Photoelectrochemical cells for hydrogen production from solar energy - Solar Energy Sciences and Engineering Applications - page 327

Equations 10.4.14 and 10.4.18 and integrating between = 0 and = S =− SC

(see Figure 10.4.2), Equation 10.4.19 is obtained (Krol and Grätzel, 2012):

eN D 1 exp e

kT

2 εε 0 A 2 − SC

0

Q SC = − −

2 εε 0 eN D A 2 SC

(10.4.19)

exp

e SC
kT

kT
e

kT
e

= + − −

Assuming that for a PEC cell under normal operating conditions the potential drop
across the space charge is at least 0.1 V, Equation 10.4.19 can be simplified to:

2 εε 0 eN D A 2 SC −

kT
e

Q SC = (10.4.20)

The accumulated total charge, Q SC , is given by the product of the amount of charges
N D and the corresponding volume ( A W ). This allows the relation between the

accumulated total charge, Q SC , to be obtained and the depletion layer thickness, W :

·

Q SC = eN D AW (10.4.21)

and so the space charge layer width can be expressed as:

2 εε 0
eN D

SC

kT
e

W = − (10.4.22)

Even though these equations have been derived for an n-type semiconductor, an anal-
ogous approach can be followed for a p-type material, where N D is replaced by the
shallow acceptor density, N A . Typically, values of space charge layer width range from
5 to 500 nm. It is notable that the accumulated charge in the depletion layer is com-
pensated by an opposite charge electrolyte layer accumulated at the surface of the
semiconductor (i.e. trapped electrons) (Krol and Grätzel, 2012).
In fact, the dopant density and the depletion layer width can be experimentally
determined by capacity measurements. The differential capacity of the space charge
layer, C SC , under certain conditions can be obtained from the total impedance of the
system by differentiating Equation 10.4.20 with respect to SC :

d Q SC
d SC

− 2

εε 0 eN D A 2 SC −

1
C SC =

2 kT

e

= (10.4.23)

This is the so-called Mott-Schottky equation, only valid for a space charge region
depleted of the majority charge carriers with respect to the bulk density. The thickness
of the space charge layer decreases as the doping concentration increases (Krol and
Grätzel, 2012). Plotting 1 /C SC

as a function of the applied potential A the donor