248
5 – Applications
with
[
] [ ]
K
P
V
h
V
O
Cu
•
Cu
2
1 4
=
#
l
l
2 Formation of neutral copper vacancies
Neutral copper vacancies result from an equilibrium between ionized
copper vacancies and holes by the reaction
V
#
Cu m V ′
Cu + h
•
with
[
]
[
] [ ]
K
V
V
h
V
Cu
Cu
•
Cu
=
#
#
l
from which the equilibrium for the formation of neutral copper vacancies is
4
1
O 2(g) m 2
1
O
#
O + V
#
Cu
with
K
P
V
V
O
Cu
Cu
2
1 4
=
#
#
6
@
and
K
K
K
V
V
V
Cu
Cu
Cu
=
# #
l
2 Formation of interstitial singly ionized oxygen
2
1
O 2(g) m O ′
i + h
•
with
[ ] [ ]
K
P
O
h
O
O
i
•
i
2
1 2
=
#
l
l
2. The expressions for the defect concentrations are deduced from the preceding
relationships:
V
h
K
P
O
h
K
P
and
Cu
V
O
i
O
O
Cu
i
2
2
1 4
1 2
=
=
:
:
#
#
l
l
l
l
6
6
6
6
@
@
@
@
Enforcing the condition of electroneutrality gives
[h
•
] = [V ′
Cu ] + [O ′
i ]
from which
h
K
P
K
P
V
O
O
O
Cu
i
2
2
1 4
1 2
=
+
:
#
#
l
l
`
`
j
j
6 @
Moreover, the p-type electronic conductivity is expressed as
σ h = u h F [h
•
]
where u h is electrical mobility of holes.
By using
k i = (u h F)
2
K i