# Electrons trapped in magnetic controlled gapped graphene-based quantum dots

Mohammed El Azar  Laboratory of Theoretical Physics, Faculty of Sciences,
Chouaïb Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed
Bouhlal  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Jellal
Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali
University, PO Box 20, 24000 El Jadida, Morocco  Canadian Quantum Research
Center, 204-3002 32 Ave Vernon, BC V1T 2L7, Canada

###### Abstract

Due to the Klein tunneling effect in graphene, it is impossible to confine or
permanently localize electrons inside the graphene quantum dot. However, an
electron can be transiently trapped inside the quantum dot under the effect of
a constant magnetic field, which will have the so-called ”quasi-bound states”
characterized by a finite lifetime. In order to improve the trapping time of
the electrons inside the quantum dot, we will add a mass term to the
Hamiltonian, creating an energy gap. We solve the Dirac equation to determine
the energy spectrum, using the continuity of the eigenspinors at the edges to
study the scattering phenomenon. We find that the energy gap can increase the
lifetime of the quasi bound states inside the quantum dot. In addition, we
show that even in the absence of the magnetic field, the scattering efficiency
can reach considerable levels when the energy gap is closed to the incident
energy of the electron passing through the quantum dot. It is found that the
density inside the quantum dot is enhanced, resulting in an improvement in the
trapping time of the electrons.

###### pacs:

81.05.ue; 81.07.Ta; 73.22.Pr
Keywords: Graphene, circular quantum dot, magnetic field, energy gap,
scattering phenomenon

##  I Introduction

A two-dimensional semi-metal called graphene is distinguished by a linear band
structure that is quite close to the Fermi level. It has a honeycomb lattice
structure comprised entirely of carbon atoms. Researchers are very driven to
enhance the conclusions established theoretically or experimentally in this
context (see [1, 2]) because of the very specific electrical characteristics
of graphene. The anomalous quantum Hall effect [3, 4], Klein tunneling [5],
high electrical conductivity, and extremely high electron mobility [6, 7] are
only a few of the extraordinary electronic characteristics of graphene.
Several scientific studies have demonstrated graphene’s interest in
fundamental physics for more technological applications in a variety of fields
(e.g., [8, 9, 10]). In-depth studies in the physics of materials like graphene
are usually based on the interactions of these materials with external fields.
They have shown that it can offer a perfect framework to study fundamental
physics and interpret physical effects and phenomena such as Landau level
quantization [11, 12, 13], Aharonov-Bohm effect [14, 15, 16]. Then graphene is
one of the strongest materials tested so far, possessing remarkable
conductivity qualities that serve to use graphene in technological
applications such as integrated circuits, light sensing devices, and
microelectronic devices [17, 18, 19].

Due to Klein’s paradox, charge carriers cannot be localized in a small,
constrained region of graphene by an electrostatic gate. The main objective of
graphene-based electronics is to confine the particles that generate quantum
dots [20, 21, 22, 23]. Since graphene does not have a band gap, there are not
any traditional quantum dots in it that can localize electrons in areas of
finite dimensions. Utilizing the quantum dot for relativistic electrons, which
behave like massless Dirac fermions, will be necessary for graphene’s future
electronic uses [2, 5]. The normal incidence issue, which is the cause of
perfect transmission (Klein paradox), is resolved by using zero-dimensional
circular quantum dots [24]. A variety of quantum dot applications in
electronics are made possible by the possibility of confining Dirac fermions
in graphene, including solar panels, lasers [25], photo-detectors [26],
quantum information processing, and quantum computers [27].

The localization of graphene quantum dots is prevented by Klein tunneling when
an electron strikes the dot with normal incidence [5]. The trapping of
electrons in quantum dots is currently a very interesting research topic in
the field of condensed matter physics. It began decades ago and continues to
receive a lot of attention from both a fundamental and an application
standpoint. The electron trapping problem was first studied in a one-
dimensional wire in the absence of a magnetic field [20, 28, 29], and then for
a quantum dot with smooth [22] and sharp [30] boundary states. The trapping
potential of a relativistic electron in graphene is dependent on several
factors, in particular the transverse momentum [20, 22, 31]. Regarding a
quantum dot, the trapping potential is closely related to the electron’s
angular momentum and becomes significantly more intense as it increases.
Another factor that influences the lifetime is the sharpness of the confining
potential. Trapping an electron in a graphene quantum dot is thus best
accomplished when the confinement potential is smooth and electronic states
have a large angular momentum. It is shown that Klein tunneling is
significantly reduced in the resonant regime, and the lifetime increases with
increasing magnetic field [32].

A gap between the two valence and conduction bands in the graphene band
structure can be made using a variety of experimental procedures [9]. As a
result of the sublattice’s symmetry violation, the highest energy gap value
might be 260 meV [33]. It is interesting to note that the energy gap value
differs among testing techniques. By manipulating the structure of the
interface between graphene and ruthenium, it has been shown that there are
alternative experimental ways besides system breaking for opening gaps [34].
Furthermore, the graphene sheet’s energy gap has been altered by altering the
substrate, SiC is one such substrate [33]. Different band gaps are created
based on depositing a graphene layer on other substrates [35, 36].

Inspired by the outcomes described above and, specifically, [32], we study the
phenomenon of incident electron scattering from a gapped circular graphene
quantum dot subjected to a constant magnetic field. This is an investigation
into scattering efficiency $Q$, probability density $\rho$, and lifetime
$\tau$. These will allow us to show how the energy gap can affect the
scattering phenomena of the present system. We first use the Dirac equation to
analytically determine the solutions of the energy spectrum, and then we use
the continuity at the interface to calculate the corresponding scattering
efficiency outside and inside the quantum dot. We find that when the energy
gap is closed to the incident energy of an electron that crosses the GQD, the
efficiency can still be quite high even in the absence of a magnetic field.
Also, we demonstrate that the probability density is strengthened, resulting
in an improvement in the lifetime of the electrons. The main characteristics
of the scattering phenomenon are analyzed in relation to the physical
parameters of our present system, with the primary goal of determining the
best situation to produce interesting and advantageous results.

The present paper is organized as follows. In Sec. II, we establish a
theoretical model describing our system and determine the solution of the
energy spectrum. After matching eigenspinors at the interface, we explicitly
determine the quantities characterizing the scattering phenomenon in Sec. III.
In Sec. IV, we numerically analyze our finding under various conditions of the
physical parameters. The scattering efficiency $Q$, probability density
$\rho$, and lifetime $\tau$ are examples. Finally, we conclude our results.

##  II Theoretical model

Let us consider a gapped circular graphene quantum dot (GQD) subjected to a
constant magnetic field, which is made of two regions as depicted in Fig. 1.

Figure 1: (color online) A graphene quantum dot of radius $R$ is placed in the
horizontal plane $xy$, with the magnetic field $B$ oriented perpendicular to
the dot plane in the direction $z$. The incident electron is represented by a
plane wave $\Phi_{k}^{i}$ with energy $E=\hbar v_{F}k$. When an electron
approaches a quantum dot, it is either reflected (wave $\Phi_{k}^{r}$) or
transmitted (wave $\Phi_{q}^{t}$).

The current system can be described by a single valley Hamiltonian as follows

$H=v_{F}\vec{\sigma}\cdot(\vec{p}-e\vec{A})+\Delta\sigma_{z}$ (1)

where $v_{F}=10^{6}$ ms-1 is the Fermi velocity,
$\vec{\sigma}=(\sigma_{x},\sigma_{y}.\sigma_{z})$ are Pauli matrices and
$\Delta$ is the energy gap resulting from the mass term. It convenient to
select the vector potential $\vec{A}=\frac{B}{2}(x,y)$ in symmetric gauge.
Because of the spherical symmetry, we can write the Hamiltonian in polar
coordinates, knowing that

$\sigma_{r}=\begin{pmatrix}0&\ e^{-i\theta}\\\ \\\ \ e^{i\theta}&0\\\
\end{pmatrix},\quad\sigma_{\theta}=\ \begin{pmatrix}0&\ -ie^{-i\theta}\\\ \\\
\ ie^{i\theta}&0\\\ \end{pmatrix}$ (2)

and therefore we have

$H=\begin{pmatrix}\Delta&-i\hbar
v_{F}e^{-i\theta}\left(\partial_{r}-\frac{i}{r}\partial_{\theta}-\frac{eBr}{2\hbar}\right)\\\
-i\hbar
v_{F}e^{-i\theta}\left(\partial_{r}+\frac{i}{r}\partial_{\theta}+\frac{eBr}{2\hbar}\right)&-\Delta\\\
\end{pmatrix}.$ (3)

Given that the total angular momentum operator
$J_{z}=-i\hbar\partial_{\theta}+\frac{\hbar}{2}\sigma_{z}$ commutes with the
Hamiltonian (1), i.e., $[H,J_{z}]=0$, the eigenspinors can be separated as

$\Phi(r,\theta)=\dbinom{\Phi^{A}(r)e^{im\theta}}{i\Phi^{B}(r)e^{i(m+1)\theta}}$
(4)

where the integer values $m$ are eigenvalue of $J_{z}$. Now using the
eigenvalue equation $H\Phi(r,\theta)=E\Phi(r,\theta)$ to obtain

$\displaystyle\left(\partial_{r}+\frac{r}{2l_{B}^{2}}-\frac{m}{r}\right)\Phi^{A}(r)=-\frac{E+\Delta}{\hbar
v_{F}}\Phi^{B}(r)$ (5a)
$\displaystyle\left(\partial_{r}+\frac{m+1}{r}-\frac{r}{2l_{B}^{2}}\right)\Phi^{B}(r)=\frac{E-\Delta}{\hbar
v_{F}}\Phi^{A}(r)$ (5b)

For instance, by injecting (5a) into (5b), we end up with a second
differential equation for $\Phi^{A}(r)$

$\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\frac{m+1}{l_{B}^{2}}-\frac{r^{2}}{4l_{B}^{2}}-\frac{m^{2}}{r^{2}}+q^{2}\right)\Phi^{A}(r)=0$
(6)

where we have set $q=\frac{\sqrt{|E^{2}-\Delta^{2}|}}{\hbar v_{F}}$ and the
magnetic length $l_{B}=\sqrt{\frac{\hbar}{eB}}$. In order to solve (6), we
start by exploring the asymptotic limits that define the necessary physical
behaviors depending on the value of $r$.

In the limit $r\rightarrow\infty$, (6) can be approximated by

$\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}-\frac{r^{2}}{4l_{B}^{2}}\right)\Phi^{A}(r)=0$
(7)

which is the updated Bessel equation for the zero-order case, therefore having
the solution

$\Phi^{A}(r)=c_{1}I_{0}\left(\frac{r^{2}}{4l_{B}^{2}}\right)+c_{2}K_{0}\left(\frac{r^{2}}{4l_{B}^{2}}\right)$
(8)

where $I_{0}(x)$ and $K_{0}(x)$ denote the zero order modified Bessel
functions of the first and second kinds, respectively. We choose $c_{1}=0$ and
$c_{2}=1$ to avoid the divergence of function $I_{0}(x)$ when $x$ reaches
infinity. Now using the asymptotic behavior $K_{0}(x)\underset{x\gg
1}{\sim}\frac{e^{-x}}{\sqrt{x}}$, to approximate $\Phi^{A}(r)$ as

$\Phi^{A}(r)\sim 2l_{B}\frac{e^{-\frac{r^{2}}{4l_{B}^{2}}}}{r}.$ (9)

In the limit $r\rightarrow 0$, (6) reduces to

$\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}-\frac{m^{2}}{r^{2}}\right)\Phi^{A}(r)=0$
(10)

which has the following solution

$\Phi^{A}(r)=\frac{c_{3}}{2}(r^{m}+r^{-m})+\frac{ic_{4}}{2}(r^{m}-r^{-m})$
(11)

where $c_{3}$ and $c_{4}$ must be chosen in such a way that the solution
adheres to the physical constraints. As a result, we examine the positive and
negative values of $m$ independently. Indeed, for $m\geq 0$, $\sim r^{-m}$
must vanish, which implies that $c_{4}=-ic_{3}$ and, again by convention, we
put $c_{3}=1$. For $m<0$, $\sim r^{m}$ must vanish, then we replace
$c_{4}=ic_{3}$ with $c_{3}=1$. Combining all to write the asymptotic behavior
of $\Phi^{A}(r)$ as

$\displaystyle\Phi^{A}(r)\sim r^{m},\qquad m\geq 0,$ (12a)
$\displaystyle\Phi^{A}(r)\sim r^{-m},\qquad m<0.$ (12b)

Using the above analysis to write the solution of (6) as

$\Phi^{A\pm}(r)=r^{\pm
m}\frac{e^{-r^{2}/4l_{B}^{2}}}{r/2l_{B}}\Xi_{q}^{A\pm}(r)$ (13)

knowing that the sign ”+” stands for $m\geq 0$, while ”-” stands for the
opposite case, $m<0$. Now, we perform the variable change
$\eta=\frac{r^{2}}{2l_{B}^{2}}$ and use the transformation
$\Xi_{q}^{A\pm}(\eta)=\sqrt{\eta}\chi_{q}^{A\pm}(\eta)$ to write (13) as

$\Phi^{A\pm}(\eta)=l_{B}^{\pm m}2^{\frac{1\pm m}{2}}\eta^{\pm
m/2}e^{-\eta/2}\chi_{q}^{A\pm}(\eta)$ (14)

which can be injected into (6) to end up with the Kummer-type differential
equations

$\displaystyle\eta\partial_{\eta}^{2}\chi_{q}^{A+}(\eta)+\left(m+1-\eta\right)\partial_{\eta}\chi_{q}^{A+}(\eta)+\frac{l_{B}^{2}q^{2}}{2}\chi_{q}^{A+}(\eta)=0$
(15a)
$\displaystyle\eta\partial_{\eta}^{2}\chi_{q}^{A-}(\eta)+\left(1-m-\eta\right)\partial_{\eta}\chi_{q}^{A-}(\eta)+\left(m+\frac{l_{B}^{2}q^{2}}{2}\right)\chi_{q}^{A-}(\eta)=0$
(15b)

showing the confluent hypergeometric functions as solutions

$\displaystyle\chi_{q}^{A+}(\eta)=\prescript{}{1}{F}_{1}\left(-\frac{l_{B}^{2}q^{2}}{2},m+1,\eta\right)$
(16a)
$\displaystyle\chi_{q}^{A-}(\eta)=\prescript{}{1}{F}_{1}\left(-l-\frac{l_{B}^{2}q^{2}}{2},1-m,\eta\right).$
(16b)

When we combine all of the above results, we get the following solutions to
the second order differential equation (6)

$\displaystyle\Phi^{A+}(r)=r^{|m|}e^{-r^{2}/4l_{B}^{2}}\prescript{}{1}{F}_{1}\left(-\frac{l_{B}^{2}q^{2}}{2},m+1,\frac{r^{2}}{2l_{B}^{2}}\right)$
(17a)
$\displaystyle\Phi^{A-}(r)=r^{|m|}e^{-r^{2}/4l_{B}^{2}}\prescript{}{1}{F}_{1}\left(-m-\frac{l_{B}^{2}q^{2}}{2},1-m,\frac{r^{2}}{2l_{B}^{2}}\right)$
(17b)

The other components of spinor (4) can be obtained by inserting (17a) and
(17b) into (5a). This process yields

$\displaystyle\Phi^{B+}(r)=\frac{q}{2(m+1)}r^{|m|+1}e^{-r^{2}/4l_{B}^{2}}\prescript{}{1}{F}_{1}\left(1-\frac{l_{B}^{2}q^{2}}{2},m+2,\frac{r^{2}}{2l_{B}^{2}}\right)$
(18a)
$\displaystyle\Phi^{B-}(r)=\frac{1}{q}r^{|m|-1}e^{-r^{2}/4l_{B}^{2}}\left[2m\prescript{}{1}{F}_{1}\left(-m-\frac{l_{B}^{2}q^{2}}{2},1-m,\frac{r^{2}}{2l_{B}^{2}}\right)+\frac{(2m+l_{B}^{2}q^{2})r^{2}}{2(1-m)l_{B}^{2}}\prescript{}{1}{F}_{1}\left(1-m-\frac{l_{B}^{2}q^{2}}{2},2-m,\frac{r^{2}}{2l_{B}^{2}}\right)\right]$
(18b)

In the forthcoming analysis, we will see how the above results can be worked
out to study the scattering phenomenon in terms of different quantities.

##  III Scattering problem

In order to examine the scattering problem, we first explain how an electron
scatters on a circular GQD of radius $R$ in the presence of a constant
magnetic field before identifying the key parameters that characterize the
scattering problem. Consider an electron moving along the $x$ direction with
energy $E=\hbar v_{F}k$, where $k$ is the corresponding wave number. As a
result, the incident electron can be described by a plane wave as follows

$\Phi_{k}^{i}(r,\theta)=\frac{1}{\sqrt{2}}e^{ikr\cos\theta}\dbinom{1}{1}=\frac{1}{\sqrt{2}}\sum_{m=-\infty}^{\infty}i^{m}\dbinom{J_{m}(kr)e^{im\theta}}{iJ_{m+1}(kr)e^{i(m+1)\theta}}$
(19)

where $J_{m}(z)$ is the Bessel function of first kind. The following equation
demonstrates how the reflected electron wave can be divided into partial waves
since it must adhere to infinite boundary requirements for the scattering
mechanism being researched [37]

$\Phi_{k}^{r}(r,\theta)=\frac{1}{\sqrt{2}}\sum_{m=-\infty}^{\infty}a_{m}^{r}i^{m}\dbinom{H_{m}(kr)e^{im\theta}}{iH_{m+1}(kr)e^{i(m+1)\theta}}$
(20)

and $H_{m}(x)$ is the Hankel function of first kind that is a linear
combinations of $J_{m}$ and Neumann $Y_{m}$, i.e.,
$H_{m}(x)=J_{m}(x)+iY_{m}(x)$. For a large value of $x$, its asymptotic
behavior is

$H_{m}(x)\underset{x\gg 1}{\sim}\sqrt{\frac{2}{\pi
x}}e^{i(x-\frac{l\pi}{2}-\frac{\pi}{4})}.$ (21)

The transmitted solution can be obtained from the previous analysis as

$\Phi_{q}^{t}(r,\theta)=\sum_{m=-\infty}^{-1}a_{m}^{t-}\dbinom{\Phi_{q}^{A-}(r)e^{im\theta}}{i\Phi_{q}^{B-}(r)e^{i(m+1)\theta}}+\sum_{m=0}^{\infty}a_{m}^{t+}\dbinom{\Phi_{q}^{A+}(r)e^{im\theta}}{i\Phi_{q}^{B+}(r)e^{i(m+1)\theta}}$
(22)

where $q$ represents the wave number associated to the electron inside the GQD
as shown in Fig. 1.

To investigate our system’s scattering problem, we must first calculate the
scattering coefficients $a_{m}^{r}$ and $a_{m}^{t}$ using the continuity of
eigenspinors at the boundary condition $r=R$. This is

$\Phi_{k}^{i}(R,\theta)+\Phi_{k}^{r}(R,\theta)=\Phi_{q}^{t}(R,\theta)$ (23)

giving rise to two conditions represented by two equations of $a_{m}^{r}$ and
$a_{m}^{t}$

$\displaystyle\frac{1}{\sqrt{2}}i^{m}J_{m}(kR)+\frac{1}{\sqrt{2}}i^{m}a_{m}^{r}H_{m}(kR)=a_{m}^{t}\Phi_{q}^{A\pm}(qR)$
(24a)
$\displaystyle\frac{1}{\sqrt{2}}i^{m+1}J_{m+1}(kR)+\frac{1}{\sqrt{2}}i^{m+1}a_{m}^{r}H_{m+1}(kR)=ia_{m}^{t}\Phi_{q}^{B\pm}(qR).$
(24b)

Consequently, we obtain

$\displaystyle a_{m}^{t\pm}$ $\displaystyle=\frac{i\sqrt{2}e^{im\pi/2}}{\pi
qR[H_{m}(kR)\Phi_{q}^{B\pm}(qR)-H_{m+1}(kR)\Phi_{q}^{A\pm}(qR)]}$ (25a)
$\displaystyle a_{m}^{r\pm}$
$\displaystyle=\frac{-J_{m}(kR)\Phi_{q}^{B\pm}(qR)+J_{m+1}(kR)\Phi_{q}^{A\pm}(qR)}{H_{m}(kR)\Phi_{q}^{B\pm}(qR)-H_{m+1}(kR)\Phi_{q}^{A\pm}(qR)}.$
(25b)

We now define the probability density function $\rho=\Phi^{\dagger}\Phi$ and
current density $j=\Phi^{\dagger}\sigma\Phi$ using Hamiltonian (1), with the
spinor $\Phi$ being dependent on the region where $\Phi=\Phi_{t}$ is inside
the GQD and $\Phi=\Phi_{i}+\Phi_{r}$ is outside. As a result, the radial
component of the current is given by

$j_{\text{rad}}^{r}(\theta)=\Phi^{\dagger}\begin{pmatrix}0&\cos\theta-i\sin\theta\\\
\cos\theta+i\sin\theta&0\end{pmatrix}\Phi.$ (26)

Taking the asymptotic behavior (21) into account, we calculate
$j_{\text{rad}}^{r}(\theta)$ as

$j_{\text{rad}}^{r}(\theta)=\frac{4}{\pi
kR}\sum_{m=-\infty}^{+\infty}|a_{m}^{r}|^{2}+\frac{8}{\pi
kR}\sum_{m<m^{\prime}}\Re(a_{m}^{r}a_{m^{\prime}}^{r})\cos[(m-m^{\prime})\theta].$
(27)

At this level, we are interested in the quantities related to our system to
determine their basic characteristics. Indeed, in the limit $kr\to\infty$,
(27) is used to calculate the effective scattering cross section $\sigma$
defined by

$\sigma=\frac{I_{\text{rad}}^{r}}{I^{\text{inc}}/A_{u}}$ (28)

knowing that $I_{\text{rad}}^{r}$ represents the total reflected flux through
the GQD of radius $R$ and that the incident flux per unit area represents the
term $I^{\text{inc}}/A_{u}$. Our calculation shows that the total reflected
flux $I_{\text{rad}}^{r}$ takes the form

$I_{\text{rad}}^{r}=\int_{0}^{2\pi}j_{\text{rad}}^{r}(\theta)rd\theta=\frac{8}{k}\sum_{m=-\infty}^{+\infty}\left|a_{m}\right|^{2}$
(29)

and $I^{i}/A_{u}=1$ for the incident wave (19).

To improve our study of the scattering problem of Dirac fermions in different
sizes of circular quantum dots, we analyze the scattering efficiency $Q$. This
is defined as the ratio of the division of the scattering cross section to the
geometrical cross section based on the lines shown in [38]

$Q=\frac{\sigma}{2R}=\frac{4}{kR}\sum_{m=-\infty}^{+\infty}|a_{m}^{r}|^{2}.$
(30)

Note that by requiring $\Delta=0$, we recover the results found in [32]. The
analytical results obtained so far will be analyzed numerically in order to
study the effect of the energy gap and other parameters on the physical
quantities that characterize the scattering phenomenon and also the
confinement of Dirac fermions in a gapped GQD. These studies are reflected in
the analysis of real space scattering and the evaluation of the lifetime
corresponding to Dirac fermions inside the GQD.

##  IV RESULTS AND DISCUSSIONS

We numerically examine the scattering phenomenon of electrons on a gapped GQD
when it is exposed to a constant magnetic field. Our studies are based on the
analysis of the magnitude in terms of scattering efficiency $Q$ and density
$\rho$ in the region close to the quantum dot and the lifetime of the quasi-
bound states. Fig. 2 represents the contour plot of $Q$ as a function of
incident energy $E$ and magnetic field $B$ for the radius $R=50$ nm and
different values of the energy gap $\Delta$. Fig. 2a shows that for
$\Delta=0$, the pattern of $Q$ is almost oscillating and consists of six bands
of high values of $Q$, each of which corresponds to a scattering mode
$m=0,1,2,3,4,5$, which is in agreement with the results found previously [32,
22, 39, 40]. As shown in Figs. 2(b,c,d), including the energy gap $\Delta$
reduces the resonance effect. As a result, we clearly observe that $Q$
diminishes as long as $\Delta$ approaches the incident energy. In addition,
the more the value of $\Delta$ is increased, the more the number of scattering
modes is gradually decreased until it becomes only four modes when $\Delta$
approaches 40 meV, as depicted in Fig. 2d. We note that when the magnetic
field corresponding to the excitation of the scattering modes decreases, the
value of $\Delta$ increases.

Figure 2: (color online) The scattering efficiency $Q$ as a function of the
incident energy $E$ and magnetic field $B$ for $R=50$ nm different values of
$\Delta$. (a): $\Delta=0$ meV, (b): $\Delta=15$ meV, (c): $\Delta=25$ meV and
(d): $\Delta=39.5$ meV

The scattering efficiency $Q$ is plotted as a function of the magnetic field
$B$ for various values of the energy gap $\Delta$ and incident energy $E$ in
Fig. 3. When $\Delta$ exceeds 6 meV for $E=7$ meV and $B=0$, $Q$ is not null,
as shown in Fig. 3a. Furthermore, we observe that $\Delta$ shifts the
resonance peaks to the right. This leads to the suppression of a resonance
peak at $\Delta\geq 6$ meV. In Figs. 3(b,c,d), we choose $E=20$, $25$ and $35$
meV where we see that the resonance peaks are always shifted to the right due
to the energy gap. Some peaks are also suppressed as $\Delta$ approaches $E$
and $Q$ takes specific values for $B=0$. This tells us that $Q$ survives and
takes important values even in the absence of the magnetic field, except that
one should be placed in the zone where $\Delta$ is closest to $E>$ 20 meV.

We show the scattering efficiency $Q$ as a function of the magnetic field $B$
in Fig. 4 for E = 20 meV, three values of the energy gap $\Delta=0$, 14, 18
meV and the quantum number $m=0,\cdots=5$ meV. The first three modes,
$m=-1,0,1$ are excited without resonance peaks in Fig. 4a for $\Delta=0$ meV.
On the other hand, for the other scattering modes $m=2,3,4$, we observe
resonance peaks for specific values of $B$. When $m$ is increased, the width
of these peaks narrows and becomes very narrow for $m=4$. These results are in
good agreement with those obtained in [32]. In Figs. 4(b,c), we see that the
behavior of $Q$ is almost similar to that found in Fig. 4a except that the
positions of the resonance peaks are shifted to the right. We also observe the
suppression of the $m=4$ scattering mode. We emphasize that in the absence of
the magnetic field, the addition of $\Delta$ leads to a scattering efficiency
$Q$ no null, which becomes very important with the increase of $\Delta$. As
shown in Figs. 4(b,c), $Q$ is solely due to the two scattering modes $m=-1,0$.

Figure 3: (color online) The scattering efficiency $Q$ as a function of the
magnetic field $B$ for different values of $\Delta$ and $E$. (a): $E=7$ meV
and $\Delta=0,4,6,6.9$ meV, (b): $E=20$ meV and $\Delta=0,14,18,19.9$ meV,
(c): $E=25$ meV and $\Delta=0,12,21,24.9$ meV, (d): $E=35$ meV and
$\Delta=0,18,28,34.9$ meV

Figure 4: (color online) The scattering efficiency $Q$ as a function of the
magnetic field $B$ for $E=20$ meV, $m=-1,0,1,2,3,4$, and three values of the
energy gap (a): $\Delta=0$ meV, (b): $\Delta=14$ meV, (c): $\Delta=18$ meV.

The scattering efficiency $Q$ as a function of the magnetic field $B$ and
radius $R$ of the GQD is shown in Fig. 5 for four different values of the
energy gap $\Delta=0$ meV, $14$, $18$ meV, $19.9$ meV. In Fig. 5a for
$\Delta=0$, we observe that starting from $R\approx 32$ nm, the wide bands
correspond to $m=1,2,3$ and narrow bands correspond to $m=4,5$ begin to show
up. These important increases of $Q$ are designated as ”scattering resonances”
and are associated with a specific scattering mode. It is clear that the
interaction is very low below below $R\approx 32$ nm. Furthermore, as shown in
Fig. 5a, the interaction is very weak inside the GQD, regardless of its
radius, below $B\approx$0.4 T. Now by increasing $\Delta$ in Figs. 5(b,c,d),
we see that bangs appear for large values of $R$ and grow as $\Delta$
increases. In particular, in Fig. 5d, the bangs appear for $R=45$ nm and an
energy gap $\Delta=19.9$ meV closes to the incident one. This shows that the
bangs corresponding to the scattering modes shift to larger values of $R$ as
long as $\Delta$ is increased.

Figure 5: (color online) The scattering efficiency $Q$ as a function of the
magnetic field $B$ and radius $R$ of the GQD for an incident energy value of
$E=20$ meV and four values of the energy gap (a): $\Delta=0$ meV, (b):
$\Delta=14$ meV, (c): $\Delta=18$ meV, (d): $\Delta=19.9$ meV.

Fig. 6 shows $Q$ as a function of $R$ for $E=20$ meV, four values of $\Delta$m
and the the magnetic field (a): $B=0.8$ T, (b): $B=2.2$ T, (c): $B=3.2$ T. In
Fig. 6a, we observe that the most relevant resonance peaks are concentrated in
the radius interval 50–100 nm. However, as $B$ increase, we see that this
interval shifts to small values of $R$, as shown in Figs. 6(b,c). More
importantly, $Q$ becomes significant when $R$ is low and $\Delta$ is close to
the incident energy. When $R$ is less than 20 nm, adding an energy gap
$\Delta$ increases scattering efficiency $Q$, but when $R$ exceeds 20 nm, $Q$
decreases as $\Delta$ increases.

Figure 6: (color online) The scattering efficiency $Q$ as a function of the
radius $R$ of the GQD for three values of the magnetic field strength (a):
$B=0.8$ T, (b): $B=2.2$ T, (c): $B=3.2$ T, and four values of the energy gap
$\Delta=$ 0, 14, 18, 19.9 meV.

In Fig. 7, each column is dedicated to a scattering mode among those studied
previously, and we examine the density in the field near the GQD for each mode
under the effect of two values of the energy gap $\Delta=0$ and 14 meV. The
boundary of the GQD is represented by the black circle. The magnetic field
values chosen correspond to the peaks of $m=0,1,2,3$ presented in Fig. 4. Line
(1) starts with the results for the four scattering modes $m=0,1,2,3$ at zero
energy gap. In Fig. 7a where $B=0.4$ T, we see that the large values of the
density are distributed in the outer part of the GQD, i.e., the electron wave
is diffracted on the GQD boundary, as can be observed in Fig. 4a where only
the $m=-1,0$ modes are non-resonantly excited with a very low value of the
scattering efficiency. Thus, in this case, we do not expect electron trapping
effects inside the GQD. The mode $m=1$ is resonantly excited with a broad peak
in Fig. 7c for $B=1.62$ T. With the presence of the diffraction bangs, we also
observe that the majority of the density is localized outside the GQD, but now
the near field density values are slightly larger. The mode $m=2$ is excited
with a narrower resonance peak in Fig. 7e for $B=2.56$ T than in the previous
case. In this case, the majority of the density is concentrated inside the GQD
with a high scattering efficiency. As a result, the small resonance peak of
the mode $m=2$ makes it more likely that the electron will be trapped inside
the GQD. The resonance of the mode $m=3$ is very clear in Fig. 7g for $B=3.8$
T, with a narrower peak than for $m=2$. We see the density concentration
inside the GQD with a higher scattering efficiency and the suppression of
diffraction bangs. Therefore, the electron trapping effect inside the quantum
dot is notable. From the aforementioned, we infer that the trapping effect
increases with the resonance peak’s narrowness. In line (2), we represent the
results obtained at an energy gap $\Delta=14$ meV for the four scattering
modes $m=0,1,2,3$ with (b): ($m=0$,$B=0.8$ T) and (d): ($m=1$,$B=2.1$ T). It
is clearly seen that the density inside and at the GQD boundary is improved
compared to those seen before, with the presence of diffraction bangs around
the GQD. Comparing (f): ($m=2$,$B=3.03$ T) and (h): ($m=3$,$B=4.3$ T) with (e)
and (g), we see that the density is more concentrated inside the GQD and has
created a clearer cloud around the core of the GQD. Therefore, when an energy
gap is added, the impact of electron trapping inside the GQD is improved.

Figure 7: (color online) A density representation for a real space examination
of electron scattering on a magnetically driven GQD. Each graph column is
devoted to a given value of the quantum number (A): $m=0$, (B): $m=1$, (C):
$m=2$, and (D): $m=3$. Each graph line represents a specific value of the
energy gap (1): $\Delta=0$ meV and (2): $\Delta=14$ meV. Each panel
corresponds to a given value of magnetic field (a): $B=0.4$ T, (b): $B=0.8$ T,
(c): $B=1.62$ T, (d): $B=2.1$ T, (e): $B=2.56$ T, (f): $B=3.03$ T, (g):
$B=3.8$ T, (h): $B=4.3$ T. A black circle indicates the geographic extent of
the GQD.

We now perform another analysis in terms of density in real space to evaluate
the trapping time (the lifetime of the quasi-bound states). The analysis of
the scattering resonances must be performed in terms of the complex incident
electronic energy $E=E_{r}-iE_{i}$, where $E_{r}$ represents the resonance
energy and $E_{i}$ gives the lifetime of the quasi-bound states $\tau$, with
$\tau=h/E_{i}$. We have previously indicated that the whole scattering process
can be decomposed into several scattering sub-processes. We will now
concentrate on the two scattering modes $m=0,3$ to see how the energy gap
$\Delta$ affects the lifetime $\tau$ of the quasi-bound states. We use
continuity to find the complex energy of the incident electron by matching the
transmitted wave function with the reflected wave function on the boundary
$r=R$ [39]. Since the incident energy of the incoming electron is not affected
by the magnetic field, we treat the following transcendental equation for $q$
and $k$

$\frac{\Phi_{q}^{A\pm}(qR)}{\Phi_{q}^{B\pm}(qR)}=\frac{H_{m}(kR)}{H_{m+1}(kR)}.$
(31)

The lifetime $\tau$ as a function of the magnetic field $B$ is shown in Fig. 8
for the two scattering modes $m=0,3$ and four values of the energy gap
$\Delta=0,14,18,19.9$ meV. The first thing we notice is that $\tau$ increases
as $B$ increases, which agrees with what we previously discovered by examining
the scattering efficiency $Q$. In Fig. 8a, $\tau$ becomes visible from $B=1.8$
T and increases more clearly from $B\approx 3.35$ T for $\Delta$ non-null. In
Fig. 8b, we see that for a given $\Delta$, $\tau$ tends to increase from small
values of $B$ when compared to the case of $\Delta=0$ for higher magnetic
fields. We conclude that when $\Delta$ is non-null, $\tau$ improves even
further.

Figure 8: (color online) The lifetime $\tau$ as a function of magnetic field
$B$ for four values of the energy gap $\Delta=0,14,18,19.9$ meV and two
scattering modes (a): $m=0$, (b): $m=3$.

##  V Conclusion

The theoretical analysis of the electron scattering mechanism on a
magnetically driven graphene quantum dot (GQD) in the presence of a mass term
generating an energy gap in the spectrum has been further investigated. In
this regard, we first established a theoretical model for the interaction of
Dirac fermions with a constant magnetic field in a circular GQD and identified
the crucial variables. To deal with the problem correctly, we first solved the
corresponding Dirac equation and analytically determined the energy spectrum.
Using the continuity of the eigenspinors, we calculated different quantities
characterizing the scattering phenomenon. In particular, the efficiency $Q$ is
found to be dependent on the magnetic field, angular moment, radius of the
GQD, incident energy, and energy gap.

We also presented numerically the obtained results to provide a general
interpretation of the scattering phenomenon and to show how an electron at
normal incidence can be trapped in a GQD for a certain period of time, with
the main objective being to improve this trapping time. We discovered that
even in the absence of a magnetic field, scattering efficiency can reach
significant values when the energy gap is close to the incident energy of the
electron crossing the GQD. For a non-null magnetic field, we showed that the
resonance peaks with a higher scattering efficiency are those corresponding to
the smallest values of radius of the GQD.

By analyzing the probability density, we showed that the diffraction
phenomenon is dominant in the domain where the scattering is non-resonant,
with a weak localization of the density in the GQD without an energy gap.
However, when an energy gap is added, it is found that the density inside the
GQD is enhanced. On the other hand, in the domain where the scattering is
resonant, we noticed the damping of the diffraction with a very important
localization of the density inside and at the boundary of the GQD, as well as
with noticeable trapping effects. The important result in our paper is to
improve the possibility of trapping electrons in the GQD under the influence
of a mass term, creating an energy gap in the energy spectrum.

###### Acknowledgements.

We warmly thank Professor Adrian Pena for his valuable support.

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#  $\EM$: Deep Transformer-based Decoding of Upperlimb sEMG for Hand Gestures
Recognition

Elahe Rahimian1, Student Member, IEEE, Soheil Zabihi2, Student Member, IEEE,
Amir Asif2, Senior Member, IEEE, Dario Farina3, Fellow, IEEE, S. Farokh
Atashzar4, Senior Member, IEEE, and Arash Mohammadi1, Senior Member, IEEE
1Elahe Rahimian and Arash Mohammadi are with Concordia Institute for
Information System Engineering (CIISE), Concordia University, Montreal, Canada
<EMAIL_ADDRESS>Zabihi and Amir Asif are with
Electrical and Computer Engineering (ECE), Concordia University, Montreal,
Canada<EMAIL_ADDRESS>Farina is with the Department of
Bioengineering, Imperial College London, London, UK, SW7-2AZ
<EMAIL_ADDRESS>Farokh Atashzar is with Electrical and Computer
Engineering and Mechanical and Aerospace Engineering, New York University
(NYU), Brooklyn, NY, USA, 11201<EMAIL_ADDRESS>project was partially
supported by the Department of National Defence’s Innovation for Defence
Excellence & Security (IDEaS), Canada.

###### Abstract

There has been a surge of recent interest in Machine Learning (ML),
particularly Deep Neural Network (DNN)-based models, to decode muscle
activities from surface Electromyography (sEMG) signals for myoelectric
control of neurorobotic systems. DNN-based models, however, require large
training sets and, typically, have high structural complexity, i.e., they
depend on a large number of trainable parameters. To address these issues, we
developed a framework based on the Transformer architecture for processing
sEMG signals. We propose a novel Vision Transformer (ViT)-based neural network
architecture (referred to as the TEMGNet) to classify and recognize upper-limb
hand gestures from sEMG to be used for myocontrol of prostheses. The proposed
TEMGNet architecture is trained with a small dataset without the need for pre-
training or fine-tuning. To evaluate the efficacy, following the recent
literature, the second subset (exercise B) of the NinaPro DB2 dataset was
utilized, where the proposed TEMGNet framework achieved a recognition accuracy
of $82.93\%$ and $82.05\%$ for window sizes of $300ms$ and $200$ms,
respectively, outperforming its state-of-the-art counterparts. Moreover, the
proposed TEMGNet framework is superior in terms of structural capacity while
having seven times fewer trainable parameters. These characteristics and the
high performance make DNN-based models promising approaches for myoelectric
control of neurorobots.

###### Index Terms:

Neurorobotics, Attention Mechanism, Myoelectric Control, Electromyogram (EMG),
Vision Transformer (ViT).

##  I Introduction

Recognizing limb motions using surface Electromyography (sEMG) signals allows
for the control of rehabilitation and assistive systems (such as bionic limbs
and exoskeletons). Surface EMG signals are obtained non-invasively by sensors
on the skin surface that measure the electrical activity of the muscle’s motor
units [1, 2]. The information obtained from sEMG signals is used to decode
discriminative and repeatable patterns that can be utilized to effectively
classify the intended motor commands of the user. In this context, several
attempts have been made to classify hand movements using traditional Machine
Learning (ML) methods. While these conventional approaches (such as Linear
Discriminant Analysis (LDA) and Support Vector Machine (SVM) [3]) have been
successfully implemented, their performance might degrade when applied to a
large-scale data set consisting of a sizable number of movements. This has
motivated a recent surge of interest in applying Deep Neural Networks (DNNs)
within this domain, with the aim of addressing the shortcomings of traditional
ML solutions.

Processing sEMG signals using DNN architectures has the potentials to provide
significantly improved performance. Convolutional Neural Network (CNN) is the
commonly used DNN architecture for recognizing upper-limb hand gestures in
which sEMG signals are translated into images [4, 5, 6]. CNN-based models,
however, target learning spatial features and are ineffective for the
extraction of temporal features from sEMG sequential data. Recurrent Neural
Networks (RNNs), such as Long Short Term Memory (LSTM), have been therefore
proposed in recent studies [7, 8] to capture the temporal information from
sEMG signals. To capture both spatial and temporal characteristics of sEMG
signals, LSTM and CNN can be combined [9], resulting in a hybrid solution. For
example, the authors in Reference [10] composed six image representations of
raw sEMG signals that were then fed as input to a hybrid CNN-LSTM
architecture. Additionally, it was shown that dilated causal convolutions in
CNN-based architectures [11, 12] have great potentials to surpass the pure RNN
architecture in term of overall accuracy. Finally, dilated LSTM models [8]
have shown the potential to enhance accuracy and reduce computational cost.

While the above-mentioned models are advanced approaches to sequence modeling,
they do not allow parallelization during the training phase due to their
sequential nature. For this reason, recurrent-based models are slow and
difficult to train. The Transformer neural network architecture [13] has been
proposed to eliminate recurrence or convolution using a self-attention
mechanism. In brief, the main building block of the transformer architecture,
making it a unique and powerful DNN model, is the attention mechanism, which
is a way to mimic and execute the function of selective focus. Transformers
were first applied to Natural Language Processing (NLP) tasks, with the goal
of solving sequence-by-sequence tasks while handling long-range dependencies
[13]. The transformer model architectures, such as Bidirectional Encoder
Representations from Transformers (BERT) [14] and Generative Pre-Training
(GPT) [15], achieved state-of-the-art results in different NLP tasks. In
addition to the NLP field, Transformers have been applied to address a variety
of other problems, including Computer Vision (CV) tasks [16];
Electroencephalogram (EEG)-based speech recognition [17]; EEG decoding [18];
Electrocardiogram (ECG)-based heartbeat classification [19]; sleep stages
classification [20], acoustic modeling [21], and ECG classification [22].

In this paper, we hypothesize that novel models, designed on the basis of
transformers, would enhance the accuracy of EMG classification, and reduce the
training process. This will be a major step toward the ultimate utilization of
deep learning models for commercial prosthetic systems. Therefore, here we
propose, design, and evaluate the performance of a novel transformer model for
hand gesture recognition. The proposed TEMGNet architecture was designed based
on the ViT architecture to improve recognition accuracy, speed of training,
and to reduce structural complexity. Generally speaking, training transformers
requires large datasets. In this study, we illustrate that transformers can be
trained with a small sEMG dataset without the need to use pre-trained models
and fine-tuning. We show that the proposed TEMGNet architecture outperforms
its state-of-the-art counterparts in terms of overall recognition accuracy and
complexity. The proposed transformer-based architecture improves the
recognition accuracy compared to its state-of-the-art counterparts (where LSTM
or hybrid LSTM-CNN are adopted) with the significantly reduced number of
trainable parameters. The proposed TEMGNet architecture was evaluated based on
the second benchmark Ninapro database, in which we aimed to classify $17$ hand
gestures from raw, sparse multichannel sEMG signals. The details of the
Ninapro database are provided in Subsection II-A. We examined the effect of
several variants of the TEMGNet architecture on both recognition accuracy and
the number of required trainable parameters. We also conducted statistical
tests to assess the significance of the proposed architecture. Finally, we
present a visualization of position embedding similarities, showing that the
model is capable of encoding position information and considering the
sequential nature of sEMG signals.

##  II Material and Methods

Figure 1: Exercise B of Ninapro database [23, 24, 25]. Figure 2: The proposed
TEMGNet architecture: (a) Each segment of sEMG signal $\bm{X}$ is split into a
sequence of fixed-size non-overlapping patches. The constructed patches are
then flattened (the blue one) and projected linearly (the green block). The
output of this step is referred to as “Patch Embedding.” Afterward, position
embeddings are added to the patch embeddings. The resulting sequence of these
embedded patches is then fed to the transformer encoder (the gray block). For
the classification, a trainable $[cls]$ token ${\mathbf{x}}^{cls}$ is added to
the sequence. (b) The transformer encoder: This module consists of $L$ layers,
each consisting of two LayerNorm modules, an MLP, and an MSA module.

First, we describe the database based on which the proposed model was
evaluated. Then, the pre-processing method for preparing the dataset is
explained.

###  II-A Database

We evaluated the proposed TEMGNet framework on the second Ninapro benchmark
database [23, 24, 25] referred to as the DB2, a well-known sparse multichannel
sEMG dataset widely used to classify hand gestures [26, 27, 8]. The DB2
database was obtained from $40$ users who repeated several movements $6$
times, each lasting $5$ seconds, followed by $3$ seconds of rest. In the DB2
dataset, muscular activities were measured using $12$ active double-
differential wireless electrodes from a Delsys Trigno Wireless EMG system at a
sampling frequency of $2$ kHz. The DB2 dataset consists of three exercises
(i.e., B, C, and D) dealing with different types of movements. In this paper,
to evaluate the proposed model, Exercise B was used, which consists of $17$
different movements. As shown in Fig. 1, the utilized dataset consists of $9$
basic wrist movements with $8$ isometric and isotonic hand configurations. In
order to compare and follow the recommendations provided by the Ninapro
database, $2/3$ of the gesture repetitions of each user (i.e., $1,3,4$, and
$6$) were used to build the training set, and the remaining repetitions (i.e.,
$2$, and $5$) were used for testing.

###  II-B Pre-processing

Before performing the classification task, the sEMG signals were pre-
processed. Following the recent literature [5, 6, 3], we applied a low-pass
Butterworth filter to the signals in order to remove high-frequency noise.
Afterward, we applied the $\mu$-law normalization to the sEMG data. This
normalization approach has been introduced in the context of sEMG processing
in Reference [4] and is defined as follows

$F(x_{t})=\text{sign}(x_{t})\frac{\ln{\big{(}1+\mu|x_{t}|\big{)}}}{\ln{\big{(}1+\mu\big{)}}},\vspace{-.1in}$
(1)

where $x_{t}$ represents the input scalar, and parameter $\mu$ denotes the new
range. Recently in [4, 26], it was shown that improved performance could be
achieved using normalization of the sEMG signals with the $\mu$-law approach.

##  III The TEMGNet Framework

In this section, we present details of the proposed TEMGNet architecture
designed for performing sEMG-based hand gesture recognition tasks. The main
fundamental block of the Transformer architecture is the attention mechanism.
It is noteworthy to mention that attention along with other architectures such
as CNN and/or LSTM has recently been utilized [10, 26, 27] to classify hand
movements based on sEMG signals, where the results showed the ability of
attention to learn the temporal information of multi-channel sEMG data. Unlike
prior works that aimed to combine CNN or LSTM architectures with self-
attention, in this paper, we will show that the proposed ViT-based
architecture, which is solely based on the attention mechanism, has the
capability to outperform the previous networks. The overall structure of the
proposed TEMGNet architecture is shown in Fig. 2. In the following, first, we
describe the building blocks of the proposed architecture, and then the
overall structure of the network.

The proposed architecture is inspired by ViT [16], which closely follows the
original transformer model. Within the TEMGNet framework, after completion of
the pre-processing step, the collected sEMG data was segmented via a sliding
window of length $200$ms with steps of $10$ms (for comparison purposes, the
results for a window of $300$ms are also provided). The segmentation step
transforms the sEMG dataset into
$\mathcal{D}=\\{(\bm{X}_{i},\mathrm{y}_{i})\\}_{i=1}^{M}$, consisting of $M$
segments, where the $i^{{}^{\text{th}}}$ segment is denoted by
$\bm{X}_{i}\in\mathbb{R}^{S\times W}$, for ($1\leq i\leq M$), with its
associated label denoted by $\mathrm{y}_{i}$. Here, $S$ denotes the number of
sensors, and $W$ shows the number of samples collected at $2$ kHz for a window
of $200$ms (or $300$ms). The main objective of the TEMGNet architecture is to
learn the mapping from the sequence of segment patches to its corresponding
label $\mathrm{y}_{i}$. As shown in Fig. 2, the TEMGNet architecture consists
of the following modules; i.e., Patch Embeddings, Position Embedding,
Transformer encoder, and finally a Multi-Layer Perceptron (MLP) head.

###  III-A Patch Embeddings

As shown in Fig. 2(a), at first, we split the segmented input $\bm{X}$ (for
simplicity, we drop the segment index $i$) into $N$ number of non-overlapping
patches. Here, we set the size of each patch to $(S\times S)$; therefore, the
number of patches will be equal to $N=W/S$. Each patch is then flattened into
a vector ${\mathbf{x}}^{p}_{j}\in\mathbb{R}^{S^{2}}$, for ($1\leq j\leq N$). A
linear projection is then applied to embed each vector into the model
dimension $d$. For the linear projection, we used a matrix
$\bm{E}\in\mathbb{R}^{S^{2}\times d}$, which is shared among different
patches. The output of this projection is called patch embeddings (Eq. 2
below).

In a similar way as in the BERT framework [14], the beginning of the sequence
of embedded patches is appended with a trainable $[cls]$ token
${\mathbf{x}}^{cls}$, to capture the meaning of the entire segmented input.
Finally, we will add position embeddings denoted by
$\bm{E}^{pos}\in\mathbb{R}^{(N+1)\times d}$ (which will be described next in
Sub-section III-B) to the patch embeddings that will allow the transformer to
capture the positional information. The formulation governing patch and
position embeddings is given by

$\displaystyle\bm{Z}_{0}=[{\mathbf{x}}^{cls};{\mathbf{x}}^{p}_{1}\bm{E};{\mathbf{x}}^{p}_{2}\bm{E};\dots;{\mathbf{x}}^{p}_{N}\bm{E}]+\bm{E}^{pos}.$
(2)

###  III-B Position Embeddings

The sEMG signal is sequential data that is presented in a specific order. If
we change this order, the meaning of the input might also change. The
transformer does not process the input sequentially and for each element, it
combines information from the other elements through self-attention. Since the
architecture of the transformer does not model the positional information,
there is a need to explicitly encode the order of the input sequence so that
the transformer knows that one piece is after another and not in any other
permutation. This is where positional embedding comes in. Positional embedding
is a form of identifier, a clear reference for the transformer that encodes
the location information within the sequence. Therefore, positional embeddings
are order or position identifiers added to the input to identify the relative
position of each element in the sequence for the transformer. There are
different ways to encode spatial information into the input of transformer
using positional embedding, e.g., Sinusoid positional embedding, Relative
positional embeddings, Convolutional embedding, 1-dimensional positional
embedding, and 2-dimensional positional embedding [21, 16]. Following [16],
the standard trainable 1-dimensional positional embeddings are used in this
study.

TABLE I: Descriptions of TEMGNet architecture variants. Window size | Model ID | Layers | Model dimension $d$ | MLP size | Heads | Params
---|---|---|---|---|---|---
200ms | 1 | 1 | 32 | 128 | 8 | 20,049
2 | 2 | 32 | 128 | 8 | 32,657
3 | 3 | 32 | 128 | 8 | 45,265
4 | 1 | 64 | 256 | 8 | 64,625
300ms | 1 | 1 | 32 | 128 | 8 | 20,593
2 | 2 | 32 | 128 | 8 | 33,201
3 | 3 | 32 | 128 | 8 | 45,809
4 | 1 | 64 | 256 | 8 | 65,713
TABLE II: classification accuracies for TEMGNet architectures variants. The
STD represents the standard variation in accuracy over the $40$ users.

Window size
200ms | Model ID | 1 | 2 | 3 | 4
---|---|---|---|---|---
Accuracy ($\%$) | $80.39$ | $81.23$ | $80.85$ | 82.05
| STD ($\%$) | $5.86$ | $6.31$ | $6.32$ | 5.78
Window size
300ms | Model ID | 1 | 2 | 3 | 4
Accuracy ($\%$) | $80.88$ | $81.54$ | $81.42$ | 82.93
| STD ($\%$) | $5.97$ | $5.99$ | $5.94$ | 5.83

###  III-C Transformer Encoder

The resulting sequence of vectors $\bm{Z}_{0}$ is fed as an input to a
standard transformer encoder. We are basically treating all of the constructed
patches as simple tokens provided into the transformer. In fact, the encoder
block is similar to the main transformer encoder block proposed by [13]. As
shown in Fig. 2(b), the transformer encoder consists of $L$ identical layers.
Each layer consists of two modules, i.e., a Multihead Self-Attention Mechanism
(MSA) and an MLP module (defined later in Eqs. 3, 4). MSA is built based on
the Self-Attention (SA) mechanism. SA and MSA are explained in Sub-sections
III-C1 and III-C2, respectively. The MLP module consists of two linear layers
and a Gaussian Error Linear Unit (GELU) activation function.

To address the degradation problem, a layer-normalization [28] is applied,
which is then followed by residual skip connections

$\displaystyle\bm{Z}^{{}^{\prime}}_{l}$ $\displaystyle=$ $\displaystyle
MSA(LayerNorm(\bm{Z}_{l-1}))+\bm{Z}_{l-1},$ (3) $\displaystyle\bm{Z}_{l}$
$\displaystyle=$ $\displaystyle
MLP(LayerNorm(\bm{Z}^{{}^{\prime}}_{l}))+\bm{Z}^{{}^{\prime}}_{l},$ (4)

for $l=1\dots L$. The final output of the transformer can be represented as
follows

$\displaystyle\bm{Z}_{L}=[{\mathbf{z}}_{L0};{\mathbf{z}}_{L1};\dots;{\mathbf{z}}_{LN}],$
(5)

where ${\mathbf{z}}_{L0}$ is used for classification purposes. Finally,
${\mathbf{z}}_{L0}$ is passed to a Linear Layer (LL), i.e.,

$\displaystyle\mathbf{y}=LL(LayerNorm({\mathbf{z}}_{L0}).$ (6)

This completes the description of the proposed TEMGNet architecture. Next, we
present the description of SA and MSA, respectively.

####  III-C1 Self-Attention (SA)

Let us define the input sequence as $\bm{Z}\in\mathbb{R}^{N\times d}$
consisting of $N$ vectors, each with an embedding dimension of $d$. The SA
mechanism was first introduced in [13]. Generally speaking, the SA mechanism
is defined with the aim of capturing the interaction between different vectors
in $\bm{Z}$. In this regard, three different matrices, namely Queries
$\bm{Q}$, Keys $\bm{K}$, and Values $\bm{V}$ are computed via a linear
transformation, i.e.,

$\displaystyle[\bm{Q},\bm{K},\bm{V}]=\bm{Z}\mathit{W}^{QKV},$ (7)

where $\mathit{W}^{QKV}\in\mathbb{R}^{d\times 3d_{h}}$ shows the learnable
weight matrix. Here, $d_{h}$ denotes the size of each vector in $\bm{Q}$,
$\bm{K}$, and $\bm{V}$. After that, the pairwise similarity between each query
and all keys is obtained using the dot-product of $\bm{Q}$ and $\bm{K}$, which
is then scaled by $\sqrt{d_{h}}$ and translated into the probabilities
$\bm{P}\in\mathbb{R}^{N\times N}$ using the softmax function as follows

$\displaystyle\bm{P}=\text{softmax}(\frac{\bm{Q}\bm{K}^{T}}{\sqrt{d_{h}}}).$
(8)

Finally, for each vector in the input sequence, the corresponding output
vector resulted from the SA mechanism is computed by taking the weighted sum
over all values $\bm{V}$ as follows

$\displaystyle SA(\bm{Z})=\bm{P}\bm{V},$ (9)

where $SA(\bm{Z})\in\mathbb{R}^{N\times d_{h}}$. Such an attention mechanism
helps the model to focus on important parts from a given sEMG input sequence.

####  III-C2 Multihead Self-Attention (MSA)

In the MSA, the SA mechanism is applied $h$ times in parallel, allowing the
model to attend to parts of the input sequence for each head differently. More
specifically, MSA consists of $h$ heads where each head has its own learnable
weight matrix $\\{\mathit{W}^{QKV}_{i}\\}^{h}_{i=1}$. For the input sequence
$\bm{Z}$, we applied the SA mechanism for each head (Eqs. (7)-(9)). Then the
outputs of $h$ heads are concatenated into a single matrix
$[SA_{1}(\bm{Z});SA_{2}(\bm{Z});\dots;SA_{h}(\bm{Z})]\in\mathbb{R}^{N\times
h.d_{h}}$ and once again projected to obtain the final values as follows

$\displaystyle
MSA(\bm{Z})=[SA_{1}(\bm{Z});SA_{2}(\bm{Z});\dots;SA_{h}(\bm{Z})]W^{MSA},$ (10)

where $\mathit{W}^{MSA}\in\mathbb{R}^{h.d_{h}\times d}$ and $d_{h}$ is set to
$d/h$, this description completes the proposed TEMGNet architecture. Next, we
present the results and experiments.

##  IV Experiments and Results

We evaluated different variants of the TEMGNet architecture. The results are
summarized in Table I, where the performance of the model for window sizes of
$200$ms and $300$ms is shown. For all model variants, we set the size of the
input patch to $12\times 12$. All models were trained using Adam optimizer
[29] with betas = ($0.9,0.999$), and the weight decay set to $0.001$. These
models were trained with a batch size of $512$. Cross-entropy loss was used
for measuring classification performance.

(a) Window size of $200$ms

(b) Window size of $300$ms

Figure 3: The accuracy boxplots for all TEMGNet architecture variants. Each boxplot shows the IQR of each model for $40$ users. The Wilcoxon signed-rank test is used to compare the Model $1$ (with a minimum number of parameters) with other models; i.e., Model $2$, Model $3$, and Model $4$ (ns: $5.00\text{e}-02<\text{p}\leq 1.00\text{e}+00$, ${}^{*}:1.00\text{e}-02<\text{p}\leq 5.00\text{e}-02$, ${}^{**}:1.00\text{e}-03<\text{p}\leq 1.00\text{e}-02$, ${}^{***}:1.00\text{e}-04<\text{p}\leq 1.00\text{e}-03$, ${}^{****}:\text{p}\leq 1.00\text{e}-04$). TABLE III: Comparison between our methodology (TEMGNet) and previous works [8]. Reference [8] |  | 200ms | 300ms
---|---|---|---
| Params | Accuracy ($\%$) | Params | Accuracy ($\%$)
4-layer 3rd Order Dilation | $\\_$ | $79.0$ | $466,944$ | $82.4$
4-layer 3rd Order Dilation (pure LSTM) | $\\_$ | $\\_$ | $\\_$ | $79.7$
SVM | $\\_$ | $26.9$ | $\\_$ | $30.7$
Our Method | Model 1 | 20,049 | $80.39$ | 20,593 | $80.88$
Model 4 | $64,625$ | 82.05 | $65,713$ | 82.93

Table II summarizes the classification accuracies for different TEMGNet
architecture variants. The computed gesture recognition accuracy was averaged
over all subjects. As shown in Table II, by increasing the model dimension $d$
from $32$ to $64$ (Model $1$ to Model $4$), the accuracy improved
approximately by $2\%$ for both window sizes. However, Model $4$ had higher
number of trainable parameters in comparison to Model $1$, resulting in higher
complexity (Table I). In a second experiment, we examined the effect of
increasing the number of layers. As shown in Table II, for both window sizes
of $200$ms and $300$ms, Model $2$ had a higher accuracy than Model $1$.
However, by increasing the number of layers to $3$, no improvements had been
observed in the classification accuracy. We used the statistical tests to show
the significance level of different model variants, i.e., Model $1$, $2$, $3$,
and $4$. Therefore, we followed [30, 26] and used the Wilcoxon signed-rank
test [31], considering each participant as a separate dataset. As shown in
Fig. 3, the difference in accuracy between Model $1$ and Model $4$, for both
window sizes of $200$ms and $300$ms was considered statistically significant
by the Wilcoxon signed-rank test as the (${}^{****}:\text{p}\leq 1.00e-4$). In
Fig. 3, a p-value is annotated by:

  * •

Not significant (ns): $5.00\text{e}-02<\text{p}\leq 1.00\text{e}+00$,

  * •

${}^{*}:1.00\text{e}-02<\text{p}\leq 5.00\text{e}-02$,

  * •

${}^{**}:1.00\text{e}-03<\text{p}\leq 1.00\text{e}-02$,

  * •

${}^{***}:1.00\text{e}-04<\text{p}\leq 1.00\text{e}-03$,

  * •

${}^{****}:\text{p}\leq 1.00\text{e}-04$.s

In Fig. 3, the performance distribution across $40$ users for each model is
shown. The boxplot for each model shows the Interquartile Range (IQR), which
is based on dividing the performance of each model for $40$ users into
quartiles. The median performance is shown by a horizontal line in each
boxplot.

(a) Model $1$ for a Window size of $300$ms

(b) Model $4$ for a Window size of $300$ms

Figure 4: Visualization of position embedding similarities for Model $1$ and
$4$ in the Window size of $300$ms. The number of samples ($W$) collected at a
frequency of $2$ kHz for a window of $300$ms is $600$. The size of each patch
is set to $(12\times 12)$; therefore, the number of patches is $N=50$; i.e.,
the input patch index is from $0$ to $49$. Each row in (a) and (b) represents
the similarity of the position embedding vector of each patch to all
positional embeddings. It is shown that position embedding vectors learn
distance in a segment. This means that the neighbors have higher similarities.

In Table III, we provide comparisons with the state-of-the-art models [8]
developed recently on the same dataset to illustrate the superior performance
of the proposed TEMGNet architecture over its counterparts. The proposed
TEMGNet method outperforms classical and state-of-the-art DNN-based models.
More specifically, for a window size of $300$ms, the classification accuracy
was $82.93\%$ with only $65,713$ number of trainable parameters, while in
Reference [8], the authors reached $82.4\%$ accuracy with $466,944$ number of
parameters. Moreover, for window size $200$ms, the accuracy for Model $1$ and
Model $4$ was $80.39\%$ and $82.05\%$, respectively. However, with the same
window size, the best accuracy reported in Reference [8] was $79.0\%$.

Fig. 4 shows the position embedding similarities for the window size of
$300$ms. As mentioned in Sub-section III-B, to encode position information,
learnable position embedding was added to the patch embeddings. This is a key
factor for the transformer to consider the sequential nature of the sEMG
signals. Fig. 4 shows that position embedding vectors learn distance in a
segment of sEMG signals. More specifically, the size of each patch was set to
$12\times 12$. Therefore, there are $N=50$ patches for a window of $300$ms,
i.e., the input patch index varied from $0$ to $49$. Each row in Fig. 4(a),
(b) represents the similarity of the position embedding vector of each patch
to all positional embeddings. It is observed that the main diagonal in Fig.
4(a), (b) shows higher similarities between the neighboring ones. Moreover,
position embedding similarities in Fig. 4(b) are greater than in Fig. 4(a),
which means that the proposed Model $4$ encode the sequential nature of sEMG
signals better than Model $1$.

##  V Conclusion

We proposed a novel transformer-based framework for the task of hand gesture
recognition from sEMG signals. We showed that the proposed architecture has
the capacity to reduce the number of trainable parameters by $7$ times with
respect to the state of the art, while improving the performance. This is a
major step toward the utilization of deep learning for the control of
prosthetic systems.

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