# Tunneling in rippled graphene superlattice with spin dependence and a mass
term

Jaouad El-hassouny  Laboratory of Nanostructures and Advanced Materials,
Mechanics and Thermofluids, Faculty of Sciences and Techniques, Hassan II
University, Mohammedia, 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  El Houssine Atmani  Laboratory of
Nanostructures and Advanced Materials, Mechanics and Thermofluids, Faculty of
Sciences and Techniques, Hassan II University, Mohammedia, Morocco

###### Abstract

The insertion of the band gap $\Delta$ in the rippled graphene superlattice
leads to new outcomes, as demonstrated. The essential thing is the appearance
of opposite-spin transmissions, which increase with $\Delta$ and vanish
without it. Furthermore, compared to the $\Delta=0$ scenario, the duration of
the suppression of the transmission with the same spin is longer, with many
peaks. The maximum value of transmissions with the same spin declines and
remains around unity. Furthermore, for particular energy values, a shift in
the behavior of the transmission channels is found. As a result, we
demonstrate that with $\Delta$, transmission filtering becomes crucial.
Finally, as a result of the band gap, distinct variations in total conductance
are discovered.

###### pacs:

72.25.-b, 71.70.Ej, 73.23.Ad
Keywords: Graphene, ripple, superlattice, mass term, spin transmission and
reflection, conductance.

##  I Introduction

Graphene is a zero-gap semiconductor [1, 2] with band structure described by a
low-energy linear dispersion relation, like massless Dirac-Weyl fermions [3,
4]. Such a band structure leads to the exceptionally high conductivity of the
charged carriers. Because of its importance in the technological field,
finding possible solutions that would manage the mobility and, therefore,
control the conductivity is a hot topic in graphene physics. Experimentally,
there are different techniques to open a gap in a graphene band structure [5]
and its maximum value could be 260 meV due to the non-symmetry of the
sublattice [6]. By manipulating the structure of the graphene-ruthenium
interface [8] or epitaxed graphene on a SiC substrate [6, 7], an energy gap
can be measured. Moreover, other methods can be used to open and adjust the
band gap of graphene, such as diffusion, which could be induced by a ripple
created in a sheet of graphene [9, 10]. Ripples act as potential barriers for
charged carriers, leading to their localization [11]. Additionally,
periodically repeated ripples can be created and controlled in suspended
graphene, notably by thermal treatment [12] or by placing the graphene in a
specifically prepared substrate.

Theoretically, alternative studies have been proposed to create an energy gap
in graphene systems. For example, the scattering of electrons through a
periodically repeated rippled chain (the superlattice) was investigated in
[13, 14]. This study demonstrated that the graphene superlattice leads to the
suppression of electron transmission with one spin orientation on the contrary
to the other. Another method investigated the scattering of electrons through
the superlattice, considering a chain formed by concave and convex undulations
to create a sinusoidal type graphene sheet [15]. As a result, the spin
filtering effect, which is weak in one ripple, becomes important with an
increasing number of connected ripples [15]. In our previous work, [16], we
demonstrated that adding a band gap to repplied graphene reduces transmission
channels with spin-up/-down but increases those with spin opposite. We
detected resonances in reflection channels with the same spin, indicating that
backscattering with a spin-up/-down is not null in ripple. It is found that
the spin filter is affected by some critical band gap values, resulting in a
reduction of the channels. Furthermore, we demonstrated that the band gap, as
opposed to the null gap [15], affects total conductance.

We extend our previous work [16] to a rippled graphene superlattice,
introducing a band gap and investigating quantum tunneling. We derive the
energy spectrum and use the matrix transfer to analytically obtain the full
transmission and reflection chancels. We show that adding a band energy acts
by decreasing the transmissions with spin-up/-down but increasing those with
spin opposite. We also observe that electron transmission is strongly
suppressed through the superlattice for some energy values. Furthermore, we
show that the total conductance gets shifted by the band gap, in contrast to
the case of gap null [13, 14]. We conclude that the presence of a band gap can
be used as a key tool to control the transport properties of our superlattice.

The following is how the present paper is organized. In section II, we develop
our theoretical model and solve the Dirac equation for the $j$-th cell to
obtain the energy spectrum for each region. In section III, using the boundary
conditions at interfaces of $2N$ points and the matrix transfer method, we
compute all the transmission and reflection channels in addition to the
corresponding conductance. We numerically analyze the transmission channels
and conductance under various conditions to highlight their basic features in
section IV. Finally, we provide a summary of our results.

##  II Theoretical model

We examine an indefinitely large corrugated graphene, as seen in Fig. 1. It
consists of $N$ elementary cells designated by $j\ (j=0,\cdots,N-1)$, each of
which is made up of a juxtaposition of two regions: one arc of a circle with
radius $r_{0}$ and angle $\phi_{j}$ involving a mass term $\Delta_{j}$, and a
flat graphene with length $d$. The $j$-th elementary cell has one internal
junction at $x_{2}j=2r_{0}\sin\phi_{j}/2+jD$ and two extreme junctions at
$x_{1}j=jD,x_{3}j=(j+1)D)$. The width of one cell is
$D=d+2r_{0}\sin\phi_{j}/2$. We assume ${W}\gg{L},$ ($W$ and $L$ being the
width along the $y$\- and length along the $x$-axes of our system,
respectively) and that the electrons are infused into the superlattice at
$k_{y}=0$ if edge impacts are ignored. According to Fig. 1, for $j$-th
elementary cell the gap $\Delta_{j}$ and the angle of ripple $\theta_{j}$ are
given by

$\Delta_{j},\theta_{j}=\left\\{\begin{array}[]{lll}\Delta_{,}\theta&\text{ if
}&jD<x<2r_{0}\sin{\phi_{j}/2}+jD\\\ 0&\text{ if
}&2r_{0}\sin{\phi_{j}/2}+jD<x<(1+j)D\end{array}\right.$ (1)

Figure 1: (color online) The profile of graphene is made up of $N$ unit cells,
each of which is an arc (ripple) of radius $r_{0}$ with a mass term $\Delta$
connected to the planar part of length $d$. The specific parts are linked at
the connection points ($1$ to $2N$).

Our graphene superlattice composed of $N$ unit cells, which are placed between
the input and output regions. In region 1 of j-th elementary cell, as depicted
in Fig. 1, has the Hamiltonian spin dependent [16, 17]

$\mathcal{H}_{1}^{j}=\begin{pmatrix}0&0&-\gamma\partial_{y}-\Delta_{j}-\mathrm{i}\frac{\gamma}{r_{0}}\partial_{\theta_{j}}&\mathrm{i}\left(\lambda_{y}+\lambda_{x}\right)\\\
0&0&\mathrm{i}\left(\lambda_{y}-\lambda_{x}\right)&-\gamma\partial_{y}+\Delta_{j}-\mathrm{i}\frac{\gamma}{r_{0}}\partial_{\theta_{j}}\\\
\gamma\partial_{y}-\Delta_{j}-\mathrm{i}\frac{\gamma}{r_{0}}\partial_{\theta_{j}}&-\mathrm{i}\left(\lambda_{y}-\lambda_{x}\right)&0&0\\\
-\mathrm{i}\left(\lambda_{y}+\lambda_{x}\right)&\gamma\partial_{y}+\Delta_{j}-\mathrm{i}\frac{\gamma}{r_{0}}\partial_{\theta_{j}}&0&0\end{pmatrix}.$
(2)

We get four bands $E_{j1s}^{s^{\prime}}$ at normal incidence, $k_{y}=0,$ by
solving the eigenvalue equation

$E_{1js}^{s^{\prime}}=s^{\prime}\sqrt{t_{m_{j}}^{2}+\lambda_{y}^{2}}+s\sqrt{\Delta_{j}^{2}+\lambda_{x}^{2}}$
(3)

where $s,s^{\prime}=\pm$ and $t_{m_{j}}=\frac{\gamma}{r_{0}}m_{j}$,
$t_{y}=\gamma k_{y}$, $\lambda_{x}=\frac{\gamma}{2r_{0}}(1+4\delta p)$,
$\lambda_{y}=\frac{\delta\gamma^{\prime}}{4r_{0}}$ have been defined. As a
result, we can calculate the angular momentum as follows:

$m_{js}^{s^{\prime}}=s^{\prime}\frac{r_{0}}{\gamma}\sqrt{\left(E_{j1s}^{s^{\prime}}-s\sqrt{\Delta_{j}^{2}+\lambda_{x}^{2}}\right)^{2}-\lambda_{y}^{2}}$
(4)

and here $m_{j}=m_{js}^{s^{\prime}}$. The eigenspinors then take the form

$\Psi_{1}^{jss^{\prime}}(\theta_{j},m_{j})=\begin{pmatrix}\cos\frac{\theta_{j}}{2}A_{ss^{\prime}}(m_{j})-\sin\frac{\theta_{j}}{2}B_{ss^{\prime}}(m_{j})\\\
\sin\frac{\theta_{j}}{2}A_{ss^{\prime}}(m_{j})+\cos\frac{\theta_{j}}{2}B_{ss^{\prime}}(m_{j})\\\
\cos\frac{\theta_{j}}{2}-\sin\frac{\theta_{j}}{2}D_{s}(m_{j})\\\
\sin\frac{\theta_{j}}{2}+\cos\frac{\theta_{j}}{2}D_{s}(m_{j})\end{pmatrix}e^{im_{j}\theta_{j}}$
(5)

and the quantities

$\displaystyle
A_{jss^{\prime}}(m_{j})=s^{\prime}\frac{\lambda_{x}\sqrt{t_{m_{j}}^{2}+\lambda_{y}^{2}}+s\lambda_{y}\sqrt{\Delta_{j}^{2}+\lambda_{x}^{2}}}{\lambda_{x}t_{m_{j}}-\lambda_{y}\Delta_{j}}$
(6) $\displaystyle
B_{jss^{\prime}}(m_{j})=-is^{\prime}\frac{\Delta_{j}\sqrt{t_{m_{j}}^{2}+\lambda_{y}^{2}}+st_{m_{j}}\sqrt{\Delta_{j}^{2}+\lambda_{x}^{2}}}{\lambda_{x}t_{m_{j}}-\lambda_{y}\Delta_{j}}$
(7) $\displaystyle C=1$ (8) $\displaystyle
D_{js}(m_{j})=-i\frac{t_{m_{j}}\Delta_{j}+\lambda_{y}\lambda_{x}+s\sqrt{(t_{m_{j}}^{2}+\lambda_{y}^{2})(\Delta_{j}^{2}+\lambda_{x}^{2})}}{\lambda_{x}t_{m_{j}}-\lambda_{y}\Delta_{j}}$
(9)

have been set. Finally, in region 1 the wave function can be composed as a
superposition of four solutions for a rippled part

$\Psi_{1}^{j}(\theta_{j})=a_{+j}\Psi_{1}^{+}\left(\theta_{j},m_{j+}^{+}\right)+b_{+j}\Psi_{1}^{+}\left(\theta_{j},m_{j+}^{-}\right)+a_{-j}\Psi_{1}^{-}\left(\theta_{j},m_{j-}^{+}\right)+b_{-j}\Psi_{1}^{-}\left(\theta_{j},m_{j-}^{-}\right)$
(10)

with $a{{}_{\pm}}_{j}$ and $b{{}_{\pm}}_{j},$ denote the coefficients of the
linear combination. It can be expressed as a matrix as

$\Psi_{1}^{j}(\theta_{j})=M_{1}(\theta_{j})\left(e^{\mathrm{i}{m_{j}}\theta_{j}}\right)D_{1j}$
(11)

such that

$\displaystyle M_{1}(\phi/2)=\Lambda{M}_{1}$ (12)
$\displaystyle\Lambda=\begin{pmatrix}\cos\frac{\phi}{4}&-\sin\frac{\phi}{4}&0&0\\\
\sin\frac{\phi}{4}&\cos\frac{\phi}{4}&0&0\\\
0&0&\cos\frac{\phi}{4}&-\sin\frac{\phi}{4}\\\
0&0&\sin\frac{\phi}{4}&\cos\frac{\phi}{4}\end{pmatrix}$ (13)
$\displaystyle{M}_{1}=\begin{pmatrix}A_{+,s^{\prime}}(m_{+})&A_{-,s^{\prime}}(m_{-})&A_{+,s^{\prime}}(-m_{+})&A_{-s^{\prime}}(-m_{-})\\\
B_{+,s^{\prime}}(m_{+})&B_{-,s^{\prime}}(m_{-})&B_{+,s^{\prime}}(-m_{+})&B_{-,s^{\prime}}(-m_{-})\\\
1&1&1&1\\\ D_{+}(m_{+})&D_{-}(m_{-})&D_{+}(-m_{+})&D_{-}(-m_{-})\end{pmatrix}$
(14)
$\displaystyle\left(e^{\mathrm{i}{m_{j}}\theta_{j}}\right)=\text{diag}\left(e^{\mathrm{i}m_{j+}\theta_{j}},e^{\mathrm{i}m_{j-}\theta_{j}},e^{\mathrm{-i}m_{j+}\theta_{j}},e^{\mathrm{-i}m_{j-}\theta_{j}}\right)$
(15) $\displaystyle D_{1j}=\begin{pmatrix}a_{+j}\\\ b_{+j}\\\ a_{-j}\\\
b_{-j}\end{pmatrix}.$ (16)

For flat part, regions 2, by solving the eigenvalue equation, we can derive
the two band energy at $k_{y}=0$ [16]

$E_{j2}=\pm\gamma|k_{j}|$ (17)

We can also write the relationship $E_{j2}=E^{s^{\prime}}_{1js}$ because of
energy conservation. The linked eigenstates have the following structure:

$\psi_{2}^{j,\tau}(x,k_{j})=\frac{1}{2}\begin{pmatrix}\text{sign}(k_{j})\\\
\text{sign}(k_{j})\ i\tau\\\ 1\\\ i\tau\end{pmatrix}e^{ik_{j}x}$ (18)

where $\tau=\pm$, $k_{j}=k_{x}$ and $\text{sign}(k)=\pm$ refers to conductance
and valence bands, respectively. As a result the wave function can be written
a superposition of four possible solutions for planar graphene

$\psi_{2}^{j}(x,k_{j})=\alpha_{j}\psi_{2}^{+}(x,k_{j})+\beta_{j}\psi_{2}^{-}(x,k_{j})+\gamma_{j}\psi_{2}^{+}(x,-k_{j})+\xi_{j}\psi_{2}^{-}(x,-k_{j})$
(19)

with $\alpha_{j},\beta_{j}$, $\gamma_{j}$ and $\xi_{j}$ indicate the
coefficients of the linear combination. It can be mapped as

$\psi_{2}^{j}(x,k_{j})=M_{2}\left(e^{\mathrm{i}{K_{j}}x}\right)D_{2j}$ (20)

such that

$\displaystyle M_{2}=\begin{pmatrix}1&1&-1&-1\\\ i&-i&-i&i\\\ 1&1&1&1\\\
i&-i&i&-i\end{pmatrix}$ (21)
$\displaystyle\left(e^{\mathrm{i}{K_{j}}x}\right)=\text{diag}\left(e^{\mathrm{i}k_{j}x},e^{\mathrm{i}k_{j}x},e^{\mathrm{-i}k_{j}x},e^{\mathrm{-i}k_{j}x}\right)$
(22) $\displaystyle D_{2j}=\begin{pmatrix}\alpha_{j}\\\ \beta_{j}\\\
\gamma_{j}\\\ \xi_{j}\end{pmatrix}.$ (23)

As we will see the above-obtained solutions will be utilized to examine
transport processes involving transmission and conductance.

##  III Transport properties

The transfer matrix is calculated using the eigenspinors’ continuity at $2N$
connection points. We recall that the $j$-th elementary cell has one internal
junction at

$\displaystyle x_{2}j=2r_{0}\sin\phi_{j}/2+jD$ (24)

and two extreme junctions at

$\displaystyle x_{1}j=jD,\qquad x_{3}j=(j+1)D).$ (25)

The width of one cell is

$\displaystyle D=d+2r_{0}\sin\phi_{j}/2.$ (26)

The wave functions of the adjoining components are equal at these connection
locations. This fact leads to the $2N$ equations system, which yields

$\displaystyle\begin{array}[]{c}M_{2}D_{in}=M_{1}\left(-\frac{\phi}{2}\right)\left(e^{-\mathrm{i}{m}\frac{\phi}{2}}\right)D_{1,1}\end{array}$
(28) $\displaystyle
M_{1}\left(\frac{\phi}{2}\right)\left(e^{\mathrm{i}{m}\frac{\phi}{2}}\right)D_{1,1}=M_{2}\
\left(e^{-\mathrm{i}{K}x_{1}}\right)D_{2,1}$ (29)
$\displaystyle\begin{array}[]{c}M_{2}\
\left(e^{\mathrm{i}{K}x_{2}}\right)D_{2,1}=M_{1}\left(-\frac{\phi}{2}\right)\left(e^{-\mathrm{i}{m}\frac{\phi}{2}}\right)D_{1,2}\end{array}$
(31)
$\displaystyle~{}~{}~{}~{}\vdots\qquad\qquad\qquad\qquad\vdots\qquad\qquad\qquad\qquad\vdots$
$\displaystyle
M_{2}\left(e^{\mathrm{i}{K}x_{N-1}}\right)D_{2,N-1}=M_{1}\left(-\frac{\phi}{2}\right)\left(e^{-\mathrm{i}{m}\frac{\phi}{2}}\right)D_{1,N}$
(32) $\displaystyle
M_{1}\left(\frac{\phi}{2}\right)\left(e^{\mathrm{i}{m}\frac{\phi}{2}}\right)D_{1,N}=M_{2}D_{out}$
(33)

where the input $D_{{in}}$ and the output $D_{{out}}$ of the superlattice are
given by

$D_{in}=\begin{pmatrix}\alpha\\\ \beta\\\ r_{\uparrow}^{\xi}\\\
r_{\downarrow}^{\xi}\end{pmatrix},\qquad
D_{out}=\begin{pmatrix}t_{\uparrow}^{\xi}\\\ t_{\downarrow}^{\xi}\\\ 0\\\
0\end{pmatrix}.$ (34)

We take the parameters $\alpha=0,\beta=1$ for spin down polarization
(${\xi}={\downarrow}$) and $\alpha=1,\beta=0$ for spin up polarization
($\xi=\uparrow$) for displacing the electron flux. Based on the preceding
arguments, we can construct the following link between $D_{in}$ and $D_{out}$:

$D_{in}=M_{2}^{-1}\Omega M_{2}D_{out}=\mathcal{M}D_{out}$ (35)

and the matrix $\Omega$ takes the form

$\Omega=M_{2}\left[C\left(e^{-i{K}d}\right)\right]^{N-1}CM_{2}^{-1}$ (36)

where

$C=M_{2}^{-1}\left[M_{1}\left(-\frac{\phi}{2}\right)\left(e^{-i{m}\phi}\right)M_{1}^{-1}\left(\frac{\phi}{2}\right)\right]M_{2}=\left({C}_{kl}\right)$
(37)

has been defined, with $k,l=1,2,3,4$.

Now by setting

$\mathcal{M}=\left[C\left(e^{-i{K}d}\right)\right]^{N-1}C=\left(\mathcal{M}_{kl}\right)$
(38)

we write (35) as follows

$D_{in}=\mathcal{M}D_{out}$ (39)

After a lengthy and straightforward algebraic calculation, we obtain the
transmission amplitudes for spin-up and spin-down

$\displaystyle
t_{\uparrow}^{\uparrow}=\frac{\mathcal{M}_{22}}{\mathcal{M}_{22}\mathcal{M}_{11}-\mathcal{M}_{21}\mathcal{M}_{12}}$
(40) $\displaystyle
t_{\downarrow}^{\downarrow}=\frac{\mathcal{M}_{11}}{\mathcal{M}_{22}\mathcal{M}_{11}-\mathcal{M}_{21}\mathcal{M}_{12}}$
(41) $\displaystyle
t_{\downarrow}^{\uparrow}=-\frac{\mathcal{M}_{21}}{\mathcal{M}_{22}\mathcal{M}_{11}-\mathcal{M}_{21}\mathcal{M}_{12}}$
(42) $\displaystyle
t_{\uparrow}^{\downarrow}=-\frac{\mathcal{M}_{12}}{\mathcal{M}_{22}\mathcal{M}_{11}-\mathcal{M}_{21}\mathcal{M}_{12}}.$
(43)

Because the input and output wave vectors are the same, the corresponding
transmission and reflection may be expressed as

$\displaystyle T_{\uparrow}^{\uparrow}=|t_{\uparrow}^{\uparrow}|^{2},\qquad$
$\displaystyle T_{\downarrow}^{\downarrow}=|t_{\downarrow}^{\downarrow}|^{2}$
(44) $\displaystyle
T_{\uparrow}^{\downarrow}=|t_{\uparrow}^{\downarrow}|^{2},\qquad$
$\displaystyle T_{\downarrow}^{\uparrow}=|t_{\downarrow}^{\uparrow}|^{2}.$
(45) $\displaystyle
R_{\uparrow}^{\uparrow}=|r_{\uparrow}^{\uparrow}|^{2},\qquad$ $\displaystyle
R_{\downarrow}^{\downarrow}=|r_{\downarrow}^{\downarrow}|^{2}$ (46)
$\displaystyle
R_{\uparrow}^{\downarrow}=|r_{\uparrow}^{\downarrow}|^{2},\qquad$
$\displaystyle R_{\downarrow}^{\uparrow}=|r_{\downarrow}^{\uparrow}|^{2}.$
(47)

We may easily verify that the probability conservation criterion

$\displaystyle\sum_{p,q=\xi}\left[\left(T_{p}^{q}\right)^{2}+\left(R_{p}^{q}\right)^{2}\right]=1.$
(48)

At zero temperature, the Landauer-Buttiker formula [18, 19, 20] can be used to
calculate the conductance of our system. It was demonstrated that the
conductance of a single multiplying channel can undoubtedly be summarized into
any number of incoming and outgoing channels [21, 22, 23]. There are various
conductances arising from the transmittances between states of definite
momentum and spin projection at the contacts [24]. Then, the total conductance
in the linear response system is given by adding up all the transmission
channels

$\displaystyle
G=\frac{e^{2}}{h}\left(T_{\uparrow}^{\uparrow}+T_{\downarrow}^{\downarrow}+T_{\uparrow}^{\downarrow}+T_{\downarrow}^{\uparrow}\right)=\frac{e^{2}}{h}\sum_{p,q=\xi}T_{p}^{q}.$
(49)

We now attempt to compute the transmission and conductance in various energy
domains numerically after getting closed form formulas. This will help us
better understand how different physical parameters affect the transmission
and conductance of rippled graphene superlattice.

##  IV Numerical results

At $k_{y}=0$, we investigate the tunneling properties of our superlattice
using appropriate values for the relevant parameters
($r_{0},\Delta,\phi,d,N$). For some values of the gap with $\phi=\pi,r_{0}=10\
\text{\AA{}},N=2$, $d=16\ \text{\AA{}}$ (up), $d=100\ \text{\AA{}}$ (bottom),
Fig. 2 illustrates the transmissions $T_{\uparrow}^{\uparrow}$ (green) and
$T_{\downarrow}^{\downarrow}$ (blue dashed) with the same spin as a function
of energy $E$. As a preliminary consequence, we see that by increasing the
gap, $T_{\uparrow}^{\uparrow}$ and $T_{\downarrow}^{\downarrow}$ decrease.
Now, in the upper panels, we choose the distance between the arcs in the
structure $d=16\ \text{\AA{}}$, and for $\Delta=0$, we see that the
$T_{\uparrow}^{\uparrow}$ and $T_{\downarrow}^{\downarrow}$ behavior varies
slowly close to the unit, as shown in Fig. 2a. When we choose $\Delta=0.05$ in
Fig. 2b, we see that it decreases $T_{\uparrow}^{\uparrow}$ but has no effect
on $T_{\downarrow}^{\downarrow}$. According to Fig. 2c, when $\Delta=0.1$,
$T_{\uparrow}^{\uparrow}$ decreases to stabilize at a minimum and
$T_{\downarrow}^{\downarrow}$ decreases as well, but by oscillating. The
bottom panels are the same as before, with the exception that the distance
$d=100\ \text{\AA{}}$ has been chosen. As a result, increasing $\Delta$ causes
both $T_{\uparrow}^{\uparrow}$ and $T_{\downarrow}^{\downarrow}$ to drop, as
seen Figs. (2d, 2e, 2f). The number of oscillations in the comparing graphs
increases as the distance between the arcs in our superlattice grows, while
the variation interval remains constant. The efficiency of the spin filter is
negligible for the parameters examined with $\Delta=0$, and the transmission
fluctuates about unity, as indicated [14]. In contrast, the insertion of the
gap $\Delta$ has an effect on transmission by lowering its maximum value and
increasing filtering efficiency, which was not discovered in the previous
study.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: (color online) Transmission $T_{\uparrow}^{\uparrow}$ (green) and
$T_{\downarrow}^{\downarrow}$ (blue dashed) at $k_{y}=0$ as a function of
energy $E$ for three values of the gap (a,d): $\Delta=0$, (b,e):
$\Delta=0.05$, (c,f): $\Delta=0.1$. We chose $\phi=\pi$, $r_{0}=10\
\text{\AA{}}$, (top panel): $d=16\ \text{\AA{}}$, (bottom panel): $d=100\
\text{\AA{}}$ and $N=2$.

Fig. 3 shows $T_{\uparrow}^{\uparrow}$ (green) and
$T_{\downarrow}^{\downarrow}$ (blue dashed) with same spin as a function of
the energy $E$ but now by increasing the cell number to $N=200$. We notice
that the transmissions completely change their behaviors by showing different
peaks. When we choose $\Delta=0$ in Fig. 3a, we see that
$T_{\downarrow}^{\downarrow}$ is strongly suppressed in comparison to
$T_{\uparrow}^{\uparrow}$. The behavior of both transmissions varies and
decreases for $\Delta=0.05$, as shown in Fig. 2b.
$T_{\downarrow}^{\downarrow}$ suppresses almost identically to
$T_{\uparrow}^{\uparrow}$, but with more peaks and a narrower interval between
peaks. In Fig. 3c, for $\Delta=0.1$, we see that $T_{\downarrow}^{\downarrow}$
is suppressed nearly as much as $T_{\uparrow}^{\uparrow}$, but with more
peaks. The interval between peaks becomes a bit wider as well as some shifts
in the transmissions at the origin are observed. The number of peaks grows as
$d$ is increased, as shown in Fig. (3d, 3e, 3f), and the space between the
peaks becomes quite small. Furthermore, notice how the increase in $\Delta$
results in the formation of additional peaks with extremely distinct spacing.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: (color online) The same as in Fig. 2, but now for $N=200$ cells.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: (color online) The same as in Fig. 3, but now for $\phi=\pi/2$.

Fig. 4 depicts $T_{\uparrow}^{\uparrow}$ (green) and
$T_{\downarrow}^{\downarrow}$ (blue dashed) with the same spin as a function
of energy $E$ for three values of gap $\Delta$, with $\phi=\frac{\pi}{2}$,
$d=16$ Å (top panel), $d=100$ Å (bottom panel), $r_{0}=10\ \text{\AA{}}$ and
$N=200$. For $\Delta=0$, the suppression of the transmission is reached for
two peaks as clearly seen in Fig. 4a. If $\Delta$ increases, however, Figs.
(4b, 4c) show the same behavior as before, with the exception that there are
numerous peaks with shifts. Furthermore, it is demonstrated that the
transmission does not achieve its minimal value as previously stated. We can
see that the transmission is canceled at multiple peaks, as illustrated in
Fig. 4d. However, if we increase $\Delta$, as shown in Figs. (4e, 4f), the
maximum value of the transmission decreases, new peaks form, but the minimum
value does not reach zero, in contrast to the case where $\phi=\pi$ was
mentioned above. The angle $\phi=\pi/2$ does not allow the total suppression
of transmission but its reduction. Similarly if we take $d=100$ Å, except that
the interval between the peaks becomes narrow Fig. (4d, 4e, 4f). As a result,
we can see that when the angle of ripple is reduced, the transmission
suppression is not complete, but the effect of $\Delta$ remains the same. We
find that $\phi$ can be utilized as a tunable to modify transmission, among
other physical parameters.

(a)

(b)

(c)

Figure 5: (color online) Transmissions of spin inversion
$T_{\downarrow}^{\uparrow}$ (green) and $T_{\uparrow}^{\downarrow}$ (blue
dashed) at $k_{y}=0$ as a function of the energy $E$ for various values of the
gap (a): $\Delta=0$, (b): $\Delta=0.1$, (c): $\Delta=0.2$ with $\phi=\pi$,
$r_{0}=10\ \text{\AA{}}$, $d=100\ \text{\AA{}}$, $N=200$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: (color online) Conductance at $k_{y}=0$ as a function of the energy
$E$ with the same parameters as Fig. 3.

We show the transmissions of spin inversion $T^{\uparrow}_{\downarrow}$
(green) and $T^{\downarrow}_{\uparrow}$ (dashed blue) as a function of the
energy $E$ for three different gap values $\Delta$ in Fig. 5 with $\phi=\pi$,
$r_{0}=10\ \text{\AA{}}$, $d=100\ \text{\AA{}}$, $N=200$. We see that the two
transmissions always behave in the same way, resulting in the relationship
$T_{\uparrow}^{\downarrow}=T_{\downarrow}^{\uparrow}$. In Fig. 5a, it is
evident that the two transmissions are null for $\Delta=0$, as shown in [15].
However, for $\Delta\neq 0$, it is clear that the transmission increases
progressively by increasing the number of peaks as $\Delta$ increases, as
shown in Fig. 5b, which shows the transmission behavior as peaks up and peaks
down. The peaks down reach the zero value, but the peaks up take the maximum
values. Surprisingly, for $\Delta=0.2$ eV, the transmission behaves radically
differently. As seen in Fig. 5c, the transmission corresponding to peaks up is
inhibited, and the suppression becomes more powerful. As a result, with the
existence of $\Delta$, we were able to retrieve the transmission of spin
inversion and control and suppress it, which was not possible in the previous
study [14, 13, 15]. Consequently, we observe that the introduction of $\Delta$
greatly changes the behavior of all transmission channels and can then be used
as a controllable parameter to tune the tunneling properties.

We show the conductance as a function of the energy $E$ in Fig. 6 for
different gap values $\Delta$ with $\phi=\pi$, $r_{0}=10\ \text{\AA{}}$,
$N=200$ and $d=16\ \text{\AA{}}$ (top panel), $d=100\ \text{\AA{}}$ (bottom
panel). Our results show that the behavior of the conductance in the form of
peaks disappears along the energy line and stabilizes. For $\Delta=0$ in Fig.
6a, the conductance reaches the value null for $E=0$, with the appearance of
other peaks which disappear. We observe that only the transmission of the same
spin dominates the conductance. Increasing $\Delta$ induces a displacement of
the conductance at the level of the origin, as well as the emergence of other
peaks, as seen in Fig. (6b, 6c). In contrast to the case where $\Delta=0$, the
cancellation of the conductance is stronger at a certain value of $E\neq 0$,
which is due to the contribution of spin inversion transmissions. The same
holds true for $d=100\ \text{\AA{}}$, as shown in Fig. (6d, 6e, 6f), except
that the number of peaks multiplies and the interval between them becomes
narrow. As a result, we conclude that the gap $\Delta$ has a significant
impact on the conductance of the rippled graphene superlattice.

##  V Conclusion

The transmission and conduction of rippled graphene superlatice with a gap
were studied. Our system was made of $N$ cells, each of which contained a
ripple and a flat sheet. In our scenario, a ripple is a curved graphene
surface with a radius of $r_{0}$ and a mass term of $\Delta$ in the shape of
an arc of a circle. The energy bands are obtained by solving the Dirac
equation in each region. As a result, we demonstrated that adding a band gap
$\Delta$ to our system appears to impact electron dispersion. To calculate all
transmissions and reflection channels, the energy spectrum is used with the
transfer matrix method.

Our numerical results reveal and confirm that the discovery of electron
transmission with opposing spin polarization is a crucial effect of $\Delta$.
This was not the case in previous $\Delta=0$ investigations [13, 14].
Furthermore, an increase in $\Delta$ causes a decrease and suppression of the
transmissions $T_{\uparrow}^{\uparrow}$ and $T_{\downarrow}^{\downarrow}$,
with the appearance of several peaks where the suppression is stronger. The
transmission behavior shifts clearly at the origin of the energy $E=0$ as well
as the interval between peaks. On the other hand, the spin inversion
transmissions of electrons increased and suppressed with a higher number of
peaks. The distance between peaks is narrowing, and peaks are also appearing
where transmission is at its highest. Our findings reveal that the conductance
is displaced at the origin level as well as the formation of other peaks. Most
notably, the add of a band gap allows transmission and conductance to be
controlled. Finally, our results can be reproduced experimentally based on the
experiment [25] proving that electron spin and orbital motion are related in
nanotubes, which may provide a solution for systems that meet technological
requirements.

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