Photoelectrochemical cells for hydrogen production from solar energy - Solar Energy Sciences and Engineering Applications - page 331

and holes G h + = G e − . On the other hand, if a hole in the valence band reacts with an

electron from the conduction band, a photon is produced - exactly the reverse process
of absorption, i.e. radiative recombination takes place. The recombination of a hole
with a conduction band electron occurs at the same rate of the spontaneous transition
of an electron from the conduction band to an unoccupied state in the valence band:

( n h + − n eq ) ( n e − − n eq )

R h + = R e − → = (10.4.34)

τ h + τ e −

The continuity equation for holes can then be written:

∂ 2 n h +
∂x 2

n e −
τ e −

∂n h +
∂t

η inj I 0 αe − αx

D h + + − = (10.4.35)

e) Hydroxyl ions in the electrolyte solution (L < x < b)

Hydroxyl ions are formed in the counter-electrode by reduction of water to hydro-
gen gas - Equation 10.2.4. Then OH − ions diffuse to the semiconductor surface
where they react with holes to produce oxygen gas - Equation 10.2.5. Only elec-
trolyte exists in the region L < x < b , so no ions are generated or lost by recombination
with holes. Accordingly, the respective continuity equation for OH − ions, applied to

this electrolyte region, can be written as:

∂ 2 n OH −
∂ x 2

∂ n OH −
∂ t

= D OH − (10.4.36)

The previous equations describe the charge transport within a PEC cell under oper-
ating light conditions. With the appropriate boundary and initial conditions and the
adequate characteristic parameters of the system, those equations can be used to model
a PEC cell (Reiss, 1978).

10.4.2.1 Energy balance

In water photoelectrolysis a net chemical reaction occurs by means of water decompo-
sition into hydrogen and oxygen by light induced processes. For photoelectrochemical
cells the energy balance behind these reactions was written by Nozik in 1976 as follow
(Andrade et al., 2011):

G o
n e F

E Bias + E g − ( x ) − E F = + η a + η c + η ohmic (10.4.37)

the term E Bias is added when an external bias applied to the PEC cell is required. E g
is the bandgap energy and E F , defined as the difference between the Fermi level and
majority carrier band edge of the semiconductor. The free energy change per electron
for the overall reaction taken as 1.23 eV for water decomposition is here represented
by G o /n e F . η a and η c are the overpotentials at the anode and cathode, respectively.