Page 157 - A Comprehensive Guide to Solar Energy Systems

158 A CoMPrehensIVe GuIde To soLAr enerGy sysTeMs

where C An is the coefficient of Auger recombination in n-type semiconductor and C Ap is
−3
the coefficient of Auger recombination in P-type. In silicon, C An ≈ 10 −31 cm and Auger
recombination may be the dominant recombination mechanism in layers with doping
19
−3
concentration higher than 10 cm .
Recombination through loca1 centers. The presence of defects within a semiconductor
crystal (from impurities or crystallographic imperfections such as dislocations) produces
discrete energy levels within the bandgap. some of energy levels lie deep in the middle
of the bandgap. These defect levels, also known as traps, greatly facilitate recombination
through a two-step process. In the first step, a free electron from the conduction band
relaxes to the defect level and then (the second step) relaxes to the valence band where it
annihilates a hole. The recombination rate is proportional to center concentration, N t . In
an approximation, carrier lifetime can be expressed by

1
τ = , (8.10)
t
τt=1CtNt, CN t
t
where C t depends on center capture cross sections for electrons σ n and holes σ p , the center
energy level W t , and it also slightly depends on carrier concentration. By controlling the
recombination center concentration N t , it is possible to control the carrier lifetime.
In general, the recombination processes can be considered to occur independently and
the resulting recombination rate is simply the sum of the individual rates. From that it fol-
lows that the resulting carrier lifetime, τ, is given by
1 = 1 + 1 + 1 . (8.11)
1τ=1τr+1τA+1τt. τ τ r τ A τ t

At the surface, a higher recombination rate can occur due to surface local states. More
details about recombination processes can be found in refs. [1,3,4].

8.2.3 Excess Carrier Concentration

As shown earlier, excess carrier generation is not uniform and an excess carrier concen-
tration gradient is caused by the decreasing generation rate with depth, below the illu-
minated surface. A flow of carriers in the direction from higher to lower concentration
is connected with the concentration gradient. The flow of charged particles is an elec-
tric current. This way, diffusion current density of electrons of the charge −q(q = 1.602 ×
10 −19 C) is given by
dn
J ndif = qD n . (8.12a)
Jndif=qDndndx. dx
The hole diffusion current density (hole charge is q) is given by

dp
J pdif =−qD p , (8.12b)
Jpdif=−qDpdpdx, dx