#  Effect of a perpendicular magnetic field on bilayer graphene under dual
gating

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

By studying the impact of a perpendicular magnetic field $B$ on AB-bilayer
graphene (AB-BLG) under dual gating, we yield several key findings for the
ballistic transport of gate $U_{\infty}$. Firstly, we discover that the
presence of $B$ leads to a decrease in transmission. At a high value of $B$,
we notice the occurrence of anti-Klein tunneling over a significant area.
Secondly, in contrast to the results reported in the literature, where high
peaks were found with an increasing in-plane pseudomagnetic field applied to
AB-BLG, we find a decrease in conductivity as $B$ increases. However, it is
worth noting that in both cases, the number of oscillations decreases compared
to the result in the study where no magnetic field was present $(B=0)$.
Thirdly, at the neutrality point, we demonstrate that the conductivity
decreases and eventually reaches zero for a high value of $B$, which contrasts
with the result that the conductivity remains unchanged regardless of the
value taken by the in-plane field. Finally, we consider the diffusive
transport with gate $U_{\infty}=0.2\gamma_{1}$ and observe two scenarios. The
amplitude of conductivity oscillations increases with $B$ for energy $E$ less
than $U_{\infty}$ but decreases in the opposite case $E>U_{\infty}$.

###### pacs:

72.80.Vp, 73.21.Ac, 73.23.Ad
Keywords: Ab-bilayer graphene, dual gates, magnetic field, Landau levels,
ballistic and diffusive transports, transmission, conductivity.

##  I Introduction

The stacking of multiple layers of graphene [1] can lead to the formation of
new materials with unique properties [2, 3, 4, 5, 6, 7, 8, 9]. We cite AB
bilayer graphene (AB-BLG), which stands out due to its ability to develop and
regulate a gap through the application of an external electric field [2, 10,
11, 12, 13, 15, 14]. In contrast to MLG, AB-BLG exhibits a parabolic
dispersion with two bands that converge at the Dirac points [12, 13, 15, 14,
16, 17] in addition to two bands separated from the Dirac points by an energy
equivalent to the interlayer coupling $\gamma_{1}\simeq 0.4$eV [11]. On the
other hand, AA-BLG demonstrates a linear energy dispersion akin to MLG,
featuring two Dirac cones shifted by an energy $\gamma\simeq 0.2$eV [18, 19,
20]. Furthermore, AB-BLG displays anti-Klein tunneling [3, 12, 13, 14, 15],
while AA-BLG is characterized by perfect electron transmission, akin to MLG
[19]. Also, AB-BLG shows a distinct quantum Hall effect, featuring a Berry
phase of $2\pi$ [21], and setting it apart from MLG.

Recent research in the field of ballistic transport has presented evidence
indicating that the conductivity at the neutrality point in AB-BLG remains
unchanged when an in-plane pseudomagnetic field is introduced [22]. This
observation aligns with the conclusions drawn in a previous study that focused
on the behavior of undoped AB-BLG [17]. These findings prompted us to explore
the influence of a perpendicular magnetic field $B$ on ballistic transmission
in AB-BLG. Our study demonstrates that the presence of $B$ results in a
reduction in transmission and the number of resonances. Notably, we observe
the occurrence of anti-Klein tunneling over a large area at a specific value
$B=14.44$T. Additionally, as $B$ increases, both the number and amplitude of
oscillations in conductivity decrease. Furthermore, at the neutrality point,
the conductivity reaches zero at $B=14.44$T. Regarding the diffusive process,
we choose $U_{\infty}=0.2\gamma_{1}$ and show that the amplitude of
conductivity oscillations increases when $B$ increases in the case
$E<U_{\infty}$, while it decreases in the opposite scenario.

The paper is organized as follows. In Sec. II, we introduce a theoretical
model describing the present system and determine the associated solutions of
the energy spectrum. Sec. III focuses on determining the transmission
probability using continuity conditions and the transfer matrix method. In
Sec. IV, we numerically present and discuss the obtained results. Finally, we
provide a summary of our findings in Sec. V.

##  II Theory and methods

Figure 1: The profile of a potential applied to graphene subjected to a
perpendicular magnetic field in the central region (region 2).

We consider AB-BLG submitted to a perpendicular magnetic field and dual gates,
as depicted in Fig. 1. Such gates manifest themselves through the application
of a potential to generate a strip of weakly doped (length $L=d_{2}-d_{1}$,
width $W$) and higher-doped contact regions for $x<d_{1}$ and $x>d_{2}$.
Otherwise, we write the potential as

$U_{j}=\begin{cases}-U_{\infty},&\text{if}~{}x<d_{1},x>d_{2}\\\
0,&\text{if}~{}d_{1}<x<d_{2},\end{cases}$ (1)

and $j=1,2,3$ labels the three regions. The magnetic field applied in region 2
takes the form

$B(x,y)=B\Theta\left[\left(d_{1}-x\right)\left(d_{2}-x\right)\right],$ (2)

with $B$ is a constant and $\Theta(x)$ is the step function. The corresponding
vector potential in the Landau gauge is given by

$A_{y}(x)=\frac{\hbar}{el_{B}^{2}}\begin{cases}d_{1},&\text{if}\quad
x<d_{1},\\\ x,&\text{if}\quad d_{1}<x<d_{2},\\\ d_{2},&\text{if}\quad
x>d_{2},\end{cases}$ (3)

and $l_{B}=\sqrt{\hbar/eB}$ is the magnetic length. To perform a theoretical
analysis of the present system, we formulate a Hamiltonian that incorporates
all the mentioned constraints. It can be expressed as follows [17, 12, 23]

$H_{j}=\begin{pmatrix}U_{j}&v_{F}\pi^{\dagger}&0&0\\\
v_{F}\pi&U_{j}&\gamma_{1}&0\\\ 0&\gamma_{1}&U_{j}&v_{F}\pi^{\dagger}\\\
0&0&v_{F}\pi&U_{j}\end{pmatrix},$ (4)

where $v_{F}=10^{6}$ m/s is the Fermi velocity and
$\pi=p_{x}+i(p_{y}+eA_{y}(x))$ is the in-plan momentum. The Hamiltonian (4)
operates on the spinor of four components, i.e.,
$\Psi_{j}(x,y)=\left(\psi_{A_{1}j},\psi_{B_{1}j},\psi_{A_{2}j},\psi_{B_{2}j}\right)^{\dagger}$,
with the symbol $\dagger$ standing for the transpose row vector. Since the
momentum is conserved in the $y$-direction, the spinor can be decomposed into

$\Psi_{j}(x,y)=e^{ik_{y}y}\left(\psi_{A_{1}j},\psi_{B_{1}j},\psi_{A_{2}j},\psi_{B_{2}j}\right)^{\dagger}.$
(5)

Now, we combine (4) with (5) into the eigenvalue equation
$H_{j}\Psi_{j}=E_{j}\Psi_{j}$ to arrive at

$\displaystyle-i\sqrt{2}\vartheta_{0}a\psi_{B_{1}j}$
$\displaystyle=\varepsilon_{j}\psi_{A_{1}j},$ (6a) $\displaystyle
i\sqrt{2}\vartheta_{0}a^{\dagger}\psi_{A_{1}j}$
$\displaystyle=\varepsilon_{j}\psi_{B_{1}j}-\gamma_{1}\psi_{A_{2}j},$ (6b)
$\displaystyle-i\sqrt{2}\vartheta_{0}a\psi_{B_{2}j}$
$\displaystyle=\varepsilon_{j}\psi_{A_{2}j}-\gamma_{1}\psi_{B_{1}j},$ (6c)
$\displaystyle i\sqrt{2}\vartheta_{0}a^{\dagger}\psi_{A_{2}j}$
$\displaystyle=\varepsilon_{j}\psi_{B_{2}j},$ (6d)

where the annihilation $a$ and creation $a^{\dagger}$ operators are
represented by

$\displaystyle
a=\frac{l_{B}}{\sqrt{2}}\left(\partial_{x}+k_{y}+\frac{eA_{y}(x)}{\hbar}\right),$
(7a) $\displaystyle
a^{\dagger}=\frac{l_{B}}{\sqrt{2}}\left(-\partial_{x}+k_{y}+\frac{eA_{y}(x)}{\hbar}\right),$
(7b)

which satisfy the commutation relation $[a,a^{\dagger}]=\mathbb{I}$. Here, we
have set the energy $\varepsilon_{j}=E_{j}-U_{j}$, with
$\vartheta_{0}=\frac{\hbar v_{F}}{l_{B}}$ denoting an energy scale. From Eqs.
(6a-6d), we can, for instance, establish a four order differential equation
for the wave function $\psi_{B_{1j}}$

$\displaystyle\left[2\vartheta_{0}^{2}aa^{\dagger}-\varepsilon_{j}^{2}\right]\left[2\vartheta_{0}^{2}a^{\dagger}a-\varepsilon_{j}^{2}\right]\psi_{B_{1j}}=\gamma_{1}^{2}\varepsilon_{j}^{2}\psi_{B_{1j}}.$
(8)

In region 2 ($d_{1}<x<d_{2}$), where $A_{y}(x)=\frac{\hbar x}{el_{B}^{2}}$ and
$U_{\infty}=0$, we solve Eq. (8) to derive $\psi_{B_{1}}$ and the result is
employed in Eqs. (6a-6d) to determine the other components of the spinor.
Thereafter, the solution of the spinor can be expressed in matrix form

$\Psi_{2}(x,y)=\mathcal{L}_{2}\mathcal{M}_{2}(x)\mathcal{C}_{2}e^{ik_{y}y},$
(9)

where $\mathcal{C}_{2}=(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2})^{\dagger}$
is a column matrix of four coefficients, $\mathcal{L}_{2}=\mathbb{I}_{4}$ is a
$4\times 4$ identity matrix, and $\mathcal{M_{\text{2}}}(x)$ is given by

$\mathcal{M_{\text{2}}}(x)=\begin{pmatrix}\eta\lambda_{+}\chi_{+,-1}^{+}&\eta^{*}\lambda_{+}\chi_{-,-1}^{+}&\eta\lambda_{-}\chi_{+,-1}^{-}&\eta^{*}\lambda_{-}\chi_{-,-1}^{-}\\\
\chi_{+,0}^{+}&\chi_{-,0}^{+}&\chi_{+,0}^{-}&\chi_{-,0}^{-}\\\
\zeta_{+}\chi_{+,0}^{+}&\zeta_{+}\chi_{-,0}^{+}&\zeta_{-}\chi_{+,0}^{-}&\zeta_{-}\chi_{-,0}^{-}\\\
\eta^{*}\zeta_{+}\chi_{+,1}^{+}&\eta\zeta_{+}\chi_{-,1}^{+}&\eta^{*}\zeta_{-}\chi_{+,1}^{-}&\eta\zeta_{-}\chi_{-,1}^{-}\end{pmatrix}.$
(10)

Here, we have introduced the Weber parabolic cylindrical function
$\chi^{\tau}_{\pm,s}=D\left[\lambda_{\tau}\pm s,\pm z\right]$ of argument
$z=\sqrt{2}(\frac{x}{l_{B}}+k_{y}l_{B})$, $s=-1,0,1$, $\tau=\pm$, and we have
set the following parameters

$\displaystyle\lambda_{\tau}=-\frac{1}{2}+\frac{E^{2}}{2\vartheta_{0}^{2}}+\tau\frac{\sqrt{\vartheta_{0}^{4}+\gamma_{1}^{2}E^{2}}}{2\vartheta_{0}^{2}},$
(11a)
$\displaystyle\eta=\frac{-i\sqrt{2}\vartheta_{0}}{E},\quad\zeta_{\tau}=\frac{E^{2}-2\vartheta_{0}^{2}\lambda_{\tau}}{\gamma_{1}E}.$
(11b)

As a result, from Eq. (11a), we derive the energy spectrum

$E^{\tau}_{n\pm}=\tau\sqrt{(2n+1)\vartheta_{0}^{2}+\frac{\gamma_{1}^{2}}{2}\pm\sqrt{(2n+1)\vartheta_{0}^{2}\gamma_{1}^{2}+\vartheta_{0}^{4}+\frac{\gamma_{1}^{4}}{4}}},$
(12)

and $n$ is an integer value, indexing the Landau levels.

Recall that in regions 1 ($x<d_{1}$) and 3 ($x>d_{2}$), the vector potential
$A_{y}(x)=\frac{\hbar}{el_{B}^{2}}d_{j}$ is constant where

$d_{j}=\left\\{\begin{array}[]{lll}d_{1},&\text{ if }&x<d_{1},\\\
d_{2},&\text{ if }&x>d_{2}.\end{array}\right.$ (13)

Upon solving Eq. (8) for $\psi_{B_{1}}$ and substituting the obtained result
into Eqs. (6a-6d), a general solution of the spinor is derived in matrix form

$\Psi_{j}(x,y)=\mathcal{L}_{j}\mathcal{M}_{j}(x)\mathcal{C}_{j}e^{ik_{y}y},$
(14)

where $\mathcal{L_{\text{$j$}}}$, $\mathcal{M_{\text{$j$}}}(x)$, and
$\mathcal{C}_{j}$ are given, respectively, by

$\displaystyle\mathcal{L}_{j}=\begin{pmatrix}f_{-j}^{+}&-f_{+j}^{+}&f_{-j}^{-}&-f_{+j}^{-}\\\
1&1&1&1\\\ -1&-1&1&1\\\
-f_{+j}^{+}&f_{-j}^{+}&f_{+j}^{-}&-f_{-j}^{-}\end{pmatrix},$ (15)
$\displaystyle\mathcal{M}_{j}(x)=\text{diag}\left[e^{ik_{j}^{+}x},e^{-ik_{j}^{+}x},e^{ik_{j}^{-}x},e^{-ik_{j}^{-}x}\right],$
(16) $\displaystyle\mathcal{C}_{j}=(a_{j},b_{j},c_{j},e_{j})^{\dagger}$ (17)

and we have defined the parameters

$f_{\pm}^{\tau}=\hbar v_{F}\frac{k_{j}^{\tau}\pm
i\left(k_{y}+\frac{d_{j}}{l_{B}^{2}}\right)}{\varepsilon_{j}}.$ (18)

We show that the associated energies are

$\varepsilon_{\pm,j}^{\tau}=\pm\sqrt{\left(\hbar
v_{F}\kappa_{j}^{\tau}\right)^{2}+\frac{\gamma_{1}^{2}}{2}+\tau\sqrt{\left(\hbar
v_{F}\kappa_{j}^{\tau}\right)^{2}\gamma_{1}^{2}+\frac{\gamma_{1}^{4}}{4}}},$
(19)

where the wave vector is

$\displaystyle\kappa_{j}^{\tau}=\sqrt{(k_{j}^{\tau})^{2}+\left(k_{y}+\frac{d_{j}}{l_{B}^{2}}\right)^{2}}$
(20)

$k_{1}^{\tau}$ represent the wave vectors of the propagating wave in the first
region, where there are two right-going (incident) propagating modes and two
left-going (reflected) propagating modes. Similarly, $k_{3}^{\tau}$ denote the
wave vectors of the propagating wave in the third region, featuring two right-
going (transmission) propagating modes.

##  III Transport properties

To determine the transmission probability, in the first step, we apply
continuity conditions at each interface of the present system. In the second
step, we use the transfer matrix method to establish a link between the
coefficients $\mathcal{C}_{\text{1}}$ in the incident region and
$\mathcal{C}_{\text{3}}$ in the transmitted region. From (17), we deduce

$\displaystyle\mathcal{C}_{\text{1}}=\left(\delta_{\tau,1},r_{+}^{\tau},\delta_{\tau,-1},r_{-}^{\tau}\right)^{\dagger},$
(21a)
$\displaystyle\mathcal{C}_{\text{3}}=\left(t_{+}^{\tau},0,t_{-}^{\tau},0\right)^{\dagger},$
(21b)

where $\delta_{\tau,\pm 1}$ denotes the Kronecker delta symbol. Then, at
interfaces $x=d_{1}$ and $x=d_{2}$, the continuity of eigenspinors (9) and
(14) allows to write

$\displaystyle\mathcal{L}_{1}\mathcal{M}_{1}(d_{1})\mathcal{C}_{1}=\mathcal{L}_{2}\mathcal{M}_{2}(d_{1})\mathcal{C}_{2},$
(22a)
$\displaystyle\mathcal{L}_{2}\mathcal{M}_{2}(d_{2})\mathcal{C}_{2}=\mathcal{L}_{3}\mathcal{M}_{3}(d_{2})\mathcal{C}_{3}.$
(22b)

After some algebra, we establish the relation

$\mathcal{C}_{\text{1}}=\mathcal{N}\mathcal{C}_{\text{3}},$ (23)

where the transfer matrix $\mathcal{N}$ is defined as

$\mathcal{N}=\prod_{j=\text{1}}^{\text{2}}\mathcal{M}_{j}^{-1}(d_{j})\mathcal{L}_{j}^{-1}\mathcal{L}_{j+1}\mathcal{M}_{j+1}(d_{j}).$
(24)

One can use Eq. (23) to derive the following transmissions coefficients

$\displaystyle
t_{+}^{\tau}=\frac{\mathcal{N}_{33}\delta_{\tau,1}-\mathcal{N}_{13}\delta_{\tau,-1}}{\mathcal{N}_{11}\mathcal{N}_{33}-\mathcal{N}_{13}\mathcal{N}_{31}},$
(25) $\displaystyle
t_{-}^{\tau}=\frac{\mathcal{N}_{11}\delta_{\tau,-1}-\mathcal{N}_{31}\delta_{\tau,1}}{\mathcal{N}_{11}\mathcal{N}_{33}-\mathcal{N}_{13}\mathcal{N}_{31}},$
(26)

where $\mathcal{N}_{ij}$ are matrix elements of $\mathcal{N}$ (24). To
completely determine the corresponding transmission probabilities, we
introduce the current density
$\textbf{j}=v_{F}{\Psi}^{\dagger}\vec{\alpha}\Psi$ associated with the present
system, and $\vec{\alpha}$ is a $4\times 4$ matrix with two Pauli matrices
$\sigma_{x}$ on its diagonal. Then we use the incident
$\textbf{j}_{\text{inc}}$ and transmitted $\textbf{j}_{\text{tra}}$ current
densities

$\displaystyle
T_{\pm}^{\tau}=\frac{\left|\textbf{j}_{\text{tra}}\right|}{\left|\textbf{j}_{\text{inc}}\right|}=\frac{k_{3}^{\tau}}{k_{1}^{\tau}}\left|t_{\pm}^{\tau}\right|^{2}.$
(27)

Concerning the conductance, we apply the Landauer-Büttiker formula [24] at
zero temperature to derive

$\mathrm{G}(E)=G_{0}\frac{W}{2\pi}\int_{-\infty}^{+\infty}\mathrm{d}k_{y}\sum_{\tau,n=\pm}T_{n}^{\tau}\left(E,k_{y}\right),$
(28)

where $G_{0}=4e^{2}/h$, the factor 4 count the valley and spin degeneracy in
graphene and $W$ is the width of the sample along the $y$-direction. The
conductivity is then expressed as [17]

$\sigma=\frac{L}{W}G(E).$ (29)

Next, we will focus on the numerical analysis of the results obtained so far.
For convenience, we will adopt the configuration of different parameters:
$\vartheta_{0}/\gamma_{1}=l/l_{B}$ and $k_{y}\equiv lk_{y}$, with $l=\hbar
v_{F}/\gamma_{1}=1.76$nm.

##  IV Results and discussions

Figure 2: (Color online) Density plot of transmission probabilities as a
function of energy $E$ and transverse wave vector $k_{y}$ for
$U_{\infty}=-50\gamma_{1}$, $L=d_{2}-d_{1}=50l$, and two values of magnetic
field (a): $B=0$ and (b): $B=3.61$T. The line plots correspond to the
transmission at fixed energy: $E=-0.05\gamma_{1}$ (blue), $E=0$ (black), and
$E=0.05\gamma_{1}$ (red).

Fig. 2 presents a comparison of the transmission probabilities under two
magnetic field conditions: (a) $B=0$ and (b) $B=3.61$T. It is evident that the
transmissions are broader for $B=0$ and narrower for $B=3.61$T. In Fig. 2b,
the channels $T^{+}_{+}$ and $T^{-}_{-}$ demonstrate that the presence of $B$
decreases the number of resonances compared to the corresponding channels for
$B=0$, see Fig. 2a. Additionally, there is an electron-hole asymmetry because
we have $T^{+}_{+}(E)\neq T^{+}_{+}(-E)$ and $T^{-}_{-}(E)\neq T^{-}_{-}(-E)$
for both values $B=0$ and $B=3.61$T, as reflected in the difference between
the blue and red lines. This result aligns with that derived by applying an
in-plane pseudomagnetic field to AB-BLG [22]. However, the correspondence
$T^{+}_{+}(-E)=T^{-}_{-}(E)$ is hold (see equivalence between blue line in
$T^{+}_{+}$ and red line in $T^{-}_{-}$). On the other hand, in channels
$T^{+}_{-}$ and $T^{-}_{+}$ for $B=0$, the transmissions exhibit electron-hole
symmetry, as indicated by the matching in blue and red lines. Contrarily, the
introduction of $B$ breaks such symmetry and that associated with normal
incidence ($k_{y}=0$) in $T^{+}_{-}$ and $T^{-}_{+}$, as clearly depicted by
the distinct behavior of the blue and red lines in Fig. 2b. This contrasts
with the result obtained in [22]. It is worthily noted that $T^{+}_{-}$ and
$T^{-}_{+}$ are equivalent under the sign change of the transverse wave vector
$k_{y}$, i.e., $T^{+}_{-}(k_{y})=T^{-}_{+}(-k_{y})$. This is in agreement with
the findings observed for diffusive transports when a bias is applied [12,
14]. Fig. 2a shows that the channels $T^{+}_{-}$ and $T^{-}_{+}$ exhibit anti-
Klein tunneling at normal and near-normal incidence for $B=0$, similar to the
results for diffusive transport [12, 13]. At $B=3.61$T, transmissions take
place at normal incidence, and anti-Klein tunneling is shifted towards the
side. As for total transmission $T_{tot}=\sum_{\tau,n=\pm} T_{n}^{\tau}$, the
electron-hole symmetry and symmetry with respect to $k_{y}=0$ are present for
both values $B=0$ and $B=3.61$T. In other words, we have the symmetry
$T^{+}_{+}(-E)=T^{-}_{-}(E)$ for all $B$, but
$T^{+}_{-}(E,k_{y})=T_{+}^{-}(E,k_{y})$ holds for $B=0$ and
$T^{+}_{-}(E,k_{y})=T^{-}_{+}(-E,-k_{y})$ is valid for non-null $B$, i.e.,
$B=3.61$T.

Figure 3: (Color online) Density plot of the total transmission as a function
of energy $E$ and transverse wave vector $k_{y}$ with the same parameters as
in Fig. 2 but for (a): $B=0.9$ T, (b): $B=7.37$ T, and (c): $B=14.44$T.

Fig. 3 shows the total transmission $T_{tot}$ for three different values of
the magnetic field (a): $B=0.9$T, (b): $B=7.37$T, and (c): $B=14.44$T. It is
observed that as we increase $B$, the transmission narrows towards the center,
and then anti-Klein tunneling increases. In addition, in Fig. 3b, the number
of resonances decreases while their thickness increases compared to Fig. 3a.
Moreover, two regions of zero transmission are found on either side of the
neutrality point ($E=0$). Furthermore, in Fig. 3c, the application of a high
magnetic field ($B=14.44$T) results in destructing tunneling effect.

Figure 4: (Color online) Conductivity as a function of energy $E$,
corresponding to the transmissions $T_{tot}$ in Figs. 2 and 3.

In Fig. 4, we plot the conductivity corresponding to each transmission
$T_{tot}$ in Figs. 2 and 3. For $B=0$ (black line), the conductivity is
equivalent to the result found in [17, 22], with $\sigma_{0}=2G_{0}/\pi$ and
the subscript 0 in $\sigma$ refers to $B=0$ at the neutrality point ($E=0$).
For $B=0.9$T (depicted by the orange line), the conductivity experiences a
slight decrease but maintains the same number of peaks as observed for $B=0$,
reaching the value $\sigma=0.98\sigma_{0}$ at the neutrality point. For
$B=3.61$T (depicted by the brown line), the conductivity experiences a
decrease, with a simultaneous reduction in the number and amplitude of the
oscillations. We also observe that its value at the neutrality point drops to
$\sigma=0.45\sigma_{0}$. Furthermore, with an increase in the magnetic field
to $B=7.37$T (represented by the cyan line), two regions of zero conductivity
become apparent, accompanied by a further reduction in the number and
amplitude of oscillations compared to the results for $B=3.61$T (brown line).
Nevertheless, the conductivity remains non-zero at the neutrality point and
takes the value $\sigma=0.43\sigma_{0}$. At $B=14.44$T (indicated by the blue
line), the conductivity drops to zero even at the neutrality point but
exhibits two peaks at the extremities $E=\pm 0.1\gamma_{1}$. It is evident
that, as the magnetic field increases, the conductivity decreases, even at the
neutrality point. This is in contrast with the result for the case of an in-
plane pseudomagnetic field [22]. Additionally, it is worth noting that, in all
cases, the conductivity is symmetric with respect to the neutrality point due
to the symmetry in the transmission probability $T_{\text{tot}}$.

Figure 5: (Color online) Conductivity as a function of energy $E$ around Dirac
point for $U_{\infty}=0.2\gamma_{1}$ and $L=100l$.

Fig. 5 shows the conductivity as a function of energy $E$ for three values of
the magnetic field: $B=0$ (black line), $0.57$T (blue line), and $0.9$T
(orange line). We reduce the potential $U_{\infty}$ to $0.2$$\gamma_{1}$ and
choose $L=100l$. We observe that the conductivity shows an asymmetry behavior
with respect to the Dirac point $E=U_{\infty}$. This asymmetry persists
irrespective of the presence or absence of $B$. However, the conductivity
behaves differently depending on the variation of $B$. Indeed, the
conductivity displays oscillations with an increasing amplitude as $B$
increases for the case $E<U_{\infty}$, and conversely, it decreases with an
increase of $B$ for $E>U_{\infty}$. In addition, at $E=U_{\infty}$, the
conductivity decreases with an increase of $B$, similarly to the result
presented in Fig. 4 at $E=0$.

##  V Conclusion

We have investigated the impact of a perpendicular magnetic field on the
ballistic transport properties of AB-BLG graphene, as well as weakly doping
the system. Our results have revealed several important findings. Firstly, we
have observed that the presence of a magnetic field leads to a narrowing of
the transmissions, a decrease in the number of resonances, and an increase in
anti-Klein tunneling. Notably, we have identified electron-hole asymmetry in
the channels $T^{+}_{+}$ and $T^{-}_{-}$, regardless of the presence or
absence of a magnetic field. However, in the channels $T^{+}_{-}$ and
$T^{-}_{+}$, we have discovered that electron-hole symmetry holds in the
absence of a magnetic field, but is broken in its presence. Additionally, the
symmetry with respect to $k_{y}=0$ is also broken. Nevertheless, due to the
equivalence between these channels under the sign change of energy $E$ or
transverse wave vector $k_{y}$, the transmission remains symmetric in
$T_{tot}$. Furthermore, we have observed the destruction of tunneling effect
as a result of considering a high magnetic field.

In the context of ballistic transport, we have found that conductivity
decreases with an increasing magnetic field. However, in the case of a weakly
doped system with a potential of $U_{\infty}=0.2\gamma_{1}$, we have observed
an asymmetry in the conductivity at the Dirac point $E=U_{\infty}$.
Additionally, we have noted that the amplitude of oscillations increases with
the magnetic field for $E<U_{\infty}$ but decreases in the opposite case,
$E>U_{\infty}$. In conclusion, our findings highlight the ability to regulate
electron transmission and conductivity in AB-BLG by applying a perpendicular
magnetic field in the context of ballistic transport. These results provide
valuable insights into the behavior of AB-BLG graphene under the influence of
a magnetic field and weak doping, which can contribute to the development of
graphene-based electronic devices with tailored transport properties.

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