Numerical study of valence band states evolution in Al_xGa_1-xAs [111] QDs systems

Theory

Here, we introduce the model with assumptions we used and implemented in the code (Lazarev & Karlsson, 2022). Portions of this text here and below were previously published as part of a thesis (Lazarev, 2019). As a basis, we start with the Kane Hamiltonian (Kane, 1957): H = H^k⋅p + H_SO, where $H^{k\cdot p}=\left[\frac{p^2}{2m_0}+V\left(r\right)\right]+\left[\frac{\hslash }{m_0}k\cdot p+\frac{{\hslash }^2{k}^2}{2m_0}\right]$ and $H_{SO}=\frac{\hslash }{4m_0^2{c}^2}\sigma \cdot \left[\nabla V\times p\right]$ thus, the Schrödinger equation looks like: (1)${\displaystyle \left(\left[\frac{p^2}{2m_0}+V\left(r\right)\right]+\left[\frac{\hslash }{m_0}k\cdot p+\frac{{\hslash }^2{k}^2}{2m_0}\right]+\frac{\hslash }{4m_0^2{c}^2}\sigma \cdot \left[\nabla V\times \left(p+\hslash k\right)\right]\right)u_k^{\left(n\right)}\left(r\right)={\varepsilon }_k^{\left(n\right)}{u}_k^{\left(n\right)}\left(r\right)}$

Where σare Paul matrixes, ∇V the gradient of atomic mean field potential, $u_k^{\left(n\right)}\left(r\right)$ the Bloch waves, and m₀ is the electron mass. Our model works with Hamiltonian (1) and solves it in a 4-band (S and P orbitals) approximation. To achieve more accurate results and consider the influence of other bands, the Löwdin perturbation theory (Löwdin, 1951) was implied. See the Supplemental Information for more details.

To each pair of electron and hole corresponds exciton energy and definer as follows:

$E_X=E_g+E_{cb}+E_{vb}-E_{Coulomb}^{e-h}$ where E_cb, E_vb are the state energies in CB and VB, and $E_{Coulomb}^{e-h}$ is Coulomb interaction energy. These values depend on QD geometry and material composition. Usually, in small QDs, confinement energy is strong compared to the Coulomb interaction. The Coulomb interaction in the case of strong confinement can be considered as a perturbation (Warburton et al., 1998). The perturbation to the Hamiltonian is then written as: (2)${\displaystyle E_{Coulomb}^{e-h}=\iint {\psi }_{cb}{\psi }_{vb}\frac{e^2}{4\pi {\varepsilon }_0\varepsilon \left|r_{cb}-r_{vb}\right|}{\psi }_{cb}^{\ast }{\psi }_{vb}^{\ast }d{r}_{cb}d{r}_{vb}}$

where ψ_vb, ψ_cb are the valence and conduction band wavefunctions, r_cb, r_vb the electron and hole positions.

Observed light intensity and polarization dependence proportional to dipole moment matrix element (Löwdin, 1951): (3)${\displaystyle I_{eh}=\sum _{S_z=\pm \frac{1}{2}}{\left|\sum _{J_z=\pm \frac{3}{2},\pm \frac{1}{2}}〈{\varphi }_e^{S_z}\left(r\right)|{\varphi }_h^{J_z}\left(r\right)〉〈u_e^{S_z}|e\cdot p|u_h^{J_z}〉\right|}^2}$

where e is polarization, p is the momentum operator, and u is Bloch wavefunction.

The final expression for polarization dependence in the case of orthogonal LH and HH wavefunctions along and perpendicular z direction is: (4)${\displaystyle I_z\propto \left(\frac{4}{3}{〈{\phi }_e\left|{\phi }_{LH}\right.〉}^2\right)I_{xy}\propto \left({〈{\phi }_e\left|{\phi }_{HH}\right.〉}^2+\frac{1}{3}{〈{\phi }_e\left|{\phi }_{LH}\right.〉}^2\right)}$

Spectral measurements often use the degree of linear polarization (DOLP) value. Here we define DOLP as $DOLP=\frac{I_z-I_{xy}}{I_z+I_{xy}}$. Thus DOLP =  − 1 corresponds to transition HH wavefunction (WF) part and DOLP = 0.6 LH WF part.

Modeling of structure composition

In our analysis, the parameters defining material distribution and geometry are derived from previous research (Zhu et al., 2006). The design of the structure’s special parameters is based on a simplified model of the “effective” Al-content resulting from the capillarity-driven Al-Ga segregation at the sharp wedges of the. The ratio between “effective” and “nominal” Al content: (5)${\displaystyle x_{eff}=\frac{x}{x+K\left(1-x\right)}}$

where x is the nominal (bulk) Al concentration. For different arias of the pyramidal nanostructure K parameter is unique, so K =8.9 for the central QWR region and K =2.1 for side QWs [24]. Since the x_eff in an Al content, by definition, is associated with the QWR spectral line (Zhu et al., 2006), it leads to an overestimation of actual Al by the blue shift of ∼30 meV (or ∼2% Al content). The overestimation is constant in all points of QWR; thus, it does not affect CB or VB structure. This statement is consistent with experimental results. Effective Al content permits calculating the bandgap profile of the corresponding components of the 3D pyramidal heterostructure. In this article, we implement an additional assumption: due to similar Al content to the bulk, we neglect LQWs regions (blue color in Fig. 1A), assuming Al content to be the same as in the bulk. More geometry details and the ratio of material distribution are also well described in previous studies (Zhu et al., 2006; Lazarev, Rudra & Kapon, 2023). The bandgap dependence over Al content in GaAs is known from the literature (Vurgaftman, Meyer & Ram-Mohan, 2001): (6)${\displaystyle E_g\left(A{l}_{x}G{a}_{1-x}As\right)=\left(1-x\right)E_g\left(GaAs\right)+x{E}_g\left(AlAs\right)-x\left(1-x\right)C}$

where C is the bowing parameter, $E_g\left(GaAs\right)=1.519$ eV and $E_g\left(AlAs\right)=3.1$ eV (low-temperature values). According to Eq. (6), each 1% of Al content gives ∼15 meV to the band gap. The band gap offsets ratio between the CB and VB was taken as ΔCB/ΔVB ∼67/33 (Kopf et al., 1992; Langbein, Gislason & Hvam, 1996).

In our computational framework, Al_xGa_1−xAs [111] nanostructures within inverted pyramids are the primary focus, although the code is versatile enough to accommodate any 3D structure specified by the user. The option to add In content to GaAs and correspondent Hamiltonian is also implemented. It is important to note that In_xGa_1−xAs compound is a material with internal strain, and the Hamiltonian presented above must be modified. While our current discussion does not delve into the strain-modified Hamiltonian, this feature is also incorporated into the code. The numerical solution of the Hamiltonian was performed by the finite difference method.

An example of the model structure geometry is presented in Fig. 2, which is a 10 nm thick GaAs QD embedded in AlGaAs barriers. The modeled area is 40 ×40 nm size which is enough to make sure that border conditions (infinite barriers) do not have an effect on the exciton structure in the area of interest. The calculation grid parameter is chosen as 0.1 nm for all simulation results presented in this article. The diameter of the QD/QWR cylinders is kept at 18 nm. Figure 2B illustrates the Al distribution in the ZX plane. To reiterate key parameters, the effective Al content within the core cylinder of the QD is approximately 2.8%, and in the vertical barrier cylinders, it stands at about ∼7.2% (for 40% nominal Al content).

This methodology was successfully applied to analyze experimental results. Lazarev, Rudra & Kapon (2023), the impact of valence band mixing and spectrum of confined states on the polarization of emitted light from QDs with tailored potential was explored as a function of confinement potential shape that allows it to carefully examine the effect of nanostructure dimensionality on nanostructure emission properties.

LH and HH states evolution in QD

Figure 3A presents the calculated energies of the lowest CB and VB (Fig. 3B) states as a function of QD thickness t, exclusive of Coulomb interaction effects. The height t of the QD cylinder (the QD “thickness“) is varied between 8 to 20 nm (see inset in Fig. 3A). The effective Al content in the core cylinder of the QD is ∼2.7% (for 20% nominal Al content. In the vertical barrier cylinders, it is ∼7.2% (for 40% nominal Al content), which is illustrated in Fig. 3E as a dark and light region in the vertical wire region (pyramid center). The LH (blue color) and HH (red color) states cross in energy for QD thickness t ∼ 13 (see Fig. 3B). Remarkably, this transition also takes places in the VB mixing characteristic (Fig. 3D), where at the crossover point, the HH and the LH WF portions are the same for the first two VB states. For t < 13 nm, the ground state is HH-like, with smaller t the structure behaves in his respect as a “thin” QD. For t > 13 nm, the ground state is LH-like, with larger t the structure behaves as a “QWR” or elongated QD. These results suggest the searching area to choose QD parameters for achieving a structure at the border between HH and LH behavior of the VB ground state. To achieve more accurate results the Coulomb interaction must be considered.

The calculated polarization-resolved optical spectra of this QD structure are shown in Fig. 3C for various thicknesses t; red and blue curves correspond to linear polarization normal and parallel to the growth direction. Figure 3E schematically presents light polarization axes with respect to the nanostructure. Line broadening of 2 meV was applied, and Coulomb interaction was taken into account as a perturbation. This set of spectra shows how the GS transition gradually changes its character from LH to HH with a reduction of QD thickness. Interestingly, the switching from HH to LH–like lowest energy state occurs near t = 14 nm (as in Fig. 3B), which is different on 1nm from switching simulated without coulomb interaction.

In a range of t ∼ 14 nm, the QD GS transition changes its polarization with the “switching” of the VB ground state character from LH to HH. Since, around this point, small variations in the confinement potential can yield such GS character “switching”, such QD can serve as a base for achieving the desired tunable transition using an electric field. Here, we analyze the effect of an electric field on such “equilibrium”: QD structure.

To illustrate the effect of an electric field, we consider the same QD structure discussed above with fixed QD thickness at t = 14 nm. The confined CB and VB states are then computed by adding an electric field E oriented in the growth direction z. From the experimental point of view, considering the structure size of L< 1µm the voltage magnitude at T =10K that can be applied is about 3V, it gives us a feasible range of electric field simulation up to E_max = 30 [kV/cm].

Figure 4A shows the calculated optical spectra of the QD structure for an electric field of amplitude E = 20 [kV/cm] oriented along the growth direction. We would like to highlight two significant differences compared to the spectrum for E = 0. First, the absorption edge is red-shifted due to a quantum confined Stark effect (Fry et al., 2000). Secondly, the lowest energy transition line with nearly equal H and V light intensity components develops for E > 0 into a richer structure with characteristic polarization features. This is illustrated by DOLP spectrum, which shows an HH-like feature at low energies instead of DOLP ∼ 0 for E = 0. The reason is that the electric field changes the QD potential profile in such a way that the lowest VB state becomes HH-like, and the Stark shifts are different for the different VB states. Thus, applying an electric field can tune the QD structure to change the polarization state of light emitted by the QD ground states. The impact of the electric field on the lowest transition state energy is displayed on the right panel of Fig. 4. The spectral lines near 1.595 eV in Fig. 4 are related to the first CB and first two VB transitions. Note that the polarization is determined by the part (LH or HH) of the VB wavefunction that best overlaps with the CB state.

QDM polarization control by barrier adjustment

QDM systems contain many parameters that affect QDs coupling strength, and thus, system’s optical properties, the main geometrical parameters, are illustrated schematically in Fig. 5. For example, here, we show the effect of barrier height on the polarization switching of the QDM GS emission. Figure 6 presents the dependence of the lowest CB and VB states on Al content (in ‘nominal’ values) in the outer barrier (nominal Al content in QD core: 20%; in external barriers: 40%, QD thickness t = 7 nm; barrier thickness d = 5 nm). The GS of the CB increases with increasing Al content as the potential energy of the barrier increases. The VB states dependence is more complex: in between the two extreme barrier heights, the VB GS changes its character from LH- to HH- like with increasing Al content (see Fig. 6B).

Figure 7A presents a comprehensive summary of the calculated optical spectra, considering Coulomb interaction as a perturbation and a line broadening of 2 meV. The lowest energy transition is mainly LH-like for low barriers and becomes degenerate with the HH-like one at ∼40% Al. The spectra show the evolution of the HH and LH transition lines, their crossing when Al content becomes ∼40%, and at 50%, the complete change of the GS to HH one. Figure 7B shows the spectral positions of the GS and the first excited transition versus the barrier Al content, indicating the VB character (red color for HH, blue for LH). Figure 7C shows the VB mixing dependence on the barrier height. In summary, the common trend for all structures is that at certain inner QDs coupling regimes (from strong to low), the GS transition changes between LH to HH-like. As a consequence, the emission polarization of the GS also changes. This figure illustrates the two regimes of behavior, with two different GS transition types and polarization of emitted light. The QDM system at the intermediate point where the HH and LH states are almost degenerate opens an opportunity to tune the GS emission character by applying an external electric field due to the sensitivity of such structure.

Next, we consider the effect of an external electric field in a QDM with the following structural parameters: QD cores of 20% Al content and thickness t = 7 nm each, QD external barriers of 40% Al content, QD inner barrier thickness d = 5 nm, and 35% Al content. Figure 8 shows the simulation results of the system without an external electric field (E = 0). Figure 8A presents the calculated optical spectra, and Fig. 8B the corresponding DOLP spectra. In this configuration, the GS LH and HH transition is almost at the same energy, with a more complex structure of the excited states transitions. The right panel of Fig. 8 shows a side view of the lowest CB and VB states WFs isosurfaces.

For the same QDM structure, the other potential asymmetry induced by an external electric field of E = 0.9 [kV/cm] aligned in the growth direction dramatically changes the optical spectra and WFs shapes (Fig. 9). As in the case of the single QD with the applied electric field, the polarization of the GS transition and its energy change. In contrast to the case E = 0, the lowest energy transition now has DOLP =  − 0.6 instead of DOLP = 0.5 to its more important HH character of the VB state involved. As usual, the electric field pushes the CB and VB WFs in the opposite direction for a Stark effect. This also leads to a reduction in the transition strength due to the smaller WF overlaps.

The case of two-QDs molecule can be extended to N QDs (QDSLs), which leads us to the formation of minibands (Sugaya et al., 2010), as was also shown in the case of QW superlattices (Dingle, 1975). Such systems can also be realized experimentally in inverted pyramids as a vertically aligned coupled QD array. In QDSLs, the barriers height and width are crucial parameters that define the coupling strength between the QDs. In such systems with modulated internal potential in a low coupling regime, GS HH WFs can have as large as LH WF in QDs with flat potential. However, the corresponding emission will have a different polarization. The consequence of low coupling is that a “forest” of close-spaced optical transition lines appears. Under the application of an electric field, the excited CB and VB WFs overlap due to tunneling to neighbor QDs, while the first CB and VB states are separate and do not overlap (Lazarev, 2019). From an experimental point of view, it’s hard to fabricate and analyze the VB structure of N QDs due to structural complexity, optical line merging, and non-linear effects during the nanostructure growth.