Photoelectrochemical cells for hydrogen production from solar energy - Solar Energy Sciences and Engineering Applications - page 325

Figure 10.4.2 Schematic representation of the band bending effect for (a) an n-type semiconductor
and (b) a p-type semiconductor.

of transferred charge to the surface of the semiconductor and the density of shallow
donors in the semiconductor material, N D .

d 2
d x 2

1
εε 0 ρ ( x )

d ξ
d x

=− =− (10.4.11)

ε and ε 0 are the dielectric constant and the permittivity of free space, respectively. ξ is
the electric field and the total charge density ρ is given by (Krol and Grätzel, 2012):

e [ N D −

ρ ( x ) = n ( x )] (10.4.12)

where x is the distance from the surface, N D is ionized donor density, given by the
doping of the semiconducting material - Equation 10.4.9. The concentration of free
electrons, n ( x ) varies with distance - Equation 10.4.5.
Since the Fermi level is expected to be constant within the space charge region,
the position of the conduction band energy E C ( x ) varies with the distance by the band
bending effect, ( x ), as follows:

N C exp n 0 exp e ( x )

kT

− −

E C ( x ) E F e ( x )

n ( x ) = − = (10.4.13)

kT