Quantum Dot Scattering in Monolayer Molybdenum Disulfide

Abstract

This chapter looks at how electrons propagate in a circular quantum dot (QD) of monolayer molybdenum disulfide (MoS2) that is exposed to an electric potential. Mathematical formulas for the eigenstates, scattering coefficients, scattering efficiency, and radial component of the reflected current and electron density are presented using the continuum model. As a function of physical characteristics such as incident electronic energy, potential barrier, and quantum dot radius, we discover two scattering regimes. We demonstrate the presence of scattering resonances for low-energy incoming electrons. We should also point out that the far-field dispersed current has unique favored scattering directions.

1. Introduction

The hunt for novel two-dimensional materials has been sparked by graphene research [1, 2]. Transition metal dichalcogenides (TMDs) are among them. TMD (${\mathrm{MX}}_2$ such as $\mathrm{M}=\mathrm{Mo}$, $\mathrm{W}$; $\mathrm{X}=\mathrm{S}$, ${\mathrm{S}}_{\mathrm{e}}$, and $\mathrm{T}$) monolayers have recently emerged as promising nanostructures for optics, electronics, and spintronic applications. For many years, molybdenum disulfide (${\mathrm{MoS}}_2$) has been a strong material that has gotten a lot of attention because of its intriguing electrical and optical characteristics [3, 4, 5]. It has a straight bandgap in the visible frequency band [6, 7, 8, 9, 10] and high carrier mobility at ambient temperature [11, 12, 13, 14, 15]. As a result, it is a strong contender for future electrical and optoelectronic technologies.

A strong spin-orbit interaction characterizes the ${\mathrm{MoS}}_2$ monolayer with its honeycomb atomic structure. As a result, totally new electron spin characteristics will emerge. On the one hand, conduction or valence electrons should be significantly less susceptible to the ultrafast spin relaxation effects observed in two-dimensional semiconductor structures such as $\text{GaAs}$ quantum wells. The absorption of circularly polarized light, on the other hand, might result in a population of spin-polarized electrons (i.e. an imbalance between the number of spin-up and spin-down electrons). Moreover, the circularly polarized excitation allows control of the distribution of these electrons in one of two valleys of reciprocal space, according to [16]. We call this valley polarization, and we are now working to better understand it so that we can use it in information storage and processing uses. The borders of the valence and conduction bands are positioned at the two corners of the Brillouin zone, i.e. the $k$ and $k^{\prime }$ points, making the ${\mathrm{MoS}}_2$ monolayer a semiconductor. This provides an additional degree of freedom for electrons and holes, which may be employed for encoding information and subsequent processing [16, 17, 18, 19].

Monolayer ${\mathrm{MoS}}_2$ quantum dots (QDs), which feature different physical and chemical characteristics, strong quantum confinement properties, edge effects [20], and a direct bandgap, are key ${\mathrm{MoS}}_2$ related nanostructures that have gotten a lot of interest in recent years. Some many approaches for preparing ${\mathrm{MoS}}_2$ QD have been suggested to date, including solvothermal treatment synthesizing [21], hydrothermal synthesis [22], grinding exfoliation [23], liquid exfoliation in organic solvents [24], electrochemical etching [25], and reaction processing [26].

The contours of the valence and conduction bands determine edge states, which are confined on the boundaries and have energies in the bandgap. The electrical behavior of ${\mathrm{MoS}}_2$ quantum dots was studied using the tight-binding model [22]. According to the orbital asymmetry [22], it was demonstrated that quantum dots with the same form but distinct electrical characteristics may be created.

In this work, we investigate electron propagation in the presence of a potential barrier in a circular electrostatically defined quantum dot monolayer molybdenium disulfide ${\mathrm{MoS}}_2$. Different scattering ranges are identified based on the quantum dot radius, potential barrier, electron spin, and electron energy.

The following is the structure of the present study. We give a theoretical investigation of electron propagation wave plane in a circular quantum dot of monolayer molybdenium disulfide ${\mathrm{MoS}}_2$ in Section 2. The spinors of the Dirac equation solutions corresponding to each area of varied scattering parameters are given. The scattering coefficients are calculated using the continuity criterion. We evaluate the scattering efficiency, square modulus of the scattering coefficients, radial component of the far-field scattered current, and electron density in Section 3, and we describe our findings using various plots. The basic findings of the study are presented in Section 4.

2. Theoretical model

As shown in Figure 1, we investigate a quantum dot of radius $\rho$ in the presence of an electric potential $U$. In the proximity of the valleys $k$ and $k^{\prime }$ by generating the wave functions via the base of conduction and valence bands, the Dirac-Weyl Hamiltonian for low-energy charge carriers in monolayer molybdenum disulfide (${\mathrm{MoS}}_2$) yields [27, 28]

$$\mathcal{H}={\mathcal{H}}_0+\frac{\kappa }{2}{\sigma }_z+\frac{{\mathrm{\Delta }}_{\mathit{so}}}{2}\tau s_z\left(1-{\sigma }_z\right)+U\left(r\right)$$E1

Figure 1.

The energy $\epsilon$ of Dirac electrons propagate in a circular quantum dot of monolayer molybdenum disulfide (${\mathrm{MoS}}_2$). The dot is defined by its radius $\rho$ and the bias introduced $U$. The incoming plane wave with energy $\epsilon >U$ (blue) belongs to a conduction band state (upper cone). The reflected wave (purple) is now in the conduction band, but the transmitting wave (red) is in the valence band (lower cone).

so the ${\mathcal{H}}_0$ is supplied by:

$${\mathcal{H}}_0=v_F\mathit{\sigma }\cdot \mathit{p}$$E2

where $v_F$, $\mathit{p}=\left(p_x,p_y\right)$ and $\mathit{\sigma }=\left({\sigma }_x,{\sigma }_y,{\sigma }_z\right)$ are, respectively, the Fermi velocity, the $2D$ momentum operator, and Pauli matrices acting on the atomic orbitals, $U\left(r\right)$ is the potential barrier, and $\kappa ={166.10}^{-3}$ eV [28] is related to the material bandgap energy, $s_z=\pm 1$ represents the electron spin-up and spin-down and $\tau =\pm 1$ denotes the $k$ and $k^{\prime }$ valleys, ${\mathrm{\Delta }}_{\mathit{so}}={75.10}^{-3}$ eV [28] is the splitting of the valence band owing to spin-orbit coupling, and $\rho$ is the dot radius. For simplicity, the values $v_F=\mathrm{\hslash }=1$ shall be assumed.

We then investigate localized-state solutions in our system, which is characterized as a circularly symmetric quantum dot, utilizing the potential barrier $U\left(r\right)$ and the energy gap $\kappa \left(r\right)$ defined by:

$$U\left(r\right)=\left\{\begin{array}{cc}0,& r>\rho \\ U,& r\le \rho \end{array}\right.,\kappa \left(r\right)=\left\{\begin{array}{cc}0,& r>\rho \\ \kappa ,& r\le \rho \end{array}\right..$$E3

We conduct our research using the polar coordinates (r, $\varpi$), so that the Hamiltonian (1) has the format:

$$\mathcal{H}=\left(\begin{array}{cc}{U}_{+}& {\pi }^{-}\\ {\pi }^{+}& U_{-}+\tau s_z{\lambda }_{\mathit{so}}\end{array}\right)$$E4

in which the two potentials and two operators have been established

$$U_{\pm }=U\pm \frac{\kappa }{2},{\pi }^{\pm }=e^{\pm \mathit{i\varpi }}\left(-i\frac{\partial }{\partial r}\pm \frac{1}{r}\frac{\partial }{\partial \varpi }\right).$$E5

The eigenvalue equation is used to determine the energy spectrum:

$$\mathcal{H}{\psi }_l\left(r,\varpi \right)=\epsilon {\psi }_l\left(r,\varpi \right).$$E6

Because $J_z=-i\mathrm{\hslash }{\partial }_{\phi }+\mathrm{\hslash }{\sigma }_z/2$ commutes with $\mathcal{H}$ by fulfilling $\left[\mathcal{H},J_z\right]=0$. The separability of ${\psi }_l$ into the radial ${\mathrm{\Phi }}^{\pm }\left(r\right)$ and angular ${\mathcal{Ϝ}}^{\pm }\left(\varpi \right)$ parts is required for this commutation, and then we obtain [29, 30]

$${\psi }_l\left(r,\varpi \right)={\mathrm{\Phi }}_l^{+}\left(r\right){\mathcal{Ϝ}}_l^{+}\left(\varpi \right)+{\mathrm{\Phi }}_{l+1}^{-}\left(r\right){\mathcal{Ϝ}}_{l+1}^{-}\left(\varpi \right)$$E7

where the two angular components are

$${\mathcal{Ϝ}}_l^{+}\left(\varpi \right)=\frac{e^{\mathit{il\varpi }}}{\sqrt{2\pi }}\left(\begin{array}{l}1\\ 0\end{array}\right),{\mathcal{Ϝ}}_{l+1}^{-}\left(\varpi \right)=\frac{e^{i\left(l+1\right)\varpi }}{\sqrt{2\pi }}\left(\begin{array}{l}0\\ 1\end{array}\right)$$E8

and $l$ is an integer, which denotes the total angular quantum number.

To obtain the energy spectrum solutions, we must first solve the eigenvalue equation:

$$\mathcal{H}{\psi }_l\left(r,\varpi \right)=\epsilon {\psi }_l\left(r,\varpi \right)$$E9

by taking into account two areas, as shown in Figure 1, outside ($r>\rho$) and within ($r\le \rho$) the quantum dot. As a result, we get an incident wave ${\psi }_i$ propagating in the x-direction, an outgoing reflected wave ${\psi }_r$, and a transmitted wave ${\psi }_t$ within the dot.

The radial components ${\mathrm{\Phi }}_l^{+}\left(r\right)$ and ${\mathrm{\Phi }}_{l+1}^{-}\left(r\right)$ satisfy two linked differential equations outside the dot ($r>\rho$):

$$-i\frac{\partial }{\partial r}{\mathrm{\Phi }}_l^{+}\left(r\right)+i\frac{l}{r}{\mathrm{\Phi }}_l^{+}\left(r\right)=\epsilon {\mathrm{\Phi }}_{l+1}^{-}\left(r\right)$$E10

$$-i\frac{\partial }{\partial r}{\mathrm{\Phi }}_{l+1}^{-}\left(r\right)-i\frac{l+1}{r}{\mathrm{\Phi }}_{l+1}^{-}\left(r\right)=\epsilon {\mathrm{\Phi }}_l^{+}\left(r\right).$$E11

This may be solved by inserting (10) into (11) to get a second differential equation that ${\mathrm{\Phi }}_l^{+}\left(r\right)$ can fulfill

$$\left(r^2\frac{{\partial }^2}{{\partial }^{2}r}+r\frac{\partial }{\partial r}+r^2{\epsilon }^2-l^2\right){\mathrm{\Phi }}_l^{+}\left(r\right)=0$$E12

where the last equation’s solutions are the Bessel functions $J_l\left(\mathit{\epsilon r}\right)$. Furthermore, the incident electron’s wave function moving along the x-direction ($x=rcos\varpi$) has the expression:

$${\psi }_l^i\left(r,\varpi \right)=\frac{1}{\sqrt{2}}\sum _l{i}^l\left[J_l\left(\mathit{kr}\right)e^{\mathit{il\varpi }}\left(\begin{array}{l}1\\ 0\end{array}\right)+i{J}_{l+1}\left(\mathit{kr}\right)e^{i\left(l+1\right)\varpi }\left(\begin{array}{l}0\\ 1\end{array}\right)\right]$$E13

as well as the reflected wave:

$${\psi }_l^r\left(r,\varpi \right)=\frac{1}{\sqrt{2}}\sum _l{i}^l{\alpha }_l\left[H_l^{\left(1\right)}\left(\mathit{kr}\right)e^{\mathit{il\varpi }}\left(\begin{array}{l}1\\ 0\end{array}\right)+i{H}_{l+1}^{\left(1\right)}\left(\mathit{kr}\right)e^{i\left(l+1\right)\varpi }\left(\begin{array}{l}0\\ 1\end{array}\right)\right]$$E14

wherein $H_l^{\left(1\right)}\left(\mathit{kr}\right)$ is the first class of Hankel function [31], ${\alpha }_l$ is the scattering parameters, and $k=\epsilon$ is the wave number.

The radial functions ${\mathrm{\Phi }}_l^{+}$ and ${\mathrm{\Phi }}_{l+1}^{-}$ are represented by the following equations inside the dot ($r\le \rho$):

$$i\left(\frac{\partial }{\partial r}-\frac{l}{r}\right){\mathrm{\Phi }}_l^{+}\left(r\right)+\left(\epsilon -U_{-}-\tau s_z{\mathrm{\Delta }}_{\mathit{so}}\right){\mathrm{\Phi }}_{l+1}^{-}\left(r\right)=0$$E15

$$i\left(\frac{\partial }{\partial r}+\frac{l+1}{r}\right){\mathrm{\Phi }}_{l+1}^{-}\left(r\right)+\left(\epsilon -U_{+}\right){\mathrm{\Phi }}_l^{+}\left(r\right)=0$$E16

Expressing (15) as

$${\mathrm{\Phi }}_{l+1}^{-}\left(r\right)=-\frac{i}{\epsilon -U_{-}-\tau s_z{\mathrm{\Delta }}_{\mathit{so}}}\left(\frac{\partial }{\partial r}-\frac{l}{r}\right){\mathrm{\Phi }}_l^{+}\left(r\right)$$E17

By substituting the Eq. (17) in (16), we find a differential equation for ${\mathrm{\Phi }}_l^{+}\left(r\right)$:

$$\left(r^2\frac{{\partial }^2}{{\partial }^{2}r}+r\frac{\partial }{\partial r}+r^2{\gamma }^2-l^2\right){\mathrm{\Phi }}_l^{+}\left(r\right)=0$$E18

where

$${\gamma }^2=\left(\epsilon -U_{+}\right)\left(\epsilon -U_{-}-\tau s_z{\mathrm{\Delta }}_{\mathit{so}}\right).$$E19

The solution of (18) can be worked out to get the transmitted wave as:

$${\psi }_l^t\left(r,\varpi \right)=\frac{1}{\sqrt{2}}\sum _l{i}^l{\beta }_l\left[J_l\left(\mathit{\gamma r}\right)e^{\mathit{il\varpi }}\left(\begin{array}{l}1\\ 0\end{array}\right)+i\mu J_{l+1}\left(\mathit{\gamma r}\right)e^{i\left(l+1\right)\varpi }\left(\begin{array}{l}0\\ 1\end{array}\right)\right]$$E20

where the ${\beta }_l$ denote the transmission coefficients and

$$\mu =\sqrt{\frac{\epsilon -U_{+}}{\epsilon -U_{-}-\tau s_z{\mathrm{\Delta }}_{\mathit{so}}}}.$$E21

Requiring the eigenspinors continuity at the boundary $r=\rho$ of the quantum dot,

$${\psi }_l^i\left(\rho \right)+{\psi }_l^r\left(\rho \right)={\psi }_l^t\left(\rho \right),$$E22

to obtain the conditions

$$J_l\left(\mathit{k\rho }\right)+{\alpha }_l{H}^{\left(1\right)}\left(\mathit{k\rho }\right)={\beta }_l{J}_l\left(\mathit{\gamma \rho }\right),$$E23

$$J_{l+1}\left(\mathit{k\rho }\right)+{\alpha }_l{H}_{l+1}^{\left(1\right)}\left(\mathit{k\rho }\right)=\mu {\beta }_l{J}_{l+1}\left(\mathit{\gamma \rho }\right).$$E24

Solving these equations to get the scattering coefficients:

$${\alpha }_l=-\frac{J_l\left(\mathit{\gamma \rho }\right)J_{l+1}\left(\mathit{k\rho }\right)-\mu J_{l+1}\left(\mathit{\gamma \rho }\right)J_l\left(\mathit{k\rho }\right)}{J_l\left(\mathit{\gamma \rho }\right)H_{l+1}^{\left(1\right)}\left(\mathit{k\rho }\right)-\mu J_{l+1}\left(\mathit{\gamma \rho }\right)H_l^{\left(1\right)}\left(\mathit{k\rho }\right)}$$E25

and the transmission coefficients by:

$${\beta }_l=\frac{J_l\left(\mathit{k\rho }\right)H_{l+1}^{\left(1\right)}\left(\mathit{k\rho }\right)-J_{l+1}\left(\mathit{k\rho }\right)H_l^{\left(1\right)}\left(\mathit{k\rho }\right)}{J_l\left(\mathit{\gamma \rho }\right)H_{l+1}^{\left(1\right)}\left(\mathit{k\rho }\right)-\mu J_{l+1}\left(\mathit{\gamma \rho }\right)H_l^{\left(1\right)}\left(\mathit{k\rho }\right)}$$E26

The density is described as $\mathit{j}={\psi }^{†}\mathit{\sigma }\psi$, with $\psi ={\psi }_r+{\psi }_i$ outside the dot and $\psi ={\psi }_t$ inside the dot.

Angular scattering is described by the far-field radial component of the reflecting current, which giving by

$$j_r={\psi }^{†}\left(cos\left(\varpi \right){\sigma }_x+sin\left(\varpi \right){\sigma }_y\right)\psi ={\psi }^{†}\left(\begin{array}{cc}0& e^{-\mathit{i\varpi }}\\ e^{\mathit{i\varpi }}& 0\end{array}\right)\psi$$E27

the corresponding radial current can be written as:

$$j_r^r=\frac{1}{2}\sum _{l=0}^{\infty }{A}_l\left(\mathit{kr}\right)\times \left(\begin{array}{cc}0& e^{-\mathit{i\varpi }}\\ e^{\mathit{i\varpi }}& 0\end{array}\right)\times \sum _{l=0}^{\infty }{A}_l^{\ast }\left(\mathit{kr}\right)$$E28

where

$$A_l\left(\mathit{kr}\right)={\left(-i\right)}^l\left[H_l^{\left(1\right)\ast }\left(\mathit{kr}\right)\left(\begin{array}{c}{\alpha }_l^{\ast }{e}^{-\mathit{il\varpi }}\\ {\alpha }_{-l-1}^{\ast }{e}^{\mathit{il\varpi }}\end{array}\right)-i{H}_{l+1}^{\left(1\right)\ast }\left(\mathit{kr}\right)\left(\begin{array}{c}{\alpha }_{-l-1}^{\ast }{e}^{i\left(l+1\right)\varpi }\\ {\alpha }_l^{\ast }{e}^{-\left(l+1\right)\varpi }\end{array}\right)\right]$$E29

By injecting the asymptotic behavior of the Hankel function of the first kind for $\mathit{kr}\gg 1$

$$H_l\left(\mathit{kr}\right)\simeq \sqrt{\frac{2}{\mathit{\pi kr}}}{e}^{i\left(\mathit{kr}-\frac{\mathit{l\pi }}{2}-\frac{\pi }{4}\right)},$$E30

into the Eq. (29), the density of the system (28) may be simplified to the following form:

$$j_r^r\left(\varpi \right)=\frac{4}{\mathit{\pi kr}}\sum _{l=0}^{\infty }\left[1+cos\left(2l+1\right)\varpi \right]{\left|c_l\right|}^2$$E31

where

$$\mid c_l\mid =\frac{1}{\sqrt{2}}{\left(\left[{\left|{\alpha }_{-\left(l+1\right)}\right|}^2+{\left|{\alpha }_l\right|}^2\right]\right)}^{12}.$$E32

The scattering cross section $\sigma$ is defined by [32]:

$$\sigma =\frac{I_r^r}{I^i/A_u}$$E33

where $I_r^r$ and $\left(I^i/A_u\right)$ denote, respectively, the total reflected flux across a concentric circle and the incident flux per unit area. Furthermore, $I_r^r$ is defined by:

$$I_r^r={\int }_0^{2\pi }{\mathit{rj}}_r^r\left(\varpi \right)\mathrm{d}\varpi =\frac{8}{k}\sum _{l=0}^{\infty }{\left|c_l\right|}^2.$$E34

For the incident wave (13), we note that the incident flux is equal to $I^i/A_u=1$.

To go deeper into our investigation of the scattering problem for a plane Dirac electron at various sizes of the circular quantum dot, we define the scattering efficiency $Q$ by splitting the scattering cross section by the geometric cross section. It is written by:

$$Q=\frac{\sigma }{2\rho }=\frac{4}{\mathit{k\rho }}\sum _{l=0}^{\infty }{\left|c_l\right|}^2.$$E35

3. Results and discussions

Figure 2a and b depict the scattering efficiency $Q$ for low energies under barriers scattering ($n$-$p$ junction) $\epsilon =0.01,0.02,0.04,0.06<U=1$, in terms of quantum dot radius $\rho$ for the spin-up state in the two valleys $k$($\tau =1$, $s_z=1$) and $k^{\prime }$($\tau =-1$, $s_z=1$) and the spin-down state in the two valleys $k$($\tau =1$, $s_z=-1$) and $k^{\prime }$($\tau =-1$, $s_z=-1$). We see that when $\rho \to 0$ implies that $Q\to 0$, and when $\rho$ rises, $Q$ raises to a maximum value $Q_{\mathit{max}}=20,25,39.5,68$ for $\epsilon =0.06,0.04,0.02,0.01$ for the state ($k^{\prime }$, $s_z$) (Figure 2a) and for the state ($k$, -$s_z$) (Figure 2b), then the scattering efficiency $Q$ has a highly damped oscillatory behavior with the appearance of net transverse resonant peaks, comparable to graphene quantum dots [32, 33, 34]. Moreover, the height of the peak decreases as the radius $\rho$ rises, but its breadth increases, which shows the peculiarities of the energy dispersion. Furthermore, by comparing Figure 2a and b, we remark that $Q$’s dependency on the spin-up and spin-down states in the two valleys is symmetrical, i.e. $Q\left(-\tau ,s_z\right)=Q\left(\tau ,-s_z\right)$.

Figure 2.

Scattering efficiency $Q$, for the potential $U=1$, in terms of the dot radius $\rho$ for different values of $\epsilon$ for the spin-up and spin-down states in two valleys $k$ and $k^{\prime }$.

Figure 2c and d present, respectively, the scattering efficiency $Q$ for energies across the barrier ($n$-$n$ junction) $\epsilon =1.16,1.2,1.3,1.5>U=1$, in terms of quantum dot radius $\rho$ for the spin-up state in the two valleys $k$ and $k^{\prime }$ as well as the spin-down state in valleys $k$ and $k^{\prime }$. However, when $\rho$ grows, the scattering efficiency $Q$ roughly linearly until it reaches a maximum value $Q_{\mathit{max}}=2.9$ relates to a certain value of $\rho$. Furthermore, by raising $\rho$ and for the four energy levels, the four curves display oscillating attitude [35]. In this domain ($\epsilon >U$), we find attitudes of symmetry $Q$ ($Q\left(-\tau ,s_z\right)=Q\left(\tau ,-s_z\right)$) with regard to $s_z$ and $\tau$ comparable with those of the previous domain ($\epsilon <U$).

To study the scattering for $\epsilon <U$ in more detail, we present in Figure 3 the scattering efficiency as a function of the electron energy $\epsilon$. In Figure 3a and b, we consider dots with small radius $\rho =0.9,01,1.2,1.3$, for the spin-up state in the two valleys $k$($\tau =1$) and $k^{\prime }$($\tau =1$) and the spin-down state in two valleys $k$($\tau =1$) and $k^{\prime }$($\tau =-1$).

Figure 3.

Scattering efficiency $Q$, for the potential $U=1$, in terms of the energy $\epsilon$ of the incident electron for distinct values of $\rho$ for the spin-up and spin-down states in two valleys $k$ and $k^{\prime }$.

Figure 3a shows that for $0\le \epsilon \le 0.6$, $Q$ exhibits a maximum for the spin-up state in the valley $k^{\prime }$($\tau =-1$, $s_z=1$) with the emergence of a single peak related to $\epsilon =0.55$ and a minima for the spin-up state in the valley $k$($\tau$, $s_z=1$ without any visible peaks. For $E>0.6$, we observe that $Q$ has a maximum for the spin-up state in the valley $k$($\tau =1$, $s_z=1$) with the emergence of a single peak appropriate for $\epsilon =0.75$ and a minima for the spin-up state in the valley $k^{\prime }$($\tau =-1$, $s_z=1$) without the emergence of peaks. For $E>0.6$, we notice that $Q$ has a maximum for the spin-up state in the valley $k$($\tau =1$, $s_z=1$) with the emergence of a single peak appropriate for $\epsilon =0.75$ and a minima for the spin-up state in the valley $k^{\prime }$($\tau =-1$, $s_z=1$) without the emergence of peaks.

Figure 3b indicates that for $0\le \epsilon \le 0.6$, $Q$ exhibits a maximum for the spin-down state in the valley $k$($\tau =1$, $s_z=-1$) with the emergence of a single peak relates to $\epsilon =0.55$ and a minima for the spin-up state in the valley $k^{\prime }$($\tau =1$, $s_z=-1$) without the emergence of peaks. We note that for $\epsilon >0.6$, $Q$ shows a maximum for the spin-down state in the valley $k^{\prime }$($\tau =-1$, $s_z=-1$) with the emergence of a single peak appropriate for $\epsilon =0.75$ and a minima for the spin-down state in the valley $k$($\tau =1$, $s_z=-1$) without the emergence of peaks. The electron scattering efficiency, on the other hand, is invariant by the transformation $Q\left(\tau ,s_z\right)\to Q\left(-\tau ,-s_z\right)$.

Figure 3c and d represent $Q$ in terms of $\epsilon$ for increasing values of $\rho =5,6.25,7,8.25$ for the spin-up state in the two valleys $k$($\tau =1$, $s_z=1$) and $k^{\prime }$($\tau =-1$, $s_z=1$) and the spin-down state in two valleys $k$($\tau =1$, $s_z=-1$) and $k^{\prime }$($\tau =-1$, $s_z=-1$), respectively. Additionally, we notice that $Q$ has significant maxima for low energies as well. However, when $\epsilon$ rises, we see the emergence of a peak with damped oscillations for both spin-up and spin-down states. The resonant excitation of the quantum dot’s normal modes causes these sharp peaks. As a result, the dependency of $Q$ on $s_z$ in the two valleys $k$ and $k^{\prime }$ is symmetric with regard to $\pm s_z$, i.e $Q\left(-\tau ,s_z\right)=Q\left(\tau ,-s_z\right)$.

The square modulus of the scattering ${\left|c_l\right|}^2$ is presented in Figure 4 for $l=0,1,2,3$ in terms of the energy $\epsilon$, for the spinup and spin-down states in the two valleys $k$($\tau =1$) and $k^{\prime }$($\tau =1$) for various size of the dot radius: Figure 4a and b: $\rho =2$, Figure 4c and d: $\rho =3$, Figure 4e and f: $\rho =4$ within all panels $U=1$. Except for the situation corresponding to $l=0$, all scattering coefficients are zero for zero or near to zero energy. Furthermore, when the energy increases, the scattering coefficients ${\left|c_l\right|}^2$ exhibit oscillatory behavior [35]. We may observe that using the spin-orbit interaction results in an increase in the number of oscillations. Furthermore, we see that some energy values, ${\left|c_l\right|}^2$, have strong peaks. Additionally, the resonances of the dot’s normal modes result in the high peaks previously seen for the scattering efficiency $Q$ in terms of energy (Figure 3). These findings indicate that the term spin-orbit interaction in Eq. (1) needs symmetry ${\left|c_l\right|}^2\left(-\tau ,s_z\right)={\left|c_l\right|}^2\left(\tau ,-s_z\right)$.

Figure 4.

Square modulus of the scattering coefficients ${\left|c_l\right|}^2$ in terms of the energy $\epsilon$ at $U=1$ for $l=0$ (blue curve), $1$ (red curve), $2$ (green curve), and $3$ (magenta curve). (a): ($k$, $k^{\prime }$, spin-up, $\rho =2$) states. (b): ($k$, $k^{\prime }$, spin-down, $\rho =2$) states. (c): ($k$, $k^{\prime }$, spin-up, $\rho =3$) states. (d): ($k$, $k^{\prime }$, spin-down, $\rho =3$) states, (e): ($k$, $k^{\prime }$, spin-up, $\rho =7.75$) states, and (f): ($k$, $k^{\prime }$, spin-down, $\rho =7.75$) states. Solid curve relates to valley $k$ and dashed curve relates to valley $k^{\prime }$.

For the spin-up and spin-down states, we graph the angular characteristic of the reflected radial component $j_r^r$ in terms of $\mathit{\text{varpi}}$ as shown in Figure 5. We note that $j_r^r$ has a maximum for $\varpi =0$ and a minimum for $\varpi =\pm \pi$. Furthermore, only forward scattering is preferred for the mode $c_0$ (Figure 5a and b). More favored scattering directions appear for higher modes. As a result, there are three preferred scattering directions for $l=1$ (Figure 5c and d). Additionally, there are five favored scattering directions for $l=2$ (Figure 5e and f) and seven preferred scattering directions for $l=3$ (Figure 5g and h). Generally, each mode has a $2l+1$ favored scattering direction that is apparent but with a different amplitude [30], although the mode ($l=0$) has a larger amplitude than the higher modes ($l>0$). The electron density profile at the dot reflects resonant scattering by only one of the normal modes. As a result, in both the up and down states, the dependency of $j_r^r$ on $\tau$ is symmetric with respect to $\pm \tau$ and $\pm s_z$, i.e. $j_r^r\left(-\tau ,s_z\right)=j_r^r\left(\tau ,-s_z\right)$.

Figure 5.

Radial component of the far-field scattered current $j_r^r$ in terms of angle $\varpi$ for various $l$ for the spin-up and spin-down states in two valleys $k$ and $k^{\prime }$ with fixed values $\rho =7.75$, $U=1$, and $\epsilon =0.0704$.

The radial component of the far-field scattered current $j_r^r$ as a function of incident energy is shown in Figure 6 for the states (a): ($k$($\tau =1$), $k^{\prime }$($\tau =-1$), spin-up), and (b): ($k$($\tau =1$), $k^{\prime }$($\tau =-1$), spin-down) for fixed values $U=1$ and $\rho =4$. In all $\varpi =0$ (red curve) and $\varpi =2\pi /3$ (blue curve) [30]. When $\epsilon \to 0$, $j_r^r$ for the two values of $\varpi$, when $\epsilon$ grows to the value $0.5$, we see the emergence of peaks of resonances with a maximum peak for $\left(\varpi ,\tau ,s_z\right)=\left(0,-1,1\right)$, as shown in Figure 6a). Although $j_r^r$ presents oscillatory behavior in the regime $0,5<\epsilon <1.5$, in the regime $\epsilon \ge 1.5$, $j_r^r$ exhibits damped oscillatory behavior with the symmetry $j_r^r\left(\varpi ,\tau ,s_z\right)=j_r^r\left(\varpi ,-\tau ,s_z\right)$. Figure 6b indicates that when $\tau \to -\tau$ and $s_z\to -s_z$, in an analogous method to write $j_r^r\left(\varpi ,-\tau ,s_z\right)=j_r^r\left(\varpi ,\tau ,-s_z\right)$, the behavior of $j_r^r$ is identical to that of Figure 6a.

Figure 6.

The radial component of the far-field scattered current $j_r^r$ in terms of the incident energy $\epsilon$ for various $\varpi$ with fixed values of $U=1$ and $\rho =2$ for the spin-up and spin-down states in two valleys $k$ and $k^{\prime }$.

The electron density profile around the dot reflects the resonant scattering of a single mode. The density inside the quantum dot is described by:

$${\psi }_t^{†}{\psi }_t={\left|{\beta }_l\right|}^2\left[{\left(j_l{\left(\mathit{\gamma r}\right)}^2+(\mu j_{l+1}\left(\mathit{\gamma r}\right)\right)}^2\right].$$E36

For the modes ${\beta }_0$, ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$, we represent the spatial density ${\psi }_t^{†}{\psi }_t$ in the quantum dot, as shown in Figure 7. For both spin-up and spin-down states, the modes ${\beta }_l$ have a maximum electron density at the center of the quantum dot, and as the size of the quantum dot rises, the electron density drops. We also show that the electron density is becoming significant as the angular monument $l$ is steadily increased. Furthermore, the spin-up and spin-down electron densities are not comparable; more interestingly, the electron density inside the dot has considerably risen, indicating momentary particle entrapment at scattering resonances.

Figure 7.

Spatial density profile ${\psi }_t^{†}{\psi }_t$ in the vicinity of the quantum dot for various $l$ for the spin-up and spin-down states in two valleys $k$ and $k^{\prime }$ for fixed values of $\rho =3$, $\epsilon =0.078$, and $U=1$.

4. Summary

The scattering of a planar Dirac electron wave on a circular quantum dot defined electrostatically in the ${\mathrm{MoS}}_2$ monolayer has been investigated. The scattering parameters ${\alpha }_l$ and ${\beta }_l$, which characterize the features of our systems, were calculated using the continuity equation at the quantum dot’s borders. The radial component of thev current density, as well as the scattering efficiency and square modulus of the scattering coefficient, were computed. In two energy regimes of the incoming electron, $\epsilon <U$ and $\epsilon \ge U$, the scattering of a planar Dirac electron wave has been examined.

In the regime $\epsilon <U$, where the incoming electron has a low energy, $Q$ exhibits a damped oscillatory behavior with emergent peaks owing to the excitation of the dot’s normal modes; tiny values of $\epsilon$ correlate with high amplitudes of $Q$. We have proven that $Q$ exhibits an oscillatory behavior in the other regime $\epsilon \ge U$. On the other hand, we discovered that $Q\left(-\tau ,s_z\right)=Q\left(\tau ,-s_z\right)$ has a remarkable valley symmetry.

To find the resonances, we looked at the energy dependence of the square module of the scattering parameters ${\left|c_l\right|}^2$. We discovered that only the lowest scattering coefficient is non-null near $\epsilon \to 0$, but as $\epsilon$ increased, the remaining coefficients began to notice significant contributions. The consecutive emergence of modes is interspersed with sudden and sharp peaks of various $\epsilon$, ${\left|c_l\right|}^2$, but for a not large $\epsilon$, we have shown that the sequential appearance of modes is interspersed with abrupt and sharp peaks of distinct ${\left|c_l\right|}^2$. We discovered that each mode has ($2l+1$) preferred paths of scattering visible with distinct amplitudes when it comes to the angular feature of the reflected radial component. Furthermore, we have demonstrated that the density of electrons inside the quantum dot has grown significantly, indicating that electrons have been temporarily trapped during the scattering resonances.