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