Page 156 - A Comprehensive Guide to Solar Energy Systems

Chapter 8 • Photovoltaics: The Basics 157

back a position in an atomic bond, which results in the annihilation of an electron–hole
pair). The recombination rate is described by
∆ 
 dn ∆n
   =− , (8.7)
 dt rec τ d∆ndtrec=−∆nτ,
where τ is the so-called carrier lifetime that characterizes the excess carrier recombination
rate.
In the recombination process, the conservation of both energy and momentum has to
be satisfied. The excess energy released as either photons (irradiative recombination) or it
is transformed into heat (nonirradiative recombination). The following are the three most
important recombination mechanisms.
Radiative recombination. This involves a conduction band electron falling from an al-
lowed conduction band state into a vacant valence band state (a hole). It is the inverse
process to optical generation, the excess energy being released mainly as a photon with
energy close to that of the bandgap. The irradiative recombination rate depends on the
concentration of free electrons, n, and free holes, p, that is, on doping concentration
(donor or acceptor) and also on excess carrier concentration ∆n. For low injection level
(∆n ≪ n 0 + p 0 ), the irradiative lifetime τ r can be expressed by

1
τ = , (8.8)
r
CN τr=1CrN,
r
where C r is a constant characteristic for the material and N is the doping concentration
(donor concentration N D in n-type or acceptor concentration N A in P-type). The irradia-
tive recombination coefficient C r depends on the band structure. For an “indirect” semi-
−3
conductor, for example, silicon, C r is very low (for si, C r ≈ 2 × 10 −13 cm ) and irradiative
recombination is not an important recombination mechanism. In the case of InGaAs and
some other semiconductors with so-called direct band structure, C r is much higher.
Auger recombination. This can be considered as a three-particle interaction where a
conduction band electron and a valence band hole recombine, with the excess energy be-
ing transferred to a third particle (free electron or hole) as kinetic energy and transferred
to heat by the thermalization process. The Auger recombination may be very important
in a highly doped semiconductor, as the carrier lifetime strongly depends on free carrier
concentration. For a low injection level (i.e., ∆n ≪ n 0 + p 0 ), the carrier lifetime due to the
Auger recombination in an n-type semiconductor is given by

1
τ = , (8.9a)
A
C An N 2 D τA=1CAnND2,
and in P-type semiconductor by

1
τ = , (8.9b)
A
C Ap N 2 A τA=1CApNA2,