Strain-induced interfacial hole localization in self-assembled quantum dots: Compressive $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ versus tensile $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$

Using an atomistic pseudopotential approach, we study how the shape of the dot (spherical vs lens shaped) affects the position-dependent strain and the electronic properties of tensile $(\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb})$ and compressive $(\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As})$ quantum dots. We compare the strain profiles, strained modified band offsets, confined levels, and atomistic wave functions of these dots. We show (i) how the existence of position-dependent strain in nonflat heterostructures can control the electronic properties, leading, for example, to interfacial localization of hole states on the interface of matrix-embedded dots and (ii) how the dots shape can control the level sequence and degeneracy. For example in spherical dots, one finds degenerate light-hole (LH) and heavy-hole (HH) states, whereas in lens-shaped dots one can have as the highest-occupied hole state either (a) a LH state inside the dot, becoming a HH state outside the dot $(\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ tensile case) or (b) a HH state inside the dot, becoming a LH states outside the dot ($\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ compressive case).

Figure 1

(Color online.) Strain profiles for embedded spherical QD’s along the [001] direction. (a) Results obtained assuming isotropic continuum elasticity (analytical), (b) atomistic strain from VFF, for tensile $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ QD’s with $\text{diameter}=5.6\mathrm{nm}$, (c) atomistic strain from VFF for compressive $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QD’s with $\text{diameter}=13.6\mathrm{nm}$. The thin vertical lines indicate the interface between the dots and their matrix materials.

Figure 2

(Color online.) Strain profiles for embedded lens-shaped QD’s: Panels (a) and (b) give the strain profiles of a lens-shaped $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ QD $(10.4\mathrm{nm}\times 2.6\mathrm{nm})$, (a) along [100] direction and (b) along [001] direction. Panels (c) and (d), give the strain profiles of a lens-shaped $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QD $(25\mathrm{nm}\times 3.5\mathrm{nm})$, (c) along [100] direction and (d) along [001] direction. The biaxial strain outside the lens in (d) decays much slower than in (b), because (d) has a larger effective radius (see text).

Figure 3

Biaxial strain (top panels), conduction band (CB), and valence band (VB1, VB2, VB3) confining potentials for a spherical $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QD with $13.6\mathrm{nm}$ diameter (left panels) and for a lens shape $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QD with $25\mathrm{nm}$ base and $3.5\mathrm{nm}$ height (right panels). The plots show a 3D rendering of the equatorial (001) plane for the sphere and a plane $1.5\mathrm{nm}$ above the base of the dot for the lens.

Figure 4

(Color online.) Strain-modified confining potentials of embedded spheres calculated from Eqs. (7, 8) for (a) tensile strain $(\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb})$ and (b) compressive strain $(\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As})$. The HH and LH bands are degenerate inside the dots, but split outside. The numbers next to CBM, VBM, and SO, are in eV with the VBM of InSb taken as reference energy. The possible hole traps are indicated by the solid circles.

Figure 5

(Color online.) Strain-modified confining potentials of lens-shaped QD’s calculated from Eqs. (7, 8) for (a) tensile strain $(\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb})$ along [100] direction, (b) tensile strain $(\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb})$ along [001] direction, (c) compressive strain $(\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As})$ along [100] direction, and (d) compressive strain $(\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As})$ along [001] direction. For the first (dashed line) and second (thin solid line) hole confining potentials, the HH, LH characters are reversed inside and outside the dots along the [001] direction. Along the [100] direction, the character of these first two hole potentials outside the dots is HH and LH mixed and no label has been assigned to them in the figure.

Figure 6

(Color online.) Decomposed HH (thick solid line), LH (dashed line), and SO (thin solid line) character of the (a) first, (b) second, and (c) third hole confining potentials for lens-shaped $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ QD’s.

Figure 7

(Color.) Square of the wave functions for spherical (a) $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ $(\text{diameter}=5.6\mathrm{nm})$ and (b) $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QD’s $(\text{diameter}=13.6\mathrm{nm})$. The half spheres indicate the position of the dots and the isosurfaces enclose 75% of the state density for the $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ dot and 25% of the state density for the $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ dot. The contour plots are slices of the density taken through the middle of the spheres. We show the first three electron and first three hole states except for the $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ QD where only one electron level is well confined.

Figure 8

(Color.) Equivalent to Fig. 7 but for lens-shaped $\mathrm{In}\mathrm{As}∕\mathrm{In}\mathrm{Sb}$ $(10.4\mathrm{nm}\times 2.6\mathrm{nm})$ and $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{Sb}$ $(25\mathrm{nm}\times 3.5\mathrm{nm})$ quantum dots. The contour plots are slices of the density taken $1\mathrm{nm}$ above the base of the lenses.