Geoscience Reference
In-Depth Information
where
μ
Fe
M
is the mobility of
Fe
•
M
and
μ
h
•
is the
mobility of electron hole. Constable and Roberts
(1997) discussed the mobility of these defects in
olivine.
Another well-documented conduction mech-
anism related to iron is ionic conduction by
the diffusion of Mg (or Fe). In addition to the
migration of electrons (or holes) just described,
ions themselves may carry the electric charge.
Because ions have electric charge, they migrate
when the electric field is applied. This is called
ionic conductivity. The relation between diffu-
sion and electrical conductivity is known as the
Nernst-Einstein relation (e.g., Mott & Gurney,
1940),
cation (
Mg
2
+
,
Fe
2
+
). Consequently ferric iron will
produce an impurity level in the band gap to
which electrons from the filled valence band can
be activated to change ferric iron to ferrous iron
(Figure 5.1). Because this energy level accepts an
electron from the valence band, it is sometimes
referred to as an ''acceptor level.'' When this tran-
sition occurs by thermal activation, then electric
current is carried by (i) hopping of an electron be-
tween ferric and ferrous iron or by (ii) the motion
of electron hole created in the valence band.
Let us now consider how the presence of such
an impurity changes electrical conductivity. In
case of iron-bearing minerals, defects such as fer-
ric iron at M-site (
Fe
•
M
(Kr oger-Vink notation)) are
formed by the chemical reaction of mineral with
the atmosphere, and the charge balance is main-
tained by atomic defects, e.g.,
q
2
f
·
D
·
n
·
σ
=
(5.7)
RT
2[
V
M
]
(
V
M
is the M-site vacancy). This leads to
[
Fe
•
M
]
[
Fe
•
M
]
=
where
n
is the concentration of ionic species
(number of species per unit volume),
q
is the
electric charge of that species,
D
is the diffu-
sion coefficient of the charged species, and
f
is
a nondimensional constant representing the ge-
ometrical factor (
f
f
1
/
6
O
2
(somewhat different relationships can
be obtained if one assumes different charge neu-
trality conditions (e.g., Karato, 1973, Schock
et al
.,
1989), but conductivity by ferric iron-related
species always increases with oxygen fugacity).
The electron holes are created from the ferric iron,
∝
∼
1) (mobility in this case is
Dq
RT
). Given the concentration and diffusion
coefficient of any charged species, one can cal-
culate the electrical conductivity. An obvious
mechanism is the electric current carried by the
diffusion of constituent ions such as
Mg
2
+
(
Fe
2
+
).
Indeed, several authors suggested that this is an
important conduction mechanism in olivine at
high temperatures (e.g., Karato, 1973; Constable,
2006). These analyses together with the latest
results on diffusion coefficients suggest that the
contribution from diffusion of Mg (Fe) is rela-
tively minor compared to the contributions from
electronic mechanisms of conduction.
In all cases for iron-related conduction, the
electrical conductivity is proportional to the con-
centration of ferric iron, viz.,
μ
=
Fe
•
M
⇔
h
•
.
Fe
M
+
(5.4)
where
Fe
M
denotes
Fe
2
+
at M-site and
h
•
is an
electron hole. Applying the law of mass action,
one finds that
[
Fe
•
M
]
[
h
•
]
=
K
4
[
Fe
M
]
·
(5.5)
where
K
4
is the reaction constant corresponding
to the Reaction (5.4) (
K
4
=
RT
). In this
case, the electrical conduction occurs either by
''hopping'' of electrons between ferric and ferrous
iron or by the conduction by electron hole, and
the electrical conductivity may be written as
K
40
exp
H
4
−
[
Fe
•
M
].
∝
σ
(5.8)
[
Fe
•
M
]
μ
Fe
M
σ
Fe
=|
e
|
for hopping
(5.6a)
This relation is different from the well-known
relationship for p- or
n-
type semiconductors (for
thesematerials
σ
∝
√
N
if impurity concentration
is high). The reason is that in iron-bearing miner-
als, the charge balance is not directly determined
and
[
h
•
]
μ
h
•
σ
Fe
=|
e
|
for free electron hole
(5.6b)
