1. Introduction
The idea of describing a quantum dot through a short-range potential brings the possibility of dealing with a confining potential with finite depth and range [1–5]. Quantum dots have a great interest in condensed matter physics with possible applications in nanotechnology [1, 2, 6]. This kind of mesoscopic device allows us to search for quantum effects such as the Aharonov–Bohm effect [7] and persistent currents [8–13]. An interesting example of a short-range potential is the attractive Gaussian potential [1]. With the aim of extending this idea of working with short-range potentials, Ciurla et al [2] proposed the power-exponential potential. In this mathematical model of a quantum dot, Ciurla et al [2] showed the attractive Gaussian potential as a particular case of it. In this way, the power-exponential potential can be an adequate mathematical model to work with quantum dots, where we can cite the works [14–25] as part of this quantum dot model.
In this work, we consider a uniform distribution of electric charges inside the spherical quantum dot. We assume that the quantum dot is a non-conducting medium. This electric charge distribution yields a linear electric field which, in turn, interacts with a spinless charged particle. Our interest is inspired by several studies in which have considered the presence of a linear electric field and raised a discussion about its influence. For instance, Edery and Audin [26] showed that the degeneracy of the Landau levels [27] are modified by a linear electric field applied in the z-direction. In another context, the interaction between the linear electric field and a spinless charged particle has been studied in the presence of disclination and spiral dislocation [28–30]. Linear electric fields have also been used to achieve the Landau quantization for neutral particles [31, 32] and in studies of the confinement of neutral particles to a quantum ring [33, 34]. Therefore, by confining a spinless charged particle to a spherical quantum dot, our interest is to search for the influence of a linear electric field near the spherical quantum dot centre. In this region, we assume that the short-range potential dominates. From the energy eigenvalues for s-waves, we thus discuss the electric field influence.
The structure of this paper is as follows: in section 2 , we introduce the spherical quantum dot model proposed by Ciurla et al [2]. Then, we consider the case p = 1 of the Ciurla et al model to study the confinement of a spinless charged particle to a quantum dot. By including the interaction between a linear electric field and a spinless charged particle, we analyse the case in which the spherical quantum dot dominates; in section 3 , we present our conclusions.
2. Influence of a linear electric field
In this section, we deal with the confinement of a spinless charged particle to a spherical quantum dot under the influence of a linear electric field. Our analysis is made based on the spherical symmetry. Hence, the Schrödinger equation is given by [35]:
$\begin{eqnarray}{ \mathcal E }\psi =-\displaystyle \frac{1}{2m}\left[\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}\left({r}^{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{r}}\right)-\displaystyle \frac{{\hat{L}}^{2}}{{r}^{2}}\right]\psi +V\left(\vec{r}\right)\psi ,\end{eqnarray}$
where ${\hat{L}}^{2}$ is the angular momentum squared operator and $V\left(\vec{r}\right)$ is the potential energy of the system.We shall work with the power-exponential potential [2] with the purpose of describing the confinement of the particle to a spherical quantum dot. In this model, the quantum dot consists in a short-range potential, whose mathematical expression is2 ), we have p ≥ 1 and V0 > 0 is the confining potential depth. In addition, r corresponds to the radial coordinate and r0 is the range of the confining potential. The potential energy (2 ) can describe several attractive short-range potentials. For instance, the attractive Gaussian potential [1, 14–18] is defined by p = 2, and the rectangular potential for p → ∞ [2]. The simplest case of the quantum dot model (2 ) is for p = 1 [23, 24]. In this work, our focus is on the spherical quantum dot model defined by p = 1 in equation (2 ), i.e.2 ) has been explored in search of global monopole topological effects [23–25, 36].
$\begin{eqnarray}{V}_{p}\left(r\right)=-{V}_{0}\,{{\rm{e}}}^{-{\left(\tfrac{r}{{r}_{0}}\right)}^{p}}.\end{eqnarray}$
In equation ( $\begin{eqnarray}{V}_{1}\left(r\right)=-{V}_{0}\,{{\rm{e}}}^{-\tfrac{r}{{r}_{0}}}.\end{eqnarray}$
Recently, the power-exponential potential (Next, let us consider the electric field produced by a uniform charge density (ρ) inside a non-conducting material. This electric field is given by $\vec{E}=\tfrac{\rho \,r}{2}\,\hat{r}$, which gives us the potential energy:
$\begin{eqnarray}{V}_{q}\left(r\right)=-q{\int }_{0}^{r}\vec{E}\cdot \,{\rm{d}}\vec{r}=-\displaystyle \frac{q\,\rho \,{r}^{2}}{4},\end{eqnarray}$
where q is the electric charge of the particle. From now on, we are considering $q=-\left|q\right|$ and ρ > 0. It is worth mentioning that the interaction of a charged particle with the radial electric field $\vec{E}=\tfrac{\rho \,r}{2}\,\hat{r}$ has been studied under the influence of the topological effects of a disclination and a spiral dislocation in [28–30].Henceforth, we consider a spinless charged particle confined to the spherical quantum dot (3 ) where it interacts with the radial electric field as described by the potential energy (4 ). With the aim of writing the radial equation, we should take into account that ${V}_{1}\left(r\right)$ and ${V}_{q}\left(r\right)$ depend only on the radial coordinate. In this way, the solution to the Schrödinger equation (1 ) is given by $\psi \left(r,\theta ,\varphi \right)=R\left(r\right){{\rm{Y}}}_{{\ell }}^{{m}_{{\ell }}}\left(\theta ,\varphi \right)=\tfrac{u\left(r\right)}{r}{{\rm{Y}}}_{{\ell }}^{{m}_{{\ell }}}\left(\theta ,\varphi \right)$ [35]. Note that the eigenvalues of the operator ${\hat{L}}^{2}$ are ℓ = 0, 1, 2, 3, …, while the eigenvalue of the operator ${\hat{L}}_{z}$ is mℓ. Hence, the radial equation is
$\begin{eqnarray}\begin{array}{rcl} & & \displaystyle \frac{{{\rm{d}}}^{2}u}{{{\rm{d}}r}^{2}}-\displaystyle \frac{\ell \left(\ell +1\right)}{{r}^{2}}\,u\\ & & +2m\,{V}_{0}\,{{\rm{e}}}^{-r/{r}_{0}}\,u-\displaystyle \frac{m\left|q\right|\,\rho }{2}\,{r}^{2}\psi +2m{ \mathcal E }\,u=0.\end{array}\end{eqnarray}$
Let us search for bound states for s-waves. Therefore, we take ℓ = 0 in equation (5 ). Moreover, let us define x = r/r0, thus, equation (5 ) becomes
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}u}{{{\rm{d}}{x}}^{2}}+2m\,{V}_{0}\,{r}_{0}^{2}\,{{\rm{e}}}^{-x}\,u-\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}\,{x}^{2}\psi +2m{ \mathcal E }\,{r}_{0}^{2}\,u=0.\end{eqnarray}$
In this work, our aim is to analyse the region x ≪ 1, i.e. the region near the quantum dot centre [2]. We should observe that the spherical quantum dot (3 ) is a short-range potential, hence we can assume that the spherical quantum dot dominates in the region x ≪ 1. In this way, we have $\left|{V}_{1}\right|\gg \left|{V}_{q}\right|$ when x ≪ 1. We also assume that ${ \mathcal E }\lt 0$ and define the parameter:6 ):8 ):10 ) is given by
$\begin{eqnarray}\beta =\sqrt{-2m{ \mathcal E }\,{r}_{0}^{2}}.\end{eqnarray}$
Besides, let us use the approximation x ≈ 1 − e−x. Thereby, we have in equation ( $\begin{eqnarray}\begin{array}{rcl} & & \displaystyle \frac{{{\rm{d}}}^{2}u}{{{\rm{d}}{x}}^{2}}+2m\,{V}_{0}\,{r}_{0}^{2}\,{{\rm{e}}}^{-x}\,u-\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}\\ & & \times \,{\left(1-{{\rm{e}}}^{-x}\right)}^{2}\psi -{\beta }^{2}\,u=0.\end{array}\end{eqnarray}$
By performing the change of variables: $\begin{eqnarray}y=\sqrt{2m\left|q\right|\,\rho \,{{r}}_{0}^{4}}\,{{\rm{e}}}^{-x},\end{eqnarray}$
we obtain in equation ( $\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}u}{{{\rm{d}}{y}}^{2}}+\displaystyle \frac{1}{y}\,\displaystyle \frac{{\rm{d}}{u}}{{\rm{d}}{y}}-\displaystyle \frac{{\lambda }^{2}}{{y}^{2}}\,u+\displaystyle \frac{\delta }{y}\,u-\displaystyle \frac{u}{4}=0.\end{eqnarray}$
The parameters λ and δ are defined as $\begin{eqnarray*}{\lambda }^{2}={\beta }^{2}+\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2};\end{eqnarray*}$
$\begin{eqnarray}\delta =\sqrt{\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}}\times \left[\displaystyle \frac{2\,{V}_{0}}{\left|q\right|\,\rho \,{r}_{0}^{2}}+1\right].\end{eqnarray}$
Therefore, the solution to equation ( $\begin{eqnarray}u\left(y\right)=\displaystyle \frac{{c}_{1}}{\sqrt{y}}\,{M}_{\delta ,\lambda }\left(y\right)+\displaystyle \frac{{c}_{2}}{\sqrt{y}}\,{W}_{\delta ,\lambda }\left(y\right),\end{eqnarray}$
where the functions ${M}_{\delta ,\lambda }\left(y\right)$ and ${W}_{\delta ,\lambda }\left(y\right)$ are the Whittaker functions of first kind and second kind, respectively [37, 38].From equation (9 ) we have that when r → ∞ ⇒ x → ∞ ⇒ y = 0. Then, a regular solution to equation (10 ) at y = 0 is obtained by taking c2 = 0 in equation (12 ). Thus, we have9 ) we have that when $r=0\Rightarrow x=0\Rightarrow y\to {y}_{0}=\sqrt{2m\left|q\right|\,\rho \,{r}_{0}^{4}}$. Thereby, we have the boundary condition:
$\begin{eqnarray}u\left(y\right)=\displaystyle \frac{{c}_{1}}{\sqrt{y}}\,{M}_{\delta ,\lambda }\left(y\right).\end{eqnarray}$
Furthermore, from equation ( $\begin{eqnarray}u\left({y}_{0}\right)=0.\end{eqnarray}$
With the purpose of exploring the boundary condition (14 ), let us write equation (13 ) in terms of the confluent hypergeometric function. According to [37, 38], we can write the function ${M}_{\delta ,\lambda }\left(y\right)$ in terms of the confluent hypergeometric function ${\,}_{1}{F}_{1}\left(a,b;y\right)$. Thereby, equation (13 ) becomes15 ) into equation (14 ), we find
$\begin{eqnarray}u\left(y\right)={c}_{1}\,{y}^{\lambda }\times {{\rm{e}}}^{-y/2}\times {\,}_{1}{F}_{1}\left(\displaystyle \frac{1}{2}+\lambda -\delta ,1+2\lambda ;y\right).\end{eqnarray}$
Note that $a=\tfrac{1}{2}+\lambda -\delta $ and b = 1 + 2λ. By substituting equation ( $\begin{eqnarray}u\left({y}_{0}\right)={c}_{1}\,{y}_{0}^{\lambda }\times {{\rm{e}}}^{-{y}_{0}/2}\times {\,}_{1}{F}_{1}\left(\displaystyle \frac{1}{2}+\lambda -\delta ,1+2\lambda ;{y}_{0}\right)=0.\end{eqnarray}$
We can explore equation (16 ) in two different cases. The first one is to consider a → −∞ (y0, b fixed). In this case, the confluent hypergeometric function ${}_{1}{F}_{1}\left(a,b;{y}_{0}\right)$ can be written as follows [37]:
$\begin{eqnarray}{}_{1}{F}_{1}\left(a,b;{y}_{0}\right)\propto \cos \left(\sqrt{\left[2b-4a\right]{y}_{0}}-\displaystyle \frac{b\pi }{2}+\displaystyle \frac{\pi }{4}\right).\end{eqnarray}$
Thereby, by substituting equation (17 ) into equation (16 ) we find the relation:18 ) is valid only if ${\left[\tfrac{\sqrt{4\delta {y}_{0}}}{\pi }-n-\tfrac{3}{4}\right]}^{2}\gt \tfrac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}$, otherwise, no bound states exist. As a consequence, we have an upper limit to the radial quantum number given by7 ) into equation (18 ), we obtain the energy levels:
$\begin{eqnarray}{\beta }^{2}={\left[\displaystyle \frac{\sqrt{4\delta {y}_{0}}}{\pi }-n-\displaystyle \frac{3}{4}\right]}^{2}-\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2},\end{eqnarray}$
where n ≫ 1 (with n = 0, 1, 2, 3, ...) is the radial quantum number. Since β2 > 0, equation ( $\begin{eqnarray}{n}_{\max }\,\lt \,\displaystyle \frac{\sqrt{4\delta {y}_{0}}}{\pi }-\displaystyle \frac{3}{4}-\sqrt{\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}}.\end{eqnarray}$
For $n\,\gt \,{n}_{\max }$, there are no bound states. Furthermore, after substituting equation ( $\begin{eqnarray}{{ \mathcal E }}_{n}=-\displaystyle \frac{1}{2m\,{r}_{0}^{2}}{\left[\displaystyle \frac{\sqrt{4\delta {y}_{0}}}{\pi }-n-\displaystyle \frac{3}{4}\right]}^{2}+\displaystyle \frac{\left|q\right|\,\rho \,{r}_{0}^{2}}{4}.\end{eqnarray}$
Hence, we have a discrete spectrum of energy for s-waves in the region x ≪ 1, i.e. in the region where the spherical quantum dot dominates. The influence of the linear electric field modifies the energy eigenvalues of the spherical quantum dot as shown in [23]. Besides, the maximal value of the radial quantum number obtained in [23] is modified, and thus, it is now given by that of equation (19 ).
The second case that can be explored from the boundary condition (16 ) is by considering y0 ≫ 1 (a, b fixed). In this case, we have that the confluent hypergeometric function ${\,}_{1}{F}_{1}\left(a,b;{y}_{0}\right)$ diverges when y0 → ∞. For this reason, we must impose that ${\,}_{1}{F}_{1}\left(a,b;{y}_{0}\right)$ becomes a polynomial of degree n (n = 0, 1, 2, 3, …). This occurs when a = −n [37, 38]. This condition guarantees that the boundary condition (14 ) is satisfied. In this way, we have $\tfrac{1}{2}+\lambda -\delta =-n$, which yields the relation:7 ) into equation (21 ). Then, we obtain the energy levels:23 ) differs from equation (20 ). Moreover, the maximal value of the radial quantum number established in equation (22 ) differs from that of equation (19 ).
$\begin{eqnarray}{\beta }^{2}={\left[\delta -n-\displaystyle \frac{1}{2}\right]}^{2}-\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}.\end{eqnarray}$
Note that n remains the radial quantum number. Due to β2 > 0, we also have an upper limit to the radial quantum number. In this case, the maximal value of n is determined by $\begin{eqnarray}{n}_{\max }\,\lt \,\delta -\displaystyle \frac{1}{2}-\sqrt{\displaystyle \frac{m\left|q\right|\,\rho \,{r}_{0}^{4}}{2}}.\end{eqnarray}$
Thus, there are bound states for $n=0,1,2,3,\ldots ,{n}_{\max }$. Let us go further by substituting equation ( $\begin{eqnarray}{{ \mathcal E }}_{n}=-\displaystyle \frac{1}{2m\,{r}_{0}^{2}}{\left[\delta -n-\displaystyle \frac{1}{2}\right]}^{2}+\displaystyle \frac{\left|q\right|\,\rho \,{r}_{0}^{2}}{4}.\end{eqnarray}$
Hence, by considering y0 ≫ 1, we also obtain a discrete spectrum of energy for s-waves in the region x ≪ 1. In this region where the spherical quantum dot dominates, the linear electric field also influences the energy eigenvalues for the case y0 ≫ 1. The presence of the linear electric field also modifies both the energy levels of the spherical quantum dot and the maximal value of n obtained in [23]. Note that equation (3. Conclusions
We have studied the confinement of a spinless charged particle to a spherical quantum dot in the presence of a linear electric field. This electric field is produced by a uniform charge density inside a non-conducting medium. We have considered a spherical quantum dot described by an attractive short-range potential [2] which has a great interest in semiconductor devices. In this study, we have focused on the case p = 1 of the power-exponential potential [2] and dealt with s-waves. Thereby, we have analyzed the region where spherical quantum dot dominates and discussed two cases where the energy levels can be achieved. In both cases studied, we have shown that a discrete spectrum of energy can be obtained, and the radial quantum number possesses a maximal value. Therefore, the presence of the linear electric field modifies the energy levels of the spherical quantum dot (3 ) in contrast to those obtained in [23].
An interesting perspective is to explore the short-range potential (3 ) in an elastic medium, but dealing with cylindrical symmetry. In this perspective, we can consider an elastic medium with disclinations and dislocations [39, 40], and then, search for analogues of the Aharonov–Bohm effect [41–47].