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In this work we present a detailed analysis of the second-order piezoelectriceffect in $c$-plane In$_{x}$Ga$_{1-x}$N/GaN quantum dots and its consequencesfor electronic and optical properties of these systems. Special attention ispaid to the impact of increasing In content $x$ on the results. We find that ingeneral the second-order piezoelectric effect leads to an increase of theelectrostatic built-in field. Furthermore, our results show that for an Incontent $\geq$30\% this increase in the built-in field has a significant effecton the emission wavelength and the radiative lifetimes. For instance, at 40\%In, the radiative lifetime is more than doubled when taking second-orderpiezoelectricity into account. Overall our calculations reveal that whendesigning and describing the electronic and optical properties of $c$-planeIn$_{x}$Ga$_{1-x}$N/GaN quantum dot based light emitters with high In contents,second-order piezoelectric effects cannot be neglected.

Impact of second-order piezoelectricity on electronic and optical  properties of $c$-plane In$_{x}$Ga$_{1-x}$N quantum dots: Consequences for  long wavelength emitters