5.3 Elasticity
125
Fig. 5.34 Strain components in an InAs pyramid (quantum dot with {101} faces), embedded in GaAs. The cross section
is through the center of the pyramid. The lattice mismatch between InAs and GaAs amounts to ≈ −7%. Reprinted with
permission from [417], c
1995 APS
Dividing the displacement by the sphere’s volume, we obtain the displacement per unit volume of the
inclusion. From the displacement we can derive the stress σ
0
i j per unit volume.
σ
0
ii =
1
4π
Y 0
1 − ν
2x
2
i − x j − x k
ρ 5
(5.75)
σ
0
i j =
3
2
1
4π
Y 0
1 − ν
x i x j
ρ 5 ,
(5.76)
where i, j and k are pairwise unequal indices. Due to the linear superposition of stresses, the stress
distribution σ
V
i j for the arbitrary inclusion of volume V can be obtained by integrating over V
σ
V
i j =
V
σ
0
i j (r − r 0 ) d
3 r .
(5.77)
The strains can be calculated from the stresses.
When 0 is constant within V , the volume integral can be readily transformed into an integral over
the surface ∂ V of V using Gauss’ theorem. With the ‘vector potentials’ A i j we fulfill divA i j = σ i j .
A ii = −
1
4π
Y 0
1 − ν
x i e i
ρ 3
(5.78)
A i j = −
1
2
1
4π
Y 0
1 − ν
x i e j + x j e i
ρ 3
.
(5.79)
Equation (5.79) is valid for the case i = j. e i is the unit vector in the i-th direction. However, special
care must be taken at the singularity r = r 0 if r 0 lies within V because the stress within the ‘δ-inclusion’
is not singular (in contrast to the electrostatic analog of a δ-charge). Thus, we find
σ
V
i j (r 0 ) =
∂ V
A i j dS + δ i j
Y 0
1 − ν
V
δ(r − r 0 ) d
3 r .
(5.80)
As an example, we show in Fig. 5.34 the numerically calculated strain components [417] (taking
into account the different elastic properties of the dot and matrix materials) in the cross section of
125
Fig. 5.34 Strain components in an InAs pyramid (quantum dot with {101} faces), embedded in GaAs. The cross section
is through the center of the pyramid. The lattice mismatch between InAs and GaAs amounts to ≈ −7%. Reprinted with
permission from [417], c
1995 APS
Dividing the displacement by the sphere’s volume, we obtain the displacement per unit volume of the
inclusion. From the displacement we can derive the stress σ
0
i j per unit volume.
σ
0
ii =
1
4π
Y 0
1 − ν
2x
2
i − x j − x k
ρ 5
(5.75)
σ
0
i j =
3
2
1
4π
Y 0
1 − ν
x i x j
ρ 5 ,
(5.76)
where i, j and k are pairwise unequal indices. Due to the linear superposition of stresses, the stress
distribution σ
V
i j for the arbitrary inclusion of volume V can be obtained by integrating over V
σ
V
i j =
V
σ
0
i j (r − r 0 ) d
3 r .
(5.77)
The strains can be calculated from the stresses.
When 0 is constant within V , the volume integral can be readily transformed into an integral over
the surface ∂ V of V using Gauss’ theorem. With the ‘vector potentials’ A i j we fulfill divA i j = σ i j .
A ii = −
1
4π
Y 0
1 − ν
x i e i
ρ 3
(5.78)
A i j = −
1
2
1
4π
Y 0
1 − ν
x i e j + x j e i
ρ 3
.
(5.79)
Equation (5.79) is valid for the case i = j. e i is the unit vector in the i-th direction. However, special
care must be taken at the singularity r = r 0 if r 0 lies within V because the stress within the ‘δ-inclusion’
is not singular (in contrast to the electrostatic analog of a δ-charge). Thus, we find
σ
V
i j (r 0 ) =
∂ V
A i j dS + δ i j
Y 0
1 − ν
V
δ(r − r 0 ) d
3 r .
(5.80)
As an example, we show in Fig. 5.34 the numerically calculated strain components [417] (taking
into account the different elastic properties of the dot and matrix materials) in the cross section of