**4 Generation and recombination**

4.1 Introduction

4.2 Generation

4.3 Recombination

4.4 Fermi's golden rule

## 4 Generation and recombination

### 4.1 Introduction

A solar cell is created with one purpose: to convert sunlight into a power source. This happens when light hits the semiconductor with an amount of energy higher then the energy bandgap. The energy that photon possess is converted into a electron-hole-pair (EHP). And as we know, a current is the sum of electrons and holes which travel through the circuit (in opposite directions, the holes with the current flow, and the electrons against). But it's seldom that all of the EHPs , which is generated from the photon, travel through the circuit. In many cases they recombine and the energy is lost to other energy forms than electricity. In this section we'll look into how an EHPs are created from photon (generation), and also how in many cases EHPs are lost in (recombination).

#### 4.1.1 Conservation of carriers and electrostatic potential

There is an important law in electronics that says that the number of carriers of each type inside a volume, must be conserved. This continuity equation is derived from two of Maxwell's equations. Under you can see the Poisson's equation and formulas for conservation of electrons and holes, and in addition an illustration of the continuity equation.

(4.1) | |

where we have that n and p satisfies: | |

(4.2) | |

(4.3) |

In (4.2) you can see that the conservation of electrons depends on the J_{n}, which is the current density in the dimension we are looking at. In 3D (dx, dy and dz), G_{n} is the
generation
of electrons inside the box (3D), and U_{n} is the
recombination of electrons inside the box (3D). The conservation law says that it can't be introduced new charges unless there is created an opposite charge, so that the summation of charge remains the same.

### 4.2 Generation

Generation occurs when an electron is excited over the bandgap and the number of free carriers increases. This is exactly what we need in solar cells (photovoltaic (PV) cells). The free carriers can carry charge around the circuit and we can make use of the electrical energy. Generation requires an input of energy. This input can be:

- Phonons - vibrational energy of the lattice
- Photons - Light, or electromagnetic waves
- Kinetic energy from another carrier

The input energy for PV cells is photons. This energy generates a EHP if the electron is excited from the valence band to the conduction band. But in some cases the electron can't get excited all the way over the bandgap, so the electron is trapped, and we only create a hole. Or we excite an electron from the trapped state into the conduction band, in which case we only create an electron.

#### 4.2.1 Photogeneration

Photogeneration, or the photocurrent, is the most important generation in PV cells. It is the process where mobile electrons and holes are created due to absorption of electromagnetic radiation in the semiconductor. The photogeneration rate is determined by the number of photons, that means the number of excited electrons, and not the energy of the photons. Though we absorb all of the photon energy, we can only use some of it. The energy used is the energy gained over the energy bandgap, the rest of the energy is lost to kinetic energy, phonons etc., generally called thermalization or cooling. The cooling happends in order of picoseconds, which is very fast. An illustration of photogeneration is shown in figure 4.2.

#### 4.2.2 Direct and indirect bandgaps

When a photon is absorbed in a solid state, we have a general rule in physics that says that momentum must be conserved. In direct bandgaps we have the smallest bandgap when k-vector is zero. This is where the transition between valence and conduction band will occur. But for indirect bandgaps, the minimum energy for the conduction band can occur at a k-vector not equal to zero. See figure 4.3. In order to get an absorption we must either absorb a phonon in addition to the photon, or emit a another phonon after the absorption of the incoming photon. Indirect bandgaps are seldom used in PV cells. The absorption coefficient has different behavior for direct and indirect bandgaps. It is larger for direct bandgaps, and is similar to a peak absorption. At higher photon input energies we also get direct optical transition in the indirect bandgap solid states. An example of materials which has direct bandgap is gallium arsenide (GaAs), and material with indirect bandgap are silicon and germanium (Si and Ge).

#### 4.2.3 Excitons and sensitisers

The absorption under the bandgap energy ,E_{g}, creates excitons. An exciton is a bound state of an electron and is a sort of a quasi-particle in a solid. Excitons are thus the main mechanism for light emission in semiconductors at low temperatures (where k_{B}T is less than the exciton binding energy), replacing the free electron-hole
recombination
at higher temperatures. The clue here is that when we are modelling the device behavior for the
generation,
we should use the net generation rate of free carriers from exciton generation and dissociation, and not simply the optical excitation rate.

Sensitiser is a chemical compound, capable of light emission after it has received energy from a molecule (which became excited previously in the chemical reaction). This is e.g. used in photography where we expose the film to light, where the "film material" in ground state stores the picture in an excited state. The film is very sensitive to light, so when we develop the film in a darkroom, we let the material go from the excited state to an ionized state, which is less sensitive to light. The final stage (used in PV cells) injects a free carrier into the semiconductor, which is very favorable for PV cells.

Absorption length of the PV material is the distance a photon (with a certain wavelength) travels before the intensity drops to 1/e. It is important that the absorption length is small, so that only a few microns is necessary to absorb the light. The reflectivity of semiconductors depends on the wavelength, and for visible light it's about 30-40%. Therefore it's important to have thin PV cells. This is the case of e.g. GaAs and InP (direct bandgap), while the cells are thicker for Si (indirect bandgap).

### 4.3 Recombination

Recombination is the opposite of generation, which means this isn't a good thing for PV cells, leading to voltage and current loss. Recombination is most common at impurities or defects of the crystal structure, but also at the surface of the semiconductor. In the latter case energy levels may be introduced inside the energy gap, which incurrage electrons to fall back into the valence band and recombine with holes. In the recombinationprocess we release energy in one of the following ways:

Did you know...

... that to prevent recombination due to energy levels inside the energy gap, the silicon has to be very pure and clean, at least 99,999%.- Non-radiative recombination - phonons, lattice vibrations
- Radiative recombination - photons, light or EM-waves
- Auger recombination - which is releasing kinetic energy to another free carrier

#### 4.3.1 Radiative recombination

Radiative recombination is when an electron goes from the conduction band down to the valence band and excites a photon (light). This is also called spontaneous emission.

If we look at the spontaneous emission rate (recombination rate), r_{sp}, for a volume (when it radiates in all directions), we get the formula shown in (4.4). In the formula we have that n_{s} is the refraction index of the surface (air = 1). E is the energy which is released, E = Δμ = E_{Fn}-E_{Fp}. α(E) is the absorption coefficient, which is given by: α(E)= α_{0}(E-E_{g})^{1/2}.

(4.4) |

Further derivation of the radiative
recombination
is done by looking at recombination in one particular direction, and integrate the energy, E, over the entire range of photon energies. We now get the total recombination, and to get the U_{n} which is expressed in formula (4.2), we have to subtract the rate at thermal equilibrium (Δμ = 0). We will not show these calculations in this text. Radiative
recombination
is more important in direct band gap materials than in indirect. This is because the radiative recombination coefficient is larger with high absorption coefficient.

#### 4.3.2 Auger recombination

Auger recombination involves three particles: an electron and a hole, which recombine in a band-to-band transition and give off the resulting energy to another electron or hole. If the energy is transferred to an electron in the conduction band, it will get an increase in kinetic energy, which normally will be lost when the electron relaxes to the band edge. This is demonstrated in figure 4.5. The energy which is released can also be given to a hole in the valence band.

#### 4.3.3 Non-radiative recombination - Shockley-Read-Hall

Non-radiative recombination or trap assisted recombination is when no photons are emitted, but the energy is lost little by little in a multiple step relaxation over the energy bandgap. The energy is trapped. In real materials this is the most common one, since real materials contains a lot of impurities. We have two types of "traps", called traps and recombination centers. Traps lie near the conduction or valence band and traps only one type of carrier, while the recombination center lies deeper into the band gap (in the middle of the band gap), and traps both types of carriers. Traps can release a carrier, or the carrier can be annihilated, which means it is recombined with the opposite type of carrier.

The generation and recombination rates, or capture and release rates, will depend on the position of the trap in the energy gap. There can also be several traps in a band gap. The derivation of these expressions will not be given in this text. The important effect of traps is that they slow down the transportation of carriers, but it doesn't remove them.

#### 4.3.4 Thermal generation and recombination

There also exists an effect called thermal generation and recombination. Thermal energy is proportional with temperature and Boltzmann's constant. At thermal equilibrium, the thermal generation is balanced with thermal recombination. Since we're only interested in the deviations, we can subtract the thermal generation and recombination and only look at the excess recombination and generation rates.

### 4.4 Fermi's golden rule

Fermi's golden rule is a way to calculate the transition rate from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to perturbation theory. The transition rate is the propability of transitions per unit time. For PV cells, Fermi's golden rule is used to e.g. calculate the absorption and emission rates, the probability of an electron in the conduction band and how many electrons there are in the conduction band. The perturbation theory is used to derive an expression for the transition rate, the generation and recombination rate.