8.7 Polarons
245
Fig. 8.22 Hole from Nb
acceptor localized on Ti
site (small polaron) in rutile
TiO 2 . Adapted from [793]
Fig. 8.23 Simulated and
experimental TEM images
of β-Ga 2 O 3 in (101)
projection. The arrow
denotes the position of a
polaron. Adapted
from [792]
simulation
experiment
(101)
called a small polaron. An example of a hole polaron in rutile TiO 2 :Nb is depicted in Fig. 8.22. In
oxides often the hole from an acceptor is bound to oxygen, e.g. in BaTiO 3 :Na, as reviewed in [791]. In
Fig. 8.23 the lattice relaxation due to a hole bound to oxygen in the monoclinc unit cell of β-Ga 2 O 3 is
depicted directly using aberration corrected TEM. The bonding of the hole to the oxygen atom breaks
the bond to a Ga atom which moves by 0.1 nm from its equilibrium position [792].
A proper theoretical analysis of a small polaron requires ab initio techniques that account for the
motion of each atom in the few unit cells nearest the electron.
7
The transport of small polarons occurs generally via thermally-activated hopping (cmp. Sect. 8.8).
Under certain conditions the following mobilities for drift and Hall effect have been given [784]:
μ d ∝ T
−1 exp(−W/(2kT )) ,
(8.43)
μ H ∝ T
−1/2 exp(−W/(6kT )) ,
(8.44)
W being the polaron binding energy. Generally, materials with small polaron transport exhibit high
carrier density, often due to structural defects, and low mobility.
8.8 Hopping Transport
Disordered solids such as amorphous semiconductors, films containing quantum dots or material with
many defects are characterized by a large density of localized states which can form band tails or a
large density of states within the band gap. Hopping conduction is the tunneling between localized
states and has been treated with various models [795–797].
7 This paragraph has been taken from the concise tutorial by S.J.F. Byrnes [794].
245
Fig. 8.22 Hole from Nb
acceptor localized on Ti
site (small polaron) in rutile
TiO 2 . Adapted from [793]
Fig. 8.23 Simulated and
experimental TEM images
of β-Ga 2 O 3 in (101)
projection. The arrow
denotes the position of a
polaron. Adapted
from [792]
simulation
experiment
(101)
called a small polaron. An example of a hole polaron in rutile TiO 2 :Nb is depicted in Fig. 8.22. In
oxides often the hole from an acceptor is bound to oxygen, e.g. in BaTiO 3 :Na, as reviewed in [791]. In
Fig. 8.23 the lattice relaxation due to a hole bound to oxygen in the monoclinc unit cell of β-Ga 2 O 3 is
depicted directly using aberration corrected TEM. The bonding of the hole to the oxygen atom breaks
the bond to a Ga atom which moves by 0.1 nm from its equilibrium position [792].
A proper theoretical analysis of a small polaron requires ab initio techniques that account for the
motion of each atom in the few unit cells nearest the electron.
7
The transport of small polarons occurs generally via thermally-activated hopping (cmp. Sect. 8.8).
Under certain conditions the following mobilities for drift and Hall effect have been given [784]:
μ d ∝ T
−1 exp(−W/(2kT )) ,
(8.43)
μ H ∝ T
−1/2 exp(−W/(6kT )) ,
(8.44)
W being the polaron binding energy. Generally, materials with small polaron transport exhibit high
carrier density, often due to structural defects, and low mobility.
8.8 Hopping Transport
Disordered solids such as amorphous semiconductors, films containing quantum dots or material with
many defects are characterized by a large density of localized states which can form band tails or a
large density of states within the band gap. Hopping conduction is the tunneling between localized
states and has been treated with various models [795–797].
7 This paragraph has been taken from the concise tutorial by S.J.F. Byrnes [794].