The concept of exciton is very important to understand the operation mode of organic electronic device when light is involved. An exciton can be defined as an electrically bound combination of an electron and a hole. This is equivalent to lifting an electron from the HOMO to the LUMO level. Excitons can emerge either by absorption of a photon, also known as photoluminescence, or a recombination of an electron and a hole, which corresponds to electroluminescence. Electroluminescense can be seen as a chemical reaction involving a negatively charged molecule, M-, and a positively charged molecule, M+, giving an excited molecule, M*, and neutral molecule, M. Because of polarization, the HOMO and LUMO levels in charged molecules tends to move closer to each other. For this reason, charged molecules are also called polarons. Hole polaron when positively charged, electron polaron when negatively charged. The reaction occurs through the transfer of an electron from the LUMO of the electron polaron, to the LUMO of the hole polaron, thus forming an excited molecule, also called exciton, and a neutral molecule. The exciton can also be seen as an electron hole pair because there is a missing electron in the normally field HOMO level, and an additional electron in the normally empty LUMO level. Because of the Coulombic force between the electron and the hole, the HOMO and LUMO levels move further towards each other. So, there is an additional reduction of the energy gap. The difference is called exciton binding energy. Going back to our table that gives the differences between organic and inorganic semiconductors, we see in the last line that excitons in organic semiconductors are called Frenkel excitons, and Wannier excitons in inorganic semiconductors. The Frenkel exciton is characterized by a strong binding energy and small spatial extension. Actually, the radius of a Frenkel exciton compares with the size of the molecule. Conversely, Wannier exciton has a weak binding energy and it extends over several molecular sites. To explain the difference, let's recall that the binding energy comes from the Coulombic force between the electron and the hole. A crude estimate is therefore given by the electrostatic energy, where e is the elemental charge, epsilon_r the dielectric constants, epsilon naught, the permittivity of vacuum and R, the radius of the exciton. We first see that the binding energy of the exciton is inversely proportional to its spatial extension, meaning that low radius is associated with high binding energy, and conversely. However, the difference between organic and inorganic excitons mainly rests on the different dielectric constant. The higher the dielectric constant, the lower the binding energy. The dielectric constant in Silicon is 11, compared to between three and four in organic semiconductors. In practice, the exciton binding energy in Silicon is only a few tens of milli-electron volts, as compared to up to around 1 eV in organic semiconductors. This table shows the exciton binding energy, E_sub_b, for various organic semiconductors. The energy ranges between half of an eV, to more than one eV. It is the difference between the energy gap and the optical gap. We have seen in the previous lecture how the optical gap can be estimated from the absorption and emission spectra. The determination of the energy gap will be explained in a lecture to come. Another important aspect of excitons come from a specificity shared by all the elementary particles, namely the spin. Quantum mechanics states that elementary particles bear an intrinsic angular momentum called spin. The spin of the electron is one half. So, its spin angular momentum can take two values, minus one half and plus one half, often named as down and up. An exciton is composed of two electrons, one in the HOMO level and one in the LUMO level. The spin of each electron is one half. So, the spin of the exciton can be either 0 or 1. When the spin in zero, the angular momentum can only take the value of zero, While with a spin of one, the angular momentum can be minus one, zero, or plus one. The state we spin zero is called singlet, because there is one value of the angular momentum. The associated wave function is a linear combination of the spin wave function of each electron. We see that because of the minus sign, the sign of the wave function changes when we permute down and up electrons, and the wave function is anti-symmetric. The state with spin one is called triplet, because there are three values of this angular momentum, minus one, zero and plus one. The corresponding wave functions are all symmetric because they stay unchanged, when we permute the down and up electrons. Note the plus sign is the wave function associated with zero angular momentum. The overall wave function of an electron is the product of its spatial wave function times its spin wave function. Electrons are fermions and Quantum mechanics states that the wave function of an assembly of fermions must be anti-symmetric. This is also known as Pauli's exclusion principle. Because the spin wave function of a singlet is anti-symmetric, its spatial wave function must be symmetric. Conversely, the spin wave function of the triplet state is symmetric, so its spatial wave function must be anti-symmetric. In this case, there is a node in the wave function, so there is much smaller overlap of the electrons in the triplet state, thus reducing the Coulombic repulsion between the electrons. As a consequence. the energy of triplet excited state is lower than that of the singlet state. The difference is termed exchange energy. We can now draw a complete diagram of the optical processes in a molecule. The ground state, S naught, with its vibrational replica, is always a singlet state. When light is absorbed, selection rules state that the electron in the ground state can only be promoted to a singlet excited states, which can be either the first excited state S1, or a higher energy state S2. In the latter case, the electron rapidly decays to S1. The lifetime of the electron in S1 is of the order of a nanosecond. Then the electron decays to the ground state either non-radiatively or radiatively, that is with emission of light. This is called fluorescence. The triplet state, T1, is located slightly below the singlet, because of the exchange energy. Although such a transition is symmetry forbidden, an excited electron can also decay to the triplet state, T1, through a process called intersystem crossing. Again, the decay of the electron from the triplet state to the single ground state is forbidden. However, it can takes place with very low probability, which means that the lifetime of the electron in the triplet state is much longer than in the singlet state. The process can be non-radiative or radiative, in which case it is called phosphorescence. The last point I would like to deal with is exciton diffusion. When molecules are embedded in a solid, they are very close to each other, and excitons can travel from molecule to molecule through a process called exciton diffusion. Exciton diffusion can be viewed as an energy transfer between a molecule one and molecule two, which can be depicted by the reaction M1* plus M2, gives M1 plus M2*. The energy transfer can occur via two mechanisms, dipolar or electron exchange. The dipole interaction is also called Förster process. During the process, the electron in the first excited state in molecule one decays to the ground states, while one electron in the ground state of molecule two is promoted to the excited state, thus resulting in the energy transfer from molecule one to molecule two. The interaction occurs via diploar interaction and does not require physical contact of the molecules. Each electron stays in the same molecule before and after the energy transfer. Consequently, the Förster process is a long range process. The exchange interaction or Dexter process involves a double substitution reaction. An electron initially on the molecule M1, jumps to the molecule M2, simultaneously with the jump of an electron from M2 to M1. The interaction requires physical contact between the molecules. So, the Dexter process is short range. This concludes my presentation of optical properties of organic semiconductors. Thank you for your attention.
