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Derization problem in conjugated polymers
ELSEVIER
Synthetic
Dimerization
fiir
Fe&k&per-
a Institute
und
of Physics,
101 (1999)
386-387
problem in conjugated polymers
S.-L. Drechsler, Institut
Metals
J. Malek”
G. Paasch, K. Hallbergb
WeTkstoffoTschung Czech
b Centro
Dresden,
D-Oll71
Academy of Sciences,
Prague,
Ato’mico,
S.$OO Bariloche,
Dresden,
Germany
Czech-Republic
Argentine
Abstract
Applying the self-consistent solitonic approach [l] to the extended Peierls-Hubbard model on odd rings, the interplay of eI-el interactions U, 1’, various off-diagonal interactions, and external dimerixation for the dimerization amplitude is studied by exact diagonalizations in the adiabatic limit. The applicability of an effective spin-Peierls Hamiltonian m the intermediate correlation regime U N 3t typical for conjugated polymers and an approximate analytical solution based on the Bethe-ansatz solution for the spin velocity and the continuum model solution for the spin-Peierls problem are discussed. Keywords:
1.
Many-body
and qua&particle
theories,
Polyacetylene
Introduction
The interplay of electron-electron (el-eZ) and electronphonon (el-ph) interaction in conjugated polymers has been the object for a longstanding debate. Starting from the pioneer work of Horsch [2] the importance of the onsite Coulomb interaction U in reproducing sizeable dimerization amplitudes ~0 for trans-polyacetylene (PA), described by the extended Peierls-Hubbard model (EPHM) at half filling, for realistic values of the off-diagonal el-ph interaction has been more and more recognized. In particular, a maximum of 2~0 at some intermediate value of U N &O comparable with the r electron bandwidth has been suggested. However, due to the involved approximations, being valid only in certain parameter regions, some details such as its dependence upon the strength of the el-ph interaction and the effect of additional terms in slightly generalized Hamiltonians remained somewhat unclear or have not been studied at all. Here we focus on two topics: how is this interplay affected by the nondegeneracy of the ground state in &-PA and other nondegenerate polymers frequently described by BrazovskiiKirova type models and (ii) to what extent can we replace the EPHM at intermediate U values typical for conjugated polymers by an effective spin-Peierls model?
2. Models
and
method
In our solitonic approach we make use of the ground state property of odd periodic rings to show maximal selfconsistent lattice deformations as explained in Ref. 1. The
and derivatives,
vinylene)
and derivatives
electronic contribution to the total energy Etot is found from exact diagonalizations performed for rings with up to IV = 13 sites for the EPHM and N =23 sites for the spin-Peierls (antiferromagnetic spin l/2 Heisenberg) model. The lattice distortions are found minimizing Etot treating the bondlength changes u~=zL~+~-u~ as classical coordinates, where i denotes the site index. The dimerization amplitude of the infinite chain is then estimated from the bond length opposite to the defect (soliton) center [l] using an appropriate extrapolation for the infinite chain (ring) IV t 0~1 (see Fig. 1). We linearize the bond-length dependent transfer integral t%,i+i and the intersite Coulomb interaction I’ _ ti,i+l
= (-qi&
- (to - Y.U*),
v,,,+1 : v - 7pi.
(1)
In order to lift the degeneracy of the ground state a constant, oscillating term has been included in Eq. (1). Due to this external alternation a bond order wave (BOW) and a finite gap occur already in both noninteracting cases. The dimensionless dimerization amplitude d and the electron phonon interaction constant g read respectively as
where K denotes the harmonic
3.
Results
and
spring cnnstant.
discussion
The dimerization amplitude d as a function of U is shown in Figs. 1 and 2, 3 for the degenerate and the nondegenerate case, respectively. The maximal dimerization occurs at
0379-6779/99/S - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)00823-6
Poly(phenylene
S.-L.
Drechsler
WN= AN= *N= VN= ON= ON=
Metals
(1999)
3X6-387
el-ph
5
0.0 Lo 10
0.4 alternation
0.3 m
Fig. 3. The same as in Fig. 1 for different e2-ph interactions g and finite external alternation 6t.
In the limit U >> to and St=0 we get a smooth transition from above to the standard adiabatic spin-Peierls limit. Our results are reasonably well described by the continuum model expression [3] rewritten as d=
-&
(g)'
($y2
+
cg2 (+g2,
(3)
where J is the NN exchange integral and ~~1.04 for ~=0.5. Adopting for arbitrary U an effective exchange integral given by Ji,i+i=2usptt,i+i/T, with the spin velocity v,,(U/to) taken from the Bethe-Ansatz solution of the infinite Hubbard ring [4] using Eqs. (1,3), we get an analytical expression in terms of modified Bessel functions (the dashed-dotted line in Fig. 1). It exhibits similar shape and magnitude as the EPHM curve. This suggests that even for conjugated polymers, being clearly outside the U>>te limit, the dimerization is mainly governed by the spin degrees of freedom and to less extent by the charge degrees of freedom. Quantitatively, our approach can be further improved if NNN U dependent frustrated exchange is taken into account. It arises from higher order transfer processes reflecting the increased itineracy with decreasing U [5]. Exact diagonalization and density matrix renormalization group studies for the frustrated spin-Peierls model show a significantly enhanced dimerization in wide parameter regions as expected [6]. Acknowledgement
!
The Deutsche Forschungsgemeinschaft for financial support. to = v=o N =
0.
10
u/to
intermediate or weak U values. Notably, its position is a nonmonotonous function of 6t. The initial enhancement od d due to U is weakened with increasing bt and g (see Fig. 3). Finally, it disappears approaching a critical value of gc(bt) (SC = 0.7 for bt=O [I]). Hence for weak or intermediate external alternation, el-ei and el-ph interactions there is a cooperative interplay whereas in all strong coupling limits the interactions do clearly compete. The correlation effect is maximal for trans-PA. For typical nondcgenerate conjugated polymers such as PPV it is somewhat reduced. Here a sizeable part of the total gap stems from the external alternation. The dimerization is further enhanced by NNN transfer, weak intersite Coulomb repulsion I/, and its derivative 7. At large U and I/=const one can replace U+ U-V.
external
5
15
u/to
0.2
0.2
r-no
Fig. 1. Dimensionless dimerization amplit,ude d vs. on-site Coulomb in- teraction U at constant intermediate el-ph interaction g and St=0 (no external alternation).
.-s -5 .Nb E s
=
to=K=l v=o N = 13
continuum (~=0.28)
I
0.6
bound
-.- - - - -i 0.0
interaction X g =
0.4
at
generalized
387
r
3 5 7 9 11 13 N=m lower l/N-approx.
I 0
et al. / Synthetic
K
=
is acknowledged
1
13
i ’ ,
Fig. 2. The same as in Fig. 1 for different external alternations 6t at constant intermediate el-p/l interaction g.
References
J. Malek et al., Phys. Rev. B 56 (1997) R8467. P. Horsch, Phys. Rev. B 24 (1981) 7351. S -.
Tnnmki ----o----
et nl. __ -__,
.T. Phvs. “. _ --j
_
c-n LJ”L.
1-7, tip1
52
(1983)
3620.
A. Ovchinnikov, Sov. Phys. JETP 30 (1970) 1160. K. Kuboki et al., J. Phys. Sot. Jpn.- JU ec t'1987) 3126. S.-L. Drechsler et al., in preparation (19 ~ 98).
Synthetic
Dimerization
fiir
Fe&k&per-
a Institute
und
of Physics,
101 (1999)
386-387
problem in conjugated polymers
S.-L. Drechsler, Institut
Metals
J. Malek”
G. Paasch, K. Hallbergb
WeTkstoffoTschung Czech
b Centro
Dresden,
D-Oll71
Academy of Sciences,
Prague,
Ato’mico,
S.$OO Bariloche,
Dresden,
Germany
Czech-Republic
Argentine
Abstract
Applying the self-consistent solitonic approach [l] to the extended Peierls-Hubbard model on odd rings, the interplay of eI-el interactions U, 1’, various off-diagonal interactions, and external dimerixation for the dimerization amplitude is studied by exact diagonalizations in the adiabatic limit. The applicability of an effective spin-Peierls Hamiltonian m the intermediate correlation regime U N 3t typical for conjugated polymers and an approximate analytical solution based on the Bethe-ansatz solution for the spin velocity and the continuum model solution for the spin-Peierls problem are discussed. Keywords:
1.
Many-body
and qua&particle
theories,
Polyacetylene
Introduction
The interplay of electron-electron (el-eZ) and electronphonon (el-ph) interaction in conjugated polymers has been the object for a longstanding debate. Starting from the pioneer work of Horsch [2] the importance of the onsite Coulomb interaction U in reproducing sizeable dimerization amplitudes ~0 for trans-polyacetylene (PA), described by the extended Peierls-Hubbard model (EPHM) at half filling, for realistic values of the off-diagonal el-ph interaction has been more and more recognized. In particular, a maximum of 2~0 at some intermediate value of U N &O comparable with the r electron bandwidth has been suggested. However, due to the involved approximations, being valid only in certain parameter regions, some details such as its dependence upon the strength of the el-ph interaction and the effect of additional terms in slightly generalized Hamiltonians remained somewhat unclear or have not been studied at all. Here we focus on two topics: how is this interplay affected by the nondegeneracy of the ground state in &-PA and other nondegenerate polymers frequently described by BrazovskiiKirova type models and (ii) to what extent can we replace the EPHM at intermediate U values typical for conjugated polymers by an effective spin-Peierls model?
2. Models
and
method
In our solitonic approach we make use of the ground state property of odd periodic rings to show maximal selfconsistent lattice deformations as explained in Ref. 1. The
and derivatives,
vinylene)
and derivatives
electronic contribution to the total energy Etot is found from exact diagonalizations performed for rings with up to IV = 13 sites for the EPHM and N =23 sites for the spin-Peierls (antiferromagnetic spin l/2 Heisenberg) model. The lattice distortions are found minimizing Etot treating the bondlength changes u~=zL~+~-u~ as classical coordinates, where i denotes the site index. The dimerization amplitude of the infinite chain is then estimated from the bond length opposite to the defect (soliton) center [l] using an appropriate extrapolation for the infinite chain (ring) IV t 0~1 (see Fig. 1). We linearize the bond-length dependent transfer integral t%,i+i and the intersite Coulomb interaction I’ _ ti,i+l
= (-qi&
- (to - Y.U*),
v,,,+1 : v - 7pi.
(1)
In order to lift the degeneracy of the ground state a constant, oscillating term has been included in Eq. (1). Due to this external alternation a bond order wave (BOW) and a finite gap occur already in both noninteracting cases. The dimensionless dimerization amplitude d and the electron phonon interaction constant g read respectively as
where K denotes the harmonic
3.
Results
and
spring cnnstant.
discussion
The dimerization amplitude d as a function of U is shown in Figs. 1 and 2, 3 for the degenerate and the nondegenerate case, respectively. The maximal dimerization occurs at
0379-6779/99/S - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SO379-6779(98)00823-6
Poly(phenylene
S.-L.
Drechsler
WN= AN= *N= VN= ON= ON=
Metals
(1999)
3X6-387
el-ph
5
0.0 Lo 10
0.4 alternation
0.3 m
Fig. 3. The same as in Fig. 1 for different e2-ph interactions g and finite external alternation 6t.
In the limit U >> to and St=0 we get a smooth transition from above to the standard adiabatic spin-Peierls limit. Our results are reasonably well described by the continuum model expression [3] rewritten as d=
-&
(g)'
($y2
+
cg2 (+g2,
(3)
where J is the NN exchange integral and ~~1.04 for ~=0.5. Adopting for arbitrary U an effective exchange integral given by Ji,i+i=2usptt,i+i/T, with the spin velocity v,,(U/to) taken from the Bethe-Ansatz solution of the infinite Hubbard ring [4] using Eqs. (1,3), we get an analytical expression in terms of modified Bessel functions (the dashed-dotted line in Fig. 1). It exhibits similar shape and magnitude as the EPHM curve. This suggests that even for conjugated polymers, being clearly outside the U>>te limit, the dimerization is mainly governed by the spin degrees of freedom and to less extent by the charge degrees of freedom. Quantitatively, our approach can be further improved if NNN U dependent frustrated exchange is taken into account. It arises from higher order transfer processes reflecting the increased itineracy with decreasing U [5]. Exact diagonalization and density matrix renormalization group studies for the frustrated spin-Peierls model show a significantly enhanced dimerization in wide parameter regions as expected [6]. Acknowledgement
!
The Deutsche Forschungsgemeinschaft for financial support. to = v=o N =
0.
10
u/to
intermediate or weak U values. Notably, its position is a nonmonotonous function of 6t. The initial enhancement od d due to U is weakened with increasing bt and g (see Fig. 3). Finally, it disappears approaching a critical value of gc(bt) (SC = 0.7 for bt=O [I]). Hence for weak or intermediate external alternation, el-ei and el-ph interactions there is a cooperative interplay whereas in all strong coupling limits the interactions do clearly compete. The correlation effect is maximal for trans-PA. For typical nondcgenerate conjugated polymers such as PPV it is somewhat reduced. Here a sizeable part of the total gap stems from the external alternation. The dimerization is further enhanced by NNN transfer, weak intersite Coulomb repulsion I/, and its derivative 7. At large U and I/=const one can replace U+ U-V.
external
5
15
u/to
0.2
0.2
r-no
Fig. 1. Dimensionless dimerization amplit,ude d vs. on-site Coulomb in- teraction U at constant intermediate el-ph interaction g and St=0 (no external alternation).
.-s -5 .Nb E s
=
to=K=l v=o N = 13
continuum (~=0.28)
I
0.6
bound
-.- - - - -i 0.0
interaction X g =
0.4
at
generalized
387
r
3 5 7 9 11 13 N=m lower l/N-approx.
I 0
et al. / Synthetic
K
=
is acknowledged
1
13
i ’ ,
Fig. 2. The same as in Fig. 1 for different external alternations 6t at constant intermediate el-p/l interaction g.
References
J. Malek et al., Phys. Rev. B 56 (1997) R8467. P. Horsch, Phys. Rev. B 24 (1981) 7351. S -.
Tnnmki ----o----
et nl. __ -__,
.T. Phvs. “. _ --j
_
c-n LJ”L.
1-7, tip1
52
(1983)
3620.
A. Ovchinnikov, Sov. Phys. JETP 30 (1970) 1160. K. Kuboki et al., J. Phys. Sot. Jpn.- JU ec t'1987) 3126. S.-L. Drechsler et al., in preparation (19 ~ 98).