132
5 Mechanical Properties
energy of the layer without any dislocation and the layer with a particular dislocation density p 1 are
identical (E − E ∞ = 0) and additionally ∂ E/∂ p| p= p 1 = 0. However, between p → ∞ and p = p 0
there is an energy barrier. The critical thickness h c2 is reached when
∂ E/∂ p | p→∞ = 0 ,
(5.98)
i.e. the energy decreases monotonically for increasing dislocation density up to the global energy
minimum at a certain equilibrium dislocation spacing p 2 . Equation (5.98) leads to the following
implicit equation for the determination of h c2 :
h c2 =
b
−16 + 3b
2
+ 8 (−4 + ν) ln (2h c2 /q)
128 f π (1 + ν)
,
(5.99)
with the length of the Burgers vector b = a 0 /
√
2.
The theoretical dependence of h c2 for Ge x Si 1−x /Si(001) with varying composition is shown in
Fig. 5.43b together with experimental data. The critical thickness for a fairly high growth temperature
is much closer to the energetic equilibrium than that deposited at lower temperature. This shows
that there are kinetic limitations for the system to reach the mechanical equilibrium state. Also, the
experimental determination of the critical thickness is affected by finite resolution for large dislocation
spacing, leading generally to an overestimate of h c .
In zincblende materials, two types of dislocations are possible, α and β, with Ga- and As-based
cores, respectively. They have [ ¯
110] and [110] line directions for a compressively strained interface.
The α dislocation has the larger glide velocity. Therefore, strain relaxation can be anisotropic with
regard to the 110 directions for zincblende material, e.g. (In,Ga)As/GaAs [441, 442].
A more complicated relaxation situation is visualized in Fig. 4.18 for an Al 0.13 Ga 0.87 N/GaN heterostructure on (30 ¯
31) substrate. Below the critical thickness only threading dislocations stemming
from the substrate are visible. Above the critical thickness, misfit dislocations have developed along
three directions which related to the intersections of the basal c-plane (00.1) and two prismatic m-plane
{10.1} glide systems with the (30 ¯
31) interface plane [324, 443]. We like to mention that cathodolumi(a)
(b)
Fig. 5.43 a Theoretical calculation for the strain energy versus inverse dislocation density for various thicknesses of
Ge 0.1 Si 0.9 layers on Si (001). The ordinate is b/2 p, b/2 being the edge component of the Burgers vector and p being the
dislocation spacing. The abscissa is the strain energy E scaled with E ∞ (5.97). (b) Critical thickness for Ge x Si 1−x layers
on Si (001). The solid line is theory (h c2 ) according to (5.99). Data points are from [440] (squares, growth temperature
of 750 ◦ C) and from [430] (triangles for growth temperature of 550 ◦ C)
5 Mechanical Properties
energy of the layer without any dislocation and the layer with a particular dislocation density p 1 are
identical (E − E ∞ = 0) and additionally ∂ E/∂ p| p= p 1 = 0. However, between p → ∞ and p = p 0
there is an energy barrier. The critical thickness h c2 is reached when
∂ E/∂ p | p→∞ = 0 ,
(5.98)
i.e. the energy decreases monotonically for increasing dislocation density up to the global energy
minimum at a certain equilibrium dislocation spacing p 2 . Equation (5.98) leads to the following
implicit equation for the determination of h c2 :
h c2 =
b
−16 + 3b
2
+ 8 (−4 + ν) ln (2h c2 /q)
128 f π (1 + ν)
,
(5.99)
with the length of the Burgers vector b = a 0 /
√
2.
The theoretical dependence of h c2 for Ge x Si 1−x /Si(001) with varying composition is shown in
Fig. 5.43b together with experimental data. The critical thickness for a fairly high growth temperature
is much closer to the energetic equilibrium than that deposited at lower temperature. This shows
that there are kinetic limitations for the system to reach the mechanical equilibrium state. Also, the
experimental determination of the critical thickness is affected by finite resolution for large dislocation
spacing, leading generally to an overestimate of h c .
In zincblende materials, two types of dislocations are possible, α and β, with Ga- and As-based
cores, respectively. They have [ ¯
110] and [110] line directions for a compressively strained interface.
The α dislocation has the larger glide velocity. Therefore, strain relaxation can be anisotropic with
regard to the 110 directions for zincblende material, e.g. (In,Ga)As/GaAs [441, 442].
A more complicated relaxation situation is visualized in Fig. 4.18 for an Al 0.13 Ga 0.87 N/GaN heterostructure on (30 ¯
31) substrate. Below the critical thickness only threading dislocations stemming
from the substrate are visible. Above the critical thickness, misfit dislocations have developed along
three directions which related to the intersections of the basal c-plane (00.1) and two prismatic m-plane
{10.1} glide systems with the (30 ¯
31) interface plane [324, 443]. We like to mention that cathodolumi(a)
(b)
Fig. 5.43 a Theoretical calculation for the strain energy versus inverse dislocation density for various thicknesses of
Ge 0.1 Si 0.9 layers on Si (001). The ordinate is b/2 p, b/2 being the edge component of the Burgers vector and p being the
dislocation spacing. The abscissa is the strain energy E scaled with E ∞ (5.97). (b) Critical thickness for Ge x Si 1−x layers
on Si (001). The solid line is theory (h c2 ) according to (5.99). Data points are from [440] (squares, growth temperature
of 750 ◦ C) and from [430] (triangles for growth temperature of 550 ◦ C)