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 paradigm for the efficient tuning of a second-harmonic generation (SHG) process in dielectric nanoantennas by integrating with chalcogenide phase-change material. In a design with ${\mathrm{Ge}}_{2}{\mathrm{Sb}}_{2}{\mathrm{Te}}_{5}$ (GST) film sandwiched between the AlGaAs nanoantennas and ${\mathrm{AlO}}_{x}$ 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 nonvolatile, multilevel 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 the potential of GST in optical nonlinearity control, but also provides a promising strategy in smart designing of tunable and reconfigurable nonlinear optical devices, e.g., light emitters, modulators, and sensors.
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