Design Rules of Crystalline Silicon 3

Continuing from Design Rules of Crystalline Silicon 2:

Besides managing charge carriers, the photons in a c-Si solar cell will also have to be managed.  The optical loss mechanisms are as shown in the diagram above.  A reduction of shading losses, as discussed in Design Rules of Crystalline Silicon 2, will have to be balanced by a reduction of resistivity losses.

The second loss of reflection at the front surface has been discussed in Light Trapping II - Anti-Reflection and Trapping Methods.  Using the Rayleigh film method, an intermediate layer (interlayer) can be placed between air and the silicon wafer.  The optimum value of the refractive index of the interlayer is as follows:

n₁ = √(n₀n_s)

where n₀, n_s are the refractive indexes of air and silicon.  Hence, the refractive index of the interlayer n₁ is about 2.1 if n₀ is 1 and n_s is 4.3.

Using the concept of destructive interference, the thickness of the interlayer for a light of 500nm can be calculated:

d = λ / 4n₁

where d becomes about 60nm.

An ideal material for the interlayer is silicon nitride (a-SiN), whose refractive index is between 2 and 2.2 for a wavelength of 500nm.

In addition, texturing of the front surface will improve the coupling of light into the wafer, thereby improving the absorption path length.  This will help absorb light with wavelengths above 900nm.

The texture on wafers can be done with wet-etching (anisotropic etching) techniques.  If a c-Si wafer at an initial 100 surface orientation is etched, textured surfaces with pyramid structures of 111 orientation will be created.  With a-SiN interlayer and textured surfaces, the wafer could be made to look dark blue or almost black (see diagram below).

Reference:
4.3 Design Rules of Crystalline Silicon, Delft University of Technology, https://www.youtube.com/watch?v=qmbrGk-c-P8

Design Rules of Crystalline Silicon 2

Continuing from Design Rules of Crystalline Silicon 1:

The metal contacts grid on solar cells is as shown in the diagram above.  The main conduit in the centre is called the busbar, with fingers going from the busbar to the edge of the solar cell.  The resistance of the fingers is also in the diagram, where ρ is the electrical sensitivity of the metal, and L, W and H are the length, width and height of the finger.  If this series resistance is high, the fill factor of the solar cell will be reduced.

As seen in the diagram above, the red colour electron can move in the spacing distance S between 2 fingers.  Since the emitter has higher resistivity than the metal contact, the power loss due to the emitter resistivity scales with finger spacing to a power of 3.

It must be noted that if there are more metal finger contacts, there will be more shading of the solar cell.  Shading means that light cannot reach the solar cell.  Hence, while having more metal fingers will lead to less loss of power generated, there cannot be too many fingers with widths too wide to block the light from the solar cell.

Similar occurences happen at the back contacts - the back surface of the solar cell.  Holes are collected at the back contact, but since electrons are the only charge carriers in metals, the holes will recombine with electrons from the back contact at the contact interface.  The distance between the p-n junction and the back contact should not be more than the average diffusion lengths of minority electrons in the p-layer.

However, defects at the metal-semiconductor interface of the back contact will still lead to SRH recombinations.  Hence, the area of the interface should be reduced by making point contacts (see diagram above).  The rest of the rear interface should be covered by an insulating passivation layer like the front surface.

In addition, a back surface field should be created by heavily p doping the point interfaces of the back contact (see diagram above).  This will create an additional space charge region at the interface that acts like a p-n junction, preventing minority electrons from moving from the p-layer to the p++ back contact region.  Hence, there is higher minority electron density in the p-layer.



Reference:
4.3 Design Rules of Crystalline Silicon, Delft University of Technology, https://www.youtube.com/watch?v=qmbrGk-c-P8

Design Rules of Crystalline Silicon 1

In a conventional p-type crystalline silicon (c-Si) wafer based solar cell (see diagram above), the yellow n-doped layer (emitter layer) is much thinner than the rest of the wafer.  This is because most light is absorbed close to the front surface of the solar cell (first 10 microns).  A thin front emitter layer allows the excited charge carriers to be within diffusion lengths of the p-n junction.

There are 3 parts to the charge collection process: the emitter layer, the metal contacts, and the back contact.  Aluminium is used as a cheap conductor for metal contacts, which is more conductive than the emitter layer.  Electrons will have to diffuse laterally through the emitter layer to the metal contacts to be collected.

In order to reduce SRH recombination for higher carrier lifetimes, the surface recombination velocity has to be reduced.  This surface recombination arises due to the many defects on a bare c-Si surface, where surface silicon atoms have valence electrons that cannot make molecular orbitals with absent neighbouring atoms.  These are called dangling bonds.  Since most carriers are generated close to the front surface, high surface recombination velocity will result in large losses of carriers, and thereby resulting in lower J_sc.

There are 2 ways to reduce surface recombination.  Firstly, a thin insulator layer can be deposited on the front surface, which restores the bonding environment of the silicon atoms and force electrons to move inside the emitter layer.  Silicon oxide and silicon nitride are the usual chemical passivation layers used.

Secondly, the minority charge carrier density at the surface region limits the surface recombination velocity, and hence has to be reduced.  This can be done by increasing the p-layer's doping, but this also reduces the diffusion lengths of minority holes in the emitter layer.  Hence, it is not viable for the whole emitter layer.

However, this can be applied at the metal-emitter interface region, where an insulating passivation layer cannot be used.  There is a high interface recombination velocity at the metal-emitter interface.  Moreover, the metal-semiconductor interface induces a barrier for majority charge carriers, which gives rise to higher contact resistance.  Hence, the area of contact between the metal and emitter should be reduced, while the emitter directly under the metal contact should be heavily doped to N+ or N++.



Reference:
4.3 Design Rules of Crystalline Silicon, Delft University of Technology, https://www.youtube.com/watch?v=qmbrGk-c-P8

Manufacturing of Crystalline Silicon

Metallurgical silicon is the lowest quality of silicon that can be produced and is 98-99% pure.  It is made from melting quartzite, a rock of pure silicon oxide, and mixing with carbon.

Polysilicon is the next lowest quality of silicon and is made by the Siemens process.  Impurities are removed by distillation in the process, and a pure silicon material is grown by chemical vapour deposition requiring high temperatures.  The Siemens process' energy consumption is very high.

An alternative to the Siemens process is the fluidized bed reactors (FBR), which uses lower temperatures and hence consume much less energy.  The purity of polysilicon can be as high as 99.9999%.

If the purity of polysilicon is not required, upgraded metallurgical silicon can be made cheaply by blowing gases through the silicon melt to remove impurities.

To make monocrystalline silicon ingots, which are solids that is one big continuous crystal, there are 2 methods.  The Czochralski processing method allows doping, orientation of the crystal to (100) or (111), and production of large crystals.

The float zone process allows the creation of ingots with very low densities of impurities like oxygen and carbon.  It also allows doping, but the size of the ingot is limited.

Polysilicon (multicrystalline silicon) can also be produced by silicon casting, where melting is done in a dedicated crucible, and the melt is poured into a growth crucible to solidify.  If melting and solidification are done in the same crucible, it is called directional solidification.

Finally, 2 methods can be used to create wafers out of ingots.  Sawing can be done, but it generates a large proportion of wasted silicon called kerf loss, and requires polishing.  Silicon ribbon is the next method.  It pulls ribbons of silicon out of silicon melt before cutting them into wafers.  Further treatment is needed, but the electronic quality of ribbon silicon is still not as good as monocrystalline silion.



Reference:
4.2 Manufacturing of Crystalline Silicon, Delft University of Technology, https://www.youtube.com/watch?v=Oqk4H2Ji79k

Properties of Crystalline Silicon

The lattice structure of crystalline silicon is cubic diamond, and has long range order and symmetry.  Different cuts can be made across and through the lattice to see different planes (see diagram above).  For the (100) surface, whose normal points in the [100] direction, the plane has 2 valence electrons pointing to the front.  For the (111) surface, the plane has 1 valence electron pointing in the direction of the normal to the plane.

The electronic band dispersion diagram (see diagram above) shows the indirect band gap of silicon.  The vertical axis shows the energy level of the valence and conduction bands, while the horizontal axis shows the crystal momentum (momentum of the charge carriers), the lattice momentum in various directions.  To get excited into the conduction band, electrons in the valence band require a change in energy and momentum.  The band gap of silicon as shown magnified in the diagram below, consists of the lowest point of the conduction band at x, which is indicative of the [100] direction, and the highest energy value of the valence band at gamma.

The indirect band gap energy required is the difference between the energy levels at x and gamma and equals 1.12eV or a maximum wavelength of 1107nm.  There can also be direct transition of electrons, where the direct band gap energy required is found at gamma.  The value is 3.4eV or a maximum wavelength of 364nm, which is the blue spectral part.  Hence, it is more difficult to excite electrons into the conduction band for silicon, as compared to direct band gap materials like GaAs and InP.  This also means that the absorption coefficient is much lower (see diagram below).

Germanium is also an indirect band gap material, but it has a low band gap of 0.67eV, so it starts absorbing light at wavelengths below 1850nm, with direct transitions at lower wavelengths at certain momentum-space directions.

Referring to the diagram above, assuming a EQE of 1, a silicon band gap of 1.12eV corresponding to a wavelength of 1107nm will lead to a theoretical maximum J_sc of about 45mA/cm².  This is the spectral utilisation consideration.

Referring to the diagram above, using Lambert-Beer's law, it can be seen that to absorb a higher wavelength of 970nm, a longer absorption path length of 230 microns is required to absorb 90% of all incident light.  This is a typical thickness of silicon wafer and shows the importance of light trapping techniques for crystalline silicon absorber layers for wavelengths above 900nm.

Finally, consider band gap utilisation as determined by recombination losses, where only Auger and SRH recombination are considered due to silicon's indirect band gap transitions.  Here, 2 types of silicon have to be considered: monocrystalline silicon (single crystalline) and polycrystalline silicon (polysilicon).  Single crystalline has an unbroken crystalline lattice up to the edges.  Polysilicon consists of many small crystalline grains in random orientations, meaning that there's many grain boundaries (see diagram below).

Hence, polysilicon has many lattice mismatches, which leads to defects at the grain boundaries, so the lifetime of charge carriers is shorter than single crystalline.  There is a lot of SRH recombinations.  Generally, the larger the grain size, the charge carrier lifetimes will be longer.  This means that band gap utilisation is better, and the V_oc will be larger.



Reference:
4.1 Properties of Crystalline Silicon, Delft University of Technology, https://www.youtube.com/watch?v=rPeBUO_08GE

Light Trapping II - Anti-Reflection and Trapping Methods

When light reaches an interface between 2 media having different refractive indices, the incident light will usually be partly reflected and partly transmitted.  Consider a light ray arriving from the left as shown in the diagram above.  The light ray's behaviour will be described as follows:

Snell's Law: n₁sinθ_i = n₂sinθ_t

θ_i = θ_r

where n₁, n₂ are the refractive indices of the 2 media, and θ_i, θ_r, θ_t are the angles of incidence, reflection, and transmission of the light ray.

If n₁ < n₂, θ_i > θ_t
If n₁ > n₂, θ_i < θ_t

Light rays are light waves with electric field oscillations as shown in the diagram above.  P-polarised light is light with the electric field oscillating in the plane of incidence.  S-polarised light is light with the electric field oscillating perpendicular to the plane of incidence.  The reflection and transmission coefficients are given by the Fresnel equations below:

When light is incident perpendicularly:
θ_i = θ_t = 0°
R_p = R_s
T_p = T_s

As seen in the diagram above, the Brewster angle is the angle of incidence where R_p = 0.  The grey line represents the average of R_p and R_s, which is randomly polarised light.

When n₁ > n₂, there will be a θ_i where θ_t = 90°.  This θ_i is known as the critical angle.  For all θ_i larger than the critical angle, there will be total internal reflection where no θ_t exists.  This is because the light rays are completely reflected.

The optical losses due to reflection between 2 media such as air and silicon is quite high because silicon has a high refractive index.  Hence, an interlayer (Rayleigh film) with a refractive index between air and silicon should be applied between air and silicon.  The diagram below shows the final outcome.  With more than one interlayer, reflection losses can be reduced further.  This is known as refractive index grading.

Another way to do anti-reflection is by super-imposing light waves.  This will lead to constructive and destructive interference of light.  Consider the diagram below.  The yellow wave is the incident light.  The green and red waves are the reflections back from the first and second interface.  Since the green and red waves appear to be in anti-phase, the total amplitude of the electric field of the final wave leaving the system is smaller.  The total irradiance lost is smaller too.

The maximum destructive interference is given by a thickness of the interlayer that is:

d = λ / 4n₂

where λ is the wavelength, and n₂ is the refractive index of the interlayer.

The last way to do anti-reflection is by using textured interfaces.  The size and scale of textured features is usually larger than the wavelength of light, which helps to couple light into the interlayer (see diagrams below).  The importance of this method cannot be over-emphasised.

Another consideration is that the absorber layer is not thick enough to absorb all light, and some light is transmitted.  Usually, absorption coefficients are higher for blue light compared to red and infrared.  The red light transmitted can be reflected back by the back reflector, or be parasitically absorbed at the back contact/reflector.



Reference:
3.3.5 Light Trapping II - Anti-Reflection and Trapping Methods, Delft University of Technology, https://www.youtube.com/watch?v=gyzzaZw6sC4

Perovskite Solar Cells - a first history as of early 2014

In 2006, it was reported that CH₃NH₃PbBr₃ cells have an efficiency of 2.2%.  In 2009, efficiency was raised to 3.8% by replacing bromine with iodine.  However, all devices were unstable and the hole transporting medium (HTM) was not solid state.

In 2011, an efficiency of 6.5% was achieved by applying TiO₂ surface treatment before deposition of perovskite as sparsely spaced hemispherical nanoparticles.  Although perovskite nanoparticles have better absorption when compared with standard N719 dye sensitizers, they dissolve in the electrolyte, so performance quickly degraded.

In 2012, a solid state HTM was finally used.  Spiro-MeOTAD was used to penetrate nanoporous TiO₂ when dissolved in an organic solvent.  After evaporation, only solute molecules are left.  Spiro-MeOTAD improved the stability and reported efficiency to 9.7%.

In 2012, another 4 developments occurred.  Spiro-MeOTAD was used with mixed halide CH₃NH₃PbI_3-xCl_x.  This provided better stability and carrier transport than using only iodine as the halogen.  The second development was to coat nanoporous TiO₂ surfaces with a thin perovskite layer.  This created extremely thin absorber (ETA) cells.  The third development was to use a non-conducting Al₂O₃ network/scaffolding instead of the conducting nanoporous TiO₂.  This improved the V_oc and efficiency rose to 10.9%.  The fourth development utilise perovskite cells without the scaffolding, demonstrating ambipolar transport with simple planar cells (see diagram below for the scaffolding).

In 2013, reported efficiency rose to 12.0% using both optional layers in the diagram.  A solid layer made of nanoporous TiO₂ infiltrated by perovskite is the scaffolding.  The best HTM used was to be poly-triarylamine.  The reported efficiency increased further to 12.3% when similar structures and mixed halide CH₃NH₃PbI_3-xBr_x is used.  Low Br content of <10% provided the best starting efficiency due to a lower bandgap.  Higher Br content of >20% gave better stability against humidity.  There is a correlation between this and a structural transition from tetragonal to pseudo-cubic.  This is due to a higher tolerance factor t due to the smaller ionic radius of Br.

In 2013, 2 further developments were made in May.  The first development improved the morphology by using TiO₂ scaffolding and a 2 step iodide deposition.  It included an independent measurement of efficiency amounting to 14.1%.  The 2nd development also improved the morphology and resulted in an efficiency of 15.4%.  It used simple but very different planar solar cells without scaffolding.  It used the 2 source thermal evaporation for CH₃NH₃PbI_3-xCl_x deposition.

Towards the end of 2013, an independently confirmed efficiency of 16.2% was reached by using mixed halide CH₃NH₃PbI_3-xBr_x with 10% to 15% Br, and a poly-triarylamine HTM.  Both optional layers shown in the diagram above were used.  The main factor for the improvement is the thickness of the perovskite-TiO₂ scaffolding relative to the continuous perovskite layer.  There are 2 unconfirmed increases in efficiencies in early 2014 - 17.9% and 19.3%.



Reference:

"The emergence of perovskite solar cells" by Martin A. Green, Anita Ho-Baillie and Henry J. Snaith, published online 27 June 2014, https://www.researchgate.net/publication/280388277

Light Trapping I - Absorption and Optical Losses

Consider light shining on a single absorbing medium of thickness d from the left (see diagram below), where the distance x starts from the left - the point of incidence of light.  2 simplifications are to be considered: that the reflection and scattering at the interfaces are ignored, and that the light is monochromatic (all photons having same wavelength/energy).  The absorption of a medium is represented by an absorption coefficient α, whose unit is cm^-1.

The reduction of light intensity as light passes through the medium, due to absorption, is described by Lambert-Beer's law:

I(λ, x) = I₀(λ)exp[-α(λ)x]

where I₀ is the full light intensity before entering the medium, and I is the intensity as light passes through the medium.

Lambert-Beer's law shows that more light is absorbed at the point of incidence/entry on the left, than on the right at the end.  It also shows that for absorption to be high, the absorption coefficient or the thickness d has to be large.  It must be noted that the absorption coefficient for each material is different at different wavelengths (see diagram below).

It can be seen from the diagram that Germanium has the lowest band gap and absorbs photons with low energies, where wavelengths are high.  Gallium arsenide (GaAs) has the highest band gap because it absorbs light with the highest photon energy, starting where wavelengths are small.  It can also be seen that for the visible spectrum from 300nm to 700nm, the absorption coefficient of Indium Phosphide (InP) and GaAs is much higher than that of silicon.  This is because InP and GaAs are direct band gap materials that have higher absorption coefficients.  This shows that silicon absorbs quite poorly compared to GaAs, and thicker absorber layers are required for the same fraction of light.

In general, the absorption of high energy photons like blue light is very much larger than low energy photons like red light.  This implies that blue light will not penetrate deeply into the absorber layer, because it will completely be absorbed once it touches the absorber layer (at the front/window layer).  Since the absorption of photons generates excited charge carriers, the diagram will also show the local generation profile of charge carriers based on the absorption coefficients and wavelengths.  More charge carriers are generated at the front layer.

This also implies that the EQE for blue light will be for charge carriers generated close to the front layer, while the EQE for red light will be for charge carriers generated throughout the entire absorber layer.

For optical losses, we consider a crystalline silicon c-Si solar cell with a p-type bulk and thin n-type layer on top (the yellow n-emitter) (see diagram below).  Metal contacts are located at the top and back.  Hence, the first optical loss mechanism comes from shading, where the top contacts block light from reaching the PV active layers.

The 2nd optical loss comes from the reflection at the front interface (between air and silicon) of the solar cell, because when light passes through an interface between 2 media with different refractive indices, light will be partly reflected and transmitted at the interface.

The 3rd optical loss comes from parasitic absorption losses in non-active PV layers, such as the green layer in the diagram.  This layer can be an anti-reflection coating or passivation layer for reducing defects at the surface of the emitter layer.  Photons absorbed at this layer will not contribute to carrier generation.  This means that this layer should preferentially have high transmissions for the spectral part utilised by the solar cell.

The last optical loss comes from the inability of the absorber layer to absorb all the light, and the light is transmitted.  This happens for thin film solar cells.



Reference:
3.3.4 Light Trapping I - Absorption and Optical Losses, Delft University of Technology, https://www.youtube.com/watch?v=FKNCF-bprhs

Spectral Utilization II - Shockley-Queisser Limit

The Shockley-Queisser limit refers to the limit imposed on the maximum conversion efficiency of solar cells, at 48%, due to 3 types of losses.  Firstly, there's optical losses, where photon energy above the band gap is lost as heat, and where energy is also lost due to non-absorption of photons having less energy than the band gap (see diagram below, assuming EQE of 100%).

Secondly, there's energy losses due to black body radiation, because at a particular temperature, light will be emitted by the solar cell in the far infrared.  The loss is about 7% of the incident AM1.5 solar spectrum's energy.

Lastly, there's recombination losses, where in deriving the Shockley-Queisser limit, only radiative recombination is considered.  This means that the temperature of the solar cell is theoretically not allowed to increase (a thermodynamic approach), and that all incoming AM1.5 spectrum energy will leave the solar cell either through generating photocurrents, or by radiative recombination.

The diagram above shows the usable energy in single junction solar cells based on the Shockley-Queisser limit.  However, there is still a gap between the Shockley-Queisser limit and the actual best efficiencies at optimum band gaps of solar cells (see diagram below).

This is due to the missing optical losses, such as reflection and parasitic absorption, and electrical losses, such as SRH and Auger recombinations.  Thus, the Shockley-Queisser limit is most valid for direct band gap materials like GaAs.

One way to overcome the Shockley-Queisser limit is to have multi-junctions, where more than one semiconductor is used.  This means that there will be multiple band gaps to improve light absorption (see diagram below).

Reference:
3.3.3 Spectral Utilization II - Shockley-Queisser Limit, Delft University of Technology, https://www.youtube.com/watch?v=x4UP9m3O99w

Spectral Utilisation I - External Quantum Efficiency

In this post and the next, we consider spectral utilisation.  For a single junction solar cell where the absorber layer has a certain band gap energy (see diagrams below), parts of the solar spectrum below the band gap energy will be ignored.  Here, the blue parts and some of the green parts are absorbed, blue being the most energetic photons.  The red part has energy below the band gap energy, so it is all left out.  In a J-V curve, the J_sc of this cell will be relatively small.

In another solar cell with a lower band gap energy (see diagrams below), the spectrum from blue to some of the red photons are absorbed.  Hence, in a J-V curve, the J_sc will be higher, while the V_oc is reduced.  This is because the separation of the quasi-Fermi levels is smaller.  The outcome of this is that V_oc and J_sc of semiconductor materials are determined by its band gap.

The external quantum efficiency is defined as the number of electrons collected at the contacts/terminals for each incoming photon at a certain wavelength.  This is a measure of the spectral utilisation of a solar cell.

Hence:
EQE(λ) = (J(λ)/q) / Φ(λ)

Due to optical and electrical losses for charge carriers, not every photon reaching the solar cell will lead to electron collection at the contacts/terminals.

The importance of EQE is because there are EQE measurement setups in labs to determine the number of electrons collected and number of photons per wavelength.  Moreover, measuring the J_sc using EQE setup will result in a current density independent of the spectral shape of the light source in use.  The EQE measurement setup using shading masks is also independent of the real contact area of solar cells.

As seen from the equations in the diagram below, the measured EQE and the spectral power density of the AM1.5 solar spectrum, will provide the J_sc of the solar cell.

Finally, it is important to note that for reporting J_sc of solar cells, it is not reliable to do a single J-V measurement.  Measurement using EQE setup is more reliable.

As seen in the diagram below, the EQE is 90% above the band gap, meaning that 10% of photons are lost.  The band gap energy in electron volts (eV) can be found from:

E = hc/λq = 1240/λ

where λ is 1100nm, which is the highest wavelength, in this graph, for photons that are absorbed.  Photons with wavelengths greater than this will not have enough photon energy to cross the band gap.

Reference:
3.3.2 Spectral Utilisation I - External Quantum Efficiency, Delft University of Technology, https://www.youtube.com/watch?v=tbOOuQFVwog

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

Series and Shunt Resistance

In a real, non-ideal solar cell, there are electrical losses, which can be represented as parasitic resistances in an ideal solar cell electrical circuit.  2 important resistances are the series and shunt resistance.

Consider a crystalline silicon c-Si solar cell.  The series resistance R_s results from the semiconductor materials of the p-n junction, the interface between the p-n junction and the metal terminals/contacts, and finally, the metal terminals/contacts themselves (see diagram below).

The shunt (or parallel) resistance R_SH results from a macroscopic defect in the solar cell, such as a crack through the semiconductor layers or a current path along the edge of the solar cell.  These are alternate paths for the photogenerated current (see diagram below).

Ideally, the series resistance should be as low as possible to minimise voltage losses, while the shunt resistance should be as high as possible to minimise leakage of photocurrent.

Hence, the circuit diagrams below show the outcome:

V = V_ideal - JR_s

J = J_PH - J_D - J_SH

J_SH = (V+JR_s) / R_SH

The effects of increasing R_s and decreasing R_SH are separately shown in the diagrams below.  The J-V curve will change.  Increasing R_s and decreasing R_SH will also reduce the maximum power point, and correspondingly, the FF, because V_oc and J_sc are unchanged.

The J-V curve of a solar cell has current density and power density varying with voltage.  To determine the point on the curve that the solar cell operates at, a load must be connected (see diagram below).

A load of low impedance means current density is high and voltage low, leading to a P_out lower than the maximum power point P_max.  A load of high impedance means the reverse: the current density is low and voltage high.  It also leading to a P_out lower than P_max (see diagrams below).  Hence, solar cells must have maximum power point trackers (MPPT) to tune the impedance of the load to ensure proximity to P_max.

In the 2 diagrams above, the lower right corner of the red rectangles point to the point on the J-V curve of the solar cell's operating parameters.  Extending a vertical from the voltage parameter will intersect the power density-voltage curve at a point to indicate the P_out.

Finally, it must be noted that the slope of the J-V curve at V_oc can be estimated as:
dJ/dV = 1/R_s

And the slope of the J-V curve at J_sc:

dJ/dV = 1/R_SH

Reference:
3.2.2 Series and Shunt Resistance, Delft University of Technology, https://www.youtube.com/watch?v=HnpDaYINCKY