15 June 1998

ELSEWER Physics Letters A 243 (1998) 91-94

Solitons in quasi-one-dimensional v-conjugated ferromagnets

W.Z. Wanga, K.L. Yaob, H.Q. Lin”

PHYSICS LETTERS A

organic

a Depa~eat of Physics, Hung Un~versi~ of Science and Tec~o~gy~ Wuhan 430074, China b CCAST (World Laboratory), Betjing 1OOO80, China

c Department of Physics, The Chinese University of Hong Kong, Shatin, m Hong Kong. China

Received 28 January 1998; accepted for publication 5 March 1998 Communicated by J. Flouquet

Abstract

Quasi-one-dimensional P-conjugated organic polymer ferromagnets are studied self-consistently. It is shown that due to the electron-phonon interaction and the Hubbard energy, there exist domain wall solitons describing the displacement of lattices along the main chain, and spin solitons describing the localization of spin at side radicals. The spin soliton corresponds to the highly degenerate band in the middle of the energy spectra, which gives the main cont~bution in the high-spin state of the system. @ 1998 Elsevier Science B.V.

PACS: 75.30.D~; 71.2O.Hk

Quantum effects in low-dimensional materials are a fascinating subject which has attracted much inter- est over the years. In local electron systems, some notable examples are ferromagnetic and antiferromag- netic chains described by the Heisenberg Hamiltonian with local spin. Linear and nonlinear excitations [ 11, namely spin waves and solitons, are obtained when spin correlation is taken into account. The soliton ex- citation arises from the spin wave interaction and has different configurations, e.g. envelope solitons [ 21 and kink solitons [3]. In noninteracting systems, a conducting polymer such as polyace~lene is a well-known representative, which is studied with the SSH Hamiltonian [4] describing electron hopping, electron-phonon coupling and the distortion of the lattice. Self-localized nonlinear excitations (solitons, polarons, and bipolarons) are obtained due to the elec~on-phonon interaction [ 5 1.

In recent years, with the discovery of organic fer- romagnets from organic molecules and hydrocarbons, such as m-PDPC [ 61 and p-NPNN [7], the search for the origin of ferromagnetism in organic ferromag- nets has become a challenge [ 8,9]. A simplified struc- ture of qu~i-one-Dimensions organic polymer ferro- magnets is shown in Fig. 1. The main chain consists of carbon atoms each with a m-electron and R is a kind of side radical containing an unpaired electron. Do solitons exist in this system? Considering the itin- eracy of zr-electrons, the Hubbard electron-electron correlation, el~~on-phonon coupling and the antifer- romagnetic spin correlation between the ?r-electrons and side radicals’ electrons, Fang et al. [ lo] obtained a ferromagnetic ground state using periodic boundary conditions. There exists a spin-density wave (SDW) with alternation of the sign and the amplitude of the spin density along the main chain. In fact, the un-

03759601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. fWSO375-9601(98)00194-7

92 WZ. Wang et al. /Physics Letters A 243 (1998) 91-94

. ..m... R R

Fig. I. A chain of a quasi-one-dimensional organic ferromagnet.

paired electrons at side radicals are not completely localized. They can hop between the main chain and the side radicals [ 111. In this Letter, we consider the hopping of the r-electrons and side radicals’ elec- trons, the Hubbard electron-electron repulsion and the strong electron-phonon (e-ph) interaction. We will use the free boundary conditions since a realistic poly- mer chain is open at two ends. With the self-consistent numerical method, we obtain a high-spin state with the static spin soliton describing the localization of the side radicals’ spin and the domain wall soliton de- scribing the distortion of the lattice along the main chain. Different e-ph interactions result in different configurations of solitons.

The Hamiltonian of the system in Fig. 1 is given by

H = - c E rc~ + ~(ur - UI+I ) 1 (&u+~a + H.c) II7

where c& (cila) denote the creation (annihilation) operator of a 7r-electron (i = 1) along each main chain or an unpaired electron (i = 2) at a side radical with spin (T on the 2th site, to is the hopping integral of the r-electron along the main chain, Tt is the hopping integral of the r-electron on the main chain and the unpaired electron at the side radical, y is the electron- lattice coupling constant, ul is the displacement of the Zth site on the main chain, K is the elastic constant of the lattice. U is the Hubbard repulsion term and nilc = cigcilc (CT = a, /3), where cy and p denote the up-spin and down-spin, respectively. We assume that the side radicals connect with the odd carbon atoms, then Al = 1 for odd sites and Al = 0 for even sites.

It is convenient to cast all quantities into dimension- less form as

YI = C-1) r(w -w+1)y

to7TK to . (2)

Since we use the tight-binding approximation in the Hamiltonian ( 1) , the wave function of the system can be expanded in site basis functions in the Wannier representation [ 121,

where IO) is the true electron vacuum state, PP denotes the ,~th eigenvector of the Hamiltonian, and Z$ is the expansion coefficient.

We will numerically solve the Schrodinger equation of the system,

-11+ 6l)‘Y~lZ,q,+, - [1 + w’-lYl-llz;*,_l

- t,AlZi2, + z4 (

c Z$,,Z$, Za = e;Z;il, >

41 P’

Co=) (4)

u ( c z$21z721

> Z12,Ar - rlZ;,,Al = E;Z;~~.

LL’ Co=)

(5)

Here, ti = CY&, + /IS,,, and we have used the mean- field approximation nilo = (nilu) + Anil,, where (. . .) is the average with respect to the ground state, Anus is the fluctuation from the average value. The electron density (nilC) and spin density Snil can be obtained self-consistently as

(CL)

h = i( (nil,) - (nirp) > . (6)

In Eqs. (4)-( 6)) (occ) means the states occupied by electrons. The dimerization parameter ye can be obtained by minimizing the total energy of the sys- tem with respect to ye. The energy eigenvalue E$ and the expansion coefficients Zz, can be obtained from Eqs. (4)-( 6) self-consistently.

We consider a chain of an organic ferromagnet as shown in Fig. 1. The chain contains N = 5 1 carbon atoms each with a 7r-electron. The side radicals, each with an unpaired electron, connect with the odd car- bon atoms. We will use the free boundary conditions

U!Z. Wang et al. /Physics Letters A 243 (1998) 91-94 93

-“.2 / <- I

_/

I’ , I

I’

- up-spin

_ _ down-spin

-31 0

I I I 20 40 60

energy level number

60

Fig. 2. The energy spectra of the system, i indicates the number of the energy levels.

(FBC) to solve Eqs. (4)-(6). In order to study the low-energy state, we always fill the 7r-electrons and side radicals’ electrons in the possible lowest levels at every iterative step.

First, we discuss energy levels of the 7r-electrons and the unpaired electrons at side radicals. Fig. 2 shows the energy spectra containing three down-spin energy bands and three up-spin energy bands. The middle two band are highly degenerate. The energy spectra are half filled since there is one electron on every site. So the ground state of the system is a high- spin ferromagnetic state.

Second, we discuss the distribution of spin density. There are alternative spins at odd sites and even sites along the main chain. For the parameters h = 0.4, u = 1.0, and tl = 0.9, the spin density is 0.04 for odd sites and -0.18 for even sites. There appears a SDW along the main chain. However, the spin density configuration at side radical sites forms a soliton with envelope shape. Fig. 3 shows that the appearance of solitons is affected by the electron-phonon interaction A. For A < 0.38, there is no soliton. The spin density is uniform at side radical sites except at two ends of the chain. For 0.38 6 A < 0.46, one soliton appears at the middle of side radical lattice. The spin density amplitude at the center of the soliton is much smaller than that elsewhere. For A > 0.46, two solitons ap- pear symmetrically about the center of the side radi- cal lattice. We also find that with increasing Hubbard

(4

IL h=0.2 -0.3

-0.2 (c)

-0.3 i A h=0.47

A I

site number i

Fig. 3. The spin density at side radical sites for different e-ph interaction A.

e-e repulsion, the soliton amplitude reaches a maxi- mum at u = 1. This means that too great or too small a Hubbard repulsion delocalizes the spin density. The calculation shows that the soliton state corresponds to the middle (N - 1) /2-fold degenerate energy band in Fig. 2. Hence, the high-spin ground state is mainly contributed by the soliton state or side radical spins.

Further, we discuss the distortion of the lattices along the main chain. In one-dimensional conducting polymers such as polyacetylene, due to the electron- phonon interaction, there exist various domain soli- tons [ 51. For the quasi- 1D organic ferromagnetic chain shown in Fig. 1, when the e-ph interaction A < 0.38, distortion occurs only at the two ends of the main chain. This result is shown in Fig. 4a. For 0.38 < A < 0.46, there appears a domain wall soliton in the middle of the main chain. For a long chain, the soliton corresponds to a displacement configuration that approaches the A phase with the dimerization yl = yo for N + +oo, and approaches the B phase with yl = -yo for N --) --oo. For A = 0.4, yo = 0.273, this configuration is shown in Fig. 4b. Fig. 4c shows that for A 2 0.46, the soliton and antisoliton appear

94 WZ. Wang et al. /Physics Letters A 243 (1998) 91-94

x=0.2

0.4 7

_ (b)

0.8 , 1

site number i

Fig. 4. The dimerlzation parameter yl for different e-ph interac- tion A\.

alternatively along the main chain and their widths de- crease. In this case, the soliton superlattice is formed along the main chain.

Comparing the result in Fig. 4 with that for poly- acetylene [ 51, we find the A phase and the B phase correspond to the two-fold degenerate ground state. In order to explain this result, we use the periodic bound- ary conditions [ 131 (PBC) corresponding to N -+ $00 and obtain a uniform state with perfect dimeriza- tion and lower energy than that for FBC. This means that the fe~omagnetic state for PBC is the ground state of the system [ 111. We also find that for PBC, the spin density on the main chain is similar to that for FBC.

However, the spin density at side radicals is dis~ibuted homogeneously and is -0.36 for the parameters A = 0.4, u = 1 .O. Comparing this result with that in Fig. 3b, we can see that the background line of the spin soliton for the FBC corresponds to the spin density at side radicals for the PBC. The solitons just described are the solutions of the nonlinear equations (4) and (5) and are the static solitons. From the point of view of physics, the domain-wall soliton results from the strong e-ph interaction, but the spin soliton originates from the Hubbard e-e correlation and e-ph coupling.

This work is supported by the National Natural Sci- ence Foundation of China and by a Direct Grant for Research from the Research Grants Council of the Hong Kong Government of China.

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