3.2 Generation and recombination of charge carriers
When an electron is lifted to the conduction band, at the same time a hole in the valence band is created. In the inverse process, the generated electron-hole pair recombines, and its energy is released. This can happen either via the emission of a photon (radiative re-combination) or recombination can take place via defect levels in the forbidden band gap, which is mainly a non-radiative process. At very high carrier densities, Auger processes prevail.
3.2.1 Electron-hole pair generation
The most important process (from the viewpoint of photovoltaics) for the generation of electron-hole pairs is the absorption of photons with energies hω≥EG. When light im-pinges on the silicon surface, a certain fraction of the light is reflected and the remainder of the photon current Jph traverses the bulk silicon. Due to absorption mainly by valence electrons but also by free carriers especially in the far infrared wavelength range (section 4.2.1), the photon current density decreases following the exponential
(
z,)
J(
z ,)
exp( ( )(
z z0) )
Jph hω = ph o hω −α hω − (3-13)
as a function of the semiconductor depth z, z0 signifying the reference point. The absorp-tion depends on the photon energy and is described by the absorpabsorp-tion coefficient α(·ω).
As a rule of thumb, the coefficient increases with increasing photon energy.
When free electron-hole pairs are generated, both the electron and the hole densities increase and the semiconductor leaves the thermodynamic equilibrium. As a result, the Fermi energy EF loses its meaning; the electron and hole densities have now to be de-scribed by the two quasi-Fermi energies EF,e for electrons and EF,h for holes: The equa-tions
replace equations (3-4) and (3-6), respectively. Equation (3-8) becomes
⎟⎟⎠
The inverse process to the electron-pair production by light is the radiative recombina-tion. In indirect semiconductors such as silicon, radiative recombination is less probable than recombination via defect levels or Auger recombination (see following sections) be-cause it requires interaction with phonons.
The total radiative ecombination rate Rrad is proportional to the concentration of free elec-trons and holes:
p n B
Rrad = (3-17)
where B is a material-related constant, B=3x10-15 cm3/s for silicon [31]. Taking into ac-count the generation rate of free carriers at the same time gives the net recombination rate
(
np n2)
B( (
n0 p0)
n n2)
B
Urad = − i = + Δ +Δ (3-18)
with Δn=Δp=n-n0 the excess carrier densities of electrons and holes which are approxi-mately equal because charge carriers are produced in pairs.
3.2.3 Recombination via defect levels – Shockley-Read-Hall theory
Impurities and crystallographic defects induce defect levels in the forbidden band gap, characterized by the trap energy Et. Figure 3.2 gives the trap levels of selected transition metals, being some of the most recombination active impurities in silicon. Extensive in-formation on transition metal properties in silicon is found in Graff [34].
Via the defect levels, electrons and holes recombine in a two- or more-step process which is described by the Shockley-Read-Hall (SRH) theory [35]. In this model, each de-fect state is either occupied by an electron or by a hole. Recombination occurs when the defect state catches a hole from the valence band in the former or an electron from the conduction band in the latter case; the probabilities of this happening are expressed with the capture cross sections σh,t and σe,t, respectively.
Figure 3.2: Trap levels of selected transition metals. The dashed line shows the center of the bandgap: Traps below ½ EG give the trap level Et from the valence band edge, traps above denote the difference to the conduction band edge. Trap levels taken from [36, 37], after [38].
However, the inverse processes – the electron being emitted from the defect level into the conduction band or the hole emitted into the valence band – have to be considered as well. The probability of re-emission increases with the proximity of the defect level to the conduction band or the valence band edge, respectively. Therefore, defect levels close to the center of the band gap generally have the largest impact on the recombina-tion activity.
The evaluation of the above mentioned processes leads to the net SRH recombination rate of the defect characterized by the density of the defect level Nt with trap level Et and capture cross sections σh,t and σe,t:
( )
Like radiative recombination, the Auger process is innate to semiconductors. Auger re-combination describes the energy transfer from a recombining electron-hole pair to a free electron or hole, which then loses its energy to the silicon lattice by thermalization. The net recombination rate is given as
( ) (
2 0 02)
gehh stand for correction factors due to the Coulombic interaction [39].3.2.5 Carrier lifetime
The annihilation of electron-hole pairs via the different recombination channels is charac-terized by the minority carrier lifetime τe in p-type (τh in n-type) silicon. In this analysis, it is supposed that the injection density is lower than the background doping (e.g.
Δn<<NA in p-type Si), and thus, the recombination rate is limited by the supply of minor-ity carriers.
The general definition of the minority carrier lifetime reads as
U n
e
= Δ
τ (3-23)
in p-type material; an analogous expression is valid in n-type silicon for τh. In the case of a homogeneous carrier distribution, the recombination equals the generation of free charge carriers under steady state conditions. The various contributions through the dif-ferent recombination channels are accounted for via
1 ...
Here, τe,surf stands for recombination at the sample surface, which can be neglected when the surface is well passivated.
An equivalent measurand is the minority carrier diffusion length L which is suitable to estimate the average distance being covered by the free minority carriers before they recombine, e.g. for electrons in the p-type base, being linked to the carrier lifetime via:
( )
Ce e ee D µ
L = , τ (3-25).
De stands for the diffusion coefficient of electrons, depending on the minority carrier con-ductivity mobility µC,e. The latter is discussed in more detail in the following section.