In this section, we present the outcome of our calculations for the impurity energy levels and the associated optical properties of interest. For the sake of illustration, the figures containing the results on the energies also show - as insets - the corresponding 2s-1s energy differences, which associate with the resonant peak positions of the absorption coefficients. Then, the discussion regarding the results in Figures
will make use of those contained in the insets of Figures
Figure 6 Ground and first excited states. Ground and first excited state energies of the electron (colored lines) as functions of the (a) inner and (b) outer radii of the QR for L=20 nm, P=0, and F=0. The impurity is placed at zi=0 and ρ (more ...)
Our results for the ground and first excited state energies of a donor impurity in a GaAs QR are shown in Figure
a,b as functions of the inner and outer radii, respectively. For a fixed value of the outer radius (Figure
a), the layer thickness W
decreases as long as the inner radius is augmented with the consequent increment in the size quantization. As a result of this, there will be an increase of both the ground and first excited state energies. Moreover, a growth in the value of the ring’s outer radius is reflected in the decrease of the energies because of the weakening of the size quantization effect (see Figure
b). As can be seen from Figure
a,b, the influence of size quantization is much stronger for the excited states. This fact was expected because the excited state is more spread out inside the ring region than the ground state.
The results regarding the effect of hydrostatic pressure on the 1s and 2s state energies of a donor impurity in a GaAs QR can be found in Figure
. It is clear that in all cases, the influence of the hydrostatic pressure has the effect of reducing the considered state energies. There are several factors which are responsible for such a behavior, namely, that as long as there is an increment in the hydrostatic pressure, the following happens: (1) the GaAs dielectric constant diminishes, (2) the electron effective mass increases, and (3) the dimensions of the structure decrease. With the increase of the effective mass, both states go down in energies. On the other hand, the reduction of the dielectric constant is related with the reinforcement of the Coulombic interaction and the diminishing of the energies. The reduction of the effective dimensions of the structure will result in a shortening of the effective electron-impurity distance, with the consequence of a decrease in the energies of both states. As it is seen from the figures, the influence of the hydrostatic pressure does not modify the overall phenomenology associated with the energy curves of the donor impurity states.
The effect of an applied electric field on the ground and first excited state energies of the on-ring-center impurity is presented in Figure
. From the figure, it can be noticed that with the increase of the electric field strength, the energies of the ground and first excited states become reduced. This fact can be explained by the following fact: with the increase of the electric field, the electron cloud is shifted far from the impurity (along −F), with a weakening of the electron localization. For this reason, there is a decrease in the energies of both states. It is also apparent that the influence of the electric field on the excited states is greater than on the ground state. This is because the electron ground state is more strongly localized.
, the results for the ground and excited state energies of an on-ring-center donor impurity in a GaAs single QR as a function of its height are depicted. With the increase of the QR’s height, the size quantization weakens which has, as a consequence, the reduction of the energy in each case.
, the calculated intraband matrix elements for several configurations of the dimensions of the structure, applied electric field, and hydrostatic pressure are reported.
Calculated intraband matrix elements
The linear, nonlinear, and total absorption coefficients for the GaAs-based QR are shown in the Figure
as functions of the energy of the incident photon for several values of the ring’s inner radius R1
. The results are for the impurity placed on the QR center. As can be seen from such figure, for the value R1
6 nm, the resonant peak of the absorption is displaced to the region of small photon energies. That is, there is a redshift of the resonant peaks of the intraband optical spectrum. For the value of the inner radius R1
12 nm, the resonant peak of the absorption coefficient is shifted back to the bigger values of the photon energy. In other words, there appears a blueshift of the resonant peaks of the intraband optical spectrum. This phenomenon is due to the fact that the difference of energies between the 1s
states is larger for R1
12 nm than for the case of R1
6 nm (see Figure
a). It should be noticed that, with the decrease of the difference of energies between the 1s
states, the dipole matrix element is larger, and for this reason, the maximum value of the linear and nonlinear absorption coefficients will have an increase.
The effect of the change in the value of the outer ring radius is presented in the Figure
. There, the variations of the linear, nonlinear, and total absorption coefficients are given as functions of the incident photon energy, with R2 as a parameter. In the calculations, the impurity is once more considered to be located at the QR center. From Figure
, it is clear that with the increase of R2, the resonant peak of the absorption spectrum will become shifted to smaller values of the incident photon energy (redshift). It is also seen that there is a growth of the maximum resonant peak value. In this case, within the whole range of increase of the outer ring, the differences between the 1s and 2s state energies are progressively reduced (see Figure
b), and for this reason, there is a redshift. Given the drop in the value of this energy difference, larger values of the dipole matrix element are obtained. So, the maximum values of the linear and nonlinear absorption resonant peaks are bigger.
The effect of the hydrostatic pressure on the linear, nonlinear, and total absorption coefficients is depicted in Figure
. It is clear that there is an appearance of a hydrostatic pressure-induced blueshift of the resonant peaks of the intraband optical spectrum. This is caused by the increment in the energy difference between the two involved states as a consequence of the increase in the pressure values (see Figure
). Again, it should be noticed that in this case, the maximum values of the amplitudes of the linear and nonlinear absorption coefficients decrease, which is caused by the decrease of the dipole matrix elements calculated between the mentioned states.
, the linear, nonlinear, and total absorption coefficients are shown as functions of the energy of the incident photon for several values of the electric field strength and height of the QR, respectively. In both cases, with the increase of electric field strength and height of QR, the localization of the electron is weakened, and as can be seen from Figures
, the energy distance between the ground and first excited states decreases. Using this fact, the redshift in the intraband absorption spectrum can be explained straightforwardly.