  In this work we investigate electron-impurity binding energy in GaN/HfO$_2$
quantum wells. The calculation considers simultaneously all energy
contributions caused by the dielectric mismatch: (i) image self-energy (i.e.,
interaction between electron and its image charge), (ii) the direct Coulomb
interaction between the electron-impurity and (iii) the interactions among
electron and impurity image charges. The theoretical model account for the
solution of the time-dependent Schr\"odinger equation and the results shows how
the magnitude of the electron-impurity binding energy depends on the position
of impurity in the well-barrier system. The role of the large dielectric
constant in the barrier region is exposed with the comparison of the results
for GaN/HfO$_2$ with those of a more typical GaN/AlN system, for two different
confinement regimes: narrow and wide quantum wells.
