Study of the magnon excitation in a quasi-one-dimensional organicpolymer ferromagnet

L. Zhao,*a K. L. Yaoabc and Y. F. Duana

a Department of Physics, Huazhong University of Science and T echnology, W uhan 430074, Chinab International Center of Material Physics, Academy of Science, Shenyang 110015, Chinac CCAST (W orld L aboratory), P.O. Box 8730, Beijing 100080, China

Received 18th May 2000, Accepted 21st July 2000Published on the Web 29th August 2000

The ground state and magnon excitation of a theoretical model proposed for a quasi-one-dimensionalp-conjugated organic polymer ferromagnet are studied. Within the mean-Ðeld approximation, consideringHubberd repulsion and the hopping of the p electrons on the main chain and the unpaired electrons on theside radicals, the ground state of the system turns out to be a high-spin ferromagnetic state. The magnonexcitation is investigated by the random-phase approximation (RPA), and a magnon excitation spectrum thatincludes an acoustic mode is obtained. The results of the calculations show that the Hubberd repulsion of theunpaired electrons on the side radicals as well as the hopping interaction between the p electrons on the mainchain and the unpaired electrons on the side radicals make an impact on the acoustic mode which holds theproperty of the ferromagnetic magnon. The large Hubberd repulsion and hopping interaction are favourableto the stability of the ferromagnetic ground state of the system.

1. IntroductionSince the discovery of organic conductors and organic super-conductors, scientists have begun to design a new class of fer-romagnetic materials based on organic molecules rather thanmetallic or ionic lattices. Several research groups have suc-cessfully synthesized quasi-one-dimensional organic ferromag-nets, such as poly-BIPO,1 m-PDPC2 and p-NPNN.3However, many things about these organic ferromagnets arenot well understood.

To obtain organic ferromagnets, one important approach,suggested by Mataga4 and Ovchinnikov,5 is the synthesis ofan organic polymer based on intramolecular ferromagneticinteractions in very large molecules. A simpliÐed structure ofthis kind of alternant hydrocarbon is shown schematically inFig. 1. The main chain consists of carbon atoms each with a pelectron and R represents free radicals attached, as sidegroups, to the carbon backbone. Each side radical R containsan unpaired electron. Recently, Fang, Liu and Yao6,7 pro-posed a theoretical model to describe this kind of quasi-one-

Fig. 1 (a) The simpliÐed structure of a quasi-one-dimensionalorganic polymer ferromagnet and (b) the arrangement of spin.

dimensional organic polymer ferromagnet and obtained a fer-romagnetic ground state. Then, Wang et al.8 studied the spin-wave properties of the system. In their model, only the pelectrons along the main carbon chain are considered to beitinerant while the unpaired electrons on the side radicals areconsidered to have no freedom except their spins.

In this paper, based on a theoretical model in which theunpaired electrons on the side radicals are no longer regardedto be totally localized, we will investigate the magnon excita-tion in a quasi-one-dimensional p-conjugated organic polymerferromagnet whose simpliÐed structure is shown in Fig. 1. InSection 2, we give the model Hamiltonian in which theHubbard repulsion and the hoppings of the p electrons on themain chain as well as the unpaired electrons on the side rad-icals are taken into account. The ground state of the system isalso discussed in Section 2. In Section 3, we study the magnonexcitation of the system using the random-phase approx-imation (RPA). The conclusions are given in Section 4.

2. The model Hamiltonian and ground stateThe model Hamiltonian employed in our study can be writtenas :

HΠ\ [ ;l, p

[(T ] *T )aül1p` aü (l~1)2p

] (T [ *T )aül1p` aü

l2p ] H.C][ T @

] ;l, p

(aül2p` aü

l3p ] H.C)

] ;l

(Unül1a

nül1b

] Unül2a

nül2b

] U@nül3a

nül3b

). (1)

Here, denotes the creation (annihilation) operator ofaülmp` (aü

lmp)an electron at a site speciÐed by l, m and spin p( \ a,b) wherea and b denote up-spin and down-spin, respectively. l(\1,2, . . . , N) labels the unit cell which contains two carbon sitesof the main chain and one side radical, m(\1, 2) labels twocarbon sites while m(\3) labels the R side radical. T is thehopping integral between two neighboring p electrons along

DOI: 10.1039/b003988h Phys. Chem. Chem. Phys., 2000, 2, 4001È4004 4001

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Fig. 2 The electronic energy bands with t@\ 0.9, *t \ 0.1 andu \ u@\ 0.8.

the main chain while T @ is the hopping integral between a pelectron on the main chain and an unpaired electron on theside radical. *T is the correction that results from dimer-ization along the main chain. U and U@ are the Hubbard e†ec-tive electronÈelectron repulsive energies of p electrons on themain chain and unpaired electrons on the side radical, respec-tively, and nü

lmp\ aülmp` aü

lmp .It is convenient to cast all quantities into dimensionless

forms as :

hü \HŒT

, *t \*TT

, t@\T @T

, u \UT

, u@\U@T

. (2)

Thus, the Hamiltonian becomes :HŒ

hü \ [ ;l, p

[(1 ] *t)aül1p` aü (l~1)2p] (1[ *t)aü

l1p` aül2p ] H.C]

[ t@ ;l, p

(aül2p` aü

l3p ] H.C)

] ;l

(unül1a

nül1b

] unül2a

nül2b

] u@nül3a

nül3b

). (3)

The term of the Hubbard electronÈelectron repulsion inHamiltonian can be dealt with by using the HartreeÈFockhüapproximation as

nülmp\ *nü

lmp ] SnülmpT , (4)

thus

nülma

nülmb

\ Snülma

Tnülmb

] Snülmb

Tnülma

[ Snülma

TSnülmb

T ] *nülma

*nülmb

, (5)

where SÉ É ÉT is the average with respect to the mean-Ðeldtheory ground state that can be determined self-consistently.

By substituting eqn. (5) into eqn. (3) and then introducingFourier transformation, within the mean-Ðeld theory, we areable to diagonalize the Hamiltonian in k-space and, accord-hüingly, obtain the energy spectrum through the self-consistentiterative numerical calculation method. The correspondingresults are shown in Fig. 2 for t@\ 0.9, *t \ 0.1 andu \ u@\ 0.8.

As seen from Fig. 2, the energy spectrum contains threeup-spin and three down-spin energy bands. The middle local-ized up-spin and down-spin bands mainly come from theunpaired electrons on the side radicals. In the ground state,the lowest two up-spin energy bands and one down-spinenergy band will be Ðlled while the higher three energy bandswill be empty and, consequently, the ground state of thesystem is a high-spin ferromagnetic state, which mainly orig-inates from the attachment of the side radicals, each contain-ing an unpaired electron, to the main chain.

3. Magnon excitationWe now proceed to the investigation of the low-lying mag-netic excitation of the system.

Transforming and into the Fourier componentsaülmp` aü

lmpwith wave vector k, the Hamiltonian [eqn. (3)] can be writtenas

hü \ ;k, p

aükp` M(k)aü

kp ]1

N

] ;k, k{

;q

;m/1

3[u ] (u@[ u)d

m, 3]

] aük`q@2, m, a` aü

k~q@2, m, aaük{~q@2, m, b` aü

k{`q@2, m, b. (6)

Here, is deÐned asaükp`

aükp` \ (aü

k1p` , aük2p` , aü

k3p` ), (7)

whose component (m\ 1, 2, 3) is the Fourier transformaükmp`

of and M(k) is a 3] 3 matrixaülmp` ,

M(k) \1 0

[[(1]*t)eik](1[*t)]0

[[(1]*t)e~ik](1[*t)]0

[t@

0

[t@0

2.

(8)

From the equation

M(k)Vi(k) \ E

i(k)V

i(k) (i \ 1, 2, 3), (9)

we can get the eigenvalue andEi(k) [E1(k) [ E2(k) [ E3(k)]

the eigenvector of M(k). is a three-dimensional rowVi(k) V

i`(k)

vector :

Vi`(k) \ (V 1i* (k), V 2i* (k), V 3i* (k)). (10)

With the transformation

aükmp` \ ;

n/1

3V

mn* (k)cü

knp` , (11a)

aükmp \ ;

n/1

3Vmn

(k)cüknp , (11b)

eqn. (6) can be written as follows :

hü \ ;k, p

;i/1

3E

i(k)cü

kip` cükip]

1

N

] ;k, k{

;q

;m/1

3[u ] (u@[ u)d

m, 3]

] ;i1, i2,

Vmi1*Ak ]

q2

B

i3, l4

] Vmi2Ak [

q2

BV

mi3*Ak@[

q2

BVmi4Ak@]

q2

B

] cük`q@2, i1` , a

cük~q@2, i2, a

cük{~q@2, i3` , b

cük{`q@2, i4, b

. (12)

As shown in Fig. 2, the ground state of the system turns outto be a six-band case with a Fermi surface in the gap. Thus,the magnon excitation with spin Ñip can be described in thefollowing forms :2

cüaq`\ ;

kA1(k)cü

k`q@2, 1, b` cü

k~q@2, 2, a

] ;k

A2(k)cük`q@2, 1, b` cü

k~q@2, 3, a

] ;k

A3(k)cük`q@2, 2, b` cü

k~q@2, 2, a

] ;k

A4(k)cük`q@2, 2, b` cü

k~q@2, 3, a

] ;k

A5(k)cük`q@2, 3, b

cük~q@2, 1, a

, (13a)

4002 Phys. Chem. Chem. Phys., 2000, 2, 4001È4004

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cübq`\ ;

kA1@ (k)cü

k~q@2, 2, a` cü

k`q@2, 1, b

] ;k

A2@ (k)cük~q@2, 3, a` cü

k`q@2, 1, b

] ;k

A3@ (k)cük~q@2, 2, a` cü

k`q@2, 2, b

] ;k

A4@ (k)cük~q@2, 3, a` cü

k`q@2, 2, b

] ;k

A5@ (k)cük~q@2, 1, a` cü

k`q@2, 3, b. (13b)

Here and (i\ 1, 2, . . . , 5) are coupling coefficients.Ai(k) A

i@(k)

The equations of motion for magnon excitation areu

a(q)cü

aq`\ [hü , cü

aq`] , (14a)

ub(q)cü

bq`\ [hü , cü

bq`] , (14b)

where and denote the magnon excitation energyua(q) u

b(q)

for the a and b branches, respectively. By introducing theRPA, we can obtain equations for (i\ 1, 2, . . . , 5) fromA

i(k)

eqn. (14a) as follows :

Ai(k)\

Cu

a(q)[ Eq

Ak ]

q2

B] El

Ak [

q2

BD~1

]C

;m/1

3[u ] (u@[ u)d

m, 3]

]GA

i(k)CV

mq*Ak ]

q2

BVmqAk ]

q2

B 1

N;k{

f1(k@ ; m)

[ Vml*Ak [

q2

BVmlAk [

q2

B 1

N;k{

f2(k@ ; m)D

] (1[ di, 5)CAi{(k)V

mq*Ak ]

q2

BVml{Ak ]

q2

B

]1

N;k{

f1(k@ ; m)

[ Ai_(k)V

mq{*Ak [

q2

BVmlAk [

q2

B

]1

N;k{

f2(k@ ; m)D

[Vmq*Ak ]

q2

BVmlAk [

q2

B 1

N;k{

f3(k@ ; m)HD

(15)

wheref1(k@ ; m)\ V

m2* (k@)Vm2(k@)] V

m3* (k@)Vm3(k@),

f2(k@ ; m)\ Vm3* (k@)V

m3(k@),

f3(k@ ; m)\ A1(k@)Vm2*Ak@[

q2

BVm1Ak@]

q2

B

] A2(k@)Vm3*Ak@[

q2

BVm1Ak@]

q2

B

] A3(k@)Vm2*Ak@[

q2

BVm2Ak@]

q2

B

] A4(k@)Vm3*Ak@[

q2

BVm2Ak@]

q2

B

[ A5(k@)Vm1*Ak@[

q2

BVm3Ak@]

q2

B,

andi@\ 3, iA \ 2, q\ 1, q@\ 3, l\ 2, l@\ 2, for i\ 1 ;

i@\ 4, iA \ 1, q\ 1, q@\ 2, l\ 3, l@\ 2, for i\ 2 ;

i@\ 1, iA \ 4, q\ 2, q@\ 3, l\ 2, l@\ 1, for i\ 3 ;

i@\ 2, iA \ 3, q\ 2, q@\ 2, l\ 3, l@\ 1, for i\ 4 ;

q\ 1, l\ 3, for i\ 5.Here, wave vectors o k o, o k@ o and o q o take their values fromzero to p. Equations for (i\ 1, 2, . . . , 5) have similarA

i@(k)

Fig. 3 The dispersion relation of a magnon with t@\ 0.9, *t \ 0.1and u \ u@\ 0.8.

forms to the above equations. These equations can be solvednumerically.

The dispersion relations of magnon excitation for di†erentbranches are shown in Fig. 3 when t@\ 0.9, *t \ 0.1 andu \ u@\ 0.8. It is demonstrated that the magnon excitationspectrum consists of Ðve energy branches : one acousticbranch and four optical branches. The four lower energybranches of magnons refer to while the highesta1, 2, 3, 4 cü

aq`

one b refers to As seen from Fig. 3, the acoustic mode ofcübq` .

the magnon excitation spectrum exhibits a quadratic disper-sion relation when q is small, which indicates thatu

a(q) P q2

the magnon excitation of the system holds the property of theferromagnetic magnons.

Fig. 4 and 5 show several acoustic modes that vary with t@and u@, respectively. It can be seen that the hopping integralbetween a p electron on the main chain and an unpaired elec-tron on the side radical and the Hubbard e†ective electronÈelectron repulsive energy of unpaired electrons on the sideradical produce an impact on the acoustic mode. The excita-tion becomes rather easy when t@ and u@ are small. It is alsonoticed that the excitation decreases more and more rapidlywith decreasing u@, whereas the decrease in the excitationslows down slightly with decreasing t@ when q increases. Thisindicates that large t@ and u@ will strengthen the ferromagnetic

Fig. 4 The acoustic modes for di†erent hopping interactions : (a)t@\ 0.9, (b) t@\ 0.7, (c) t@\ 0.6, (d) t@\ 0.5 and (e) t@\ 0.4 when*t \ 0.1 and u \ u@\ 0.8.

Phys. Chem. Chem. Phys., 2000, 2, 4001È4004 4003

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Fig. 5 The acoustic modes for di†erent Hubbard repulsions : (a)u@\ 0.8, (b) u@\ 0.6, (c) u@\ 0.5, (d) u@\ 0.4 and (e) u@\ 0.3 whent@\ 0.9, *t \ 0.1 and u \ 0.8.

correlation in the system and, hence, be favourable to the sta-bility of the high-spin ferromagnetic ground state of thesystem.

4. ConclusionsWe have studied the ferromagnetic properties of a quasi-one-dimensional p-conjugated organic polymer ferromagneticmodel based on a simpliÐed structure as shown in Fig. 1. Theground state of the system is studied by the mean-Ðeld

approximation. Attachment of side radicals each containingan unpaired electron to the main chain makes the groundstate of the system a stable high-spin ferromagnetic state.

The magnon excitation is studied in the random-phaseapproximation. A magnon excitation spectrum that containsan acoustic branch and four optical branches is obtained. Theacoustic branch has been found to demonstrate the propertyof the ferromagnetic magnon. The hopping interaction (t@)between a p electron on the main chain and an unpaired elec-tron on the side radical and the Hubbard repulsion (u@) ofunpaired electrons on the side radical signiÐcantly a†ect theacoustic mode. The larger t@ and u@, the more stable the high-spin ferromagnetic ground state of the system.

AcknowledgementThis work was supported by the National Natural ScienceFoundation of China under the grant No. 19777101 and19774023.

References1 Y. V. Korshak, T. V. Medvedeva, A. A. Ovchinnikov and V. N.

Spector, Nature (L ondon), 1987, 326, 370.2 K. Nasu, Phys. Rev. B, 1986, 33, 330.3 M. Takahashi, P. Turk, Y. Nakazawa, M. Tamura, K.Nozawa,

D. Shioni, M. Ishikawa and M. Kinoshita, Phys. Rev. L ett., 1991,67, 746.

4 N. Mataga, T heor. Chim. Acta, 1968, 10, 273.5 A. A. Ovchinnikov, T heor. Chim. Acta, 1978, 47, 297.6 Z. Fang, Z. L. Liu and K. L. Yao, Phys. Rev. B, 1994, 49, 3916.7 Z. Fang, Z. L. Liu and K. L. Yao, Phys. Rev. B, 1995, 51, 1304.8 Y. Q. Wang, Y. S. Xiong, L. Yi and K. L. Yao, Phys. Rev. B,

1996, 53, 8481.

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