In order to simulate the events in a organic solar cell or a light emitting diode (LED) there are two processes that is needed to be taken into consideration: First the absorption of an photon which, in the case of the solar cell, translates to the excitation of a molecule with a ray of light. In the LED the process in reversed with the emitting of an photon by "de-excitation" of a molecule where an excited electron falls down to a lower orbital. The next step is the actual transport or the electron (or hole) which in the case of a solar cell means that we need to transport electrons away when they have been excited in the molecule to create a currect. Again for the LED the process is reversed and the electrons (or holes) are fed into the molecule to create light with a certain wavelength. The latter process is called a charge transfer (CT) and together with the excitation they make up the foundations of light absorption and emission. These processes will be described below in a quantum mechanical and physical sound way by introducing a model including these both events. In the end this model will be compared to experimental results for verification.

Frenkel Model

The Frenkel, named after the Russian physicist Yakov Frenkel, exciton can be applied when the materials dielectric function is very small so that the Coulomb interaction between electrons and holes are very strong. This results in that the exciton size stays relative small, in the same order as a unit cell, and therefore the electron and hole resides on the same molecule with the binding energy in the order of 1.0 eV. In the Frenkel model this is the only allowed exciton and will therefore be best suited for organic materials where there are no strong covalent bindings.

The starting point of the Frenkel model is a crystal in its lowest energy state where all molecules in the system are in their electronic and vibronic ground state. Within the effective mode model, a consequence of BO-approximation, it is possible to factorize the electronic ground state of the crystalline phase into the product of the ground state wave function of the constituting molecule. The operation of the Frenkel exciton operator on a system in its ground state results in a factorization of the excited state molecules wave function:

With this operator it is possible to build a Hamiltonian describing the energies of the system. The resulting pure Frenkel exciton Hamiltonian for the 3D molecular crystal, with two molecules per unit cell reads
where the neutral transfer is described as

where the T is the transfer integral of a neutral exciton between two molecules and the S are the Huang-Rhys factors meantioned in the "Vibrational Theory" section. The Hamiltonian can be schematically illustrated with figure 1.1 below

Figure 1.1 Visualization of the Frenkel operator operating on a PTCDA crystal in its ground state |g>. The molecule in unit cell n in stack A is excited to the first excited state with a given vibroninc level. The molecules still in their ground states are depicted with wires and the excited molecule is visualized by a ball-and-stick scheme.

Fourier Transform

As the system is periodic we can perform a total Fourier transformation into wave-vector representation to be able to include all molecules in the infinite stack, the k-space operator can be written as
The resulting Fourier transformed Frenkel Hamiltonian
The Hamiltonian can now be diagonalized using for example the LAPACK package of FORTRAN. This model has been shown to work for molecules with small charge transfer interaction such as PTCDA. However to be able to model molecules with large charge transfer mixing we need to introduce another possible exciton: The charge transfer (CT) exciton which includes electron or hole transfers between molecules. This exciton will be introduced in the next section.

Frenkel and Charge Transport (CT) Model

The full Hamiltonian including Frenkel, CT and their mixing can summed up as
where the first Hamiltoninan is given in the section before. We need to introduce an operator which excite the ground state molecules into one electron excited state and one hole excited state. The new operator working on a system in its ground state can be written as
A way to understand this is by looking at the figure 3.1 below
Figure 3.1 Visualization of the CT operator operating on a crystal in its ground state. The A stack molecule in the nth unit cell is excited to a cationinc state with a given vibronic level and the A stack molecule in the n-1th unit cell is excited to the anionic state with a certain vibronic level. The molecules still in their ground state are depicted with wires and the excited molecules are visualized by a ball-and-stick scheme.

By using the same logics as before the CT Hamiltonian can be written as

and the Frenkel and CT mixing Hamiltonian follows:
where the electron transfer integral is
and the same logics will lead to the hole transfer intetral.

For a summation of allowed transfers that are included in the model, see figure 3.2 below

Figure 3.2 Schematic description of allowed (black arrows) and non-allowed (grey dotted arrows) transfers for the model. The double arrows indicates that the transfer is possible both ways. An arrow crossing the middle dotted line represents a transfer between different aligned molecules in the unit cells and a horizontal arrow represents transfers in the same stack.

To summarize one can write the Hamiltonians in their matrix representation. The hopping of CT-states are neglected here since that would imply transfers of two charge carrier simultaneously which are much less probable so we do not consider this. Under the given conditions we have periodic boundary conditions and as before trasformation via a Fourier scheme is performed into the momentum space representation.

Optical Properties

The transition dipole moment for the ground state to the excited state is given by the relevent off-diagonal elements of the dipole matrix. We neglect exchange of electrons between CT dimers and the other molecules in their ground states so we get for charge transfer dipole moment. The total expression can be written as
To obtain the frequency dependent dielectric tensors foth the x and y directions correctly it is necessary to include the broadening in a physical meaningful manner. It has been shown that Gaussians describes the nature of the lifetime broadening, which occurs from mainly electron-phonon and electron-defect interaction, very well. Therefore we use Gaussian functions which are Kramers-Kronig consistent for the imaginary part. The imaginary part of the dielectric tensor can then be written as
and to get the real part we apply the Kramers-Kronig relation
where P denotes the Cauchy principal value.

F-CT Model Results

The F-CT model optical spectra can be compared to experimental results. Below are some of the molecule spectras that were presented in my PhD. thesis. For all the molecules and their optical lines shapes please consult my PhD. thesis or article [1].
Figure 5.1 Comparsion of a experimental PTCDA spectrum and the F-CT model.

Figure 5.1 Comparsion of a experimental Me-PTCDI spectrum and the F-CT model.

Figure 5.1 Comparsion of a experimental PB31 spectrum and the F-CT model.

Figure 5.1 Comparsion of a experimental DIP spectrum and the F-CT model.

[1] L. Gisslen and R. Scholz, Submitted (2009).