Inter-chain coupling effects on the electronic structure of alkali doped trans-polyacetylene

Synthetic Metals, 55-57 (1993) 4278-4283

4278

INTER - CHAIN COUPLING EFFECTS ON THE ELECTRONIC STRUCTURE OF ALNALI DOPED

T R A N S - P O L Y A C E T Y L E N E

S. TABOR

Institute of Physics, A. Mickiewicz University, Poznafi (Poland) S. STAFSTROM Departmellt of Physics, Link6ping University, Link6ping (Sweden)

ABSTRA('T The ell'oct of inter-chain coupling Oll the electronic properties of alkali doped /ra~S-l)olyacetylene has heen studied. Systems of three and four coupled chains at. different dopant concentrations (4-17%) and differenl distributions of countel'ions have been considered. As |heoretical model, the Peierls-Huhbard Hamiltonian in the Hartree-Fock a pproxinlatiou has been used. Tile bond lenghts of solitons of each chain have been self-consislently optimized. For both the three and four chains systems, a two-fold periodicity distribution of ions changes the electronic properties of the system remarkably, l)articularly tile euergy gap around the Fermi level. We also discuss the ol)timal distribution of counterions fl'om the point of view of the total energy.

INTRODUCTION Tile insulator (semiconductor)-metal transition in highly doped

trans

-

polyacetylene

(/raT~s - [CH]j.), (in particular tile arising Pauli susceptibility [1]) is one of the lnost interesting phenomena ill tile field of conducting polymers. The Peierls effect, whiela is particularly strong ill quasi-one dimensional systems like t r a n s

-

[CH]~:, creates a gap around the Fermi

level in the undoped state and a corresponding dimerization of the lattice. The material is thus non - metallic. The strong electron-phonon coupling also results in the creation of localized charged soliton defects upon doping of the system [2]. The doping process therefore changes the electronic structure of a single chain and a metallic state is reached asymptoticly for heavely doped samples. Consequently there is no phase transition. Inter - chain couplings have been used [3-5] to explain the metallic behaviour of heavily doped

trat,

s -

[6'H]~. These interactions reduce the Peierls effect, and lead to a band

broadening and consequently to a reduction of the energy forbidden region, i.e. the gap. Elsevier Sequoia

4279

In Ibis l)aper we study some aspects of the geometrica.l distribution of COUlderions.

A

i)ossible desertion fio,n an ideal ordered (uniform) distribution is considered, namely a mixed density of ions along the systenl of chains (see Fig.l). Our studies include mixtures of lwo di|[erenl densilies of counterions. The lwo halfs of each chain have been dol)ed with different ion COllcentrations (two fold ion periodicity). This type of the distribution of counlerious has been applied Io systems of lhree and four interacting chains, corresponding 1o the case of

A.(Li) and l((Rh. ('.q) doping, resl)eclively [6].

~ O O C C O 0 0 0 0 0 0 0 0 0 0 0 C O O O C O 0 0 C O C O C O C C C O O C

high ion concentralion area

low ion concentration area

I"igure 1: lligh and low counlerion concentration in h',n.q- [('H], (elnpty circles-- ('1t unils, filled circles

counterions).

MI:TIIODOLOGY The following Hamiltonian is applied to the system of interacting chains:

(1

H = H s s n + H~l-,t + Hpot + H±

ltssn is the well known Su-Shrieffer-Heeger Hamiltonian [2]. H,t-,a is the electron~lectron interactions operator: H,t-,l

=

1/2

:,,,,,j.j ..q ,+j . ~ c , . j•, . q , , , ,+, . , c , , , y . . ,•

~

•

(2)

i,i',j,jLa,a'

where Ui.i,jj, is the effective Coulonfl) integral between 2p: orbitals attached to sites i and i' on tile j - - t l l and j ' - t h polynler chain. The exponentially screened Olmo expression [7] is used for this integral:

Uoea:p(-71r,,e,j.j,I)

Uci,,j,j, = (I + 0.6117r~.i,j.y)l/2

(3)

Hpo, describes the potential energy due to dopand ions. It is chosen in the form of the screened (!oulomb potentiah

H,o, = ~ l;c+j,.c,,j. t .3.¢r

(4)

4280 v = -~

~

1 - , > , , , l l - ~ . , v ( - - , I,,.,,.

.

I)

(r,)

e ( r i , , + d 2)

where ~' measures a distance along tile chain, d is a tl'ansverse ion chain distance, the screening I)al'anleler

3 is identical to that used ill Eq.3 and tile cut-off function q is chosen Io fit Ill<'

resulls of Hef.[3]. Finally. lhe hoplfing l)erpendicular to the chain is included in the term H±: H± =

~ + ~±.i(e~.j.,.c,./.. + q+ . , . . .c. ~.,.) i..t..i',e~

(6)

The value of tj_,i is mo(lulated along the chain in order to take inlo account the enhancenlent of the inlerchain hoplling due to counterions [8, 9]. The vahle for lj_,i is set, to 0.1 eV for sites nol coinciding with the i)osition of a counterion and 0.24 eV for sites laying opposite to a count(-rion. These vahles of t±, are sufficient to reach the metallic state[10]. The llarlree Fock al)l),'oxinlalion is applied to Ill(" four l)arlicle term in Eq.2. T h e tolal energy o[" the sysleul is minimized v
disF, lacenwnt. A

F,roce,.lure v.'hich has beell F,resented in detail in Ref.[11]. T h e cal,:utat ions are performed on h ' a , s -

[( "H].,. system, folmed by three and four chains

each conlaining I l 1 ('11 units surrounding tile linear array of counlerions. Periodic boundary condilions have been assulned. I{ I:~S1r I:I'S VG' have conq,ared the results obtained for tile nonuniform ion distribution with lhose obtained in the case of the uniform distribution. As a nonuniform distribution we consider the systenl, where the concentration in the first half is equal ,qx and ill tile second half is .q2- Two 91 :!12 prol)ortions have been studied, namely, 1:2 and 2:3. In Fig.2 is shown tile electronic gap as a fimction of the dopant concentration for a system of 3 chains. The mixed dopant concentration results in a smaller gap, in particular in the high doping regime (above 8%), where the gap is reduced by a factor of two. Note that the gap for the nonuniformly doped system is smaller at. intermediate doping levels then for the uniform distribution at, high doping levels. This effect, of the different periodicities of the counterions distribution is more evident for a system of 4 interacting chains (see Fig.3). In this case the gap is wider than for three inleracting chains and it. is necessary to include a desertion fi'om uniform distribution 1o reach tile metallic state. From the point of view of the total energy, this non - uniform distribution is prefered: the reduction of total energy (including the ions repulsion energy) is in range of 2 ~ of tile energy of the uniform distribution of counterions. The spat, ial distribution of solitons in the case of a non - uniform distribntion of counterions is remarkably different from that obtained in the case of a uniform distribution of counterions:

4281

I

A/- [,,v] L

I).:/

11.2

\" \'tl

0.1 L I I1

5

I

I

10

15

,-pT,,]

F tmre "2: l(h'ctronic gap as a function of dol)ant concentration for the :1 chains svsWm. Solid lira'

unifol'nl distl'il)ulion: dashed line

proportion of densilies 1:2: ellll)ly circles

i)l,q)orl iOli of dellsit ies 2::J.

AI:"

[e\']

(I.1

0.:/ o- . . . . . \

0.:2

%

\

°'I

\ \

\ \ I

0

\

•

5

b

I

10

15

,-['7<,]

Figure 3: Electronic gap a.s a. function of dopant concentration for the 4 chains systeln: the same symbols as in Fig.2.

the centers of tile solitons are not located ol)posite to the counterions positions, instead the solilo,> a c c u m u l a t e in the region of a higher counlerion density (see Fig.4).

This effect

is explailled hv the i'act that. tile ])otelllia] which is strollger ill the region of high dopin?£. a t t r a c t s the ol)positcly charged solilons.

4282

0.]

0.0: (

-0.0:

I

I

i

I

I

I

I

20

40

60

80

100

120

140

0.1

0.05

£

-0.05

I

20

I

i

i

~

~

~

i

1~

I

1~

I

1~

Figure 4: Soliton distributions for two-fold ion periodicity in a system of 3 chains; ul)per ctu've corresl)o,ds lo a uniform counterion distril)ntion and the lower to a nonuniform counlerion (listribution: dopant concentration y = 6.67%. Counterion./positions signed by +.

CONCLUSIONS Tile inter - chain interactions have been shown to lead to a reduction in the energy gap around the Fermi level for doped t r a n s - [ C H ] ~ .

Howeover, for t , ' a n ~ - [ C H ] ~ doped with

K or a heavier alkali metal (4 chains system) tile phase transition is not evident for uniform counterions distribution. Ill this case a mixed density of dopants are shown to lead to ~ closure of the energ3' gal). The effect of a closing is weak at low dopant concentrations but become strong above a critical concentration. The self - consistent method of the geometry optimisation used in this study made it possible to find tile additional effect - an unusually high concentration of solitons in the region of the high dol)ant concentration of the system. We believe that this structural effect is responsible for changing of electronic properties under the doping with locally mixed densities.

4283 ACKNOWLEDGEMENTS

S.Tabor would like Io thank LinkSping Ihfix'ersitv for lhe finaucial SUl)l)ort. This work is SUl)l)orl¢'cl I)v the Swedish Natural Science Foundation, grant no. F - F I I 8785

305.

l{ I':FEI{ EN( 'ES [1] 11. Shirikawa i S. Ikeda. Syl~lhetic Mela.ls 1 (1979/1980) 175. [2] \V.-I'. Su, .I.I:{. Shl'ieffer i A..I. lteeger, Phys. Rev. B 22 (1980) 2099. [:3] 1". M. ('ouwelt, I-I. A. Mizes and S. Jeyadev, Phys. Rev. B 40 (1989) 16:30. [1] S. Tabor and S. Stafslr6uu J. Magu. Ma.gn. Mat. 104 107 (1992) 2099. [5] E..1. Mele and M..]. Rice, Phys. Rev. B 23 (1981) 5397. [6] N. S. lklu,'thy, L. \V. Shacklette and R. H. 13aughman. Phys. Rev. B 41 (1990)3708. [7] I1. Fukutome and M. Sasai, Pros. Theor. Phys. 69 (1983) 337. [8] I{..I. ('ohe,l aud A. J. Glick, Phys. Rev. B 42 (1990) 7658. [9] P. Ikl. ( h a n t and I. P. Batra, Solid State (',ommun. 29 (1979) 225. [10] It. A. [\Iizes aud E. M. Couwell. Phys. Rex'. B 43 (1991) 9053. [11] S. SlafstrSm, Phys. Rev. B 43 (1991) 9158.