The ability to engineer nonlinear optical processes in all-dielectric nanostructures is both of fundamental interest and highly desirable for high-performance, robust, and miniaturized nonlinear optical devices. Herein, we propose a novel paradigm for the efficient tuning of second-harmonic generation (SHG) process in dielectric nanoantennas by integrating with chalcogenide phase change material. In a design with Ge$_{2}$Sb$_{2}$Te$_{5}$ (GST) film sandwiched between the AlGaAs nanoantennas and AlO$_{x}$ substrate, the nonlinear SHG signal from the AlGaAs nanoantennas can be boosted via the resonantly localized field induced by the optically-induced Mie-type resonances, and further modulated by exploiting the GST amorphous-to-crystalline phase change in a non-volatile, multi-level manner. The tuning strategy originates from the modulation of resonant conditions by changes in the refractive index of GST. With a thorough examination of tuning performances for different nanoantenna radii, a maximum modulation depth as high as 540$%$ is numerically demonstrated. This work not only reveals out the potential of GST in optical nonlinearity control, but also provides promising strategy in smart designing tunable and reconfigurable nonlinear optical devices, e.g., light emitters, modulators, and sensors.
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