Photoelectrochemical cells for hydrogen production from solar energy - Solar Energy Sciences and Engineering Applications - page 330

b) Electrons in the semiconductor bulk (0 < x < a)

In the semiconductor bulk the effect of the macroscopic electric field is negligible. Thus,
the charge transport within the semiconductor takes place only via diffusion; the flux
equation only considers the transport driven by the concentration gradient as follows:

∂n e −
∂x

j e − =− D e − (10.4.29)

In this region, outside the depletion layer zone, semiconductor bulk recombination of
conduction band electrons with holes in the valence band is taken into account, since
the contribution of the macroscopic electric field responsible for charge separation is
negligible. The electron recombination is given by a first-order reaction based on the
electron density excess regarding its equilibrium value, n e − =

n e − ( x , t ) − n eq :

R e − = k e − n e − (10.4.30)

where k e − = 1 /τ e − is the first-order reaction rate constant, τ e − is the electron lifetime

and n eq is the dark equilibrium electron density, which can be calculated using Equa-
tion 10.4.5. The density-of-states of the conduction band in Equation 10.4.5 can be
determined from Equation 10.4.3.
The continuity equation for electrons in the bulk semiconductor is written as:

∂ 2 n e −
∂x 2

∂n e −
∂t

( n e − − n eq )

η inj α ( λ ) I 0 e − α ( λ )(1 − x )

= D e − + − (10.4.31)

τ e −

c) Holes in the semiconductor depletion layer (a < x < L)

The holes produced in the depletion layer region are perhaps the only ones capable of
reacting with the hydroxyl ions in the semiconductor surface adjacent to the electrolyte
solution. The continuity equation for the holes is similar to the one written for electrons
in the depletion layer:

∂ 2 n h +
∂x 2

∂n h +
∂t

∂ ( n h + ξ )
∂x

η inj α ( λ ) I 0 e − α ( λ )(1 − x ) = D h + − µ h + + (10.4.32)

When the semiconductor absorbs solar radiation with enough energy to excite an elec-
tron from the valence band to the conduction band, an electron-hole pair is generated.
Thus, holes and electrons are generated at the same rate:

G h + = G e − (10.4.33)

and then the generation terms are equal for the electrons and holes species.

d) Holes in the semiconductor bulk (0 < x < a)

Similar to what happens with the electrons in the semiconductor bulk, the macroscopic
electric field is negligible and the transport of holes to the surface of the semiconductor
takes place via diffusion. As mentioned the generation term is equal for both electrons