[en] To improve the light extraction efficiency of a GaN-based light-emitting diode (LED), a new complex periodic photonic crystal structure containing two kinds of hemispheres was designed. This complex photonic crystal structure is arrayed on the surface of the planar LED, and a finite-difference time-domain method was used to optimize the hemisphere materials (ZnO, GaN, and SiC) and the structural parameters of the complex photonic crystal (the size of the hemisphere radius and the lattice constant of complex structure). A high improvement of light extraction efficiency is achieved in complex photonic crystal LED, translated into an 8.3-fold enhancement for the case of a planar LED. (paper)