3.7 Alloys
63
Table 3.8 Probability p n (3.23) and symmetry of an A atom being surrounded by n B atoms in a tetrahedrally configured
B x A 1−x random alloy
n
p n
Symmetry
0
x 4
T d
1
4 x 3 (1 − x)
C 3v
2
6 x 2 (1 − x) 2
C 2v
3
4 x (1 − x) 3
C 3v
4
(1 − x) 4
T d
found, corresponding to an effective repulsive pair interaction energy of 0.1 eV for the nearest neighbor
In–In pairs along the [110]-direction due to strain effects [248].
If the binary end components have different crystal structure, the alloy shows a transition (or
compositional transition range) from one structure to the other at a particular concentration. An example
is the alloy between wurtzite ZnO and rocksalt MgO. Mg x Zn 1−x O alloy thin films exhibits wurtzite
structure up to about x = 0.5 and rocksalt structure for x > 0.6 [249] (cmp. Fig. 3.43).
If the alloy contains four atom species it is called quaternary. A quaternary zincblende alloy can
have the mixing of three atom species on one sublattice, such as Al x Ga y In 1−x−y As or GaAs x P y Sb 1−x−y
or the mixing of two atom species on both of the two sublattices, such as In x Ga 1−x As y N 1−y .
The random placement of different atoms on the (sub)lattice in an alloy represents a perturbation of
the ideal lattice and causes additional scattering (alloy scattering). In the context of cluster formation,
the probability of an atom having a direct neighbor of the same kind on its sublattice is important.
Given a A x B 1−x C alloy, the probability p S to find a single A atom surrounded by B atoms is given by
(3.24a). The probability p D 1 to find a cluster of two neighbored A atoms surrounded by B atoms is
given by (3.24b).
p S = (1 − x)
12
(3.24a)
p D 1 = 12 x (1 − x)
18
.
(3.24b)
These formulas are valid for fcc and hcp lattices. For larger clusters [250, 251], probabilities in fcc
and hcp structures differ.
3.7.2 Phase Diagram
The mixture A x B 1−x with average composition x between two materials A and B can result in a single
phase (alloy), a two-phase system (phase separation) or a metastable system. The molar free enthalpy
G of the mixed system is approximated by
G = x (1 − x) + kT [x ln(x) + (1 − x) ln(1 − x)] .
(3.25)
The first term on the right-hand side of (3.25) is the (regular solution) enthalpy of mixing with the
interaction parameter , which can depend on x. The second term is the ideal configurational entropy
based on a random distribution of the atoms. The function is shown for various ratios of kT // in
Fig. 3.41a. In an equilibrium phase diagram (see Fig. 3.41b) the system is above the binodal curve in
one phase (miscible). On the binodal line T b (x) in the (x, T ) diagram the A- and B-rich disordered
phases have equal chemical potentials, i.e. ∂G/∂ x = 0. For independent of x the temperature T b
63
Table 3.8 Probability p n (3.23) and symmetry of an A atom being surrounded by n B atoms in a tetrahedrally configured
B x A 1−x random alloy
n
p n
Symmetry
0
x 4
T d
1
4 x 3 (1 − x)
C 3v
2
6 x 2 (1 − x) 2
C 2v
3
4 x (1 − x) 3
C 3v
4
(1 − x) 4
T d
found, corresponding to an effective repulsive pair interaction energy of 0.1 eV for the nearest neighbor
In–In pairs along the [110]-direction due to strain effects [248].
If the binary end components have different crystal structure, the alloy shows a transition (or
compositional transition range) from one structure to the other at a particular concentration. An example
is the alloy between wurtzite ZnO and rocksalt MgO. Mg x Zn 1−x O alloy thin films exhibits wurtzite
structure up to about x = 0.5 and rocksalt structure for x > 0.6 [249] (cmp. Fig. 3.43).
If the alloy contains four atom species it is called quaternary. A quaternary zincblende alloy can
have the mixing of three atom species on one sublattice, such as Al x Ga y In 1−x−y As or GaAs x P y Sb 1−x−y
or the mixing of two atom species on both of the two sublattices, such as In x Ga 1−x As y N 1−y .
The random placement of different atoms on the (sub)lattice in an alloy represents a perturbation of
the ideal lattice and causes additional scattering (alloy scattering). In the context of cluster formation,
the probability of an atom having a direct neighbor of the same kind on its sublattice is important.
Given a A x B 1−x C alloy, the probability p S to find a single A atom surrounded by B atoms is given by
(3.24a). The probability p D 1 to find a cluster of two neighbored A atoms surrounded by B atoms is
given by (3.24b).
p S = (1 − x)
12
(3.24a)
p D 1 = 12 x (1 − x)
18
.
(3.24b)
These formulas are valid for fcc and hcp lattices. For larger clusters [250, 251], probabilities in fcc
and hcp structures differ.
3.7.2 Phase Diagram
The mixture A x B 1−x with average composition x between two materials A and B can result in a single
phase (alloy), a two-phase system (phase separation) or a metastable system. The molar free enthalpy
G of the mixed system is approximated by
G = x (1 − x) + kT [x ln(x) + (1 − x) ln(1 − x)] .
(3.25)
The first term on the right-hand side of (3.25) is the (regular solution) enthalpy of mixing with the
interaction parameter , which can depend on x. The second term is the ideal configurational entropy
based on a random distribution of the atoms. The function is shown for various ratios of kT // in
Fig. 3.41a. In an equilibrium phase diagram (see Fig. 3.41b) the system is above the binodal curve in
one phase (miscible). On the binodal line T b (x) in the (x, T ) diagram the A- and B-rich disordered
phases have equal chemical potentials, i.e. ∂G/∂ x = 0. For independent of x the temperature T b