Quantum Rényi entropy of hydrogenic impurity states in the GaAsxP1-x semiconductor quantum dot

Study of quantum Rényi entropy of hydrogenic impurity states in quantum dots is a fascinating research area in semiconductor physics and information theory. In this work, we investigate the quantum Rényi entropy of hydrogenic impurity states in the GaAs_xP_1-x semiconductor quantum dot. By calculating the Rényi entropy in both the position and momentum space ($R_r^{\alpha }$ and $R_p^{\beta }$), we uncover some novel phenomenon. For a given quantum state, both $R_r^{\alpha }$ and $R_p^{\beta }$ change monotonically with the quantum dot radius, however, the Rényi entropy sum ($R_t^{\left(\alpha ,\beta \right)}$) shows extreme points in its variation with the radius. This behavior is related to the localization-delocalization transitions of the hydrogenic impurity states. Additionally, for different order of $\alpha$ and $\beta$, the variations of $R_t^{\left(\alpha ,\beta \right)}$ with the quantum dot radius follow a similar pattern for the same quantum state. Furthermore, the effect of the As-doping on the Rényi entropy of this system is also discussed. It is interesting to find though $R_r^{\alpha }$ and $R_p^{\beta }$ changes with the As-doping content, their sum ($R_t^{\left(\alpha ,\beta \right)}$) exhibits the property of translation invariant, with only the positions of the extreme points shifting toward higher As-doping content. Since the Rényi entropy offers valuable insights into the complexity of quantum states, this allows us to explore the localization-delocalization characteristics of the hydrogenic impurity states by changing the content of the doping element in the semiconductor quantum dots. This study provides a rich framework for understanding the quantum properties of semiconductor material, and has potential applications for advancements in quantum computing and optoelectronic devices.