GaAs Quantum Dot Confined with a Woods–Saxon Potential: Role of Structural Parameters on Binding Energy and Optical Absorption

Abstract

We present the first detailed study of optical absorption coefficients (OACs) in a GaAs quantum dot confined with a Woods–Saxon potential containing a hydrogenic impurity at its center. We use a finite difference method to solve the Schrödinger equation within the framework of the effective mass approximation. First, we compute energy levels and probability densities for different parameters governing the confining potential. We then calculate dipole matrix elements and energy differences, $E_{1p}-E_{1s}$, and discuss their role with respect to the OACs. Our findings demonstrate the important role of these parameters in tuning the OAC to enable blue or red shifts and alter its amplitude. Our simulations provide a guided path to fabricating new optoelectronic devices by adjusting the confining potential shape.

1. Introduction

The tunability of energy levels in low dimensional systems such as quantum wells (QWs), quantum wires (QWRs), and quantum dots (QDs) enable a multitude of optoelectronic devices, such as quantum cascade lasers, optical modulators, optical switches, and infrared photodetectors [1,2,3,4]. In addition, QDs are used in the creation of universal memory elements due to their spatial distribution of free carriers that are confined in three dimensions [5,6,7,8]. Generally, the position of different energy levels is determined via the geometrical shape of the confining potential of the quantum structure, such as square, parabolic, semi-parabolic, Gaussian, Razavy, Konwent, and Manning shapes [9,10,11,12,13,14]. QDs are of particular interest in optical applications due to their luminescence, potential to emit different frequencies with intense efficacies, high extinction, and prolonged lifetimes [15,16,17]. For these reasons, QDs are used in other technological applications such as light-emitting diodes (LEDs), electronic transistors, medical laser imaging, biosensors, quantum cascade lasers, and quantum computing architectures [18,19,20,21,22,23,24,25].

QDs generally show larger energetic separations between different levels compared to QWs and QWRs due to the three-dimensional confinement of carriers. They also give more intense density of states (DOS) than other quantum systems, which enables them to be used in amplifier applications. In addition to the geometry and shape of the confining potential, the incorporation of a hydrogenic impurity in QDs can modulate electronic and optical absorption coefficients (OACs) due to the electrostatic attraction between the free electrons and the impurity [26,27,28,29,30,31,32]. Previous work on OACs in QDs has focused on both theoretical and experimental studies [33,34,35,36]. For instance, Schrey et al. studied the optical absorption of quantum dots in photodetectors and analyzed the effect of QD size on their minibands [37]. Bahar et al. calculated OACs of a QD with a hydrogenic impurity in a Mathieu potential and found that the OAC and refractive index were affected by hydrostatic pressure and temperature variations [38]. Batra and coauthors examined structural parameters and the optical response of a QD with a tunable Kratzer confining potential [39]. Bassani et al. treated the effects of donor and acceptor impurities on OACs in a spherical QD [40]. The process of intraband and interband absorption in an InGaAs/GaAs QD was studied by Narvaez et al. [41], and the effects of size and distance separating QDs were evaluated by Stoleru et al. [42] The oscillator strengths between lower energy state transitions in a spherical QD with a hydrogenic impurity were calculated by Yilmaz et al. [43]. Kirak and coauthors evaluated the effect of an applied electric field on the OAC in a spherical QD with a parabolic potential under the influence of a hydrogenic impurity [44]. Fakkahi et al. studied OACs and oscillator strengths in multilayer spherical QDs under the influence of a radial electric field and hydrogenic impurities. Other works on OACs in multiple spherical QDs are also discussed in references [45,46,47,48].

Motivated by these studies, we investigate the electronic and optical properties of electrons confined in a GaAs quantum dot with a radial confinement described by the Woods–Saxon confining potential. The functional form of this potential was first proposed to describe and interpret interactive forces in the nuclear shell model [49]. Furthermore, this confining potential describes a smooth interface structure and gives an accurate description of aluminum diffusion from the AlGaAs barrier towards the GaAs quantum well. Our study commences with a calculation of the 1s and 1p energy levels and their probability densities as a function of structural parameters in the Woods–Saxon potential. We then analyze the dipole matrix elements (DMEs) and OACs as the parameters of the Woods–Saxon potential are varied in the presence of a hydrogenic impurity. Further details and approximations of our theoretical model are given in Section 2. In Section 3, our findings and the resulting physical observables are discussed. Finally, Section 4 summarizes our results.

2. Theoretical Modeling

2.1. Woods–Saxon Potential Form

We begin this section by discussing the confining potential and its structural parameters. These parameters alter the geometrical form of the potential and affect the position of different energy levels. When the confining potential is spherically symmetric, the carrier’s motion is quantized and described by angular and magnetic quantum numbers, with the associated wave functions being expressed as a function of the well-known spherical harmonics. The form of the radial electronic wavefunctions is mainly determined by the geometrical shape of the confining potential. Since the energy separation,$E_{1p}-E_{1s}$, between the initial and final states, plays a major role in the OAC expression, we examine its dependence on QD size, the structural parameters of the confining potential, or both.

We first examine the radial Woods–Saxon potential, which is given by [45]

$$V_{\mathrm{w}\mathrm{s}}\left(r\right)=\frac{V_0}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[\right(R_0-r)/\gamma ]}+\frac{V_0}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[\right(R_0+r)/\gamma ]}.$$

(1)

$V_0$ is the height of the Woods–Saxon potential and $R_0=R/2$, where $R$ denotes the QD radius, and $\gamma$ is a parameter characterizing the slope between the well and barrier regions.

Figure 1 depicts a schematic of the quantum dot, which consists of a GaAs core with radius $R=25 \mathrm{n}\mathrm{m}$ surrounded by an AlGaAs barrier. This latter has an external radius of $R_{\mathrm{e}\mathrm{x}\mathrm{t}}=2R.$

Before studying the optical properties of our structure, we plot the geometrical dependence of the Woods–Saxon potential on the parameter $\gamma$ in Figure 2a–d. The radius of the QD is $R=25 \mathrm{n}\mathrm{m}$. For $\gamma =5 \mathrm{\AA },$ the Woods–Saxon potential resembles a square quantum well since it takes a flat form between 0 and $5 \mathrm{\AA }$. However, when $\gamma$ increases, the bottom of the potential becomes more parabolic. Furthermore, the top of the well becomes more curved as $\gamma$ increases. For instance, the potential reaches $1500 \mathrm{m}\mathrm{e}\mathrm{V}$ at $r=10 \mathrm{n}\mathrm{m}$ for $\gamma =5 \mathrm{\AA }$ (Figure 2a); however, it reaches this value at $r=15 \mathrm{n}\mathrm{m}$ for $\gamma =20 \mathrm{\AA }$ in Figure 2d. Increasing the parameter $\gamma$ influences the distribution of the confined energy levels and consequently affects the energy separation and OAC.

2.2. Calculation of Electronic and Optical Properties

An electron in a spherical QD with a hydrogenic impurity within the effective mass approximation can be completely described by solving the radial Schrödinger equation [10,43,44]:

$$\left[-\frac{{\hslash }^2}{2}\overrightarrow{{\nabla }_r}\left(\frac{1}{m^{*}\left(r\right)}\overrightarrow{{\nabla }_r}\right)+\frac{\mathcal{l}\left(\mathcal{l}+1\right){\hslash }^2}{2m^{*}\left(r\right){ r}^2}-\frac{{Z e}^2}{\epsilon r}+V_{\mathrm{w}\mathrm{s}}\left(r\right)\right]R_{n\mathcal{l}}\left(r\right)={E_{n\mathcal{l}} R}_{n\mathcal{l}}\left(r\right),$$

(2)

where $m^{*}\left(r\right)$ is the position-dependent mass of the electron, $\hslash$ represents the reduced Planck constant, $\epsilon$ is the dielectric constant, and $\mathcal{l}$ is the angular quantum number. Furthermore, $R_{n\mathcal{l}}\left(r\right)$ and $E_{n\mathcal{l}}$ are the radial wavefunction and energy eigenvalue, respectively.

The first term in Equation (2) represents the kinetic energy, whereas the second term containing $\mathcal{l}\left(\mathcal{l}+1\right)$ denotes the centrifugal contribution of the potential due to the spherical symmetry of the Woods–Saxon potential. The third term represents the electron-impurity attraction. The two cases, $Z=0$ and $Z=1$, correspond to the absence and presence of the hydrogenic impurity, respectively. $V_{\mathrm{w}\mathrm{s}}\left(r\right)$ represents the Woods–Saxon potential which is a radial confinement term. To compute $E_{n\mathcal{l}}$ and $R_{n\mathcal{l}}\left(r\right)$, we discretized Equation (2) using the finite difference method and transformed it into a linear eigenvalue equation of the form $AX=\lambda X$, where $A$ is a tridiagonal matrix, $X$ represents $R_{n\mathcal{l}}\left(r\right)$, and $\lambda$ denotes $E_{n\mathcal{l}}$. The 1D discretization of the radial Schrödinger equation was carried out with a finite difference method (FDM). Thus, Equation (2) takes the linear form:

$$\begin{gathered} R_{n\mathcal{l}}\left(j+1\right)\left[-\frac{{\hslash }^2}{2m^{*}{r}_j\left(∆r\right)}-\frac{{\hslash }^2}{2m^{*}{\left(∆r\right)}^2}\right]+R_{n\mathcal{l}}\left(j\right)\left[\frac{{\hslash }^2}{m^{*}{\left(∆r\right)}^2}+\frac{\mathcal{l}\left(\mathcal{l}+1\right)}{m^{*}{\left(r_j.∆r\right)}^2}+{ V}_{WS}\left(j\right)\right] \\ +R_{n\mathcal{l}}\left(j-1\right)\left[\frac{{\hslash }^2}{2m^{*}{r}_j\left(∆r\right)}-\frac{{\hslash }^2}{2m^{*}{\left(∆r\right)}^2}\right]=E_{n\mathcal{l}}{R}_{n\mathcal{l}}\left(j\right), \end{gathered}$$

(3)

where $r_j=j∆r \left(j=1,\dots ,N\right)$ and $∆r=\frac{R}{N}$ is the mesh discretization. Equation (3) is of the form $Hx=\lambda x$, where $\lambda$ is the energy $E_{n\mathcal{l}}$, $x$ is the radial wavefunction $R_{n\mathcal{l}}\left(j\right)$, and $H$ is a tridiagonal matrix with elements given by

$$H_{ij}=\left\{\begin{array}{c}\frac{{\hslash }^2}{m^{*}{\left(∆r\right)}^2}+\frac{\mathcal{l}\left(\mathcal{l}+1\right)}{m^{*}{\left(r_j.∆r\right)}^2}+V_{\mathrm{W}\mathrm{S}}\left(j\right), \mathrm{i}\mathrm{f} j=i\\ \frac{{\hslash }^2}{2m^{*}{r}_j\left(∆r\right)}-\frac{{\hslash }^2}{2m^{*}{\left(∆r\right)}^2}, \mathrm{i}\mathrm{f} j=i-1\\ \begin{array}{c} -\frac{{\hslash }^2}{2m^{*}r\left(∆r\right)}-\frac{{\hslash }^2}{2m^{*}{\left(∆r\right)}^2}, \mathrm{i}\mathrm{f} j=i+1\\ 0, \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\end{array}\right..$$

(4)

In our study, we assume that the radial wavefunction at the external boundary point $\left(N+1\right)$ is zero. The dimension of matrix $H$ is $(N\times N)$, and in all of our calculations, we set $N=1200$ with the boundary condition $R_{n\mathcal{l}}\left(r=R_{\mathrm{e}\mathrm{x}\mathrm{t}}\right)=0.$

Optical absorption in the QD occurs when an electron in its initial level $E_i$ is excited to a final energy $E_f$ after absorption of a photon with energy $\hslash \omega =(E_f-E_i)$. According to Fermi’s golden rule, the OAC can be written as [45]

$$\alpha \left(\hslash \omega \right)=\frac{16{\pi }^2{\delta }_{\mathrm{F}\mathrm{S}}{P}_{if}}{n_r{V}_{\mathrm{c}\mathrm{o}\mathrm{n}}}\hslash \omega {\left|M_{if}\right|}^2\delta (E_f-E_i-\hslash \omega ),$$

(5)

where $P_{if}$, ${\delta }_{\mathrm{F}\mathrm{S}}$, and $V_{\mathrm{c}\mathrm{o}\mathrm{n}}$ represent the electron population difference, the fine structure, and the confinement volume, respectively. $n_r$ represents the refractive index of the GaAs semiconductor, and $\left|M_{if}\right|$ denotes the DME of the transition. Furthermore, the $∆\mathcal{l}=\pm 1$ selection rule satisfied by the quantum number $\mathcal{l}$ is taken into consideration.

In the present paper, we address only the transition between the 1s and 1p states. Furthermore, the $\delta$-function in the previous equation is substituted with a Lorentzian profile:

$$\delta \left(E_f-E_i-\hslash \omega \right)=\frac{\hslash \Gamma }{\pi \left[{\left(E_f-E_i-\hslash \omega \right)}^2+{\left(\hslash \Gamma \right)}^2\right]},$$

(6)

where $\hslash \omega$ is the energy of the incident photon, and $\hslash \Gamma$ is the width at half height of the Lorentzian function. In the next section, and for simplicity of notation, we consider the initial state $(i=1)$ to be 1s and the final state $(f=2)$ to be the 1p state, so the term ${\left|M_{if}\right|}^2$ in Equation (3) is simply designated as ${\left|M_{12}\right|}^2$. In our study, the electromagnetic radiation is polarized along the z-axis, and ${\left|M_{12}\right|}^2$ is given by the following expression [50,51,52]:

$${\left|M_{12}\right|}^2=\frac{1}{3}{\left|{\int }_0^{\infty }{R}_{1s}\left(r\right)r^3{R}_{1p}\left(r\right)dr\right|}^2,$$

(7)

where the $\frac{1}{3}$ pre-factor arises from integration of the spherical harmonics. In addition to the optical absorption, we have evaluated the impurity binding energy of the neutral donor, defined as $E_b=E_{n,l}^{z=0}-E_{n,l}^{z=1}$, where $E_{n,l}^{z=0}$ and $E_{n,l}^{z=1}$ denote the energy levels for QDs without and with the impurity, respectively.

3. Results and Discussion

Atomic units $\left(\hslash =e=m_0=1\right)$ are used throughout the rest of this work, which defines the Rydberg energy ($1R_y\cong 5.6 \mathrm{m}\mathrm{e}\mathrm{V}$) and Bohr radius ($1a_B\cong 100 Å$). In addition, $V_0$ is set at $0.228 \mathrm{e}\mathrm{V}$, which corresponds to the band offset between GaAs and ${\mathrm{A}\mathrm{l}}_x{\mathrm{G}\mathrm{a}}_{(1-x)}\mathrm{A}\mathrm{s}$ with x = 0.3. Additional physical parameters used in our simulation are $\hslash \Gamma =3 \mathrm{m}\mathrm{e}\mathrm{V}, m^{*}=0.067m_0, \mathrm{a}\mathrm{n}\mathrm{d} \epsilon =13.11{\epsilon }_0$. The radius of the QD is fixed at $R=25 \mathrm{n}\mathrm{m}$.

Figure 3a–d displays the probability densities of the 1s and 1p states with the confining potential in the absence of the hydrogenic impurity (i.e., $Z=0$) for four values of the structural parameter $(\gamma =5, 10, 15, \mathrm{a}\mathrm{n}\mathrm{d} 20 \mathrm{n}\mathrm{m})$ with $R_0=R/2.$ Increasing $\gamma$ also increases the amplitudes of the probability densities of the 1s and 1p states and widens the spatial extent of the wavefunctions. For instance, when $\gamma =5 \mathrm{n}\mathrm{m}$, the 1s and 1p densities decay to zero at $r=15$ and $20 \mathrm{n}\mathrm{m}$, respectively; however, when $\gamma =20 \mathrm{n}\mathrm{m}$, both densities decrease to zero at $r=24 \mathrm{n}\mathrm{m}$. This behavior is due to the slope of the Woods–Saxon potential decreasing with increasing $\gamma$ (see Figure 2a–d). The spread in $V_{\mathrm{w}\mathrm{s}}\left(r\right)$, especially near its top, enhances the amplitudes of the densities and enlarges their geometrical distribution along the $r$ axis. This, in turn, modifies the energy levels and DMEs between the 1s and 1p wavefunctions since their overlap is now modified. Figure 4a–d plots these densities with an on-center hydrogenic impurity. In this case, there are two confining contributions. The first one is due to the geometrical behavior of the $V_{\mathrm{w}\mathrm{s}}\left(r\right)$ potential due to the increase in the parameter $\gamma$, and the second one arises from the electrostatic attraction between the hydrogenic impurity and the electron in different states. This is reflected in the decrease in the amplitudes in the 1s and 1p probabilities. Note that the amplitude for the 1s density is less sensitive than that of 1p for $\gamma =15$ and $20 \mathrm{n}\mathrm{m}$. For these values, the impact of geometrical confinement becomes negligible compared to that of the electrostatic attraction, and no additional changes are observed for $\gamma >20 \mathrm{n}\mathrm{m}.$

Figure 5 plots the energy levels $E_{1p}$ and $E_{1s}$, which increase with $\gamma$. At low values of $\gamma$, the energy levels are well separated from each other; however, for higher values of $\gamma$, their separation is considerably reduced. The decrease in the energy difference between $E_{1p}$ and $E_{1s}$, with and without the presence of hydrogenic impurity, is responsible for the red shift of the OAC, which we discuss later. Furthermore, for all values of $\gamma$, the energy levels in the presence of the hydrogenic impurity are less than those without the hydrogenic impurity. This is due to the attraction between the electron and the impurity, which causes the electron to be near the impurity at the center of the QD.

From Equation (5), the OAC is proportional to ${\left|M_{12}\right|}^2$, which controls the amplitude of the OAC and explains the overlap between the $1s$ and $1p$ wavefunctions. Figure 6 plots its variation with the energy separation $\left(E_{1p}-E_{1s}\right)$ as a function of $\gamma$. For $\gamma =5 \mathrm{n}\mathrm{m}$, the values of ${\left|M_{12}\right|}^2$ with $(Z=1$) and without impurity $(Z=0$) are similar. However, when $\gamma$ is increased, ${\left|M_{12}\right|}^2$ increases and takes higher values for $Z=0$ than for $Z=1$. This result is due to the change in the overlap between the 1s and 1p wavefunctions. In addition, Figure 5 shows that the energy separation $\left(E_{1p}-E_{1s}\right)$ decreases for both cases (with and without impurity), resulting in a red shift in the OAC.

Figure 7 plots the OAC as a function of photon energy for $\gamma =5, 10, \mathrm{a}\mathrm{n}\mathrm{d} 20 \mathrm{n}\mathrm{m}$. We report results for two cases: with $(Z=1$) and without $(Z=0$) the hydrogenic impurity. The OAC amplitudes move towards lower energies (red shift) with increasing $\gamma$. This variation is in accordance with the variation of $\left(E_{1p}-E_{1s}\right)$, previously shown in Figure 5. In addition, we note that the OAC amplitudes in the presence of the hydrogenic impurity are always smaller than those without the hydrogenic impurity. This is due to the difference in the DME with and without the presence of the impurity, as shown in Figure 5.

Figure 8 plots the binding energies of the 1s and 1p states as a function of $\gamma$. Both states gradually decrease with $\gamma$. For lower values, they decrease rapidly; however, for higher values $\left(\gamma >15 \mathrm{n}\mathrm{m}\right)$, the binding energies show a small variation. This behavior in binding energy for the 1s and 1p states is explained by the strong attraction near the center of the QD; however, for higher values of $r$, this attraction is reduced compared to the geometrical confinement, and consequently, the binding energy remains constant for all higher values of $r.$

We now turn our attention to the effect of $R_0$. Figure 9a–d plots the probability densities of the lowest electronic states 1s and 1p with the confining potential in the absence of the hydrogenic impurity (i.e., $Z=0$) for $R_0=7, 12, 17, \mathrm{a}\mathrm{n}\mathrm{d} 22 \mathrm{n}\mathrm{m},$ with $\gamma =10 \mathrm{n}\mathrm{m}.$ Increasing $R_0$ enlarges the potential and minimizes its values at the center and surface of the quantum dot. Consequently, the two probability densities maintain the same spread; however, their amplitudes increase with $R_0$. The amplitude of the 1p density is more sensitive than that of 1s when $R_0$ increases. The influence of the hydrogenic impurity on these densities is shown in Figure 10a–d. The densities have the same spread along the radius of the quantum dot, but their amplitudes are reduced due to the electrostatic attraction introduced by the hydrogenic impurity. To evaluate the effect of the on-center impurity on the OAC, Figure 10 plots its variation as a function of the incident energy for three values of $R_0$. The OAC peak moves towards higher energy (blue shift) when $R_0$ increases from $8$ to $18 \mathrm{n}\mathrm{m}$. Subsequently, it moves in the direction of low energies, exhibiting a red shift. This double behavior can be interpreted via the variation in the energy separation between the 1s and 1p energy levels.

Figure 11 plots the variation of the 1s and 1p energy levels as a function of $R_0$, which shows a gradual decrease for the two cases (with and without impurity). This decrease is due to the enlargement of the confining potential with $R_0$ as shown in Figure 9 and Figure 10. However, the slope of this decrease is slightly different. Figure 12 plots the energy separation $E_{1p}-E_{1s}$ as a function of the parameter $R_0$, which shows that this separation increases up to $R_0=16 \mathrm{n}\mathrm{m}$ but subsequently decreases. This behavior confirms the red and blue shift shown in the OAC variation in Figure 13. In addition, Figure 12 shows the variation of the dipole matrix element ${\left|M_{12}\right|}^2$ as a function of $R_0$. This physical quantity decreases up to $R_0=16 \mathrm{n}\mathrm{m}$ and then subsequently increases. This arises from the variation of the overlap between the $R_{1p}$ and $R_{1s}$ wave functions, which agree with the OAC trends shown in Figure 12. For $R_0<16 \mathrm{n}\mathrm{m}$, the OAC amplitude diminishes; however, for $R_0>16 \mathrm{n}\mathrm{m}$, the amplitude subsequently increases.

Finally, Figure 14 displays the binding energy as a function of $R_0$. The binding energy increases up to $R_0=16 \mathrm{n}\mathrm{m}$ and subsequently diminishes gradually. Consequently, this critical value of ${ R}_0$ can play an important role in shifting the OAC from red to blue as well as controlling the variation of the binding energy toward high or low values.

4. Conclusions

In summary, we have presented the first study of the optical and electronic properties of a GaAs spherical QD with a Woods–Saxon potential in the presence of a hydrogenic impurity. By solving the radial part of the Schrödinger equation using the finite difference method, we obtain energy levels of $1s$ and $1p$ states and their probability densities. These quantities allow us to calculate dipole matrix elements, energy separations, OACs, and binding energies as a function of the parameters $R_0$ and $\gamma$. Our results indicate that increasing $\gamma$ leads to a red shift of the OAC; however, an increase in $R_0$ initially gives rise to a blue shift and, subsequently, a red shift. We also demonstrated that the variation of the OAC amplitude is determined via the dipole matrix element, which effectively captures the overlap between $R_{1p}$ and $R_{1s}$. Moreover, our findings indicate that the insertion of a hydrogenic impurity at the center of the QD considerably decreases the energy levels due to the strong attraction between the free electrons and the hydrogenic impurity. Our numerical calculations provide mechanistic insight into the electronic transport and optical properties of spherical QDs.