InAs quantum dots embedded in InP nanowires form an important platform for basic research studies, as well as for quantum dot applications. Notably, understanding of nanowire quantum dot spectral properties is essential in both cases. Therefore, in this work we use atomistic theory to study spectra of the single exciton $(X)$, the biexciton $(XX)$, the triexciton $(XXX)$, and the positively and negatively charged trions $({X}^{+}$ and ${X}^{\ensuremath{-}})$ confined in these nanostructures. We focus on the role of vertical and lateral confinement, therefore, we systematically study a large family of quantum dots with different heights and diameters, and find the important role of correlations due to presence of higher states. We find that the order of excitonic binding energies is a characteristic feature of InAs/InP nanowire quantum dots being (ordered from negative to positive values): ${X}^{\ensuremath{-}},XX$, and ${X}^{+}$, with strongly bound ${X}^{\ensuremath{-}}$, rather weakly bound $XX$, and typically unbound ${X}^{+}$. Next, we determine the key role of alloy randomness due to intermixing, which turns out to especially important for larger quantum dot heights and phosphorous contents over 40%. In selected cases, the alloying can lead to an unbound biexciton, and can even reverse ordering of excitonic lines.
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