Effects of linear and nonlinear piezoelectricity on the electronic properties of $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots

The existence of enormous strain fields in self-assembled quantum dots has led to the expectation of dramatic effects of piezoelectricity. However, only linear piezoelectric tensors were used in all previous calculations. We calculate the piezoelectric properties of self-assembled quantum dots using the linear and quadratic piezoelectric tensors derived from first-principles density functional theory. We find that the previously ignored quadratic term has similar magnitude as the linear term and the two terms tend to cancel each other. We show the effect of piezoelectricity on electron and hole energy levels and wave functions as well as on correlated absorption spectra.

Figure 1

(Color online) (a) Piezoelectric density of an InAs quantum dot with $25\mathrm{nm}$ base and $5\mathrm{nm}$ height. The isosurfaces represent $\pm 0.003e∕{\mathrm{a.u.}}^3$. (b) Corresponding piezoelectric potential with isosurfaces at $\pm 35\mathrm{mV}$. (c) Contour plots of the piezoelectric potential $1\mathrm{nm}$ above the wetting layer. The maximum value of the potential is $28\mathrm{mV}$ and the minimum is $29\mathrm{mV}$. The three columns correspond to (i) neglect of ${\tilde{B}}_{\mu jk}$, (ii) neglect of ${\tilde{e}}_{\mu j}$, and (iii) retention of both ${\tilde{e}}_{\mu j}$ and ${\tilde{B}}_{\mu jk}$.

Figure 2

(Color online) Single-particle eigenvalues, with respect to the GaAs valence band edge, of the electron [(a),(b)] and hole states [(c),(d)] in lens-shaped $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dots with different heights [(b),(d)] keeping the base diameter at $25\mathrm{nm}$ and different base diameters [(a),(c)] keeping the height at $3.5\mathrm{nm}$. Solid lines are results with piezoelectric effect and dashed lines without.

Figure 3

(Color online) Splitting of the electron $P$ states for a lens-shaped $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot decomposed into the (i) atomistic, (ii) linear piezoelectric, and (iii) quadratic piezoelectric contributions. The upper panel shows the height dependence of the splitting for a dot with $25\mathrm{nm}$ base diameter and the lower panel shows the base dependence for a dot with $3.5\mathrm{nm}$ height. The dashed lines give the contributions of the linear term when the experimental values of ${\tilde{e}}_{\mu j}$ are used.

Figure 4

(Color online) Square of the electron (left) and hole (right) single-particle wave functions for an $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ lens-shaped quantum dot with $25\mathrm{nm}$ base and $5\mathrm{nm}$ height. The inner (outer) isosurface encloses 45% (75%) of the state density. The second and fifth columns show the charge transfer upon inclusion of piezoelectricity and the numbers give the percentages of one electron charge transferred.

Figure 5

Optical absorption [(b),(d)] and anisotropy $\lambda$ [(a),(c)] of a lens-shaped quantum dot with $b=25\mathrm{nm}$ and $h=5\mathrm{nm}$, with [(c),(d)] and without [(a),(b)] piezoelectric effect.