3. Results and discussions

V rð Þ¼ <sup>1</sup> 2 μω<sup>2</sup>

ffiffiffiffiffiffiffiffi μω<sup>0</sup> ħ

r

α ¼ iR

g1ð Þ¼ <sup>r</sup> μω<sup>0</sup>

2.1.4. The inverse lateral bi-parabolic (hill-like) potential

α ¼ R

function (Eq. (5)) [24] with

110 Heterojunctions and Nanostructures

and the arguments

but with [24]

and the arguments

<sup>0</sup>ð Þ <sup>r</sup> � <sup>R</sup>=<sup>2</sup> <sup>2</sup>

, <sup>γ</sup> ¼ � 4ikee<sup>2</sup> εħ

ffiffiffiffiffiffiffiffi μ ħω<sup>0</sup> r

> ffiffiffiffiffiffiffiffi μω<sup>0</sup> ħ

<sup>2</sup> <sup>ħ</sup>ω<sup>0</sup> (17)

<sup>2</sup> � �, rð Þ <sup>&</sup>lt; <sup>R</sup> (18)

ffiffiffiffiffiffiffiffiffi -μ iħω<sup>0</sup>

<sup>2</sup> <sup>ħ</sup>ω<sup>0</sup> (21)

r: (20)

r

, <sup>γ</sup> ¼ � 4ikee2 εħ

r

ffiffiffiffiffiffiffiffiffiffiffi -iμω<sup>0</sup> ħ

r

and infinity elsewhere, in the presence of the impurity, is in terms of the Heun biconfluent

<sup>2</sup><sup>ħ</sup> ð Þ <sup>r</sup> � <sup>R</sup> <sup>r</sup>, and g2ð Þ¼� <sup>r</sup> <sup>i</sup>

Requiring that the electron wave function should vanish at the walls of the SQD avails the

The hill-like potential has a concave parabolic increase in the radial distance from the center to reach maximum at a radial distance half the radius ð Þ r ¼ R=2 , after which a concave parabolic

<sup>o</sup> Rr � r

and infinity elsewhere. The radial component of the Schrödinger equation for this potential in the presence of the impurity is also solvable in terms of the Heun biconfluent function (Eq. (5))

> <sup>0</sup>R<sup>2</sup> � 8Elm � � 4iħω<sup>0</sup>

2i<sup>ħ</sup> ð Þ <sup>R</sup>-<sup>r</sup> <sup>r</sup> and g2ð Þ¼ <sup>r</sup>

Application of the boundary conditions at the walls of the SQD avails the energy spectrum as

<sup>0</sup>R<sup>2</sup> � <sup>i</sup>β<sup>E</sup>

energy spectrum for an electron in an SQD with an intrinsic bi-parabolic potential as

where β<sup>E</sup> is the value of β that satisfies the condition stipulated in Eq. (8).

decrease brings it to a minimum at the walls of the SQD ð Þ r ¼ R

ffiffiffiffiffiffiffiffi μω<sup>0</sup> iħ

g1ð Þ¼ <sup>r</sup> μω<sup>0</sup>

with β<sup>E</sup> being the value of β that satisfies the condition set in Eq. (8).

, <sup>β</sup> <sup>¼</sup> μω<sup>2</sup>

Elm <sup>¼</sup> <sup>1</sup> 8 μω<sup>2</sup>

r

V rð Þ¼ <sup>1</sup> 2 μω<sup>2</sup>

Elm ¼ � <sup>β</sup><sup>E</sup>

, <sup>β</sup> ¼ � 2Elm ħω<sup>0</sup>

, (14)

r: (16)

(15)

(19)

The parameters used in these calculations are relevant to GaAs quantum dots: effective electronic mass μ ¼ 0:067me, me being the free electron mass and ε ¼ 12:5. The impurity linewidth has been taken such that ħΓ ¼ 0:1 meV [18, 19]. The spatial variation of the confining electric potentials across the SQD is illustrated in Figure 1, where <sup>κ</sup> <sup>¼</sup> <sup>2</sup><sup>=</sup> μω<sup>2</sup> <sup>0</sup>R<sup>2</sup> . Figure 2 displays the effects of these potential geometries on the ground-state radial electron wave functions across an SQD of radius R = 250 Å in the absence of the hydrogenic impurity. The parabolic potential shifts the electron wave functions toward the center of the SQD, while the shifted parabolic potential (SPP) shifts the electron wave functions toward the walls of the SQD. As stated earlier, the cup-like is zero at r ¼ 0:5R but maximum at both the center and at the walls of the SQD. Thus, this potential tends to "concentrate" the electron's wave functions of the excited states to regions near r ¼ 0:5R but diminish the ground-state wave functions near regions where it is maximum. By contrast, the hill-like potential is maximum at r ¼ 0:5R and thus has the opposite effect on the respective electron's wave functions.

Figure 3 depicts the variation of the first-order ð Þ s ! p and second-order ð Þ p ! d transition energies as functions of the strengths of the potentials, viz: the parabolic potential (PP), shifted parabolic potential (SPP), the cup-like potential (CPP), and the hill-like potential (HPP). These are the differences in the energies of states between which an electron is allowed to make transitions within the dipole approximation during photoionization. Now, in the absence of the impurity, the first-order transition energies ΔEsp are always lower than those of secondorder transition ΔEpd, that is, for all values of nano-dot radius. In the presence of the impurity, however, there is some characteristic radius R0 at which the first-order and the second-order

Figure 2. The effect of the different potentials on the ground-state radial electron wave function for an SQD of radius R = 250 Å. The potentials, parabolic (PP), shifted parabolic (SPP), cup-like (CPP), and the hill-like (HPP) all have strength ħω<sup>0</sup> ¼ 10 meV. The dashed curve represents ground-state electron wave function in an ISSQW (ħω<sup>0</sup> ¼ 0 meV).

Figure 3. The dependence of the first- ð Þ s ! p and second ð Þ p ! d -order transition energies on the strengths of the different potentials, viz.: the parabolic potential (PP), shifted parabolic potential (SPP), the cup-like potential (CPP), and the hill-like potential (HPP), for an SQD of radius R = 250 Å.

of the parabolic potential blueshifts the peaks of the PCS, simultaneously moving them apart. This can be beneficial in cases where transitions between different states (e.g., the s ! p and the p ! d transitions) need to be isolated and distinct, for research or practical purposes.

Figure 5. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves) and for an SQD with the shifted parabolic potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW, for a

Figure 4. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves) and for an SQD with the parabolic potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW (solid plots),

Photoionization Cross Section in Low-Dimensional Systems

http://dx.doi.org/10.5772/intechopen.75736

113

for a radius R = 250 Å.

radius R = 250 Å.

transition energies coincide. For the system investigated here, this radius is in the neighborhood of R0 = 171 Å. For SQDs with radii less (greater) than R0, the second-order transition energies are more (less) than the first-order transition energies. The parabolic potential and hill-like potentials reduce the value of this radius as they intensify. On the contrary, increasing the strengths of the shifted parabolic potential and the cup-like potentials increases R0, sending it to infinity as it intensifies further. In this case, ΔEsp and ΔEpd would never coincide and ΔEpd > ΔEsp. The parabolic potential widens the gap between the energies of the initial and final states, regardless of the order of transition. The increase is more pronounced in transitions involving the lower states than in transitions involving the higher states. The shifted parabolic potential decreases transition energies also regardless of the order of transition, and with the reduction being more pronounced for transitions involving the lower states than in those involving the higher states. However, the situation is not so straightforward with the cup-like and the hill-like potentials. The cup-like potential decreases transition energies of only transitions involving the ground (s) state and enhances transition energies involving higher states. The hill-like potential increases only the transition energies involving the ground state but decreases transition energies involving higher states.

Figure 4 shows the sum of the s ! p and p ! d normalized photoionization cross sections for an SQD of radius R = 250 Å, where the dashed curve is for an ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid curve corresponds to the parabolic potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on the ISSQW. Here, as in subsequent figures, the radius of the SQD is greater than R0, thus the s ! p peak occurs at larger beam energies than the second-order peak. Increasing the strength

Figure 4. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves) and for an SQD with the parabolic potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW (solid plots), for a radius R = 250 Å.

of the parabolic potential blueshifts the peaks of the PCS, simultaneously moving them apart. This can be beneficial in cases where transitions between different states (e.g., the s ! p and the p ! d transitions) need to be isolated and distinct, for research or practical purposes.

transition energies coincide. For the system investigated here, this radius is in the neighborhood of R0 = 171 Å. For SQDs with radii less (greater) than R0, the second-order transition energies are more (less) than the first-order transition energies. The parabolic potential and hill-like potentials reduce the value of this radius as they intensify. On the contrary, increasing the strengths of the shifted parabolic potential and the cup-like potentials increases R0, sending it to infinity as it intensifies further. In this case, ΔEsp and ΔEpd would never coincide and ΔEpd > ΔEsp. The parabolic potential widens the gap between the energies of the initial and final states, regardless of the order of transition. The increase is more pronounced in transitions involving the lower states than in transitions involving the higher states. The shifted parabolic potential decreases transition energies also regardless of the order of transition, and with the reduction being more pronounced for transitions involving the lower states than in those involving the higher states. However, the situation is not so straightforward with the cup-like and the hill-like potentials. The cup-like potential decreases transition energies of only transitions involving the ground (s) state and enhances transition energies involving higher states. The hill-like potential increases only the transition energies involving the ground state but

Figure 3. The dependence of the first- ð Þ s ! p and second ð Þ p ! d -order transition energies on the strengths of the different potentials, viz.: the parabolic potential (PP), shifted parabolic potential (SPP), the cup-like potential (CPP), and

Figure 4 shows the sum of the s ! p and p ! d normalized photoionization cross sections for an SQD of radius R = 250 Å, where the dashed curve is for an ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid curve corresponds to the parabolic potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on the ISSQW. Here, as in subsequent figures, the radius of the SQD is greater than R0, thus the s ! p peak occurs at larger beam energies than the second-order peak. Increasing the strength

decreases transition energies involving higher states.

the hill-like potential (HPP), for an SQD of radius R = 250 Å.

112 Heterojunctions and Nanostructures

Figure 5. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves) and for an SQD with the shifted parabolic potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW, for a radius R = 250 Å.

Figure 5 depicts the summed normalized PCS for the s ! p and p ! d transitions in an SQD of radius R = 250 Å. The dashed curve is associated with the ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid plot corresponds to PCS for an SQD with a shifted parabolic potential of the so-called strength such that ħω<sup>0</sup> ¼ 5 meV. Overall, the shifted parabolic potential redshifts the resonance peaks of the PCSs. It is interesting to note, however, that the first-order resonance peak redshifted to a much greater extent than that of the second order. These results suggest that the shifted parabolic potential can be utilized to manipulate the first-order and second-order transitions according to their corresponding photon energy of excitation [23].

Figure 6 illustrates the normalized s ! p and p ! d PCSs as functions of the photon energy for an SQD of radius R = 250 Å. The dashed curve is for the purely ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid plot is for the cup-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on the ISSQW. As can be clearly seen from the figure, the cup-like potential redshifts peaks of the s ! p PCS while it blueshifts the peaks of the p ! d PCS. This potential also blueshifts peaks of PCS of transitions involving higher states (d ! f , f ! g and so forth).

Figure 7 depicts the variation of the normalized s ! p and p ! d PCSs with the photon energy for an SQD of radius R = 250 Å. Here also, the dashed curve represents the purely ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid plot is for the hill-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on the ISSQW. Increasing the strength of the hill-like potential blueshifts the peaks of the s ! p PCS while it redshifts those of the p ! d PCS. Although not shown here, the hill-like potential also redshifts peaks of the PCS associated with transitions from higher states (d ! f , f ! g and so forth).

4. Conclusions

radius R = 250 Å.

Conflict of interest

The electron's wave functions in a spherical quantum dot with a centered donor impurity have been obtained, and these were utilized to evaluate the effects of the geometry of confining electric potentials on PCS in an SQD. The parabolic potential enhances photoionization transition energies independent of the initial or the final state, while the shifted parabolic potential decreases the transition energies, also independent of the order of transition. As a result, the parabolic potential blueshifts the peaks of the PCS, while the shifted parabolic potential redshifts the peaks, for all transitions. The cup-like and the hill-like potentials exhibit a selective enhancement or a reduction of transition energies. The hill-like parabolic potential enhances the transition energies involving the ground state but dwindles those involving higher states. A consequence of this effect is that the hill-like parabolic potential blueshifts peaks of s ! p PCS but redshifts those involving higher states. The situation is the other way around in the case of the cup-like parabolic potential. The results presented here also suggest that nano-patterning techniques may offer yet another method of tuning the process of photo-

Figure 7. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves in both) and for an SQD with the hill-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW, for a

Photoionization Cross Section in Low-Dimensional Systems

http://dx.doi.org/10.5772/intechopen.75736

115

ionization to resonance, through tailored electric potentials.

The authors have no conflict of interest to declare.

Figure 6. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves) and for an SQD with the cup-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW, for a radius R = 250 Å.

Figure 7. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves in both) and for an SQD with the hill-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW, for a radius R = 250 Å.

## 4. Conclusions

Figure 5 depicts the summed normalized PCS for the s ! p and p ! d transitions in an SQD of radius R = 250 Å. The dashed curve is associated with the ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid plot corresponds to PCS for an SQD with a shifted parabolic potential of the so-called strength such that ħω<sup>0</sup> ¼ 5 meV. Overall, the shifted parabolic potential redshifts the resonance peaks of the PCSs. It is interesting to note, however, that the first-order resonance peak redshifted to a much greater extent than that of the second order. These results suggest that the shifted parabolic potential can be utilized to manipulate the first-order and second-order

Figure 6 illustrates the normalized s ! p and p ! d PCSs as functions of the photon energy for an SQD of radius R = 250 Å. The dashed curve is for the purely ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid plot is for the cup-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on the ISSQW. As can be clearly seen from the figure, the cup-like potential redshifts peaks of the s ! p PCS while it blueshifts the peaks of the p ! d PCS. This potential also blueshifts peaks of

Figure 7 depicts the variation of the normalized s ! p and p ! d PCSs with the photon energy for an SQD of radius R = 250 Å. Here also, the dashed curve represents the purely ISSQW (ħω<sup>0</sup> ¼ 0 meV) while the solid plot is for the hill-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on the ISSQW. Increasing the strength of the hill-like potential blueshifts the peaks of the s ! p PCS while it redshifts those of the p ! d PCS. Although not shown here, the hill-like potential also redshifts peaks of the PCS associated with transitions from higher states

Figure 6. The sum of the first- and second-order normalized PCSs as functions of beam energy for the ISSQW (dashed curves) and for an SQD with the cup-like potential of strength ħω<sup>0</sup> ¼ 5 meV superimposed on an ISSQW, for a radius

transitions according to their corresponding photon energy of excitation [23].

PCS of transitions involving higher states (d ! f , f ! g and so forth).

(d ! f , f ! g and so forth).

114 Heterojunctions and Nanostructures

R = 250 Å.

The electron's wave functions in a spherical quantum dot with a centered donor impurity have been obtained, and these were utilized to evaluate the effects of the geometry of confining electric potentials on PCS in an SQD. The parabolic potential enhances photoionization transition energies independent of the initial or the final state, while the shifted parabolic potential decreases the transition energies, also independent of the order of transition. As a result, the parabolic potential blueshifts the peaks of the PCS, while the shifted parabolic potential redshifts the peaks, for all transitions. The cup-like and the hill-like potentials exhibit a selective enhancement or a reduction of transition energies. The hill-like parabolic potential enhances the transition energies involving the ground state but dwindles those involving higher states. A consequence of this effect is that the hill-like parabolic potential blueshifts peaks of s ! p PCS but redshifts those involving higher states. The situation is the other way around in the case of the cup-like parabolic potential. The results presented here also suggest that nano-patterning techniques may offer yet another method of tuning the process of photoionization to resonance, through tailored electric potentials.

#### Conflict of interest

The authors have no conflict of interest to declare.
