Dependence of exciton energy on dot size in $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}$ quantum dots

Rapid Communication

Dependence of exciton energy on dot size in $Ga N ∕ Al N$ quantum dots

D. P. Williams, A. D. Andreev, and E. P. O’Reilly

Phys. Rev. B 73, 241301(R) – Published 6 June 2006

Abstract

Using previously derived analytical expressions for the polarization field in nitride quantum dots (QDs), we show that the potential in the growth direction can be approximated as linear in such dots, and that the slope of this linear potential depends only on the aspect ratio (height/radius) of the dot. We demonstrate how the large polarization field leads to a linear dependence of the exciton energy on dot size, provided the aspect ratio of the dot is conserved while the size is varied. We also present a useful analytical approximation for the electron and hole wave functions in nitride QDs in terms of Airy functions, which compares well with the solutions of a numerical computation. We note that the disagreement concerning the sign of the shear piezoelectric coefficient $e_{15}$ leads to a significant uncertainty in the calculated potential.