248
8 Transport
The transport in delocalized states is similar to band transport. The conductivity (for electrons) is
given as
σ = −e
∞
E C
D e (E) μ e (E) f e (E) dE .
(8.51)
If the Fermi energy is close to the middle of the gap, pinned to deep states, the Fermi-Dirac distribution
can be replaced by the Boltzmann factor. Assuming a constant density of states and mobility for the
delocalized states,
σ = −e D e (E C ) μ e (E C ) kT exp
E C − E F
kT
.
(8.52)
Charge carriers from localized states in the tails can be thermally excited into delocalized states and
contribute to conductivity (thermally activated hopping). The mobility then contains an exponential
thermal activation term [203].
8.10 Ionic Transport
Ionic transport is the movement of ions upon application of a voltage. Here, we discuss only solid
electrolytes. The transport can include the motion of one or several of the constituents of the lattice
and the transport of other ions (e.g. hydrogen ions (protons), oxygen ions) through the crystal. Related
to this is the diffusive ionic movement of impurities or defects through the crystal (cmp. Sect. 4.2.3).
Ionic conduction of the lattice constituents under dc voltage will eventually destroy the crystal.
In typical semiconductors like silicon or gallium arsenide, the conductivity is entirely due to electronic conduction. A typical solid electrolyte is zirconia (ZrO 2 ) doped with yttria, so-called yttriastabilized zirconia (YSZ) that takes on a cubic fluorite lattice (see Sect. 3.4.8). It can conduct oxygen
ions via the mobility of oxygen vacancies for use in solid-oxide fuel cells (SOFC) [803]. The conductivity is about 0.01 S/cm at a temperature around 1000 K, almost entirely due to ionic transport.
Doping with calcium oxide results in an oxygen conductor that is used in oxygen sensors in automobiles (lambda sensor). The ionic conductivity can be significantly increased, compared to bulk material,
along interfaces [804, 805].
Other typical solid electrolytes are copper iodide (CuI) [568] and also AgI. In the high temperature
cubic phase (α-polymorph), the iodide ions form a fairly rigid cubic framework and the metal ions are
mobile; the copper diffusion pathways have been discussed [806, 807]. The temperature dependence
of conductivity of CuI is shown in Fig. 8.27.
8.11 Diffusion
A gradient of a particle concentration n leads to a particle current proportional to −∇n. This diffusion
law (Fick’s law) corresponds microscopically to a random walk. The gradients of the semiconductor
carrier densities ∇n or ∇ p thus lead to electron and hole currents, respectively:
j n = eD n ∇n
(8.53a)
j p = −eD p ∇ p .
(8.53b)
The coefficients D n and D p are called the electron and hole diffusion coefficient, respectively. Thus
the total electron and hole currents in the presence of an electric field E and diffusion are
8 Transport
The transport in delocalized states is similar to band transport. The conductivity (for electrons) is
given as
σ = −e
∞
E C
D e (E) μ e (E) f e (E) dE .
(8.51)
If the Fermi energy is close to the middle of the gap, pinned to deep states, the Fermi-Dirac distribution
can be replaced by the Boltzmann factor. Assuming a constant density of states and mobility for the
delocalized states,
σ = −e D e (E C ) μ e (E C ) kT exp
E C − E F
kT
.
(8.52)
Charge carriers from localized states in the tails can be thermally excited into delocalized states and
contribute to conductivity (thermally activated hopping). The mobility then contains an exponential
thermal activation term [203].
8.10 Ionic Transport
Ionic transport is the movement of ions upon application of a voltage. Here, we discuss only solid
electrolytes. The transport can include the motion of one or several of the constituents of the lattice
and the transport of other ions (e.g. hydrogen ions (protons), oxygen ions) through the crystal. Related
to this is the diffusive ionic movement of impurities or defects through the crystal (cmp. Sect. 4.2.3).
Ionic conduction of the lattice constituents under dc voltage will eventually destroy the crystal.
In typical semiconductors like silicon or gallium arsenide, the conductivity is entirely due to electronic conduction. A typical solid electrolyte is zirconia (ZrO 2 ) doped with yttria, so-called yttriastabilized zirconia (YSZ) that takes on a cubic fluorite lattice (see Sect. 3.4.8). It can conduct oxygen
ions via the mobility of oxygen vacancies for use in solid-oxide fuel cells (SOFC) [803]. The conductivity is about 0.01 S/cm at a temperature around 1000 K, almost entirely due to ionic transport.
Doping with calcium oxide results in an oxygen conductor that is used in oxygen sensors in automobiles (lambda sensor). The ionic conductivity can be significantly increased, compared to bulk material,
along interfaces [804, 805].
Other typical solid electrolytes are copper iodide (CuI) [568] and also AgI. In the high temperature
cubic phase (α-polymorph), the iodide ions form a fairly rigid cubic framework and the metal ions are
mobile; the copper diffusion pathways have been discussed [806, 807]. The temperature dependence
of conductivity of CuI is shown in Fig. 8.27.
8.11 Diffusion
A gradient of a particle concentration n leads to a particle current proportional to −∇n. This diffusion
law (Fick’s law) corresponds microscopically to a random walk. The gradients of the semiconductor
carrier densities ∇n or ∇ p thus lead to electron and hole currents, respectively:
j n = eD n ∇n
(8.53a)
j p = −eD p ∇ p .
(8.53b)
The coefficients D n and D p are called the electron and hole diffusion coefficient, respectively. Thus
the total electron and hole currents in the presence of an electric field E and diffusion are