Utilisation of Band Gap Energy

There are 3 engineering design tools: utilisation of band gap energy, spectral utilisation, and light trapping.  In this post, utilisation of band gap energy will be discussed.

The actual energy for voltage generation comes from qV_oc.  This energy arises from the splitting of the quasi Fermi levels, and is always smaller than the total band gap energy under normal operation conditions.  So, V_oc is dependent on doping.

Previously, when J = 0 (see diagram above):
V_oc = (kT/q) ln (J_PH/J₀ + 1)

where J_PH is the photo current density arising from irradiance, and J₀ is the current density arising from the diode leakage current in the dark.

This equation can also be expressed as follows:

V_oc = (2kT/q) ln (G_Lτ₀/n_i)

where G_L is the generation rate, τ₀ is the minority charge carriers lifetime, and n_i is the intrinsic density of charge carriers in the semiconductor material.

These indicate that increasing irradiance (the generation rate of charge carriers) will increase V_oc.  Increasing the lifetime of minority charge carriers will also increase V_oc.  The τ₀ depends on the recombination rate.  In recombination, energy and momentum are transferred from charge carriers to phonons (lattice vibrations) or photons.

Considering SRH recombination (the first recombination to consider), it depends on the defect density.  τ₀ should be reciprocally dependent on defect density N_t, meaning that a large N_t will reduce τ₀ and V_oc (see diagram above).  These defects may be found in the bulk of semiconductor materials, and may also be inside the various interfaces between the materials used in solar cells, such as semiconductors, transparent conductive oxides (TCO), and metal contacts.

When solar cells are without or with little defects, radiative and auger recombination takes precedence.  Consider Auger recombination, which is a process where momentum and energy of the recombining electron hole pair is conserved by transferring energy and momentum to another electron or hole.  After transfer, the excited charge carrier will relax again, losing the energy as phonon modes, which are lattice vibrations (heat).

Due to the 3 particle interaction, the Auger recombination rate R is strongly dependent on charge carrier densities (see diagram above).  n and p are the densities of electrons and holes respectively, while R_electron is dominant when electrons are the majority charge carriers.  Similarly for R_hole, which applies to holes.  Hence, lifetime is approximately 1 over density of charge carriers squared, and Auger recombination will dominate when charge carrier density is high.

Lastly, there's radiative recombination.

There are 2 types of band gap: direct and indirect.  This arises due to the difference in positions of the valence and conduction bands in different directions of the lattice coordination (see diagram below).  In the energy-momentum space of the electrons, there is the vertical axis for the energy state in the electronic bands, and the horizontal axis for the momentum of charge carriers (the crystal momentum).

For indirect band gaps, the highest point of the valence band is not directly under (vertically aligned with) the lowest point of the conduction band.  When electrons are excited from valence to conduction band, energy provided by a photon and momentum provided by a phonon are required.

For direct band gaps, there is vertical alignment between the highest point of the valence band and lowest point of the conduction band, thereby requiring only the photon's energy for electron excitation.  Hence, excitation of electrons by photon absorption is more likely for direct band gap materials, meaning that the absorption coefficient will be significantly higher than indirect band gap materials.  This also means that radiative recombination will more likely happen.

Crystalline silicon, being an indirect band gap material, will be dominated by Auger recombination if SRH recombination can be ignored.  Gallium Arsenide (GsAs), being a direct band gap material, will be dominated by radiative recombination if there is moderate illumination conditions.  If illumination is strong, Auger recombination will have to be considered.

In conclusion, in defect rich solar cells, V_oc is limited by SRH recombination.  In defect free solar cells using indirect band gap materials, V_oc is limited by Auger recombination.  In defect free solar cells using direct band gap materials, V_oc is limited by radiative recombination.

An additional point to note for recombination is the maximum thickness for the absorber layer of solar cells.  Since recombination affects the diffusion length of minority charge carriers, the absorber layer of solar cells cannot be thicker than the usual diffusion length.  This is to prevent minority charge carriers from recombining before reaching the p-n junction or back contacts.  If they recombine, the charge carriers will not be collected.



Reference:
3.3.1 Utilisation of Band Gap Energy, Delft University of Technology, https://www.youtube.com/watch?v=OGbzore-ebo