Open access peer-reviewed chapter

Photoionization Cross Section in Low-Dimensional Systems

By Moletlanyi Tshipa and Monkami Masale

Submitted: October 19th 2017Reviewed: February 20th 2018Published: July 25th 2018

DOI: 10.5772/intechopen.75736

Downloaded: 234

Abstract

A theoretical investigation of the effects of the parabolic, shifted parabolic, hill-like, and cup-like parabolic confining electric potentials on photoionization cross section (PCS) in a spherical quantum dot is presented. Each of the parabolic potentials is superimposed on an infinite spherical square quantum well (ISSQW) potential. The parabolic potential blueshifts the peaks of the PCS, while the shifted parabolic potential causes a redshift. As the so-called strength of cup-like parabolic potential is increased, the peak of the PCS becomes redshifted for the s → p transition, but blueshifted for the p → d , d → f (and so forth) transitions. On the contrary, an increase in the strength of the hill-like parabolic potential blueshifts peaks of the PCS for s → p transitions, while it redshifts those of transitions between higher states.

Keywords

  • photoionization cross section
  • confining electric potential
  • spherical quantum dot
  • hydrogenic impurity

1. Introduction

Recent advances in nanofabrication technology have made it possible to fabricate nanostructures of different sizes and geometries [1, 2, 3]. Nanostructures have a wide range of applications including in nanomedicine [4, 5], optoelectronics [6, 7], energy physics [8, 9, 10, 11, 12], and gas sensing [13]. Now, even with utmost care and employing the most advanced techniques, it is not possible to fabricate nanostructures which are free of impurities. It may be advantageous, however, to introduce impurities into a nanostructure at the fabrication stage. The presence of such deliberately introduced impurities can lead to improved performance of nanodevices, for example, enhancement of electrical conductivity of semiconducting materials [14]. The impurity may actually be positively charged, in which case an electron may become bound to it, thus forming an electron-hole pair. Photoionization is one of the useful probes for the particular nature of electron-impurity interactions in low-dimensional systems. In the process of photoionization, upon absorbing sufficiently enough energy from the irradiating electromagnetic field, the electron can break free from the impurity. In a sense, photoionization is the classical analog of the binding energy problem. Certainly, the subtlety in photoionization effects is in the variety of conditions in low-dimensional systems. These conditions include quantization of the electron’s energy levels as well as the optical properties of the specimen.

In this regard, photoionization studies on nanostructures could offer insight into the electron-impurity interaction in a wide variety of conditions. These photoionization effects have fueled significant interest in the processes of photoionization in low-dimensional systems. The effects of geometry and hydrostatic pressure on photoionization cross section (PCS) have been reported in concentric double quantum rings [15]. The effect of applied electric field on photoionization cross section has also been probed in cone-like quantum dots [16]. The role that impurity position plays in modifying the PCS in a core/shell/shell quantum nanolayer [17] and a purely spherical quantum has been investigated [18]. Overall, it has been found that photoionization transitions are independent of the photon polarization for a centered impurity, while the transitions are dependent on the photon polarization when the impurity is off-centered. Influences of intense laser field and hydrostatic on PCS in pyramid-shaped quantum dots have also been reported [19]. There also have been studies of PCS in spherical core/shell zinc blende quantum structures under hydrostatic pressure and electric field [20].

In this chapter, the effect of geometry of confining electric potential on centered donor-related PCS in spherical quantum dots is investigated. The electric potentials considered are the parabolic, shifted parabolic, cup-like, and the hill-like potentials, all of which have a parabolic dependence on the radial distance of the spherical quantum dot. To start with, the Schrödinger equation is solved for the electron’s eigenfunctions and energy eigenvalues within the effective mass approximation. It is emphasized that the treatment of photoionization process given here is limited only to isotropic media.

2. Theory

The basic problem of photoionization involves an electron deemed to be bound to a donor charge or indeed a center of positive charge embedded in a semiconductor specimen. An electron, upon absorbing sufficiently enough energy from the irradiating electromagnetic field, can be “liberated” from the electrostatic field of the positive charge. Now, in low-dimensional systems, the energy of an electron is quantized into different energy levels. The process of photoionization can thus involve intermediate transitions wherein an electron in some initial state iabsorbs a photon of energy ħωand thereby makes a transition to a final state f. It is worth noting that in photoionization calculations, the initial states of the electron are described by wave functions taking into account the presence of the impurity. The final states, however, are described by the wave functions in the absence of the impurity. This notion of taking the initial and final quantum states of the electron, in a sense, is a simulation of calculations of the binding energies in classical mechanics. The energies of the corresponding initial and final states are Eiand Ef, respectively. The system investigated here is a spherical quantum dot (SQD) of refractive index nand relative dielectric constant ε, which may be a GaAs material embedded in a Ga1-xAlxAs matrix, with a donor impurity embedded at its center. Now, one of the physical quantities that are useful in the description of this binding energy-like problem is called photoionization cross section. This quantity may be regarded as the probability that a bound electron can be liberated by some appropriate radiation per unit time per unit area, given by [15, 16, 17, 18, 19, 20]

σlm=σoħωffri2δEfEiħωE1

where friis the interaction integral coupling initial states to final states, αFSis the fine structure constant and ris the electron position vector. Finally, the amplitude of the PCS is given by σo=4π2αFSnEin2/3Eav2εin which Einis the effective incident electric field and Eavthe average electric field inside the quantum dot. Evaluation of the matrix elements for an SQD leads to the selection rules Δl=±1[21], that is, the allowed transitions are only those for which the l values of the final and initial states will be unity. In the investigations carried out here, the evaluations of the PCS are for transitions only between two electron’s energy subbands. For purposes of computation, therefore, the Dirac delta function in Eq. (1) is replaced by its Lorentzian equivalent given by

δEfEiħω=ħΓEfEiħω2+ħΓ2,E2

where this is the so-called Lorentzian linewidth.

Now, in view of spherical symmetry, the solutions of the Schrödinger wave equation are sought in the general form Ψlmrθφ=ClmYlmθφχr, where Clmthe normalization constant, Ylmθφthe spherical harmonics of orbital momentum and magnetic quantum numbers l and m, respectively. The radial part of the total wave function, χρ, is found to be the following linear second-order differential equation

1r2ddrr2rdr+2μħ2Elm+kee2εrVrll+1r2χr=0E3

where μis the effective mass of electron (of charge e) and keis the Coulomb constant.

2.1. The electron’s wave functions

The specific forms of the solutions of the differential equation described above depend on the particular electric confining potential considered. Here, the different radially dependent forms of the so-called intrinsic electric confinement potential of the spherical quantum dot, in turn, taken into account in solving Eq. (3) are (shown in Figure 1) (1) simple parabolic, (2) shifted parabolic, (3) bi-parabolic (cup-like), and (4) inverse bi-parabolic (hill-like), each superimposed on an infinite spherical square quantum well (ISSQW).

Figure 1.

The spatial variation of the confining electric potentials across the SQD: simple parabolic potential (PP), shifted parabolic potential (SPP), cup-like potential (CPP), and the hill-like potential (HPP).

2.1.1. Parabolic potential

When the parabolic potential (PP), which has the form

Vr=12μω02r2,r<RE4

and infinity elsewhere, is inserted into the Schrödinger equation (Eq. (2)) in the presence of the donor impurity, then the second-order differential equation is solvable in terms of the Heun biconfluent function [22, 23].

χρ=C1lmeg1rrlHeunB2l+1αβγg2r+C2lmeg1rrl+1HeunB2l+1αβγg2rE5

with

α=0,β=2Elmħω0,γ=4kee2εħμħω0E6

and the arguments

g1r=μω02ħr2,andg2r=2g1r.E7

Eq. (5) is the complete solution of the differential equation given earlier; however, the second solution diverges at the origin and so C2lmmust be taken as zero. The application of the standard boundary condition of continuity of the wave function at the walls (r=R) of the SQD leads to the following electron’s energy eigenvalue equation:

HeunB2l+1αβEγg2R=0.E8

The electron’s energy spectrum is derived from numerically solving Eq. (8) for its roots βEaccording to

Elm=βE2ħω0.E9

2.1.2. Shifted parabolic potential

This potential is convex: maximum at the center and decreases parabolically to assume a minimum value (here taken as zero) at the radius

Vr=12μωo2rR2,r<RE10

and infinity elsewhere. The solution to the radial component of the Schrödinger equation (Eq. (3)) corresponding to this potential is also in terms of the Heun biconfluent function (Eq. (5)) but with [23]

α=2μω0R2ħ,β=2Elmħω0,γ=4kee2εħμħω0E11

and the arguments

g1r=μω02ħr2Rrandg2r=iμω0ħrE12

The energy spectrum is given by the usual boundary conditions at the walls of the SQD as

Elm=βE2ħω0E13

where βEis the value of βthat satisfies the condition given in Eq. (8).

2.1.3. The bi-parabolic (cup-like) potential

The solution to the Schrödinger equation for the bi-parabolic potential

Vr=12μω02rR/22,E14

and infinity elsewhere, in the presence of the impurity, is in terms of the Heun biconfluent function (Eq. (5)) [24] with

α=iRμω0ħ,β=2Elmħω0,γ=4ikee2εħμħω0E15

and the arguments

g1ρ=μω02ħρRρ,andg2ρ=iμω0ħρ.E16

Requiring that the electron wave function should vanish at the walls of the SQD avails the energy spectrum for an electron in an SQD with an intrinsic bi-parabolic potential as

Elm=βE2ħω0E17

where βEis the value of βthat satisfies the condition stipulated in Eq. (8).

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

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 r=R/2, after which a concave parabolic decrease brings it to a minimum at the walls of the SQD r=R

Vr=12μωo2Rrr2,r<RE18

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)) but with [24]

α=Rμω0iħ,β=μω02R28Elm4iħω0,γ=4ikee2εħμiħω0E19

and the arguments

g1r=μω02iħRrrandg2r=iμω0ħr.E20

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

Elm=18μω02R2E2ħω0E21

with βEbeing the value of βthat satisfies the condition set in Eq. (8).

3. Results and discussions

The parameters used in these calculations are relevant to GaAs quantum dots: effective electronic mass μ=0.067me, mebeing the free electron mass and ε=12.5. The impurity linewidth has been taken such that ħΓ=0.1meV[18, 19]. The spatial variation of the confining electric potentials across the SQD is illustrated in Figure 1, where κ=2/μω02R2. 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=0.5Rbut 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=0.5Rbut diminish the ground-state wave functions near regions where it is maximum. By contrast, the hill-like potential is maximum at r=0.5Rand thus has the opposite effect on the respective electron’s wave functions.

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 ħ ω 0 = 10 meV. The dashed curve represents ground-state electron wave function in an ISSQW ( ħ ω 0 = 0 meV).

Figure 3 depicts the variation of the first-order spand second-order pdtransition 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 ΔEspare always lower than those of second-order 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 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, ΔEspand ΔEpdwould 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 3.

The dependence of the first- s → p and second 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.

Figure 4 shows the sum of the spand pdnormalized photoionization cross sections for an SQD of radius R = 250 Å, where the dashed curve is for an ISSQW (ħω0=0meV) while the solid curve corresponds to the parabolic potential of strength ħω0=5meV superimposed on the ISSQW. Here, as in subsequent figures, the radius of the SQD is greater than R0, thus the sppeak occurs at larger beam energies than the second-order peak. Increasing the strength 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 spand the pdtransitions) need to be isolated and distinct, for research or practical purposes.

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 ħ ω 0 = 5 meV superimposed on an ISSQW (solid plots), for a radius R = 250 Å.

Figure 5 depicts the summed normalized PCS for the spand pdtransitions in an SQD of radius R = 250 Å. The dashed curve is associated with the ISSQW (ħω0=0meV) while the solid plot corresponds to PCS for an SQD with a shifted parabolic potential of the so-called strength such that ħω0=5meV. 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 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 ħ ω 0 = 5 meV superimposed on an ISSQW, for a radius R = 250 Å.

Figure 6 illustrates the normalized spand pdPCSs as functions of the photon energy for an SQD of radius R = 250 Å. The dashed curve is for the purely ISSQW (ħω0=0meV) while the solid plot is for the cup-like potential of strength ħω0=5meV superimposed on the ISSQW. As can be clearly seen from the figure, the cup-like potential redshifts peaks of the spPCS while it blueshifts the peaks of the pdPCS. This potential also blueshifts peaks of PCS of transitions involving higher states (df, fgand so forth).

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 ħ ω 0 = 5 meV superimposed on an ISSQW, for a radius R = 250 Å.

Figure 7 depicts the variation of the normalized spand pdPCSs with the photon energy for an SQD of radius R = 250 Å. Here also, the dashed curve represents the purely ISSQW (ħω0=0meV) while the solid plot is for the hill-like potential of strength ħω0=5meV superimposed on the ISSQW. Increasing the strength of the hill-like potential blueshifts the peaks of the spPCS while it redshifts those of the pdPCS. Although not shown here, the hill-like potential also redshifts peaks of the PCS associated with transitions from higher states (df, fgand so forth).

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 ħ ω 0 = 5 meV superimposed on an ISSQW, for a radius R = 250 Å.

4. Conclusions

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 spPCS 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.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Moletlanyi Tshipa and Monkami Masale (July 25th 2018). Photoionization Cross Section in Low-Dimensional Systems, Heterojunctions and Nanostructures, Vasilios N. Stavrou, IntechOpen, DOI: 10.5772/intechopen.75736. Available from:

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