Atomistic theoretical study of electronic and polarization properties of single and vertically stacked elliptical InAs quantum dots

The demonstration of isotropic polarization response from semiconductor quantum dots (QDs) is a crucial step towards the design of several optoelectronic technologies. Among many parameters that impact the degree of polarization (DOP${}_{\left[\vec{n}\right]}$) of a QD system, the shape asymmetry is a critical factor. We perform multi-million-atom simulations to study the impact of the elliptical shapes on the electronic and polarization properties of single and vertically stacked InAs QDs. The comparison between a low aspect ratio (AR) and a high AR QD reveals that the electronic and the polarization properties strongly depend on the AR of the QD; the elongation of a tall QD allows tuning of the DOP${}_{\left[\vec{n}\right]}$ over a much wider range. We then extend our analysis to an experimentally reported vertical stack of nine QDs (9-VSQDs) that has shown significant potential to achieve isotropic polarization properties. We analyze the contribution from the shape asymmetry in the large, experimentally measured, in-plane polarization anisotropy. Our analysis shows that the orientation of the base elongation controls the sign of the DOP${}_{\left[\vec{n}\right]}$; however, the magnitude of the base elongation has only a very little impact on the magnitude of the DOP${}_{\left[\vec{n}\right]}$. We further predict that the elliptical shape of the 9-VSQDs can only tune either DOP${}_{\left[110\right]}$ or DOP${}_{\left[\overline{1}10\right]}$ for the isotropic response. Our model results, in agreement with the TEM findings, suggest that the experimentally grown 9-VSQDs has either a circular or a slightly [$\overline{1}$10] elongated base. Overall, the detailed investigation of DOP${}_{\left[\vec{n}\right]}$ as a function of the QD shape asymmetry provides a theoretical guidance for the continuing experimental efforts to achieve tailored polarization properties from QD nanostructures for the design of optical devices.

Figure 1

Schematic diagrams of quantum dots are shown for theoretical modeling. (a) A low aspect ratio (flat) dome-shaped InAs quantum dot with the base diameter and height of 20 and 4.5 nm, respectively. (b) A high aspect ratio (tall) dome-shaped InAs quantum dot with the base diameter and height of 20 and 8 nm, respectively. (c) A vertical stack consisting of nine InAs QDs (9-VSQDs), each with the base diameter and height of 20 and 4 nm, respectively. The geometry parameters are directly taken from the experimental study.[9] (d) Top view illustrating the base elongations for the type-I elongation. The elliptical shape is formed by decreasing the diameter either along the [110] or along the [$\overline{1}$10] direction. (e) Top view illustrating the base elongations for the type-II and type-IIv elongations. The elliptical shape is formed by simultaneously decreasing (increasing) the base diameter along the [110] direction and increasing (decreasing) the base diameter along the [$\overline{1}$10] direction. For the type-IIv elongation, we select values for the $d_{\left[110\right]}$ and $d_{\left[\overline{1}10\right]}$ such that to keep the overall volume of the QD fixed. (f) Plots of the products of the QD diameters along the [110] and the [$\overline{1}$10] directions, $d_{\left[110\right]}\times d_{\left[\overline{1}10\right]}$, as a function of the elongation factor ($\eta$). For a fixed QD height, this product is directly proportional to the volume of the QD.

Figure 2

(a)–(c) The lowest three conduction band energy levels (e1–e3) are plotted as a function of the elongation factor ($\eta$) for the (a) type-I, (b) type-IIv, and (c) type-II elongations. (d)–(f) The highest five valence band energy levels (h1–h5) are plotted as a function of the QD elongation factor ($\eta$) for the (d) type-I, (e) type-IIv, and (f) type-II elongations. The corresponding increase/decrease in the optical gap energy ($E_g$) is also specified in each case by using the vertical arrows.

Figure 3

The top view of the wave function plots for the lowest three conduction band (e1–e3) and the highest five valence band (h1–h5) states are shown for the circular-based QD and for the selected elongations of the QD. The intensity of the colors in the plots represents the magnitude of the wave functions, with the dark red color indicating the largest magnitude and the light blue color indicating the smallest magnitude. The boundaries of the QDs are also shown to guide the eye.

Figure 4

The plots of the total piezoelectric potentials (linear+quadratic) are shown for (a) circular-based QD, (b), (d), and (f) [110]-elongated QDs, and (c), (e), and (g) [$\overline{1}$10]-elongated QDs. In each case, the type and magnitude of elongation are specified. The solid red lines are plotted along the [$\overline{1}$10] direction through the center of the QD, ≈ 0.5 nm above its base. The dotted (broken) black lines are plotted along the [110] direction through the center of the QD, ≈ 0.5 nm above its base. The boundaries of the QD region are also marked in each case by specifying the lengths of the QD along the [110] and [$\overline{1}$10] directions, $d_{\left[110\right]}$ and $d_{\left[\overline{1}10\right]}$.

Figure 5

The polarization-dependent optical transition modes TE${}_{\left[110\right]}$, TE${}_{\left[\overline{1}10\right]}$, and TM${}_{\left[001\right]}$ are plotted as a function of the (a) type-I, (b) type-IIv, and (c) type-II QD elongations. The figures are plotted using the same scales to facilitate easy mutual comparison. (d) The plots of the in-plane polarization anisotropy (Pol${}_{\left|\right|}$) as defined by Eq. (2) are shown for the three types of elongations, exhibiting a quadratic polynomial dependence on ${\eta }^{-1}$. Four cases are marked by using green ovals indicating that the two different types of elongations, with roughly similar values of $\eta$, exhibit similar values of the Pol${}_{\left|\right|}$.

Figure 6

The plots of the DOP${}_{\left[110\right]}$ and DOP${}_{\left[\overline{1}10\right]}$ are shown as a function of the (a) type-I, (b) type-IIv, and (c) type-II QD elongations. The values of the DOP decrease irrespective of the direction of the QD elongation.

Figure 7

(a)–(c) The lowest three conduction band energy levels (e1–e3) are plotted as a function of the elongation factor ($\eta$) for the (a) type-I, (b) type-IIv, and (c) type-II elongations. (d)–(f) The highest five valence band energy levels (h1–h5) are plotted as a function of the QD elongation factor ($\eta$) for the (d) type-I, (e) type-IIv, and (f) type-II elongations. The corresponding increase/decrease in the optical gap energy ($E_g$) is also specified in each case by using the vertical arrows.

Figure 8

The top view of the wave-function plots for the lowest three conduction band (e1–e3) and the highest five valence band (h1–h5) states are shown for the circular-base of the QD and for the selected elongations of the QD. The intensity of the colors in the plots represents the magnitude of the wave functions, with the dark red color indicating the largest magnitude and the light blue color indicating the smallest magnitude. The boundaries of the QDs are also shown to guide the eye.

Figure 9

The plots of the total (linear+quadratic) piezoelectric potentials are shown for (a) circular-based QD, (b), (d), and (f) [110]-elongated QDs, and (c), (e), and (g) [$\overline{1}$10]-elongated QDs. In each case, the type and the magnitude of the elongation is specified. The solid red lines are plotted along the [$\overline{1}$10] direction through the center of the QD, 0.5 nm above its base. The dotted (broken) black lines are plotted along the [110] direction through the center of the QD, 0.5 nm above its base. The boundary of the QD region is also marked in each case by specifying the lengths of QD along the [110] and [$\overline{1}$10] directions, $d_{\left[110\right]}$ and $d_{\left[\overline{1}10\right]}$.

Figure 10

The polarization-dependent optical transition modes TE${}_{\left[110\right]}$, TE${}_{[\overline{1}10}]$, and TM${}_{\left[001\right]}$ are drawn as a function of the (a) type-I, (b) type-IIv, and (c) type-II QD elongations. The figures are plotted using the same scales to facilitate easy mutual comparison. (d) The plots of the in-plane polarization anisotropy (Pol${}_{\left|\right|}$) as defined by Eq. (2) are shown for the three type of the elongations, exhibiting a quadratic polynomial dependence on ${\eta }^{-1}$. Four cases are also marked by using green ovals indicating that the two different types of elongations, with roughly similar values of $\eta$, exhibit similar values of the Pol${}_{\left|\right|}$.

Figure 11

The plots of the DOP${}_{\left[110\right]}$ and DOP${}_{\left[\overline{1}10\right]}$ are shown as a function of the (a) type-I, (b) type-IIv, and (c) type-II elongations. A large decrease in the value of the DOP${}_{\left[\vec{n}\right]}$ along the minor axis of the elliptical QDs is calculated.

Figure 12

The plots of (a) the lowest three conduction band energies (e1–e3) and (b) the highest five valence band energies (h1–h5) for the 9-VSQDs as a function of the type-II elongation factor $\eta$. All of the energies increase irrespective of the orientation of the elongation. (c) and (d) The plots of the electron and hole energies as in (a) and (b), but without including the effect of strain. In the absence of strain, only $\Delta d$ and $\Delta V$ contribute in the energy shifts. (e) The plots of the biaxial strain component (${\epsilon }_{xx}+{\epsilon }_{yy}-2{\epsilon }_{zz}$) along the [110] direction through the center of the 9-VSQDs with circular-based ($\eta =1$) and the [110] elongated base ($\eta =1.75$). (f) The plots of the biaxial strain component (${\epsilon }_{xx}+{\epsilon }_{yy}-2{\epsilon }_{zz}$) along the [$\overline{1}$10] direction through the center of the 9-VSQDs with circular-based ($\eta =1$) and the [$\overline{1}$10] elongated base ($\eta =0.57$).

Figure 13

Top views of the wave function plots for the lowest three conduction band (e1–e3) and the highest five valence band (h1–h5) states are shown for the circular-based 9-VSQDs and the two selected type-II elongations of the 9-VSQD. The intensity of the colors in the plots represent the magnitude of the wave functions, with the dark color indicating the smallest magnitude and the light green color indicating the largest magnitude. The boundaries of the QDs are also shown to guide the eye.

Figure 14

(a) The plots of the polarization-dependent TE and TM modes as a function of the type-II elongation factor $\eta$ for the 9-VSQDs. (b) Plot of the Pol${}_{\left|\right|}$ as a function of the type-II elongation factor for the 9-VSQDs. The plots indicate a high degree of in-plane anisotropy for the 9-VSQDs, which is very sensitive to the elongation direction. (c) Plots of the DOP${}_{\left[110\right]}$ and DOP${}_{\left[\overline{1}10\right]}$ as a function of the type-II elongation factor for the 9-VSQDs. For the circular-based 9-VSQDs, $\eta =1.0$, the DOP${}_{\left[110\right]}$ is negative and the DOP${}_{\left[\overline{1}10\right]}$ is positive. As we elongate the 9-VSQDs along the [110] direction by 1 nm (marked by blue oval), the DOP${}_{\left[\overline{1}10\right]}$ becomes close to zero, however, the DOP${}_{\left[110\right]}$ drastically increases to $+$0.60, suggesting that it is not possible to simultaneously engineer both DOPs below or close to zero for this 9-VSQDs using [110] elongation engineering.

Figure 15

Normalized polar plots for the 9-VSQDs system with (a) circular-based, (b) type-II [110] elongation ($\eta =1.095$), and (c) type-I [110] elongation ($\eta =1.047$). In each case, the polarization direction of the incident light ($\vec{n}$) is kept along the [110] direction (red squares) for the TE${}_{\left[\overline{1}10\right]}$ mode and along the [$\overline{1}$10] direction (blue circles) for the TE${}_{\left[110\right]}$ mode. The polar plots based on the cumulative sum of the optical transition strengths between the lowest conduction band state and the highest five valence band states are drawn with respect to the angle $\theta$ between the [001] direction and either the [110] or the [$\overline{1}$10] direction.