Effect of magnetic field on the binding energy of a hydrogenic donor in a |z|^2/3 quantum well

We have calculated the binding energy of a hydrogenic donor in a quantum well with potential shape proportional to |z|^2/3 as a function of the width of the quantum well and the barrier height under an applied uniform magnetic field along the z axis. As the well width decreases, the binding energy increases initially up to a critical well width (which is nearly the same for all magnetic fields) at which there is a turnover. The results are qualitatively similar to those of a hydrogenic donor in a rectangular well. We have also calculated ⟨p²⟩^1/2 and ⟨z²⟩^1/2 for the donor electron. ⟨p²⟩^1/2 is found to be strongly dependent on the magnetic field for a given well width and weakly dependent on the well width and the barrier height, for a given value of magnetic field. ⟨z²⟩^1/2 is weakly dependent on the applied magnetic field. The probability of finding the donor electron inside the well shows a rapid decrease as the well width is reduced at nearly the well width at which the binding energy shows a maximum.