCAST(World Laboratory), Beijing, People's Republic of China /International Center of Material Physics, Academy Sinica, Shenyang 110015, People's Republic of China, 3Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China

Received: 10 January 1995/Revised version: 5 April 1995

Abstract. The next-nearest neighbor hopping interactions of n-electrons in quasi-one-dimensional organic polymer ferromagnets are considered. Within the mean-field the- ory and allowing for full lattice relaxation, a set of self- consistent equations is established to study the system. It is found that with increasing of the next-nearest neighbor interaction a charge-density-wave (CDW) transition will happen. At the CDW state, a strong charge density distri- bution along the main chain will appear, and the spin density-wave (SDW) along the main chian will be tuned by the CDW. Consequently the ferromagnetic state, in which all the spins of the unpaired electrons at side free- radicals are arranged parallely, will be no longer the stable ground state of the system.

PACS: 71.20.Hk; 75.50.Dd

I. Introduction

In recent years, with the discovery of organic conductors and organic superconductors, the design and synthesis of a new clas sof ferromagnetic materials based on organic molecules rather than metallic or ionic lattices have be- come a challenging problem that has attracted consider- able attention [1-4]. Since organic conductors have been found to possess peculiar properties, which have been exploited in devices, it is hoped that similar discoveries can be made for organic ferromagnets. The possibility of the existence of such organic polymer ferromagnetic ma- terials was demonstrated by the observation of ferromag- netic characteristic in poly-BIPO [5], m-PDPC [6] and p-NPNN [7]. Various structures have been proposed as possible organic polymer ferromagnets [8-10].

The simple n-conjugated structure (shown in Fig. 1), which shows interesting electrical and potentially fer- romagnetic properties, is a prototypical structure for quasi-one-dimensional organic polymer ferromagnets, such as poly-BIPO and its derivatives [11]. The main zigzag chain consists of n-conjugated carbon atoms, and

R' is a kind of side radicals containing an unpaired elec- tron. The ferromagnetic properties of the organic fer- romagnetic polymers have been discovered by Ovchinnikov et al. [11] and Cao et al. 1-12]. In this kind of quasi-one-dimensional organic polymer ferromagnets, one important problem is the strong (electron-electron) interaction and (electron-phonon) coupling, which might change the magnetic properties of the system dramati- cally. Considering the strong e-e interaction, the e-ph coupling and the antiferromagnetic correlation between itinerant n-electrons and the localized unpaired electrons at side radicals, we have proposed a theoretical model for the quasi-one-dimensional organic polymer ferromagnets [13]. We found that the n-electrons along the main chain play an important role on the ferromagnetic order in the system. Mediated by the itinerant n-electrons along the main chain [14], a parallel spin arrangement of the un- paired electrons at side free-radicals can be obtained. However, we must notice that the theoretical model only gives a tight binding description of the carbon main chain and only includes the first-neighbor hopping interactions. In fact, owing to the peculiar geometry (the zigzag struc- ture of the main carbon chain) and the short C-C bond lengths, the next-nearest neighbor hopping interactions of n-electrons will take effects just as in polyacetylene [15]. This next-nearest neighbor hopping interactions should affect the n-electron properties and consequently should affect the ferromagnetic state of the system. Thus, in this article, we will try to add the long-range neighbor hopping interactions into the theoretical model. In part II we will give a theoretical model and computational method. Re- sults and discussions will be given in part llI.

II. Theoretical model and numerical method

We add the next-nearest neighbor hopping interaction terms into the generalized SSH hamiltonian to describe the main chain and the e-ph coupling. The e-e interaction is introduced by the Hubbard model. Then the Hamil- tonian that has been employed for modeling the polymer

426

is given by

H = - - ~ [to - - ~ o ( V / + , - V~)](C++L.,C,,,. + H.C)

K ~/(V/+ 1 - - V/)2 +~-

- - E [ t l - - ~ l ( g / + 2 - - V/)] (Ci+2.~r Ci. a + H . C ) i.o.

+ U ~ n ~ n ~ + Jf ~.6~S~R'S~ (1) i i

where, K, U, Jr, 6~, S~ and S~ have the same meaning as in our previous paper [13], C +,,. and Ci,. are creation and annihilation operators for a n-electron with spin a on the ith site respectively, V~ is the displacement of the ith lattice site from the equilibrium position, to and tx are the over- lap integral between the original and the nearest neighbor carbon sites and between the original and the next-nearest neighbor carbon sites for the undimerized chain respec- tively, ~o and ~ are the corresponding rate of change of the overlap integral with distance.

The Hamiltonian does not include the long-range elas- tic interaction. It is easy to show that they can be taken into account by using an effective elastic constant. We use mean-field approximation to deal with the last term of the Hamiltonian (I), then the Hamiltonian can be written as

H = - ~ [to - ao(V~+~ - V~)] (CL, , . C,,. + H.C) i,a

- Z[t, - ~ , ( V ~ + z - V ~ ) ] ( C ~ + ~ , , , C , , , + H.C) i,a

K + ~ E(v,+,. - v y + v y n , , n , ,

i 2 hi4) + & (2)

Using the self-consistent' iterative method and self- consistent field Hartree-Fock approximation [ 132, we can obtian the eigenenergies ei, the expansion coefficients Z~ of the molecular orbitals and the optimized geometry V~+I- V~ from the following self-consistent iterative equations:

- [to - ~o(V~+1 - V3] Z~,,,+ 1

- [ to - So (V , - V ~ - ~ ) ] Z L , _ ,

- I t , - ~ ( ~ + 2 - v~)] Z L , + ~

- [t~ - ~ , (v~ - V~_~)]ZL,_~

+ [ U E #* # Jsbi<-S~">lz" z<,,z~.i ~ = 4 zL, (3) L

(occ) -- [t O-~0(v~+I -- V3]Z~,i+ ,

- [ t o - s o ( V / - V/ - I ) ] Z#,/_ 1

- [ t , - "t(Vi+2 - V i ) ] Z ~ . + 2

- [ t , - ~ , ( v , - v ~ _ ~ ) ] z S _ ~

+ V y. "* ~ & z,,., = e.Z.,i (4) ~' Z,..iZ~.i - 2 (occ)

v,+, - v,= -- f f [,oZL,ZL,+,

( )

+ ~(zL~zL~+~ + z L ~ - , zL~+,)]

N ~i ~ [ ~ 1 7 6 ladr

(occ)

-}- ~ l ( Z p , iZ l t , i+2 + Zlt , i - 1 Z u , i + 1) l (5) J

Here, periodic boundary conditions are used, and (occ) means those states occupied by electrons. New values of the dimerization order parameter Vi+~- Vi are cal- culated by minimizing the total energy E of the system with respect to Vi+l - V~,

E ( E + I - V 3 = - ~ {[to--O~o(Vf+l--Vi)l

(oct)

Z ~ ~ "" ~ ]} [Zt~,i + l Z , . i -~- Z~t.i Z , . i + 1

la (occ)

-- ~., { [tl -- ~l(Vi+2 - Vi)] (oct)

[ Z , . i + 2 Zct.i + Z , . i Z # . i + 2 i.t

(occ)

K + y ~ (E+ ~ - V,#

+ ~ Z E E Iz~,,I ~ Iz~,,,I ~ i ~ I a"

(occ) (occ)

2 [IZL~I2 - I z~A~] i p

(occ) (6)

And the distribution of the spin density and the charge density of ~r-electrons along the main chain can be ob- tained self-consistently as:

1 ( "" ~ ~ ,., , . , } (7) 6ni = (n~ - n~i )/2 2 \(~ (o~c) / = - ~ Z . , i Z . , i - V W ' . Z p .'~

= = Z ~ , i Z # , i - - Z # , i Z t t , i

k.(ocr (oct) j

In our calculations, we choose the parameters of the next-nearest neighbor hopping term as to =2.5ev, So =4.1 ev/~,, K = 21 evfA, which corresponds to the e-ph coupling constant

2 = = 0.2 rcKto

Since the hopping integral is the exponential function vs the length of the hopping, we have taken the parameters

R R' R

Fig. 1. The simplified structure of poly-BIPO. Rmeans a kind of side free-radical contains an unpaired electron

(E-Eli u )tev} 0 . 0 0

~ - 0 . 5 0

c.

,= o &

~ - ~ . o o

o.

b-

- 1 . 5 0 0 .0 0)2 0)4 0)6 0)8

P 1.0

N e x t - n e a r e s t n e i g h b o r h o p p i n g m t e r a c h o n

Fig. 2. The total energy of the ferromagnetic state referenced from the nonmagnetic state as a function of the next-nearest neighbor hopping interaction p

427

03

C) ffl

0 20 40 60 8'0 tOO Site n u m b e r i

"1.0

0 . 9

0.1

0.0

- 0 . !

O.t

0 .0

- 0 . 1

Fig. 3. The BOW, SDW and CDW along the main chain for p = 0.5

L)

r r.D

! . 0 '

~ 'l O.t

0 . 0

- 0 . 1

p = h / t o = al/C~o to describe the next-nearest neighbor hopping interactions. We choose U = 1.0 to, JF = 0.5 to and use

Yi = ( -- 1) I ~___qO (V i+ 1 - - Vi) to

as the order parameter of demarization

(9)

0 . !

0 .0

-0.2t

o 20 4o so 8o too Site n u m b e r i

Fig. 4. The BOW, SDW and CDW along the main chain for p = 0.6

III. Results and discussions

We consider a chain of 100 carbon atoms (N = 100) with 50 side free-radicals (shown in Fig. 1), and periodic bound- ary condition is used. In order to study the ground state, we always fill the rt-electrons in the possible lowest levels in every iterative step. Owing to the last two terms of Hamiltonian (2), 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 7z- electrons along the main chain will form an antiferromag- netic SDW. And mediated by the SDW, a ferromagnetic order of the unpaired electrons of side radicals can be obtained [14].

With adding of the next-neighbor hopping interaction, the properties of the main chain and the ~-electrons will be affected. Consequently the ferromagnetic state (where the spins of the unpaired electrons at side free-radicals are parallel) of the system will also be affected. Figure 2 shows

the total energy of the ferromagnetic state referenced from the nonmagnetic state ( J s = 0) as a ruction of the next- nearest neighbor hopping interaction p. We can see clearly that there appears an abrupt increasing of total energy at about p = 0.55, al though for smaller p the total enery of the system will decrease a little with increasing of p. This means that when p is larger than a critical value (about 0.55) the stability of the ferromagnetic state will be destroyed. We may convince that a transition will take place at about p = 0.55.

To make things more clear, Figs. 3, 4 and 5 give the spin-density-wave (SDW), the charge-density-wave (CDW) and the bond-order-wave (BOW) along the main chain, expressed as 6n i , ni and y~, for p = 0.5, 0.6 and 0.7 respectively. At first, when p = 0.5 (shown in Fig. 3) there is a perfect dimerization for the main chain, all the n-electrons will form a perfect antiferromagnetic SDW, and there is almost no C D W along the main chain. This case is almost the same as the case of p = 0.0,

428

(D

0.9

0 . !

0 .0

- 0 . !

0.1

0 0.0

- 0 . !

0 20 40 60 80 I00 Site n u m b e r i

Fig. 5. The BOW, SDW and CDW along the main chain for p = 0.7

0.04

0.03

r~

co "0

0 .02 oA

r..)

O . O l

0 .00

J j J

0.0

f /

i

0.2 O.J4 0.'6 0 .8 The n e x t - n e a r e s t ne ighbor hopping znterac[ion

P

Fig. 6. The amplitude of the CDW as a function of the next-nearest neighbor hopping interaction p

although the SDW is slightly stronger. When p = 0.6 (shown in Fig. 4), it changes greatly. There appears a C D W along the main chain..The main chain is no longer perfectly dimerized, and there appears a BOW. The SDW is also no longer perfectly antiferromagnetic, and it is tuned by the CDW. The case o fp = 0.7 is shown in Fig. 5. At this time, the C D W also exists, and the period of the C D W has become smaller than the case of p = 0.6. Com- paring all the three figures, we may convince that it is a C D W transition. It can be demonstrated more clearly by Fig. 6, which gives the amplitude of the CDW, expressed as (n~ + n~) in (8), vs the next-nearest neighbor hopping interactions. We can see clearly that when the the next- nearest neighbor interaction p reaches a critical value (about 0.55), the charge density along the main chain will increase abruptly. Consequently a strong C D W state will appear.

At the C D W state, there is a strong charge density distribution along the main chain, and the SDW along the

main chain will be tuned by the CDW. Consequently, the SDW along the main chain will be no longer perfectly antiferromagnetic. Since the magnetic interaction between the unpaired electrons at side free-radicals is mediated by the SDW. Owing to the topological structure of the sys- tem, only when the SDW is antiferromagnetic, the interac- tion between two neighbor side free-radicals can be ferromagnetic. So in this case the ferromagnetic state, where all the spins of the unpaired electrons at side free- radicals are arranged parallelly, is no longer the stable ground state of the system. This causes an abrupt increas- ing of the total energy from the nonmagnetic state as shown in Fig. 2.

In summary, within the mean-field theory, we have found a C D W transition with increasing of the next- nearest neighbor interaction. At the C D W state, a s trong C D W along the main chain will appear, and the S D W along the main chain will be tuned by the CDW. Conse- quently the ferromagnetic state of the system will be no longer the stable ground state of the system.

As is already well known, there is no purely one- dimensional ferromagnet. To get real ferromagnetism, the interchain spin coupling must also be ferromagnetic. However, this depends on the three-dimensional structure of polymer. These questions can only be answered after a synthesis of the polymer. On the other hand, in real system the effective conjugation is always about some ten repeat units. In our calculation, if the cha'in length is of the order of some ten units, only part of wave length will be shown, which is not so evident as long chain one, since periodic boundary condition has been used. For finite sys- tem without periodic boundary condition, a new method should be adopted, which will be studied in the future.

Project supported by National Natural Science Foundation of China.

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