# Tuned gap in graphene through laser barrier

Hasna Chnafa  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco  Miloud Mekkaoui
Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali
University, PO Box 20, 24000 El Jadida, Morocco  Ahmed Jellal
<EMAIL_ADDRESS>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  Abdelhadi
Bahaoui  Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb
Doukkali University, PO Box 20, 24000 El Jadida, Morocco

###### Abstract

We study the effect of the energy gap on the transmission of fermions in
graphene exposed to linearly polarized light as a laser barrier. We determine
the energy spectrum, apply boundary conditions at interfaces, and use the
transfer matrix approach to obtain transmissions for all energy modes. We show
that when the energy gap increases, the oscillations of transmissions decrease
dramatically until they vanish entirely. However, when the barrier width
varies, the oscillations become more significant and exhibit sharp peaks. By
increasing the incident energy, the laser field suppresses the Fabry-Pérot
resonance, and the transmissions move to the right when the energy gap is
tuned.

###### pacs:

78.67.Wj, 05.40.-a, 05.60.-k, 72.80.Vp
Keywords: Graphene, energy gap, laser field, Dirac equation, Floquet theory,
transmission.

##  I Introduction

Graphene is a single sheet of carbon atoms organized in a two-dimensional
honeycomb arrangement [1], which is regarded as a miracle material because of
its exceptional physical properties. It has a high resistivity [2], a high
thermal conductivity [3], a Hall effect [4, 5, 6], a Klein tunneling [7, 8], a
mechanical strength [9], and an electronic mobility that is $100$ times
greater than the most efficient material [10]. It is also extremely
transparent and light, making it an intriguing material for a broad range of
applications, including electronics [2], photonics [11], and other fields. At
weak energies, the behavior of massless particles in graphene can be described
by the Dirac equation, which gives rise to a linear dispersion relation [12].
Nevertheless, the lack of a band gap in graphene limits its use in specific
kinds of electronic devices. There have been several approaches developed to
open such a band gap, for example, doping graphene with atoms such as nitrogen
[13] or boron [14, 15]. Depositing graphene on substrates generates a
staggered potential in the graphene layer. Practically, this means that the
carbon atoms in sublattices A and B have different values for the energy
levels, designated by $\Delta$ and $-\Delta$, respectively [16, 17].
Consequently, an energy gap emerges between the conduction and valence bands
in the form of a mass term, which will be carried by the $\sigma_{z}$
component of the Pauli matrices. The use of hexagonal boron nitride (hBN) as
substrate, which possesses a lattice constant that closely resembles that of
graphene, could result in the creation of a small band gap around $30$ meV
[18]. Silicone carbide (SiC) is another substrate used to produce a band gap
$260$ meV in graphene [19]. Also, a band gap can be opened by engineering
graphene under suitable deformations [20, 21, 22, 23], and experimentally a
band gap $300$ meV can be attained by applying $1\%$ uniaxial strain to
graphene [20].

Floquet theory has grown enormously in recent years due to its ability to
explain the electronic behavior of periodically driven quantum mechanical
systems subjected to a transient periodic field [24, 25]. It should be
emphasized that recent advances in optical and microwave physics, as well as
technological advances in lasers and emerging applications of quantum optics
in condensed matter, have facilitated direct experimental verification of
these theoretical predictions. This has resulted in concrete applications in
real optoelectronic devices [26, 27, 28, 29, 30]. Several studies have been
done, examining the application of Floquet theory to different materials [31,
32], notably nanotubes [33, 34], graphene [35, 36, 37], $\alpha-\text{T}_{3}$
[38, 39] and tilted [40, 41] materials, and so on. It has been shown that the
use of Floquet theory under the influence of periodic electromagnetic fields
can provide information about the electron (or optical) dressed states and
control the electronic [36, 42, 43, 44], topological [45, 46, 47, 48] and
transport [49, 50, 51, 52, 53, 54] properties of these materials, opening up
new potential applications in quantum electronics and spintronics fields.

The study of laser effects on graphene-based systems is a well-established
research field in modern physics. Lately, it has been demonstrated that the
laser field is applied to tune the electronic properties of graphene and
create controllable and dynamic gaps in the quasi-energy spectrum based on the
intensity and polarization of the field [55, 56, 57, 58, 59, 60]. For
circularly polarized light, it becomes possible to produce a band-gap at the
Dirac point and generate dynamic gaps at various momenta. Moreover, these gaps
can be adjusted by manipulating the field intensity [61]. In contrast, when
linearly polarized light interacts with graphene, it affects its electrons in
different ways, such as by absorbing or emitting photon energy. As a result,
the energy levels of the electrons in graphene can vary depending on the
direction of linear polarization, forming an anisotropic quasi-energy spectrum
that displays dynamic gaps only in specific directions at non-zero momentum,
while no gap is observed at the Dirac point. Thus, light can be considered an
optimal tool for regulating the transport properties of graphene systems.
Furthermore, it is found that a strong monochromatic field induces asymmetric
Klein tunneling at normal incidence [62, 51, 52, 63, 64, 65]. Indeed, when the
laser field is present, the Hamiltonian takes on a non-commutative form with a
chirality operator, and the resulting broken chirality leads to the loss of
the symmetric Klein phenomenon [52]. Other results seen in the presence of a
vector potential oscillating in time include the fact that the irradiation
area of the graphene sheet acts as a barrier, making it likely to confine the
Dirac fermions [51], the transmission via the electrostatic barrier in
irradiated graphene can display the oscillations of the Fano kind [52, 66],
and the electron conductance can exhibit specific patterns by adjusting the
laser field parameters [67]. One of the effects observed is the significant
increase in conductance when the graphene is illuminated by the laser. Another
important effect is the modulation of electronic conductance, where the
conductance varies periodically. Even though there have been several works on
the subject, we think that tuning an energy gap in graphene under linearly
polarized laser light could be of interest.

We study the effect of the energy gap on the transmission of the Dirac
fermions in graphene subjected to a laser field of amplitude $F$ and frequency
$\omega$ that acts a as barrier of height and width. We apply the Floquet
approach and solve the Dirac equation to derive the eigenspinors, satisfying
the continuity at interfaces. By using the matrix transfer method, we
explicitly determine the transmission probabilities for energy modes
$\varepsilon+m\omega$ ($m=0,\pm 1,\pm 2{,\cdots}$) [68]. By restricting
ourselves to the central band $m=0$ and two sidebands $m=\pm 1$, we show that
the amplitudes and peaks of transmissions are strongly affected by the energy
gap. Indeed, it is found that the transmissions over all channels exhibit
oscillations with an amplitude that decreases gradually for a small energy gap
and vanishes completely for a large one. This behavior changes with the
increase in barrier width, resulting in more oscillations with sharp peaks.
Furthermore, the Fabry-Pérot resonance is found to be suppressed when incident
energy is increased.

The paper is organized as follows. In Sec. II, we set a theoretical formalism
including a Hamiltonian describing gapped graphene subjected to a time-
dependent vector potential. Subsequently, we solve the eigenvalue equation and
use the Floquet theory to get the solutions of energy spectrum. These
solutions at boundary conditions, together with the transfer matrix method,
can be used to determine the transmissions associated with energy modes in
Sec. III. To provide a better understanding, we numerically analyze and
discuss transmissions for the central band and the first sidebands in Sec. IV.
We summarize our work in Sec. V.

##  II Theoretical formulation

We study the effect of energy gap engineering on transmission in a laser
barrier made of gapped graphene subjected to a time-varying vector potential.
This system is composed of regions labeled
$\text{j}=\text{I},\text{II},\text{III}$ with a tuned gap $\Delta$ and a
linearly polarized light applied to II, as depicted in Fig.1.

Figure 1: (color online) Schematic illustration of gapped graphene exposed to
a linearly polarized laser field.

The present system can be described by the following Hamiltonian

$\displaystyle\mathcal{H}=v_{F}\bm{\sigma}\cdot\left[\bm{p}-e\bm{A}\right]+{\Delta}\sigma_{z}$
(1)

where $v_{F}\approx 10^{6}$ m/s is the Fermi velocity, $\bm{p}=(p_{x},p_{y})$
is the momentum, $e$ is the electron charge, and $\Delta$ is the energy gap.
In the dipole approximation [69], the laser light is represented by the vector
potential $\bm{A}=\left(0,A\cos{\omega t}\right)$ with $\omega$ is the
frequency of a wave, and the corresponding laser field is
$\bm{E}=-\dfrac{\partial}{\partial t}\bm{A}=\left(0,F\sin{\omega t}\right)$,
$F={\omega}{A}$ with $F$ being the amplitude.

According to the Floquet approach [70, 71, 72], we write the eigenspinors
$\psi_{\text{j}}(x,y,t)=\left[\psi_{\text{A},\text{j}}(x,y,t),\psi_{\text{B},\text{j}}(x,y,t)\right]^{\text{T}}=e^{-i\mathcal{\varepsilon}t}\Phi_{\text{j}}(x,y,t)$
and therefore the eigenvalue equation gives

$\displaystyle\begin{pmatrix}\Delta&-i{\partial_{x}}-{\partial_{y}}+i{\dfrac{F}{\omega}}\cos{\omega
t}\\\ -i{\partial_{x}}+{\partial_{y}}-i{\dfrac{F}{\omega}}\cos{\omega
t}&-\Delta\end{pmatrix}\begin{pmatrix}\psi_{\text{A},\text{j}}(x,y,t)\\\
\psi_{\text{B},\text{j}}(x,y,t)\end{pmatrix}=i\partial_{t}\begin{pmatrix}\psi_{\text{A},\text{j}}(x,y,t)\\\
\psi_{\text{B},\text{j}}(x,y,t)\end{pmatrix}$ (2)

where the dimensionless quantities are defined $F\equiv{F}/{{F_{0}}}$,
$\textbf{r}\equiv{\textbf{r}}/{L_{0}}$,
$\mathcal{\varepsilon}\equiv{\mathcal{\varepsilon}}/{\mathcal{\varepsilon}_{0}}$,
${\Delta}\equiv{\Delta}/{\mathcal{\varepsilon}_{0}}$, $t\equiv{t}/{t_{0}}$,
$\omega\equiv{\omega}/{\omega_{0}}$ with scales: length
$L_{0}=\sqrt{\dfrac{\hbar v_{F}}{eF_{0}}}$, time $t_{0}=\dfrac{L_{0}}{v_{F}}$,
energy $\mathcal{\varepsilon}_{0}=\dfrac{\hbar{\tiny}v_{F}}{L_{0}}$, frequency
$\omega_{0}=\dfrac{v_{F}}{L_{0}}$ and electric field
$F_{0}=\dfrac{\mathcal{\varepsilon}_{0}}{eL_{0}}$. By taking into account the
conservation of $p_{y}$ and using the periodicity
$\Phi_{\text{j}}(x,y,t+2\pi/\omega)=\Phi_{\text{j}}(x,y,t)$, we can write the
Fourier series [51, 73]

$\displaystyle\Phi_{\text{j}}(x,y,t)=e^{ik_{y}y}\Phi_{\text{j}}(x)e^{-i\frac{F}{\omega^{2}}\sin{\omega
t}}=e^{ik_{y}y}\Phi_{\text{j}}(x)\sum^{+\infty}_{n=-\infty}J_{n}\left(\frac{F}{\omega^{2}}\right)e^{-in\omega
t}$ (3)

where $J_{n}$ the first-order Bessel function of argument
$\frac{F}{\omega^{2}}$. Note that a linear combination of spinors of energies
$\varepsilon+m\omega$ ($m=0,\pm 1,\pm 2,\cdots$) should be considered since
the laser field can change the amount of energy that electrons have in units
of $\omega$. As a result, the solution of (2) can be written as

$\displaystyle\psi_{\text{j}}(x,y,t)=e^{ik_{y}y}\sum^{+\infty}_{m,n=-\infty}\Phi^{m}_{\text{j}}(x)J_{n-m}\left(\frac{F}{\omega^{2}}\right)e^{-i\left(\mathcal{\varepsilon}+n\omega\right)t}.$
(4)

As it is clearly seen, to get a complete solution, one has to determine
$\Phi^{m}_{\text{j}}(x)$, needs to be done in the next step by considering
each region separately.

In region-I $(x<0)$, without energy gap ($\Delta=0$) and laser field ($F=0$),
the eigenspinors can be written by exchanging $J_{n-m}(0)$ by $\delta_{nm}$

$\displaystyle\psi_{\text{I}}(x,y,t)$
$\displaystyle={e^{ik_{y}y}}\sum^{+\infty}_{m,n=-\infty}\left[\delta_{m0}\begin{pmatrix}1\\\
\gamma_{m}\end{pmatrix}e^{i{k_{m}}x}+r_{m}\begin{pmatrix}1\\\
-\dfrac{1}{\gamma_{m}}\end{pmatrix}e^{-i{k_{m}}x}\right]\delta_{nm}e^{-i\left(\mathcal{\varepsilon}+n\omega\right)t}$
(5)

where $r_{m}$ is the reflection coefficient, the complex number and the angle
outside the barrier are given by

$\displaystyle\gamma_{m}=s_{m}\dfrac{k_{m}+ik_{y}}{\sqrt{k_{m}^{2}+k^{2}_{y}}}=s_{m}e^{i\theta_{m}},\quad\theta_{m}=\tan^{-1}\left(\frac{k_{y}}{k_{m}}\right)$
(6)

with $s_{m}=\text{sgn}(\varepsilon+m\omega)$ is the sign function and we have
set $k_{x,m}=k_{m}$. The corresponding energies and $x$-direction component of
the wave vector read as

$\displaystyle{\mathcal{\varepsilon}+m\omega=s_{m}\sqrt{k_{m}^{2}+k^{2}_{y}}}$
(7) $\displaystyle
k_{m}=\sqrt{\left(\mathcal{\varepsilon}+m\omega\right)^{2}-k^{2}_{y}}.$ (8)

In region-II ($0\leq x\leq L$), $F\neq 0$ and $\Delta\neq 0$, then we have

$\displaystyle\begin{pmatrix}-i\left(\varepsilon+m\omega{-\Delta}\right)&\left(\partial_{x}+k_{y}-m\omega\right)\\\
\left(\partial_{x}-k_{y}+m\omega\right)&-i\left(\varepsilon+m\omega{+\Delta}\right)\end{pmatrix}\begin{pmatrix}\Phi^{m}_{\text{A},{\text{II}}}(x)\\\
\Phi^{m}_{\text{B},{\text{II}}}(x)\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix}$ (9)

which can be solved to end up with the eigenspinors

$\displaystyle\Phi^{m}_{\text{II}}(x)$
$\displaystyle=\alpha_{m}\begin{pmatrix}\xi^{+}_{m}\\\
\xi^{-}_{m}\gamma^{\prime}_{m}\end{pmatrix}e^{i{q_{m}}x}+\beta_{m}\begin{pmatrix}\xi^{+}_{m}\\\
-\dfrac{\xi^{-}_{m}}{\gamma^{\prime}_{m}}\end{pmatrix}e^{-i{q_{m}}x}$ (10)

where we have set

$\displaystyle\xi^{\pm}_{m}=\left(1\pm\dfrac{s^{\prime}_{m}\Delta}{\sqrt{q_{m}^{2}+(k_{y}-m\omega)^{2}+\Delta^{2}}}\right)^{\frac{1}{2}}$
(11)

as well as the quantities

$\displaystyle\gamma^{\prime}_{m}=s^{\prime}_{m}\dfrac{q_{m}+i\left(k_{y}-m\omega\right)}{\sqrt{{q_{m}^{2}}+(k_{y}-m\omega)^{2}}}=s^{\prime}_{m}e^{i\theta^{\prime}_{m}},\quad\theta^{\prime}_{m}=\tan^{-1}\left(\frac{k_{y}-m\omega}{q_{m}}\right)$
(12)

with $s^{\prime}_{m}=\text{sgn}(\varepsilon+m\omega)$, $q_{x,m}=q_{m}$,
$\alpha_{m}$ and $\beta_{m}$ are two constants. We obtain the following
eigenvalues and the wave component

$\displaystyle{\mathcal{\varepsilon}+m\omega=s^{\prime}_{m}\sqrt{{q_{m}^{2}+(k_{y}-m\omega)^{2}{+\Delta^{2}}}}}$
(13) $\displaystyle
q_{m}=\sqrt{\left(\mathcal{\varepsilon}+m\omega\right)^{2}-(k_{y}-m\omega)^{2}{-\Delta^{2}}}.$
(14)

Combining all to get the eigenspinors in region-II ($0\leq x\leq L$)

$\displaystyle\psi_{\text{II}}(x,y,t)$
$\displaystyle={e^{ik_{y}y}}\sum^{+\infty}_{m,n=-\infty}\left[\alpha_{m}\begin{pmatrix}\xi^{+}_{m}\\\
\xi^{-}_{m}\gamma^{\prime}_{m}\end{pmatrix}e^{i{q_{m}}x}+\beta_{m}\begin{pmatrix}\xi^{+}_{m}\\\
-\dfrac{\xi^{-}_{m}}{\gamma^{\prime}_{m}}\end{pmatrix}e^{-i{q_{m}}x}\right]J_{n-m}\left(\frac{F}{\omega^{2}}\right)e^{-i\left(\mathcal{\varepsilon}+n\omega\right)t}.$
(15)

For region-III ($x>L$), the eigenspinors can be worked out as in region-I to
get

$\displaystyle\psi_{\text{III}}(x,y,t)$
$\displaystyle={e^{ik_{y}y}}\sum^{+\infty}_{m,n=-\infty}\left[t_{m}\begin{pmatrix}1\\\
\gamma_{m}\end{pmatrix}e^{i{k_{m}}x}+\lambda_{m}\begin{pmatrix}1\\\
-\dfrac{1}{\gamma_{m}}\end{pmatrix}e^{-i{k_{m}}x}\right]\delta_{nm}e^{-i\left(\mathcal{\varepsilon}+n\omega\right)t}$
(16)

where $t_{m}$ is the transmission coefficient and $\lambda_{m}$ is the null
vector. We will then look at how the results described above can be used to
determine the transmission probabilities for all energy modes.

##  III Transmission Probabilities

To determine the transmission probabilities in gapped graphene through a laser
assisted barrier, we first introduce the continuity of eigenspinors at the
interfaces $x=0,{L}$, i.e.,
${\psi_{\text{I}}(0,y,t)}={\psi_{\text{II}}(0,y,t)}$ and
${\psi_{\text{II}}({L},y,t)}={\psi_{\text{III}}({L},y,t)}$. Second, by taking
into account that $e^{in{\omega}t}$ as an orthogonal basis, then at $x=0$ we
have

$\displaystyle\delta_{n0}+r_{n}=\sum^{+\infty}_{m=-\infty}\left[\alpha_{m}\xi^{+}_{m}+\beta_{m}\xi^{+}_{m}\right]J_{n-m}{{\left(\frac{{F}}{{\omega}^{2}}\right)}}$
(17)
$\displaystyle\delta_{n0}{\gamma_{n}}-{r_{n}}\dfrac{1}{\gamma_{n}}=\sum^{+\infty}_{m=-\infty}{\left[\alpha_{m}\xi^{-}_{m}{\gamma^{\prime}_{m}}-\frac{\beta_{m}\xi^{-}_{m}}{{\gamma^{\prime}_{m}}}\right]}J_{n-m}{{\left(\frac{{F}}{{\omega}^{2}}\right)}}$
(18)

and at $x=L$, we get

$\displaystyle
t_{n}{e^{i{k_{n}}{L}}}+{\lambda_{n}}{e^{-i{k_{n}}{{L}}}}=\sum^{+\infty}_{m=-\infty}\left[\alpha_{m}\xi^{+}_{m}{e^{i{q_{m}}{L}}}+\beta_{m}\xi^{+}_{m}{e^{-i{q_{m}}{L}}}\right]J_{n-m}{{\left(\frac{{F}}{{\omega}^{2}}\right)}}$
(19) $\displaystyle
t_{n}{\gamma_{n}}{e^{i{k_{n}}{{L}}}}-{{\lambda_{n}}}\dfrac{1}{\gamma_{n}}{e^{-i{k_{n}}{L}}}=\sum^{+\infty}_{m=-\infty}\left[\alpha_{m}\xi^{-}_{m}{\gamma^{\prime}_{m}}{e^{i{q_{m}}{L}}}-\frac{\beta_{m}\xi^{-}_{m}}{{\gamma^{\prime}_{m}}}{e^{-i{q_{m}}{L}}}\right]J_{n-m}{{\left(\frac{{F}}{{\omega}^{2}}\right)}}.$
(20)

These can be written in matrix form as follows:

$\displaystyle\begin{pmatrix}{\mathbb{E}_{1}}\\\
{\mathbb{E}^{\prime}_{1}}\end{pmatrix}={\mathbb{V}}\begin{pmatrix}\mathbb{E}_{2}\\\
\mathbb{E}^{\prime}_{2}\end{pmatrix}$ (21)

where the transfer matrix ${\mathbb{V}}$ is given by

$\displaystyle{\mathbb{V}}=\begin{pmatrix}{\mathbb{V}_{11}}&{\mathbb{V}_{12}}\\\
{\mathbb{V}_{21}}&{\mathbb{V}_{22}}\end{pmatrix}={\mathbb{V}^{-1}_{1}}(0)\cdot{\mathbb{V}_{2}}(0)\cdot{\mathbb{V}^{-1}_{2}}(L)\cdot{\mathbb{V}_{1}}(0)\cdot{\mathbb{V}_{3}}(L)$
(22)

and different matrices

$\displaystyle{\mathbb{V}_{1}}(0)=\begin{pmatrix}{\mathbb{I}}&{\mathbb{I}}\\\
{{\mathbb{U}^{+}}}&{{\mathbb{U}^{-}}}\\\
\end{pmatrix},\quad{\mathbb{V}_{2}}(L)=\begin{pmatrix}{{\mathbb{W}^{+}(L)}}&{{\mathbb{W}^{-}(L)}}\\\
{{\mathbb{Y}^{+}(L)}}&{{\mathbb{Y}^{-}(L)}}\end{pmatrix},\quad{\mathbb{V}_{3}}(L)=\begin{pmatrix}{{\mathbb{Z}^{+}(L)}}&{\mathbb{O}}\\\
{{\mathbb{O}}}&{{\mathbb{Z}^{-}(L)}}\\\ \end{pmatrix}$ (23)

involving the set of elements

$\displaystyle\left({{\mathbb{U}^{\pm}}}\right)_{nm}=\pm\left({\gamma_{n}}\right)^{\pm
1}\delta_{nm}$ (24)
$\displaystyle\left({{\mathbb{W}^{\pm}(L)}}\right)_{nm}=\xi^{+}_{m}e^{\pm
iq_{m}{L}}J_{n-m}{{\left(\frac{{F}}{{\omega}^{2}}\right)}}$ (25)
$\displaystyle\left({{\mathbb{Y}^{\pm}(L)}}\right)_{nm}=\pm\xi^{-}_{m}({\gamma^{\prime}_{m}})^{\pm
1}e^{\pm iq_{m}{L}}J_{n-m}{{\left(\frac{{F}}{{\omega}^{2}}\right)}}$ (26)
$\displaystyle\left({\mathbb{Z}^{\pm}(L)}\right)_{nm}=e^{\pm
i{k_{n}}{L}}\delta_{nm}$ (27)

with ${\mathbb{O}}$ is the null matrix, ${\mathbb{I}}$ is the unit matrix,
${\mathbb{E}_{1}}=\\{{\delta_{0m}}\\}$,
${\mathbb{E}^{\prime}_{1}}=\\{r_{m}\\}$, $\mathbb{E}_{2}=\\{t_{m}\\}$ and
$\mathbb{E}^{\prime}_{2}=\\{\lambda_{m}\\}=0$. These allow us to establish the
relation

$\displaystyle\mathbb{E}_{2}={\mathbb{V}_{11}^{-1}\cdot\mathbb{E}_{1}}.$ (28)

This involves several modes that cause a problem in determining the
transmission coefficient $t_{m}$. To overcome such a situation, we truncate
these infinite series into a limited number within the range of $-N$ to $N$
with $N>F/\omega^{2}$ [75, 68, 74, 76]. Consequently, we write

$t_{-N+k}={\mathbb{V}^{\prime}}\left[k+1,N+1\right],\quad k=0,1,\cdots,2N$
(29)

where ${\mathbb{V}^{{}^{\prime}}}$ is the inverse matrix
${\mathbb{V}_{11}^{-1}}$.

To derive the transmission probabilities $T_{m}$, we use the transmitted
${\bm{J}_{\textbf{tran},m}}$ and incident ${\bm{J}_{\textbf{inc},0}}$ current
densities, which can obtained from

$\displaystyle\bm{J}=\psi^{\dagger}(x,y,t)\sigma_{x}\psi(x,y,t).$ (30)

to derive the transmission probabilities
$T_{m}=\left|\dfrac{{\bm{J}_{\textbf{tran},m}}}{{\bm{J}_{\textbf{inc},0}}}\right|$
as follows

$\displaystyle
T_{m}=\dfrac{s_{m}}{s_{0}}\dfrac{\cos\theta_{m}}{\cos\theta_{0}}{\left|{t_{m}}\right|^{2}}.$
(31)

In illustrating the above results and due to numerical complexity, we can
limit ourselves to three transmission channels: the central band ($m=0$) and
the two first sidebands ($m=\pm 1$)

$\displaystyle t_{-1}={\mathbb{V}^{{}^{\prime}}}[1,2],\quad
t_{0}={\mathbb{V}^{{}^{\prime}}}[2,2],\quad
t_{1}={\mathbb{V}^{{}^{\prime}}}[3,2].$ (32)

Next, we will analyze and discuss the transmissions under various conditions
to underline the basic features of the present system.

##  IV Results and discussions

Fig. 2 shows the transmission probabilities versus the energy gap $\Delta$ for
$L=1.2$, $k_{y}=2$. In Fig. 2(a), we present the transmission for the static
barrier $T_{s0}$ ($F/\omega^{2}=0$), central band $T_{0}$ and the two first
sidebands $T_{\pm 1}$ ($F/\omega^{2}=0.6$). We observe that $T_{s0}$ starts
from unity (Klein tunneling) and exhibits oscillations when
$\Delta<\varepsilon$, where the maxima of their amplitude remain constant at
$T_{s0}=1$ (transmission mode), as seen also in [77, 78]. It vanishes in the
opposite case $\Delta>\varepsilon$ because the wave vector $q_{0}$ inside the
barrier becomes imaginary, resulting in the emergence of evanescent states. By
choosing $F/\omega^{2}=0.6$, we see that $T_{0}$ displays the same behavior as
for $T_{s0}$ except that its magnitude drops sharply. Moreover, $T_{\pm 1}$ is
constant until $\Delta\simeq 3$ after which it shows peaks in the zone
$3\lesssim\Delta\lesssim 16$ with different amplitudes and becomes null for
larger $\Delta$. Contrary to a barrier oscillating in time [76], we notice
that the transmission for single photon absorption and emission do not
coincide at lower energy gaps. Fig. 2(b) presents the transmission for no
photon exchange $T_{0}$ versus $\Delta$ with three values of incident energy
$\varepsilon=15$ (blue line), $\varepsilon=20$ (green line), and
$\varepsilon=25$ (red line). By increasing $\varepsilon$, we see that $T_{0}$
starts to fluctuate from different points and moves to the right, as well as
that the number of its oscillations augments rapidly. It is also worth noting
that the amplitude of transmission reduces dramatically until it disappears at
$\Delta>\varepsilon$ corresponding to an evanescent mode. These results show
that $\Delta$ affects the transmissions, and then it can be used to tune the
tunneling effect.

(a)

(b)

Figure 2: (color online) Transmission probabilities versus the energy gap
$\Delta$ for ${L}=1.2$, $k_{y}=2$. (a): $\varepsilon=15$, $F=0$, $\omega=0$
for static barrier $T_{s0}$ (magenta line), and $F=0.6$, $\omega=1$ for
central band $T_{0}$ (blue line) and two first sidebands $T_{-1}$ (green
line), $T_{+1}$ (red line). (b): Transmission probability for the central band
$T_{0}$ with $\varepsilon=15$ (blue line), $\varepsilon=20$ (green line),
$\varepsilon=25$ (red line).

Fig. 3 shows the effect of amplitude $F$ and frequency $\omega$ of the laser
field on the transmission $T_{0}$ as a function of $\Delta$ for ${L}=1.2$,
$k_{y}=2$, $\varepsilon=15$. In Fig. 3(a), we choose
$\\{\omega=1,F=0.3,0.6,0.9\\}$ and observe that $T_{0}$ decreases smoothly
when $F$ increases. By fixing $F$ and varying $\omega$ as shown in Fig. 3(b)
with $\\{F=0.6,\omega=1,1.3,2.6\\}$, we notice that $T_{0}$ increases rapidly
by increasing $\omega$ until the maxima of its oscillation become constant at
$T_{0}=1$ (red line), which is in agreement with the results obtained in [73,
79]. Consequentially, the sharp peaks of the central band are greatly
influenced by the laser field.

(a)

(b)

Figure 3:  (color online) Transmission probability for the central band
$T_{0}$ versus the energy gap $\Delta$ for ${L}=1.2$, $k_{y}=2$,
$\varepsilon=15$. (a): $\omega=1$, $F=0.3$ (blue line), $F=0.6$ (green line),
$F=0.9$ (red line). (b): $F=0.6$, $\omega=1$ (blue line), $\omega=1.3$ (green
line), $\omega=2.6$ (red line).

The transmission probabilities of a static barrier $T_{s0}$ ($F/\omega^{2}=0$)
and three first sidebands $T_{m}$ ($F/\omega^{2}=0.6$) versus the barrier
width $L$ are shown in Fig. 4 for $k_{y}=2$, $\varepsilon=15$ and
$\Delta=0,7$. It is clearly seen that in Fig. 4(a) $T_{s0}$ is unity, and
there is a perfect transmission as noted in [80]. For $F/\omega^{2}=0.6$, we
observe that the transmission $T_{0}$ of the central band initiates at unity
and periodically oscillates, which it reduces under the increase of $L$ as
found in [81], contrary to a scenario in which the barrier oscillates over
time and the maximum value of the peaks remains constant at unity [74].
However, the transmissions ${T_{\pm 1}}$ for the two sidebands start at zero
and exhibit a sinusoidal shape. For $\Delta\neq 0$ in Fig. 4(b), we see that
$T_{s0}$ oscillates with an unchanging amplitude as long as $L$ increases. The
amplitudes of the transmissions $T_{m}$ change dramatically and become much
more oscillatory. The observation that ${T_{\pm 1}}$ oscillate less than
$T_{0}$ of the central band is intriguing. We see an increase in the total
number of peaks for all modes, a steady decrease in the $T_{0}$ maxima, and a
downward shift of the $L$-dependent minima. These are a consequence of the
presence of an energy gap $\Delta$ and laser field in the intermediate region
of the present system.

(a)

(b)

Figure 4:  (color online) Transmission probabilities versus the barrier width
$L$ for $k_{y}=2$, $\varepsilon=15$, $F=0$, $\omega=0$ for static barrier
$T_{s0}$ (magenta line), and $F=0.6$, $\omega=1$ for central band $T_{0}$
(blue line) and two first sidebands $T_{-1}$ (green line), $T_{+1}$ (red
line). (a): $\Delta=0$. (b): $\Delta=7$.

In Fig. 5, we present the transmission $T_{-1}$ versus the barrier width $L$.
We choose $k_{y}=2$, $\varepsilon=15$, $\Delta=7$ with $\omega=1$, $F=0.25$
(blue line), $F=0.45$ (green line), $F=0.65$ (red line) in Fig. 5(a) and
$F=0.6$, $\omega=1$ (blue line), $\omega=1.15$ (green line), $\omega=1.3$ (red
line) in Fig. 5(b). We observe that $T_{-1}$ exhibits oscillating behavior
with sharp peaks as a result of $\Delta\neq 0$. This kind of oscillation
resembles a plane wave modulated by higher frequency [82], which has been
attributed to the presence of the Bessel function in the transmission
expansion. Physically, it can be attributed to constructive or destructive
quantum interference phenomena between the transmitted and reflected electron
waves. More specifically, when the width of the barrier is altered, the
electron waves propagate across the barrier and interact with each other. This
can lead to enhancements or cancellations of the transmission amplitude,
resulting in periodic oscillations. In Fig. 5(a), we observe that the
amplitude of $T_{-1}$ increases slowly with the increase of $F$, as seen in
[73]. Now we plot $T_{-1}$ in Fig. 5(b) by fixing $F$ and varying $\omega$. We
observe that the magnitude of oscillations decreases quickly, and its number
increases, shifting to the left when $L$ increases [73].

(a)

(b)

Figure 5: (color online) Transmission probability for single photon emission
$T_{-1}$ versus the barrier width $L$ for $k_{y}=2$, $\varepsilon=15$,
$\Delta=7$. (a): $\omega=1$, $F=0.25$ (blue line), $F=0.45$ (green line),
$F=0.65$ (red line). (b): $F=0.6$, $\omega=1$ (blue line), $\omega=1.15$
(green line), $\omega=1.3$ (red line).

Fig. 6 shows the effect of $\Delta$ on the transmission probabilities versus
the incident energy $\varepsilon$ with ${L}=1.2$, $k_{y}=2$. For $\Delta=7$ in
Fig. 6(a), we observe that the transmission of the static barrier (magenta
line) is perfectly zero when $\varepsilon\leq 7$, i.e., total reflection. It
exhibits the Fabry-Pérot oscillations, which appear at $\varepsilon\geq 7$,
while its amplitude almost disappears, nearing ($T_{s0}\sim 1$) for larger
$\varepsilon$ as seen in [66, 83]. When the laser field is activated, we
clearly see that $T_{0}$ is more dominant than ${T_{\pm 1}}$ and displays the
same behavior as $T_{s0}$, with the exception that the maxima of its amplitude
become less than one, as shown in [66]. Furthermore, we notice that ${T_{\pm
1}}$ are null for small $\varepsilon$, which occurs because the charge
carriers cannot cross the barrier due to the presence of a forbidden region
where no electronic states are allowed in the conduction or valence band.
Afterwards, they display several peaks with variant amplitudes in the region
$6\leq\varepsilon\leq 30$ and become nearly linear, especially at larger
$\varepsilon$. This can be explained by noting that as the incident energy of
the electrons increases, some electrons acquire sufficient energy to cross the
energy gap and enter the conduction band. This leads to a decrease in the
probability of transmission through graphene, as some of the energy is used to
excite the electrons above the gap and create electron-hole pairs. As a
result, transmission oscillations decrease in amplitude. Fig. 6(b), we depict
${T_{+1}}$ versus $\varepsilon$ for $\Delta=6$ (blue line), $\Delta=7$ (green
line), $\Delta=8$ (red line). One sees that the amplitude and shape of
$T_{+1}$ are affected by $\Delta$, because the maxima of oscillations increase
as $\Delta$ increases and decrease for larger $\varepsilon$. Another
observation discovered is the presence of an asymmetric Fano resonance in the
transmission [84, 85] around $\varepsilon=10$. These peaks may be attributed
to quantum interference phenomena between discrete and continuous states that
arise here due to the interaction between the laser field and the Dirac
fermions. The interferences can be either destructive, leading to a decrease
in transmission amplitude, or constructive, resulting in an increase in
transmission amplitude, as shown in Fig. 6(b).

(a)

(b)

Figure 6: (color online) Transmission probabilities versus the incident energy
$\varepsilon$ for ${L}=1.2$, $k_{y}=2$. (a): $\Delta=7$, $F=0$, $\omega=0$ for
static barrier $T_{s0}$ (magenta line), and $F=0.6$, $\omega=1$ for central
band $T_{0}$ (blue line) and two first sidebands $T_{-1}$ (green line),
$T_{+1}$ (red line). (b): $T_{+1}$ with $\Delta=6$ (blue line), $\Delta=7$
(green line), $\Delta=8$ (red line).

##  V Conclusion

We have studied the effect of an energy gap $\Delta$ on the tunneling
transport of Dirac fermions in graphene subjected to a linearly polarized
laser field of amplitude $F$ and frequency $\omega$. The eigenvalues and
eigenvectors were obtained by applying the Floquet method and solving the
Dirac equation in three regions composing the present system. These have been
merged at interfaces, and by using the transfer matrix method and current
densities transmission probabilities for all energy modes have been
established. For simplicity and a better understanding, we have restricted
ourselves to three transmission modes: central band ($m=0$), first sideband of
photon absorption ($m=+1$) and first sideband of photon emission ($m=-1$).

Subsequently, we have numerically analyzed the transmission probabilities
$T_{m}$ in appropriate conditions of $\Delta$, $F$, $\omega$, barrier width
$L$ and incident energy $\varepsilon$. It has been found that the
transmissions exhibit oscillations that gradually decrease at
$\varepsilon<\Delta$ and disappear entirely at $\varepsilon>\Delta$, which
corresponds to an evanescent mode. We have found that the transmissions of
photon absorption $T_{+1}$ and emission $T_{-1}$ do not meet for a small
$\Delta$ as opposed to a barrier that oscillates in time, where $T_{+1}$ and
$T_{-1}$ are identical. We have shown that $F$ and $\omega$ have a
considerable effect on the behavior of the transmission $T_{m}$ $(m=0,\pm 1)$.
Indeed, they reduce or grow slightly as long as $F$ varies, and they also rise
or drop drastically with the change of $\omega$. Another important finding
with $\Delta$ is that the transmissions become significantly more oscillatory
as the barrier width increases, in addition to an increase in the number of
oscillations. Moreover, their maxima also show resonance that resembles sharp
peaks. Furthermore, we have discovered that by changing the incident energy
the laser field suppresses the Fabry-Pérot oscillations and the transmissions
$T_{m}$ shift to the right as $\Delta$ varies.

On the basis of the experiment [86] showing that the oxidation in irradiation
regions via a two-photon mechanism results in the production of a band gap in
the graphene, our conclusions can also be confirmed experimentally. It is
important to note that the energy gap can be tuned via several parameters,
such as the amplitude of the laser field and its frequency, the barrier width,
and the incident electron energy. The findings presented in this study
demonstrate a novel approach to tuning the energy gap of graphene, offering
exciting possibilities to generate optoelectronic devices based on graphene
for technological applications in the future.

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