For higher index substrate ([112]), faceting effects become visible on the edges of quantum dots and are very well pronounced for high-[118] index substrate. The characteristic steps in quantum dot strain distribution due to faceting effects are also well visible in Figure 1 (lower row) where hydrostatic strain is plotted on a [1

1

Figure 2 shows different functions of strain calculated at the geometric dot center. The absolute of (a) the hydrostatic strain Tr(ε)=_εxx + _εyy + _εzzfor [111] substrate (3.8%) is noticeably larger than [001] substrate (3.1%). We speculate that one could expect even larger difference between [001] and [111] substrates for highly strained InAs/GaAs quantum dots. With increasing substrate index, the magnitude of the hydrostatic strain goes down reaching, for [119] case, the value comparable to that of non-tilted [001] system. Similar trend is observed for (b) biaxial strain defined as B ( ε ) = ε x − ε y ) 2 + ( ε y − ε z ) 2 + ( ε z − ε y ) 2 , where x,y, and z correspond to directions defined by crystal axes [100], [010] and [001], respectively. Biaxial strain defined as B(ε) is largest for flat and disk-like quantum dot located on [001] substrate, while it is exactly zero (by definition and the quantum dot symmetry) for [111] case. On the contrary, function of shear (off-diagonal) strain defined as S ( ε ) = ε xy + ε yz + ε zy / 3 is exactly zero (at the dot center) for [001] substrate quantum dot and reaches maximum for [111], demonstrating the significant role of the [111] biaxial strain for systems grown on [111] substrate.

Different functions of strain at the geometric dot center.

The spectrum of confined hole states is more susceptible to a choice of substrate than that of electron states, yet well-visible shell structure is still present as seen in Figure 4b. The splitting of hole p shell varies from ≈7 meV for [111] substrate and goes down to 2 meV for high-index and [001] substrates. s-p hole level spacing is in the order of 20 meV, while p-d level spacing is about 15 meV, both smaller than that of electrons due to larger hole effective mass. Splittings within p and d shells reach maximum for the most anisotropic quantum dot grown on [112] substrate and are generally systematically larger than for electrons, as holes are more affected by anisotropy of confining potential through biaxial strain terms in TB Hamiltonian 25. Energies corresponding to electron levels, being formed predominately from atomic s levels, follow similar trends as hydrostatic strain in Figure 2a that can be understood in terms of simple Bir-Pikus model 22 and single a^c deformation potential. Holes, on the other hand, are build mostly from atomic p orbitals and are susceptible to both hydrostatic and biaxial [001] and [111] strains that enter Hamiltonian via different deformation potentials (a^vb and d).

Figure 4 shows that, when going from [001] to [111] substrate, the ground hole state is energetically shifted up by 44 meV, while electron ground state is shifted by a smaller amount of about 37 meV only. This difference results in overall reduction of the single particle gap by 7 meV (Figure 4c). Our recent abinitio calculations of band deformation potentials 27 suggest gap reduction for [111]-grown quantum dots compared with [001] case, yet the effect should be more pronounced for highly strained InAs/GaAs quantum dot.

Increasing substrate index from [111] to [119] reduces the single particle gap as seen in Figure 4, but even for [119] case, the gap is far from being converged and is larger by about 25 meV compared to [001] system. Interestingly, the gap value reveals oscillations due to faceting effects that were not well pronounced in the electron and hole spectra separately.

In this work, we neglect effects of piezoelectricity. Such approach is well justified 28 for InAs/InP systems due to small strain magnitude as compared to InAs/GaAs systems and partial cancelation of first order piezoelectric terms by second order contributions 29. More importantly, contrary to straightforward k.p approach 26, piezoelectricity would not alter symmetry of the Hamiltonian which is already well defined by atomistic strained positions entering the TB calculation.

Figure 5b shows XXX^- and X⁺ binding energies calculated according to above formulas, suggesting that both X⁺ and XX are unbound (have positive binding energy). In HF picture, the X⁺ binding energy can be roughly estimated to vary between 4 and 6 meV; XX binding energies are also positive and reached up to 4 meV, while only X^- is bound with bounding energy ≈−3 meV. This characteristic ordering of emission lines with increasing energy: X^-XXX and X⁺ is analogous to that reported in the study of Gong et al. 17 for InAs/InP lens-shaped quantum dot.

However, when correlation effects due to the configuration interaction (mixing) with higher shells (pd) are included 153031, the binding energies are shifted towards lower energies as shown in Figure 5c. In particular, X^- binding energy now reaches ≈−4 meV, while X⁺ binding energy is significantly reduced by a semi-rigid shift (≈3.5 meV) for all considered substrate indices. Most importantly, similar correlation correction (≈3.5 meV) results in binding of XX complex with binding energy of ≈−1.1 meV for [001] substrate and, interestingly, very small (-0.12 meV) binding energy for quantum dot grown on [111] substrate. With correlation effects accounted for emission lines show following order: X^-XXX and X⁺. It must be pointed here that detailed ordering of these levels may depend on dot diameter and height 1617 and should be subject to further studies. The hidden correlation parameter (a measure of correlation effect) 17, defined as Δ = Δ E HF ( XX ) − Δ E HF X − − Δ E HF X + for nanowire InAs/InP quantum dot studied in this paper, varies from 1.2 to 1.5 meV, a much larger value than 0.7-0.9 meV reported for self-assembled InAs/InP quantum dots, proving the necessity of the full configuration interaction 1531 approach for the studies of excitonic complexes in nanowire quantum dots. We point here that small XX binding energy for [111]-grown system can be advantageous for certain entangled photon pair generation schemes based on tuning biexciton and exciton energies to resonance 32.

Interestingly, splitting of electron and hole p-shell cannot be used as simple measure of quantum dot anisotropy and for the straightforward estimation of the fine structure splitting. As mentioned before, electron p-shell splitting does not change significantly with substrate index in contrast to bright exciton splitting. The hole p-shell changes are more pronounced, and indeed, this splitting reaches a maximum (8 meV) for quantum dot grown on [112] substrate (Figure 3b). Yet, both electron and hole p-shell splittings are non-zero for quantum dots on [001] and [111] substrates which have zero bright exciton splitting. Thus a more complicated character the exciton fine structure splitting is revealed, rather than a simple shape anisotropy and the splitting of single particle states.

Last but not least, in Figure 6b, we present dark exciton fine structure splitting as a function of substrate orientation. There is quantitative difference between dark and bright FSS. As predicted by symmetry analysis, dark exciton splitting is exactly zero for C^3v quantum dot on [111] substrate, but non-zero (0.1 μeV) for D_2d quantum dot on [001] substrate. In analogy to bright FSS, the dark FSS reaches maximum (0.5 μeV) for quantum dot on [112] substrate; however, it varies rapidly and stabilizes for high-index substrate at the value (0.15 μeV) similar to [001] case. We note that dark excitons gains non-negligible oscillator strengths for quantum dots on [112]-[115] substrates, but the detailed analysis of dark exciton lifetimes goes beyond the scope of this work.