Environmental Engineering Reference
In-Depth Information
under dark conditions. Charge transport and relaxation rates of the carrier populations
to re-establish the former equilibrium can be described using the continuity equation
(Andrade et al., 2011):
∂j i
∂x +
∂n i
∂t
OH
e , h +
G i ( x )
R i ( x )
=
,
i
=
and
(10.4.25)
The first term on the left-hand side of the equation represents the carrier flux defined
by the respective transport equation. The second and third terms are the generation
and relaxation rates of species i , respectively. Both reaction kinetics, are characterized
by a time constant. The term on the right-hand side of the equation corresponds to the
carrier concentration history of species i .
Equation 10.4.25 will be applied to the different PEC system components and for
the different species that exist on each region: semiconductor bulk, depletion layer or
electrolyte solution. For simplicity, a flat structure of semiconductor material stable at
an alkaline media was considered.
a)
Electrons in the depletion layer (a < x < L)
In the depletion layer there are two contributions for the electron flux, j e : the dif-
fusive transport of electrons and the convection transport driven by a macroscopic
electric field. Equation 10.4.26 relates the electron flux at any position x with the
gradient of electrons concentration across the depletion layer, n e , by means of the elec-
tron diffusion coefficient, D e , the electron mobility, µ e , and the macroscopic electric
field, ξ :
∂n e
∂x
j e =−
D e
µ e n e ξ
(10.4.26)
The generation term for electrons in Equation 10.4.25 is given by the Beer Lambert
equation, which relates the absorption of light to the properties of the material through
which the light is travelling (Andrade et al., 2011):
η inj α ( λ ) I 0 e α ( λ )(1 x )
G e =
(10.4.27)
which assumes that each photon, with energy hv
E g , absorbed by the semiconductor
results in an injected electron into its conduction band. α ( λ ) is the wavelength-
dependent absorption coefficient, which is a material property; I 0 is the incident photon
flux, corrected for reflection losses of the TCO glass; and η inj is the electron injection
efficiency. The expression (1
x ) in the exponential term indicates that front illumina-
tion of the semiconductor is considered. The electron recombination term is neglected,
it being assumed that in the depletion layer electrons and holes are efficiently separated
(Reichman, 1980).
The continuity equation for electrons in the depletion layer between x
=
a and
x
=
L can be written as:
2 n e
∂x 2
∂n e
∂t
( n e ξ )
∂x
η inj α ( λ ) I 0 e α ( λ )(1 x )
=
D e
+
µ e
+
(10.4.28)
 
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