Effect of a spatially dependent effective mass on the hydrogenic impurity binding energy in a finite parabolic quantum well

The effect of a spatially dependent effective mass in a finite ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ parabolic quantum well on hydrogenic impurity ground state $\mathrm{}\left(1s\right)\mathrm{}$ binding energies and transition energies between a $1s$ state and low-lying excited state (${2p}_{\pm }-\mathrm{l}\mathrm{i}\mathrm{k}\mathrm{e})$ has been calculated, respectively, as a function of well width and impurity position by using the one-dimensional method. Our results are compared with Niculescu’s results of constant effective mass. We find that the $1s$ state binding energies and the $1\overrightarrow{s}{2p}_{\pm }$ transition energies are greater than Niculescu results for the same well width, respectively. These results are obtained as the impurity is located at the well center. At the same time the well widths corresponding to maximum values of the $1s$ state binding energies and the $1\overrightarrow{s}{2p}_{\pm }$ transition energies are less than those of Niculescu, respectively. The physical meaning of $1/\left(1-\lambda \right)$ is discussed.