Size dependency of strain in arbitrary shaped anisotropic embedded quantum dots due to nonlocal dispersive effects

Both quantitative and qualitative knowledge of strain and strain distributions in quantum dots are essential for the determination and tailoring of their optoelectronic properties. Typically strain is estimated using classical elasticity and then coupled to a suitable band structure calculation approach. However, classical elasticity is intrinsically size independent. This is in contradiction to the physical fact that at the size scale of a few nanometers, the elastic relaxation is size dependent and a departure from classical mechanics is expected. First, in the isotropic case, based on the physical mechanisms of nonlocal interactions, we herein derive (closed-form) scaling formulas for strain in embedded lattice-mismatched spherical quantum dots. In addition to a size dependency, we find marked differences in both spatial distribution of strain as well as in quantitative estimates especially in cases of extremely small quantum dots. Fully recognizing that typical quantum dots are neither of idealized spherical shape nor isotropic, we finally extend our results to cubic anisotropy and arbitrary shape. In particular, an exceptionally simple expression is derived for the dilation in an arbitrary shaped quantum dot. For the more general case (incorporating anisotropy), closed-form results are derived in the Fourier space while numerical results are provided to illustrate the various physical insights. Apart from qualitative and quantitative differences in strain states due to nonlocal effects, an aesthetic by-product for the technologically important polyhedral shaped quantum dots is that strain singularities at corners and vertices (which plague the classical elasticity formulation) are absent. Choosing GaAs as an example material, our results indicate that errors as large as hundreds of meV may be incurred upon neglect of nonlocal effects in sub-10-nm quantum dots.

Figure 1

(Color online) Strain dilatation (normalized by classical elasticity result) is plotted as a function of position and size.

Figure 2

(Color online) Dilatational strain as a function of size for fixed position $(r=0)$. The results are normalized with respect to the classical elasticity solution.

Figure 3

(Color online) Deviation ratio is defined as $[\mathrm{Tr}{\left({\epsilon }_{ij}\right)}_{\text{classical}}-\mathrm{Tr}{\left({\epsilon }_{ij}\right)}_{\text{surface}∕\text{nonlocal}}]∕\mathrm{Tr}{\left({\epsilon }_{ij}\right)}_{\text{classical}}$. The mismatch strain is 0.067 nm (typical for the $\mathrm{InAs}∕\mathrm{GaAs}$ system). Internal length scale is 0.82 nm. Interfacial tension is $1\mathrm{J}∕{\mathrm{m}}^2$. Material properties are based on isotropic data in Table .

Figure 4

(Color online) Strain dilation of a spherical QD based on combination of nonlocal and surface effects. Normalized strain is defined as $\mathrm{Tr}\left({\epsilon }_{ij}\right)(3K+4\mu )∕9K{\epsilon }^m$. The materials properties are based on the isotropic case in Table . The QD is chosen with radius as $R=4l^{\prime }$ and the internal length scale is chosen as $l^{\prime }=0.82\mathrm{nm}$.

Figure 5

(Color online) Strain dilation at center of QD with mismatch strain equal to 0.067. Material properties are based on Table . Internal length scale is $l^{\prime }=0.82\mathrm{nm}$.

Figure 6

(Color online) Strain gradient results for dilation strain of cuboidal quantum dot. Dilation strain is normalized as $\mathrm{Tr}\left({\epsilon }_{ij}\right)\left(2\mu +\lambda \right)∕P^m$ Light area represents high dilation. Internal length scale is $l^{\prime }=0.82\mathrm{nm}$.

Figure 7

(Color online) Normalized strain dilation is plotted for two directions on the crossing plane $z=0$. The subscript “sg” indicates strain gradient while “cl” indicates classical. Vertical dotted lines mark the boundary of QD at two directions ($x$ and $y$). QD1’s size is $a_1=5\mathrm{nm}$, $a_2=7\mathrm{nm}$, and $a_3=5\mathrm{nm}$. QD2’s size is $a_1=7\mathrm{nm}$, $a_2=9\mathrm{nm}$, and $a_3=7\mathrm{nm}$.

Figure 8

(Color online) Strain components ${\epsilon }_{11}$ and ${\epsilon }_{33}$ of cuboidal QD with $a_1=5\mathrm{nm}$, $a_2=7\mathrm{nm}$, and $a_3=5\mathrm{nm}$. Vertical dotted line is the boundary of the QD. The plot is along (010).

Figure 9

(Color online) Strain components ${\epsilon }_{13}$ along (111) direction of cuboidal QD with edge length equal to 5 nm.

Figure 10

(Color online) Dilation strain for finite cylindrical QD1, with diameter $D=5\mathrm{nm}$ and height $h=8\mathrm{nm}$, and QD2, with $D=7\mathrm{nm}$ and $h=10\mathrm{nm}$.

Figure 11

(Color online) Strain components for finite cylinder QD with $D=5\mathrm{nm}$ and $h=8\mathrm{nm}$. [${\epsilon }_{33}$ (cl) overlaps with ${\epsilon }_{33}$ (sg).]

Figure 12

(Color online) Size sensitivity to height of cylindrical QD. [$h=8\mathrm{nm}$ (cl) overlaps with $h=8\mathrm{nm}$ (sg).]

Figure 13

Schematic figure of orthogonal polyhedral QD.

Figure 14

(Color online) Strain dilation along $y$ axis.

Figure 15

(Color online) Strain components along (010) of orthogonal polyhedron with $a_1=5\mathrm{nm}$, $a_2=6\mathrm{nm}$, $a_3=4\mathrm{nm}$.

Figure 16

(Color online) Strain component of finite cylinder QD for different cases (a) ${\epsilon }_{11}$, (b) ${\epsilon }_{22}$, (c) ${\epsilon }_{33}$. Strain components of sphere QD for different cases (d) ${\epsilon }_{11}$, (e) ${\epsilon }_{22}$, (f) ${\epsilon }_{33}$. (In the classical isotropic case, ${\epsilon }_{33}$ should be equal to ${\epsilon }_{11}$ due to symmetry, which is indicated here. However, it is different for the anisotropic case.)

Figure 17

Unstrained band profile for $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dot heterogeneous system. Figures are chosen from Ref. 82, while $E_v^{\mathrm{Ga}\mathrm{As}}$ is set to 0 eV.

Figure 18

(Color online) Band energy for different size QDs.

Figure 19

Band energies and carrier density ${\mid \psi \mid }^2$ for QD with radius equal to 5 nm. The central areas (dark gray) indicate higher numbers for the contour plot for carrier density. These results are for a slice of the $y\text{−}z$ plane.

Figure 20

(Color online) Dilation strain of a cylindrical quantum wire.