w. z. wang, k. l. yao and h. q. lin- charge density wave transition and instability in interchain...

8/3/2019 W. Z. Wang, K. L. Yao and H. Q. Lin- Charge density wave transition and instability in interchain coupled organic fer

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Charge density wave transition and instability in interchain coupledorganic ferromagnets with next-nearest-neighbor hopping interaction

W. Z. WangDepartment of Physics, Huazhong University of Science and Technology, Wuhan 430074,Peoples Republic of China

K. L. YaoCCAST (World Laboratory), Beijing, P.O. Box 8730, 100080 China

H. Q. LinDepartment of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Received 28 April 1997; accepted 5 November 1997

Considering interchain interaction and the intrachain next-nearest-neighbor hopping interaction,

within the self-consistent-field HartreeFock approximation, we deal with two neighboring organic

ferromagnetic chains. We take into account the -electrons, the Hubbard repulsion, and

antiferromagnetic correlation between -electrons and side radical electrons. It is shown that there

appear to be two kinds of charge density waves successively when the next-nearest-neighbor

hopping interaction is greater than a critical value, which decreases with the interchain coupling.

The first charge density wave is accompanied by a bond order wave and the second one has no

lattice distortion. The spin density wave along the main chain is modulated by the charge density

wave. It is also found that the interchain coupling and the next-nearest-neighbor hopping integral

destabilizes the ferromagnetic ground states. 1998 American Institute of Physics.

S0021-96069852106-X

I. INTRODUCTION

Recently, a new class of ferromagnetic materials based

on molecular rather than metallic or ionic lattices has at-

tracted considerable attention.1 6 Scientists have successfully

synthesized several quasi-one-dimensional organic ferro-

magnetics, such as m-PDPC Ref. 7 and p-NPNN.8 How-

ever, about these quasi-one-dimensional organic polymer

ferromagnets, many things are still unclear.McConnell9 first proposed a intermolecular ferro-

magnetic interaction in organic molecules in 1963.

Ovchinnikov10 and Mataga11 reported another strategy to

prepare organic ferromagnets based on the intramolecular

ferromagnetic interaction in very large molecules. Ovchinni-

kov et al.12 proposed a simplified structure of a quasi-1D

organic polymer ferromagnet a single chain in Fig. 1. The

main chain consists of carbon atoms each with a -electron

and R is a kind of side radical containing an unpaired elec-

tron. They treated the -electrons along the main carbon

chain as an antiferromagnetic spin chain and assumed that

there is an antiferromagnetic correlation between the -

electron spin and the residual spin of the side radical. Re-cently, Fang et al.13,14 proposed a theoretical model to de-

scribe this kind of organic ferromagnet. They considered the

-electrons, the Hubbard electron-electron correlation, and

the antiferromagnetic spin correlation between the -

electrons and unpaired electrons on side radicals. Within the

self-consistent-field Hartree Fock approximation, they cal-

culated the energy levels which split off with respect to dif-

ferent spins. In the ferromagnetic ground state, there exists

an antiferromagnetic spin-density-wave SDW along the

main chain. Mediated by the SDW, ferromagnetic order of

the unpaired electrons of side radicals can be obtained. How-

ever, in previous works, the system was treated as isolated

chains. Since there is no purely one-dimensional system, the

interchain interaction can have a significant effect as will be

demonstrated in this paper. As is well known, for one-

dimensional systems, like conducting polymers, charge-

transfer solids and transition-metal linear-chain complexes,

can show a variety of symmetry-broken ground states like

bond-ordering-waves BOW, charge density waves CDW,

spin density waves, and even the superconductor state. These

phase transitions originate from the interplay between the

electronelectron and electronphonon interaction. So we

believe that for the organic ferromagnets it is more reason-

able to consider the intrachain non-nearest-neighbor electron

hopping integral and interchain coupling. In our model, the

interchain coupling is considered as an interchain electron-

transfer between the corresponding sites on nearest chains.

Due to the topological structure of the system, the interchain

couplings are different with respect to different sites in

chains. This is similar with the situations in several typical

organic polymers with chainlike structure, such aspolyacetylene1517 and polyacene,18 in which the interchain

interaction is the interchain hopping of electrons, but

phonons are strictly one-dimensional. For some organic fer-

romagnets with linear chains such as m-PDPC (Ref. 7) and

p-NPNN,8 we conceive that the interchain electron transfer

interaction can be achieved if the distance between two

neighboring chains is not great.

In this paper, we consider two neighboring chains shown

in Fig. 1. Our generalized model for interchain coupling is

introduced in Sec. II. For some low-dimensional systems

JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 7 15 FEBRUARY 1998

28670021-9606/98/108(7)/2867/6/$15.00 1998 American Institute of Physics


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such as polyacetylene19,20 and organic ferromagnets,13,21 the

mean field theory is a reasonable method in the studies of

energy band and excitation. We will use this method to study

the stability of the ferromagnetic ground state and the con-

figuration of spin density and charge density. We find that

there appears to be two kinds of CDW transitions accompa-

nied with BOW and SDW successively when the next-

nearest-neighbor hopping integral is greater than a critical

value, which varies with the interchain coupling. The resultsand discussions are given in Sec. III.

II. THE MODEL HAMILTONIAN AND COMPUTATIONALMETHOD

We consider two neighboring chains shown in Fig. 1.

Based on the discussion in Sec. I, the Hamiltonian employed

in our study can be written as

HH0H, 1

where H0 is the Hamiltonian of two isolated chains with the

next-nearest-neighbor hopping interaction, H

describes theinterchain coupling. We can write H0 explicitly as follows:

H0H1H2H3 , 2

H1jl

t00ujlujl1 cjl

cjl1H.c.

2 jl ujlujl12

jl t1ujlujl2

cjl

cjl2H.c., 3

H2Ujl

njlnjl , 4

H3J0jl

SjlRSjl l . 5

The first term H1 describes the intrachain -electron

hopping, the electronphonon interaction, and the distortion

of the lattice. Where cjl (cjl ) denote the creation annihi-

lation operator of a -electron with spin on the l th site of

the j th chain, t0(t) is the nearestnext-nearest-neighbor

hopping integral when there is no distortion of the lattice,

0(1) is the electronphonon coupling constant, ujl is the

displacement of the lth carbon atom of the j th chain, and is

the elastic constant of the lattice.

The second term H2 describes the Hubbard repulsion

between two -electrons when they are on the same carbon

atom, and njl cjl

cjl (,), where and denote

up-spin and down-spin, respectively.

The third term H3 describes the antiferromagnetic corre-

lations between the spin Sjl of -electrons and the residual

spin SjlR of the side radical R . We assume the coupling J00 between the main chain and side radical, and the radical

R connects with the even carbon atom, then

l1, l is even6

l0, l is odd.

The term SjlRSjl can be rewritten as

SjlRSjlSjlRz

Sjl12 SjlR

SjlSjlR

Sjl, 7

where Sjlz and Sjl

denote the Pauli spin matrix,13,21

Sjlz

12 njl njl,

Sjlcjl

cjl ,

Sjlcjl

cjl . 8

Since the side radicals only connect with even sites on

each chain shown in Fig. 1, the interchain transfer integrals

would be unsymmetrical with respect to even sites and odd

sites. So we assume the interchain coupling has the follow-

ing oscillatory form:

H

lT

11 lT

2 c

1l

c

2lc

2l

c

1l, 9

where T1 and T2 are the hopping integrals from site l of the

first chain to the corresponding site l of the second one.

Then we use the mean-field approximation to divide njl and SjlR

z as follows:13,21

njl njl njl ,

SjlRzSjlR

z SjlRz . 10

Here G G is the average with respect to theground state G , n

jl and S

jlR

z are fluctuations from the

average values.

It is convenient to cast all quantities into dimensionless

forms as

hH

t0, u

U

t0, j 0

J0

t0, t1

T1

t0, t2

T2

t0,

11

20

2

t0, yjlujlujl10 /t0 .

We assume t/t01 /0 , then the Hamiltonian h be-

comes

FIG. 1. A simplified structure of chain for a quasi-one-dimensional organicferromagnet.

2868 J. Chem. Phys., Vol. 108, No. 7, 15 February 1998 Wang, Yao, and Lin


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hh eh, 12

hejl

1yjl cjl

cjl1H.c.ujl

njl njl

j 0

2 jl SjlRz njl njll

jl 1yjl

yjl1cjl

cjl2H.c.l t11lt2

c1l

c 2lc2l

c 1l, 13

h1

jl yjl2 , 14

where h e is the electronic part of the Hamiltonian, and h

describes the elastic energy of lattice.

Since we use the tight-binding approximation in the

Hamiltonian Eq. 1, the wave function of the system can

be expanded in site basis states,

jl Zjl cjl

0 , 15

where 0 is true electron vacuum state, denotes the theigenvector of the Hamiltonian, and Zjl

is the expansion

coefficient.

Using a self-consistent iterative method,13 we can nu-

merically solve the Schrodinger equation,

he . 16

With the initial value ofSjlRz , and the lattice configuration,

electronic energy levels and the expansion coefficientsZjl can be obtained from the eigenvalue equations of the

system,

1yjl Zjl1

1yjl1Zjl1

t11lt2

Z2l

j1Z1l

j21yjl1yjl Zjl2

1yjl1yjl2Zjl2

u occ

Zjl

Zjl

*1

2j 0lSjlR

z Zjl

Zjl

j1,2. 17

The total energy of the system with the Hamiltonian Eq.

13 is

Eyjl jl

1yjl

occ

Zjl1* Zjl

Zjl

*Zjl1

1

jl yjl2

jl ,occ

Zjl 2Zjl

2

1

2jl

occ

j 0lSjlRz Zjl

2Zjl 2

locc

t11lt2Z1l

* Z2lZ2l

* Z1l

jl

1yjlyjl1

occ

Zjl2* Zjl

Zjl

*Zjl2 . 18

The dimerization yjl can be obtained by minimizing the total

energy of the system with respect to yjl ,

yjl

occ

Zjl

Zjl1

Zjl

Zjl2

Zjl1 Zjl1

1

N

locc

Zjl Zjl1

Zjl Zjl2

Zjl1

Zjl1 . 19

Here, we have used the periodic boundary condition, N is the

number of sites in each chain, and occ means those states

occupied by electrons. The charge density njl and spindensity njl of -electrons along the chain can be obtained

self-consistently as

njl njlnjlocc

Zjl 2, 20

njl1

2njlnjl

1

2 occ

Zjl 2

occ

Zjl

2 . 21

The starting geometry in the iterative optimization pro-

cess is usually the one with zero dimerization and njl njl1/2. The stability of the optimized geometry is al-ways tested by using another starting configuration and per-

forming the optimization once again. The criterion for termi-

2869J. Chem. Phys., Vol. 108, No. 7, 15 February 1998 Wang, Yao, and Lin


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nating the optimization is that between two successive

iterations, the differences are less than 106 for the dimer-

ization and spin density.

III. RESULTS AND DISCUSSION

We consider two neighboring periodic chains shown in

Fig. 1, each chain with 60 carbons along the main chain and

30 side radicals. From Eq. 17, we know that the eigenvalue

equation is unsymmetrical about spin owing to the Hubbard

electronelectron repulsion and the antiferromagnetic corre-

lation between -electrons and unpaired electrons at the side

radicals. So in the system, the spin degeneracy has beenlifted, and we must solve the given equations with different

spins. In order to study the ground state, we always fill the

-electrons in the possible lowest levels in every iterative

step.

In our calculation, we assume the parameters 0.2, u

1.0, j 00.5, and SjlRz 1/2. For t10.1, t20.05, we

calculate the energy levels of the -electrons shown in Fig.

2, where i indicates the ith energy level. From Fig. 2, we can

see clearly that the energy levels belonging to different spins

split off. However, owing to the interchain coupling, the de-

generacy of the energy levels with respect to different chains

has been lifted. The energy levels of one chain increase

while those of the other chain decrease. In quasi-one-dimensional systems, due to dimerization there exists an en-

ergy gap. We calculate the energy gap with different intrac-

hain next-nearest-neighbor hopping integral and interchain

coupling t1 and t2 shown in Fig. 3. Curves a, b, c corre-

sponds to the interchain coupling t1t20; t10.1, t20.2 and t10.1, t20, respectively. It is shown that for

definite interchain coupling, the energy gap is constant as the

next-nearest hopping integral is small, and when is larger

than a certain value the gap decreases rapidly. There exists a

critical value c1 at which the gap disappears. We can

easily see that the energy gap and the critical value c1 re-

duces with an increase of interchain coupling t1compare

curves a and c, but increases with an increase of interchain

coupling t2 compare curves b and c. The results of the

numerical calculations show that when the gap exists, the

charge density is homogeneous along each main chain. How-

ever, when c1 , the gap disappears, and there appears to

be two kinds of charge density waves along each main chain

successively, which is separated at the second critical value

c2 . For t10.2, t20, c10.45, c20.52, we obtain

the CDW, SDW, and BOW along the main chain in Fig. 4

with a 0.50 and b 0.55. Figure 4a shows that forc2c1 , the CDW is not accompanied with perfect

dimerization along the main chain. There appears the bond-

order-wave BOW expressed as yjl in Eq. 19. The SDW isno longer perfectly antiferromagnetic, and it is modulated by

the CDW. From Fig. 4b, we can see that for c2 , the

CDW has a period of two sites, but the dimerization along

the main chain is zero. In this case, there are no distortion of

lattice, but the transfer of the -electrons along the main

chain is similar with that in the situation with perfect dimer-

ization. With an increase of the electronphonon interaction,

the dimerization increases22 and consequently the energy gap

increases, while with an increase of the next-nearest-

neighboring hopping interaction the gap and dimerization de-

crease; we are convinced that the CDW transition results

from the competition between the electronphonon coupling

and the next-nearest neighbor hopping interaction. We alsofind that the interchain coupling t1 and t2 have a different

effect on the above CDW transition. With an increase of t1 ,

the critical value c1 and c2 decrease. To make things more

clear, Fig. 5 gives the critical next-nearest neighboring hop-

ping interaction c as a function of the interchain coupling t1with t20. Curves a and b correspond to cc2 and cc1 , respectively. We also find that for t10, when we

add the interchain coupling t2 , there exists only the second

CDW with the period of two sites for all c1 . Because t2represents the difference of the interchain coupling at odd

sites and even sites see Eq. 9, the behavior of t2 distin-

guish the odd sites from even sites. This is similar with the

FIG. 2. The energy levels of the -electrons for j00.5, 0.2, t10.1,

t20.05, and u1.0. Curves A and B correspond to two chains, respec-

tively.

FIG. 3. The energy gap vs the intrachain next-nearest-neighbor hopping

integral for different interchain coupling a t1t20; b t10.1, t20.2, and c t10.1, t20.



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effect of the dimerization, and consequently the interchain

coupling t2 makes the CDW have the period of two sites.

Then we discuss the ferromagnetic order of the system.

In our calculation, we assume that the unpaired electrons at

side radicals have up-spin positive-spin. When the energy

gap exists, the up-spin electrons and down-spin electrons oc-

cupying the energy levels have the same number, all the

-electrons along each main chain form an antiferromagnetic

SDW while the unpaired electrons of side radicals form aferromagnetic order Refs. 13 and 14. However, when the

next-nearest-neighbor hopping interaction is greater than a

critical value, the energy gap disappears, owing to the split-

ting of energy levels with respect to different spins, and there

are more electrons occupying the lower down-spin energy

levels than the higher up-spin energy levels shown in Fig.

2. So, in the ground state, there exists a kind of SDW along

the main chain, whose total spin is negative shown in Fig.

4. We find that the SDWs are modulated by corresponding

CDWs. Since in the absence of the energy gap, the total spin

along the main chain is negative while the spins at side radi-

cals is positive, the CDW transition weakens the ferromag-

netism of the system. Because the interchain coupling de-

creases the energy gap and makes the CDW transition

happen easily, the interchain coupling destabilizes the ferro-

magnetic ground state of the system.

In conclusion, we have studied numerically an interchain

coupling model for a quasi-one-dimensional organic ferro-

magnet. It is shown that there exists two kinds of CDW

sequentially when the next-nearest-neighbor hopping interac-

tion is greater than a critical value, which decreases with an

increase of the interchain coupling. The SDW along the main

chain is modulated by the CDW. The interchain interaction

and the interchain next-nearest-neighbor hopping integral de-

stabilize the ferromagnetic ground state of the system.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science

Foundation of China and the Direct Grant for Research from

the Research Council of the Hong Kong Government.

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2871J. Chem. Phys., Vol. 108, No. 7, 15 February 1998 Wang, Yao, and Lin


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w. z. wang, k. l. yao and h. q. lin- charge density wave transition and instability in interchain...

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