Green function of conjugated oligomers: the exact analytical solution with an application to the molecular conductance

Materials Science and Engineering C 8–9 Ž1999. 273–281 www.elsevier.comrlocatermsec

Green function of conjugated oligomers: the exact analytical solution with an application to the molecular conductance Alexander Onipko a

a,)

, Yuri Klymenko b, Lyuba Malysheva

c

Department of Physics, IFM, Linkoping UniÕersity, 581 83 Linkoping, Sweden ¨ ¨ b Space Research Institute, KieÕ, 252022, Ukraine c BogolyuboÕ Institute for Theoretical Physics, KieÕ, 252143, Ukraine

Abstract The exact analytical expression of the Green function of oligomers M–M– . . . –M and M 1 –M 2 –M 1 – . . . –M 2 –M 1, where M, M 1, and M 2 are monomers of arbitrary p electronic structure described by the tight-binding Hamiltonian, is derived for the first time. This result makes possible to address relevant spectral and electron transport properties of large linear molecules on the basis of realistic exactly solvable models. The power of the approach is exemplified by obtaining a number of explicit relations between the transport related quantities, in particular, through molecule tunneling decay constant, and the molecular electronic structure for a wide family of potential molecular wires. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Green function; Conjugated oligomers; Molecular conductance

1. Introduction Much attention has recently been paid to the application of Green function method in calculations of spectroscopic and electron transport related quantities for polyatomic molecules w1–12x. For instance, the Green function technique has been extensively used for a perturbative description of bridge-mediated electron transfer between donor and acceptor w2x, as well as for obtaining model exact analytical expressions of the electronic factor in the intramolecular electron-transfer rate w4x and linear-response molecular conductance w5x. However, mostly, the Green function approach has been used for the development of efficient schemes to compute the electronic factor w3,7–9x and molecular conductance w10–12x. The reported analytical treatments of long-distance electron transferrtransmission across molecules have been essentially restricted to the molecular chain modeled by a linear sequence of one-site one level subunits coupled via a constant nearest-neighbor hopping integral w1,4,5x. This model is known to give a single band of one electron states, that is nothing else but 1-D metal. On the other hand, the molecules tried in electrical measurements as ) Corresponding author. IFM, Linkoping University, 581 83 Linkoping, ¨ ¨ Sweden. Tel.: q46-13-281226; fax: q46-13-132285; E-mail: [email protected]

molecular wires w11–15x are inherently either dielectrics or semiconductors but not metals. Moreover, as it is evidenced experimentally, the Fermi energy of metals used to contact these molecules falls somewhere within the band gap of the molecular spectrum. Under such conditions, the molecular electron states from both valence and conduction bands are expected to contribute the electron tunneling across molecules connecting two metals. For these reasons, the above mentioned model of molecular chain, though useful, is not adequate to reproduce the performance of real molecular wires. To take into account the molecular band gap, a more sophisticated model has recently been proposed w16x that refers to CH- and SN-chains. A far more reach variety of linear organic molecules can be approached analytically by the Green function technique. To describe the electrical properties of metal–molecular–metal heterojunctions, the Green function of the molecule spanned between metal electrodes has to be found with account to the molecule–metal interaction w5,10,17x. The perturbation of molecular levels can be included by a proper definition of the self-energy operator. An important point to note is that the effect of molecule– metal interaction is taken into account not in a perturbative manner but exactly w17x. Thereby, the problem to solve is finding the resolvent of a complex Žbecause of adding self energy to the molecular Hamiltonian. hermitian matrix of the same order as the matrix of non-perturbed molecular

0928-4931r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 8 - 4 9 3 1 Ž 9 9 . 0 0 0 3 8 - 7

274

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

Hamiltonian. By further redefinition of the self energy the electron–phonon interaction as well as charging effects can also be included into consideration, at least approximately w12,17x. In all the applications mentioned Žand many others, e.g., in the chemisorption theory w18x., the knowledge of unperturbed molecular Green function greatly facilitates calculations. The formal reason for the simplification is that provided the molecular Green function is known the problem of inverting the matrix of the order determined by the basis set of the molecule is reduced to the order determined by the number of atoms interacting with the metal. The latter is usually much less than the total number of atoms in the molecule. For instance, it can be taken as small as two w5,10–12x. Furthermore, the unperturbed Green function matrix elements referred to the end binding sites of the molecule Žwhich connectsrbridges electrodes in metal–molecular heterojunctions or donor and acceptor in donor–bridge– acceptor systems. determine the through-molecule tunneling rate w4,5,19,20x. In particular, from the Green function expression the tunneling decay constant can be deduced. By using the latter quantity, the charge transmitting abilities of different molecules can be compared with a clear reference to the molecular electronic structure. Otherwise, the estimate of the tunneling decay constant, not to mention a detailed analysis of molecular conductance, requires a large volume of computational work. In view of the system complexity, this kind of calculations is usually performed on the basis of approximate semi-empirical methods w10,21,22x with much uncertainty left with regard to their accuracy. It makes therefore advisable a comparison with rigorous exact results which can be obtained in the framework of simplified but realistic tight-binding models. The conjugated oligomers are considered to be the most likely candidates to act as molecular wires w10–15x. They possess a number of other potentials for molecular electronics w23x. The relevant Green functions are known only for the simplest finite linear sequences of repeating units, such as simple tight-binding chains with constant w24x and alternating w6,25x value of hopping integral. The purpose of this paper is to give the exact analytical solution of the Green function problem for two broad families of conjugated oligomers covered by structural formulas M–M– . . . –M Žoligomers of polyene, polythiophene, polyparaphenylene, etc.. and M 1 –M 2 –M 1 – . . . –M 2 –M 1 Žoligomers of polyaniline, polyparaphenylenevinylene, etc... As an illustration, the solution obtained is then used to describe the molecular tunnel conductance of metal–molecular heterojunctions.

oligomers., where M denotes a monomer of arbitrary chemical structure can be represented as N

Hˆ O M s

Ý HˆnM q Vˆ ,

Ž 1.

ns1

where HˆnM is the Hamiltonian operator of the nth monomer, N is the number of monomers in the molecule, and Vˆ is the energy operator of inter-monomer interaction, N

Vˆ s b int

Ý < Ž n q 1. a :² na < q < Ž n y 1. a :² na < l

r

r

In one-electron approximation, the p electron Hamiltonian of linear conjugated molecules M–M– . . . –M ŽM-

.

Ž 2. In Eq. Ž2., <0a r : s <Ž N q 1.a l : s 0, the ket < na : has its usual meaning of the p orbital of the a th atom in the nth monomer, b int is the energy of resonance p electron transfer Žhopping integral. between the neighboring sites Ždenoted as a r and a l , respectively. of the nth and Ž n q 1.th monomers. Since the further derivation is not restricted to the particular form of operator HˆnM its form will not be specified at this stage. To find the Green function for the Hamiltonian defined in Eqs. Ž1. and Ž2., it is convenient to use the Dyson equation

ˆ ˆO M Ž E . , Gˆ O M Ž E . s Gˆ O Ž E . q Gˆ O Ž E . VG

Ž 3.

which in the matrix representation reads Žhenceforth, the indication of explicit dependence of the Green function on electron energy E is omitted. GnOn M, mm s dn , m GnM, m M q b int GnM, a lGŽOny1. a

r , mm

M q GnM, a rGŽOnq1. a

l , mm

,

Ž 4. M G 0OaM, m s 0, GŽONq1. a r

m

l , mm

s 0,

Ž 5.

where GnM, m s ² n <Ž EIˆy HˆnM .y1 < m : is the matrix element of the monomer Green function operator; indexes n and m take all possible values from 1 to N, while indexes n and m identify the location of atoms within the monomer. In particular, setting in Eq. Ž4. n s a l , a r , and m s a l , a r , we obtain four equations for the matrix elements referring to the binding sites of monomers. Finding the solutions to Eqs. Ž4. and Ž5. is substantially simplified by using the dispersion relation between the energy E and quasi-impulse j of electron in M-oligomers that has the form w26x 2cos j s f Ž E . , f Ž E . s Ž b int GaMl , a r .

y1

Ž 1 y GDM . , Ž 6.

where 2

2. Green function of M-oligomers

l

ns1

2 GDM ' b int GaMl , a lGaMr , a r y Ž GaMl , a r . .

Ž 7.

Representing the Green function matrix elements appeared in Eq. Ž4. for n s a l , a r , and m s a r , as a superposition of incident and reflected waves and exploiting Eq.

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

Ž6., after some lengthy but straightforward algebra we arrive at the following expressions of GnOaM, m a and GnOaM, m a 2 GnOn M, m a s b int GaMl , a rD M sin j

Ž

l

°b

M int Ga l , a l sin j

l

y1

l

r

l

.

Ž N q1y m .

= w sin j Ž ny1. y b int GaMl , a rsin j Ž ny2. x , 2 b int GaMl , a lGaMr , a rsin j

if n s a l ,

Ž N q1y m . sin j n,

if n s a r ,

where operator Vˆ is defined in Eq. Ž2. with N replaced by N y 1; all the other notations have already been introduced. The matrix elements of operator Gˆ O M 1 – M 2 Ž E . s Ž EIˆy O Hˆ M 1 – M 2 .y1 can be found from GnOn M, m1 m– M 2 s Gn0n , mm q b int Gn0n ,Ž Ny1.a GNOaM,1m– M 2 r

n- m, = b int GaMl , a l w sin j Ž my1.

~

y b int GaMl , a rsin j

l

if n s a l ,

Ž my1. y b int GaMl , a rsin j

Ž my2. x w sin j M = w sin j Ž N y n . y b int Ga , a sin j Ž N y1y n . x , l

r

if n s a r ,

nG m.

Ž 8. The expressions of matrix elements GnOaM, m a can be found r

r

° ~

r

GaMr , a r

sin j Ž N y n . y b int GaMl , a rsin j Ž N y ny1.

b int GaMl , a r

sin j

=

¢l n

sin j m DM

Ž 12 .

r

r

G 1OaM,1N– aM 2 s b int l r

s

m

ˆ y Hˆ O M y HˆNM 1 .y1 < mm : and operawhere Gn0n , mm s ² nn <Ž IE O tor Hˆ M is given by Eq. Ž1. with N replaced by N y 1, and M by M 1 –M 2 . For the Green function matrix elements referred to the end atoms of the ŽM 1 –M 2 .-oligomer, it follows from Eq. Ž12.

in a similar way, GnOaM, m a

l

M1 – M2 qGn0n , Na GŽONy1. , a , mm

Ž my2. x sin j Ž N q1y n . ,

¢

275

G 1OaM,11 –aM 2 s l

,

l

if n) m,

m,

if nF m.

Ž 9.

GNOaM,1N– aM 2 s r r

GaMl ,1a rG1OaM,Ž Ny1.a l

2 M 1 y b int GaMl ,1a lGŽONy1. a

r

,

,Ž Ny1. a r r

G 1OaM,1a y GaMl ,1a lGDO M l

l

2 M 1 y b int GaMl ,1a lGŽONy1. a M GaMr ,1a r y GŽONy1. a

r

,

,Ž Ny1. a r r

M1 ,Ž Ny1. a rGD

2 M 1 y b int GaMl ,1a lGŽONy1. a

r

Ž 13 .

,Ž Ny1. a r

In Eqs. Ž8. and Ž9., a new notation is introduced D M s sin j N y b int GaMl , a rsin j Ž N y 1 . .

Ž 10 .

If the electron energy satisfies Eq. Ž6., the solutions to equation D M s 0 determine the eigenvalues of quasi-impulse j in M-oligomers w26x. The equations represented above give the exact explicit expression for any component of the M-oligomer Green function operator Gˆ O M in terms of the matrix elements of monomer Green function operator Gˆ M . In Section 3, they are used to derive the Green function for another wide class of linear organic compounds with a somewhat more complex structure, M 1 –M 2 –M 1 – . . . –M 2 –M 1 , called here for brevity ŽM 1 –M 2 .-oligomers.

where GDM is defined in Eq. Ž7., and 2 M GDO M s b int G1OaM,1a GŽONy1. a l

s

l

r

,Ž Ny1. a r y

Ž G1O ,Ž Ny1. .

2

M

l

sin j Ž N y 1 . y b int GaMl , a rsin j N sin j Ž N y 1 . y b int GaMl , a rsin j Ž N y 2 .

r

.

Ž 14 .

The dispersion relation for ŽM 1 –M 2 .-oligomers has the form of Eq. Ž6., where M is of course a compound monomer M 1 –M 2 . It may be instrumental to use the Green function components of monomers M 1 and M 2 which have a more simple structure than those of the compound monomer. For this purpose one can use the following expressions for the matrix elements of operator Gˆ M 1 – M 2

3. Green function of (M 1 –M 2 )-oligomers The ŽM 1 –M 2 .-oligomers can be considered as Moligomers, M s M 1 –M 2 , where at the chain end M is substituted by M 1. The molecular Hamiltonian is now conveniently represented as

GaMl ,1a–rM 2 s b int

GaMl ,1a–l M 2 s

Ny1

Hˆ O M 1 – M 2 s

Ý HˆnM – M 1

2

GaMl ,2a rGaMl ,1a r 2 1 y b int GaMl ,2a lGaMr ,1a r

GaMl ,1a l y GaMl ,2a lGDM 1 2 1 y b int GaMl ,2a lGaMr ,1a r

,

,

q Vˆ

ns1

q b int < Ž N y 1 . a r :² Na l < q HˆNM 1 ,

Ž 11 .

GaMr ,1a–rM 2 s

GaMr ,2a r y GaMr ,1a rGDM 2 2 1 y b int GaMl ,2a lGaMr ,1a r

.

Ž 15 .

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

276

Expressed in terms of the matrix elements of Green function operators Gˆ M 1 and Gˆ M 2 function f Ž E . in the dispersion relation for ŽM 1 –M 2 .-oligomers takes the form f Ž E. s

1

2 1 y b int Ž GaMl ,1a lGaMr ,2a r

2 b int GaMl ,1a rGaMl ,2a r

gs

qGaMr ,1a rGaMl ,2a l . q GDM 1 GDM 2 .

Ž 16 .

For applications, it is also useful to have the expressions of the components of ŽM 1 –M 2 .-oligomer Green function as functionals of j and matrix elements of operators Gˆ M 1 and Gˆ M 2 . Making use of Eqs. Ž8., Ž9. and Ž14. in Eq. Ž13., and taking into account the relations Ž15. and Ž16. one can find G 1OaM,1N– aM 2 s l

r

GaMl ,1a rGaMl ,2a rsin j GaMl ,2a rsin j N q GaMl ,1a rGDM 2 sin j Ž N y 1 .

,

Ž 17 . G 1OaM,11 –aM 2 l

s

l

GaMl ,1a lGaMl ,2a rsin j N q GaMl ,1a rGaMl ,2a l sin j Ž N y 1 . GaMl ,2a rsin j N q GaMl ,1a rGDM 2 sin j Ž N y 1 .

,

Ž 18 . GNOaM,1N– aM 2 r

s

r

GaMr ,1a rGaMl ,2a rsin j N q GaMl ,1a rGaMr ,2a rsin j Ž N y 1 . GaMl ,2a rsin j N q GaMl ,1a rGDM 2 sin j Ž N y 1 .

It is worth noting that as in the case of M-oligomers, for the given kind of linear molecules the bilinear combination of Green function matrix elements 2 b int G 1OaM,11 –aM 2 GNOaM,1N– aM 2 y G 1OaM,1N– aM 2

s

l

r

r

ž

l

r

2

/

GDM 1 GaMl ,2a rsin j N q GaMl ,1a rsin j Ž N y 1 . GaMl ,2a rsin j N q GaMl ,1a rGDM 2 sin j Ž N y 1 .

,

8 e2 h

¦

1a l <

2

Im S l Ž EF . Im Sr Ž EF . E F Iˆy Hˆ O M y Sˆ l Ž E F . y Sˆr Ž EF .

< Na

; r

,

Ž 21 . where the self energy operator Sˆ lŽ EF . q Sˆr Ž EF ., which takes into account the interaction between molecule and metal pads, is defined as ² n < Sˆ lŽ r . Ž E F . < m : s 2 Ž . dn , m dn ,1a Ž Na . b lŽr . G E F , b lŽr . is the molecule–metal coul r pling constant, and GŽ E F . is the surface diagonal matrix element of the Green function operator for a free electron in semi-infinite cubic lattice. In this simplest model of metal–molecule–metal heterojunction, the shift and broadening of molecular levels that interact with metal are determined by the real and imaginary parts of product 2 b lŽr. GŽ EF .. Eq. Ž21. has been used to examine the molecular wire properties modeled by chains with a constant w5x and alternating w27x value of the hopping integral. According to experimental data w11–15x, the Fermi energy is usually far from the molecular levels and the metal–molecule interaction is weak. Hence, to estimate the molecular conductance in the first approximation, one can use Eq. Ž21. with the Green function of bare molecule, i.e.,

.

Ž 19 .

l

the molecule wM- or ŽM 1 –M 2 .-oligomerx is coupled electronically with metal pads only via its end atoms, it can be shown that the molecular conductance is given by

Ž 20 .

which is an analogue of Eq. Ž14., has poles of the same order as that of the Green function itself.

4. Molecular tunnel conductance The ohmic conductance of a molecule that connects two metal pads is determined by the Green function of the whole system w5,10x Žconsisting of the source and drain electrodes and the molecule itself. which is taken at the Fermi energy E F . Under the assumption that the electrodes can be modelled by a semi-infinite 3-D cubic lattice with the nearest-neighbor electron-transfer interaction and that

gs

8e 2 h

2

Im S l Ž E F . Im Sr Ž EF . G 1OaM, Na Ž E F . . l

r

Ž 22 .

This expression makes apparent the fact that, in addition to the effective metal–molecule coupling constants Im S lŽ EF . and Im Sr Ž EF ., the molecular electronic structure and particularly, the width of the band gap, and the relative position of the Fermi energy with respect to the molecular levels play a decisive role in determining the efficiency of the through band gap electron transmission between metal electrodes. The gross structure of the one-electron spectrum of conjugated oligomers is described by Eq. Ž6.. In particular, it determines the boundaries of allowed Žreal values of j in the interval w0,p x. and forbidden Žcomplex values of j , j s "i d or j s p " i d . zones and also the dispersion relations EŽ j . and EŽ d . that correspond to these zones. The band boundaries determine the energy intervals of delocalized electron states of the oligomer. The number of such states in the band is proportional to the molecular length, that is to the number of monomers. It follows from Eq. Ž6. that the band boundaries which correspond to j s 0 and to j s p are determined by equations f Ž E . s 2 and f Ž E . s y2, respectively.

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

In the case of periodic boundary conditions, the band gaps would be free of electron states. The chain ends, however, play the role of defects. Under certain conditions specified below, even in ideal oligomers consisting of identical monomers, there exist in-gap Žlocal. electron levels. Strictly speaking, approximation Ž22. is not valid, if the Fermi energy is near in-gap states. Therefore, whether or not the local levels are present in the molecular spectrum is an important question to answer. The presence or absence of in-gap states, as well as the band joining, band degeneracy and other important features of the gross structure of oligomer electron spectrum

277

can be overseen at the monomer level by examining the monomer Green function properties. In particular, the p electron spectrum of M-oligomers does not contain in-gap states, if w26x < b int GaM, a < - 1 l

r

Ž 23 .

at energies that satisfy equation GaMl , a lGaMr , a r s 0,

Ž 24 .

the solution to which may exist within the band gap energy intervals only.

Fig. 1. Examples of chemical structure of conjugated oligomers Žfrom top to bottom.: M s C 2 H 2 ; M s C 6 H 4 ; M 1 s C 6 H 4 , M 2 s C 2 H 2 ; M 1 s C 6 H 4 , M 2 s C 4 H 4 ; M s C 4 H 2 X; M 1 s C 4 H 2 X, M 2 s C 2 H 2 ; M 1 s C 4 H 2 X, M 2 s C 4 H 4 . X denotes heteroatom, for instance, S. Oligomers of comparable length have the same number of carbons in row Ž NC .. N and N y 1 indicate the number of monomers M and M 1 –M 2 in M- and ŽM 1 –M 2 .-oligomers, respectively.

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

278

An extension of this result to the case of ŽM 1 –M 2 .oligomers leads to the following condition: GaMl ,1a r GaMl ,2a r

GDM 2 - 1

Ž 25 .

at the energies determined by

Ž

GaMr ,2a r y GaMr ,1a rGDM 2

.Ž

ž

GaMl ,2a l y GaMl ,1a lGDM 2

. s 0,

Ž 26 .

which ensures the absence of in-gap states in the ŽM 1 – M 2 .-oligomer spectrum. Obviously, the through band gap electron tunneling is substantially dependent on whether the in-gap levels exist or not. In what follows we restrict our consideration by the assumption that, in the respective cases, inequalities Ž23. and Ž25. are valid. To calculate the molecular tunnel conductance given by Eq. Ž22., one needs only the expression of the Green function matrix element, G 1OaM, Na or G1OaM,1N–aM 2 . As explained l r l r above, these must be taken at energies within the band gap, where < f Ž E .
d s ln < f Ž E .
ž

( f Ž E . r2

2

/

y1 ,

Ž 27 .

where the choice of the sign in front of the square root accounts to condition Ž23. wor Eq. Ž25.x. For complex values of j s p " i d and j s "i d Eq. Ž8. for n s N and m s 1, and Eq. Ž17. give G 1OaM, Na s l r

related to the molecular electronic structure. Specifically, we have Žin units 2e 2rh. g s g 0 g 0mol ey2 d N , Ž 30 . 2 Ž . Ž . where g 0 s 4 Im S l EF Im Sr E F rb int , and factors 4sinh2d mol g0 s Ž 31 . 2 y1 b int GaMl , a r " ey d

N Ž .1. GaMl , a rsinh d

sinh Ž d N . " b int GaMl , a rsinh d Ž N y 1 .

/

and g 0mol s

4 Ž b int GaMl ,2a rsinh d .

Ž GaM, a rGaM, a . GDM eyd . 2

l

eh ,

g int DM

,

M sC 2 H 2 ,

eh ,

M sC 4 H 4 ,

~2coshh , X

M sC 6 H 4 ,

X

eh Ey ´ X .

¢ Žg ° q

2 X

X

Ž E 2 yey2 h . ,

M sC 4 H 2 X,

E,

g int DM

M sC 2 H 2 ,

2 E Ž E 2 ye 2 h yg int .,

M sC 4 H 4 ,

~E w E y2coshŽ2h .y1x , X

2

M sC 6 H 4 , X

E Ž Ey ´ X . w E 2 y2cosh Ž 2h . x

¢ °

Ž .1. GaMl ,1a rGaMl ,2a rsinh d , sinh Ž d N . . GaMl ,1a rGDM 1 sinh d Ž N y 1 . Ž 29 .

respectively, where the upper sign corresponds to j s p " i d , and the lower sign corresponds to j s "i d Ž d ) 0.. The above expressions determine the molecular tunnel conductance in the case of weak molecule–metal coupling w28x. Also, as mentioned in Section 1, Eqs. Ž28. and Ž29. can be used for estimating the electronic factor in the non-adiabatic oligomer-mediated electron transfer rate w19,20x. The strict proof of these relations is given here for the first time. Under the condition expŽ d N . 4 1, Eqs. Ž28., Ž29. and Ž22. recover the exponential dependence of the tunnel conductance on the oligomer length with the pre-exponential factor and the tunneling decay constant explicitly

r

°

N

r

l

For the monomers in focus the Green function matrix elements GaMl , a r and GaMl , a l s GaMr , a r, and bilinear combination GDM calculated in the tight-binding approximation take the form

b int GaMl , a l s

l

r

Ž 32 .

2

2

5. Tunnel conductance of aromatic-ring based oligomers

Ž 28 .

G 1OaM,1N– aM 2 s

1

multiplied by expŽy2 d N . represent a purely molecular contribution into the conductance of metal–molecular heterojunctions based on M- and ŽM 1 –M 2 .-oligomers, respectively. In Section 5, we specify further this contribution in the particular case of aromatic-ring based oligomers without ŽM-oligomers. and with C 2 H 2 or C 4 H 4 group ŽŽM 1 – M 2 .-oligomers. between the five- and six-membered rings, C 4 H 2 X and C 6 H 4 , see Fig. 1 where X stands for heteroatom.

b int GaMl , a rs

and

2

X

yg X2 Ž E 2 yey2 h . ,

M sC 4 H 2 X,

1,

M gy1 int GD s

g int

M sC 2 H 2 ,

2 E 2 yg int ,

~Ž E ye 2

DM

¢Ž

2h

M sC 4 H 4 , X

X

.Ž E 2 yey2 h .

X = w E 2 y2cosh Ž 2h . q1 x

y1

,

M sC 6 H 4 ,

X

Ey ´ X . Ž E 2 yey2 h . ,

M sC 4 H 2 X,

Ž 33 . where

°

E 2 ye 2 h , 2

~

Ž E ye

M sC 2 H 2 ,

2h 2

.

2 y E 2g int , X

D M s w E 2 y2cosh Ž 2h . q1 x Ž E 2 y4cosh2hX . ,

M sC 4 H 4 , M sC 6 H 4 ,

Ž E 2 q Eey hX ye 2 hX . w Ž Ey ´ X .

¢

X X X = Ž E 2 y Eey h ye 2 h . y2g X2 Ž E 2 yey h . x ,

M sC 4 H 2 X.

Ž 34 .

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

279

Table 1 Minimal conductance in units 2e 2rh, g min rg 0 s g 0m o l Ž dmax .ey2 d max N , and maximal resistance in units hrŽ2e 2 . f 12.9 k V, R max s g 0 rg min calculated for oligomers listed in Fig. 1 Žrepresented in the same order as in Fig. 1.. The Fermi energy corresponds to the maximum of tunneling decay constant dma x within the HOMO–LUMO gap D HL . The oligomer length is indicated as NC s 16, 20, 22, and 28, NC is the number of C atoms in row, see Fig. 1. The X parameters of the molecular electronic structure are: for polyene h s 0.1333 w30a,30bx, and h s 0.1 in all other cases; for M s C 6 H 4 h s 0, g int s 0.92, X and for M s C 4 H 2 S h s 0.1, g int s 1; g X s 0.38, ´ X s 0.25 w31x; b s y3.757 eV w30a,30bx Oligomer

ŽM. MsC2 H2 M s C6 H 4 MsC4 H2S ŽM 1 –M 2 . M1 s C 6 H 4 M2 sC2 H2 M2 sC4 H4 M1 s C 4 H 2 S M2 sC2 H2 M2 sC4 H4

g 0mo l Ž dmax .

dmax

D HL ŽeV.

16

20

22

28

16

20

0.48 0.04 0.35

0.28

0.06

0.71 5 0.91

2.08 25 2.86

1 1 1

0.27 0.78 0.56

2.0 3.4 2.0

1.4 0.20 1.1

1.5 2.2

0.98 1.18

2.7 2.4

0.43

1.5 2.2

0.77 0.97

1.9 1.9

1.5

0.06 0.02 0.32

3.57

0.94

28 16.7

16.7 5.26

0.67

0.66

Ž 35 .

22

2.33

0.19

The notations appeared in Eqs. Ž33. and Ž34. have the following meaning. The hopping integrals associated with double and single C–C bonds in the polyene chain and between the rings are X denoted as X b eh and b eyh , respectively. Similarly, b eh and b eyh distinguish double and single C–C bonds within aromatic rings. b´ X is the heteroatom site energy, bg X is the hoping integral of C–X bond, and g int s b intrb . All energies are in units b , and the electron site energy at carbon is taken as the reference point. Further details on the monomer Hamiltonians used to obtain the above relations are given in w26,29x. It can be seen from the dispersion relations Ž6. and Ž16. wthey are simplified considerably by using Eqs. Ž33. and Ž34.x that in the case of aromatic ring based oligomers function f Ž E . may have singularities at some energies and, hence, the tunneling decay constant d may go to infinity. Physically, this means that such oligomers can act as nearly ideal switchers provided the Fermi energy is properly tuned. If the properties of the monomer Green function are such that f Ž E . has no singularities, the tunneling decay constant remains finite, and its maximal value dmax within the given band gap is reached at the extremum of this function, where f X Ž E . s 0. Table 1 represents the values of dmax for oligomers shown in Fig. 1. It is of interest to find the expression of the decay constant at the energies which are solutions to Eqs. Ž24. and Ž26.. In the case of alternant, all carbon oligomers, within the HOMO–LUMO Žhighest occupied molecular orbital–lowest unoccupied molecular orbital. gap such a solution corresponds to the extremum of f Ž E . at E s 0. At the energies which satisfy Eq. Ž24. for M-oligomers and Eq. Ž26. for ŽM 1 –M 2 .-oligomers, the expression of tunneling decay constant takes an especially simple form

d s ln < b int GaMl , a r
R max Ž=10 2 .

g min Ž=10y2 .

50 3.13

1.5

1.06

and GaMl ,1a r

d s ln

GaMl ,2a r

y1

GDM 2

,

Ž 36 .

respectively. To recall, the case of < b int GaMl , a r < ) 1 and < GaM,1a Ž GaM,2a .y1 GDM 2 < ) 1 is excluded of the consideration. l r l r Finally, using in Eqs. Ž35. and Ž36. the above given explicit expressions of GaMl , a r and GDM for alternant oligomers one gets

°h

2 ,

4h , X ln  2coshh w 2cosh X = Ž 2h . y1x rg int 4 , dmax s X ln  2coshh w 2cosh X = Ž 2h . y1x rg int 4 q2h , X ln  2coshh w 2cosh X = Ž 2h . y1x rg int 4 q4h ,

~

¢

M sC 2 H 2 , M sC 4 H 4 , M sC 6 H 4 , M 1 sC 6 H 4 ,M 2 sC 2 H 2 , M 1 sC 6 H 4 ,M 2 sC 4 H 4 .

Ž 37 . The maximal value of the tunneling decay constant Ž37. can be easily estimated by using the known data: the value of h is close to 0.1 w30a,30bx; this same value is characteristic for parameter hX in the heterocyclic rings w31x while C–C bonds within phenyl ring are nearly equal to each other, i.e., hX f 0; and expected values of g int are close to unity. As an example, for oligomers of polyparaphenylenevinylene these data yield the value of dimensional tunneling decay constant in the middle of the gap ˚ y1 Ž k max s 2 dmax NrL, L is the oligomer k max s 0.306 A ˚ . length in A . For this estimate, a commonly accepted geometry was assumed, namely: the length of M 1 s C 6 H 4 , M 2 s C 2 H 2 , and C–C bond between M 1 and M 2 was ˚ respectively; taken to be equal to 2.78, 1.32, and 1.48 A, and the angle between double and single C–C bonds was set to be 1208. The estimated value of k max is in a good

280

A. Onipko et al.r Materials Science and Engineering C 8–9 (1999) 273–281

agreement with that calculated by Magoga and Joachim ˚ y1 , and also by Larsson et al. w22x 0.303 A˚ y1 . w21x 0.278 A ˚ y1 which is It is also consistent with the value of 0.2 A considered as a typical characteristic of linear conjugated molecules in the bridge-mediated electron transfer w32x.

6. Conclusion The exact solution of the Green function problem for M-oligomers described by the tight-binding Hamiltonian represents the central result of the paper. In the solution found, the matrix elements of the oligomer Žlinear macromolecule. Green function are explicitly expressed in terms of monomer Green function. The calculation of the latter quantity is a far more easy task. As is demonstrated in the above illustrative examples, in many cases of interest, it can be found in an analytical form. A number of physical properties can be conveniently described if the Green function of the system is known. The solution found makes the p electronic structures of two families of conjugated oligomers covered by formulas M–M– . . . –M and M 1 –M 2 –M 1 – . . . –M 2 –M 1 available for a comprehensive Green function analysis. Actually, due to early works of Sandorfy w33x, and Pople and Santry w34x, with an appropriate choice of the system parameters, the same type of Hamiltonian, as that has been used here, is also applicable for the description of s electrons. The formal analogy between p and s electron system has recently been used for a comparison of the electron-transmitting abilities of conjugated and saturated carbon chains w27x. Furthermore, by the standard methods, the knowledge of M-oligomer Green function can be extended to the case of molecular chains with side andror end groups whose Green functions are to be found separately. For instance, the Green function of oligomers terminated by arbitrary end groups can be obtained essentially in the same way as it has been derived above for ŽM 1 –M 2 .-oligomers. In this work, the knowledge of the Green function has been used to examine the electron tunneling through the molecular band gap. So far, the molecular tunnel conductance has been studied mostly by numerical methods. For a number of conjugated oligomers, the tunneling decay constant has been calculated on the basis of the extended Huckel model w21x. It is not our intention to make a ¨ detailed comparison with these and similar results but rather to notice an obvious advantage of the analytical definition given in Eq. Ž27.. Firstly, this and subsidiary equations derived make equally simple the estimate of the tunneling decay constant at any energy within the band gap. This is a useful application of the theory in view of uncertainty with regard to the relative position of the Fermi energy which in numerical models is usually fixed w21,22x. Secondly, the analytical expression of the tunneling decay constant establishes the relationship between the through band gap tunneling efficiency and the parameters

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