6.5 Alloy Semiconductors
153
Table 6.3 Comparison of band gap, lattice constant and ionicity of gallium–group V semiconductors for various anions.
Lattice constant for GaN has been recalculated for a cubic cell
Anion
E g (eV)
a 0 (nm)
f i
N
3.4
0.45
0.50
P
2.26
0.545
0.33
As
1.42
0.565
0.31
Sb
0.72
0.61
0.26
the band gap is written as
E g (A x B 1−x C) = E g (BC) + x
E g (AC) − E g (BC)
− b x (1 − x) .
(6.30)
Even on the virtual crystal approximation (VCA) level (Sect. 3.7.3) a nonzero bowing parameter b
is predicted. However, a more thorough analysis shows that the bowing cannot be treated adequately
within VCA and is due to the combined effects of volume deformation of the band structure with
the alloy lattice constant, charge exchange in the alloy with respect to the binary end components, a
structural contribution due to the relaxation of the cation–anion bond lengths in the alloy and a small
contribution due to disorder [485]. The discussion of Sect. 6.12.3 is related.
The Si x Ge 1−x alloy has diamond structure for all concentrations and the position of the conductionband minimum in k-space switches from L to X at about x = 0.15 (Fig. 6.24a). However, for all
concentrations the band structure is indirect. The In x Ga 1−x As alloy has zincblende structure for all
compositions. The band gap is direct and decreases with a bowing parameter of b = 0.6 eV [486]
(Fig. 6.24b). This means that for x = 0.5 the band gap is 0.15 eV smaller than expected from a linear
interpolation between GaAs and InAs, as reported by various authors [487].
If one binary end component has a direct band structure and the other is indirect, a transition occurs
from direct to indirect at a certain composition. An example is Al x Ga 1−x As where GaAs is direct and
AlAs is indirect. For all concentrations the crystal has zincblende structure. In Fig. 6.24c, the , L and
X conduction-band minima for ternary Al x Ga 1−x As are shown. Up to an aluminum concentration of
x = 0.4 the band structure is direct. Above this value the band structure is indirect with the conductionband minimum being at the X-point. The particularity of Al x Ga 1−x As is that its lattice constant is almost
independent of x. For other alloys lattice match to GaAs or InP substrates is only obtained for specific
compositions, as shown in Fig. 6.25. The band gap bowing in the group-III–nitride system has been
discussed in [488].
If the two binary end components have different crystal structure, a phase transition occurs at a
certain composition (range). An example is Mg x Zn 1−x O, where ZnO has wurtzite structure and MgO
has rocksalt structure. The band gap is shown in Fig. 6.24d. In this case, each phase has its own bowing
parameter.
All alloys of Fig. 6.24b–d have mixed cations. The band gap also varies upon anion substitution in
a similar way as shown in Fig. 6.26 for ternary alloys with the cation Zn and the chalcogenides S, Se,
Te and O.
6.6 Amorphous Semiconductors
Since the crystal lattice in an amorphous semiconductor is not periodic, the concept of k-space and
the related concepts such as band structure E(k) break down at least partially. The density of states,
however, remains a meaningful and useful quantity (Sect. 6.13.2).
153
Table 6.3 Comparison of band gap, lattice constant and ionicity of gallium–group V semiconductors for various anions.
Lattice constant for GaN has been recalculated for a cubic cell
Anion
E g (eV)
a 0 (nm)
f i
N
3.4
0.45
0.50
P
2.26
0.545
0.33
As
1.42
0.565
0.31
Sb
0.72
0.61
0.26
the band gap is written as
E g (A x B 1−x C) = E g (BC) + x
E g (AC) − E g (BC)
− b x (1 − x) .
(6.30)
Even on the virtual crystal approximation (VCA) level (Sect. 3.7.3) a nonzero bowing parameter b
is predicted. However, a more thorough analysis shows that the bowing cannot be treated adequately
within VCA and is due to the combined effects of volume deformation of the band structure with
the alloy lattice constant, charge exchange in the alloy with respect to the binary end components, a
structural contribution due to the relaxation of the cation–anion bond lengths in the alloy and a small
contribution due to disorder [485]. The discussion of Sect. 6.12.3 is related.
The Si x Ge 1−x alloy has diamond structure for all concentrations and the position of the conductionband minimum in k-space switches from L to X at about x = 0.15 (Fig. 6.24a). However, for all
concentrations the band structure is indirect. The In x Ga 1−x As alloy has zincblende structure for all
compositions. The band gap is direct and decreases with a bowing parameter of b = 0.6 eV [486]
(Fig. 6.24b). This means that for x = 0.5 the band gap is 0.15 eV smaller than expected from a linear
interpolation between GaAs and InAs, as reported by various authors [487].
If one binary end component has a direct band structure and the other is indirect, a transition occurs
from direct to indirect at a certain composition. An example is Al x Ga 1−x As where GaAs is direct and
AlAs is indirect. For all concentrations the crystal has zincblende structure. In Fig. 6.24c, the , L and
X conduction-band minima for ternary Al x Ga 1−x As are shown. Up to an aluminum concentration of
x = 0.4 the band structure is direct. Above this value the band structure is indirect with the conductionband minimum being at the X-point. The particularity of Al x Ga 1−x As is that its lattice constant is almost
independent of x. For other alloys lattice match to GaAs or InP substrates is only obtained for specific
compositions, as shown in Fig. 6.25. The band gap bowing in the group-III–nitride system has been
discussed in [488].
If the two binary end components have different crystal structure, a phase transition occurs at a
certain composition (range). An example is Mg x Zn 1−x O, where ZnO has wurtzite structure and MgO
has rocksalt structure. The band gap is shown in Fig. 6.24d. In this case, each phase has its own bowing
parameter.
All alloys of Fig. 6.24b–d have mixed cations. The band gap also varies upon anion substitution in
a similar way as shown in Fig. 6.26 for ternary alloys with the cation Zn and the chalcogenides S, Se,
Te and O.
6.6 Amorphous Semiconductors
Since the crystal lattice in an amorphous semiconductor is not periodic, the concept of k-space and
the related concepts such as band structure E(k) break down at least partially. The density of states,
however, remains a meaningful and useful quantity (Sect. 6.13.2).