126
5 Mechanical Properties
a pyramidal InAs quantum dot in a GaAs matrix on top of a two-dimensional InAs layer. The strain
component zz is positive in the 2D layer, as expected from (5.66). However, in the pyramid zz exhibits
a complicated dependence and even takes negative values at the apex.
5.3.5 Substrate Bending
If a lattice-mismatched layer is pseudomorphically grown on top of a substrate it suffers biaxial strain.
For finite substrate thickness part of the strain will relax via substrate bending. If the substrate is circular,
a spherical cap is formed. If the lattice constant of the film is larger (smaller) than that of the substrate,
the film is under compressive (tensile) strain and the curvature is convex (concave) with respect to the
outward normal given by the growth direction (Fig. 5.35a). Substrate bending can also be induced by a
mismatch of the thermal expansion coefficients α
f
th and α
s
th of the film and substrate, respectively. If a
film/substrate system is flat at a given temperature, e.g. growth temperature, a decrease of temperature,
e.g. during cooling, will lead to compressive (tensile) strain if α
f
th is smaller (larger) than α
s
th .
In a curved structure, the lattice constant in the tangential direction increases from a
t
i at the inner
surface (r = R = κ
−1 ) to a
t
u at the outer surface (r = R + d). Thus, the tangential lattice constant
varies with the radial position
a
t
(r ) = a
t
i (1 + r κ) ,
(5.81)
where d is the layer thickness (Fig. 5.35b). Therefore a u = a i (1 + d/R). We note that (5.81) holds in
all layers of a heterostructure, i.e. the film and the substrate.
The lattice constant in the radial direction a
r , however, depends on the lattice constant a 0 of the
local material and is calculated from the biaxial strain condition, such as (5.66). The in-plane strain is
= (a
t
− a 0 )/a 0 (we assume a spherical cap with = θθ = φφ ). For an isotropic material we find
a
r
= a 0 (1 + ⊥ ) with ⊥ = −2νν /(1 − ν). The local strain energy density U is given by
U =
Y
1 − ν
2
.
(5.82)
The total strain energy per unit area U
of a system of two layers with lattice constants a 1 , a 2 , Young’s
moduli Y 1 , Y 2 and thickness d 1 , d 2 (we assume the same Poisson constant ν in both layers) is
U
=
d 1
0
U 1 dr +
d 2
d 1
U 2 dr .
(5.83)
The total strain energy needs to be minimized with respect to a i and R in order to find the equilibrium
curvature κ. We find
(a)
tensile
compressive
(b)
a i
a u
R
d
Fig. 5.35 a Schematic bending of a film/substrate system for compressive (left) and tensile (right) film strain. b Schematic
deformation of curved film of thickness d. The lattice constants at the inner and outer surface are a i and a u , respectively
5 Mechanical Properties
a pyramidal InAs quantum dot in a GaAs matrix on top of a two-dimensional InAs layer. The strain
component zz is positive in the 2D layer, as expected from (5.66). However, in the pyramid zz exhibits
a complicated dependence and even takes negative values at the apex.
5.3.5 Substrate Bending
If a lattice-mismatched layer is pseudomorphically grown on top of a substrate it suffers biaxial strain.
For finite substrate thickness part of the strain will relax via substrate bending. If the substrate is circular,
a spherical cap is formed. If the lattice constant of the film is larger (smaller) than that of the substrate,
the film is under compressive (tensile) strain and the curvature is convex (concave) with respect to the
outward normal given by the growth direction (Fig. 5.35a). Substrate bending can also be induced by a
mismatch of the thermal expansion coefficients α
f
th and α
s
th of the film and substrate, respectively. If a
film/substrate system is flat at a given temperature, e.g. growth temperature, a decrease of temperature,
e.g. during cooling, will lead to compressive (tensile) strain if α
f
th is smaller (larger) than α
s
th .
In a curved structure, the lattice constant in the tangential direction increases from a
t
i at the inner
surface (r = R = κ
−1 ) to a
t
u at the outer surface (r = R + d). Thus, the tangential lattice constant
varies with the radial position
a
t
(r ) = a
t
i (1 + r κ) ,
(5.81)
where d is the layer thickness (Fig. 5.35b). Therefore a u = a i (1 + d/R). We note that (5.81) holds in
all layers of a heterostructure, i.e. the film and the substrate.
The lattice constant in the radial direction a
r , however, depends on the lattice constant a 0 of the
local material and is calculated from the biaxial strain condition, such as (5.66). The in-plane strain is
= (a
t
− a 0 )/a 0 (we assume a spherical cap with = θθ = φφ ). For an isotropic material we find
a
r
= a 0 (1 + ⊥ ) with ⊥ = −2νν /(1 − ν). The local strain energy density U is given by
U =
Y
1 − ν
2
.
(5.82)
The total strain energy per unit area U
of a system of two layers with lattice constants a 1 , a 2 , Young’s
moduli Y 1 , Y 2 and thickness d 1 , d 2 (we assume the same Poisson constant ν in both layers) is
U
=
d 1
0
U 1 dr +
d 2
d 1
U 2 dr .
(5.83)
The total strain energy needs to be minimized with respect to a i and R in order to find the equilibrium
curvature κ. We find
(a)
tensile
compressive
(b)
a i
a u
R
d
Fig. 5.35 a Schematic bending of a film/substrate system for compressive (left) and tensile (right) film strain. b Schematic
deformation of curved film of thickness d. The lattice constants at the inner and outer surface are a i and a u , respectively