# Scattering in Monolayer Molybdenum Disulfide Quantum Dot

Abdelhadi Belouad<EMAIL_ADDRESS>L.P.M.C. Laboratory, Theoretical
Physics Group, Faculty of Sciences, Chouaïb Doukkali University, PO Box 20,
24000 El Jadida, Morocco  Abdellatif Kamal<EMAIL_ADDRESS>L.P.M.C. Laboratory, Theoretical Physics Group, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Department of
Mechanical Engineering, National Higher School of Arts and Crafts, Hassan II
University, Casablanca, Morocco  Rachid Houça<EMAIL_ADDRESS>L.P.M.C.
Laboratory, Theoretical Physics Group, Faculty of Sciences, Chouaïb Doukkali
University, PO Box 20, 24000 El Jadida, Morocco  Equipe de Physique Théorique
et Hautes Energies, Faculté des Sciences, université Ibn Zohr, PO Box 8106,
Agadir, Maroc  El Bouâzzaoui Choubabi<EMAIL_ADDRESS>L.P.M.C. Laboratory,
Theoretical Physics Group, Faculty of Sciences, Chouaïb Doukkali University,
PO Box 20, 24000 El Jadida, Morocco  Mohammed El Bouziani<EMAIL_ADDRESS>L.P.M.C. Laboratory, Theoretical Physics Group, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco

We investigate the propagation of electrons in a circular quantum dot of
monolayer Molybdenium disulfide $\mathrm{MoS_{2}}$, subjected to an electric
potential. Using the continuum model, we present analytical expressions for
the eigenstates, scattering coefficients, scattering efficiency, and radial
component of the reflected current and electron density. We identify two
scattering regimes as a function of physical parameters such as incident
electronic energy, potential barrier, and quantum dot radius. For the incident
electron low energy, we show that there is an appearance of scattering
resonances. Also, we note that the Far-field scattered current has distinct
preferred scattering directions.

PACS numbers: 73.21.La, 72.80.Vp, 73.20.At, 03.65.Nq
Keywords: Scattering, monolayer molybdenium disulfide, quantum dot, electric
potential, electron density.

##  1 Introduction

Graphene research [1, 2] has stimulated the search for new two-dimensional
materials. Among them, the transition-metals dichalcogenides (TMD). The
monolayers of metal dichalcogenides of TMD ($\mathrm{MX_{2}}$ such as
$\mathrm{M=Mo}$, $\mathrm{W}$; $\mathrm{X=S}$, $\mathrm{S_{e}}$, $\mathrm{T}$)
have very recently appeared as very promising nanostructures for various
applications both in the fields of optics, electronics, and spintronics.
Molybdenum disulfide($\mathrm{MoS_{2}}$) has been a strong material known for
many years, which has attracted significant attention because of its
interesting electrical and optical properties [3, 4, 5]. It is characterized
by a direct band gap [6, 7, 8, 9, 10] in the visible frequency range and by
excellent carrier mobility at room temperature [11, 12, 13, 14, 15]. This
makes it a good candidate for future electronic and optoelectronic
applications.

The $\mathrm{MoS_{2}}$ monolayer with its honeycomb atomic lattice is
characterized by a strong spin-orbit interaction. This will lead to completely
new electron spin properties. On the one hand, one would expect to have
conduction or valence electrons that are much less sensitive to the ultrafast
spin relaxation effects known for two-dimensional semiconductor structures
like $\mathrm{GaAs}$ quantum wells. On the other hand, the absorption of
circularly polarized light can generate a population of spin-polarized
electrons (i.e. with an imbalance between the number of _spin up_ and _spin
down_ electrons). Furthermore, the circularly polarized excitation makes it
possible to control the distribution of these electrons in one valley or
another of reciprocal space [16]. We thus speak of _valley polarization_ , and
it is this degree of freedom that we are currently seeking to better
characterize to possibly exploit it in information storage or processing
applications. The $\mathrm{MoS_{2}}$ monolayer can be thought of as a
semiconductor, the edges of the conduction and valence band are located at the
two corners of the Brillouin zone, i.e. the $k$ and $k^{\prime}$ points. This
gives electrons and holes an additional degree of freedom which can be used
for encoding information and further processing [16, 17, 18, 19].

Monolayer $\mathrm{MoS_{2}}$ quantum dots (QDs) possess distinct physical and
chemical properties, strong quantum confinement properties, edge effects [20]
and direct band gap, are important $\mathrm{MoS_{2}}$ based nanostructures
that have attracted considerable attention from research in recent years. To
date, several strategies for preparing $\mathrm{MoS_{2}}$ QD have been
proposed, solvothermal treatment synthesis [21], hydrothermal synthesis [22],
grinding exfoliation [23], liquid exfoliation in organic solvents [24],
electrochemical etching [25], and reaction processing [26].

The QDs exhibit edge states, which are determined by the curvatures of
conduction and valence bands, localized on the edges and with energies lying
in the band gap. The tight-binding model [22] was used to study the electronic
structure of $\mathrm{MoS_{2}}$ quantum dots. It was shown that it is possible
to build quantum dots with the same shape but having different electronic
properties due to the orbital asymmetry [22].

In this work, we study the electron propagation in a circular
electrostatically defined quantum dot monolayer molybdenium disulfide
$\mathrm{MoS_{2}}$, in the presence of a potential barrier. We identify
different scattering regimes depending on the radius of the quantum dot,
potential barrier and electron spin as well as the electron energy.

The present paper is organized as follows. In section 2, we present a
theoretical study of propagation wave plane of electrons in a circular quantum
dot of monolayer molybdenium disulfide $\mathrm{MoS_{2}}$. We give the
solutions of the spinors of the Dirac equation corresponding to each region of
different scattering parameters. We use the continuity condition in order to
calculate the scattering coefficients. In section 3, we analyze the scattering
efficiency, square modulus of the scattering coefficients, radial component of
the far-field scattered current and electron density, we discuss our results
by presenting different plot. In section 4, we present the basic conclusions
of the paper.

##  2 Theoretical model

Figure 1: The energy $E$ of the Dirac electrons propagating in monolayer
molybdenium disulfide $\mathrm{MoS_{2}}$ in a circular quantum dot. The dot is
characterised by its radius $R$ and the applied bias $V$. The incident plane
wave with energy $E>V$ (blue) corresponds to a state in the conduction band
(upper cone). The reflected wave (purple) also lies in the conduction band,
while the transmitted wave inside the dot (red) occupies the valence band
(lower cone).

We consider a quantum dot of radius $R$ in the presence of a electric
potentiel $V$, as illustrated in Figure 1. In the vicinity of the
$k$($\tau=1$) and $k^{\prime}$($\tau=-1$) valleys, by constructing the wave
functions through the basis of conduction and valence bands, the Dirac-Weyl
Hamiltonian for low-energy charge carriers in monolayer molybdenium disulfide
$\mathrm{MoS_{2}}$ reads [27, 28]

$H=H_{0}+\frac{\Delta}{2}\sigma_{z}+\frac{\lambda_{so}}{2}\tau
s_{z}(1-\sigma_{z})+V(r)$ (1)

such that $H_{0}$ is given by

$H_{0}=v_{F}\,\bm{\sigma}\cdot\bm{p}$ (2)

where $v_{F}$ is the Fermi velocity, $\bm{p}\\!\\!\\!=\\!\\!\\!(p_{x},p_{y})$
is the two-dimensional momentum operator,
$\bm{\sigma}\\!=\\!(\sigma_{x},\sigma_{y},\sigma_{z})$ are Pauli matrices
acting on the atomic orbitals, V(r) is the potential barrier, $\Delta=166$ meV
[28] is related to the material band gap, $s_{z}=\pm 1$ stands for the
electron _spin up_ and _spin down_ , $\tau=\pm 1$ stands for $k$ and
$k^{\prime}$ valleys, $\lambda_{so}=75$ meV [28] is the splitting of the
valence band due to spin-orbit coupling and $R$ is the dot radius. We use
units such that $\hbar=1$ and the Fermi velocity $v_{F}=1$.

We now discuss localized-state solutions in our system which is modeled as a
circularly symmetric quantum dot using the potential barrier $V(r)$ and the
energy gap $\Delta(r)$ respectively defined by

$V(r)=\left\\{\begin{array}[]{cc}0,&r>R\\\ V,&r\leq
R\end{array}\right.,\qquad\Delta(r)=\left\\{\begin{array}[]{cc}0,&r>R\\\
\Delta,&r\leq R\end{array}\right..$ (3)

We carry out our work by considering the polar coordinates (r, $\theta$), such
that the Hamiltonian (1) takes the form

$H=\begin{pmatrix}V_{+}&\pi^{-}\\\ \pi^{+}&V_{-}+\tau s_{z}\lambda_{so}\\\
\end{pmatrix}$ (4)

where we have set the two potentials and the two operators

$V_{\pm}=V\pm\dfrac{\Delta}{2},\qquad\pi^{\pm}=e^{\pm
i\theta}\left(-i\,\frac{\partial}{\partial
r}\pm\frac{1}{r}\frac{\partial}{\partial\theta}\right).$ (5)

The energy spectrum is determined by solving the eigenvalue equation

$H\,\psi_{m}(r,\theta)=E\,\psi_{m}(r,\theta).$ (6)

Since the operator for the total angular momentum,
$J_{z}=-i\,\hbar\,\partial_{\varphi}+\hbar\,\sigma_{z}/2$, commutes with $H$
by satisfying $[H,\leavevmode\nobreak\ J_{z}]=0$. This commutation requires
the separability of $\psi_{m}$ into the radial $\Phi^{\pm}(r)$ and angular
$\digamma^{\pm}(\theta)$ parts and then we have [29, 30]

$\psi_{m}(r,\theta)=\Phi_{m}^{+}(r)\,\digamma_{m}^{+}(\theta)+\Phi_{m+1}^{-}(r)\,\digamma_{m+1}^{-}(\theta)$
(7)

where the two angular components are

$\digamma_{m}^{+}(\theta)=\frac{e^{im\theta}}{\sqrt{2\pi}}\begin{pmatrix}1\\\
0\\\
\end{pmatrix},\qquad\digamma_{m+1}^{-}(\theta)=\frac{e^{i(m+1)\theta}}{\sqrt{2\pi}}\begin{pmatrix}0\\\
1\\\ \end{pmatrix}$ (8)

and $m=0,\leavevmode\nobreak\ \pm 1,\leavevmode\nobreak\ \pm
2,\leavevmode\nobreak\ \cdots$, is the total angular quantum number.

In order to get the solutions of the energy spectrum, we have to solve the
eigenvalue problem

$H\,\psi_{m}(r,\theta)=E\,\psi_{m}(r,\theta)$ (9)

by considering two regions according to Figure 1 : outside ($r>R$) and inside
($r\leq R$) the quantum dot. Thus, we have an incident wave $\psi_{i}$
propagation in the $x$-direction, the reflected wave $\psi_{r}$ is an outgoing
wave and a transmitted wave $\psi_{t}$ inside the dot.

Outside the dot ($r>R$), the radial components $\Phi_{m}^{+}(r)$ and
$\Phi_{m+1}^{-}(r)$ satisfy two coupled differential equations

$\displaystyle-i\,\frac{\partial}{\partial
r}\Phi_{m}^{+}(r)+i\,\frac{m}{r}\,\Phi_{m}^{+}(r)=E\,\Phi_{m+1}^{-}(r)$ (10)
$\displaystyle-i\,\frac{\partial}{\partial
r}\Phi_{m+1}^{-}(r)-i\,\frac{m+1}{r}\,\Phi_{m+1}^{-}(r)=E\,\Phi_{m}^{+}(r).$
(11)

which can be handled by injecting (10) into (11) to derive a second
differential equation satisfied by $\Phi_{m}^{+}(r)$

$\left(r^{2}\frac{\partial^{2}}{\partial^{2}r}+r\frac{\partial}{\partial
r}+r^{2}E^{2}-m^{2}\right)\Phi_{m}^{+}(r)=0$ (12)

where the solutions are the Bessel functions $J_{m}(Er)$. Moreover, the wave
function of the incident electron, propagating along $x$-direction
($x=r\cos\theta$), takes the form

$\displaystyle\psi^{i}_{m}(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sum_{m}i^{m}\,\left[J_{m}(kr)\,e^{im\theta}\begin{pmatrix}1\\\
0\\\ \end{pmatrix}+i\,J_{m+1}(kr)\,e^{i(m+1)\theta}\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}\right]$ (13)

as well as the reflected wave

$\psi^{r}_{m}(r,\theta)=\frac{1}{\sqrt{2}}\sum_{m}i^{m}\,\alpha_{m}\left[H^{(1)}_{m}(kr)\,e^{im\theta}\begin{pmatrix}1\\\
0\\\ \end{pmatrix}+i\,H^{(1)}_{m+1}(kr)\,e^{i(m+1)\theta}\begin{pmatrix}0\\\
1\\\ \end{pmatrix}\right]$ (14)

wehere $H^{(1)}_{m}(kr)$ is the Hankel functions of the first kind [31] and
$\alpha_{m}$ is the scattering coefficients and the wave number $k=E$.

Inside the dot ($r\leq R$), we obtain the following equations corresponding to
the radial functions $\Phi_{m}^{+}$ and $\Phi_{m+1}^{-}$

$\displaystyle i\left(\frac{\partial}{\partial
r}-\frac{m}{r}\right)\Phi_{m}^{+}(r)+\left(E-V_{-}-\tau
s_{z}\lambda_{so}\right)\Phi_{m+1}^{-}(r)=0$ (15) $\displaystyle
i\left(\frac{\partial}{\partial
r}+\frac{m+1}{r}\right)\Phi_{m+1}^{-}(r)+\left(E-V_{+}\right)\Phi_{m}^{+}(r)=0$
(16)

Expressing (15) as

$\Phi_{m+1}^{-}(r)=-\frac{i}{E-V_{-}-\tau
s_{z}\lambda_{so}}\left(\frac{\partial}{\partial
r}-\frac{m}{r}\right)\Phi_{m}^{+}(r)$ (17)

and replacing it in (16) we get a differential equation for $\Phi_{m}^{+}(r)$

$\left(r^{2}\frac{\partial^{2}}{\partial^{2}r}+r\frac{\partial}{\partial
r}+r^{2}\gamma^{2}-m^{2}\right)\Phi_{m}^{+}(r)=0$ (18)

where

$\gamma^{2}=\left(E-V_{+}\right)(E-V_{-}-\tau s_{z}\lambda_{so}).$ (19)

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

$\psi^{t}_{m}(r,\theta)=\frac{1}{\sqrt{2}}\,\sum_{m}i^{m}\,\beta_{m}\left[J_{m}(\gamma
r)\,e^{im\theta}\begin{pmatrix}1\\\ 0\\\ \end{pmatrix}+i\,\mu\,J_{m+1}(\gamma
r)\,e^{i(m+1)\theta}\begin{pmatrix}0\\\ 1\\\ \end{pmatrix}\right]$ (20)

where the $\beta_{m}$ denote the transmission coefficients and

$\mu=\sqrt{\frac{E-V_{+}}{E-V_{-}-\tau s_{z}\lambda_{so}}}.$ (21)

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

$\psi^{i}_{m}(R)+\psi^{r}_{m}(R)=\psi^{t}_{m}(R),$ (22)

to obtain the conditions

$\displaystyle J_{m}(kR)+\alpha_{m}\,H^{(1)}(kR)=\beta_{m}\,J_{m}(\gamma R),$
(23) $\displaystyle
J_{m+1}(kR)+\alpha_{m}\,H_{m+1}^{(1)}(kR)=\mu\,\beta_{m}\,J_{m+1}(\gamma R).$
(24)

Solving these equations to get the scattering coefficients

$\alpha_{m}=-\frac{J_{m}(\gamma R)\,J_{m+1}(kR)-\mu\,J_{m+1}(\gamma
R)\,J_{m}(kR)}{J_{m}(\gamma R)\,H^{(1)}_{m+1}(kR)-\mu\,J_{m+1}(\gamma
R)\,H^{(1)}_{m}(kR)}$ (25)

and the transmission coefficients by

$\beta_{m}=\frac{J_{m}(kR)\,H^{(1)}_{m+1}(kR)-J_{m+1}(kR)\,H_{m}^{(1)}(kR)}{J_{m}(\gamma
R)\,H^{(1)}_{m+1}(kR)-\mu\,J_{m+1}(\gamma R)\,H^{(1)}_{m}(kR)}$ (26)

The current density is defined by ${\bm{j}}=\psi^{{\dagger}}{\bm{\sigma}}\psi$
where inside the dot we have $\psi=\psi_{t}$, however, outside the dot we have
$\psi=\psi_{i}+\psi_{r}$.

The far-field radial component of the reflected current, which characterizes
angular scattering, reads

$j_{r}=\psi^{{\dagger}}\left(\sigma_{x}\cos\theta+\sigma_{y}\sin\theta\right)\psi=\psi^{{\dagger}}\begin{pmatrix}0&e^{-i\theta}\\\
e^{i\theta}&0\\\ \end{pmatrix}\psi$ (27)

the corresponding radial current can be written as

$\displaystyle
j^{r}_{r}=\frac{1}{2}\sum^{\infty}_{m=0}A_{m}(kr)\times\begin{pmatrix}0&e^{-i\theta}\\\
e^{i\theta}&0\\\ \end{pmatrix}\times\sum^{\infty}_{m=0}A_{m}^{*}(kr)$ (28)

where

$A_{m}(kr)=(-i)^{m}\left[H_{m}^{(1)*}(kr)\begin{pmatrix}\alpha_{m}^{*}\,e^{-im\theta}\\\
\alpha^{*}_{-m-1}\,e^{im\theta}\end{pmatrix}-i\,H_{m+1}^{(1)*}(kr)\begin{pmatrix}\alpha^{*}_{-m-1}\,e^{i(m+1)\theta}\\\
\alpha_{m}^{*}\,e^{-(m+1)\theta}\end{pmatrix}\right]$ (29)

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

$\displaystyle\displaystyle H_{m}(kr)\simeq\sqrt{\dfrac{2}{\pi
kr}}\,e^{i\left(kr-\frac{m\pi}{2}-\frac{\pi}{4}\right)},$ (30)

into (29), the current density (28) can be reduced to the following

$j^{r}_{r}(\theta)=\frac{4}{\pi
kr}\sum^{\infty}_{m=0}\left[1+\cos\left(2m+1\right)\theta\right]\,|c_{m}|^{2}$
(31)

where

$|c_{m}|^{2}=\frac{1}{2}\left[{|\alpha_{m}|^{2}+|\alpha_{-(m+1)}|^{2}}\right].$
(32)

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

$\displaystyle\sigma=\dfrac{I_{r}^{r}}{I^{i}/A_{u}}$ (33)

where $I_{r}^{r}$ is the total reflected flux through a concentric circle and
$(I^{i}/A_{u})$ is the incident flux per unit area. Moreover, $I_{r}^{r}$ is
given by

$I^{r}_{r}=\int_{0}^{2\pi}j^{r}_{r}(\theta)\,r\,\mathrm{d}\theta=\frac{8}{k}\sum^{\infty}_{m=0}|c_{m}|^{2}.$
(34)

We note that for the incident wave (13), we have $I^{i}/A_{u}=1$.

To go deeply in our study for the scattering problem for a plane Dirac
electron for different size of the circular quantum dot, we define the
scattering efficiency $Q$ by dividing the scattering cross section by the
geometric cross section. It is given by

$Q=\frac{\sigma}{2R}=\frac{4}{kR}\sum^{\infty}_{m=0}|c_{m}|^{2}.$ (35)

##  3 Results and discussions

In Figures 2(a) and 2(b), we show the scattering efficiency $Q$ for low
energies under the barrier scattering ($n$-$p$ junction)
$E=0.01,\leavevmode\nobreak\ 0.02,\leavevmode\nobreak\
0.04,\leavevmode\nobreak\ 0.04<V=1$, as a function of radius of quantum dot
$R$ 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
notice that as $R\rightarrow 0$, $Q\rightarrow 0$, when $R$ increases, $Q$
increases to a maximum value $Q_{max}=20,\leavevmode\nobreak\
25,\leavevmode\nobreak\ 39.5,\leavevmode\nobreak\ 68$ for
$E=0.06,\leavevmode\nobreak\ 0.04,\leavevmode\nobreak\
0.02,\leavevmode\nobreak\ 0.01$ respectively, for the state ($k^{\prime}$,
$s_{z}$) (Figure 2(a)) and for the state ($k$, -$s_{z}$) (Figure 2(b)), then
the scattering efficiency $Q$ has a strongly damped oscillatory behavior with
the appearance of net transverse resonant peaks, similar to graphene quantum
dots [32, 33, 34]. As the radius $R$ increases, the height of the peak is
reduced but its width becomes larger which shows the peculiarity of the energy
dispersion becomes apparent. Moreover, by comparing Figures 2(a) and 2(b), we
find that the dependence of $Q$ for the _spin up_ and _spin down_ states in
the two valleys is symmetrical i.e. $Q(-\tau,s_{z})=Q(\tau,-s_{z})$.

Figure 2: Scattering efficiency $Q$, for the potential $V=1$, as a function of
the dot radius $R$ at $E=0.01,\leavevmode\nobreak\ 0.02,\leavevmode\nobreak\
0.04,\leavevmode\nobreak\ 0.06$ for (a): the _spin up_ state in two valleys
$k$($\tau=1$, $s_{z}=1$) and $k^{\prime}$($\tau=-1$, $s_{z}=1$) and (b): the
_spin down_ state in two valleys $k$($\tau=1$, $s_{z}=-1$) and
$k^{\prime}$($\tau=-1$, $s_{z}=-1$) and at $E=1.16,\leavevmode\nobreak\
1.2,\leavevmode\nobreak\ 1.3,\leavevmode\nobreak\ 1.5$ for (c): the _spin up_
state in two valleys k($\tau=1$, $s_{z}=1$) and $k^{\prime}$($\tau=-1$,
$s_{z}=1$) and (d): the _spin down_ state in two valleys $k$($\tau=1$,
$s_{z}=-1$) and $k^{\prime}$($\tau=-1$, $s_{z}=-1$).

Figures 2(c) and 2(d) show, respectively, the scattering efficiency $Q$ for
energies over the barrier ($n$-$n$ junction) $E=1.16,\leavevmode\nobreak\
1.2,\leavevmode\nobreak\ 1.3,\leavevmode\nobreak\ 1.5>V=1$, as a function of
radius of quantum dot $R$ 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
$k^{\prime}$($\tau=-1$, $s_{z}=1$). Furthermore, by increasing $R$, the
scattering efficiency $Q$ increases almost linearly until a maximum value
$Q_{max}=2.9$ corresponds to a specific value of $R$. However, by increasing
$R$ and for the four values of energy, the four curves are exhibits
oscillatory behavior [35]. In this regime ($E>V$), we find behaviors of
symmetry $Q$ ($Q(-\tau,s_{z})=Q(\tau,-s_{z})$) with respect to $s_{z}$ and
$\tau$ similar to those of the preceding regime ($E<V$).

Figure 3: Scattering efficiency $Q$, for the potential $V=1$, as a function of
the energy E of the incident electron at $R=0.9,\leavevmode\nobreak\
1,\leavevmode\nobreak\ 1.2,\leavevmode\nobreak\ 1.3$ for (a): the _spin up_
state in two valleys $k$($\tau=1$, $s_{z}=1$) and $k^{\prime}$($\tau=-1$,
$s_{z}=1$) and (b): the _spin down_ state in two valleys $k$($\tau=1$,
$s_{z}=-1$) and $k^{\prime}$($\tau=-1$, $s_{z}=-1$) and
$R=5,\leavevmode\nobreak\ 6.25,\leavevmode\nobreak\ 7.25,\leavevmode\nobreak\
8.25$ for (c): the _spin up_ state in two valleys $k$($\tau=1$, $s_{z}=1$) and
$k^{\prime}$($\tau=1$, $s_{z}=1$) and (d): the _spin down_ state in two
valleys $k$($\tau=1$, $s_{z}=-1$) and $k^{\prime}$($\tau=-1$, $s_{z}=-1$).

To study the scattering for $E<V$ in more detail, we present in Figure 3 the
scattering efficiency as a function of the electron energy E. In figures 3(a)
and 3(b), We consider dots with small radii $R=0.01,\leavevmode\nobreak\
0.02,\leavevmode\nobreak\ 0.03,\leavevmode\nobreak\ 0.05$, 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 $k^{\prime}$($\tau=-1$).

In Figure 3(a), for $0\leq E\leq 0.6$ we show that $Q$ presents a maxima for
the state of _spin up_ in the valley $k^{\prime}$($\tau=-1$, $s_{z}=1$) with
the appearance of a single peak corresponding to $E=0.55$ and a minimum for
the _spin up_ state in the valley $k$($\tau$, $s_{z}=1$) without the
appearance of peaks. For $E>0.6$ we show that $Q$ presents a maxima for the
_spin up_ state in the valley $k$($\tau=1$, $s_{z}=1$) with the appearance of
a single peak suitable for $E=0.75$ and a minima for the _spin-up_ state in
the valley $k^{\prime}$($\tau=-1$, $s_{z}=1$) without the appearance of peaks.

In Figure 3(b), for $0\leq E\leq 0.6$ we show that $Q$ presents a maxima for
the state of spin-down in the valley $k$($\tau=1$, $s_{z}=-1$) with the
appearance of a single peak corresponds to $E=0.55$ and a minimum for the
spin-up state in the valley $k^{\prime}$($\tau=1$, $s_{z}=-1$) without the
appearance of peaks. for $E>0.6$ we show $Q$ presents a maxima for the _spin
down_ state in the valley $k^{\prime}$($\tau=-1$, $s_{z}=-1$) with the
appearance of a single peak suitable for $E=0.75$ and a minima for the _spin-
down_ state in the valley $k$($\tau=1$, $s_{z}=-1$) without the appearance of
peaks. However, the electron scattering efficiency is invariant under the
transformation $Q(\tau,\leavevmode\nobreak\ s_{z})\rightarrow
Q(-\tau,\leavevmode\nobreak\ -s_{z})$.

In Figures 3(c) and 3(d) we present $Q$ as a function of $E$ for large values
of $R=5,\leavevmode\nobreak\ 6.25,\leavevmode\nobreak\
7.25,\leavevmode\nobreak\ 8.25$ 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, we observe that $Q$ also
shows large maxima for low energies. But when $E$ increases, we observe the
appearance of a peak emerging with damped oscillations for both _spin up_ and
_spin down_ states. These sharp peaks are due to the resonant excitation of
the normal modes of the quantum dot. Consequently, in the two valleys $k$ and
$k^{\prime}$, the dependence of $Q$ on $s_{z}$ is symmetric with respect to
$\pm s_{z}$ i.e $Q(-\tau,\leavevmode\nobreak\
s_{z})=Q(\tau,\leavevmode\nobreak\ -s_{z})$.

Figure 4: Square modulus of the scattering coefficients $|c_{m}|^{2}$, with
the potential $V=1$, for $m=0$ (blue line), $1$ (red line), $2$ (green line),
$3$ (magenta line) as a function of the energy $E$ at $V=1$ for (a): ($k$,
$k^{\prime}$, _spin up_ , $R=2$) states, (b): ($k$, $k^{\prime}$, _spin down_
, $R=2$) states, (c): ($k$, $k^{\prime}$, _spin up_ , $R=3$) states, (d):
($k$, $k^{\prime}$, _spin down_ , $R=3$) states, (e): ($k$, $k^{\prime}$,
_spin up_ , $R=7.75$) states and (f): ($k$, $k^{\prime}$, _spin down_ ,
$R=7.75$) states. Solid line corresponds to valley $k$ and dashed line
corresponds to valley $k^{\prime}$ for all.

In Figure 4, we plot the square modulus of the scattering coefficients
$|c_{m}|^{2}$ for $m=0,\leavevmode\nobreak\ 1,\leavevmode\nobreak\
2,\leavevmode\nobreak\ 3$ as a function of the energy $E$, for the _spin up_
state ($s_{z}=1$) in the two valleys $k$($\tau=1$) and $k^{\prime}$($\tau=1$)
and the _spin-down_ state in the two valleys $k$($\tau=1$) and
$k^{\prime}$($\tau=1$) for different size of the dot radius : 4(a) and 4(b):
$R=2$, 4(c) and 4(d): $R=3$, 4(e) and 4(f): $R=4$ with in all panels $V=1$. We
observe that for zero energy or close to zero energy all scattering
coefficients are zero except the case corresponding to $m=0$. Also, by
increasing the energy the scattering coefficients $|c_{m}|^{2}$ tend to
oscillatory behavior [35]. We can see that the application of the spin-orbit
interaction leads to an increase in the number of oscillations. Moreover, we
note the some values of the energy, $|c_{m}|^{2}$ present sharp peaks.
Furthermore, these resonances of normal modes of the dot lead to the sharp
peaks already observed for the scattering efficiency $Q$ as a function of the
energy (Figure 3). These results show that the term spin-orbit interaction of
equation (1) requires symmetry $|c_{m}|^{2}(-\tau,\leavevmode\nobreak\
s_{z})=|c_{m}|^{2}(\tau,\leavevmode\nobreak\ -s_{z})$.

Figure 5: Radial component of the far-field scattered current $j^{r}_{r}$ as a
function of the angle $\theta$ for (a): ($k$, $k^{\prime}$, _spin up_ , $m=0$)
and (b): ($k$, $k^{\prime}$, _spin down_ , $m=0$) states, (c): ($k$,
$k^{\prime}$, _spin up_ , $m=1$) and (d): ($k$, $k^{\prime}$, _spin down_ ,
$m=1$) states, (e): ($k$, $k^{\prime}$, spin up, $m=2$)and (f): ($k$,
$k^{\prime}$, spin down, $m=2$) states, (g): ($k$, $k^{\prime}$, _spin up_ ,
$m=3$) and (h): ($k$, $k^{\prime}$, _spin down_ , $m=3$) states with $R=7.75$,
$V=1$ and $E=0.0704$.

In Figure 5, we plot the angular characteristic of the reflected radial
component $j^{r}_{r}$ as a function of $\theta$ for the _spin up_ and _spin
down_ states. We show that $j^{r}_{r}$ presents a maxima for $\theta=0$ and a
minima for $\theta=\pm\pi$. Moreover, for the mode $c_{0}$ (Figures 5(a) and
5(b)) only forward scattering is favored. While for higher modes more
preferred scattering directions emerge. There by, for $m=1$ (Figures 5 (c) and
5(d)) three preferred scattering directions.

However, for $m=2$ (Figures 5 (e) and 5(f)) five preferred scattering
directions and for $m=3$ (Figures 5 (g) and 5(h)) seven preferred scattering
directions. In general, each mode has $2m+1$ preferred scattering direction
observable but with different amplitude [30], however the mode ($m=0$)
presents a greater amplitude than the higher modes ($m>0$). Resonant
scattering by only one of the normal modes is also reflected by the electron
density profile near the dot. Consequently, in both the up and down states,
the dependence of $j^{r}_{r}$ on $\tau$ is symmetric with reference to
(regarding) $\pm\tau$ and $\pm s_{z}$, i.e.
$j^{r}_{r}(-\tau,s_{z})=j^{r}_{r}(\tau,-s_{z})$.

Figure 6: The radial component of the far-field scattered current $j_{r}^{r}$
as a function of the incident energy $E$ for the angles $\theta=2\pi/3$ (blue
line) and $\theta=0$ (red line) with $V=1$ and $R=2$. (a): ($k$, $k^{\prime}$,
_spin up_) and (b): ($k$, $k^{\prime}$, _spin down_) states.

In figure 6 we present the radial component of the far-field scattered current
$j_{r}^{r}$ as a function of the incident energy for the states (a):
($k$($\tau=1$), $k\textquoteright$($\tau=-1$), _spin up_) and (b):
($k$($\tau=1$), $k^{\prime}$($\tau=-1$), _spin down_) with $V=1$ and $R=4$. In
all $\theta=0$ (red line) and $\theta=2\pi/3$ (blue line) [30]. In Figure 6(a)
we show that when $E\rightarrow 0$, $j_{r}^{r}$ for the two values of
$\theta$, when $E$ increases to the value $0.5$, we observe the appearance of
peaks of resonances with a maximum peak for $(\theta,\leavevmode\nobreak\
\tau,\leavevmode\nobreak\ s_{z})=(0,\leavevmode\nobreak\
-1,\leavevmode\nobreak\ 1)$. While in the regime $05<E<1.5$, $j_{r}^{r}$ shows
an oscillatory behavior, moreover, in the regime $E\geq 1.5$, $j_{r}^{r}$
presents a damped oscillatory behavior with the symmetry
$j_{r}^{r}(\theta,\leavevmode\nobreak\ \tau,\leavevmode\nobreak\
s_{z})=j_{r}^{r}(\theta,\leavevmode\nobreak\ -\tau,\leavevmode\nobreak\
s_{z})$. In Figure 6(b) we show that the behavior of $j_{r}^{r}$ is similar to
that of Figure 6(a) when $\tau\rightarrow-\tau$ and $s_{z}\rightarrow-s_{z}$,
in an equivalent way to write $j_{r}^{r}(\theta,\leavevmode\nobreak\
-\tau,\leavevmode\nobreak\ s_{z})=j_{r}^{r}(\theta,\leavevmode\nobreak\
\tau,\leavevmode\nobreak\ -s_{z})$.

The resonant scattering of a single mode is also reflected by the electron
density profile near the dot. Inside the quantum dot, the density is given by

$\psi_{t}^{\dagger}\psi_{t}=|\beta_{m}|^{2}\left[\left(j_{m}(\gamma
r)^{2}+(\mu\,j_{m+1}(\gamma r)\right)^{2}\right].$ (36)

Figure 7: Spatial density profile $\psi_{t}^{\dagger}\psi_{t}$ in the vicinity
of the quantum dot for (a): ($k$, $k^{\prime}$, _spin up_ , $m=0$) and (b):
($k$, $k^{\prime}$, _spin down_ , $m=0$) states (c): ($k$, $k^{\prime}$, _spin
up_ , $m=1$) and (d): ($k$, $k^{\prime}$, _spin down_ , $m=1$) states (c):
($k$, $k^{\prime}$, _spin up_ , $m=2$) and (d): ($k$, $k^{\prime}$, _spin
down_ , $m=2$) states for $R=3$, $E=0.078$ and $V=1$.

In figure 7, we plot the spatial density $\psi_{t}^{\dagger}\psi_{t}$ in the
quantum dot for the modes $\beta_{0}$, $\beta_{1}$, $\beta_{2}$ and
$\beta_{3}$. The modes $\beta_{m}$ has a maximum electron density in the
center of the quantum dot and when the size of quantum dot increases the
electron density decreases for both _spin up_ and _spin down_ states. By
gradually increasing the angular monument m, we also show that the electron
density becomes important. In addition, the _spin up_ electron density and the
_spin down_ electron density are not similar, more importantly, the electron
density inside the dot is greatly increased, which is a sign of temporary
trapping from particles to scattering resonances.

##  4 Conclusion

We have studied the scattering of a plane Dirac electron wave on a circular
quantum dot defined electrostatically in the monolayer of $\mathrm{MoS_{2}}$.
We used the boundary conditions at the edges of the quantum dot to determine
the scattering coefficients $\alpha_{m}$ and $\beta_{m}$ which describe the
characteristics of our systems. The scattering efficiency, the square modulus
of the scattering coefficient, and the radial component of the current density
were calculated. The scattering of a plane Dirac electron wave has been
studied in two energy regimes of the incident electron $E<V$, and $E\geq V$.

For the regime $E<V$, low energy of the incident electron, $Q$ presents a
damped oscillatory behavior with the appearance of emerging peaks due to the
excitation of the normal modes of the dot appear, small values of $E$
correspond to large amplitudes of $Q$. For the other regime $E\geq V$, we have
shown that $Q$ has an oscillatory behavior. On the other hand, we observed a
remarkable valley symmetry $Q(-\tau,\leavevmode\nobreak\
s_{z})=Q(\tau,\leavevmode\nobreak\ -s_{z})$.

To identify the resonances, we have studied the energy dependence of the
square module of the scattering coefficients $|c_{m}|^{2}$ it was found that
near $E\rightarrow 0$, only the lowest scattering coefficient is non-null but
with the increase of $E$ the remaining coefficients started to show some
contributions. For larger $E$, $|c_{m}|^{2}$ tend to have oscillatory
behavior, but for a not too large $E$ we have seen that the successive
appearance of modes is interspersed with sudden and sharp peaks of different
$|c_{m}|^{2}$. Regarding the angular characteristic of the reflected radial
component, we have found that each mode has ($2m+1$) preferred directions of
scattering observable with different amplitudes. Moreover, we have shown that
the density of electrons inside the quantum dot are considerably increased,
which is a sign of temporary trapping of electrons during the scattering
resonances.

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[table]capposition=top

# Wireless Powering Internet of Things with UAVs: Challenges and Opportunities

Yalin Liu, Hong-Ning Dai, Qubeijian Wang, Muhammad Imran, Nadra Guizani Y.
Liu, is with Faculty of Information Technology, Macau University of Science
and Technology, Macau. e-mail<EMAIL_ADDRESS>Dai is with
Department of Computing and Decision Sciences, Lingnan University, Hong Kong.
email<EMAIL_ADDRESS>Wang is with School of Cyber Security, Northwestern
Polytechnical University, China. email<EMAIL_ADDRESS>Imran is
with the School of Engineering, Information Technology & Physical Sciences,
Federation University Australia, Brisbane, Australia. email:
<EMAIL_ADDRESS>Guizani is with the School of Electrical and Computer
Engineering at Washington State University, U.S.A. email:
<EMAIL_ADDRESS>
###### Abstract

Unmanned aerial vehicles (UAVs) have the potential to overcome the deployment
constraint of Internet of Things (IoT) in remote or rural area. Wirelessly
powered communications (WPC) can address the battery limitation of IoT devices
through transferring wireless power to IoT devices. The integration of UAVs
and WPC, namely UAV-enabled Wireless Powering IoT (Ue-WPIoT) can greatly
extend the IoT applications from cities to remote or rural areas. In this
article, we present a state-of-the-art overview of Ue-WPIoT by first
illustrating the working flow of Ue-WPIoT and discussing the challenges. We
then introduce the enabling technologies in realizing Ue-WPIoT. Simulation
results validate the effectiveness of the enabling technologies in Ue-WPIoT.
We finally outline the future directions and open issues.

##  I Introduction

We have witnessed the proliferation of IoT, which has been widely adopted in
diverse urban applications, such as smart home, smart healthcare, smart
industry, smart grid. IoT has been typically deployed in the scenarios with
the availability of communications infrastructure, such as base stations,
access points and IoT gateways. However, the deployment and maintenance of
infrastructure nodes inevitably bring huge operational expenditure. Moreover,
it is difficult to deploy wireless infrastructure nodes at remote or rural
area (e.g., forest surveillance and livestock monitoring). In addition, IoT
devices are also life-limited due to their built-in battery limitations. These
two fundamental constraints of IoT prevent its wide deployment in remote
scenarios.

The recent advances in UAVs bring opportunities to overcome the limitations of
IoT. Related work has regard UAVs as elastic communication nodes to increase
the communication coverage and enhance the network capacity [1]. Particularly,
UAVs have the advantage in aerial mobility in contrast to ground vehicles. For
example, UAVs can be flexibly dispatched to remote, rural or disaster area
where ground vehicles can hardly or cannot directly reach [1]. Hence, UAVs can
potentially foster remote IoT applications such as environment monitoring,
farm observation and emergency communications.

Radio frequency (RF)-based Wireless Power Transfer (WPT) is a promising method
to prolong the life of the battery-limited nodes. In previous literature, RF-
WPT has been presented to support wireless communications. The integration of
RF-WPT and wireless communications leads to a new type of wireless
communications called WPC [2]. Particularly, WPC has two main application
scenarios: WPC networks and simultaneous wireless information and power
transfer (SWIPT). In WPC networks, IoT nodes first harvest wireless energy,
which is then used for data transmission. SWIPT aiming to achieve WPT and
information transmission simultaneously in the same channel has a critical
hardware requirement on IoT nodes. Therefore, WPC is more preferred for
energy-limited IoT devices, thereby overcoming the second constraint of IoT.

[16.2cm] [][]  []  []  Figure 2: Wake-up procedure Figure 3: Wireless powering
and data transmission

Figure 1: Applications of Ue-WPIoT Figure 4: An overview of Ue-WPIoT

The integration of UAVs with WPC brings opportunities to extend the IoT
applications from cities to remote or rural areas. Combining with WPC, a UAV
has the capability to supply energy to IoT nodes for their data transmission.
We name such UAV-enabled wireless powering Internet of Things as Ue-WPIoT.
However, the realization of Ue-WPIoT suffers a number of challenges including
the limited powering range, energy efficiency optimization, flying trajectory
design. This article aims at investigating solutions to those challenges when
realizing Ue-WPIoT. As a summary, this article has the following
contributions.

We present Ue-WPIoT to support remote IoT applications (in Section II). In Ue-
WPIoT, IoT nodes that sleep (or hibernate) to save the energy can be activated
by the wake-up signals emitted from a UAV. Consequently, the UAV can locate
the IoT nodes and then transmit wireless energy to IoT nodes. Thereafter, IoT
nodes can transmit the collected data to the UAV via using the harvested
energy. Ue-WPIoT can essentially enable a plethora of applications including
forest deforestation monitoring, livestock monitoring, water inspection, farm
observation and forest fire surveillance as shown in Fig. 4.

We then discuss the challenges before the realization of Ue-WPIoT. Firstly,
the RF-based WPT usually experiences high attenuation over a distance, thereby
limiting the effective communication range. Secondly, the overall energy
efficiency during WPT and data transmission needs to be optimized. Thirdly,
the trajectory design of UAVs needs to be optimized after the joint
consideration of multi-node communications and the travelling path length
together.

We next introduce enabling technologies to address the challenges in realizing
Ue-WPIoT (in Section III). Specifically, we adopt the adaptive energy beam-
forming technologies in the UAV, thereby extending the powering distance.
Moreover, we formulate a common optimization model to optimize the energy
efficiency in Ue-WPIoT. Finally, we design the optimal flying trajectory of
the UAV. The numerical results also validate the effectiveness of the
promising technologies (in Section IV). We also outline the future directions
in Ue-WPIoT (in Section V).

##  II UAV-enabled Wireless Powering IoT

###  II-A Working flow

Fig. 4 presents the applications of Ue-WPIoT and the working procedure of Ue-
WPIoT. As shown in Fig. 4, we mainly consider a fixed-wing UAV with sufficient
battery capacity or fuels to support a long flight to remote or rural area.
Ue-WPIoT can support a diversity of applications including deforestation
monitoring, livestock monitoring, water inspection, farm observation and
forest fire surveillance. In these scenarios, a UAV can first transfer
wireless energy to charge IoT nodes which send back the collected data to the
UAV. One UAV-IoT communication procedure consists of two stages as shown in
Fig. 4 and Fig. 4.

We next describe the working flow of one Ue-WPIoT communication as follows: 1)
Wake-up procedure. Since IoT nodes have usually been in sleeping or
hibernating mode to save energy, the UAV wakes up IoT nodes by transmitting
the Wake-up Radio (WuR) signals [3] (i.e., Step  ❶ as shown in Fig. 4). Then
IoT nodes can be activated when they detect enough power from the wake-up
radio signals. The activated IoT nodes transmit the feedback signals to the
UAV which can then identify the activated IoT nodes (i.e., Step  ❷ as shown in
Fig. 4). In the Ue-WPIoT system, each UAV is equipped with an antenna array,
which can effectively differentiate arrivals of multiple feedback signals and
locate the precise orientation of each IoT node [4]. 2) Wireless powering and
data transmission. The UAV next transfers wireless energy toward the activated
IoT nodes. With an antenna array, the UAV is capable of generating a sharp
beam toward IoT nodes, thereby improving the wireless powering efficiency of
IoT nodes [5] (i.e., Step  ❸ as shown in Fig. 4). IoT nodes can leverage the
harvested energy to transmit the data to the UAV (i.e., Step  ❹ as shown in
Fig. 4).

During the above procedure, Steps  ❶ and  ❷ can activate and then locate the
IoT nodes, thereby paving the way for the following wireless power transfer
and data transmission. It is worth mentioning that the performance of data
transmission greatly relies on the harvested energy in the previous wireless
powering step. Thereafter, IoT nodes can harvest enough energy from the UAV
(i.e., Step  ❸ ) to support the data transmission in Step  ❹ . Ue-WPIoT can
essentially support multiple concurrent communications since multiple nodes
can be activated as the same time when they are close to each other. In this
case, a multi-access mechanism (e.g., time-division-multiple-access or spatial
division-multiple-access) shall be adopted to support multiple concurrent
communications with different IoT nodes.

###  II-B Design Challenges

Ue-WPIoT can promote the wide adoption of UAVs to connect IoT nodes in remote
or rural area while the realization of Ue-WPIoT is faced with the following
design challenges.

  * •

Feasible Communication Range. Generally, WPT has a much shorter range than
data transmission as indicated in [6]. Thus, the feasible communication range
of Ue-WPIoT is essentially limited by the achievable WET range. The
communication range constraint may limit the wide application of Ue-WPIoT in
different scenarios.

  * •

Energy-efficiency optimization. UAVs are also suffering from the limited
energy to support both flight propulsion and Ue-WPIoT communications. The
energy consumption of the whole flight occupies the major proportion of the
entire energy consumption. Even though UAVs can harvest energy from the
ambience (e.g., solar panels), the ambience energy sources being susceptible
to environment fluctuations cannot be the stable energy sources for UAVs.
Thus, it is a critical issue to optimize the energy-efficiency with
consideration of flight time, multi-access and trajectory design.

  * •

Flying Trajectory Design. In Ue-WPIoT, a UAV is expected to conduct tasks to
visit multiple IoT nodes along a given trajectory. It is challenging to design
the optimal trajectory, which requires the joint consideration of multiple
factors, such as multi-node access, multi-task scheduling and the total flight
time.

TABLE I: Summary of Enabling Technologies to Address Ue-WPIoT Challenges Challenges |  Enabling Technologies
---|---
Feasible Communication Range |  $\cdot$ Adaptive energy beam-forming
$\cdot$ IoT EH Circuit Design
Energy-efficiency optimization |  Resource allocation and optimization
Flying trajectory design |  Joint trajectory optimization of multi-access and WPC

##  III Enabling Technologies

This section discusses several enabling technologies to address the above
challenges of Ue-WPIoT. Table I summarizes the state-of-the-art solutions in
different aspects.

###  III-A Adaptive Energy Beamforming and IoT EH Circuit Design

Ue-WPIoT is suffering from the limited communication range, which is mainly
constrained by the WET range. The fundamental limitation of the WET range lies
in much higher threshold for energy harvesting (EH) than that for information
decoding [6]. The advent of energy beam-forming (BF) technology brings
opportunities to overcome this limitation. In particular, the UAV employing a
BF antenna can generate the BF energy toward a certain direction. The focused
energy can significantly extend the powering distance.

In Ue-WPIoT, the adoption of adaptive energy BF technology (AE-BF) can greatly
improve the WET efficiency at IoT nodes. Specifically, a UAV with AE-BF
capability can generate BF radio signal toward a certain orientation thereby
improve the WET efficiency due to the focused energy. The adaptive BF is
usually constructed via the prior-knowledge of channel state information
(CSI), which can be obtained after analysing the feedback signals from the
activated IoT nodes (i.e., Step  ❷ ). In addition, the adaptive BF vector can
be optimized by maximizing the harvested power of the IoT nodes as in [5].

With respect to the hardware design of IoT nodes, it may not be feasible to
directly adopt the adaptive BF technology at IoT devices because it is
difficult to equip expensive and bulky BF antennas at IoT nodes. However,
appropriate configurations of IoT devices can extend the EH distance. For
example, the sensitivity adjustment of the EH circuits can extend the powering
distance [6]. Another design consideration is the radio frequency. In previous
studies, most of prototypes of WuRs/energy harvesters are based on the
frequency about 2.4GHz, 400MHz, and 900MHz, where 2.4GHz-based circuits are
more compatible for 802.11-based wireless communication networks, while 400MHz
or 900MHz based circuits may result in longer communication range.

###  III-B Resource Allocation and Optimization

In Ue-WPIoT, practical powering process is susceptible to both the channel
fluctuation and path loss effect since the IoT nodes can only transmit their
data after harvesting enough wireless energy from the UAV. Meanwhile,
different IoT nodes may gain a varied portion of wireless energy due to
variations of air-to-ground (A2G) channels. Thus, it is necessary to design
optimal resource allocation strategies for both UAVs and IoT nodes to optimize
the energy efficiency of the entire Ue-WPIoT system.

To solve the resource allocation and optimization, we put forth an
optimization framework with the objective of minimizing the joint energy-and-
latency cost during a Ue-WPIoT communication period. In this optimization
problem, the energy cost is represented by the supplied energy emitted from
the UAV, and the latency cost is essentially the overall time spending on a
Ue-WPIoT communication process, depending on the summation of powering time
and data transmission time (i.e., Steps  ❸ and  ❹ ). It is worth mentioning
that the UAV may consume a large portion of the hovering propulsion energy
during the Ue-WPIoT communication process. The optimization is subject to the
constraints including the sufficient harvested energy to ensure the successful
data transmission, the limited latency to ensure the communication task being
completed within a given time, and also the feasible values for system
parameters (e.g., the WPT power). It is non-trivial to solve this optimization
problem due to the multiple factors, convexity and dynamicity of the Ue-WPIoT
communication. The solutions to this optimization problem may depend on
several optimal variables, such as BF vector, powering duration time, the data
transmission power emitted from IoT nodes.

Moreover, this optimization framework shall also consider multiple concurrent
communications. In particular, the optimal resource allocation scheme needs to
be adjusted to power multiple activated nodes which can then transmit the data
back to the UAV. Regarding the concurrent multi-node powering strategy, the
energy BF vector at the UAV needs to be optimized, thereby maximizing the
received power at multiple nodes. In the data transmission procedure, the UAV
then needs to conduct a fine-grained task scheduling to fulfill the multi-
access data transmission. Particularly, time-division-multiple-address (TDMA),
space-division multiple access (SDMA) and orthogonal frequency-division
multiple access (OFDMA) schemes can be adopted to support multiple accesses
[7]. In TDMA, each IoT node will be assigned with a time slot, during which
the data can be transmitted to the UAV. Consequently, the data transmission
from multiple activated IoT nodes will be scheduled one after the other. The
SDMA scheme leverages the BF technology to spatially separate the multiple
data transmissions. Specifically, the UAV can adjust a dedicated BF vector for
each IoT node through the position estimation of IoT nodes from the feedback
signals in the wake-up procedure (i.e., Step  ❷ ).

###  III-C UAV Trajectory Optimization

The UAV trajectory in Ue-WPIoT needs to be carefully designed with
consideration of multiple factors together. Different from the UAV trajectory
design in existing studies, which merely consider the shortest-path solutions
to cover all the IoT nodes [8], Ue-WPIoT needs to a joint consideration of
multi-node access, multi-task scheduling and energy efficiency optimization.
First, the UAV trajectory should be as short as as possible to save the total
flight time in Ue-WPIoT. Moreover, multiple nodes can be simultaneously
visited by the UAV thanks to the multi-node access mechanisms as discussed in
Section III-B. In this way, the total flight time can be saved.

The optimal trajectory in Ue-WPIoT can be designed accordingly. In particular,
the UAV can activate and power a set of nodes; this set of nodes is called a
WPC group. In the optimal trajectory design, the UAV selects one node from a
WPC group as a _traversal_ point. The UAV can simultaneously power and
communicate with other nodes within in the same WPC group. Then, the optimal
trajectory becomes the shortest path to visit every traversal point (instead
of visiting every IoT node). Intuitively, this trajectory design can achieve a
shorter travelling path length than that of a pure shortest-path strategy,
which visits every node in one-by-one manner. Ref. [9] presents a similar
trajectory design, which also partitions the whole network into several
regions. With each region, one node is selected as the visited point in each
region though this design only considers geographical partitions of nodes. In
contrast, our design optimizes the trajectory with consideration of multi-node
access and WPC.

In our optimal trajectory design, the number of nodes within a WPC group has
the impact on the overall travelling path length. For example, more nodes
within a WPC group may lead to a shorter travelling path. The number of nodes
within a WPC group depends on the flying height of the UAV. In particular, we
denote the flying height of the UAV by $H$, the maximum powering distance by
$d_{EH}$, which is essentially the distance between the UAV and the IoT node.
We then project the powering distance to the 2D plane in order to calculate
the coverage area in the plane. The horizontal coverage range (i.e., the
projection $d_{EH}$ on the plane) is denoted by a ground circle with radius
$R$, which needs to fulfill the condition $R=\sqrt{d_{EH}^{2}-H^{2}}$
according to the triangular relation. When the WPT power is fixed (i.e.,
$d_{EH}$ is fixed), the lower flying height of the UAV leads to more IoT nodes
to be covered, implying a shorter travelling path. The numerical results as
shown in Section III-C will further confirm this observation.

##  IV Numerical results and analysis

This section provides numerical results to demonstrate the effectiveness of
enabling technologies mainly from the following two aspects.

###  IV-A Achievable EH distance and Data rate

We first present simulation results on the achievable EH distance and the data
transmission rate in Ue-WPIoT. We consider that an IoT node can harvest the
wireless energy emitted by one UAV and consequently use the harvested energy
to transmit the sensory data to the UAV. It is worth mentioning that the
achievable EH distance (i.e., the maximum WET range) can be derived by
analyzing the radio propagation condition between the UAV and one IoT node.
The data rate can be obtained by solving the optimization problem of
minimizing overall energy consumption of the UAV for WET. In particular, the
achievable data rate is derived after substituting the optimal data
transmission power into Shannon capacity, where the optimal data transmission
power power is mainly determined by the harvested power at the IoT node.

In our simulations, the A2G channel model is determined by the probability-
based Line of Sight (LoS)/Non Line of Sight (NLoS) path loss under suburban
geographical parameters. It is worth mentioning that the LoS component usually
dominates the A2G channel when the UAV is in the high altitude [1] while the
NLoS component may have higher impact than the LoS component especially in the
low altitude, where there are obstacles such as trees and bushes [10]. Thus,
the probability-based LoS and NLoS model can take both the cases into account,
especially considering the remote or rural area. Moreover, the energy
transmitting power at the UAV is fixed at $10\mathrm{W}$. The energy
conversion efficiency at the IoT node is set as $0.3$. The communication
bandwidth for data transmission is $15\mathrm{MHz}$. We choose different AE-BF
settings (as discussed in Section III-A) to evaluate the achievable powering
distance and data rate.

  1. 1.

We consider that a UAV is equipped with an omnidirectional antenna or a BF
antenna consisting of multiple antenna elements. In particular, we denote the
number of antenna elements by $N$, which is equal to 1, 16, and 32, where 1
antenna element denotes an omnidirectional antenna (or an isotropic antenna);
$N=16$ and $N=32$ represent 16 antenna elements and 32 antenna elements,
respectively. Generally, the more antenna elements imply the higher antenna
gain (the more directivity toward a direction).

  2. 2.

We adopt three carrier frequencies, 400MHz, 900MHz, and 2.4GHz for WPT. Note
that each IoT receiver (i.e., energy harvester) has its input power threshold
corresponding to the different carrier frequencies. In particular, as
indicated in the latest study [11], the 400MHz-based energy harvester requires
the input power at least $-20\mathrm{dBm}$, while the 900MHz-based energy
harvester and the 2.4GHz-based energy harvester can support the input power
threshold of $-23\mathrm{dBm}$ and $-50\mathrm{dBm}$, respectively.

(a) Under carrier frequency 400MHz

(b) Under 32 antenna elements

Figure 5: Harvested power at the IoT node versus the distance, carrier
frequency and number of antenna elements, with the energy transmitting power
being $10\mathrm{W}$ and the energy conversion efficiency being $0.3$.

The above BF antenna can be realized by a uniform planar array (UPA) [12]. For
instance, a 32-element UPA in $4\times 8$ element-array with 400MHz carrier
frequency has an approximate size about $1.125\times 2.625\mathrm{m}^{2}$.
Equivalently, 32-elements UPA antennas with 900MHz carrier frequency and
2.4GHz have approximate sizes about $0.5\times 1.16\mathrm{m}^{2}$ and
$0.1875\times 0.4375\mathrm{m}^{2}$, respectively. Large size UAVs such as
MQ-1 Predator drones (used for US Army) and civilian drones like Zipline
drones111https://spectrum.ieee.org/robotics/drones/in-the-air-with-ziplines-
medical-delivery-drones. can be potentially adopted for WPC tasks in the
future.

Fig. 5 presents the harvested power at IoT node versus the distance with
varied carrier frequencies and different numbers of antenna elements. In
particular, Fig. 5a plots the harvested power at IoT node versus the distance
when the carrier frequency is fixed at 400MHz. We observe from Fig. 5a that
the harvested power drops dramatically with the increased distance due to the
path loss effect over the long distance. Moreover, Fig. 5a also shows that the
increased number of antenna elements can counteract the path loss effect. For
example, when the number of antenna elements is 32, the harvested power at
$10\mathrm{m}$ is still above $-20\mathrm{dBm}$ (i.e., the threshold of input
circuit at the harvester) while omnidirectional antenna does not reach this
threshold. This is because the more antenna elements implies the higher
antenna gain (i.e., the more directivity of an antenna).

Moreover, the carrier frequency also affects the harvested power. Fig. 5b
plots the harvested power at IoT node versus the distance when the number of
antenna elements is fixed at 32. We observe from Fig. 5b that the lower
frequency can lead to a longer achievable energy harvested range. For example,
the achievable EH distance is $13\mathrm{m}$ when the carrier frequency is
400MHz and the minimum input power $-19\mathrm{m}$.

(a) Under carrier frequency 400MHz

(b) Under 32 antenna elements

Figure 6: Achievable data rate versus the distance, carrier frequency and
antenna elements, with the bandwidth being 15MHz.

Fig. 6 presents the achievable data rate versus the distance with varied
carrier frequencies and different numbers of antenna elements. Obviously, the
achievable data rate has a similar trend to the harvested power. In
particular, Fig. 6a plots the achievable data rate when the carrier frequency
is fixed at 400MHz. We also find that the achievable data transmission rate
decreases with the increased distance due the path loss effect while the more
antenna elements can compensate for the path loss. Fig. 6b plots the
achievable data rate versus the distance when the number of the antenna
elements is fixed to 32. We have similar findings to the harvested power, that
is, the lower carrier frequency can compensate for the data rate loss. It is
worth mentioning that data rates in an order of magnitude above 50Mbps are
achievable in most of settings as shown in Fig. 6. For instance, the data rate
of $65\mathrm{Mbps}$ is achieved under the setting of $900\mathrm{MHz}$ and 32
elements as shown in Fig. 6b. It implies that Ue-WPIoT can potentially support
high data-rate applications.

(a) Shortest Path in one-by-one manner

(b) Shortest Path with multi-node communications with the UAV Height
$H=10\text{m},5\text{m}$, respectively

Figure 7: The UAV trajectory in the $100\text{m}\times 100\text{m}$ area with
the nodes density being $0.25$, where the blue circle denotes the achievable
powering range.

###  IV-B Trajectory Design for multi-node communications

We next conduct simulations to analyze the UAV trajectory with consideration
of multi-node communications. In particular, our experiments were conducted in
a $100\mathrm{m}\times 100\mathrm{m}$ area where the IoT nodes are randomly
distributed with density of $0.25$, as shown in Fig. 7. Implied by the results
in Section IV-A, we consider the following optimal settings: 1) the maximum EH
distance being given by $d_{EH}=13\mathrm{m}$, 2) the antenna consisting of 32
antenna elements, 3) the carrier frequency being fixed at 400MHz. In addition,
to ensure a feasible powering range, the UAV’s flying height $H$ requires to
fulfill the triangular relation as shown in Fig. 7b.

In our simulation, we consider three trajectory strategies: 1) the shortest-
path trajectory when the UAV covers every IoT node in one-by-one manner; 2 the
designed shortest-path trajectory with multi-node communications when the UAV
hovers at height being $10\textrm{m}$ to serve multiple IoT nodes within the
same achievable EH distance $d_{EH}$; 3) the designed shortest-path trajectory
with multi-node communications when the UAV hovers at height being
$5\textrm{m}$ to serve multiple IoT nodes within the same achievable EH
distance $d_{EH}$. Regarding strategies 2) and 3), the horizontal coverage
range of the UAV can be derived by the triangular relation. Specifically,
given $H=10\mathrm{m},5\mathrm{m}$ in strategies 2) and 3), the corresponding
horizontal coverage circles have the radius $R_{2}=\sqrt{13^{2}-10^{2}}\approx
8.31\mathrm{m}$ and $R_{3}=\sqrt{13^{2}-5^{2}}=12\mathrm{m}$, respectively. We
observe that strategy 3 has a larger coverage area than strategy 2, implying
more IoT nodes potentially falling into a WPC group.

Fig. 7 compares three trajectory strategies. It is observed that strategies 2)
and 3) achieve the travelling path length values with $427.7429\mathrm{m}$ and
$400.8843\mathrm{m}$, respectively when the flying height is
$H=10\mathrm{m},5\mathrm{m}$, respectively. In contrast, the travelling path
length in strategy 1) is $446.0828\mathrm{m}$, which is much longer than those
in strategies 2) and 3). This is because our trajectory design with
consideration of multi-node communications can serve for multiple nodes within
one WPC group while the conventional one-by-one shortest path strategy can
only cover one node at a time. In addition, we can observe that the travelling
path length of strategy 3) is even shorter than that of strategy 2). The
reason may lie in more nodes to be served in strategy 3) when the horizontal
coverage range $R=12\mathrm{m}$, which is longer than $R\approx
8.31\mathrm{m}$ of strategy 2), thereby further shortening the travelling
path.

Although a lower flying height can lead to a larger coverage area, the low-
flying UAV may be susceptible to obstacles (such as trees or rocks) at a low
altitude. The recent advances in autonomous UAV manipulation and auto-
navigation bring the opportunities to address this issue [13]. For example,
Artificial Intelligence (AI) assisted computer vision technologies can help
UAVs detect obstacles.

##  V Conclusion and future directions

In this article, we present an overview of UAV-enabled wireless powering
Internet of Things (Ue-WPIoT), which can potentially overcome two major
constraints of IoT: 1) energy constraint of IoT nodes and 2) difficulty in
deploying and maintaining infrastructure nodes in remote or rural area. We
then elaborate the design challenges of Ue-WPIoT and discuss the enabling
technologies to address these challenges. Simulation results validate the
effectiveness of the presented solutions. We outline several future directions
in Ue-WPIoT as follows.

###  V-A Resource limitation of UAVs

In Ue-WPIoT, UAVs are serving as both energy suppliers and data collectors.
Although the optimization of wireless energy transferring and data collection
processes can somehow save energy of UAVs, UAVs still suffer from a
substantial energy consumption. Energy charging or fuel filling may severely
affect the trajectory and the coverage of UAVs, especially in the remote and
rural area. Thus, energy-harvesting UAVs from ambience will be an important
future direction. The possible energy-harvesting technologies for UAVs include
energy harvesting from solar panels and the adoption of windmilling
propellers.

###  V-B Trajectory Privacy Protection

The trajectory information of UAVs is a prerequisite for the effective
control, efficient route planning and navigation, especially in adverse
weathers or disaster situations. However, UAVs can be vulnerable to malicious
attacks such as hijacking after stealing or intercepting the trajectory
information [14]. For example, UAVs can be tracked, intercepted and even
hijacked once the trajectories of UAVs are exposed to malicious users [15].
Moreover, the behaviours of UAV users can be tracked and deduced through
analysing the trajectories of UAVs. Therefore, the trajectory privacy
protection of UAVs will be an important future direction.

###  V-C Intelligent Algorithms for Trajectory Design of UAVs

As analyzed in the earlier part of this article, the trajectory of UAVs in Ue-
WPIoT needs to consider multiple factors, such as multi-node communications
and priority of data collection tasks. However, it is challenging to design
optimal trajectories for UAVs with consideration of all these factors
together. In addition, the dynamicity of IoT (e.g., the failure of some IoT
nodes) makes the situation even worse, i.e., the pre-designed trajectory needs
to be adjusted. The advent of AI, deep learning, reinforcement learning brings
the opportunities to address this rising challenge. The lightweight or
portable AI models are expected to be designed for UAVs in the future.

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Yalin Liu is pursuing her Ph.D. degree in Faculty of Information Technology at
Macau University of Science and Technology. Her current research interests
include Internet of Things and Unmanned aerial vehicle communications.
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Qubeijian Wang is currently with in School of Cyber Security, Northwestern
Polytechnical University, Taicang, China. His current research interests
include Internet of Things, wireless security and Unmanned aerial vehicle
communications.
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Hong-Ning Dai [SM’16] is currently with the Department of Computing and
Decision Sciences, Lingnan University, Hong Kong, as an associate professor.
He obtained the Ph.D. degree in Computer Science and Engineering from
Department of Computer Science and Engineering at the Chinese University of
Hong Kong. His current research interests include the Internet of Things and
blockchain technology. He has served as associate editors of IEEE Transactions
on Industrial Informatics, IEEE Systems Journal, and IEEE Access. He is also a
senior member of the ACM.
---
Muhammad Imran is working as a senior lecturer in the School of Engineering,
Information Technology & Physical Sciences, Federation University, Australia.
His research interests include mobile and wireless networks, Internet of
Things, cloud and edge computing, and information security. He has published
more than 200 research articles in reputable international conferences and
journals. His research is supported by several grants. He serves as an
associate editor for many top ranked international journals. He has received
various awards.
---
Nadra Guizani is a Clinical Assistant Professor at the School of Electrical
and Computer Engineering at Washington State University. She received her PhD
from Purdue University, West Lafayette, USA in 2020. Her research interests
include machine learning, mobile networking, big data analysis and prediction
techniques. She is an active member of both the Women in Engineering program
and the Computing Research Association.
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