64
3 Crystals
is given by (3.26a). A critical point is at the maximum temperature T mg and concentration x mg of the
miscibility gap. For independent of x it is given by T mg = /2 and x mg = 1/2. In the region
below the spinodal boundary, the system is immiscible and phases immediately segregate (by spinodal
decomposition). On the spinodal line T sp (x) the condition ∂
2 G/∂ x
2
= 0 is fulfilled. For independent
of x the temperature T sp is given by (3.26b). The region between the binodal and spinodal curves is
the metastable region, i.e. the system is stable to small fluctuations of concentration or temperature but
not for larger ones.
kT b (x) =
2x − 1
ln(x) − ln(1 − x)
(3.26a)
kT sp (x) = 2 x (1 − x) .
(3.26b)
In Fig. 3.42 calculated diagrams for GaAs-AlAs and GaAs-GaP [252] are shown. The arrows denote
the critical point. These parameters and the interaction parameters for a number of ternary alloys are
given in Table 3.9. For example, for Al x Ga 1−x As complete miscibility is possible for typical growth
temperatures (> 700 K), but for In x Ga 1−x N the In solubility at a typical growth temperature of 1100 K
is only 6% [253].
The alloy system (Al,Ga,In)(As,P,Sb) always crystallizes in the zincblende structure and (Al,Ga,In)N
always in the wurtzite structure. If the binary end components of a ternary alloy have a different crystal
structure things become more complicated and the crystal phase has to be determined experimentally
(and modelled) for each composition. As an example the energy of the wurtzite, hexagonal and rocksalt
(a)
(b)
Fig. 3.41 a Free enthalpy of mixed binary system (3.25) in units of for =const. and various values of kT //
as labeled. b Schematic phase diagram for binary mixture. The temperature is given in units of The solid (dashed)
line denotes the binodal (spinodal) line
100
90
80
70
60
50
40
AlAs
Al Ga As
x
x
-
1
(a)
350
300
250
200
150
100
GaP As 1-x
x
GaAs
GaP
(b)
spinodal
x=0.5
binodal
GaAs
x=0.5
Fig. 3.42 Calculated phase diagrams for a Al x Ga 1−x As and b GaP x As 1−x . The binodal (spinodal) curve is shown as
solid (dashed) line. Adapted from [252]
3 Crystals
is given by (3.26a). A critical point is at the maximum temperature T mg and concentration x mg of the
miscibility gap. For independent of x it is given by T mg = /2 and x mg = 1/2. In the region
below the spinodal boundary, the system is immiscible and phases immediately segregate (by spinodal
decomposition). On the spinodal line T sp (x) the condition ∂
2 G/∂ x
2
= 0 is fulfilled. For independent
of x the temperature T sp is given by (3.26b). The region between the binodal and spinodal curves is
the metastable region, i.e. the system is stable to small fluctuations of concentration or temperature but
not for larger ones.
kT b (x) =
2x − 1
ln(x) − ln(1 − x)
(3.26a)
kT sp (x) = 2 x (1 − x) .
(3.26b)
In Fig. 3.42 calculated diagrams for GaAs-AlAs and GaAs-GaP [252] are shown. The arrows denote
the critical point. These parameters and the interaction parameters for a number of ternary alloys are
given in Table 3.9. For example, for Al x Ga 1−x As complete miscibility is possible for typical growth
temperatures (> 700 K), but for In x Ga 1−x N the In solubility at a typical growth temperature of 1100 K
is only 6% [253].
The alloy system (Al,Ga,In)(As,P,Sb) always crystallizes in the zincblende structure and (Al,Ga,In)N
always in the wurtzite structure. If the binary end components of a ternary alloy have a different crystal
structure things become more complicated and the crystal phase has to be determined experimentally
(and modelled) for each composition. As an example the energy of the wurtzite, hexagonal and rocksalt
(a)
(b)
Fig. 3.41 a Free enthalpy of mixed binary system (3.25) in units of for =const. and various values of kT //
as labeled. b Schematic phase diagram for binary mixture. The temperature is given in units of The solid (dashed)
line denotes the binodal (spinodal) line
100
90
80
70
60
50
40
AlAs
Al Ga As
x
x
-
1
(a)
350
300
250
200
150
100
GaP As 1-x
x
GaAs
GaP
(b)
spinodal
x=0.5
binodal
GaAs
x=0.5
Fig. 3.42 Calculated phase diagrams for a Al x Ga 1−x As and b GaP x As 1−x . The binodal (spinodal) curve is shown as
solid (dashed) line. Adapted from [252]