Tunneling of Electrons in Graphene via Double Triangular Barrier in External
Fields

Miloud Mekkaouia, Ahmed<EMAIL_ADDRESS>– a.jellal@ucd.ac.maa,b and
Hocine Bahloulic

aLaboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali
University,

PO Box 20, 24000 El Jadida, Morocco

bCanadian Quantum Research Center, 204-3002 32 Ave Vernon,
BC V1T 2L7, Canada

cPhysics Department, King Fahd University of Petroleum $\&$ Minerals,
Dhahran 31261, Saudi Arabia

We study the transmission probability of Dirac fermions in graphene scattered
by a triangular double barrier potential in the presence of an external
magnetic field. Our system made of two triangular potential barrier regions
separated by a well region characterized by an energy gap. Solving our Dirac-
like equation and matching the solutions at the boundaries allowed us to
express our transmission and reflection coefficients in terms of transfer
matrix. We show in particular that the transmission exhibits oscillation
resonances that are manifestations of the Klein tunneling effect. The
triangular barrier electrostatic field was found to play a key role in
controlling the peak of tunneling resistance. However, it only slightly
modifies the resonances at oblique incidence and leaves Klein paradox
unaffected at normal incidence.

PACS numbers: 72.80.Vp, 73.21.-b, 71.10.Pm, 03.65.Pm

Keywords: graphene, double barriers, scattering, transmission.

##  1 Introduction

Graphene [1] remains among the most fascinating and attractive subjects in
contemporary condensed matter physics. This is because of its exotic physical
and transport properties and the apparent similarity of its mathematical model
to the one describing massless relativistic fermions in two dimensions. As a
consequence of this relativistic-like behavior, particles could tunnel through
very high barriers in contrast to the conventional tunneling of non-
relativistic particles, an effect known in relativistic field theory as Klein
tunneling. This tunneling effect has already been observed experimentally [2]
in graphene systems. However, there are various ways for creating barrier
structures in graphene [3, 4]. For instance, it can be realized by applying a
gate voltage, cutting the graphene sheet into a finite width to create a
nanoribbons, using doping or through the creation of a magnetic barrier. In
the case of graphene, results of the transmission coefficient and the
tunneling conductance were already reported for the electrostatic barriers [5,
6, 7], magnetic barriers [6, 8, 9], potential barrier [10], linear [11] and
triangular [12] barriers.

One of the present author [10] theoretically studied the electronic transport
properties of Dirac fermions through one and double triangular barriers in
graphene nanoribbon. The transmission, conductance and Fano factor are
obtained to be various parameters dependent such as well width, barrier height
and barrier width. Therefore, different discussions are given and comparison
with the previous significant works is done. In particular, it is shown that
at Dirac point the Dirac fermions always own a minimum conductance associated
with a maximum Fano factor and change their behaviors in an oscillatory way
(irregularly periodical tunneling peaks) when the potential of applied voltage
is increased.

In our present work we study the transmission probability of Dirac fermions in
graphene scattered by a triangular double barrier potential in the presence of
a uniform magnetic fields $B$ confined to the well region between the two
barriers. We emphasis that $B$-field discussed in our manuscript is applied
externally and is perpendicular to the graphene sheet. It can be created for
instance by depositing a type-I superconducting film on top of the system
except for a strip $|x|<d_{1}$ which is isolated in order to apply a
perpendicular magnetic field in its domain. This patterning technique of
creating the desired magnetic field profile was proposed in [13]. One of the
interesting features of such a magnetic field profile is that it can bind
electrons, contrary to the usual potential step profile. Such a step magnetic
field will indeed result in electron states that are bound to the step
$B$-field and that move in one direction along the step. Thus there is a
current along the $y$-direction but it is a very small effect and is not
relevant for our problem (those electrons have $k_{x}=0$). Indeed, we consider
free electron states that have in general non zero $k_{x}$, because otherwise
they will not tunnel. A recent work studied double barriers with magnetic
field in graphene without mass term [14].

The paper is organized as follows. In section 2, we formulate our model by
setting the Hamiltonian system describing particles scattered by a triangular
double barrier whose well potential zone is subject to a magnetic field in the
presence of a mass term. In section 3, we consider the case of static double
barriers and derive the energy spectrum to finally determine the transmission
and reflection probabilities. Their behaviors are numerically investigated and
in particular resonances were seen in different regions while Klein tunneling
was observed to persist at normal incidence. In section 4, we study the same
system but this time taking into account the presence of a uniform magnetic
field confined to the barrier region. Using boundary conditions, we split the
energy into three domains and then calculate the transmission probability in
each case. In each situation, we discuss the transmission resonances that
characterize each region and stress the importance of our results. We conclude
our work in the final section.

##  2  System setting

Our system is a flat sheet of graphene subject to the barrier $V(x)=V_{\sf j}$
along the $x$-direction while particles are free in the $y$-direction and a
mass term is applied in region 3 as graphically represented in Figure 1.
Formally, we have regions ${\sf{j=1}},{\sf{\cdots}},{{\sf 5}}$ characterized
by the electrostatic potential

$V(x)=\left\\{\begin{array}[]{ll}\Lambda(d_{2}+\gamma
x),&\hbox{$d_{1}\leq|x|\leq d_{2}$}\\\ V_{2},&\hbox{$|x|\leq d_{1}$}\\\
0,&\hbox{otherwise}\\\ \end{array}\right.$ (1)

where $\gamma=1$ for $x\in[-d_{2},-d_{1}]$, $\gamma=-1$ for
$x\in[d_{1},d_{2}]$ and the parameter $\Lambda=\frac{V_{1}}{d_{2}-d_{1}}$
gives the slope of triangular potentials which represents the strength of the
applied electrostatic filed.

Figure 1: Schematic diagram for the monolayer graphene double barrier.

The Hamiltonian system can be written as

$H=v_{F}{\bm{\sigma}}\cdot\left(\textbf{p}+\frac{e}{c}\textbf{A}\right)+V(x){\mathbb{I}}_{2}+G_{p}\Theta\left(d_{1}^{2}-x^{2}\right)\sigma_{z}$
(2)

with the Fermi velocity ${v_{F}\approx 10^{6}m/s}$, the Pauli matrices
$\sigma_{i}$ in pseudospin space, the momentum operator
$\textbf{p}=(p_{x},p_{y})$, the $2\times 2$ unit matrix ${\mathbb{I}}_{2}$ and
$\Theta$ is the Heaviside step function. The vector potential associated to
perpendicular constant magnetic field $B$ will be chosen the Landau gauge
$\textbf{A}=(0,Bx)$. The energy gap $G_{p}=mv_{F}^{2}$ is originating from the
sublattice symmetry breaking, which also can be seen as $G_{p}=G_{p,so}$
originating from spin-orbit interaction.

##  3 Static double barrier

###  3.1 Energy spectrum: electrostatic double barrier

We consider a Hamiltonian describing Dirac fermions in graphene scattered by
an electrostatic double barrier potential in the absence of magnetic field,
$\textbf{A}=0$. In this case (2) reduces to

$H_{s}=v_{F}{\bm{\sigma}}\cdot\textbf{p}+V(x){\mathbb{I}}_{2}+G_{p}\Theta\left(d_{1}^{2}-x^{2}\right)\sigma_{z}$
(3)

where ${\sf{j}}$ labels the five regions indicated schematically in Figure 1
showing the space configuration of the potential profile. Due to inversion and
time reversal symmetries we therefore need to study our system only near the K
point. The time-independent Dirac equation for the spinor
$\Phi(x,y)=\left(\varphi^{+},\varphi^{-}\right)^{T}$ at energy
$E=v_{F}\epsilon$ then reads, in unit system $\hbar=m=c=1$, as

$\left[{\bm{\sigma}}\cdot\textbf{p}+{v_{j}}{\mathbb{I}}_{2}+\mu\Theta\left(d_{1}^{2}-x^{2}\right)\sigma_{z}\right]\Phi(x,y)=\epsilon\Phi(x,y)$
(4)

where $V_{\sf j}=v_{F}v_{\sf j}$ and $G_{p}=v_{F}\mu$. Our system is supposed
to have finite width $W$ with infinite mass boundary conditions on the
wavefunction at the boundaries $y=0$ and $y=W$ along the $y$-direction [15,
16]. These boundary conditions result in a quantization of the transverse
momentum along the $y$-direction as

$k_{y}=\frac{\pi}{W}\left(n+\frac{1}{2}\right),\qquad n=0,1,2\cdots.$ (5)

One can therefore assume a spinor solution of the following form $\Phi_{\sf
j}=\left(\varphi_{\sf j}^{+}(x),\varphi_{\sf
j}^{-}(x)\right)^{\dagger}e^{ik_{y}y}$ and the subscript ${\sf j}=1,2,3,4,5$
refers to the space region while the superscripts indicate the two spinor
components. Solving the eigenvalue equation to obtain the upper and lower
components of the eignespinor in the incident and reflection region 1
($x<-d_{2}$) gives

$\displaystyle\Phi_{\sf 1}=\left(\begin{array}[]{c}{1}\\\ {z_{1}}\\\
\end{array}\right)e^{i(k_{1}x+k_{y}y)}+r_{s,n}\left(\begin{array}[]{c}{1}\\\
{-z_{1}^{-1}}\\\ \end{array}\right)e^{i(-k_{1}x+k_{y}y)},\qquad
z_{1}=s_{1}\frac{k_{1}+ik_{y}}{\sqrt{k_{1}^{2}+k_{y}^{2}}}$ (10)

associated to the dispersion relation
$\epsilon=s_{1}\sqrt{k_{1}^{2}+k_{y}^{2}}$ with $s_{\sf
j}={\mbox{sign}}{\left(E\right)}$.

In regions 2 and 4 ($d_{1}<|x|<d_{2}$), the general solution can be expressed
in terms of the parabolic cylinder function [17, 18, 11] as

$\chi_{\gamma}^{+}=c_{n1}D_{\nu_{n}-1}\left(Q_{\gamma}\right)+c_{n2}D_{-\nu_{n}}\left(-Q^{*}_{\gamma}\right)$
(11)

where $c_{n1}$ and $c_{n2}$ are constants,
$\nu_{n}=\frac{ik_{y}^{2}}{2\varrho}$ and
$Q_{\gamma}(x)=\sqrt{\frac{2}{\varrho}}e^{i\pi/4}\left(\gamma\varrho
x+\epsilon_{0}\right)$, with $\epsilon_{0}=\epsilon-v_{1}$,
$\Lambda=v_{F}\varrho$, $V_{1}=v_{F}v_{1}$. The lower spinor component is
given by

$\displaystyle\chi_{\gamma}^{-}=-\frac{c_{n2}}{k_{y}}\left[2(\epsilon_{0}+\gamma\varrho
x)D_{-\nu_{n}}\left(-Q^{*}_{\gamma}\right)+\sqrt{2\varrho}e^{i\pi/4}D_{-\nu_{n}+1}\left(-Q^{*}_{\gamma}\right)\right]-\frac{c_{n1}}{k_{y}}\sqrt{2\varrho}e^{-i\pi/4}D_{\nu_{n}-1}\left(Q_{\gamma}\right).$
(12)

The components of the spinor solution of the Dirac equation (3) in regions 2
and 4 can be obtained from (11) and (12) with
$\varphi_{\gamma}^{+}(x)=\chi_{\gamma}^{+}+i\chi_{\gamma}^{-}$ and
$\varphi_{\gamma}^{-}(x)=\chi_{\gamma}^{+}-i\chi_{\gamma}^{-}$. Then, in
regions 2 and 4 we have the eigenspinors

$\displaystyle\Phi_{\sf j}$ $\displaystyle=$ $\displaystyle
a_{\sf{j}-1}\left(\begin{array}[]{c}\eta^{+}_{\gamma}(x)\\\
\eta^{-}_{\gamma}(x)\\\ \end{array}\right)e^{ik_{y}y}+a_{\sf
j}\left(\begin{array}[]{c}\xi^{+}_{\gamma}(x)\\\ \xi^{-}_{\gamma}(x)\\\
\end{array}\right)e^{ik_{y}y}$ (17)

where $\gamma=\pm 1$ and we have set

$\displaystyle\eta^{\pm}_{\gamma}(x)=D_{\nu_{n}-1}\left(Q_{\gamma}\right)\mp\frac{1}{k_{y}}\sqrt{2\varrho}e^{i\pi/4}D_{\nu_{n}}\left(Q_{\gamma}\right)$
(18)
$\displaystyle\xi^{\pm}_{\gamma}(x)=\pm\frac{1}{k_{y}}\sqrt{2\varrho}e^{-i\pi/4}D_{-\nu_{n}+1}\left(-Q_{\gamma}^{*}\right)\pm\frac{1}{k_{y}}\left(-2i\epsilon_{0}\pm
k_{y}-\gamma 2i\varrho x\right)D_{-\nu_{n}}\left(-Q_{\gamma}^{*}\right).$ (19)

In region 3, we use (3) to obtain the eigenspinor

$\displaystyle\Phi_{\sf 3}=b_{1}\left(\begin{array}[]{c}{\alpha}\\\ {\beta
z_{3}}\\\
\end{array}\right)e^{i(k_{3}x+k_{y}y)}+b_{2}\left(\begin{array}[]{c}{\alpha}\\\
{-\beta z_{3}^{-1}}\\\ \end{array}\right)e^{i(-k_{3}x+k_{y}y)}$ (24)

with the parameters

$z_{3}=s_{3}\frac{k_{3}+ik_{y}}{\sqrt{k_{3}^{2}+k_{y}^{2}}},\qquad{\alpha=\left({1+\frac{\mu}{\epsilon-
v_{2}}}\right)}^{1/2},\qquad{\beta=\left({1-\frac{\mu}{\epsilon-
v_{2}}}\right)}^{1/2}$ (25)

and the wave vector $k_{3}=\sqrt{(\epsilon-v_{2})^{2}-\mu^{2}-{k_{y}}^{2}}$,
$s_{3}=\mbox{sign}(\epsilon-v_{2})$. The eigenspinor in region 5 can easily be
obtained

$\Phi_{\sf 5}=t_{s,n}\left(\begin{array}[]{c}{1}\\\ {z_{1}}\\\
\end{array}\right)e^{i(k_{1}x+k_{y}y)}$ (26)

in similar way to region 1. These results will be embedded to deal with
tunneling properties of our system.

###  3.2  Transmission and reflection: electrostatic double barrier

We determine the transmission and reflection probabilities for an
electrostatic double barrier alone by requiring the continuity of the
eigenspinor at junction interfaces. This yields

$\displaystyle\Phi_{\sf 1}(-d_{2})=\Phi_{\sf 2}(-d_{2}),\qquad\Phi_{\sf
2}(-d_{1})=\Phi_{\sf 3}(-d_{1}),\qquad\Phi_{\sf 3}(d_{1})=\Phi_{\sf
4}(d_{1}),\qquad\Phi_{\sf 4}(d_{2})=\Phi_{\sf 5}(d_{2}).$ (27)

We prefer to express these relationships in terms of $2\times 2$ transfer
matrices between different regions, then we write

$\left(\begin{array}[]{c}a_{\sf j}\\\ b_{\sf j}\\\ \end{array}\right)=M_{{\sf
j},{\sf j}+1}\left(\begin{array}[]{c}a_{{\sf j}+1}\\\ b_{{\sf j}+1}\\\
\end{array}\right)$ (28)

where $M_{{\sf j},{\sf j}+1}$ are the transfer matrices that couple the
wavefunction in the $j$-th region to the wavefunction in the $({\sf j}+1)$-th
region. Finally, we obtain the full transfer matrix $M$ over the whole double
barrier which can be written, in an obvious notation, as follows

$\left(\begin{array}[]{c}1\\\ r_{s,n}\\\ \end{array}\right)=\prod_{{\sf
j}=1}^{4}M_{{\sf j},{\sf j}+1}\left(\begin{array}[]{c}t_{s,n}\\\ 0\\\
\end{array}\right)=M\left(\begin{array}[]{c}t_{s,n}\\\ 0\\\
\end{array}\right)$ (29)

where $M=M_{12}\cdot M_{23}\cdot M_{34}\cdot M_{45}$ is given by

$M=\left(\begin{array}[]{cc}m_{11}&m_{12}\\\ m_{21}&m_{22}\\\
\end{array}\right)\\\ $ (30)

and the remaining matrices read as

$\displaystyle
M_{12}=\left(\begin{array}[]{cc}e^{-\textbf{\emph{i}}k_{1}d_{2}}&e^{\textbf{\emph{i}}k_{1}d_{2}}\\\
z_{1}e^{-\textbf{\emph{i}}k_{1}d_{2}}&-z^{\ast}_{1}e^{\textbf{\emph{i}}k_{1}d_{2}}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}\eta_{1}^{+}(-d_{2})&\xi_{1}^{+}(-d_{2})\\\
\eta_{1}^{-}(-d_{2})&\xi_{1}^{-}(-d_{2})\\\ \end{array}\right)$ (35)
$\displaystyle
M_{23}=\left(\begin{array}[]{cc}\eta_{1}^{+}(-d_{1})&\xi_{1}^{+}(-d_{1})\\\
\eta_{1}^{-}(-d_{1})&\xi_{1}^{-}(-d_{1})\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}\alpha
e^{-\textbf{\emph{i}}k_{3}d_{1}}&\alpha e^{\textbf{\emph{i}}k_{3}d_{1}}\\\
\beta z_{3}e^{-\textbf{\emph{i}}k_{3}d_{1}}&-\beta
z^{\ast}_{3}e^{\textbf{\emph{i}}k_{3}d_{1}}\\\ \end{array}\right)$ (40)
$\displaystyle M_{34}=\left(\begin{array}[]{cc}\alpha
e^{\textbf{\emph{i}}k_{3}d_{1}}&\alpha e^{-\textbf{\emph{i}}k_{3}d_{1}}\\\
\beta z_{3}e^{\textbf{\emph{i}}k_{3}d_{1}}&-\beta
z^{\ast}_{3}e^{-\textbf{\emph{i}}k_{3}d_{1}}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}\eta_{-1}^{+}(d_{1})&\xi_{-1}^{+}(d_{1})\\\
\eta_{-1}^{-}(d_{1})&\xi_{-1}^{-}(d_{1})\\\ \end{array}\right)$ (45)
$\displaystyle
M_{45}=\left(\begin{array}[]{cc}\eta_{-1}^{+}(d_{2})&\xi_{-1}^{+}(d_{2})\\\
\eta_{-1}^{-}(d_{2})&\xi_{-1}^{-}(d_{2})\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}e^{\textbf{\emph{i}}k_{1}d_{2}}&e^{-\textbf{\emph{i}}k_{1}d_{2}}\\\
z_{1}e^{\textbf{\emph{i}}k_{1}d_{2}}&-z_{1}^{\ast}e^{-\textbf{\emph{i}}k_{1}d_{2}}\\\
\end{array}\right).$ (50)

Consequently, the transmission and reflection amplitudes take the forms

$t_{s,n}=\frac{1}{m_{11}},\qquad r_{s,n}=\frac{m_{21}}{m_{11}}.$ (51)

Some symmetry relationship between the parabolic cylindric functions are worth
mentioning, they are given by

$\eta_{-1}^{\pm}(d_{1})=\eta_{1}^{\pm}(-d_{1}),\qquad\eta_{-1}^{\pm}(d_{2})=\eta_{1}^{\pm}(-d_{2})$
(52)
$\xi_{-1}^{\pm}(d_{1})=\xi_{1}^{\pm}(-d_{1}),\qquad\xi_{-1}^{\pm}(d_{2})=\xi_{1}^{\pm}(-d_{2}).$
(53)

We should point out at this stage that we were unfortunately forced to adopt a
somehow cumbersome notation for our wavefunction parameters in different
potential regions due to the relatively large number of necessary subscripts
and superscripts. Before matching the eigenspinors at the boundaries, let us
define the following shorthand notation

$\eta_{1}^{\pm}(-d_{1})=\eta_{11}^{\pm},\qquad\eta_{1}^{\pm}(-d_{2})=\eta_{12}^{\pm}$
(54)
$\xi_{1}^{\pm}(-d_{1})=\xi_{11}^{\pm},\qquad\xi_{1}^{\pm}(-d_{2})=\xi_{12}^{\pm}.$
(55)

After some lengthy algebra, one can solve the linear system given in (29) to
obtain the transmission and reflection amplitudes in closed forms. As for the
former one, we find

$t_{s,n}=\frac{\alpha\beta
e^{2i(k_{1}d_{2}+k_{3}d_{1})}\left(1+z_{1}^{2}\right)\left(1+z_{3}^{2}\right)}{z_{3}\left(e^{4ik_{3}d_{1}}-1\right)\left(\alpha^{2}\mathcal{Y}_{2}+\beta^{2}\mathcal{Y}_{1}\right)+\alpha\beta\mathcal{Y}_{3}}\left(\xi_{11}^{+}\eta_{11}^{-}-\xi_{11}^{-}\eta_{11}^{+}\right)\left(\xi_{12}^{-}\eta_{12}^{+}-\xi_{12}^{+}\eta_{12}^{-}\right)$
(56)

where we have defined the following quantities

$\displaystyle\mathcal{Y}_{1}=\left(\xi_{12}^{-}\eta_{11}^{+}-\xi_{11}^{+}\eta_{12}^{-}-\xi_{12}^{+}\eta_{11}^{+}z_{1}+\xi_{11}^{+}\eta_{12}^{+}z_{1}\right)\left(\xi_{11}^{+}\eta_{12}^{+}+\xi_{11}^{+}\eta_{12}^{-}z_{1}-\eta_{11}^{+}(\xi_{12}^{+}+\xi_{12}^{-}z_{1}\right)$
(57)
$\displaystyle\mathcal{Y}_{2}=\left(\xi_{11}^{-}\eta_{12}^{+}-\xi_{11}^{-}\eta_{12}^{-}z_{1}-\eta_{11}^{-}(\xi_{12}^{+}+\xi_{12}^{+}z_{1}\right)\left(-\xi_{12}^{-}\eta_{11}^{-}+\xi_{12}^{+}\eta_{11}^{-}z_{1}-\xi_{11}^{-}(\eta_{12}^{-}+\eta_{12}^{+}z_{1}\right)$
(58)
$\displaystyle\mathcal{Y}_{3}=\Gamma_{0}\left(1+z_{1}^{2}z_{3}^{2}\right)+\Gamma_{1}z_{1}\left(1-z_{3}\right)+\Gamma_{2}\left(z_{1}^{2}+z_{3}^{2}\right)+e^{4id_{1}k_{3}}\left(\Gamma_{3}+\Gamma_{4}\right)$
(59)

together with

$\displaystyle\Gamma_{0}$ $\displaystyle=$
$\displaystyle-\xi_{12}^{+}\xi_{12}^{-}\eta_{11}^{+}\eta_{11}^{-}+\xi_{11}^{+}\xi_{12}^{-}\eta_{11}^{-}\eta_{12}^{+}+\xi_{11}^{-}\xi_{12}^{+}\eta_{11}^{+}\eta_{12}^{-}-\xi_{11}^{+}\xi_{11}^{-}\eta_{12}^{+}\eta_{12}^{-}$
(60) $\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle\left(\xi_{12}^{+}\right)^{2}\eta_{11}^{+}\eta_{11}^{-}-\left(\xi_{12}^{-}\right)^{2}\eta_{11}^{+}\eta_{11}^{-}-\xi_{11}^{-}\xi_{12}^{+}\eta_{11}^{+}\eta_{12}^{+}-\xi_{11}^{+}\xi_{12}^{+}\eta_{11}^{-}\eta_{12}^{+}$
$\displaystyle+\xi_{11}^{+}\xi_{11}^{-}\left(\eta_{12}^{+}\right)^{2}-\xi_{11}^{+}\xi_{11}^{-}\left(\eta_{12}^{-}\right)^{2}+\xi_{11}^{-}\xi_{12}^{-}\eta_{11}^{+}\eta_{12}^{-}+\xi_{11}^{+}\xi_{12}^{-}\eta_{11}^{-}\eta_{12}^{-}$
$\displaystyle\Gamma_{2}$ $\displaystyle=$
$\displaystyle\xi_{12}^{+}\xi_{12}^{-}\eta_{11}^{+}\eta_{11}^{-}-\xi_{11}^{-}\xi_{12}^{-}\eta_{11}^{+}\eta_{12}^{+}-\xi_{11}^{+}\xi_{12}^{+}\eta_{11}^{-}\eta_{12}^{-}+\xi_{11}^{+}\xi_{11}^{-}\eta_{12}^{+}\eta_{12}^{-}$
(62) $\displaystyle\Gamma_{3}$ $\displaystyle=$
$\displaystyle\left(\xi_{12}^{+}\right)^{2}\eta_{11}^{+}\eta_{11}^{-}\left(z_{3}^{2}-1\right)-\xi_{11}^{-}\xi_{12}^{-}\eta_{11}^{+}\left[\eta_{12}^{+}\left(1+z_{1}^{2}z_{3}^{2}\right)-\eta_{12}^{-}z_{1}\left(z_{3}^{2}-1\right)\right]$
$\displaystyle+\xi_{11}^{-}\xi_{11}^{+}\left[\left(\eta_{12}^{+}\right)^{2}z_{1}-\left(\eta_{12}^{-}\right)^{2}z_{1}+\eta_{12}^{+}\eta_{12}^{-}\left(z_{1}^{2}-1\right)\left(z_{3}^{2}-1\right)\right]$
$\displaystyle\Gamma_{4}$ $\displaystyle=$
$\displaystyle\xi_{12}^{-}\eta_{11}^{-}\left[-\xi_{12}^{-}\eta_{11}^{+}z_{1}\left(z_{3}^{2}-1\right)+\xi_{11}^{+}\left(\eta_{12}^{-}z_{0}\left(z_{3}^{2}-1\right)+\eta_{12}^{+}\left(z_{1}^{2}+z_{3}^{2}\right)\right)\right]$
$\displaystyle\xi_{12}^{+}\xi_{12}^{-}\eta_{11}^{+}\eta_{11}^{-}\left(z_{1}^{2}+1\right)\left(z_{1}^{3}-1\right)-\xi_{12}^{+}\xi_{11}^{+}\eta_{11}^{-}\left(\eta_{12}^{-}\left(1+z_{1}^{2}z_{3}^{2}\right)+\eta_{12}^{+}z_{1}\left(z_{1}^{3}-1\right)\right)$
$\displaystyle+\xi_{12}^{+}\xi_{11}^{-}\eta_{11}^{+}\left[\eta_{12}^{-}\left(z_{1}^{2}+z_{3}^{2}\right)+\eta_{12}^{+}z_{1}\left(1-z_{3}^{2}\right)\right].$

Now we are ready for the computation of the transmission $T_{s}$ and
reflection $R_{s}$ probabilities. For this purpose, we introduce the
associated current density $J$, which defines $R_{s}$ and $T_{s}$ as follows

$T_{s}=\frac{J_{\sf{tra}}}{J_{\sf{inc}}},\qquad
R_{s}=\frac{J_{\sf{ref}}}{J_{\sf{inc}}}$ (65)

where $J_{\sf{\sf{inc}}}$, $J_{\sf{ref}}$ and $J_{\sf{\sf{tra}}}$ stand for
the incident, reflected and transmitted components of the current density,
respectively. It is easy to show

$J=e\upsilon_{F}\Phi^{\dagger}\sigma_{x}\Phi$ (66)

which gives rise to the following results

$\displaystyle J_{\sf{inc}}=e\upsilon_{F}(\Phi_{\sf
1}^{+})^{\dagger}\sigma_{x}\Phi_{\sf 1}^{+}$ (67) $\displaystyle
J_{\sf{ref}}=e\upsilon_{F}(\Phi_{\sf 1}^{-})^{\dagger}\sigma_{x}\Phi_{\sf
1}^{-}$ (68) $\displaystyle J_{\sf{tra}}=e\upsilon_{F}\Phi_{\sf
5}^{\dagger}\sigma_{x}\Phi_{\sf 5}$ (69)

and allow us to express the transmission and reflection probabilities as

$T_{s}=|t_{s}|^{2},\qquad R_{s}=|r_{s}|^{2}.$ (70)

We need to point out that we were able to consider a separate transmission
probability for each of the propagating modes in the leads, because the
matching condition does not mix these modes. Under such circumstances,
physical measurable quantities such as conductance will be obtained using
Landauer formula where summation of over all modes is performed.

###  3.3 Numerical results: electrostatic double barrier

The transmission $T_{s}$ of Dirac electrons in graphene scattered by a
triangular double barrier potential are evaluated numerically as a function of
our system parameters: the energy $\epsilon$, the $y$-direction wave vector
$ky$, the energy gap $\mu$, the barriers widths ($d_{1},d_{2}$), the strength
of potential barriers ($v_{1},v_{2}$).

(a)

(b)

(c)

Figure 2: (color online) Contour plot for $T(v_{2},k_{y})$ with the following
parameters: $d_{1}=1$, $d_{2}=2$, $\epsilon=3$, $\mu=0$. Let $v_{2}$ changes
from 0 to 6 and $k_{y}$ from -2 to -0.001 and 0.001 to 2 that is to say
$k_{y}$ from -2 to 2 and $k_{y}\neq 0$. (a): $v_{1}=0.5$. (b): $v_{1}=4$. (c):
$v_{1}=8$.

In Figure 2 we present the contour plot of the transmission probability
$T=tt^{\dagger}$ for a triangular double barrier structure as a function of
well potential $v_{2}$ and the transverse wave vector $k_{y}$. This has been
performed by fixing the parameters $d_{1}=1$, $d_{2}=2$, $\epsilon=3$, $\mu=0$
for three different values of the potential height $v_{1}=0.5$, $v_{1}=4$ and
$v_{1}=8$ corresponding to figures 2a, 2b and 2c, respectively. Note that the
strength of the potential height $v_{1}$ in this figure is proportional to the
strength of the applied electrostatic field. The different colors from purple
to red correspond to different values of the transmission continuously from
$0$ to $1$. The number of oscillations increases as the strength of the
electric field increases. The resonances that are clear in the transmission
probability show up as peaks, we also observe the occurrence of a new strong
and sharp resonance around $v_{2}=\epsilon$.

(a)

(b)

(c)

Figure 3: (color online) Contour plot for $T(\epsilon,k_{y})$ with the
following parameters: $d_{1}=1$, $d_{2}=2$, $\mu=0$, $v_{2}=0$. Let $\epsilon$
changes from 0 to 4 and $k_{y}$ from -2 to -0.001 and then from 0.001 to 2
that is to say $k_{y}$ from -2 to 2 and $k_{y}\neq 0$. (a): $v_{1}=2$. (b):
$v_{1}=3$. (c): $v_{1}=5$.

In Figure 3 we show the contour plot of the transmission probability as
function of energy $\epsilon$ and the transfer wave vector $k_{y}$ for
$d_{1}=1$, $d_{2}=2$, gapless graphene $\mu=0$ and three different values for
potential $v_{1}$. Here, we take the case with $v_{1}=2$ in figure 3a
initially and then increase $v_{1}$ from $v_{1}=3$ in figure 3b to $v_{1}=5$
in figure 3c and notice that the number of oscillations increases, when the
incidence is almost normal, the wave vector is close to $k_{y}\approx 0$
associated with total transmission. We see that full transmission is squeezed
into a very narrow angle region around normal incidence, $k_{y}\approx 0$,
when we increase the strength of the electrostatic field make it favorable
application to tunneling collimation.

(a)

(b)

(c)

Figure 4: (color online) Contour plot for $T(\epsilon,k_{y})$ with the
following parameters: $v_{1}=2$, $d_{2}=2$, $\mu=0$, $v_{2}=0$. Let $\epsilon$
changes from 0 to 4 and $k_{y}$ from -2 to -0.001 and then from 0.001 to 2
that is to say $k_{y}$ from -2 to 2 and $k_{y}\neq 0$. (a): $d_{1}=0.1$. (b):
$d_{1}=0.9$. (c): $d_{1}=1.5$.

In Figure 4 we show the contour plot of the transmission probability as
function of energy $\epsilon$ and the transverse wave vector $k_{y}$ from -2
to 2 and $k_{y}\neq 0$, we fix the potential $v_{1}=2$, $v_{2}=0$, the barrier
width $d_{2}=2$, for gapless graphene $\mu=0$ and three different values of
the barrier width $d_{1}=0.1$ in figure 4a, $d_{1}=0.0.9$ in figure 4b and
$d_{1}=1.5$ in figure 4c. By increasing the width $d_{1}$ the number of
oscillation peaks increases. The parameters of the triangular potential
($v_{1}$, $d_{2}-d_{1}$), whose ratio represents the strength of the applied
electrostatic field, are found to play a key role in controlling the tunneling
resistance peak. All these predicted attractive transport properties are
expected to be extremely useful for designing both novel electronic and
optical graphene-based devices and electronic lenses in ballistic-electron
optics.

(a)

(b)

Figure 5: (color online) (a): Transmission (red color) and reflection (green
color) probabilities $(T_{s},R_{s})$ as a function of energy $\epsilon$ with
$d_{1}=0.6$, $d_{2}=2.5$, $\mu=4$, $k_{y}=2$, $v_{1}=60$ and $v_{2}=30$. (b):
Transmission probability $T_{s}$ as a function of energy gap $\mu$ with
$d_{1}=0.5$, $d_{2}=1.5$, $k_{y}=1$, $v_{1}=50$, $v_{2}=40$, $\epsilon=15$
(green color), $\epsilon=25$ (red color) and $\epsilon=35$ (blue color).

In Figure 5a we show the transmission and reflection probabilities versus the
energy $\epsilon$. Obviously, we can check that the probability conservation
condition $T_{s}+R_{s}=1$ is well satisfied. In the first energy interval
$\epsilon\leq k_{y}$ we have no transmission because it is a forbidden zone.
However, in the second energy intervals $k_{y}\leq\epsilon\leq
v_{2}-k_{y}-\frac{\mu}{2}$ and $v_{2}+k_{y}+\frac{\mu}{2}\leq\epsilon\leq
v_{1}$, we observe resonance oscillations due to the Klein regime. We have no
transmission (gap region) when $v_{2}-k_{y}-\frac{\mu}{2}\leq\epsilon\leq
v_{2}+k_{y}+\frac{\mu}{2}$. Finally, in the interval where $\epsilon>v_{1}$,
there exist usual high energy oscillations, which saturate asymptotically.
Note that (25) implies that for certain energy gap $\mu$, there is no
transmission. In fact, under the condition $\mu>|v_{2}-\epsilon|$ every
incoming wave is reflected. In Figure 5b we see that the transmission vanishes
for values of $\epsilon$ below the critical value $\mu=|v_{2}-\epsilon|$.

(a)

(b)

Figure 6: (color online) Transmission probability for the static barrier
$T_{s}$ as a function of energy $\epsilon$ with $d_{1}=0.3$ (red color),
$d_{1}=1$ (blue color), $d_{2}=2.5$, $\mu=4$ and $k_{y}=2$. (a): for
$v_{1}\leq v_{2}$: $v_{1}=30$, $v_{2}=60$. (b): for $v_{1}\geq v_{2}$:
$v_{1}=60$, $v_{2}=30$.

Figure 6 shows the transmission $T_{s}$ as a function of incident electron
energy $\epsilon$ for the Dirac fermion scattered by a double triangular
barriers with $d_{2}=2.5$, $\mu=4$, $k_{y}=2$ and two values of barrier width
$d_{1}=\\{0.3,1\\}$. In Figure 6a we consider the parameters
$v_{1}=\frac{v_{2}}{2}=30$ for the Dirac fermion scattered by a double barrier
triangular potential where we distinguish five different zones. Indeed, the
first is a forbidden zone where 0$\leq\epsilon\leq k_{y}$. The second zone
$k_{y}\leq\epsilon\leq v_{1}$ is the lower Klein energy zone with transmission
resonances. The third zone contains transmission oscillations. The fourth one
$v_{2}-k_{y}-\frac{\mu}{2}\leq\epsilon\leq v_{2}+k_{y}+\frac{\mu}{2}$ is a
window where the transmission is zero, the wavefunction is damped and
transmission decays exponentially. The fifth zone $\epsilon\geq
v_{2}+k_{y}+\frac{\mu}{2}$ contains over barrier oscillations, the
transmission converges to unity at high energies similarly to the non-
relativistic result. We also notice from Figure 6a that resonances move to the
right as we increase the width of the field region (reduction of the
electrostatic field). In Figure 6b we consider the opposite situation, that is
$v_{1}=2v_{2}=60$, the results show that as long as the well width $d_{1}$
increases the transmission resonances shift to the left and the width of the
resonances increase in regions $k_{y}\leq\epsilon\leq
v_{2}-k_{y}-\frac{\mu}{2}$ and $v_{2}+k_{y}+\frac{\mu}{2}\leq\epsilon\leq
v_{1}$.

(a)

(b)

Figure 7: (color online) (a): Transmission probability through the static
barrier $T_{s}$ as a function of the potential $v_{2}$ for $d_{1}=0.2$ (green
color), $d_{1}=0.6$ (red color), $d_{1}=1.2$ (blue color), $d_{2}=2$, $\mu=3$,
$k_{y}=1$, $\epsilon=40$ and $v_{1}=60$. (b): Transmission probability through
the static barrier $T_{s}$ as a function of the potential $v_{1}$ with
$d_{1}=0.05$ (green color), $d_{1}=0.7$ (red color), $d_{1}=2$ (blue color),
$d_{2}=2.5$, $\mu=4$, $k_{y}=2$, $\epsilon=30$ and $v_{2}=60$.

We present in Figure 7a the transmission versus the potential $v_{2}$. It is
clear that the two transmission curves are almost symmetric with respect to
the point $v_{2}=\epsilon$. While an increase in the value $d_{1}$ widens the
bowl width. As we increase the well width we see an increase in the number of
resonances while a zero transmission region appears only for small values of
$d_{1}$ (large values of the electrostatic field). Figure 7b shows the
transmission probability for a static barrier $T_{s}$ as function of the
strength of the applied voltage $v_{1}$ for different widths of the electric
field region. Full transmission is observed for small values of $v_{1}$, less
than the energy of the incident fermion. It then decreases sharply for
$v_{1}>\epsilon-(2k_{y}+\mu)$ until it reaches a relative minimum and then
begins to increase in an oscillatory manner. However, we noticed that a gap
region appeared for energies around the value of the elevated well potential,
$v_{2}-k_{y}-\frac{\mu}{2}\leq\epsilon\leq v_{2}+k_{y}+\frac{\mu}{2}$,
independently of the ordering strength of the potentials $v_{1}$ and $v_{2}$.
As a results, we observe that the double barrier strongly affect the
transmission and then it can be used a tunable parameter to control the
tunneling in our system, which can be used in technological applications.

##  4 Magnetic double barrier

Consider a two-dimensional system of Dirac fermions forming a graphene sheet.
This sheet is subject to a double barrier potential in addition to a mass term
and an externally applied magnetic field. Particles and antiparticles moving
in the positive and negative energy regions, respectively, have a conserved
component of the wave vector along the $y$-direction (mode) due to the fact
that each of the $N$ propagating modes in the leads is independent because the
matching condition does not mix these modes. A uniform perpendicular magnetic
field is applied, along the $z$-direction and confined to the well region
between the two electrostatic barriers. It is defined by

$B(x,y)=B\Theta(d_{1}^{2}-x^{2})$ (71)

where $B$ is the strength of the magnetic field within the strip located in
the region $|x|<d_{1}$ and $B=0$ otherwise, $\Theta$ is the Heaviside unit
step function. Choosing the Landau gauge and imposing continuity of the vector
potential at the boundary to avoid unphysical effects, we end up with the
following vector potential

$\qquad
A_{y}(x)=A_{j}=\frac{c}{e}\times\left\\{\begin{array}[]{ll}-\frac{1}{l_{B}^{2}}d_{1},&\hbox{$x<-d_{2}$}\\\
\frac{1}{l_{B}^{2}}x,&\hbox{$\mid x\mid<d_{1}$}\\\
\frac{1}{l_{B}^{2}}d_{1},&\hbox{$x\geq d_{2}$}\\\ \end{array}\right.$ (72)

with the magnetic field length $l_{B}=\sqrt{1/B}$ in the unit system
($\hbar=c=e=1$). The system is composed of five regions denoted ${\sf
j=1,2,3,4,5}$. The left region (${\sf j=1}$) describes the incident electron
beam with energy $E=v_{F}\epsilon$ at an incident angle $\phi_{1}$ where
$v_{F}$ is the Fermi velocity. The extreme right region (${\sf j=5}$)
describes the transmitted electron beam at an angle $\phi_{5}$. The
Hamiltonian for describing our system reads as

$H_{m}=v_{F}{\bm{\sigma}}\cdot\left(\textbf{p}+\frac{e}{c}\textbf{A}\right)+V(x){\mathbb{I}}_{2}+G_{p}\Theta\left(d_{1}^{2}-x^{2}\right)\sigma_{z}.$
(73)

To proceed further, we need to find the solutions of the corresponding Dirac
equation and their associated energy spectrum.

###  4.1 Energy spectrum: presence of magnetic field

We are set to determine the eigenvalues and eigenspinors of the Hamiltonian
$H_{m}$. Indeed, the Dirac Hamiltonian describing regions 1 and 5, is obtained
from (73) as

$H_{m}=\left(\begin{array}[]{cc}0&\upsilon_{F}\left(p_{x{\sf
j}}-i\left(p_{y}+\frac{e}{c}A_{\sf j}\right)\right)\\\
\upsilon_{F}\left(p_{x{\sf j}}+i\left(p_{y}+\frac{e}{c}A_{\sf
j}\right)\right)&0\\\ \end{array}\right).$ (74)

The corresponding time independent Dirac equation for the spinor $\psi_{\sf
j}(x,y)=(\varphi_{\sf j}^{+},\varphi_{\sf j}^{-})^{T}$ at energy
$E=\upsilon_{F}\epsilon$ is given by

$H_{m}\left(\begin{array}[]{c}\varphi_{\sf j}^{+}\\\ \varphi_{\sf j}^{-}\\\
\end{array}\right)=\epsilon\left(\begin{array}[]{c}\varphi_{\sf j}^{+}\\\
\varphi_{\sf j}^{-}\\\ \end{array}\right).$ (75)

This eigenvalue problem can be written as two linear differential equations of
the from

$\displaystyle p_{x{\sf j}}-i\left(p_{y}+\frac{e}{c}A_{\sf
j}\right)\varphi_{\sf j}^{-}=\epsilon\varphi_{\sf j}^{+}$ (76) $\displaystyle
p_{x{\sf j}}+i\left(p_{y}+\frac{e}{c}A_{\sf j}\right)\varphi_{\sf
j}^{+}=\epsilon\varphi_{\sf j}^{-}$ (77)

which gives the energy eigenvalue

$\epsilon=s_{\sf j}\sqrt{p_{x{\sf j}}^{2}+\left(p_{y}+\frac{e}{c}A_{\sf
j}\right)^{2}}$ (78)

where $s_{\sf j}=\mbox{sign}(\epsilon)$, the incoming momentum ${\bm{p_{\sf
j}}}=(p_{x{\sf j}},p_{y})$ and position ${\bm{r}}=(x,y)$. The incoming spinor
is

$\displaystyle\psi_{in}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
z_{p_{x{\sf j}}}\end{array}\right)e^{\textbf{\emph{i}}{\bm{p_{\sf
j}}}\cdot{\bm{r}}},\qquad z_{p_{x{\sf j}}}=z_{\sf j}=s_{\sf
j}e^{\textbf{\emph{i}}\phi_{\sf j}}$ (81)

with $s_{0}=\mbox{sgn}(\epsilon)$ and $\phi_{\sf
j}=\arctan\left(\frac{p_{y}-\frac{e}{c}A_{\sf j}}{p_{x{\sf j}}}\right)$ is the
angle that the incident electron beam makes with the $x$-direction, $p_{x1}$
and $p_{y}$ are the $x$ and $y$-components of the electron wave vector,
respectively. The eigenspinors are given by

$\displaystyle\psi_{\sf j}^{+}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
z_{\sf j}\end{array}\right)e^{\textbf{\emph{i}}(p_{x{\sf j}}x+p_{y}y)}$ (84)
$\displaystyle\psi_{\sf j}^{-}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-z^{*}_{\sf j}\end{array}\right)e^{\textbf{\emph{i}}(-p_{x{\sf j}}x+p_{y}y)}.$
(87)

It is straightforward to solve the tunneling problem for Dirac fermions. We
assume that the incident wave propagates at the angle $\phi_{1}$ with respect
to the $x$-direction and write the components, of the Dirac spinor
$\varphi_{\sf j}^{+}$ and $\varphi_{\sf j}^{-}$, for each region. For
$x<-d_{2}$ (region 1):

$\displaystyle\epsilon=\left[p_{x1}^{2}+\left(p_{y}-\frac{1}{l_{B}^{2}}d_{1}\right)^{2}\right]^{\frac{1}{2}}$
(88) $\displaystyle\psi_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
z_{1}\end{array}\right)e^{\textbf{\emph{i}}(p_{x1}x+p_{y}y)}+r_{m}\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-z^{*}_{1}\end{array}\right)e^{\textbf{\emph{i}}(-p_{1x}x+p_{y}y)}$ (93)
$\displaystyle
z_{1}=s_{1}\frac{p_{x1}+i\left[p_{y}-\frac{1}{l_{B}^{2}}d_{1}\right]}{\sqrt{p_{x1}^{2}+\left[p_{y}-\frac{1}{l_{B}^{2}}d_{1}\right]^{2}}}.$
(94)

In the barrier $x>d_{2}$ (region 5):

$\displaystyle\epsilon=\left[p_{x5}^{2}+\left(p_{y}+\frac{1}{l_{B}^{2}}d_{1}\right)^{2}\right]^{\frac{1}{2}}$
(95)
$\displaystyle\psi_{5}=\frac{1}{\sqrt{2}}t_{m}\left(\begin{array}[]{c}1\\\
z_{5}\end{array}\right)e^{\textbf{\emph{i}}(p_{x5}x+p_{y}y)}$ (98)
$\displaystyle
z_{5}=s_{5}\frac{p_{x5}+i\left[p_{y}+\frac{1}{l_{B}^{2}}d_{1}\right]}{\sqrt{p_{x1}^{2}+\left[p_{y}+\frac{1}{l_{B}^{2}}d_{1}\right]^{2}}}.$
(99)

In region 2 and 4 ($d_{1}<|x|<d_{2}$): The general solution can be expressed
in terms of the parabolic cylinder function [17, 18, 11] as

$\chi_{\gamma}^{+}=c_{1}D_{\nu_{\gamma}-1}\left(Q_{\gamma}\right)+c_{2}D_{-\nu_{\gamma}}\left(-Q^{*}_{\gamma}\right)$
(100)

where
$\nu_{\gamma}=\frac{i}{2\varrho}\left(k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}\right)^{2}$,
$\epsilon_{0}=\epsilon-v_{1}$ and
$Q_{\gamma}(x)=\sqrt{\frac{2}{\varrho}}e^{i\pi/4}\left(\gamma\varrho
x+\epsilon_{0}\right)$, $c_{1}$ and $c_{2}$ are constants. It gives the other
component

$\displaystyle\chi_{\gamma}^{-}$ $\displaystyle=$ $\displaystyle-
c_{2}\frac{1}{k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}}\left[2(\epsilon_{0}+\gamma\varrho
x)D_{-\nu_{\gamma}}\left(-Q^{*}_{\gamma}\right)+\sqrt{2\varrho}e^{i\pi/4}D_{-\nu_{\gamma}+1}\left(-Q^{*}_{\gamma}\right)\right]$
(101)
$\displaystyle-\frac{c_{1}}{k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}}\sqrt{2\varrho}e^{-i\pi/4}D_{\nu_{\gamma}-1}\left(Q_{\gamma}\right).$

The components of the spinor solution of the Dirac equation (3) in region 2
and 4 can be obtained from (100) and (101) with
$\varphi_{\gamma}^{+}(x)=\chi_{\gamma}^{+}+i\chi_{\gamma}^{-}$ and
$\varphi_{\gamma}^{-}(x)=\chi_{\gamma}^{+}-i\chi_{\gamma}^{-}$. We have the
eigenspinor

$\displaystyle\psi_{\sf j}$ $\displaystyle=$ $\displaystyle a_{{\sf
j}-1}\left(\begin{array}[]{c}u^{+}_{\gamma}(x)\\\ u^{-}_{\gamma}(x)\\\
\end{array}\right)e^{ik_{y}y}+a_{\sf
j}\left(\begin{array}[]{c}v^{+}_{\gamma}(x)\\\ v^{-}_{\gamma}(x)\\\
\end{array}\right)e^{ik_{y}y}$ (106)

where ${\sf j=2,4}$ and $\gamma=\pm 1$, the function $u^{\pm}_{\gamma}(x)$ and
$v^{\pm}_{\gamma}(x)$ are given by

$\displaystyle u^{\pm}_{\gamma}(x)$ $\displaystyle=$ $\displaystyle
D_{\nu_{\gamma}-1}\left(Q_{\gamma}\right)\mp\frac{1}{k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}}\sqrt{2\varrho}e^{i\pi/4}D_{\nu_{\gamma}}\left(Q_{\gamma}\right)$
(107) $\displaystyle v^{\pm}_{\gamma}(x)$ $\displaystyle=$
$\displaystyle\pm\frac{1}{k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}}\sqrt{2\varrho}e^{-i\pi/4}D_{-\nu_{\gamma}+1}\left(-Q_{\gamma}^{*}\right)$
(108)
$\displaystyle\pm\frac{1}{k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}}\left(-2i\epsilon_{0}\pm\left(k_{y}-\gamma\frac{d_{1}}{l_{B}^{2}}\right)-\gamma
2i\varrho x\right)D_{-\nu_{\gamma}}\left(-Q_{\gamma}^{*}\right).$

In region 2:

$\displaystyle\psi_{\sf 2}$ $\displaystyle=$ $\displaystyle
a_{1}\left(\begin{array}[]{c}u^{+}_{1}(x)\\\ u^{-}_{1}(x)\\\
\end{array}\right)e^{ik_{y}y}+a_{2}\left(\begin{array}[]{c}v^{+}_{1}(x)\\\
v^{-}_{1}(x)\\\ \end{array}\right)e^{ik_{y}y}$ (113)

In region 4:

$\displaystyle\psi_{\sf 4}$ $\displaystyle=$ $\displaystyle
a_{3}\left(\begin{array}[]{c}u^{+}_{-1}(x)\\\ u^{-}_{-1}(x)\\\
\end{array}\right)e^{ik_{y}y}+a_{4}\left(\begin{array}[]{c}v^{+}_{-1}(x)\\\
v^{-}_{-1}(x)\\\ \end{array}\right)e^{ik_{y}y}$ (118)

In the region $|x|\leq d_{1}$: From the nature of the system under
consideration, we write the Hamiltonian corresponding to region ${\sf 3}$ in
matrix form as

$H_{m}=v_{F}\left(\begin{array}[]{cc}\frac{V_{2}}{v_{F}}+\frac{G_{p}}{v_{F}}&-i\frac{\sqrt{2}}{l_{B}}\left(\frac{l_{B}}{\sqrt{2}}\left(\partial_{x}-i\partial_{y}+\frac{e}{c}A_{3}\right)\right)\\\
i\frac{\sqrt{2}}{l_{B}}\left(\frac{l_{B}}{\sqrt{2}}\left(-\partial_{x}-i\partial_{y}+\frac{e}{c}A_{3}\right)\right)&\frac{V_{2}}{v_{F}}-\frac{G_{p}}{v_{F}}\\\
\end{array}\right)$ (119)

Note that, the energy gap $G_{p}$ behaves like a mass term in Dirac equation.
Certainly this will affect the above results and leads to interesting
consequences on the transport properties of our system. We determine the
eigenvalues and eigenspinors of the Hamiltonian $H_{m}$ by considering the
time independent equation for the spinor
$\psi_{3}(x,y)=(\psi_{3}^{+},\psi_{3}^{-})^{T}$ using the fact that the
transverse momentum $p_{y}$ is conserved, we can write the wave function
$\psi_{3}(x,y)=e^{ip_{y}y}\varphi_{3}(x)$ with
$\varphi_{3}(x)=(\varphi_{3}^{+},\varphi_{3}^{-})^{T}$, the energy being
defined by $E=\upsilon_{F}\epsilon$ leads to

$H_{m}\left(\begin{array}[]{c}\varphi_{3}^{+}\\\ \varphi_{3}^{-}\\\
\end{array}\right)=\epsilon\left(\begin{array}[]{c}\varphi_{3}^{+}\\\
\varphi_{3}^{-}\\\ \end{array}\right)$ (120)

At this stage, it is convenient to introduce the annihilation and creation
operators. They can be defined as

$\displaystyle
a=\frac{l_{B}}{\sqrt{2}}\left(\partial_{x}+k_{y}+\frac{e}{c}A_{3}\right),\qquad
a^{\dagger}=\frac{l_{B}}{\sqrt{2}}\left(-\partial_{x}+k_{y}+\frac{e}{c}A_{3}\right)$
(121)

which obey the canonical commutation relations $[a,a^{\dagger}]={\mathbb{I}}$.
Rescaling our energies $G_{p}=\upsilon_{F}\mu$ and $V_{2}=\upsilon_{F}v_{2}$,
(120) can be written in terms of $a$ and $a^{\dagger}$ as

$\left(\begin{array}[]{cc}v_{2}+\mu&-i\frac{\sqrt{2}}{l_{B}}a\\\
+i\frac{\sqrt{2}}{l_{B}}a^{\dagger}&v_{2}-\mu\\\
\end{array}\right)\left(\begin{array}[]{c}\varphi_{3}^{+}\\\
\varphi_{3}^{-}\\\
\end{array}\right)=\epsilon\left(\begin{array}[]{c}\varphi_{3}^{+}\\\
\varphi_{3}^{-}\\\ \end{array}\right)$ (122)

which gives

$\displaystyle(v_{2}+\mu)\varphi_{3}^{+}-i\frac{\sqrt{2}}{l_{B}}a\varphi_{3}^{-}=\epsilon\varphi_{3}^{+}$
(123) $\displaystyle
i\frac{\sqrt{2}}{l_{B}}a^{\dagger}\varphi_{3}^{+}+(v_{2}-\mu)\varphi_{3}^{-}=\epsilon\varphi_{3}^{-}.$
(124)

Injecting (124) in (123), we obtain a differential equation of second order
for $\varphi_{3}^{+}$

$\left[(\epsilon-
v_{2})^{2}-\mu^{2}\right]\varphi_{3}^{+}=\frac{2}{l_{B}^{2}}aa^{\dagger}\varphi_{3}^{+}.$
(125)

It is clear that $\varphi_{3}^{+}$ is an eigenstate of the number operator
$\widehat{N}=a^{\dagger}a$ and therefore we identify $\varphi_{3}^{+}$ with
the eigenstates of the harmonic oscillator $|n-1\rangle$, namely

$\varphi_{3}^{+}\sim\mid n-1\rangle$ (126)

which gives for (85)

$\left[(\epsilon-v_{2})^{2}-\mu^{2}\right]\mid
n-1\rangle=\frac{2}{l_{B}^{2}}n\mid n-1\rangle$ (127)

and the associated energy spectrum is

$\epsilon-v_{2}=s_{3}\epsilon_{n}=s_{3}\frac{1}{l_{B}}\sqrt{(\mu
l_{B})^{2}+2n}$ (128)

where we have set $\epsilon_{n}=s_{3}(\epsilon-v_{2})$ and
$s_{3}=\mbox{sign}(\epsilon_{n}-v_{2})$ correspond to positive and negative
energy solutions. For this reason we write the eigenvalues as

$\epsilon=v_{2}+s_{3}\frac{1}{l_{B}}\sqrt{(\mu l_{B})^{2}+2n}$ (129)

The second eigenspinor component then can be obtained from

$\varphi_{3}^{-}=\frac{i\sqrt{2}a^{\dagger}}{(\epsilon-v_{2})l_{B}+\mu
l_{B}}\mid n-1\rangle=\frac{i\sqrt{2n}}{(\epsilon-v_{2})l_{B}+\mu l_{B}}\mid
n\rangle$ (130)

where $\sqrt{2n}=\sqrt{(\epsilon_{n}l_{B})^{2}-(\mu l_{B})^{2}}$. We find

$\varphi_{3}^{-}=s_{3}i\sqrt{\frac{\epsilon_{n}l_{B}-s_{3}\mu
l_{B}}{\epsilon_{n}l_{B}+s_{3}\mu l_{B}}}\mid n\rangle$ (131)

After normalization we arrive at the following expression for the positive and
negative energy eigenstates

$\varphi_{3}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\sqrt{\frac{\epsilon_{n}l_{B}+s_{3}\mu
l_{B}}{\epsilon_{n}l_{B}}}\mid n-1\rangle\\\
s_{3}i\sqrt{\frac{\epsilon_{n}l_{B}-s_{3}\mu l_{B}}{\epsilon_{n}l_{B}}}\mid
n\rangle\\\ \end{array}\right)$ (132)

Introducing the parabolic cylinder functions
$D_{n}(x)=2^{-\frac{n}{2}}e^{-\frac{x^{2}}{4}}H_{n}\left(\frac{x}{\sqrt{2}}\right)$,
we express the solution in region 3 as

$\psi_{\sf 3}(x,y)=b_{1}\psi_{3}^{+}+b_{2}\psi_{3}^{-}$ (133)

with its two components

$\psi_{3}^{\pm}(x,y)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\sqrt{\frac{\epsilon_{n}l_{B}+s_{3}\mu
l_{B}}{\epsilon_{n}l_{B}}}D_{\left(\left(\epsilon_{n}l_{B}\right)^{2}-(\mu
l_{B})^{2}\right)/2-1}\left(\pm\sqrt{2}\left(\frac{x}{l_{B}}+k_{y}l_{B}\right)\right)\\\
\pm
i\frac{s_{3}\sqrt{2}}{\sqrt{\epsilon_{n}l_{B}\left(\epsilon_{n}l_{B}+s_{3}\mu
l_{B}\right)}}D_{\left(\left(\epsilon_{n}l_{B}\right)^{2}-\left(\mu
l_{B}\right)^{2}\right)/2}\left(\pm\sqrt{2}\left(\frac{x}{l_{B}}+k_{y}l_{B}\right)\right)\\\
\end{array}\right)e^{ik_{y}y}$ (134)

As usual the coefficients $(a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},r,t)$ can be
determined using the boundary conditions, continuity of the eigenspinors at
each interface.

###  4.2  Transmission and reflection: presence of magnetic field

We will now study some interesting features of our system which are reflected
in the corresponding transmission probability. Before doing so, let us
simplify our writing using the following shorthand notation

$\displaystyle\vartheta_{\tau 1}^{\pm}=D_{\left[(\epsilon_{n}l_{B})^{2}-(\mu
l_{B})^{2}\right]/2-1}\left[\pm\sqrt{2}\left(\frac{\tau
d_{1}}{l_{B}}+k_{y}l_{B}\right)\right]$ (135) $\displaystyle\zeta_{\tau
1}^{\pm}=D_{\left[(\epsilon_{n}l_{B})^{2}-(\mu
l_{B})^{2}\right]/2}\left[\pm\sqrt{2}\left(\frac{\tau
d_{1}}{l_{B}}+k_{y}l_{B}\right)\right]$ (136) $\displaystyle
f_{1}^{\pm}=\sqrt{\frac{\epsilon_{n}\pm\mu}{\epsilon_{n}}},\qquad
f_{2}^{\pm}=\frac{\sqrt{2/l_{B}^{2}}}{\sqrt{\epsilon_{n}(\epsilon_{n}\pm\mu)}}$
(137) $\displaystyle u^{\pm}_{\gamma}(\tau d_{1})=u^{\pm}_{\gamma,\tau
1},\qquad u^{\pm}_{\gamma}(\tau d_{2})=u^{\pm}_{\gamma,\tau 2}$ (138)
$\displaystyle v^{\pm}_{\gamma}(\tau d_{1})=v^{\pm}_{\gamma,\tau 1},\qquad
v^{\pm}_{\gamma}(\tau d_{2})=v^{\pm}_{\gamma,\tau 2}$ (139)

where $\tau=\pm$. Dirac equation requires the following set of continuity
conditions

$\displaystyle\psi_{\sf 1}(-d_{2})=\psi_{\sf 2}(-d_{2}),\qquad\psi_{\sf
2}(-d_{1})=\psi_{\sf 3}(-d_{1}),\qquad\psi_{\sf 3}(d_{1})=\psi_{\sf
4}(d_{1}),\qquad\psi_{\sf 4}(d_{2})=\psi_{\sf 5}(d_{2})$ (140)

That is, requirement of the continuity of the spinor wave functions at each
junction interface gives rise to the above set of equations. We prefer to
express these relationships in terms of $2\times 2$ transfer matrices between
j-th and (j+1)-th regions, $\mathcal{M}_{{\sf j},{\sf j}+1}$, we obtain the
full transfer matrix over the whole double barrier which can be written, in an
obvious notation, as follows

$\left(\begin{array}[]{c}1\\\ r_{m}\\\ \end{array}\right)=\prod_{{\sf
j}=1}^{4}\mathcal{M}_{{\sf j},{\sf j}+1}\left(\begin{array}[]{c}t_{m}\\\ 0\\\
\end{array}\right)=\mathcal{M}\left(\begin{array}[]{c}t_{m}\\\ 0\\\
\end{array}\right)$ (141)

where the total transfer matrix
$\mathcal{M}=\mathcal{M}_{12}\cdot\mathcal{M}_{23}\cdot\mathcal{M}_{34}\cdot\mathcal{M}_{45}$
are transfer matrices that couple the wave function in the ${\sf j}$-th region
to the wave function in the (${\sf j}+1$)-th region. These are given
explicitly by the forms

$\displaystyle\mathcal{M}=\left(\begin{array}[]{cc}\tilde{m}_{11}&\tilde{m}_{12}\\\
\tilde{m}_{21}&\tilde{m}_{22}\\\ \end{array}\right)$ (144)
$\displaystyle\mathcal{M}_{12}=\left(\begin{array}[]{cc}e^{-\textbf{\emph{i}}p_{x1}d_{2}}&e^{\textbf{\emph{i}}p_{x1}d_{2}}\\\
z_{1}e^{-\textbf{\emph{i}}p_{x1}d_{2}}&-z^{\ast}_{1}e^{\textbf{\emph{i}}p_{x1}d_{2}}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}u_{1,-2}^{+}&v_{1,-2}^{+}\\\
u_{1,-2}^{-}&v_{1,-2}^{-}\\\ \end{array}\right)$ (149)
$\displaystyle\mathcal{M}_{23}=\left(\begin{array}[]{cc}u_{1,-1}^{+}&v_{1,-1}^{+}\\\
u_{1,-1}^{-}&v_{1,-1}^{-}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}\vartheta_{1}^{+}&\vartheta_{1}^{-}\\\
\zeta_{1}^{+}&\zeta_{1}^{-}\\\ \end{array}\right)$ (154)
$\displaystyle\mathcal{M}_{34}=\left(\begin{array}[]{cc}\vartheta_{-1}^{+}&\vartheta_{-1}^{-}\\\
\zeta_{-1}^{+}&\zeta_{-1}^{-}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}u_{-1,1}^{+}&v_{-1,1}^{+}\\\
u_{-1,1}^{-}&v_{-1,1}^{-}\\\ \end{array}\right)$ (159)
$\displaystyle\mathcal{M}_{45}=\left(\begin{array}[]{cc}u_{-1,2}^{+}&v_{-1,2}^{+}\\\
u_{-1,2}^{-}&v_{-1,2}^{-}\\\
\end{array}\right)^{-1}\left(\begin{array}[]{cc}e^{\textbf{\emph{i}}p_{x5}d_{2}}&e^{-\textbf{\emph{i}}p_{x5}d_{2}}\\\
z_{5}e^{\textbf{\emph{i}}p_{x5}d_{2}}&-z_{5}^{\ast}e^{-\textbf{\emph{i}}p_{x5}d_{2}}\\\
\end{array}\right).$ (164)

These will enable us to compute the reflection and transmission amplitudes

$t_{m}=\frac{1}{\tilde{m}_{11}},\qquad
r_{m}=\frac{\tilde{m}_{21}}{\tilde{m}_{11}}.$ (165)

More explicitly, we have for transmission

$t_{m}=\frac{e^{id_{2}\left(p_{x1}+p_{x5}\right)}\left(1+z_{5}^{2}\right)\left(\vartheta_{1}^{-}\zeta_{1}^{+}+\vartheta_{1}^{+}\zeta_{1}^{-}\right)}{f_{2}^{+}\left(f_{1}^{-}\mathcal{L}_{1}+if_{2}^{-}\mathcal{L}_{2}\right)+f_{1}^{+}\left(f_{2}^{-}\mathcal{L}_{3}+if_{1}^{-}\mathcal{L}_{4}\right)}\mathcal{D}$
(166)

where the quantities $\mathcal{D}$, $\mathcal{L}_{1}$, $\mathcal{L}_{2}$,
$\mathcal{L}_{3}$ and $\mathcal{L}_{4}$ are defined by

$\displaystyle\mathcal{D}$ $\displaystyle=$
$\displaystyle\left(u_{-1,1}^{-}v_{-1,1}^{+}-u_{-1,1}^{+}v_{-1,1}^{-}\right)\left(u_{1,-2}^{+}v_{1,-2}^{-}-u_{1,-2}^{-}v_{1,-2}^{+}\right)$
(167) $\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle\vartheta_{-1}^{-}\zeta_{1}^{+}\mathcal{F}\mathcal{G}-\vartheta_{1}^{-}\zeta_{-1}^{+}\mathcal{K}\mathcal{J}$
(168) $\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle\left(\zeta_{1}^{+}\zeta_{-1}^{-}-\zeta_{1}^{-}\zeta_{-1}^{+}\right)\mathcal{F}\mathcal{J}$
(169) $\displaystyle\mathcal{L}_{3}$ $\displaystyle=$
$\displaystyle\vartheta_{-1}^{+}\zeta_{1}^{-}\mathcal{F}\mathcal{G}-\vartheta_{1}^{+}\zeta_{-1}^{-}\mathcal{K}\mathcal{J}$
(170) $\displaystyle\mathcal{L}_{4}$ $\displaystyle=$
$\displaystyle=\left(\vartheta_{1}^{+}\vartheta_{-1}^{-}-\vartheta_{1}^{-}\vartheta_{-1}^{+}\right)\mathcal{K}\mathcal{G}$
(171)

and

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\left[u_{1,-1}^{+}v_{1,-2}^{-}-u_{1,-2}^{-}v_{1,-1}^{+}-z_{1}\left(u_{1,-1}^{+}v_{1,-2}^{+}-u_{1,-2}^{+}v_{1,-1}^{+}\right)\right]$
(172) $\displaystyle\mathcal{G}$ $\displaystyle=$
$\displaystyle\left[u_{-1,1}^{-}v_{-1,2}^{+}-u_{-1,2}^{+}v_{-1,1}^{-}+z_{5}\left(u_{-1,1}^{-}v_{-1,2}^{-}-u_{-1,2}^{-}v_{-1,1}^{-}\right)\right]$
(173) $\displaystyle\mathcal{K}$ $\displaystyle=$
$\displaystyle\left[u_{1,-1}^{-}v_{1,-2}^{-}-u_{1,-2}^{-}v_{1,-1}^{-}-z_{1}\left(u_{1,-1}^{-}v_{1,-2}^{+}-u_{1,-2}^{+}v_{1,-1}^{-}\right)\right]$
(174) $\displaystyle\mathcal{J}$ $\displaystyle=$
$\displaystyle\left[u_{-1,1}^{+}v_{-1,2}^{+}-u_{-1,2}^{+}v_{-1,1}^{+}+z_{5}\left(u_{-1,1}^{+}v_{-1,2}^{-}-u_{-1,2}^{-}v_{-1,1}^{+}\right)\right]$
(175)

Actually the transmission probabilities $T_{m}$ and reflection $R_{m}$ for
each independent mode can be obtained using the corresponding electric current
density $J$ for our system. From our previous Hamiltonian, we can show that
incident, reflected and transmitted currents take the following forms

$\displaystyle
J_{\sf{inc,m}}=e\upsilon_{F}(\psi_{1}^{+})^{\dagger}\sigma_{x}\psi_{1}^{+}$
(176) $\displaystyle
J_{\sf{ref,m}}=e\upsilon_{F}(\psi_{1}^{-})^{\dagger}\sigma_{x}\psi_{1}^{-}$
(177) $\displaystyle
J_{\sf{tra,m}}=e\upsilon_{F}\psi_{5}^{\dagger}\sigma_{x}\psi_{5}.$ (178)

These can be used to write the reflection and transmission probabilities as

$T_{m}=\frac{p_{x5}}{p_{x1}}|t_{m}|^{2},\qquad R_{m}=|r_{m}|^{2}.$ (179)

###  4.3 Numerical results: presence of magnetic field

The physical outcome of particle scattering through the double triangular
barrier depends on the energy of the incoming particle. We numerically
evaluate the transmission probability $T_{m}$ as a function of structural
parameters of the graphene triangular double barrier in the presence of a
perpendicular magnetic field, in terms of the energy $\epsilon$, the
$y$-component of the wave vector $k_{y}$, the magnetic field $B$, the energy
gap $\mu$ and the applied potentials $v_{1}$ and $v_{2}$. The results are
shown in Figures 8, 9 and 10. In addition to the expected above-barrier full
transmission for some values of $\epsilon l_{B}$ and $v_{2}l_{B}$. We note
that in Figure 8a, when the energy is less than the height of the potential
barrier $\epsilon l_{B}<k_{y}l_{B}+\frac{d_{1}}{l_{B}}$, we have zero
transmission. In the second interval
$k_{y}l_{B}+\frac{d_{1}}{l_{B}}\leq\epsilon l_{B}\leq v_{1}l_{B}$ the third
zone contains oscillations. Finally the interval $\epsilon l_{B}>v_{1}l_{B}$
contains the usual over barrier oscillations and asymptotically goes to unity
at high energy. Figure 8b shows the transmission spectrum for different
transverse wave vector $k_{y}l_{B}$, the energy gap $\mu l_{B}$ is zero and
$v_{2}l_{B}=0$. We see that if we increase the wave vector $k_{y}l_{B}$ the
zone of zero transmission increases according to the condition $\epsilon
l_{B}<k_{y}l_{B}+\frac{d_{1}}{l_{B}}$. In the second interval the transmission
oscillates get very close to sharp Fabry–Pérot type of resonances and zero
transmission regions appear as $k_{y}l_{B}$ increases. Finally in the interval
$\epsilon l_{B}>v_{1}l_{B}$ the transmission increases to its asymptotic
value.

(a)

(b)

Figure 8: (color online) Transmission probability $T_{m}$ for the magnetic
barrier as a function of energy $\epsilon l_{B}$ with
$\frac{d_{2}}{l_{B}}=1.5$, $v_{1}l_{B}=60$, $v_{2}l_{B}=0$ and $\mu l_{B}=0$.
(a) for $k_{y}l_{B}=2$, $\frac{d_{1}}{l_{B}}=0.12$ (blue color),
$\frac{d_{1}}{l_{B}}=0.24$ (green color) and $\frac{d_{1}}{l_{B}}=0.6$ (red
color). (b) for $\frac{d_{1}}{l_{B}}=0.12$, $k_{y}l_{B}=1$ (blue color),
$k_{y}l_{B}=2$ (red color), $k_{y}l_{B}=3$ (green color) and $k_{y}l_{B}=5$
(magenta color).

On the other hand, if we keep the same well region and cancel both the applied
magnetic field and mass term in the well region, the series of potentials
behave like a simple double barrier with the same effective mass $k_{y}$.
Thus, in this case we reproduce exactly the transmission obtained in
[Alhaidari], for the massive Dirac equation with $m=k_{y}$. Let us treat the
triangular double barrier case when $v_{2}<v_{1}$ and $v_{2}>v_{1}$. In both
cases, the transmission is plotted in Figure 9 and then in Figure 9a
$v_{2}>v_{1}$ we distinguish five different zones characterizing the behavior
of the transmission. Indeed, the first zone is determined by the greater
effective mass, namely $\epsilon l_{B}<k_{y}l_{B}+\frac{d_{1}}{l_{B}}$. The
second zone identifies with the lower Klein energy zone characterized by
resonances in the interval $k_{y}l_{B}+\frac{d_{1}}{l_{B}}<\epsilon
l_{B}<v_{1}l_{B}$. Here we have full transmission at some specific energies
despite the fact that the particle energy is less than the height of the
barrier. As $d_{1}/l_{B}$ increases, the oscillations in the Klein zone get
reduced. This strong reduction in the transmission in the Klein zone seem to
suggest the potential suppression of Klein tunneling as we increase the width
of the well region, $d_{1}/l_{B}$. The third zone $v_{1}l_{B}<\epsilon
l_{B}<v_{2}l_{B}-k_{y}l_{B}-\frac{\mu l_{B}}{2}$ is a window where the
transmission oscillates around the value of the full transmission. The fourth
zone defined by $v_{2}l_{B}-k_{y}l_{B}-\frac{\mu l_{B}}{2}<\epsilon
l_{B}<v_{2}l_{B}+k_{y}l_{B}+\frac{\mu l_{B}}{2}$ is a window where the
transmission is almost zero. The fifth zone $\epsilon
l_{B}>v_{2}l_{B}+k_{y}l_{B}+\frac{\mu l_{B}}{2}$ contains oscillations, the
transmission converges towards unity. Contrary to the case $v_{1}>v_{2}$, the
situation where $v_{2}>v_{1}$ is shown in Figure 9b we distinguish fourth
different zones characterizing the behavior of the transmission. Indeed,
compared to Figure 9a, the behavior in the first zone is the same. Concerning
the zones $k_{y}l_{B}-\frac{d1}{l_{B}}<\epsilon
l_{B}<v_{2}l_{B}-k_{y}l_{B}-\frac{\mu l_{B}}{2}$ and
$v_{2}l_{B}+k_{y}l_{B}+\frac{\mu l_{B}}{2}<\epsilon l_{B}<v_{1}l_{B}$ the
transmission oscillates similarly to Figure 9a. In the zone
$v_{2}l_{B}-k_{y}l_{B}-\frac{\mu l_{B}}{2}<\epsilon
l_{B}<v_{2}l_{B}+k_{y}l_{B}+\frac{\mu l_{B}}{2}$, one can see that both curves
start from zero transmission and oscillate while the valley gets wider as
$d_{1}/l_{B}$ decreases. Finally zone $\epsilon l_{B}>v_{1}l_{B}$ the
transmission oscillate to reach full transmission asymptotically.

(a)

(b)

Figure 9: (color online) Transmission probability $T_{m}$ for the magnetic
barrier as a function of energy $\epsilon l_{B}$ with
$\frac{d_{1}}{l_{B}}=0.1$ (red color), $\frac{d_{1}}{l_{B}}=0.5$ (blue color),
$\frac{d_{2}}{l_{B}}=1.5$, $\mu l_{B}=4$ and $k_{y}l_{B}=2$. (a):
$v_{1}l_{B}=30$ , $v_{2}l_{B}=60$. (b): $v_{1}l_{B}=60$ , $v_{2}l_{B}=30$.
Figure 10: (Color online) Transmission probability $T_{m}$ for the magnetic
barrier as a function of the potential $v_{2}l_{B}$ with
$\frac{d_{1}}{l_{B}}=0.1$ (green color), $\frac{d_{1}}{l_{B}}=0.2$ (red
color), $\frac{d_{1}}{l_{B}}=0.34$ (blue color), $\frac{d_{2}}{l_{B}}=1.5$,
$\mu l_{B}=4$, $k_{y}l_{B}=2$, $v_{1}l_{B}=60$ and $\epsilon l_{B}=30$.

It is worth analyzing the transmission versus the potential $v_{2}l_{B}$. In
doing so, we choose a fixed value of $d_{1}/l_{B}$ as shown in Figure 10. It
is clear that as we increase the well width $d_{1}/l_{B}$, hence for a fixed
$v_{1}l_{B}$ and $d_{2}/l_{B}$ we will be increasing the strength of the
electrostatic field, the number of oscillations increases and a zero
transmission region appears.

##  5 Conclusion

We have considered a model that describes electron transmission in monolayer
graphene through a triangular electrostatic double barriers in the presence of
a magnetic field localized in the well region. To investigate the transport
properties of our system, we have considered separately the two effects: first
we considered an electrostatic double barrier alone then we added a uniform
magnetic field effect in the well region. In both cases, we have investigated
analytically and numerically the transmission probability. This has been
achieved through solving the corresponding eigenvalue equation which resulted
in the energy spectrum in terms of different physical parameters that control
the system Hamiltonian.

Using the continuity of the wavefunctions at the interfaces between different
regions inside and outside the barriers we have ensured conservation of the
local current density and derived the relevant transport coefficients of the
present system. Specifically, using the transfer matrix method, we have
deduced the corresponding transmission coefficient and determined how the
transmission probability is affected by various physical parameters. In
particular for the electrostatic double barrier in the absence of magnetic
field, the resonances were seen to appear in different Klein tunneling
regions. A gap region appeared for energies around the value of the elevated
well potential, independently of the strength of the electrostatic field.

Subsequently, we have analyzed the same system but this time taking into
account the presence of a static magnetic field localized in the well region.
Using boundary conditions, we have split the energy space into three domains
and then calculated the transmission probability in each domain. In each
situation, we have discussed the transmission and resonances that characterize
each region and stressed the importance of our results.

Our results indicate that the Klein paradox, full transmission close to normal
incidence, persists for all values the electrostatic field or equivalently the
double barrier-tilting angle. From the application point of view, our current
investigation suggests that tunable filtering of Dirac electrons by an
electrostatic double barrier is possible for near normal incidence at all
values of the electrostatic field or tilting angle. This observation could be
very beneficial in designing graphene based electronic lenses.

## Acknowledgments

The generous support provided by the Saudi Center for Theoretical Physics
(SCTP) is highly appreciated. AJ and HB acknowledge the support of King Fahd
University of Petroleum and minerals.

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