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 ∣ii absorbs a photon of energy ħω and thereby makes a transition to a final state ∣fi. 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 Ei and Ef , respectively. The system investigated here is a spherical quantum dot (SQD) of refractive index n and 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 [15–20]

$$
\sigma\_{\rm lm} = \sigma\_o \hbar \omega \sum\_f \left| \left< f \right| \begin{array}{c} \overrightarrow{r} \end{array} \left| \dot{\mathbf{i}} \right> \right|^2 \delta \left( E\_f - E\_i - \hbar \omega \right) \tag{1}
$$

where f j r ! <sup>j</sup><sup>i</sup> �D E � � � � � is the interaction integral coupling initial states to final states, <sup>α</sup>FS is the fine structure constant and r ! is the electron position vector. Finally, the amplitude of the PCS is given by <sup>σ</sup><sup>o</sup> <sup>¼</sup> <sup>4</sup>π<sup>2</sup>αFSnE2 in<sup>=</sup> 3E<sup>2</sup> av<sup>ε</sup> � � in which Ein is the effective incident electric field and Eav the average electric field inside the quantum dot. Evaluation of the matrix elements for an SQD leads to the selection rules Δl ¼ �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

$$\delta \left( \mathbf{E}\_f - \mathbf{E}\_i - \hbar \omega \right) = \frac{\hbar \Gamma}{\left( \mathbf{E}\_f - \mathbf{E}\_i - \hbar \omega \right)^2 + \left( \hbar \Gamma \right)^2},\tag{2}$$

where this is the so-called Lorentzian linewidth.

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 electronimpurity 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 offcentered. 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].

is limited only to isotropic media.

106 Heterojunctions and Nanostructures

2. Theory

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

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 ∣ii absorbs a photon of energy ħω and thereby makes a transition to a final state ∣fi. 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 Now, in view of spherical symmetry, the solutions of the Schrödinger wave equation are sought in the general form Ψlmð Þ¼ r; θ;φ ClmYlmð Þ θ;φ χð Þr , where Clm the normalization constant, Ylmð Þ θ;φ the spherical harmonics of orbital momentum and magnetic quantum numbers l and m, respectively. The radial part of the total wave function, χð Þ r , is found to be the following linear second-order differential equation

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d\chi(r)}{dr}\right) + \left\{\frac{2\mu}{\hbar^2}\left[E\_{lm} + \frac{k\_e e^2}{\varepsilon r} - V(r)\right] - \frac{l(l+1)}{r^2}\right\}\chi(r) = 0\tag{3}$$

where μ is the effective mass of electron (of charge -e) and ke is 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

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).

parabolic, (3) bi-parabolic (cup-like), and (4) inverse bi-parabolic (hill-like), each superimposed on an infinite spherical square quantum well (ISSQW).

#### 2.1.1. Parabolic potential

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

$$V(r) = \frac{1}{2}\mu\omega\_0^2r^2, \qquad (r < R) \tag{4}$$

g1ð Þ¼ <sup>r</sup> μω<sup>0</sup>

SQD leads to the following electron's energy eigenvalue equation:

V rð Þ¼ <sup>1</sup> 2 μω<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � μω0R2 ħ

according to

2.1.2. Shifted parabolic potential

(Eq. (5)) but with [23]

and the arguments

minimum value (here taken as zero) at the radius

α ¼ 2

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

s

g1ð Þ¼ <sup>r</sup> μω<sup>0</sup>

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

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

<sup>2</sup><sup>ħ</sup> <sup>r</sup> 2 , and g2ð Þ¼ r

Eq. (5) is the complete solution of the differential equation given earlier; however, the second solution diverges at the origin and so C2lm must be taken as zero. The application of the standard boundary condition of continuity of the wave function at the walls (r ¼ R) of the

The electron's energy spectrum is derived from numerically solving Eq. (8) for its roots β<sup>E</sup>

This potential is convex: maximum at the center and decreases parabolically to assume a

<sup>o</sup> ð Þ <sup>r</sup> � <sup>R</sup> <sup>2</sup>

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

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

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

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

Elm ¼ � <sup>β</sup><sup>E</sup>

, <sup>γ</sup> <sup>¼</sup> 4kee<sup>2</sup> εħ

Elm ¼ � <sup>β</sup><sup>E</sup>

ffiffiffiffiffiffiffiffiffiffiffiffi 2g1ð Þr q

HeunB 2l <sup>þ</sup> <sup>1</sup>; <sup>α</sup>; <sup>β</sup>E; <sup>γ</sup>; g2ð Þ <sup>R</sup> � � <sup>¼</sup> <sup>0</sup>: (8)

<sup>2</sup> <sup>ħ</sup>ω0: (9)

Photoionization Cross Section in Low-Dimensional Systems

, rð Þ < R (10)

(11)

r (12)

ffiffiffiffiffiffiffiffiffiffiffiffi � <sup>μ</sup> ħω<sup>0</sup>

r

ffiffiffiffiffiffiffiffi μω<sup>0</sup> ħ

<sup>2</sup> <sup>ħ</sup>ω<sup>0</sup> (13)

r

: (7)

109

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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].

$$\chi(\rho) = \mathbb{C}\_{1\text{lm}} \mathfrak{s}^{\mathfrak{s}\_1(r)} r^l \text{HeunB}\{2l+1, a, \beta, \gamma, \mathfrak{g}\_2(r)\} + \mathbb{C}\_{2\text{lm}} \mathfrak{e}^{\mathfrak{g}\_1(r)} r^{\tau(l+1)} \text{HeunB}\{- (2l+1), a, \beta, \gamma, \mathfrak{g}\_2(r)\} \tag{5}$$

with

$$
\alpha = 0, \quad \beta = -\frac{2E\_{lm}}{\hbar \omega\_0}, \gamma = \frac{4k\_c e^2}{\varepsilon \hbar} \sqrt{-\frac{\mu}{\hbar \omega\_0}} \tag{6}
$$

and the arguments

Photoionization Cross Section in Low-Dimensional Systems http://dx.doi.org/10.5772/intechopen.75736 109

$$\mathcal{g}\_1(r) = \frac{\mu \omega\_0}{2\hbar} r^2, \text{and } \mathcal{g}\_2(r) = \sqrt{2\mathcal{g}\_1(r)}. \tag{7}$$

Eq. (5) is the complete solution of the differential equation given earlier; however, the second solution diverges at the origin and so C2lm must be taken as zero. The application of the standard boundary condition of continuity of the wave function at the walls (r ¼ R) of the SQD leads to the following electron's energy eigenvalue equation:

$$\text{Heun}B\left(2l+1, \alpha, \beta\_E, \gamma, \mathbf{g}\_2(\mathbb{R})\right) = 0.\tag{8}$$

The electron's energy spectrum is derived from numerically solving Eq. (8) for its roots β<sup>E</sup> according to

$$E\_{\rm lm} = -\frac{\beta\_E}{2}\hbar\omega\_0.\tag{9}$$

#### 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

$$V(r) = \frac{1}{2}\mu\omega\_o^2 (r - R)^2, \qquad (r < R) \tag{10}$$

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]

$$\alpha = 2\sqrt{-\frac{\mu\omega\_0 R^2}{\hbar}}, \quad \beta = -\frac{2E\_{\text{lm}}}{\hbar\omega\_0}, \gamma = \frac{4k\_e e^2}{\varepsilon\hbar}\sqrt{-\frac{\mu}{\hbar\omega\_0}}\tag{11}$$

and the arguments

parabolic, (3) bi-parabolic (cup-like), and (4) inverse bi-parabolic (hill-like), each superimposed

Figure 1. The spatial variation of the confining electric potentials across the SQD: simple parabolic potential (PP), shifted

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

g1ð Þ<sup>r</sup> r

ffiffiffiffiffiffiffiffiffiffiffiffi � <sup>μ</sup> ħω<sup>0</sup>

r

, <sup>γ</sup> <sup>¼</sup> 4kee<sup>2</sup> εħ

, rð Þ < R (4)


(5)

(6)

on an infinite spherical square quantum well (ISSQW).

parabolic potential (SPP), cup-like potential (CPP), and the hill-like potential (HPP).

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

V rð Þ¼ <sup>1</sup> 2 μω<sup>2</sup> 0r 2

HeunB 2l <sup>þ</sup> <sup>1</sup>; <sup>α</sup>; <sup>β</sup>; <sup>γ</sup>; g2ð Þ<sup>r</sup> � � <sup>þ</sup> <sup>C</sup>2lme

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

ħω<sup>0</sup>

2.1.1. Parabolic potential

108 Heterojunctions and Nanostructures

biconfluent function [22, 23].

g1ð Þ<sup>r</sup> r l

χð Þ¼ r C1lme

and the arguments

with

$$g\_1(r) = \frac{\mu \omega\_0}{2\hbar}(r - 2\mathcal{R})r \text{ and } g\_2(r) = -i\sqrt{\frac{\mu \omega\_0}{\hbar}}r\tag{12}$$

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

$$E\_{lm} = -\frac{\beta\_E}{2}\hbar\omega\_0\tag{13}$$

where β<sup>E</sup> is the value of β that 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

$$V(r) = \frac{1}{2}\mu\omega\_0^2(r - R/2)^2,\tag{14}$$

3. Results and discussions

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

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

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).

<sup>0</sup>R<sup>2</sup> . Figure 2 displays

Photoionization Cross Section in Low-Dimensional Systems

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111

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>

thus has the opposite effect on the respective electron's wave functions.

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

$$\alpha = \text{iR}\sqrt{\frac{\mu\omega\_0}{\hbar}}, \beta = -\frac{2E\_{\text{lm}}}{\hbar\omega\_0}, \gamma = -\frac{4\text{ik}\_\epsilon e^2}{\varepsilon\hbar}\sqrt{\frac{\mu}{\hbar\omega\_0}}\tag{15}$$

and the arguments

$$\mathcal{g}\_1(\rho) = \frac{\mu \omega\_0}{2\hbar} (\rho - R)\rho, \text{and } \mathcal{g}\_2(\rho) = -i\sqrt{\frac{\mu \omega\_0}{\hbar}}\rho. \tag{16}$$

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

$$E\_{lm} = -\frac{\beta\_E}{2}\hbar\omega\_0\tag{17}$$

where β<sup>E</sup> is the value of β that 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 ð Þ r ¼ R=2 , after which a concave parabolic decrease brings it to a minimum at the walls of the SQD ð Þ r ¼ R

$$V(r) = \frac{1}{2}\mu\omega\_o^2(\mathcal{R}r - r^2), \qquad (r < \mathcal{R}) \tag{18}$$

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]

$$\alpha = \mathcal{R}\sqrt{\frac{\mu\omega\_0}{i\hbar}}, \beta = \frac{\left(\mu\omega\_0^2 \mathcal{R}^2 - 8E\_{lm}\right)}{4i\hbar\omega\_0}, \gamma = -\frac{4i\mathcal{k}\_\epsilon\mathcal{c}^2}{\varepsilon\hbar}\sqrt{\frac{\cdot\mu}{i\hbar\omega\_0}}\tag{19}$$

and the arguments

$$\mathcal{g}\_1(r) = \frac{\mu \omega\_0}{2i\hbar} (\text{R-}r) \text{r and } \mathcal{g}\_2(r) = \sqrt{\frac{\text{-} i\mu \omega\_0}{\hbar}} r. \tag{20}$$

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

$$E\_{lm} = \frac{1}{8} \mu \omega\_0^2 \mathcal{R}^2 - \frac{i\beta\_E}{2} \hbar \omega\_0 \tag{21}$$

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