In the applied physics of semiconductors, two types of carriers with similar properties but with opposite charge, take part in the optical and electrical phenomena: electrons and holes. In halide perovskite, as well as in organic semiconductors, the main carrier contributions arise from excitons. One may think about these as bound state of coupled electrons and holes moving together through the material.
The role of excitons in semiconductors
To understand excitons, one must think about the band structure of a semiconductor. In the ideal case, absorption of a phonon may raise an electron from the valence band to the conduction band, leaving a hole in the valence band. When the density of defects is large enough, there may be so-called exciton lines in the optical spectrum. These are intermediate levels located in between the conduction and valence bands, that may trap electrons or holes. Since the excitation is electrically neutral, the exciton lines do not contribute directly to electrical conduction.
Two models describe these excitons: the Wannier exciton and the Frenkel exciton.
By doping a semiconductor with acceptor impurities, it is possible to increase the number of holes to a requested amount, thus making a p-type semiconductor. Thus, acceptor impurities are negatively charged and may attract holes, forming an impurity level just above the top of the valence band. Reciprocally, doping with donor impurities may give rise to n-type semiconductors. In this case, impurity levels are formed just below the bottom of the conduction band, by attracting the charged impurity for extra electrons. These two kinds of impurity levels create bound states for holes and at small distances below ore above the zero of their kinetic energy.
In a compensated sample, the number of donor and acceptor impurities compensate, meaning that some of the electrons may fall from the donor to the acceptor impurity levels, annihilating the holes on this bottom level. Thus, the empty impurities are no more electrically neutral: the donors are positively charged and the acceptor negatively charged. Hence these impurities will attract or repel the remaining carriers. In this case, one may set up a hydrogen-like wave function in which electrons and holes circle about their joint centre of mass, creating a Wannier exciton. The bound carriers interact through a Coulomb interaction, screened by the dielectric constant ϵ of the material. When this dielectric constant decreases, the exciton becomes more localised, breaking up this Wannier exciton model.
The Frenkel exciton
An alternative way of thinking the exciton is in terms of excited atoms. Let’s take one of the electrons on an atom, and raise it to the excited state ϕe(𝙧). We make here a Bloch function of the form Ψ𝙠 = ∑𝙡 ei(𝙠·𝙡) ϕe(𝙧-𝙡) to allow the excitation to travel through the crystal.
This exciton can move through the crystal in a process that does not require the vacancies, interstitials or substitutional impurities to move. This model does not take into account the electron-electron interaction, and the exciton state is far more complicated in practice that the Bloch function I presented. With it, the travelling electron carries a local polarisation of the other electrons existing in the valence band. We can see it as a travelling electron carrying its own hole with it. A subsequent amount of energy would be required to separate the exciton into independent carriers, that can be collected in a solar cell.
It is important to consider that the Wannier exciton and the Frenkel excitons are opposite descriptions of the same phenomenon. Whereas the Frenkel exciton describes a single excited ionic level where the electron and hole are localised sharply on the atomic scale, the Wannier exciton represent localised electron and hole levels that extend over many lattice constant.
Excitons in halide perovskite
Given the very high density of defects in halide perovskites, is has been reported that the excitons are the main charge carrier. The dissociation energy is about 50eV, that is large enough to prevent the dissociation at RT. In these materials, the static dielectric constant ϵ is about 70kHz. Therefore we may consider the Wannier model to describe these charge carriers.
Moreover, a small effective mass of charge carriers favours large exciton radii, adequate with the Wannier model where the exciton radius is larger than the lattice constant. Because of this, it becomes clear that the exciton does not play an important role in the performance of perovskite solar cells. However, because excitons are the main carrier to travel through this material, proper understanding of their behaviour may allow progress in perovskite solar cells efficiencies.
- Theory of Solids, JM Ziman, Cambridge University Press
- Solid State Physics, Ashcroft/Meermin, Cengage Learning
- Electro-optics of perovskite solar cells, Lin Q et al, Nature Photonics 9, 2015