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Physical review. B, Condensed matter

Binding energy of 1Bu singlet excitons in the one-dimensional extended Hubbard-Peierls model

Zhigang Shuai, Swapan Kumar Pati, W. P. Su, Jean‐Luc Brédas, S. Ramasesha

Published Jun 15, 1997• 49 citations
Publisher

Abstract

Using a symmetrized density-matrix renormalization-group formulation, we have investigated the electronic binding energy of the lowest optically allowed exciton (${1\mathrm{B}}_{\mathrm{u}}$) within an extended Hubbard-Peierls model (with parameters U, V, and \ensuremath{\delta}) for conjugated chains with N=80 sites. Three symmetries, the ${\mathrm{C}}_{2}$, spin parity, and electron-hole symmetries, have been applied to construct the projector operator. Analysis of our results on the ratios between exciton binding energies and exciton energies, sheds light on the current experimental and theoretical controversy on exciton binding for polyacetylene and polyparaphenylene vinylene. We show that in the absence of dimerization, the exciton binding energy is vanishingly small for U/t up to 5 and V/t up to 2.3; for a finite dimerization, it is only when V is large enough that the exciton gets to be significantly bound. This result questions the applicability to conjugated polymers of the strong correlation picture, in which the exciton binding energy is equal to V.

DOI
10.1103/physrevb.55.15368

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Using a symmetrized density-matrix renormalization-group formulation, we have investigated the electronic binding energy of the lowest optically allowed exciton (1Bu{1\mathrm{B}}_{\mathrm{u}}1Bu​) within an extended Hubbard-Peierls model (with parameters U, V, and \ensuremath{\delta}) for conjugated chains with N=80 sites. Three symmetries, the C2{\mathrm{C}}_{2}C2​, spin parity, and electron-hole symmetries, have been applied to construct the projector operator. Analysis of our results on the ratios between exciton binding energies and exciton energies, sheds light on the current experimental and theoretical controversy on exciton binding for polyacetylene and polyparaphenylene vinylene. We show that in the absence of dimerization, the exciton binding energy is vanishingly small for U/t up to 5 and V/t up to 2.3; for a finite dimerization, it is only when V is large enough that the exciton gets to be significantly bound. This result questions the applicability to conjugated polymers of the strong correlation picture, in which the exciton binding energy is equal to V.
By applying the numerically accurate symmetrized density-matrix renormalization-group method coupled with the extended Hubbard-Peierls model, we find that (i) the on-site Hubbard repulsion energy U dramatically reduces the binding energy of the lowest optically allowed 1Bu{1B}_{u}1Bu​ exciton; (ii) in the zero-dimerization limit, there exists a critical value of V at which the 1Bu{1B}_{u}1Bu​ exciton becomes bound; the critical value Vc=2t{V}_{c}=2tVc​=2t is fully in agreement with the recent analytical results at the infinite-U limit by Gallagher and Mazumdar [Phys. Rev. B 56, 15 025 (1997)], furthermore, this critical value decreases appreciably for weaker on-site correlation strengths, when the dimerization amplitude $(\ensuremath{\delta})$ is nonzero. The present accurate numerical results contradict those obtained recently by Yu, Saxena, and Bishop [Phys. Rev. B 56, 3697 (1997)] both qualitatively and quantitatively. We also present first-order perturbation plus random-phase-approximation and single configuration-interaction analyses to rationalize the numerical calculations.
We show that charge excitations within one-dimensional extended Hubbard models in the limit of large on-site Coulomb interaction can be mapped onto pairs of spin reversals from the ferromagnetic Ising-Heisenberg spin Hamiltonian. The exciton in the extended Hubbard models is equivalent to the bound two-magnon state of the spin Hamiltonian. From exact results for the spin Hamiltonian, for the intersite Coulomb interactions limited between nearest neighbors, the lowest optical exciton is separated in energy from the electron-hole continuum only for the intersite interaction larger than half the one-electron bandwidth. The commutation relationship between spin operators allows us to numerically investigate exciton binding energies and optical absorptions from the ground state in the limit of large on-site Coulomb interaction even for cases where analytic results are not available. For intersite Coulomb interaction limited between nearest neighbors, the binding energy of the lowest optical exciton changes very weakly for realistic Peierls bond dimerization, although there is a profound change in the optical absorption. While in the undimerized chain a single optical exciton can occur, two additional excitons that occur above the threshold of the electron-hole continuum become optically visible in the dimerized case. For long-range intersite Coulomb interactions, multiple excitons can occur even for the undimerized chain, and additional exciton states become optically visible upon dimerization. With the 1/r1/r1/r form for the intersite Coulomb interaction, our numerical calculations suggest that the formation of at least one optical exciton is unconditional, unlike the case of nearest-neighbor interaction. The binding energy of the lowest optical exciton again depends weakly on the dimerization for long-range interaction. Our theoretical results can explain the occurrence of two charge-transfer absorption bands in half-filled band segregated stack charge transfer solids below the spin-Peierls dimerization temperature. Our results also suggest that the lowest optical state of \ensuremath{\pi}-conjugated polymers, in which Coulomb interactions are long ranged, is necessarily an exciton.
This work addresses the question of whether low-lying excitations in conjugated polymers are comprised of free charge carriers or excitons. States are characterized as bound or unbound according to the scaling of the average particle-hole separation with system size. We critically examine other criteria commonly used to characterize states. The polymer is described by an extended Hubbard model with alternating transfer integrals. The model is solved by exact diagonalization and the density-matrix renormalization-group (DMRG) method. We demonstrate that the DMRG accurately determines excitation energies, transition dipole moments and particle-hole separations of a number of dipole-forbidden (Ag){(A}_{g})(Ag​) and dipole-allowed (Bu){(B}_{u})(Bu​) states. Within a parameter regime considered reasonable for polymers such as polyacetylene, it is found that the charge gap, often used to define the exciton binding energy, is not a good criterion by which to decide whether a state is bound or unbound. The essential nonlinear optical state mAg{\mathrm{mA}}_{g}mAg​ is found to mark the onset of unbound excitations in the Ag{A}_{g}Ag​ symmetry sector. In the Bu{B}_{u}Bu​ symmetry sector, on the other hand, it is found that all low-lying states are unbound and that there is no well-defined nBu{\mathrm{nB}}_{u}nBu​ state. That is, the 1Bu{1B}_{u}1Bu​ state marks the onset of unbound excitations in this sector.
We study the elementary excitations of a model Hamiltonian for the $\ensuremath{\pi}$ electrons in polydiacetylene chains. In these materials, the bare band gap is only half the size of the observed single-particle gap and the binding energy of the exciton of 0.5 eV amounts to 20% of the single-particle gap. Therefore, exchange and correlations due to the long-range Coulomb interaction require a numerically exact treatment which we carry out using the density-matrix renormalization group method. Employing both the Hubbard-Ohno potential and the screened potential in one dimension, we reproduce the experimental results for the binding energy of the singlet exciton and its polarizability. Our results indicate that there are optically dark states below the singlet exciton, in agreement with experiment. In addition, we find a weakly bound second exciton with a binding energy of 0.1 eV. The energies in the triplet sector do not match the experimental data quantitatively, probably because we do not include polaronic relaxation effects.
For the one-dimensional, extended Peierls–Hubbard model we calculate analytically the ground-state energy and the single-particle gap to second order in the Coulomb interaction for a given lattice dimerization. The comparison with numerically exact data from the density-matrix renormalization group shows that the ground-state energy is quantitatively reliable for Coulomb parameters as large as the bandwidth. The single-particle gap can almost triple from its bare Peierls value before substantial deviations appear. For the calculation of the dominant optical excitations, we follow two approaches. In Wannier theory, we perturb the Wannier exciton states to second order. In two-step perturbation theory, similar in spirit to the GW -BSE approach (where the Bethe–Salpeter equation is solved for quasi-particles which are calculated in the GW -approximation), we form excitons from dressed electron–hole excitations. We find the Wannier approach to be superior to the two-step perturbation theory. For singlet excitons, Wannier theory is applicable up to Coulomb parameters as large as half the bandwidth. For triplet excitons, second-order perturbation theory quickly fails completely.
We employ dynamical density-matrix renormalization-group (DDMRG) and field-theory methods to determine the frequency-dependent optical conductivity in one-dimensional extended, half-filled Hubbard models. The field-theory approach is applicable to the regime of ``small'' Mott gaps which is the most difficult to access by DDMRG. For very large Mott gaps the DDMRG recovers analytical results obtained previously by means of strong-coupling techniques. We focus on exciton formation at energies below the onset of the absorption continuum. As a consequence of spin-charge separation, these Mott-Hubbard excitons are bound states of spinless, charged excitations (``holon-antiholon'' pairs). We also determine exciton binding energies and sizes. In contrast to simple band insulators, we observe that excitons exist in the Mott-insulating phase only for a sufficiently strong intersite Coulomb repulsion. Furthermore, our results show that the exciton binding energy and size are not related in a simple way to the strength of the Coulomb interaction.