DONOR STATES IN A QUANTUM WELL

Abstract

We present calculations of the binding energy of the ground state and some low-lying excited states of a shallow, hydrogenic donor in a quantum well. Because the problem is not analytically tractable, two different approximation methods are used in our studies. Unlike previous studies which have traditionally employed variational methods sometimes involving a large number of linear parameters in the trial function (as many as 13), our first approach is to use a suitable trial function involving a few nonlinear parameters. There is only one paper based on this choice of trial function, which has proved to give results at least comparable to earlier variational methods involving a large set of linear parameters. In the first part of the study, the method of variation is applied to calculate the binding energy of a donor electron in 1s and 2p states in a Ga0.47In0.53As/AI0.48In0.52As quantum well for varying well size and impurity position. Our results show that the donor binding energy decreases with well width. For a given well width, the donor electron has the highest binding energy when the impurity is at the well centre and this decreases when the impurity moves towards the well edge. Previous work employing a large number of variational parameters has centred mainly on donor binding energies and there is relatively little discussion on the donor states. Also with a larger number of parameters, the problem of obtaining accurate estimates of the parameters characterising the donor states becomes increasingly difficult since many different sets of parameters may well give energy minima which are very close in energy. Our results also show that there are effectively only two nonlinear parameters, ? and ?1 characterising the donor state and therefore the trends in the donor binding energy may be correlated with changes in the trial function unlike other trial functions with many parameters. For the ls-like state, the variational parameters, ? and ?1 decrease with well width and they also decrease when the impurity moves from the well center to the well edge, implying greater localisation or the radial wave function about the z-axis. As shown in the study, the trends of these parameters as a function of well width or impurity position are physically consistent with the trends or the binding energy. For the 2p. -like state, ? is found to be less significant than ?1, thus emphasising the similarity of the radial wave function of the donor state with the two-dimensional hydrogenic state. Similar to the 1s state, ?1 decreases with well width and the impurity position as the impurity moves away from the well center. With relatively few parameters, one can reduce the problem of finding accurate estimates of these parameters. Therefore it becomes possible to study the trends affecting ? and ?1 and hence those of the donor wave function in a systematic way, which is not possible with a large set of parameters. Throughout our computations, proper values of the dielectric constants and the effective masses of the electron for the well and barrier media are used. We find that the use of the actual values of the dielectric constants has increased the binding energy significantly. In addition, the effect on the donor state is studied when a uniform magnetic field is applied in the direction of the growth axis of the quantum well. The calculations reveal that for a central donor, the magnetic field increases the binding energy of the donor electron. In the next part of the thesis, a perturbation-variational method is proposed to compute the binding energy of the 1s state. A comparison of results obtained by this method with those from the method of variation shows good quantitative agreement. We also apply this perturbation-variational method to calculate the binding energy of the 2po state in a 1OOA quantum well for varying impurity position. This study reveals a peak in the binding energy when the impurity donor is midway between the well center and well edge. This is not found in the 1s and 2p states. For a central donor in a l00A quantum well the binding energy of the 2po state is lower than the binding energy of the 1s state but much higher than that of the 2p state. Furthermore, a uniform magnetic field applied along the growth axis of the quantum well increases the binding energy of the 2po state. Lastly we have also used the perturbation-variational method to compute the transition energies for Is-2p±, I s-3p, and Is-4p, transitions in a GaAs/Ga0.7Al0.3As quantum well. Our results are found to agree well with experiment.