# Tunneling through ABC-ABA-ABC trilayer graphene junction

Abderrahim El Mouhafid<EMAIL_ADDRESS>Laboratory of Theoretical
Physics, Faculty of Sciences, Chouaïb Doukkali University, PO Box 20, 24000 El
Jadida, Morocco  Mouhamadou Hassane Saley<EMAIL_ADDRESS>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

###### Abstract

The electronic tunneling properties of trilayer graphene (TLG) are
significantly influenced by the specific stacking order. To illustrate this
stacking influence, we investigate the transport properties of a p-n-p
junction formed with ABC-ABA-ABC stacking TLG. Utilizing the transfer matrix
method and considering continuity conditions at the junction boundaries, we
establish transmission, reflection probabilities, and conductance. Remarkably,
electron transport through the ABC-ABA-ABC junction demonstrates Klein
tunneling, resulting in high conductance even in the absence of a potential
barrier $V_{0}$. This phenomenon is attributed to the effective barrier
induced by our stacking, facilitating the passage of a maximum number of
electrons. However, the presence of $V_{0}$ diminishes Klein tunneling,
leading to conductance minima. Furthermore, we reveal that the interlayer bias
$\delta$ causes hybridization of the linear and parabolic bands of ABA-TLG in
the junction, reducing resonances. In cases where $\delta\neq 0$ and
$V_{0}\neq 0$, a suppression of the gap is observed, contrary to the results
obtained in ABC tunneling studies where a gap exists.

###### pacs:

##  1 Introduction

Graphene, a gapless semiconductor composed of a single layer of carbon atoms,
exhibits massless charge carriers with a linear energy band, governed by the
Dirac equation for relativistic particles [1, 2, 3, 4, 5]. This unique
characteristic has sparked considerable interest among researchers, theorists,
and experimenters. For instance, in graphene, the electron transport across a
potential barrier is marked by perfect transmission, known as Klein tunneling
[6, 7]. Additionally, graphene displays an unconventional quantum Hall effect
[2, 3, 8]. Moreover, graphene possesses transparency [9], high electronic
mobility [4, 10], impressive thermal conductivity [11, 12], and a remarkable
strength of 130 GPa [13, 14]. These diverse properties underscore the
significance of graphene across a broad spectrum of applications.

In addition to monolayer graphene (MLG), multilayer graphene, involving the
stacking of two or more layers, exhibits appealing properties dependent on the
stacking type and layer count. Specifically, Bernal (AB) bilayer graphene
(BLG) features four parabolic energy bands, allowing for gap opening through
the application of a perpendicular electric field [15, 16, 17, 18].
Conversely, AA-BLG displays a linear energy dispersion akin to MLG but with a
shift in the positions of the two Dirac cones [19, 20]. Unlike AB-BLG, the
application of an electric field in AA-BLG does not result in a band gap [21,
22]. Furthermore, AA-BLG exhibits Klein tunneling similar to MLG, while AB-BLG
demonstrates perfect reflection [20, 22, 23, 6, 24, 25, 26, 27]. Trilayer
graphene comes in three distinct configurations: ABC (rhombohedral), ABA
(bernal), and AAA stackings [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].
The ABC stacking displays a cubic dispersion with conduction and valence bands
touching at low energy, while the ABA stacking exhibits both MLG linear bands
and BLG parabolic bands [28, 31, 33, 40]. The AAA stacking comprises three
copies of MLG linear bands [39]. Studies have shown that the presence of an
electric field breaks the inversion symmetry of ABC-TLG, leading to the
appearance of a band gap, while ABA-TLG remains metallic [32, 35, 41, 42, 43].
In ABA-TLG, the mirror symmetry is broken in the presence of an electric
field, resulting in the hybridization of the linear and parabolic bands [44].

In a recent experimental investigation, Yin et al. documented the emergence of
a domain wall in the transitional zone between ABA and ABC stacking [45].
Building on this research, we explore electron tunneling through an ABC-ABA-
ABC junction with the goal of evaluating its impact on the transport
properties of Dirac fermions. Our analysis involves determining transmission
and reflection probabilities along with the corresponding conductance. In
scenarios without a potential barrier ($V_{0}$), we demonstrate that electrons
can undergo perfect tunneling through the ABC-ABA-ABC junction under normal
incidence, a phenomenon known as Klein tunneling. As a result, maximal
conductance is achieved, attributed to the junction inducing an effective
potential that facilitates the optimal flow of electrons, leading to total
transmission. However, the introduction of a potential barrier diminishes
Klein tunneling, as evident in the observed minima in conductance.
Furthermore, we show that the presence of an interlayer bias ($\delta$)
significantly reduces transmission in specific regions, primarily due to the
hybridization of MLG-like and AB-BLG-like bands in ABA-TLG. Notably, contrary
to the findings in reference [46], no gap is observed for $V_{0}\neq 0$ and
$\delta\neq 0$. Our study includes various comparisons of these results with
findings related to the p-n-p junction (potential barrier) using either ABC-
TLG or ABA-TLG exclusively.

The present paper is organized as follows. In Sec. 2, we introduce the
theoretical model for our system and calculate the transmission and reflection
probabilities. The numerical results obtained are presented and analyzed in
Sec. 3. Ultimately, a summary of the key findings is presented in Sec. 4.

##  2 Model and method

We recall that trilayer graphene is characterized by a specific arrangement of
carbon atoms forming a structure composed of three layers of single-layer or
monolayer graphene sheets. Each monolayer exhibits a hexagonal crystal
structure with atoms labeled A and B in its unit cell, and their interatomic
distance is $a_{0}=0.142$ nm [47], with an intra-layer coupling of
approximately $\gamma_{0}\approx 3$ eV [48]. The trilayer structure consists
of a first monolayer followed by a second monolayer, aligning with the first
in an alternating pattern known as ABA stacking. In ABA stacking, the second
monolayer is directly positioned on top of the first monolayer, and the third
monolayer aligns with the first monolayer in the same pattern, creating a
repeating ABA arrangement throughout the trilayer. Alternatively, trilayer
graphene can adopt another stacking configuration called ABC stacking. In ABC
stacking, the carbon atoms in the third monolayer are aligned with those in
the second monolayer in a different pattern. Specifically, in ABC stacking,
the third monolayer is rotated by $60^{\circ}$ degrees relative to the first
monolayer, introducing a stagger between the layers.

###  2.1 Hamiltonian model

Figure 1: (Color online) Profile illustrating the trilayer graphene junction
in an ABC-ABA-ABC configuration, along with a potential barrier characterized
by a height $V_{0}$ and a width $d$.

Fig. 1 depicts two regions of ABC-TLG ($j$=I, III) characterized by zero
potential, alongside an intermediate region of ABA-TLG ($j$=II) with a nonzero
potential. The potential on the top and bottom layers is defined as

$\displaystyle V_{j}(x)=\left\\{\begin{array}[]{l}V_{0}\pm\delta,\ \ \
j=\text{II}\\\ 0,\ \ \ \ \ \ \ \ \ \ j=\text{I, III}\end{array}\right.$ (3)

where $V_{0}$ represents the height of the potential barrier and $\delta$
denotes the electrostatic potential difference (bias).

In order to obtain the solution within each region $j$, we solve the
eigenvalue equation $H_{j}\psi_{j}=E_{j}\psi_{j}$, where the effective
Hamiltonians $H_{j}$ near the Dirac point for ABC-TLG (regions I, III) and
ABA-TLG (region II) can be derived from the tight-binding approximation,
respectively, [41, 47]

$\displaystyle\mathcal{H}_{ABC}(\vec{p})=\begin{pmatrix}\mathcal{H}_{1}(\vec{p})&\Gamma&0\\\
\Gamma^{{\dagger}}&\mathcal{H}_{2}(\vec{p})&\Gamma\\\
0&\Gamma^{{\dagger}}&\mathcal{H}_{3}(\vec{p})\end{pmatrix},$ (4)
$\displaystyle\mathcal{H}_{ABA}(\vec{p})=\begin{pmatrix}\mathcal{H}_{1}(\vec{p})&\Gamma&0\\\
\Gamma^{{\dagger}}&\mathcal{H}_{2}(\vec{p})&\Gamma^{{\dagger}}\\\
0&\Gamma&\mathcal{H}_{3}(\vec{p})\end{pmatrix},$ (5)

where the interlayer coupling $\Gamma$ is given by

$\displaystyle\Gamma=\begin{pmatrix}0&0\\\ \gamma_{1}&0\end{pmatrix},$ (6)

and $\mathcal{H}_{i}(\vec{p})=\hbar
v_{F}\vec{\sigma}\cdot\vec{p}+V_{i}(x)\mathbb{I}_{2}$ is the SLG Hamiltonian
of the i-th layer $(i=1,2,3)$, with $\mathbb{I}_{2}$ is a unit matrix,
$v_{F}\approx 1.01\times 10^{6}$m/s is the Fermi velocity,
$\vec{\sigma}=(\sigma_{x},\sigma_{y})$ are the Pauli matrices,
$\vec{p}=(p_{x},p_{y})$ denotes the in-plane momentum, $\gamma_{1}=0.4$ eV is
the nearest neighbor coupling term between adjacent layers. These Hamiltonians
are formulated based on the eigenfunctions of atomic orbitals

$\Psi_{ABC/ABA}=\left(\psi_{A_{1}},\psi_{B_{1}},\psi_{A_{2}},\psi_{B_{2}},\psi_{A_{3}},\psi_{B_{3}}\right)^{\dagger},$
(7)

where the indices signify the sublattice linked to the corresponding
eigenfunction and the symbol $\dagger$ stands for transpose.

A simplification can be achieved by converting the Hamiltonian of ABA-TLG (5)
into a block-diagonal structure through a unitary transformation that
symmetrically and antisymmetrically combines the orbital eigenfunctions, as
outlined in [44]. This transformation can be expressed as

$U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&0&0&-1&0\\\ 0&1&0&0&0&-1\\\
1&0&0&0&1&0\\\ 0&1&0&0&0&1\\\ 0&0&\sqrt{2}&0&0&0\\\ 0&0&0&\sqrt{2}&0&0\\\
\end{pmatrix},$ (8)

which leads to the following Hamiltonian

$\displaystyle\mathcal{H}_{ABA}(\vec{p})=\begin{pmatrix}\mathcal{H}_{1}(\vec{p})&\delta\mathbb{I}_{2}&0\\\
\delta\mathbb{I}_{2}&\mathcal{H}_{2}(\vec{p})&\sqrt{2}\Gamma\\\
0&\sqrt{2}\Gamma^{{\dagger}}&\mathcal{H}_{3}(\vec{p})\end{pmatrix}.$ (9)

We consider the domain wall oriented in the $y$-direction with infinite
length. Consequently, the system exhibits translational invariance, and the
momentum $p_{y}$ is conserved. This allows us to formulate the eigenfunctions
for ABC ($j=$I, III) and ABA ($j=\text{II}$), respectively, as

$\displaystyle\psi_{j}(x,y)=e^{ik_{y}y}({\phi}_{A_{1}j},{\phi}_{B_{1}j},{\phi}_{A_{2}j},{\phi}_{B_{2}j},{\phi}_{A_{3}j},{\phi}_{B_{3}j})^{{\dagger}},$
(10)
$\displaystyle\psi_{\text{II}}(x,y)=e^{ik_{y}y}({\chi}_{A_{1}},{\chi}_{B_{1}},{\chi}_{A_{2}},{\chi}_{B_{2}},{\chi}_{A_{3}},{\chi}_{B_{3}})^{{\dagger}}.$
(11)

To streamline the notation, we introduce the length scale $a=\frac{\hbar
v_{F}}{\gamma_{1}}\approx 1.64$ nm, representing the interlayer coupling
length, together with the dimensionless quantities: $E_{j}\rightarrow\
\frac{E_{j}}{\gamma_{1}}$, $V_{0}\rightarrow\ \frac{V_{0}}{\gamma_{1}}$,
$\delta_{j}\rightarrow\ \frac{\delta_{j}}{\gamma_{1}}$, $k_{y}\rightarrow\
ak_{y}$ and $\vec{r}\rightarrow\ \frac{\vec{r}}{a}$.

###  2.2 Solutions of energy spectrum

####  2.2.1 ABC-TLG in regions I and III

Concerning regions I and III, the eigenvalue equation associated with the
Hamiltonian (4) leads to a system of six-coupled differential equations

$\displaystyle-i(\partial_{x}+k_{y})\phi_{B_{1}j}=E_{j}\phi_{A_{1}j},$ (12a)
$\displaystyle-i(\partial_{x}-k_{y})\phi_{A_{1}j}=E_{j}\phi_{B_{1}j}-\phi_{A_{2}j},$
(12b)
$\displaystyle-i(\partial_{x}+k_{y})\phi_{B_{2}j}=E_{j}\phi_{A_{2}j}-\phi_{B_{1}j},$
(12c)
$\displaystyle-i(\partial_{x}-k_{y})\phi_{A_{2}j}=E_{j}\phi_{B_{2}j}-\phi_{A_{3}j},$
(12d)
$\displaystyle-i(\partial_{x}+k_{y})\phi_{B_{3}j}=E_{j}\phi_{A_{3}j}-\phi_{B_{2}j},$
(12e) $\displaystyle-i(\partial_{x}-k_{y})\phi_{A_{3}j}=E_{j}\phi_{B_{3}j},$
(12f)

with $k_{y}$ represents the wave vector in the $y$-direction. To uncouple
these equations, let us initially derive $\phi_{A_{1}j}$ and $\phi_{B_{3}j}$
in relation to $\phi_{B_{1}j}$ and $\phi_{A_{3}j}$ using Eqs. (12a) and (12f).
Subsequently, we substitute these expressions into the second and fifth
equations, respectively, yielding

$\displaystyle(\partial^{2}_{x}-k^{2}_{y})\phi_{B_{1}j}+E_{j}\phi_{A_{2}j}$
$\displaystyle=$ $\displaystyle E_{j}^{2}\phi_{B_{1}j},$ (13a)
$\displaystyle(\partial^{2}_{x}-k^{2}_{y})\phi_{A_{3}j}-E_{j}\phi_{B_{2}j}$
$\displaystyle=$ $\displaystyle-E_{j}^{2}\phi_{A_{3}j}.$ (13b)

We can obtain $\phi_{A_{3}j}$ by transforming Eqs. (13a) and (13b) into a
single sixth-order equation

$\displaystyle\left[\frac{d^{6}}{dx^{6}}+\left(\alpha_{j}-3k_{y}^{2}\right)\frac{d^{4}}{dx^{4}}-\left(2\alpha_{j}k_{y}^{2}-3k_{y}^{4}-\beta_{j}\right)\frac{d^{2}}{dx^{2}}-k_{y}^{6}+\alpha_{j}k_{y}^{4}-\beta_{j}k_{y}^{2}+\omega_{j}\right]\phi_{A_{3}j}=0,$
(14)

with the parameters $\alpha_{j}=3E_{j}^{2},\beta_{j}=E^{2}_{j}(3E_{j}^{2}-2)$,
and $\omega_{j}=E_{j}^{2}(1-E_{j}^{2})^{2}$. Following extensive algebraic
manipulations, the solution to Eq. (14) and therefore the system of equations
(12) can be expressed as a linear combination of plane waves, represented in
terms of matrices specific to each region ($j$=I, III):

$\psi_{j}(x,y)=Q_{j}M_{j}(x)C_{j}e^{ik_{y}y}$ (15)

where the three matrices are given by

$\displaystyle
Q_{j}=\begin{pmatrix}\frac{\lambda_{1j}f_{1j}}{E_{j}^{2}g_{1j}}&\frac{\lambda_{1j}g_{1j}}{E_{j}^{2}f_{1j}}&\frac{\lambda_{2j}f_{2j}}{E_{j}^{2}g_{2j}}&\frac{\lambda_{2j}g_{2j}}{E_{j}^{2}f_{2j}}&\frac{\lambda_{3j}f_{3j}}{E_{j}^{2}g_{3j}}&\frac{\lambda_{3j}g_{3j}}{E_{j}^{2}f_{3j}}\\\
\frac{\lambda_{1j}}{E_{j}g_{1j}}&-\frac{\lambda_{1j}}{E_{j}f_{1j}}&\frac{\lambda_{2j}}{E_{j}g_{2j}}&-\frac{\lambda_{2j}}{E_{j}f_{2j}}&\frac{\lambda_{3,j}}{E_{j}g_{3j}}&-\frac{\lambda_{3j}}{E_{j}f_{3j}}\\\
\frac{\mu_{1j}}{E_{j}g_{1j}}&-\frac{\mu_{1j}}{E_{j}f_{1j}}&\frac{\mu_{2j}}{E_{j}g_{2j}}&-\frac{\mu_{2j}}{E_{j}f_{2j}}&\frac{\mu_{3j}}{E_{j}g_{3j}}&-\frac{\mu_{3j}}{E_{j}f_{3j}}\\\
\frac{\rho_{1j}}{E_{j}}&\frac{\rho_{j}}{E_{j}}&\frac{\rho_{2j}}{E_{j}}&\frac{\rho_{2j}}{E_{j}}&\frac{\rho_{3j}}{E_{j}}&\frac{\rho_{3j}}{E_{j}}\\\
1&1&1&1&1&1\\\
\frac{g_{1j}}{E_{j}}&-\frac{f_{1j}}{E_{j}}&\frac{g_{2j}}{E_{j}}&-\frac{f_{2j}}{E_{j}}&\frac{g_{3j}}{E_{j}}&-\frac{f_{3j}}{E_{j}}\\\
\end{pmatrix},$ (16) $\displaystyle
M_{j}=\text{diag}(e^{ik_{1j}x},e^{-ik_{1j}x},e^{ik_{2j}x},e^{-ik_{2j}x},e^{ik_{3j}x},e^{-ik_{3j}x}),$
(17) $\displaystyle
C_{j}=(a^{+}_{1j},b^{-}_{1j},a^{+}_{2j},b^{-}_{2j},a^{+}_{3j},b^{-}_{3j})^{\dagger},$
(18)

where $g_{nj}=k_{nj}+ik_{y}$, $f_{nj}=k_{nj}-ik_{y}$,
$\rho_{nj}=E_{j}^{2}-k_{nj}^{2}-k_{y}^{2}$, $\mu_{nj}=E_{j}(\rho_{nj}-1)$,
$\lambda_{nj}=\rho_{nj}^{2}-E^{2}_{j}$, and $k_{nj}$ ($n=1,2,3$) are the wave
vectors along the $x$-direction in regions $j$=I and III, solutions to the
following cubic equation

$\displaystyle\left(k_{nj}^{2}+k^{2}_{y}\right)^{3}-\alpha_{j}\left(k_{nj}^{2}+k^{2}_{y}\right)^{2}+\beta_{j}\left(k_{nj}^{2}+k^{2}_{y}\right)-\omega_{j}=0,$
(19)

and the superscript plus/minus indicates the right/left propagation or
evanescent states.

Figure 2: (Color online) TLG energy spectrum without bias $\delta=0$ ((a),
(b)) and with bias $\delta=0.3\gamma_{1}$.

It is worth noting that in regions $j=\text{I}$ and III where the potential is
zero, the equality
$Q_{\text{I}}M_{\text{I}}(x)=Q_{\text{III}}M_{\text{III}}(x)$ holds. As the
wave vectors $k_{n,\text{I}}$ are equivalent to those in region III,
$k_{n,\text{III}}$, we will simplify our notation in the subsequent analysis
by using $k_{n}$ and omitting the subscript $j$. It is observed that the
energy spectrum of the Hamiltonian (4) reveals six bands corresponding to the
six wave vectors $\pm k_{1}$, $\pm k_{2}$, and $\pm k_{3}$, solutions of Eq.
(19). This is illustrated in Fig. 2(a) and (c), both without ($\delta=0$) and
with bias ($\delta=0.3\gamma_{1}$). These wave vectors play a crucial role in
comprehending the distinct propagation modes across our junction.
Specifically, for $E<\gamma_{1}^{\prime}$
($\gamma_{1}^{\prime}=0.918\gamma_{1}$) [46, 37], there is only one mode of
propagation corresponding to the wave vector $k_{1}$. However, for
$E>\gamma_{1}^{\prime}$, three propagation modes emerge, associated with the
wave vectors $k_{1}$, $k_{2}$, and $k_{3}$, as depicted in Fig. 3 (for further
details, refer to [46, 37]).

####  2.2.2 ABA-TLG in region II

To obtain the energy spectrum solution for region II (ABA-TLG), one can
substitute Eqs. (9) and (11) into $\mathcal{H}_{ABA}\chi(x)=E\chi(x)$ to
derive the coupled equations

$\displaystyle-i(\partial_{x}+k_{y})\chi_{B_{1}}=\varepsilon\chi_{A_{1}}-\delta\chi_{A_{2}},$
(20a)
$\displaystyle-i(\partial_{x}-k_{y})\chi_{A_{1}}=\varepsilon\chi_{B_{1}}-\delta\chi_{B_{2}},$
(20b)
$\displaystyle-i(\partial_{x}+k_{y})\chi_{B_{2}}=\varepsilon\chi_{A_{2}}-\delta\chi_{A_{1}},$
(20c)
$\displaystyle-i(\partial_{x}-k_{y})\chi_{A_{2}}=\varepsilon\chi_{B_{2}}-\delta\chi_{B_{1}}-\sqrt{2}\chi_{A_{3}},$
(20d)
$\displaystyle-i(\partial_{x}+k_{y})\chi_{B_{3}}=\varepsilon\chi_{A_{3}}-\sqrt{2}\chi_{B_{2}},$
(20e)
$\displaystyle-i(\partial_{x}-k_{y})\chi_{A_{3}}=\varepsilon\chi_{B_{3}},$
(20f)

where we have set $\varepsilon=E-V_{0}$. The system of coupled first-order
differential equations 20(a-f) can be transformed into a sixth-order
differential equation for $\chi_{B_{1}}$ as follows

$\displaystyle\bigg{[}\frac{d^{6}}{dx^{6}}-(\omega_{1}-3k_{y}^{2})\frac{d^{4}}{dx^{4}}-(2\omega_{1}k_{y}^{2}-3k_{y}^{4}-\omega_{2})\frac{d^{2}}{dx^{2}}+k_{y}^{6}-\omega_{1}k_{y}^{4}-\omega_{2}k_{y}^{2}+\omega_{3}\bigg{]}\chi_{B_{1}}=0,$
(21)

with
$\omega_{1}=3\varepsilon^{2}+2\delta^{2},\omega_{2}=3\varepsilon^{4}-4\varepsilon^{2}+\delta^{4},\omega_{3}=\varepsilon^{2}(\varepsilon^{2}-\delta^{2})(4-\varepsilon^{2}+\delta^{2})$.
The solution of Eq. (21) can be written as a linear combination of plane waves

$\chi_{B_{1}}=\sum^{3}_{n=1}\left(a_{n}e^{iq_{n}x}+b_{n}e^{-iq_{n}x}\right),$
(22)

where $a_{n}$ and $b_{n}$ $(n=1,2,3)$ are coefficients of normalization and
the wave vectors $q_{n}$ along the x-direction in each region are the
solutions of the cubic equation:

$\displaystyle
q_{n}^{6}-(\omega_{1}-3k_{y}^{2})q_{n}^{4}-(2\omega_{1}k_{y}^{2}-3k_{y}^{4}-\omega_{2})q_{n}^{2}+k_{y}^{6}-\omega_{1}k_{y}^{4}-\omega_{2}k_{y}^{2}+\omega_{3}=0,$
(23)

In Figs. 2(b) and (d), we present the energy spectrum of the ABA-TLG
Hamiltonian (5) both without ($\delta=0$) and with bias
($\delta=0.3\gamma_{1}$), respectively. The energy spectrum comprises a
combination of a pair of nearly linear bands, reminiscent of monolayer
graphene, and two pairs of parabolic bands, resembling those of bilayer
graphene. The influence of this energy spectrum behavior significantly affects
the transport properties of graphene, as will be discussed in Sec. 3.

The rest of the spinor component solutions of the system 20(a-f) are given in
terms of the matrices as

$\displaystyle
Q_{\text{II}}=\begin{pmatrix}\frac{\lambda_{1}f_{1}}{\alpha_{1}\alpha_{2}g_{1}}&\frac{\lambda_{1}g_{1}}{\alpha_{1}\alpha_{2}f_{1}}&\frac{\lambda_{2}f_{2}}{\alpha_{1}\alpha_{2}g_{2}}&\frac{\lambda_{2}g_{2}}{\alpha_{1}\alpha_{2}f_{2}}&\frac{\lambda_{3}f_{3}}{\alpha_{1}\alpha_{2}g_{3}}&\frac{\lambda_{3}g_{3}}{\alpha_{1}\alpha_{2}f_{3}}\\\
\frac{\lambda_{1}}{\alpha_{1}g_{1}}&-\frac{\lambda_{1}}{\alpha_{1}f_{1}}&\frac{\lambda_{2}}{\alpha_{1}g_{2}}&-\frac{\lambda_{2}}{\alpha_{1}f_{2}}&\frac{\lambda_{3}}{\alpha_{1}g_{3}}&-\frac{\lambda_{3}}{\alpha_{1}f_{3}}\\\
\frac{\mu_{1}}{\alpha_{1}g_{1}}&-\frac{\mu_{1}}{\alpha_{1}f_{1}}&\frac{\mu_{2}}{\alpha_{1}g_{2}}&-\frac{\mu_{2}}{\alpha_{1}f_{2}}&\frac{\mu_{3}}{\alpha_{1}g_{3}}&-\frac{\mu_{3}}{\alpha_{1}f_{3}}\\\
\frac{\rho_{1}}{\alpha_{1}}&\frac{\rho_{1}}{\alpha_{1}}&\frac{\rho_{2}}{\alpha_{1}}&\frac{\rho_{2}}{\alpha_{1}}&\frac{\rho_{3}}{\alpha_{1}}&\frac{\rho_{3}}{\alpha_{1}}\\\
1&1&1&1&1&1\\\
\frac{g_{1}}{\alpha_{1}}&-\frac{f_{1}}{\alpha_{1}}&\frac{g_{2}}{\alpha_{1}}&-\frac{f_{2}}{\alpha_{1}}&\frac{g_{3}}{\alpha_{1}}&-\frac{f_{3}}{\alpha_{1}}\\\
\end{pmatrix},$ (24) $\displaystyle
M_{\text{II}}=\text{diag}(e^{iq_{1}x},e^{-iq_{1}x},e^{iq_{2}x},e^{-iq_{2}x},e^{iq_{3}x},e^{-iq_{3}x}),$
(25) $\displaystyle
C_{\text{II}}=(c^{+}_{1},d^{-}_{1},c^{+}_{2},d^{-}_{2},c^{+}_{3},d^{-}_{3})^{\dagger},$
(26)

where $g_{n}=q_{n}+ik_{y}$, $f_{n}=q_{n}-ik_{y}$,
$\rho_{n}=\alpha_{1}^{2}-q_{n}^{2}-k_{y}^{2}$,
$\eta_{n}=\varepsilon^{2}-q_{n}^{2}-k_{y}^{2}$,
$\mu_{n}=\varepsilon\rho_{n}-\alpha_{1}$,
$\lambda_{n}=\rho_{n}\eta_{n}-\alpha_{1}\varepsilon$,
$\alpha_{1}=\varepsilon-\delta$, $\alpha_{2}=\varepsilon+\delta$. As a result,
the general solution in region II can be written as

$\psi_{\text{II}}(x,y)=Q_{\text{II}}M_{\text{II}}(x)C_{\text{II}}\
e^{ik_{y}y}.$ (27)

We will explore how the aforementioned outcomes are applied to ascertain
various physical quantities. Our particular emphasis will be on examining the
transmission and reflection probabilities, as well as the conductance.

###  2.3 Transmission and conductance

As we employ the transfer matrix approach [49], our focus lies on the
normalization coefficients, which are the components of $C_{j}$, on both sides
of the ABC-ABA-ABA junction. In order to provide clarity, we must establish
the definition of our spinor within regions I

$\displaystyle\phi_{A_{1}\text{I}}=\sum^{3}_{n=1}\left(\delta_{s,n}Q_{1,2n-1}e^{ik_{n}x}+r^{s}_{n}Q_{1,2n}e^{-ik_{n}x}\right),$
(28a)
$\displaystyle\phi_{B_{1}\text{I}}=\sum^{3}_{n=1}\left(\delta_{s,n}Q_{2,2n-1}e^{ik_{n}x}+r^{s}_{n}Q_{2,2n}e^{-ik_{n}x}\right),$
(28b)
$\displaystyle\phi_{A_{2}\text{I}}=\sum^{3}_{n=1}\left(\delta_{s,n}Q_{3,2n-1}e^{ik_{n}x}+r^{s}_{n}Q_{3,2n}e^{-ik_{n}x}\right),$
(28c)
$\displaystyle\phi_{B_{2}\text{I}}=\sum^{3}_{n=1}\left(\delta_{s,n}Q_{4,2n-1}e^{ik_{n}x}+r^{s}_{n}Q_{4,2n}e^{-ik_{n}x}\right),$
(28d)
$\displaystyle\phi_{A_{3}\text{I}}=\sum^{3}_{n=1}\left(\delta_{s,n}Q_{5,2n-1}e^{ik_{n}x}+r^{s}_{n}Q_{5,2n}e^{-ik_{n}x}\right),$
(28e)
$\displaystyle\phi_{B_{3}\text{I}}=\sum^{3}_{n=1}\left(\delta_{s,n}Q_{6,2n-1}e^{ik_{n}x}+r^{s}_{n}Q_{6,2n}e^{-ik_{n}x}\right),$
(28f)

and III

$\displaystyle\phi_{A_{1}{\text{III}}}=t^{s}_{1}Q_{1,1}e^{ik_{1}x}+t^{s}_{2}Q_{1,3}e^{ik_{2}x}+t^{s}_{3}Q_{1,5}e^{ik_{3}x},$
(29a)
$\displaystyle\phi_{B_{1}{\text{III}}}=t^{s}_{1}Q_{2,1}e^{ik_{1}x}+t^{s}_{2}Q_{2,3}e^{ik_{2}x}+t^{s}_{3}Q_{2,5}e^{ik_{3}x},$
(29b)
$\displaystyle\phi_{A_{2}{\text{III}}}=t^{s}_{1}Q_{3,1}e^{ik_{1}x}+t^{s}_{2}Q_{3,3}e^{ik_{2}x}+t^{s}_{3}Q_{3,5}e^{ik_{3}x},$
(29c)
$\displaystyle\phi_{B_{2}{\text{III}}}=t^{s}_{1}Q_{4,1}e^{ik_{1}x}+t^{s}_{2}Q_{4,3}e^{ik_{2}x}+t^{s}_{3}Q_{4,5}e^{ik_{3}x},$
(29d)
$\displaystyle\phi_{A_{3}{\text{III}}}=t^{s}_{1}Q_{5,1}e^{ik_{1}x}+t^{s}_{2}Q_{5,3}e^{ik_{2}x}+t^{s}_{3}Q_{5,5}e^{ik_{3}x},$
(29e)
$\displaystyle\phi_{B_{3}{\text{III}}}=t^{s}_{1}Q_{6,1}e^{ik_{1}x}+t^{s}_{2}Q_{6,3}e^{ik_{2}x}+t^{s}_{3}Q_{6,5}e^{ik_{3}x},$
(29f)

where $\delta_{s,i}$ $(i=1,2,3)$ is the Kronecker symbol, $s=1,2,3$ indicate
the mode of propagation or evanescent waves characterized by three distinct
wave vectors: $k_{1}$, $k_{2}$, and $k_{3}$. $Q_{u,v}$ are the elements of the
matrix (16).

By incorporating the suitable boundary condition within the framework of the
transfer matrix approach [49], one can derive the transmission and reflection
probabilities. The preservation of spinor continuity at the boundaries
$(x=0,d)$ yields the components of the vector $C_{j}$ in regions I and III

$\displaystyle
C^{s}_{\text{I}}=(\delta_{s,1},r^{s}_{1},\delta_{s,2},r^{s}_{2},\delta_{s,3},r^{s}_{3})^{\dagger}$
(30) $\displaystyle
C^{s}_{\text{III}}=(t^{s}_{1},0,t^{s}_{2},0,t^{s}_{3},0)^{\dagger},$ (31)

which can be connected via the transfer matrix $M$

$C^{s}_{\text{I}}=MC^{s}_{\text{III}},$ (32)

where $M$ is solution of the continuity equations

$\displaystyle
Q_{\text{I}}M_{\text{I}}(0)C_{\text{I}}=Q_{\text{II}}M_{\text{II}}(0)C_{\text{II}},$
(33) $\displaystyle
Q_{\text{II}}M_{\text{II}}(d)C_{\text{II}}=Q_{\text{III}}M_{\text{III}}(d)C_{\text{III}}.$
(34)

Consequently, getting $M$ we can determine the complex coefficients of
transmission $t^{s}_{i}$ $(i=1,2,3)$ and reflection $r^{s}_{i}$. Given that
the velocities of waves scattered through three different modes are not
identical, it is advantageous to employ the current density $\vec{\mathbf{J}}$
to calculate the transmission $T$ and reflection $R$ probabilities. This is
achieved by

$\vec{\mathbf{J}}=v_{F}{\Psi}^{\dagger}\vec{\alpha}\Psi,$ (35)

where $\Psi$ represents the spinor solutions in regions (I, III), and
$\vec{\alpha}$ is a $6\times 6$ diagonal matrix with the diagonal elements
being the three Pauli matrices $\sigma_{x}$. Utilizing Eq. (35), we deduce the
incident $\mathbf{J}_{\text{inc}}$, reflected $\mathbf{J}_{\text{ref}}$, and
transmitted $\mathbf{J}_{\text{tra}}$ currents that lead to [46, 37]

$T=\frac{|\mathbf{J}_{\text{tra}}|}{|\mathbf{J}_{\text{inc}}|},\quad
R=\frac{|\mathbf{J}_{\text{ref}}|}{|\mathbf{J}_{\text{inc}}|},$ (36)

and therefore we end up with

$T^{s}_{i}=\frac{A^{x}_{i,i}}{A^{x}_{s,s}}|t^{s}_{i}|^{2},\quad
R^{s}_{i}=\frac{A^{x}_{i,i}}{A^{x}_{s,s}}|r^{s}_{i}|^{2},$ (37)

where $A^{x}_{i,i}$ and $A^{x}_{s,s}$ are the elements of the diagonal matrix
$\vec{\mathbf{A}}=Q^{\dagger}\vec{\alpha}Q$ consisting of traceless $2\times
2$ blocks, and each one corresponds to a propagating mode $s$. These
expressions can be elucidated as follows: given the presence of six bands,
electrons can undergo scattering between them, necessitating consideration of
the change in their velocities. Consequently, we identify nine channels in
transmission and reflection, corresponding to three modes of propagation:
$k_{1}$, $k_{2}$, and $k_{3}$ solutions of Eq. (19).

Figure 3: (Color online) Illustration of various modes along with their
corresponding transmission and reflection probabilities through the ABC-ABA-
ABC junction. Dots and arrows indicate the energy regions wherein electrons,
incident perpendicular to the barrier, will experience transmission or
reflection. Here, $\gamma_{1}^{\prime}$ represents the minimum energy
necessary for the emergence of propagating modes $k_{2}$ and $k_{3}$.

In Fig. 3, we illustrate these different modes and the associated transmission
and reflection probabilities within the ABC-ABA-ABC junction. Specifically, at
low energies $(E<\gamma_{1}^{\prime})$, a single propagation mode $k_{1}$
exists, resulting in one transmission $T^{1}_{1}$ and one reflection
$R^{1}_{1}$ channel across the two conduction bands touching at zero energy on
both sides of the junction. Conversely, at higher energies
$(E>\gamma_{1}^{\prime})$, three propagation modes, $k_{1}$, $k_{2}$, and
$k_{3}$, give rise to nine transmission $T^{1}_{i}$, $T^{2}_{i}$, $T^{3}_{i}$
$(i=1,2,3)$ and reflection $R^{1}_{i}$, $R^{2}_{i}$, $R^{3}_{i}$ channels
across the six conduction bands. Concerning transmission, these include three
non-scattered channels denoted as $T^{1}_{1}$, $T^{2}_{2}$, and $T^{3}_{3}$
for propagation via $k_{1}$, $k_{2}$, and $k_{3}$, respectively. Additionally,
there are six scattered channels where the particle enters through one channel
and exits through another. These are denoted as $T^{1}_{3,2}$, $T^{2}_{3,1}$,
and $T^{3}_{2,1}$ for scattering from the $k_{1}$ band to the $k_{2,3}$, from
the $k_{2}$ band to the $k_{1,3}$, and from the $k_{3}$ band to the $k_{1,2}$
bands, respectively. A parallel definition applies to the $R^{1,2,3}_{1,2,3}$
reflection channels. The schematic representation of these nineteen channels
is depicted in Fig. 3. It’s worth noting that $T^{1,2,3}_{1,2,3}$ and
$R^{1,2,3}_{1,2,3}$ adhere to the following equation:

$\sum_{{s}=1}^{3}\left(T^{{s}}_{1,2,3}+R^{{s}}_{1,2,3}\right)=1.$ (38)

As illustration, for the lower band in Fig. 3, we have
$T^{1}_{1}+R^{1}_{1}+T^{1}_{2}+R^{1}_{2}+T^{1}_{3}+R^{1}_{3}=1.$

Having determined the transmission probabilities, let’s examine their impact
on the conductance of our system. The conductance can be obtained through the
Landauer-Büttiker formula [50] by summing over all channels, resulting in

$G(E)=G_{0}\frac{L_{y}}{2\pi}\int_{-\infty}^{+\infty}dk_{y}\sum^{3}_{s,n=1}T^{s}_{n}(E,k_{y}),$
(39)

where $L_{y}$ represents the width of the sample in the $y$-direction, and
$G_{0}=4e^{2}/h$, where the factor $4$ accounts for the valley and spin
degeneracy in graphene. The subscript $s=1,2,3$ denotes the propagation mode.
The results obtained will undergo numerical analysis to explore the
fundamental characteristics of our system and establish connections with other
published findings. Given the nature of our system, the analysis will
differentiate between two cases based on band tunneling.

##  3 Numerical analysis

####  3.0.1 Transmission and reflection probabilities

To assess transport properties across the ABC-ABA-ABC junction, we will
numerically analyze our theoretical results. We begin by examining the low-
energy case ($E<\gamma_{1}$), where the corresponding transmission is depicted
in Fig. 4 as a function of the incident energy $E$ at normal incidence
($k_{y}=0)$ for $V_{0}=\delta=0$. In Fig. 4 (a), the transmission is shown for
various junction widths: $d=10$nm, $d=25$ nm, and $d=100$nm. For small $d$
(blue), the transmission is low for $E<0.1\gamma_{1}$, but it increases and
approaches unity for $E>0.1\gamma_{1}$. As the junction width is increased to
$d=25$nm (red), the transmission is higher even for $E<0.1\gamma_{1}$, and
resonance peaks emerge, surpassing those observed in AB-BLG [18, 51, 52]. With
further increases in junction width, such as $d=100$nm (green), the number of
resonance peaks also increases. In Fig. 4 (b), the transmission is plotted
against the junction width $d$ for different incident energy values:
$E=\frac{1}{5}\gamma_{1}$ (blue), $E=\frac{2}{5}\gamma_{1}$ (red), and
$E=\frac{8}{5}\gamma_{1}$ (green). Notably, as the energy increases, the
number of resonance peaks also increases, and the transmission exhibits
numerous oscillations, surpassing those reported in [18, 51], even in the
absence of $V_{0}$ and $\delta$.

Figure 4: (Color online) The transmission probability at normal incidence
$(k_{y}=0)$ through an ABC-ABA-ABC junction where $V_{0}=\delta=0$. (a): The
energy dependence of the transmission probability for junction width $d=10$ nm
(blue), $d=25$ nm (red), and $d=100$ nm (green). (b): The width junction
dependence of the transmission probability for Fermi energy of
$E=\frac{1}{5}\gamma_{1}$ (blue), $E=\frac{2}{5}\gamma_{1}$ (red) and
$E=\frac{8}{5}\gamma_{1}$ (green). Figure 5: (Color online) Density plot of
transmission and reflection probabilities for ABC-ABA-ABC junction with
$V_{0}=\delta=0$ and $d=25\ $nm. The dashed white and black lines represent
the band inside (ABA-TLG) and outside (ABC-TLG) the barrier, respectively.
Figure 6: (Color online) The same as in 5 but now with $V_{0}=1.5\
\gamma_{1}$, $\delta=0$. Figure 7: (Color online) The same as in 5 but now
with $V_{0}=0$, $\delta=0.3\gamma_{1}$. Figure 8: (Color online) The same as
in 5 but now with $V_{0}=1.5\gamma_{1}$, $\delta=0.3\gamma_{1}$ .

For energies beyond $\gamma_{1}^{\prime}=0.918\gamma_{1}$
$(E>\gamma_{1}^{\prime})$, we depict the transmission and reflection
probabilities as a function of energy $E$ and the transverse wave vector
$k_{y}$. The outcomes are presented for $V_{0}=0$ in Fig. 5 and for a barrier
of height $V_{0}=1.5\gamma_{1}$ in Fig. 6. It is worth noting that there are
nine transmission and nine reflection channels. In the $T^{1}_{1}$ channel for
$V_{0}=0$, transmission resonances manifest at normal incidence ($k_{y}=0$),
as shown in Fig. 5. This occurrence, recognized as Klein tunneling, is a
distinctive feature of the specific junction under consideration and stands in
contrast to findings in [6], which require the presence of a potential
barrier. Nevertheless, some narrow anti-resonances are identified for
$0<E<\gamma_{1}^{\prime}$ at normal incidence, whereas at non-normal
incidence, they become broader and even manifest for $E>\gamma_{1}^{\prime}$.
In the case of the potential barrier $V_{0}=1.5\gamma_{1}$ depicted in Fig. 6,
we observe a reduction in transmission, and anti-Klein tunneling is evidently
close to normal incidence in the energy region below $V_{0}$ in the
$T^{1}_{1}$ channel. At normal incidence, it occurs for
$\gamma_{1}^{\prime}<E<V_{0}$ and at certain energies smaller than
$\gamma_{1}^{\prime}$. This behavior could be attributed to the presence of
BLG-like propagating states inside the barrier, while the minimal transmission
at normal incidence may be linked to MLG-like propagating states [33, 46].
However, transmission resonances emerge at non-normal incidence, where the
modes $k_{1}$ outside and $q_{1,3}$ inside the barrier are coupled,
reminiscent of BLG tunneling [27, 53, 54, 51, 55]. For the $T^{2}_{2}$
channel, in both Fig. 5 and Fig. 6, the transmission is null in the energy
range $0<E<\gamma_{1}^{\prime}$ where the propagating mode $k_{2}$ is
unavailable on the left and right sides of the junction. As
$E>\gamma_{1}^{\prime}$, the $k_{2}$ mode becomes accessible, leading to
resonances attributed to the propagating modes $q_{1,2,3}$ inside the junction
in Fig. 5 and $q_{1,3}$ (for $\gamma_{1}^{\prime}<E<V_{0}$) or $q_{1,2}$ (for
$E>V_{0}$) in Fig. 6. Notably, the outcome for the $T^{2}_{2}$ channel in Fig.
6 closely resembles the result for the $T^{-}_{-}$ channel through the AB-2SL-
AB junction with $V_{0}=1.5\gamma_{1}$ [56]. However, here, we observe non-
zero transmission at a specific incidence due to the presence of the
propagating mode $q_{3}$ inside the barrier. In Fig. 5 for channel
$T^{3}_{3}$, resonances emerge for $E>\gamma_{1}^{\prime}$ where the
propagating mode $k_{3}$ is present outside the junction. Conversely, when
$V_{0}=1.5\gamma_{1}$, resonances are observed for
$\gamma_{1}^{\prime}<E<V_{0}$, in contrast to the result in ABC-TLG [46], as
seen in Fig. 6. This difference arises from the presence of the propagating
modes $q_{1}$ and $q_{3}$ inside the junction, while in [46] the propagating
mode $q_{3}$ is absent. However, no transmission is observed for
$0<E<\gamma_{1}^{\prime}$ and $V_{0}<E<2V_{0}$ due to cloaking [18], while a
sharp resonance appears for $E>2V_{0}$. For $V_{0}=0$ in Fig. 5, the
transmission and reflection channels exhibit symmetry with respect to
$k_{y}=0$, akin to the cases of unbiased ABC-TLG [46] and unbiased AB-BLG [18,
27] tunneling. However, for $V_{0}=1.5\gamma_{1}$ in Fig. 6, the scattered
transmission channels, i.e., $T^{i}_{j}$ with $i\neq j$, and all reflection
channels are no longer symmetric with respect to $k_{y}=0$, similar to the
outcome for the double barrier in biased ABC-TLG [37]. This asymmetrical
behavior can be attributed to the non-alignment of ABC-TLG’s bands outside and
ABA-TLG’s bands inside the ABC-ABA-ABC junction due to the presence of a
potential barrier [56]. Additionally, in Fig. 6, the transmission decreases in
the $T^{1}_{2}$ and $T^{2}_{1}$ channels when compared with the results in
Fig. 5, while it remains weak in the other scattered channels in both Fig. 5
and Fig. 6.

With the same parameters as in Fig. 5, we present the transmission and
reflection probabilities in Fig. 7 in the presence of an interlayer bias
$\delta=0.3\gamma_{1}$. It is crucial to acknowledge that within the junction,
the interlayer bias transforms ABA-TLG’s linear bands into parabolic bands,
forming a gap between them while two of the BLG-like bands still touch each
other at the Dirac points [42, 46]. Due to the absence of MLG-like propagating
states (linear band) inside the junction, the transmission experiences a
significant decrease for energy regions greater than $\gamma_{1}^{\prime}$ in
the $T^{i}_{i}$ $(i=1,2,3)$, $T^{1}_{2}$ and $T^{2}_{1}$ channels compared to
those seen in Fig. 5. Additionally, the transmission is influenced by $\delta$
in channel $T^{1}_{1}$ for energies smaller than $\gamma_{1}^{\prime}$.
Specifically, for $0<E<\delta$, there is no propagating mode $q_{1}$ since it
has shifted from $E=0$ to $E=\delta$, as observed in [46] for ABA-BLG.
Consequently, there are only two sharp resonances resulting from the
propagating mode $q_{2}$. For $\delta<E<\gamma_{1}^{\prime}$, the change of
propagating mode $q_{1}$ from MLG-like to parabolic reduces the resonances, as
mentioned in [46]. Moreover, in the scattered channels, asymmetry is observed
with respect to normal incidence, and here, it arises from the presence of the
interlayer bias $\delta$, as shown in [37]. It is important to note that, as
in Fig. 5, the transmission is weak in channels $T^{1}_{3}=T^{3}_{1}$ and
$T^{2}_{3}=T^{3}_{2}$. Another noteworthy feature is that in scattered
transmission channels, we have $T^{i}_{j}(k)=T^{j}_{i}(-k)$ as a consequence
of the time-reversal symmetry of the system [46].

In Fig. 8, we employ the same parameters as in Fig. 6, but with
$\delta=0.3\gamma_{1}$. Similar to Fig. 7, we observe a decrease in
transmission due to the presence of the interlayer bias, as mentioned
previously. Specifically, in the $T^{1}_{1}$ and $T^{3}_{3}$ channels, no
resonance of transmission is found for $E>V_{0}$, while in the $T^{2}_{2}$
channel, only a few sharp resonances occur at non-normal incidence.
Additionally, in $T^{2}_{2}$ and $T^{3}_{3}$ channels, there is no resonance
in the region beneath $V_{0}$, but the transmission differs from zero,
contrasting with the result in [37]. In the same energy region, in the
$T^{1}_{1}$ channel, the result is similar to that obtained in Fig. 6. On the
other hand, we observe non-zero transmission inside the gap due to the
presence of the propagating mode $q_{2}$ in both Fig. 7 and Fig. 8. The
scattered channels exhibit a similar behavior as in Fig. 7, such as asymmetry
with respect to $k_{y}=0$ and $T^{i}_{j}(k)=T^{j}_{i}(-k)$ as a consequence of
the interlayer bias and the time-reversal symmetry of the system [46]. It is
worth noting that in ABC tunneling, in the presence of a potential barrier and
for a non-zero interlayer bias, a gap is typically observed [37]. However,
here, we observe the opposite phenomenon, namely the suppression of the gap
when both a potential barrier and an interlayer bias are present.

####  3.0.2 Conductance

Figure 9: (Color online) Conductance of the ABC-ABA-ABC junction as a function
of energy with $d=25\ $nm and for (a) $V_{0}=\delta=0$, (b)
$V_{0}=1.5\gamma_{1}$, $\delta=0$, (c) $V_{0}=0$, $\delta=0.3\gamma_{1}$ and
(d) $V_{0}=1.5\ \gamma_{1}$, $\delta=0.3\gamma_{1}$.

In Fig. 9, we explore the energy dependence of the conductance through the
ABC-ABA-ABC junction under the same conditions as those in the systems studied
above. Fig. 9(a) illustrates the conductance associated with the transmission
channels in Fig. 5. For $E<\gamma_{1}^{\prime}$, we have
$G_{\text{Total}}=G_{1}^{1}$ since in this region electrons can only propagate
via mode $k_{1}$. However, $G^{2}_{2}$, $G^{1}_{2}=G^{2}_{1}$, and
$G^{2}_{3}=G^{3}_{2}$ begin conducting at $E=\gamma_{1}^{\prime}$ where more
than one propagation mode is available, whereas $G^{3}_{3}$ and
$G^{1}_{3}=G^{3}_{1}$ start conducting for $E>\gamma_{1}^{\prime}$. The
observed equivalence in conductance channels $G^{i}_{j}=G^{j}_{i}$ ($i\neq j)$
aligns with the results in [46, 37, 51]. Additionally, a linear-like behavior
of $G^{1}_{1}$ is noticeable as a consequence of the absence of the potential
barrier ($V_{0}=0$), similar to result observed in [56]. Furthermore, due to
the resonances displayed in Fig. 5, the contributions from $G^{2}_{2}$,
$G^{3}_{3}$, and $G^{1}_{2}=G^{2}_{1}$ significantly increase the total
conductance $G_{\text{Total}}$, in contrast to the results seen in [37]. The
conductance is notably higher, despite the absence of a potential barrier.
However, the contributions from $G^{1}_{3}=G^{3}_{1}$ and
$G^{2}_{3}=G^{3}_{2}$ are low due to the weak transmission observed in these
channels, see Fig. 5. In Fig. 9(b), we plot the conductance associated with
the transmission channels in Fig. 6. The presence of the potential barrier
$V_{0}=1.5\gamma_{1}$ alters the conductance behavior in contrast to the
result marked in Fig. 9(a). More precisely, it behaves similarly to that
observed in [27]. Indeed, the conductance $G_{\text{Total}}$ exhibits minima
at $E=\gamma_{1}^{\prime}$ and even for $E>\gamma_{1}^{\prime}$. Additionally,
there is a decrease in $G_{\text{Total}}$, resulting from the reduced
transmission found in Fig. 6. In Fig. 9(c), we illustrate the conductance
corresponding to Fig. 7 with $V_{0}=0$ and $\delta=0.3\gamma_{1}$. Upon
comparing the conductance $G_{\text{Total}}$ with the result in Fig. 9(a), it
is evident that, for $E<\gamma_{1}$, the number of peaks decreases, and
$G_{\text{Total}}$ tends to zero at $E=0.1\gamma_{1}$. This outcome arises
from the shift of the propagating mode $q_{1}$ inside the junction in the
$T^{1}_{1}$ channel (Fig. 7), induced by $\delta$. For
$E>\gamma_{1}^{\prime}$, there is a decrease in $G^{1}_{1}$ and $G^{2}_{2}$
compared to the result in Fig. 9(a). This is a consequence of the absence of
the MLG-like channel due to the presence of $\delta$, in contrast to Fig.
9(a). In Fig. 9(d), we present the conductance associated with the
transmission in Fig. 8 ($V_{0}=1.5$ and $\delta=0.3\gamma_{1}$). We observe
that the peaks of $G_{\text{Total}}$ are decreasing in height and increasing
in number compared to the result in Fig. 9(b), where $\delta=0$. Additionally,
there is a decrease in $G^{1}_{1}$, $G^{2}_{2}$, and $G^{3}_{3}$ channels
compared to Fig. 9(b), which is a manifestation of $\delta$. We stress that
despite the presence of a potential barrier $V_{0}$ and an interlayer bias
$\delta$, the gap no longer exists in the ABC-ABA-ABC junction, contrary to
the ABC stacking studied in [46]. Another notable observation is the
convergence of $G^{2}_{2}$ towards $G^{1}_{1}$ with increasing energy in both
Figs. 9(b) and (d), which differs from the outcomes reported in [46, 37].

##  4 Conclusion

We have studied the transport properties of Dirac fermions traversing in the
ABC-ABA-ABC junction. Initially, we determined the eigenvectors and the
propagating modes associated with the Hamiltonians $H_{ABC}$ and $H_{ABA}$,
describing, respectively, the ABC-TLG and ABA-TLG stackings. Subsequently,
employing continuity conditions at the boundaries and the transfer matrix
method, we computed the transmission and reflection probabilities. Among the
interesting results, we found that even in the absence of a potential barrier
(i.e., $V_{0}=0$), the transport of charge carriers through ABC-ABA-ABC
manifests Klein tunneling phenomena at normal incidence. This came as a
consequence of the junction under consideration, and it can be understood that
there is an induced potential that favors maximum electrons traveling to the
transmitted region III.

On the contrary, our findings indicate that the introduction of a potential
barrier $V_{0}$ can effectively diminish Klein tunneling and regulate the
transport of charge carriers in our junction. The inclusion of an interlayer
bias $\delta$ results in a reduction of resonances for both cases, i.e.,
$V_{0}=0$ and $V_{0}=1.5\gamma_{1}$. Despite the presence of a potential
barrier and an interlayer bias, there is an observed suppression of the gap,
in contrast to the outcomes presented in [46]. Additionally, we observed an
evident asymmetry in the transmission channels at normal incidence. Moreover,
employing Landauer-Buttiker’s formula allowed us to compute the conductance of
the junction. In the absence of $V_{0}$ and $\delta$, a notably high
conductance was observed. Furthermore, it became evident that the introduction
of a potential barrier altered the behavior of the total conductance
$G_{\text{Total}}$. It is noteworthy that the suppression of the gap in
conductance observed with the application of a barrier potential and an
interlayer contrasts with the findings in [46]. These results suggest that
electron transport can be effectively controlled in ABC-ABA-ABC junctions,
providing a promising avenue for designing spin-gapless semiconductor devices.

## References

  * [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science 306, 666 (2004).
  * [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature 438, 197 (2005).
  * [3] Y. Zhang, Y. Tan, H. L. Stormer and P. Kim, Nature 438, 201 (2005).
  * [4] S. V. Morozov, K. S. Novoselov, M. I. Katsnelson, F. Schedin, D. C. Elias, J. A. Jaszczak and A. K. Geim, Phys. Rev. Lett. 100, 016602 (2008).
  * [5] A. K. Geim., Science 324, 1530 (2009).
  * [6] M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nat, Phys. 2, 620 (2006).
  * [7] A. F. Young and P. Kim, Nat. Phys. 5, 222 (2009).
  * [8] V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005).
  * [9] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres and A. K. Geim, Science 320, 1308 (2008).
  * [10] R. S. Shishir and D. K. Ferry, J. Phys.: Condens. Matter 21, 232204 (2009).
  * [11] A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao and C. N. Lau, Nano Lett. 8, 902 (2008).
  * [12] D. S. Ghosh, I. Calizo, D. Teweldebrhan, E. P. Pokatilov, D. L. Nika, A. A. Balandin, W. Bao, F. Miao and C. N. Lau, Appl. Phys. Lett. 92, 151911 (2008).
  * [13] C. Lee, X. Wei, J. W. Kysar and J. Hone, Science 321, 385 (2008).
  * [14] D. G. Papageorgiou, I. A. Kinloch and R. J. Young, Prog. Mater. Sci. 90, 75 (2017).
  * [15] E. McCann Phys. Rev. B 74, 161403 (2006).
  * [16] E. McCann, D. S. L. Abergel and V. I. Fal’ko , Eur. Phys. J.: Spec. Top. 148, 91 (2007).
  * [17] Y. Zhang, T.-T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen and F. Wang, Nature 459, 820 (2009).
  * [18] B. Van Duppen and F. M. Peeters, Phys. Rev. B 87, 205427 (2013).
  * [19] I. Lobato and B. Partoens, Phys. Rev. B 83, 165429 (2011).
  * [20] M. Sanderson, Y. Sin Ang and C. Zhang, Phys. Rev. B 88, 245404 (2013).
  * [21] C. J. Tabert and E. J. Nicol, Phys. Rev. B 86, 075439 (2012).
  * [22] H. M. Abdullah, M. A. Ezzi and H. Bahlouli, J. Appl. Phys. 124, 204303 (2018).
  * [23] H. M. Abdullah and H. Bahlouli, J. Comput. Sci. 26, 135 (2018).
  * [24] T. Tudorovskiy, K. J. A. Reijnders and M. I. Katsnelson, Phys. Scr. T146, 014010 (2012).
  * [25] N. Gu, M. Rudner and L. Levitov, Phys. Rev. Lett. 107, 156603 (2011).
  * [26] B. Van Duppen and F. M. Peeters, EPL 102, 27001 (2013).
  * [27] M. Hassane Saley, A. El Mouhafid, A. Jellal and A. Siari, Ann. Phys. (Berlin) 534, 2200308 (2022).
  * [28] F. Guinea, A. H. Castro Neto and N. M. R. Peres, Phys. Rev. B 73, 245426 (2006).
  * [29] M. Aoki and H. Amawashi, Solid State Commun. 142, 123 (2007).
  * [30] M. F. Craciun, S. Russo, M. Yamamoto, J. B. Oostinga, A. F. Morpurgo and S. Tarucha, Nat Nanotechnol. 4, 383 (2009).
  * [31] W. Bao, L. Jing, J. Velasco Jr, Y. Lee, G. Liu, D. Tran, B. Standley, M. Aykol, S. B. Cronin, D. Smirnov, M. Koshino, E. McCann, M. Bockrath and C. N. Lau, Nat. Phys. 7, 948 (2011).
  * [32] C. Lui, Z. Li, K. Mak, E. Cappelluti and T. F. Heinz, Nat. Phys. 7, 944 (2011).
  * [33] L. C. Campos, A. F. Young, K. Surakitbovorn, K. Watanabe, T. Taniguchi and P. Jarillo-Herrero, Nat Commun. 3, 1239 (2012).
  * [34] S. H. R. de Sena, J. M. Pereira Jr., F. M. Peeters and G. A. Farias Phys. Rev. B 89, 035420 (2014)
  * [35] Y. P. Wang, X. G. Li, J. N. Fry and H. P. Cheng, Phys. Rev. B 94, 165428 (2016)
  * [36] K. Sugawara, N. Yamamura, K. Matsuda, W. Norimatsu, M. Kusunoki, T. Sato and T. Takahashi, NPG Asia Mater. 10, e466 (2018).
  * [37] A. El Mouhafid, A. Jellal and M. Mekkaoui, Ann. Phys. (Berlin) 534, 2100513 (2022).
  * [38] Y. Mohammadi, R. Moradian and F. Shirzadi Tabar, Solid State Commun. 193, 1 (2014).
  * [39] I. Redouani, A. Jellal, A. Bahaoui and H. Bahlouli, Superlattices Microstruct. 116, 44 (2018).
  * [40] L. Zhang, Y. Zhang, J. Camacho, M. Khodasa and I. Zaliznyak, Nat. Phys. 7, 953 (2011).
  * [41] F. Zhang, B. Sahu, H. Min and A. H. MacDonald, Phys. Rev. B 82, 035409 (2010).
  * [42] M. Yankowitz, F. Wang, C. N. Lau and B. J. LeRoy, Phys. Rev. B 87, 165102 (2013).
  * [43] K. Zou, F. Zhang, C. Clapp, A. H. MacDonald and J. Zhu, Nano Lett. 13 369 (2013).
  * [44] M. Koshino and E. McCann, Phys. Rev. B 79, 125443 (2009).
  * [45] L.-J. Yin, W.-X. Wang, Y. Zhang, Y.-Y. Ou, H.-T. Zhang, C.-Y. Shen and L. He, Phys. Rev. B 95, 081402(R) (2017).
  * [46] B. Van Duppen, S. H. R. Sena and F. M. Peeters, Phy. Rev. B 87, 195439 (2013).
  * [47] B. Partoens and F. M. Peeters, Phys. Rev. B 75, 193402 (2007).
  * [48] F. Zhang, J. Jung, G. A. Fiete, Q. Niu and A. H. Mac- Donald, Phys. Rev. Lett. 106, 156801 (2011).
  * [49] M. Barbier, P. Vasilopoulos, and F.M. Peeters, Physical Review B 82, 235408 (2010).
  * [50] Ya. M. Blanter and M. Büttiker, Physics Reports 336, (2000).
  * [51] N. Benlakhouy, A. El Mouhafid and A. Jellal, Physica E 134, 114835 (2021).
  * [52] H. Bahlouli, H. M. Abdullah, A. El Mouhafid and A. Jellal, Mater. Res. Express 4, 025009 (2017).
  * [53] A. Jellal, I. Redouani and H. Bahlouli, Physica E 72, 149 (2015).
  * [54] I. Redouani, A. Jellal and H. Bahlouli, J. Low Tem. Phys. 181, 197 (2015).
  * [55] M. Hassane Saley and A. Jellal, Magnetic field effect on tunneling through triple barrier in AB bilayer graphene, arXiv:2301.12479 (2023)m to appear in Computational Materials Science.
  * [56] H. M. Abdullah, B. Van Duppen, M. Zarenia, H. Bahlouli and F. M. Peeters, J. Phys.: Condens. Matter 29, 425303 (2017).