Electrostatic built-in fields in wurtzite III-N nanostructures: Impact of growth plane on second-order piezoelectricity

Abstract

In this work we present a detailed analysis of the second-order piezoelectric effect in wurtzite III-N heterostructures, such as quantum wells and quantum dots, grown on different substrate orientations. Our analysis is based on a continuum model using a here derived analytic expression for the second-order piezoelectric polarization vector field as a function of the incline angle $\ensuremath{\theta}$ to the wurtzite $c$ axis. This expression allows for a straightforward implementation in existing quantum well and quantum dot codes. Our calculations on III-N quantum well systems reveal that especially for semipolar structures with high incline angle values (${55}^{\ensuremath{\circ}}\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}{80}^{\ensuremath{\circ}}$ and ${105}^{\ensuremath{\circ}}\phantom{\rule{0.16em}{0ex}}\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}\ensuremath{\theta}\phantom{\rule{0.16em}{0ex}}\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}{120}^{\ensuremath{\circ}}$), second-order piezoelectricity noticeably contributes to the overall electric built-in field. For instance, in an InGaN/GaN multiple quantum well system with 22% In, the electric field increases by approximately 20% due to second-order piezoelectricity. Overall, when including second-order piezoelectric effects in the calculation of electric fields in GaN/AlN and InGaN/GaN quantum well systems an improved agreement between our theory and experimental literature data is observed. When studying quantum dots, at least for the here considered model geometry and growth planes, we observe that for GaN/AlN structures second-order effects are of secondary importance. The situation is different for non-$c$-plane ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{N}$/GaN quantum dots. For example, inside a nonpolar ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{N}$/GaN dot the built-in potential arising from second-order piezoelectricity is comparable in magnitude to the built-in potential originating from spontaneous and first-order piezoelectric polarization, but opposite in sign. This feature leads to a change in the built-in potential profile both in and around the ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{N}$/GaN quantum dot structure, which in general is relevant for electronic and optical properties of these systems.