174
6 Band Structure
E and E + δ E is D(E)δ E. In the vicinity of the extrema of the band structure many states are at the
same energy such that the density of states is high.
The dispersion relation of a band will be given as E = E(k). If several bands overlap, the densities
of state of all bands need to be summed up. The density of states at the energy ˜
E for the given band is
D( ˜
E) dE = 2
d
3 k
(2π/L) 3 δ( ˜
E − E(k)) ,
(6.63)
where, according to (5.5), (2π/L)
3 is the k-space volume for one state. The factor 2 is for spin
degeneracy. The integral runs over the entire k-space and selects only those states that are at ˜
E.
The volume integral can be converted to a surface integral over the isoenergy surface S( ˜
E) with
E(k) = ˜
E. The volume element d
3 k is written as d
2 S dk ⊥ . The vector dk ⊥ is perpendicular to S( ˜
E)
and proportional to ∇ k E(k), i.e. dE = |∇ k E(k)| dk ⊥ .
D( ˜
E) = 2
S( ˜
E)
d
2 S
(2π/L) 3
1
|∇ k E(k)|
.
(6.64)
In this equation, the dispersion relation is explicitly contained. At band extrema the gradient diverges,
however, in three dimensions the singularities are integrable and the density of states takes a finite
value. The corresponding peak is named a van-Hove singularity. The concept of the density of states
is valid for all possible dispersion relations, e.g. for electrons, phonons or photons.
The density of states for the silicon band structure (see Fig. 6.9a) is shown in Fig. 6.52.
6.13.2 Amorphous Semiconductors
If disorder is introduced, the density of states is modified as shown in Fig. 6.53 for amorphous germanium using a calculation with complex eigenenergies. The defects, as compared to the perfect lattice,
introduced states in the band gap and generally wash out the sharp features from the crystalline DOS.
Several models exist for the defect level distributions within the band gap. The first model was
the Mott model which has band tails at the valence and conduction band edges [547]. In the CohenFritzsche-Ovshinsky (CFO) model [548], the band tails are more severe and overlap; the Fermi energy
lies at the minimum of the density of states. In the Davis–Mott model [549] deep states were added in
the gap and eventually the Marshall-Owen model [550] assumes band tails and donor- and acceptorlike deep states. The four models are schematically shown in Fig. 6.54. These model densities of states
allow also the interpretation of carrier transport in amorphous semiconductors, taking into account
localized and delocalized states (see Sect. 8.9).
The density of states for an amorphous semiconductor is best calculated from atomistic models,
possibly averaging over many configurations. The typical features, compared to the clear band gap of
a similar ordered material, are band tails due to disorder (cmp. Sect. 5.2.9) and deep levels within the
gap due to specific atomic arrangements not present in ordered bulk. The most investigated system is
amorphous silicon; in Fig. 6.55 a numerical calculation of the density of states is shown together with
charge distribution of four states at selected energies [551]. The further the states are in the band tail,
the stronger their localization is. The two most right states shown in Fig. 6.55 are not conducting.
As another example, simulations of ZnSnO 3 are shown in Fig. 6.56. The band tail between 0 and
0.5 eV is due to disorder of oxygen 2p orbitals [552]. At 0.9 eV a level due to under-coordinated oxygen
appears. Deep levels are due to metal-metal bonds. Band tails due to chemically disordered oxygen
have been experimentally observed for amorphous GIZO [553].
6 Band Structure
E and E + δ E is D(E)δ E. In the vicinity of the extrema of the band structure many states are at the
same energy such that the density of states is high.
The dispersion relation of a band will be given as E = E(k). If several bands overlap, the densities
of state of all bands need to be summed up. The density of states at the energy ˜
E for the given band is
D( ˜
E) dE = 2
d
3 k
(2π/L) 3 δ( ˜
E − E(k)) ,
(6.63)
where, according to (5.5), (2π/L)
3 is the k-space volume for one state. The factor 2 is for spin
degeneracy. The integral runs over the entire k-space and selects only those states that are at ˜
E.
The volume integral can be converted to a surface integral over the isoenergy surface S( ˜
E) with
E(k) = ˜
E. The volume element d
3 k is written as d
2 S dk ⊥ . The vector dk ⊥ is perpendicular to S( ˜
E)
and proportional to ∇ k E(k), i.e. dE = |∇ k E(k)| dk ⊥ .
D( ˜
E) = 2
S( ˜
E)
d
2 S
(2π/L) 3
1
|∇ k E(k)|
.
(6.64)
In this equation, the dispersion relation is explicitly contained. At band extrema the gradient diverges,
however, in three dimensions the singularities are integrable and the density of states takes a finite
value. The corresponding peak is named a van-Hove singularity. The concept of the density of states
is valid for all possible dispersion relations, e.g. for electrons, phonons or photons.
The density of states for the silicon band structure (see Fig. 6.9a) is shown in Fig. 6.52.
6.13.2 Amorphous Semiconductors
If disorder is introduced, the density of states is modified as shown in Fig. 6.53 for amorphous germanium using a calculation with complex eigenenergies. The defects, as compared to the perfect lattice,
introduced states in the band gap and generally wash out the sharp features from the crystalline DOS.
Several models exist for the defect level distributions within the band gap. The first model was
the Mott model which has band tails at the valence and conduction band edges [547]. In the CohenFritzsche-Ovshinsky (CFO) model [548], the band tails are more severe and overlap; the Fermi energy
lies at the minimum of the density of states. In the Davis–Mott model [549] deep states were added in
the gap and eventually the Marshall-Owen model [550] assumes band tails and donor- and acceptorlike deep states. The four models are schematically shown in Fig. 6.54. These model densities of states
allow also the interpretation of carrier transport in amorphous semiconductors, taking into account
localized and delocalized states (see Sect. 8.9).
The density of states for an amorphous semiconductor is best calculated from atomistic models,
possibly averaging over many configurations. The typical features, compared to the clear band gap of
a similar ordered material, are band tails due to disorder (cmp. Sect. 5.2.9) and deep levels within the
gap due to specific atomic arrangements not present in ordered bulk. The most investigated system is
amorphous silicon; in Fig. 6.55 a numerical calculation of the density of states is shown together with
charge distribution of four states at selected energies [551]. The further the states are in the band tail,
the stronger their localization is. The two most right states shown in Fig. 6.55 are not conducting.
As another example, simulations of ZnSnO 3 are shown in Fig. 6.56. The band tail between 0 and
0.5 eV is due to disorder of oxygen 2p orbitals [552]. At 0.9 eV a level due to under-coordinated oxygen
appears. Deep levels are due to metal-metal bonds. Band tails due to chemically disordered oxygen
have been experimentally observed for amorphous GIZO [553].