Chapter 2 Fundamental Theory
2.2 Strain effects
Initial strain in InAs/GaAs self-assembled QDs
InAs/GaAs self-assembled QDs have built-in strains due to the lattice mismatching of the two materials InAs and GaAs, this effect induces initial strains. Here, this induced strain is treated as an initial strain in the analysis[8].
The in-plane mismatch is defined as
GaAs InAs represents the lattice mismatching percentage. The negative sign refers to the compression in the x and y directions, in which the compression of xy-plane results the tension in z-direction.
This case involves a very large and thin plate, explaining why a general formalism that changes with size is derived here. Refrences [10] and [11]
provide the general formulae. Prof. Cheng derived the following application analysis formulae. Notably, the strain elements are precisely at the center of cubic shaped quantum dots of isotropic material. For an isotropic crystal, the elastic matrix iso
c
ij is set to be:11 12 12
Analytical solution for initial strain Symmetry dots
Fig. 2.2.1 shows a cubic shaped quantum dot
2 2 2
Analytical solution for initial strain
cubic shaped quantum dot and its corresponding length.
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(0,0,0)= (0,0,0)= (0,0,0)
xy xz yz
ε ε ε (2.2.11)
The following section summarizes those results and compare them with the results of the finite element software package Comsol multiphysics.
Software comsol manipulation of the initial strain in self-assembled InAs/GaAs cubic shaped quantum dot
Symmetry cubic QDs
(a) (b)
Symmetry
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(c) (d)
Fig. 2.2.2 Strain quantities of symmetry cubic QDs (a) εxx, εyy,(b) εzz, (c)
xx yy 2 zz
ε +ε − ε , (d) εxx +εyy +εzz from Lx =10 ~ 100 nm,Lz =2 nm.
When the plate is extremely large as Lx =Ly =100 nm, εxxand εyy approach
0 6.69%
ε = − ,εzzis close to 7.27%. For a symmetry QD, εxx =εyy. Shear strain tensors are zero in the simulation of comsol, which consists of the analytical solution. As is well known, the reason that εxx > εyy is very similar to that for rubber bands, in which longer the bands, more easier to extended the bands.
Therefore, the strain tensors decrease as the cubits become diminish in size.
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Asymmetric cubic QDs
(a) (b)
(c) (d)
(e)
Asymmetry
26 like rubber bands, the longer side is easily extended; thus, the length variation in x-direction is greater than that in y-direction. Still, the shear strain tensors are zero as in the symmetric case.
The strain quantities related to Hamiltonian are introduced as follows.
Q = ( 2 )
Applied stress induced strain
A force acts on a body might cause a displacement in shape. Stress resembles force, and the displacement resembles strain. However, in this study, strain is not a specific length of displacement but the ratio of displacement compared with the original length.
Definition of strain
Origin point O is fixed in the end of string, and a point P moves on the string that moves at a distance u to P′, O P = xand O P′ = +x u .
Consider two close points P and Qi, in which distance between P and Qi
denoted as ∆xi = P Qi (∆x1 = P Q1, ∆x2 = P Q2 ...). After deformation,
i i
The normal strain tensor is given by
1 direction relative to its original lattice length.
For the angle
Since we restrict the discussion to small displacements, than the displacements
small as well. Thus, the Eq.(2.1.5)
2
The normal strain tensor is given by
Fig. 2.2.4 The displacement along the parallel direction and perpendicular direction relative to its original lattice length.
2
28 shear strain and as well the change in angle. The definition of three-dimensional strain resembles the two-dimensional strain. The strain tensor can be presented as:
Similar to the symmetrical property in two-dimensional form, the stress tensor of the symmetrical part of
e
ij is represented asε
ij.( )
1 + ( , 1, 2, 3)
ij 2 eij eji i j
ε = = (2.2.21)
For the symmetrical strain tensor,
ij ji
ε ε =
(2.2.22)Hooke’s Law states that
σ = c ε
(2.2.23)ε = s σ
(2.2.24)29
s represents the elastic compliance constant. The compliance is related to the elastic stiffness constant, c. c is known as Young’s Modulus. Also, stress σ is applied homogeneously to a crystal results a strain
ε
respectively. The relationship between stress and strain in tensor form is given by( , , , 1,2,3)
ij sijkl kl i j k l
ε
=σ
= (2.2.25)the Einstein notation is used here.
Stress induced strain in any reference frame Hooke’s law can be written in tensor form as
ij
s
ijkl klε = σ
(2.2.26)The strain tensor is expanded as
11 1111 11 1112 12 1113 13
1121 21 1122 22 1123 23
1131 31 1132 32 1133 33
2321 21 2322 22 2323 23
2331 31 2332 32 2333 33
Tensor into matrix form is followed by the notation rule.
yy zz yz,zy zx,xz xy,yx tensor notation (ij) 11 22 33 23,32 31,13 12,21 matrix notation (m) 1 2
xx
3 4 5 6
30
For strain, from tensor notation to matrix notation
1
From tensor notation to matrix notation when m and n are 1, 2, 3
The stress induced strain thus becomes
1 1
31
Furthermore, the matrix notation is transformed into Cartesian notation.
xx xx
For the cubic crystal in which the main axis is along [100], the compliance can be obtained from the elastic stiffness constant, c.[7]
The compliance matrix is precisely for cubic crystal along [100][7]:
[100 ] [100 ] [100 ]
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[100] [100] [100] [100] [100] [100] [100]
[100] [100] [100] [100] [100] [100] [100]
[100] [100] [100] frame, stress may occasionally occur that misaligns the main axes.
For a second-rank stress tensor that works misaligned with the main axes, the stress is transformed by two transformation matrixes and the vector component in the main reference frame is obtained. Here, the double-prime label represents the frame of applied stress, and the raw label states its component in the reference frame.[7]
mn a aom pn op
σ = σ′′ (2.2.35)
The angles are redefined here, in which the stress is assumed to be misaligned with the main axes with an angle
φ
σto x , Appendix I describes the choice and settlement of the rotation matrix.33 perpendicular to each other:
11 1 1
With the same method, all of the stress tensor elements are obtained. Finally, the stress becomes:
2 2
sin cos sin cos
′′ ′′ second-rank tensor method and
2 2
sin cos sin cos
This equation is universally used for with an angle
φ
σ to the reference frame (a)Fig. 2.2.5 (a) stresses in given coordinate system. (b) stress applied with an angle φσ
For the following usage, the matrix notations
notation, subsequently allowing for additional discussion via direct observation.
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sin cos sin cos
′′ ′′ rank tensor method and represented as
2 2
sin cos sin cos
This equation is universally used for stress that is misaligned with the main axes to the reference framex .
(b)
Fig. 2.2.5 (a) stresses in given coordinate system. (b) stress applied to the main reference coordinate.
For the following usage, the matrix notations are presented
allowing for additional discussion via direct observation.
(2.2.40)
The stress components in the reference frame can be obtained via the
(2.2.41)
misaligned with the main axes
Fig. 2.2.5 (a) stresses in given coordinate system. (b) stress applied
are presented as the Cartesian allowing for additional discussion via direct observation.
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sin cos sin cos
xx
Also, insert Eq. (2.2.42) into Eq. (2.2.30).
The strain induced by a set of bi-axial stress and applied with an angle
φ
σto x and y axes for any reference frame is shown below:( )
cos sin sin cos
cos sin sin cos
2 0 0
Thus, for the main reference frame along [110], the stress induced strain can be written as
cos sin sin cos
cos sin sin