Cc. Yang et al., EIGENSTATES AND FINE-STRUCTURE OF A HYDROGENIC IMPURITY IN A SPHERICAL QUANTUM-DOT, Physical review. B, Condensed matter, 58(4), 1998, pp. 1954-1961
The fine structure of the energy levels for a hydrogenic impurity loca
ted in the center of a spherical quantum dot is calculated using a sim
pler exact solution for the potential well. The results reveal that wh
en the dot radius approaches zero, the eigenenergies are just like a f
ree-space hydrogenic atom. When the doe radius is large enough, then t
he eigenenergies approach a free-space hydrogenic atom but are shifted
by the confining potential. Also we find that the radial expectation
values will be equal to a free-space hydrogenic atom, when the dot rad
ius is extremely small and extremely large. Between these two situatio
ns, the radial expectation values are smaller than those of a free spa
ce because of the pressing of the confining potential. Not every dot r
adius influences the eigenenergy to the same degree. It is decided by
the bumps of the electron's wave function and the place of the potenti
al well's margin. When the margin of the well begins to push the bumps
of the wave then the eigenenergy will increase more quickly. Because
of the changing of the electron distribution probability, the degenera
cy of the different I value in a free-space hydrogenic atom is removed
by the confining potential. The total-energy shifts of the fine struc
ture of the impurity could be six times larger than the total energy s
hifts of a free-space atom.