152
6 Band Structure
(a)
(b)
(c)
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
2md E/h
2
2
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
P
E g
Fig. 6.22 Kronig-Penney model (along 111, b/a = 3) for a a IV–IV semiconductor and b for a III–V (or II–VI)
semiconductor, c resulting band structure (P 0 = −3). d denotes the lattice constant (d = b + a). Adapted from [481]
Fig. 6.23 Optical image of
two inch wafers of GaAs
(left), GaP (center) and
ZnO (right). A GaN wafer
would look like the ZnO
wafer
This behavior can be understood within the framework of a modified Kronig-Penney model [481]
(Appendix F). Double potential wells (b/a = 3) are chosen to mimic the diatomic planes along the
111 direction in the zincblende structure (Fig. 6.22a). The first investigation of such diatomic onedimensional bandstructure was reported in [482]. Symmetric wells (depth P 0 ) are chosen to model
covalent semiconductors and asymmetric wells with depths P 0 ± to model partially ionic semiconductors. Results are shown in Fig. 6.22a for P 0 = −3. With increasing asymmetry, i.e. increasing
ionicity, the band gap increases, mostly due to a downward shift of the valence band. The case of III–V
(II–VI) semiconductors is reached for ≈ 2 (4). The calculation of effective masses in [481] is
incorrect and has been rectified in [483]; the effective mass increases monotonically with
In Fig. 6.23, the visual impression of 2" wafers of GaAs, GaP and GaN on white paper is shown.
GaAs (and GaSb) is opaque since the band gap is below the visible spectral range. GaP has a band
gap in the green and appears red, GaN has a band gap in the ultra-violet and thus appears transparent.
As can be seen from Table 6.3, the anion sequence Sb, As, P, and N leads to smaller lattice constant
and higher ionicity. A notable deviation from this rule is InN whose band gap (0.7 eV) is much smaller
than that of InP [484].
6.5 Alloy Semiconductors
In alloy semiconductors [166], the size of the band gap and the character of the band gap will depend on
the composition. The dependence of the band gap on the ternary composition is mostly nonlinear and
can usually be expressed with a bowing parameter b that is mostly positive. For a compound A x B 1−x C
6 Band Structure
(a)
(b)
(c)
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
2md E/h
2
2
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
P
E g
Fig. 6.22 Kronig-Penney model (along 111, b/a = 3) for a a IV–IV semiconductor and b for a III–V (or II–VI)
semiconductor, c resulting band structure (P 0 = −3). d denotes the lattice constant (d = b + a). Adapted from [481]
Fig. 6.23 Optical image of
two inch wafers of GaAs
(left), GaP (center) and
ZnO (right). A GaN wafer
would look like the ZnO
wafer
This behavior can be understood within the framework of a modified Kronig-Penney model [481]
(Appendix F). Double potential wells (b/a = 3) are chosen to mimic the diatomic planes along the
111 direction in the zincblende structure (Fig. 6.22a). The first investigation of such diatomic onedimensional bandstructure was reported in [482]. Symmetric wells (depth P 0 ) are chosen to model
covalent semiconductors and asymmetric wells with depths P 0 ± to model partially ionic semiconductors. Results are shown in Fig. 6.22a for P 0 = −3. With increasing asymmetry, i.e. increasing
ionicity, the band gap increases, mostly due to a downward shift of the valence band. The case of III–V
(II–VI) semiconductors is reached for ≈ 2 (4). The calculation of effective masses in [481] is
incorrect and has been rectified in [483]; the effective mass increases monotonically with
In Fig. 6.23, the visual impression of 2" wafers of GaAs, GaP and GaN on white paper is shown.
GaAs (and GaSb) is opaque since the band gap is below the visible spectral range. GaP has a band
gap in the green and appears red, GaN has a band gap in the ultra-violet and thus appears transparent.
As can be seen from Table 6.3, the anion sequence Sb, As, P, and N leads to smaller lattice constant
and higher ionicity. A notable deviation from this rule is InN whose band gap (0.7 eV) is much smaller
than that of InP [484].
6.5 Alloy Semiconductors
In alloy semiconductors [166], the size of the band gap and the character of the band gap will depend on
the composition. The dependence of the band gap on the ternary composition is mostly nonlinear and
can usually be expressed with a bowing parameter b that is mostly positive. For a compound A x B 1−x C