Electronic Structure of Graphene with two Strains and Double Barrier

TL;DR

This paper investigates how strain affects the electronic properties of graphene with double barriers, revealing strain-induced phenomena like fermion beam collimation, surface states, and confinement, with detailed transmission and conductance analysis.

Contribution

It demonstrates that strain can be used to control electronic transport and surface states in graphene with double barriers, providing new insights into strain engineering.

Findings

Strain induces fermion beam collimation in graphene.

Surface states and confinement are tunable via strain.

Transmission probability and conductance depend on strain configurations.

Abstract

We study the electronic structure of Dirac fermions scattered by double barrier potential in graphene under strain effect. We show that traction and compression strains can be used to generate fermion beam collimation, 1D channels, surface states and confinement. The corresponding transmission probability and conductance at zero temperature are calculated and their numerical implementations taking into account different configurations of physical parameters enabled us to analyze some features of the system.

Equations

$$t({\mathbf{R}}_i,\mathbf{n})=t+\delta t({\mathbf{R}}_i,\mathbf{n}),t({\mathbf{R}}_i,\mathbf{n})=t-\delta t({\mathbf{R}}_i,\mathbf{n})$$

$$H=v_F\int d\text{r}{\Psi }^{†}\left[\begin{array}{cc}\mathbf{\sigma }\cdot \left(\mathbf{p}-\frac{1}{v_F}\mathcal{A}\right)& 0\\ 0& -\mathbf{\sigma }\cdot \left(\mathbf{p}+\frac{1}{v_F}\mathcal{A}\right)\end{array}\right]\Psi$$

$$\mathcal{A}(\mathbf{r})={\mathcal{A}}_x(\mathbf{r})-i{\mathcal{A}}_y(\mathbf{r})=\sum _{\mathbf{n}}\delta t(\mathbf{r},\mathbf{n})e^{i\mathbf{K}\cdot \mathbf{n}}.$$

$$\delta t_j({\mathbf{R}}_{\mathbf{i}},\mathbf{n})=\delta t_j{\delta }_{\mathbf{n},0}\theta (X_i-x_j)\theta (w-X_i+x_j)$$

$${\mathcal{A}}_j(\mathbf{r})=\delta t_j\theta (x-x_j)\theta (w-x+x_j){\mathbf{u}}_y.$$

$$\displaystyle\bigl{[}-i(\partial_{x}-\mathcal{A}_{yj}(x))-\partial_{y}-\mathcal{A}_{xj}(y)\bigl{]}\psi^{B}_{j}(x,y)$$

$$\displaystyle\bigl{[}-i(\partial_{x}+\mathcal{A}_{yj}(x))+\partial_{y}-\mathcal{A}_{xj}(y)\bigl{]}\,\psi^{A}_{j}(x,y)$$

$$\displaystyle-i\bigl{[}(\partial_{x}-\delta t_{j})+k_{y}\bigl{]}\varphi^{B}_{j}(x)e^{\textbf{\emph{i}}k_{y}y}$$

$$\displaystyle-i\bigl{[}(\partial_{x}+\delta t_{j})-k_{y}\bigl{]}\,\varphi^{A}_{j}(x)e^{\textbf{\emph{i}}k_{y}y}$$

$${\psi }_j(x,y)={\psi }_j(x)e^{\text{i}{k}_{y}y}=w_j(x)D_j{e}^{\text{i}{k}_{y}y}$$

$$w_{j}(x)=\left(\begin{array}[]{cc}e^{\textbf{\emph{i}}k_{j}x}&e^{-\textbf{\emph{i}}k_{j}x}\\ s_{j}z_{j}e^{\textbf{\emph{i}}k_{j}x}&-s_{j}z_{j}^{-1}e^{-\textbf{\emph{i}}k_{j}x}\\ \end{array}\right),\qquad D_{j}=\left(\begin{array}[]{c}\alpha_{j}\\ \beta_{j}\\ \end{array}\right)$$

$$\epsilon -{\mathbb{V}}_j=s_j\sqrt{(k_{j}d{)}^2+{\left(k_{y}d-\delta {\tau }_j\right)}^2}$$

$$M_{j}=\left(\begin{array}[]{cc}e^{\textbf{\emph{i}}k_{j}j\varrho}&e^{-\textbf{\emph{i}}k_{j}j\varrho}\\ s_{j}z_{j}e^{\textbf{\emph{i}}k_{j}j\varrho}&-s_{j}z_{j}^{-1}e^{-\textbf{\emph{i}}k_{j}j\varrho}\\ \end{array}\right)^{-1}\left(\begin{array}[]{cc}e^{\textbf{\emph{i}}k_{j+1}j\varrho}&e^{-\textbf{\emph{i}}k_{j+1}j\varrho}\\ s_{j+1}z_{j+1}e^{\textbf{\emph{i}}k_{j+1}j\varrho}&-s_{j+1}z_{j+1}^{-1}e^{-\textbf{\emph{i}}k_{j+1}j\varrho}\\ \end{array}\right)$$

$$k_j$$

$$M=M_1{M}_2{M}_3{M}_4=\left(\begin{array}{cc}{M}_{11}& M_{12}\\ M_{11}& M_{12}\end{array}\right)$$

$$D_{1}=\left(\begin{array}[]{c}1\\ r\\ \end{array}\right),\qquad D_{5}=\left(\begin{array}[]{c}t\\ 0\\ \end{array}\right)$$

$$D_1=M{D}_5.$$

$${\psi }_j(x)={\alpha }_j{\psi }_j^{+}(x)+{\beta }_j{\psi }_j^{-}(x)$$

$$\displaystyle\psi_{j}^{+}(x)=\left(\begin{array}[]{c}1\\ s_{j}z_{j}\\ \end{array}\right)e^{\textbf{\emph{i}}k_{j}x},\qquad\psi_{j}^{-}(x)=\left(\begin{array}[]{c}1\\ -s_{j}z_{j}^{-1}\\ \end{array}\right)e^{-\textbf{\emph{i}}k_{j}x}$$

$$\sin {\theta }_j=\frac{1}{|\epsilon -{\mathbb{V}}_j|}\left(\epsilon \sin \theta -\delta {\tau }_j\right)$$

$$J_x^{\mathsf{in}}={{\psi }_1^{+}(x)}^{†}{\sigma }_x{\psi }_1^{+}(x)=2s_1\frac{k_1}{\sqrt{k_1^2+k_y^2}}$$

$$J_x^{\mathsf{re}}={{\psi }_1^{-}(x)}^{†}{\sigma }_x{\psi }_1^{-}(x)=2s_1|r{|}^2\frac{k_1}{\sqrt{k_1^2+k_y^2}}$$

$$J_x^{\mathsf{tr}}={{\psi }_5^{+}(x)}^{†}{\sigma }_x{\psi }_5^{+}(x)=2s_5|t{|}^2\frac{k_5}{\sqrt{k_5^2+k_y^2}}$$

$$T=\frac{\left|J_x^{\mathsf{tr}}\right|}{\left|J_x^{\mathsf{in}}\right|}=\frac{s_5{k}_5}{s_1{k}_1}\frac{\sqrt{k_1^2+k_y^2}}{\sqrt{k_5^2+k_y^2}}|t{|}^2,R=\frac{\left|J_x^{\mathsf{re}}\right|}{\left|J_x^{\mathsf{in}}\right|}=|r{|}^2.$$

$$T=|t{|}^2=\frac{1}{|M_{11}{|}^2}.$$

$$G$$

$$=$$

$$T(\theta ,\delta \tau ,\delta \tau )=T(-\theta ,-\delta \tau ,-\delta \tau )$$

$$T(\theta ,\delta \tau ,-\delta \tau )=T(\theta ,-\delta \tau ,\delta \tau )$$

$$T(\theta ,\pm \delta \tau ,\mp \delta \tau )=T(-\theta ,\pm \delta \tau ,\mp \delta \tau ).$$

$${\epsilon }^2=d^2{k}_{2,4}^2+(d{k}_y+\delta \tau {)}^2,{\epsilon }^2=d^2{k}_{2,4}^2+(d{k}_y-\delta \tau {)}^2.$$

Full text

**Electronic Structure of Graphene with two Strains

and Double Barrier **

El Bouâzzaoui Choubabia, Ahmed Jellala.jellal@ucd.ac.maa,b***, Abdellatif Kamala and Hocine Bahloulib,c

a**Laboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali University,

PO Box 20, 24000 El Jadida, Morocco

b**Saudi Center for Theoretical Physics, Dhahran, Saudi Arabia

*c**Physics Department, King Fahd University of Petroleum $\&$ Minerals,

Dhahran 31261, Saudi Arabia*

We study the electronic structure of Dirac fermions scattered by double barrier potential in graphene under strain effect. We show that traction and compression strains can be used to generate fermion beam collimation, 1D channels, surface states and confinement. The corresponding transmission probability and conductance at zero temperature are calculated and their numerical implementations taking into account different configurations of physical parameters enabled us to analyze some features of the system.

PACS numbers: 81.05.ue, 81.07.Ta, 73.22.Pr

Keywords: graphene, strain effect, double barrier, transmission, conductance.

1 Introduction

Recently, investigations of the strain effect on the electronic band structure, optical properties, electronic transport, spin transport and valley transport in graphene nanostructures and devices has attracted much attention from theoretical and experimental sides [1, 2, 3, 4, 5]. These studies were based on generalized Hamiltonian describing the quasiparticles in graphene subject to arbitrary strain effect. In particular, it turned out that the linear elasticity theory applied to graphene is a reasonable approximation for relatively small deformations, since mechanical properties demonstrate that graphene can sustain a linear elasticity regime up to $20\%$ [6, 7, 8].

We intend to study the electronic structure of graphene ribbon deposited on a controllable silicate dioxide substrate. Such ribbon is of infinite length along the propagating $x$-direction of Dirac fermions with armchair boundaries (Figure 1), which is subject to two mechanical strains in two separate regions such that strain results in a contraction in one region and extension in the other. We can achieve this setting by depositing graphene onto substrates that allow the control of the strain effect [3].

In the framework of the tight-binding model, Figure 1 shows the effect of modifying the horizontal inter-carbon bonds in the ribbon due to changes in nearest-neighbor hopping amplitude $t$ ($\approx 3\,\text{eV}$). The space-dependent compression and contraction influence the hopping amplitude through introducing modulations

[TABLE]

where $\delta t(\bm{R}_{i},\bm{n})$ is a perturbation, $\bm{n}$ is the nearest neighbor vectors and $\bm{R}_{i}$ the $i$-th site position. The ribbon becomes subdivided into five different regions that we index according to the positive propagation direction as ①, ②, ③, ④ and ⑤, successively. In addition to this ribbon configuration, we apply three different potentials $V_{2}$, $V_{3}$ and $V_{4}$ in the three regions ②, ③ and ④ of width $d_{2}$, $d_{3}$ and $d_{4}$, respectively. The input and output regions, ① and ⑤, are made of pristine graphene and the potential is zero. The strained graphene ribbon and the profile potential constitute a double barrier of width $d=d_{2}+d_{3}+d_{4}$ as indicated in Figure 1.

The present paper is organized as follows. In section 2, we set the theoretical model and solve the eigenvalue equation to obtain the solutions of the energy spectrum. We use the boundary conditions together with the continuity equations to determine the transmission and reflection probabilities as well as the conductance in section 3. We numerically analyze the main results and present different discussions in section 4. We conclude our work in the final section.

2 Theoretical model

In strained graphene, near the valleys $K$ and $K^{\prime}$ of the first Brillouin zone, the modification of hopping energies between different sites, through smooth perturbations, is governed by the following low energy Hamiltonian [9]

[TABLE]

with the momentum operator $\bm{p}=(p_{x},p_{y})$, the Fermi velocity $v_{F}=3ta/2\hbar\approx 10^{6}m/s$, the Pauli matrices $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$, the gauge field $\bm{\mathcal{A}}$, acting on the electrons dynamics described by a Dirac equation, and the spinor $\Psi=\left[\psi^{A}_{K}(\bm{r}),\,\psi^{B}_{K}(\bm{r}),\,\psi^{B}_{K^{\prime}}(\bm{r}),\,\psi^{A}_{K^{\prime}}(\bm{r})\right]^{\dagger}$. The gauge field comes from the perturbation of homogeneous amplitude jump $\delta t(\bm{R},\bm{n})$ and they are related via

[TABLE]

In the double barrier system, regions ② and ④ have the same width $\varrho=d_{2}=d_{3}=d_{4}$, the perturbations $\delta t_{j}$ of horizontal hopping are constant, the index $j$ labels the different regions and runs from $1$ to $5$. Such perturbations and the associated \bm{\mathcal{A}}_{j}$$(\bm{r}) can be written as

[TABLE]

Note that the unit vector $\bm{u}_{y}$ is collinear to the gauge field $\bm{\mathcal{A}}_{j}$ and is perpendicular to the propagating $\bm{u}_{x}$-direction. We can generalize the Hamiltonian to describe all regions composing our system with different couplings. Then adopting the unit system $v_{F}=\hbar=1$ and allowing for the presence of an electrostatic potential $V_{j}(x,y)$ in the barrier region [3], the wave equations for the $K$ valley can then be cast into

[TABLE]

Taking into account the potential profile, translation invariance in the $y$-direction and using (4), we write (5) as

[TABLE]

with $\psi^{A/B}_{j}(x,y)=\varphi^{A/B}_{j}(x)e^{\textbf{\emph{i}}k_{y}y}$. It is convenient to introduce in $j$-th region the dimensionless quantities $\mathbb{V}_{j}=V_{j}/E_{F}$, $\varepsilon_{j}=E_{j}/E_{F}$, $\delta\tau_{j}=\delta t_{j}/E_{F}$ with $E_{F}=\hbar v_{F}/d$. Thus, the solutions in each $j$-region are given by

[TABLE]

where the two matrices read as

[TABLE]

$\psi_{j}(x)$ is a spinor of components $\varphi^{A}_{j}(x)$ and $\varphi^{B}_{j}(x)$, $\alpha_{j}$ and $\beta_{j}$ being the amplitudes of positive and negative propagation wave functions inside the $j$-th region, respectively. We show that the associated eigenvalues are

[TABLE]

where $s_{j}=\text{sign}(\varepsilon-\mathbb{V}_{j})$ is the usual sign function. It is clearly seen that in the energy spectrum of the $j$-th strained region (9), the component $k_{y}d$ of the wave vector is shifted by $\delta\tau_{j}$ compared to that of the pristine graphene. This behavior was already encountered in our previous work [10] where we had a magnetic field that also shifted the wave vector $k_{y}$ by $\frac{d}{l_{B}^{2}}$, $l_{B}=1/\sqrt{B_{0}}$ is the magnetic length and $B_{0}$ is the strength of the magnetic field. Hence from this behavior we can say that the deformation (strain effect) behaves like an effective magnetic field, and the two shifts play the role of a mass term.

3 Transmission and conductance

To determine the transmission probability, we use the boundary conditions applied successively at the interfaces along the $x$-direction and evaluate the current densities in the incident, reflected and transmitted regions. Indeed, at interfaces $x_{j}=j\varrho$, we obtain the transfer matrix $D_{j}=M_{j}D_{j+1}$ with the matrix

[TABLE]

and we have

[TABLE]

with $d=3\varrho$, $s_{1}=s_{5}$, $k_{1}=k_{5}$, $z_{1}=z_{5}$, $\delta\tau_{j}=(0,\delta\tau_{2},0,\delta\tau_{4},0)$ and $\mathbb{V}_{j}=(0,\mathbb{V}_{2},0,\mathbb{V}_{4},0)$. After some lengthy algebras, we obtain the transfer matrix

[TABLE]

Since the incident and reflected amplitudes are defined by

[TABLE]

then the wavefunctions in the input and output regions are connected by the matrix $M$ as

[TABLE]

On the other hand, the eigenspinors $\psi_{j}(x)$, appearing in (7), can be decomposed as

[TABLE]

where the two components are given by

[TABLE]

which are the positive and negative propagating spinors associated to their amplitudes $\alpha_{j}$ and $\beta_{j}$, respectively. In input region, conservation of the mode $k_{y}$ gives the relation

[TABLE]

where $\theta_{j}$ is the angle of propagation of Dirac fermions within $j$-region, $\theta=\theta_{1}$ is the incident angle and $\varepsilon$ is the incident energy. From (21) we notice that the strain directly affects the transmission angle of Dirac fermions [11].

At this stage, we can introduce the current density corresponding to our system. Indeed, we show that the incident, reflected and transmitted current densities can be written as

[TABLE]

giving rise to the transmission and reflection probabilities

[TABLE]

Due to the symmetry of the double barrier configuration in the input and output regions, we have the relations $s_{1}=s_{5}$ and $k_{1}=k_{5}$. Thus, using (14) to obtain

[TABLE]

Since the transmission probability $T$ is determined for each mode $k_{y}$, then we derive the conductance $G$ at zero temperature. Indeed, using the definition [12] we find the conductance in the unit systems

[TABLE]

where $L_{y}$ is the sample size along the $y$-direction and $G_{0}=\frac{e^{2}\varepsilon L_{y}}{\pi^{2}d}$ is the unit conductance, $d$ is the total barrier width. These results will numerically be investigated to analyze the system behavior and underline its basic features.

4 Results and discussions

In the beginning we analyze only the double strain effects on the graphene systems, which means that we keep strain parameter in two regions (②, ④) and forget about the double barrier potential $\mathbb{V}_{2}=\mathbb{V}_{4}=0$. Note that, the conservation imposes $k_{1}=k_{3}=k_{5}$, $s_{1}=s_{3}=s_{5}$, which will be considered in the forthcoming analysis.

In Figure 2, we present density plots of transmission probability as function of incident energy $\varepsilon$ and incident angle $\theta$. We show that the white zones that are limited in Figure 2LABEL:sub@fig01:SubFigA by ${\varepsilon=\frac{\delta\tau}{1-\sin\theta}}$ (orange dashed line), in Figure 2LABEL:sub@fig01:SubFigB by ${\varepsilon=\frac{\delta\tau}{1-\sin\theta}}$ and ${\varepsilon=\frac{\delta\tau}{1+\sin\theta}}$ (green dashed line), and in Figure 2LABEL:sub@fig01:SubFigC by ${\varepsilon=\frac{\delta\tau}{1+\sin\theta}}$ represent forbidden zones (FZs). Otherwise, the FZs are determined by angles ${\theta=\arcsin\left(\frac{\pm(\varepsilon-\delta\tau)}{\varepsilon}\right)}$, which give rise to a phenomenon resembling beam collimation in optics. These relations can be obtained either from Figure 3 (see below) or directly from the conservation of the mode $k_{y}$ (21). We notice that the tunneling effect is always completely suppressed: in Figures (2LABEL:sub@fig01:SubFigA, 2LABEL:sub@fig01:SubFigC) if $\varepsilon<\frac{\delta\tau}{2}$ (dashed line) which was found in the case of a single barrier [3], and in Figure 2LABEL:sub@fig01:SubFigB if $\varepsilon<\delta\tau$ (red dashed line) corresponding to the two collimations generated mutually by contraction and compression strains. Figure 2 tells us that the Klein paradox does not always exist at normal incidence angle $\theta=0$. Furthermore, the transmission is not symmetrical with respect to the normal incidence angle in Figures (2LABEL:sub@fig01:SubFigA, 2LABEL:sub@fig01:SubFigC), but it restores its symmetry in Figure 2LABEL:sub@fig01:SubFigB. Note that in Figure 2 the dashed yellow line is a symmetry curve of transmission. We observe that in Figure 2LABEL:sub@fig01:SubFigB there is a perfect symmetry, on the other in Figures (2LABEL:sub@fig01:SubFigA, 2LABEL:sub@fig01:SubFigC) there is absence of symmetry. In Figures 2, purple zones represent transmission gaps ($T=0$) and red zones represent total transmission ($T=1$) i.e. Klein tunneling effect. Inspecting the last Figures, exposes the following symmetry relations

[TABLE]

Collimation is due to filtering effect at certain incidence angles, Figure 3 shows this effect which is best understood by inspecting the phase-spaces [3]. The Fermi surface of pristine graphene is highlighted in yellow and has a dispersion given by $\varepsilon^{2}=d^{2}(k_{1}^{2}+k_{y}^{2})$. The strained graphene, on the other hand, exposes a green and orange Fermi surface for compression and contraction strain whose equations are, respectively,

[TABLE]

Conservation of energy and momentum $k_{y}$ immediately leads to a sector of allowed incident angles (red surfaces) whose openings are limited by the red arrows in Figure 3 and corresponding exactly to the allowed incident angles shown in Figure 2. If $\delta\tau$ exceeds $2\varepsilon$, the red surface will be omitted and therefore tunneling in strained double barrier is suppressed. Figures (3LABEL:sub@fig02:SubFigA, 3LABEL:sub@fig02:SubFigC) show that the two red surfaces are symmetric with respect to the normal incidence angle but in Figure 3LABEL:sub@fig02:SubFigB the red surface is symmetric with respect to the normal incidence angle. By combining Figures (2, 3) we can explain easily the Dirac fermions like collimation in strained double barrier.

Considering the symmetry between $T(\theta,\delta\tau,\delta\tau)$ and $T(\theta,-\delta\tau,-\delta\tau)$ with respect to $\theta=0$, we will only study ($T(\theta,\delta\tau,\delta\tau)$, $T(\theta,\delta\tau,0)$) and deduce the other symmetrical cases. In Figure 4, we evaluate the transmission difference between a strained single barrier with ($\delta\tau_{2}=\delta\tau,\delta\tau_{4}=0$) and the strained double barrier with ($\delta\tau_{2}=\delta\tau_{4}=\delta\tau$), the two Figures (4LABEL:sub@fig03:SubFigA, 4LABEL:sub@fig03:SubFigC) have the same white forbidden zone and therefore the same allowed incident angles. The double barrier introduces a radical change as compared to in single barrier transmission. In Figure 4LABEL:sub@fig03:SubFigA the single barrier is not symmetrical (no yellow curve), it possess separate total energy bands (red color) having parabolic form that start with peaks at grazing incidence $\theta=\frac{\pi}{2}$ for $\delta\tau$ ($\theta=-\frac{\pi}{2}$ for $-\delta\tau$), these peaks are intercalated by transmission gaps (purple zones). The first transmission gap starts its location between energy $\frac{\delta\tau}{2}$ (blue dashed line) and first peak, it coats the transmission density plot on the green dashed line side ${\varepsilon=\frac{\delta\tau}{1+\sin\theta}}$ and becomes thin when the energy increases. Transmission gaps between peaks decreases in width (compared to $\theta$) and increases in height (compared to $\varepsilon$) when the energy increases. The double barrier effect is explained in Figure 4LABEL:sub@fig03:SubFigC, the density plot has a symmetry deformed with respect to the yellow line. Indeed the total transmission bands (red zones) become fragmented and contain islands of different transmission values, transmission gaps intercalated between the peaks (at the grazing angle) are duplicated and asymmetric with respect to the yellow line, and the peaks are duplicated and asymmetric as well. We observe that the energy $\varepsilon=\frac{\delta\tau}{2}$ plays the role of mass term whose lower energy is forbidden. In the presence of a mass term comes the first transmission gap followed by the first peak.

To show difference between peaks behavior of the transmissions in strained single and double barriers at the grazing incidence angle $\theta=\frac{\pi}{2}$ for $\delta\tau$ ($\theta=-\frac{\pi}{2}$ for $-\delta\tau$), we present Figure 5 with the same conditions as in Figure 4. In Figure 5LABEL:sub@fig04:SubFigA, the transmission probability $T$ versus incident energy $\varepsilon$ at grazing incidence $\theta$ contains a series of peaks intercalated with transmission gaps that increase as $\varepsilon$ increases. In Figure 5LABEL:sub@fig04:SubFigC, $T$ behaves like the one in the Figure 5LABEL:sub@fig04:SubFigA but with double resonance peaks, the separation between peak doublet increases also with energy. Such resonance peaks can easily be explained by the doubling of the resonator model of Fabry-Perot in graphene [13, 14, 15]. The two barrier types (single and double) have no effect on the width of the transmission gaps between transmission peaks. Figure 5 shows also the forbidden transmission zone between the zero energy $\varepsilon=0$ and the mass-like term $\frac{\delta\tau}{2}$.

Figure 6 shows the transmission density plot versus incident energy $\varepsilon$ and deformation $\delta\tau$ where white zones correspond to forbidden transmission, purple zones correspond to transmission gaps and red zones correspond to total transmission. The allowed transmission corresponds to the energies ${\varepsilon\geq\frac{\delta\tau}{1-\sin\theta}}$. In Figures (6LABEL:sub@fig05:SubFigA, 6LABEL:sub@fig05:SubFigB), the single and double barrier transmissions are illustrated respectively at normal incidence $\theta=0$, we observe that near $\delta\tau=0$ (i.e. the pristine graphene is slightly deformed) always there is the effect of Klein paradox (red color). As long as the deformation increases, the different transmission values appear and we end up with transmission gaps just before the energy ${\varepsilon=\frac{\delta\tau}{1-\sin\theta}}$. The contribution of the double barrier compared to the single one is marked by the appearance of the islands of transmissions, which are surrounded by total transmission zones and multiplication of transmission gap zones on the sides of ${\varepsilon=\frac{\delta\tau}{1-\sin\theta}}$. The number of these islands and transmission gaps increase as long as energy increases. Figure 6LABEL:sub@fig05:SubFigC $(\theta=\frac{\pi}{4})$ shows that if the angle of incidence increases the forbidden zone increases, islands and transmission gaps decrease, which telling us the system behaves like a slightly strained pristine graphene.

Beside the strained graphene, a symmetrical potential profile $(\mathbb{V}_{2},\mathbb{V}_{4})=(\mathbb{V},\mathbb{V})$ has been applied to study its additional effect compounded to that of strain. Figure 7 shows the transmission density plot versus incident energy $\varepsilon$ and incident angle $\theta$. The allowed transmissions correspond, respectively to the energy zones: $[\varepsilon\geq(\varepsilon^{+}_{-}=\varepsilon^{-}_{-})\wedge\varepsilon\geq(\varepsilon^{+}_{-}=\varepsilon^{-}_{+})]\vee[\varepsilon\leq(\varepsilon^{+}_{-}=\varepsilon^{-}_{-})\wedge\varepsilon\leq(\varepsilon^{+}_{-}=\varepsilon^{-}_{+})]$ Figure 7LABEL:sub@fig06:SubFigA, $[\varepsilon\geq\varepsilon^{-}_{-}\wedge\varepsilon\geq\varepsilon^{+}_{+}]\vee[\varepsilon\leq\varepsilon^{-}_{-}\wedge\varepsilon\leq\varepsilon^{+}_{+}]$ Figure 7LABEL:sub@fig06:SubFigB, $[\varepsilon\geq\varepsilon^{-}_{+}\wedge\varepsilon\geq\varepsilon^{+}_{-}]\vee[\varepsilon\leq\varepsilon^{-}_{+}\wedge\varepsilon\leq\varepsilon^{+}_{-}]$ Figure 7LABEL:sub@fig06:SubFigC, $[\varepsilon\geq\varepsilon^{+}_{+}\wedge\varepsilon\geq\varepsilon^{+}_{-}]\vee[\varepsilon\leq\varepsilon^{-}_{+}\wedge\varepsilon\leq\varepsilon^{-}_{-}]$ Figure 7LABEL:sub@fig06:SubFigD, $[\varepsilon\geq\varepsilon^{+}_{+}]$ Figure 7LABEL:sub@fig06:SubFigE, and $[\varepsilon\geq\varepsilon^{+}_{+}]$ Figure 7LABEL:sub@fig06:SubFigF. The forbidden transmission zones (white color) correspond respectively to the complementary allowed transmission zones in each systems as illustrated in Figure 7, with ${\varepsilon^{\pm}_{+}=\frac{\mathbb{V}\pm\delta\tau}{1+\sin\theta}}$ and ${\varepsilon^{\pm}_{-}=\frac{\mathbb{V}\pm\delta\tau}{1-\sin\theta}}$. Figure 7LABEL:sub@fig06:SubFigA shows the transmission probability $T(\varepsilon,\theta)$ for a strainless double barrier with potential ($\mathbb{V}_{2}=\mathbb{V}$), the energy spectrum is subdivided into three energy domains. The first domain is $0\leq\varepsilon\leq\frac{\mathbb{V}}{2}$ where the transmission is allowed for all incidence angles $-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$, the double barrier behaves as a more refractive medium than pristine graphene. The second domain is $\frac{\mathbb{V}}{2}\leq\varepsilon\leq\mathbb{V}$, which shows that at each energy there are two critical angles symmetrical with respect to the normal incidence, they come closer when the energy tends to $\mathbb{V}$. The third domain corresponds to $\varepsilon\geq\mathbb{V}$, when the energy increases the critical angles remain always symmetrical but they move away parabolically. In the two last domains, the double barrier behaves like a less refractive medium than pristine graphene. In vicinity of normal incidence, we have a total transmission (red color) whatever the propagating energy of Dirac fermions (Klein Paradox). The transmission probability is symmetrical with respect to normal incidence angle $\theta=0$, it vanishes when approaching the limit angles to give transmission gaps (purple color) intercalated between resonance peaks. The transmission, in Figure 7LABEL:sub@fig06:SubFigA, has the same form as that obtained by the transmission through a potential barrier in monolayer graphene studied in [16]. In Figures (7LABEL:sub@fig06:SubFigB, 7LABEL:sub@fig06:SubFigC), we illustrate the double barrier, which becomes strained by adding $(\delta\tau_{2}=\delta\tau_{4}=\delta\tau=7\pi,\delta\tau<\mathbb{V})$ and $(\delta\tau_{2}=\delta\tau_{4}=\delta\tau=-7\pi,\delta\tau<\mathbb{V})$, respectively. With respect to normal incidence, the two transmissions in these last two Figures are symmetrical to each other, the transmission lose its symmetry (as compared to Figure 7LABEL:sub@fig06:SubFigA) with respect to the normal incidence. The transmission has an additional domain compared to that of strainless double barrier, which just appeared between the first and the second zones. In this domain, the transmission in 7LABEL:sub@fig06:SubFigB (7LABEL:sub@fig06:SubFigC) is allowed in the positive (negative) incidence $\theta>0$ ($\theta<0$) and limited by critical angles in the negative (positive) incidence $\theta<0$ ($\theta>0$). In Figures (7LABEL:sub@fig06:SubFigE, 7LABEL:sub@fig06:SubFigF) plotted for $(\delta\tau_{2}=\delta\tau_{4}=\delta\tau=12\pi,\delta\tau>\mathbb{V})$ and $(\delta\tau_{2}=\delta\tau_{4}=\delta\tau=-12\pi,\delta\tau>\mathbb{V})$, respectively, there remains only one transmission domain for energies $\varepsilon>\frac{\mathbb{V}+\delta\tau}{2}$. The system behaves as a strained double barrier, with no potential profile, with a like compression strain $\delta\tau^{\prime}=\mathbb{V}+\delta\tau$. Figure 7LABEL:sub@fig06:SubFigD for a double barrier composed by compression and traction ($\delta\tau_{2}=-\delta\tau_{4}=\delta\tau=\pm 7\pi$) with a predicted potential $(\mathbb{V}_{2},\mathbb{V}_{4})=(\mathbb{V},\mathbb{V})$, shows an energy gap between $\varepsilon=\mathbb{V}+\delta\tau$ and $\varepsilon=\mathbb{V}-\delta\tau$ of width $2\delta\tau$. For energy $0\leq\varepsilon\leq\frac{\mathbb{V}-\delta\tau}{2}$, the double barrier behaves like a more refractive medium than pristine graphene. On the other hand, for the energy $\frac{\mathbb{V}-\delta\tau}{2}\leq\varepsilon\leq\mathbb{V}-\delta\tau$, it behaves like a less refractive medium and the critical angles are symmetric with respect to $\theta=0$. The critical angles for energy $\varepsilon\geq\mathbb{V}+\delta\tau$ are also symmetrical and increase as a function of energy in a parabolic way.

By adding the potential to strained graphene (Figure 8), the Fermi surfaces of each region ② and ④ change their radii and their equations become respectively $(\varepsilon-\mathbb{V})^{2}=d^{2}k_{2}^{2}+(dk_{y}\pm\delta\tau)^{2}$ and $(\varepsilon-\mathbb{V})^{2}=d^{2}k_{4}^{2}+(dk_{y}\pm\delta\tau)^{2}$. Since the radius becomes variable with the potential $\mathbb{V}$, Figure 8 generalizes Figure 3. The collimation realized by filtering of certain angles of incidence [3] for different configurations of the double strained barrier, makes it possible to have the contours delimiting the allowed and forbidden transmission zones (red zones) in Figure 8. Figure 8LABEL:sub@fig07:SubFigA illustrates the behavior of Fermi surfaces for a double barrier without strain, the allowed angles are symmetric with respect to the normal incidence angle and correspond to those of the Figure 7LABEL:sub@fig06:SubFigA. Figure 8LABEL:sub@fig07:SubFigB illustrates the behavior of the Fermi surfaces for a double barrier with two strains of different nature (one of contraction and the other of compression), the allowed angles have also a symmetry with respect to the normal incidence angle and correspond to those of Figure 7LABEL:sub@fig06:SubFigD. Finally Figure 8LABEL:sub@fig06:SubFigC (8LABEL:sub@fig06:SubFigD) shows the Fermi surfaces behavior of a barrier with two strains of the same nature compression (traction). The displacement of the origin of Fermi surface corresponding to the strained zone, omitted the symmetry with respect to the normal incidence angle. But the permitted angles of Figure 8LABEL:sub@fig06:SubFigC (compression strain) are symmetrical compared to the allowed angles of Figure 8LABEL:sub@fig06:SubFigD (traction strain) with respect to the normal angle of incidence.

A geometric manipulation is used to determine easily the limiting angles $\theta^{s}_{s^{\prime}}=\arcsin\left(\frac{\mathbb{V}-\varepsilon+s\delta\tau}{s^{\prime}\varepsilon}\right)$ corresponding to energy contours $\varepsilon^{s}_{s^{\prime}}=\frac{\mathbb{V}+s\delta\tau}{1+s^{\prime}\sin\theta}$ separating different transmission zones in Figure 8, with $s=\pm$ and $s^{\prime}=\pm$. These angles are well illustrated in Figure 9 under suitable conditions. Then we have easily obtained such angles delimiting different transmission zones such as $\varepsilon^{+}_{+}\longrightarrow\theta^{+}_{+}$ (magenta dashed line), $\varepsilon^{+}_{-}\longrightarrow\theta^{+}_{-}$ (brown dashed line), $\varepsilon^{-}_{-}\longrightarrow\theta^{-}_{-}$ (orange dashed line), $\varepsilon^{-}_{+}\longrightarrow\theta^{-}_{+}$ (green dashed line).

At this level, we show that the corresponding conductance is affected by physical parameters and depends on the nature of system. Indeed, Figure 10 shows the conductance as a function of incident energy $\varepsilon$ for different configurations of a strained double barrier which is not subject to any potential, i.e. ($\mathbb{V}_{2}=\mathbb{V}_{4}=0$). In Figures (10LABEL:sub@fig10:SubFigA, 10LABEL:sub@fig10:SubFigB, 10LABEL:sub@fig10:SubFigC), we present the conductance for strained single barrier (($\delta\tau_{2}=\pm\delta\tau,\delta\tau_{3}=\delta\tau_{4}=0$) or ($\delta\tau_{2}=\delta\tau_{3}=0,\delta\tau_{4}=\pm\delta\tau$)), double barrier with the same strain ($\delta\tau_{2}=\delta\tau_{4}=\pm\delta\tau,\delta\tau_{3}=0$) and double barrier with different strains ($\delta\tau_{2}=-\delta\tau_{4}=\pm\delta\tau,\delta\tau_{3}=0$), respectively. We observe that under some physical conditions there are conductances for $\delta\tau=0$ (orange color), $\delta\tau=3\pi$ (red color), $\delta\tau=7\pi$ (black color), $\delta\tau=11\pi$ (blue color) and $\delta\tau=15\pi$ (green color). These Figures show that each conductance starts at the energy $\varepsilon=\delta\tau$ and there is an ordering such that $G/G_{0}(\delta\tau=0)>G/G_{0}(\delta\tau=3\pi)>G/G_{0}(\delta\tau=7\pi)>G/G_{0}(\delta\tau=11\pi)>G/G_{0}(\delta\tau=15\pi)$. Figure 10LABEL:sub@fig10:SubFigD illustrates the comparison between conductances corresponding to strained single barrier (SSB) (blue color) and strained double barriers for both: same strain (SSDB) (red color) and different strains (DSDB) (black color) with $\delta\tau=7\pi$. Then it is clearly seen that the result $G/G_{0}$ (SSB) $>$ $G/G_{0}$ (SSDB) $>$ $G/G_{0}$ (DSDB) holds.

Figures 11 shows the effect of strain together with applied potential on the conductances with ($\mathbb{V}_{2}=\mathbb{V}_{4}=10\pi$) and ($\mathbb{V}_{2}=10\pi,\mathbb{V}_{4}=0$ or $\mathbb{V}_{2}=0,\mathbb{V}_{4}=10\pi$). We observe that the behavior of the conductance changed absolutely compared to Figure 10. The strainless double barrier conductance in Figures 11LABEL:sub@fig11:SubFigB, 11LABEL:sub@fig11:SubFigC, 11LABEL:sub@fig11:SubFigD and single barrier conductance in Figure 11LABEL:sub@fig11:SubFigA (orange color) have even minimum located at $\varepsilon=\mathbb{V}=10\pi$. These two conductances differ in resonance peaks, which due to the double barrier (Fabry-Perot resonator model). Figures 11LABEL:sub@fig11:SubFigA (SSB) and 11LABEL:sub@fig11:SubFigB (SSDB) show that the conductances, for $\delta\tau=0$ (orange color), $\delta\tau=3\pi$ (red color), $\delta\tau=7\pi$ (black color), are almost comparable in values and only differ in their mass terms corresponding to the limit of the forbidden zones, respectively, $\varepsilon=0,3\pi$ and $7\pi$. But the conductances for $\delta\tau=11\pi$ (blue color) and $\delta\tau=15\pi$ (green color) remain separated and only differs by oscillations. In Figure 11LABEL:sub@fig11:SubFigC of DSDB the three conductances fell at energy $\varepsilon=\mathbb{V}=\delta\tau$ until the value zero, then they resume after an increase in energy. A comparison between the three kinds of strained double barriers subject to a potential $\mathbb{V}$ is illustrated in Figure 11LABEL:sub@fig11:SubFigC. The conductance of SSB has the same minimum as that of simple potential barrier $\mathbb{V}=10\pi$ at $\varepsilon=\mathbb{V}=10\pi$, but the conductance of the SSDB and DSDB start with zero at $\varepsilon=\mathbb{V}=\delta\tau$. By increasing their Fermi energies, SSB resumes its conductance before DSDB.

5 Conclusion

We have studied the electronic structure of Dirac fermions through a double barrier potential in a strained graphene ribbon. Our system is a graphene chip made up of five regions where the second and fourth subjected to the strain effect and double barrier potential. In such system the armchair configuration along the $x$-direction imposes a gauge field $\bm{\mathcal{A}}_{j}$ perpendicular to the direction of propagation, which only affected horizontal inter-carbon links. After writing the Hamiltonian governing Dirac fermions in each region, we have determined the energies spectrum and associated eigenspinors. It was shown that the transverse wave vector is shifted by the strain term compared to that of the pristine graphene and the strain directly affects the transmission angle of Dirac fermions. The continuity of the eigenspinors at each interface separating two consecutive regions, allowed us to build a transfer matrix connecting the amplitudes of propagation in the input and output regions. We have calculated the transmission and reflection probabilities together with the conductance at zero temperature.

Subsequently, we have numerically presented the transmission probability density plots versus the incident energy $\varepsilon$ and incident angle $\theta$ under suitable conditions. These density plots showed the forbidden and permitted zones of transmission delimited by the energy contours ${\varepsilon=\frac{\delta\tau}{1\pm\sin\theta}}$, where Klein paradox zones are highlighted in red (total transmission), transmission gaps in purple and forbidden zones in white in all generated Figures. It has been seen that the Klein paradox is not always verified at normal incident angle and the transmissions satisfy three symmetry relations related to the incident angle and strain effect (deformation), see (27a-27c). The Fermi surfaces, corresponding to the dispersion relation, allowed us to determine the collimation angles for different configurations of the strain effect and double barrier. These angles can also be obtained using the conservation of the $k_{y}$ wave-vector component throughout the Dirac fermions propagation. We have showed that the transmission probability exhibited double resonance peaks, which can be explained by Fabry-Perot resonator model. At grazing incidence angle $\theta=\frac{\pi}{2}$, we have seen that the transmission showed only resonance peaks and transmission gaps.

By considering double barrier potential together with strain effect, we have showed that the transmission behavior changed completely. Indeed, the allowed and forbidden zones of transmission were now delimited by energy contours $\varepsilon^{s}_{s^{\prime}}=\frac{\mathbb{V}+s\delta\tau}{1+s^{\prime}\sin\theta}$ with $s=\pm$ and $s^{\prime}=\mp$. Always under the conservation constraint of the $k_{y}$ wave-vector component, and taking into account the Fermi surfaces, we have the following correspondences between the delimiting energies contours and the collimation angles: $\varepsilon^{s}_{s^{\prime}}=\frac{\mathbb{V}+\alpha\delta\tau}{1+s^{\prime}\sin\theta}\rightarrow\theta^{s}_{s^{\prime}}=\arcsin\left(\frac{\mathbb{V}-\varepsilon+s\delta\tau}{s^{\prime}\varepsilon}\right)$ Finally, we have performed a comparative study of the conductances between different configurations of strained double barrier with or without potential profile.

Acknowledgments

The generous support provided by the Saudi Center for Theoretical Physics (SCTP) is highly appreciated by all authors. AJ and HB acknowledge the support of KFUPM under research group project RG181001.