Conjugation Length Distribution in Poly(p-phenylenevinylene) (PPV) Films.

We studied the absorption line-shape of poly(p-phenylenevinylene) (PPV) films deposited via spin coating and Langmuir-Blodgett techniques with the intent of identifying the conjugation length distribution in these two types of films, a key morphological aspect of conjugated polymer films. We treated the excitons in the polymer as independent oligomer excitons and modeled the absorption spectra of the individual oligomers using simple expressions for the oligomer size dependence of the gap energy, the line-broadening factor, the transition dipole moment and the Huang-Rhys parameter. We validated these expressions by independent measurements on phenyl-based oligomers and Density Functional Theory calculations. Our results show clear evidence that, for both types of PPV films, the conjugation length distribution depends exponentially on the segment size. Our results also set a lower limit, of about ten repeat units, for the maximum exciton length of three different phenyl-based oligomers.

Impact of the intermixed phase and the channel network on the carrier mobility of nanostructured solar cells.

We analyzed the impact of the complex channel network of donor and acceptor domains in nanostructured solar cells on the mobility of the charge carriers moving by thermally activated hopping. Particular attention was given to the so called intermixed phase, or interface roughness, that has recently been shown to promote an increase in the cell efficiency. The domains were obtained from a Monte Carlo simulation of a two-species lattice gas. We generated domain morphologies with controllable channel size and interface roughness. The field and density dependence of the carrier hopping mobility in different morphologies was obtained by solving a master equation. Our results show that the mobility decreases with roughness and increases with typical channel sizes. The deleterious effect of the roughness on the mobility is quite dramatic at low carrier densities and high fields. The complex channel network is shown to be directly responsible for two potentially harmful effects to the cell performance: a remarkable decrease of the mobility with increasing field and the accumulation of charge at the domains interface, which leads to recombination losses.

Relaxation time in disordered molecular systems.

Relaxation time is the typical time it takes for a closed physical system to attain thermal equilibrium. The equilibrium is brought about by the action of a thermal reservoir inducing changes in the system micro-states. The relaxation time is intuitively expected to increase with system disorder. We derive a simple analytical expression for this dependence in the context of electronic equilibration in an amorphous molecular system model. We find that the disorder dramatically enhances the relaxation time but does not affect its independence of the nature of the initial state.

Extended Hückel method calculation of polarization energies: the case of a benzene dimer.

We adapted Hoffmann's extended Hückel method to an interacting molecular system and use this approach to compute the electron affinity and ionization potential of benzene dimers. We restrict the added charge to one of the molecules and argue that the dimer energy computed in this manner is the relevant energy in any meaningful thermally activated hopping rate expression. The dimer electron affinity and ionization potential differs from the isolated molecule corresponding quantity by what is called polarization energy. The polarization energy normally stabilizes the anion and this is particularly relevant for benzene, given that its isolated anion is unstable with respect to charge detachment. We found that the anionic benzene dimer is only stabilized in certain conformations, suggesting that the stabilization of a benzene anion in an amorphous environment is very unlikely. The modest computational cost of the method makes it a viable alternative to compute the energy of charged molecules in amorphous molecular films, a central issue in the problem of charge transport in organic electronics.

The theoretical current-voltage dependence of a non-degenerate disordered organic material obtained with conductive atomic force microscopy.

We develop a simple continuum model for the current voltage characteristics of a material as measured by the conducting atomic force microscopy, including space charge effects. We address the effect of the point contact on the magnitude of the current and on the transition voltages between the different current regimes by comparing these with the corresponding expressions obtained with planar electrodes.

Reformulated space-charge-limited current model and its application to disordered organic systems.

We have reformulated a traditional model used to describe the current-voltage dependence of low mobility materials sandwiched between planar electrodes by using the quasi-electrochemical potential as the fundamental variable instead of the local electric field or the local charge carrier density. This allows the material density-of-states to enter explicitly in the equations and dispenses with the need to assume a particular type of contact. The diffusion current is included and as a consequence the current-voltage dependence obtained covers, with increasing bias, the diffusion limited current, the space-charge limited current, and the injection limited current regimes. The generalized Einstein relation and the field and density dependent mobility are naturally incorporated into the formalism; these two points being of particular relevance for disordered organic semiconductors. The reformulated model can be applied to any material where the carrier density and the mobility may be written as a function of the quasi-electrochemical potential. We applied it to the textbook example of a nondegenerate, constant mobility material and showed how a single dimensionless parameter determines the form of the I(V) curve. We obtained integral expressions for the carrier density and for the mobility as a function of the quasi-electrochemical potential for a Gaussianly disordered organic material and found the general form of the I(V) curve for such materials over the full range of bias, showing how the energetic disorder alone can give rise, in the space-charge limited current regime, to an I∝V(n) dependence with an exponent n larger than 2.

The mobility in disordered molecular systems with energies given by a charge-induced dipoles interaction.

We investigate the field dependence of the mobility in a model for a disordered molecular system containing spatial and energetic disorders. In this model we assign an isotropic polarizability to each site and take the site energies to be the site polarization energies, the interaction energy of a charge in the given site with the induced dipoles in the neighboring sites. This model was shown, in a previous publication, to contain short-ranged energetic correlations and we show in this work that this correlation produces a charge mobility proportional to the exponential of the square root of the applied field, the Poole-Frenkel dependence observed in various disordered organic materials, over a significant range of fields. We present an expression for the field dependence of the mobility in terms of the average intersite separation and of the isotropic polarizability of the electronic states, the two model parameters.

Density of states and energetic correlation in disordered molecular systems due to induced dipoles.

We have considered two models for a system of disordered organic molecules: one based on a regular lattice with Gaussian site displacements and another based on a hard sphere distribution. The site energies were given by a charge-induced dipole interaction (the polarization energy). We obtained the density of states of both models and observed that it changes from a Gaussian to the density of states of a uniform site distribution, whose form was obtained analytically, depending on the degree of disorder in one model or the packing fraction in the other model. The site energy distribution is short-ranged correlated in both models since nearby molecules polarize basically the same disordered environment.

The role of short-ranged energetic correlations in the mobility field dependence of disordered organic materials.

We studied the mobility of charge carriers in a model for disordered organic solids where the energies of the localized states are Gaussianly distributed with short-ranged correlations. We obtained an expression for the mobility as a function of electric field, temperature, energetic variance, and correlation radius. The temperature dependence obtained with short-ranged energetic correlations is different from that obtained with power-law decaying energetic correlations and suggests a possible way to distinguish the two types of correlations from the measured mobility. This work also presents a practical way of computing the mobility, applicable to any transport model based on a linear master equation, directly from the matrix of the hopping rates.

Master equation approach to charge injection and transport in organic insulators.

We develop a master equation model of a disordered organic insulator sandwiched between metallic electrodes by treating as rate processes both the injection and the internal transport. We show how the master equation model allows for the inclusion of crucial correlation effects in the charge transport, particularly of the Pauli exclusion principle and of space-charge effects, besides, being dependent on just the microscopic form of the transfer rate between the localized electronic states, it allows for the investigation of different microscopic scenarios in the organic, such as polaronic hopping, correlated energy levels, interaction with image charge, etc. The model allows for a separate analysis of the injection and the recombination currents. We find that the disorder, besides increasing the injection current, eliminates the possibility of observation of a Fowler-Nordheim injection current at zero temperature, and that it does not alter the Schottky barrier size of the zero-field thermionic injection current from the value based on the energy difference between the electrode Fermi level and the highest occupied molecular orbital/lowest unoccupied molecular orbital levels in the organic, but it makes the Arrhenius temperature dependence appear at larger temperatures. We investigate how the I(V) characteristics of a device is affected by the presence of correlations in the site energy distribution and by the form of the internal hopping rate, specifically the Miller-Abrahams rate and the Marcus or small-polaron rate. We show that the disorder does not modify significantly the ebeta square root E field dependence of the net current due to the Schottky barrier lowering caused by the attraction between the charge and its image in the electrode.

Electron transfer in proteins: nonorthogonal projections onto donor-acceptor subspace of the Hilbert space.

We develop nonorthogonal projectors, called Löwdin projectors, to construct an effective donor-acceptor system composed of localized donor (D) and acceptor (A) states of a long-distance electron transfer problem. When these states have a nonvanishing overlap with the bridge states these projectors are non-Hermitian and there are various possible effective two-level systems that can be built. We show how these can be constructed directly from the Schrödinger or Dyson equation projected onto the D-A subspace of the Hilbert space and explore these equations to determine the connection between Hamiltonian and Green function partitioning. We illustrate the use of these effective two-level systems in estimating the electron transfer rate in the context of a simple electron transfer model.