Quantum transport across van der Waals domain walls in bilayer graphene

The eigenfunctions of the 2SL Hamiltonian are those of the isolated graphene sheet, [47]

$$\begin{align} \displaystyle \label{eq02} {\bf \Phi}=\left(\begin{array}{@{}cccc@{}} \phi_{1}\ \nonumber \\ \phi_{2}\ \end{array} \!\!\!\right),\phi_j=\left(\begin{array}{@{}cccc@{}} \mu^{-}_j & -\mu^{+}_j \nonumber \\ 1 & 1 \end{array} \!\!\!\right)\left(\begin{array}{@{}cccc@{}} {\rm e}^{{\rm i}k_jx} \nonumber \\ {\rm e}^{-{\rm i}k_jx} \end{array} \!\!\!\right), \nonumber \end{align} \tag{ 2 }$$

where $j=1, 2$ is the layer index, $k_j=\sqrt{(\epsilon+s_{j}\delta){\hspace{0pt}}^2-k_y^2}$ with $s_j=$sgn$(j-1.5 )$, $\mu^\pm_ j=(k_j\pm\ {\rm i}k_y)/(\epsilon+s_{j}\delta)$, $\epsilon=E-v_0,$$\delta=(V_1-V_2)/2, \ v_0=(V_1+V_2)/2$. Introducing the length scale $l=\hbar v_{\rm F}/\gamma_{1}$, which represents the inter-layer coupling length, allows us to define the following dimensionless quantities:

$$\begin{align} \displaystyle \epsilon\rightarrow\frac{\epsilon}{\gamma_1},\ v_0\rightarrow\frac{v_0}{\gamma_1},\ \delta\rightarrow\frac{\delta}{\gamma_1},\ k_y\rightarrow lk_y,\ {\rm and}\ \vec r\rightarrow\frac{\vec r}{l}. \label{dimensionless} \nonumber \end{align} \tag{ 3 }$$

Notice that for the two stacking configurations, $\gamma_{1}$ was found to be different. For the AB-BL the value is $\gamma_1\approx0.4$ eV while for AA-BL it is $\gamma_1\approx0.2$ eV [48, 50, 51].

In order to discuss the different scattering modes, we introduce the notation $A_{\rm incoming}^{\rm outgoing}$, where A can stand for transmission (T) or reflection (R) probabilities and the indexes denote the mode by which the particles are incoming or outgoing. Figure 2 depicts all possible transitions that are considered in the present work. Figure 2(a) shows all possible transmission processes in a 2SL-BL-2SL system where t denotes the top layer on either side and b the bottom layer. For example, $T^{b}_{t}$ denotes a particle coming through the top layer and exiting on the bottom layer.

For AB-BL there are two branches corresponding to propagating modes. These branches correspond to the wave vector $k^{\pm}$ given by

$$\begin{align} \displaystyle \label{eq03} k^{\pm}=\left[-k^{2}_{y}+\epsilon^{2}+\delta^{2} \pm \sqrt{\epsilon^{2}(1+ 4\delta^{2})-\delta^{2}}\right]^{1/2}\!\!\!\!\!\!\!\!. \nonumber \end{align} \tag{ 4 }$$

The modes presented in equation (4) labeled by '$k^{+}$' correspond to eigenstates that are odd under layer inversion, while the '$k^{-}$' modes are even. These modes are shown, respectively, in blue and red in figure 2(b). This means that there are two available channels for transmission at a given energy, and an additional two for the reflection probabilities. Note that for energies $0<E<\gamma_1$, there is only one propagating mode and one transmission and reflection channel. Similarly, the wave function of AB-BL can be written as [49]

$$\begin{align} \displaystyle \label{eq04} \Psi(x,y)=G M(x)C {\rm e}^{{\rm i}k_{y}y}, \nonumber \end{align} \tag{ 5 }$$

where $M(x)$ corresponds to a 44 diagonal matrix consisting of exponential terms, while the components of the constant vector C depend on the propagating region, and G is given by

$$\begin{align} \displaystyle \label{eq05} G=\left(\begin{array}{@{}cccc@{}} \xi^{+}_{-} & -\xi^{+}_{+} & \xi^{-}_{-} & -\xi^{-}_{+} \nonumber \\ 1 & 1 & 1 & 1 \nonumber \\ \rho^{+} & \rho^{+} & \rho^{-} & \rho^{-} \nonumber \\ \zeta^{+}_{+} & -\zeta^{+}_{-} & \zeta^{-}_{+} & -\zeta^{-}_{-} \nonumber \\ \end{array} \right), \nonumber \end{align} \tag{ 6 }$$

where $\xi^{\pm}_{\pm}=(k^{\pm}\pm {\rm i}k_{y})/E-\delta, \ \rho^{\pm}=(\epsilon-\delta)\left[1-((k^{\pm}){\hspace{0pt}}^{2}\,+\right.$$\left.k^{2}_{y})/(\epsilon-\delta){\hspace{0pt}}^{2}\right]$ and $\zeta^{\pm}_{\pm}=(\epsilon-\delta)\rho^{\pm}\xi^{\pm}_{\pm}/(\epsilon+\delta)$.

The use of the matrix notation will prove to be very useful to construct the transfer matrix as outlined below.

In the case of an AA-BL, the corresponding wave function can be written similar to equation (5) but now with the matrix G given by

$$\begin{align} \displaystyle \label{eq07} G=\left(\begin{array}{@{}cccc@{}} \xi^{-}_{+} & \xi^{+}_{+} & \xi^{-}_{-} & \xi^{+}_{-} \nonumber \\ 1 & 1 & 1 & 1 \nonumber \\ \zeta^{-}_{+} & \zeta^{+}_{+} & \zeta^{-}_{-} & \zeta^{+}_{-} \nonumber \\ \rho^{+} & \rho^{+} & \rho^{-} & \rho^{-} \nonumber \\ \end{array} \right), \nonumber \end{align} \tag{ 7 }$$

where $\rho^{\pm}=\frac{1}{2\epsilon}\left[-(k_y^2+(k^\pm){\hspace{0pt}}^2)+(\epsilon^{}-\delta){\hspace{0pt}}^2+1)\right]$, $\xi^{\pm}_{\pm}=(\rho^{\pm}+$$\delta+\epsilon)({\rm i}k_y\pm k^\pm)/(\delta^2-\epsilon^2+1)$ and $\zeta^{\pm}_{s}=(\xi^{\pm}_{\pm}-\rho^{\pm}\zeta^{\pm}_{s}=$$(\xi^{\pm}_{\pm}-\rho^{\pm}({\rm i}k_y \pm k^\pm)/(\epsilon+\delta$). To investigate when scattering between the Dirac cones of AA-BL is allowed or forbidden, one can apply a unitary transformation that forms symmetric and anti-symmetric combinations of the top and bottom layer. This yields a Hamiltonian in the basis ${\bf \Psi}=2^{-1/2}(\Psi_{\alpha 2}+\Psi_{\alpha 1}, \Psi_{\beta 2}+\Psi_{\beta 1}, \Psi_{\alpha 2}-\Psi_{\alpha 1}, \Psi_{\beta 2}-\Psi_{\beta 1}){\hspace{0pt}}^{T}$ of the form:

$$\begin{align} \displaystyle \label{eq10} H_{AA}=\left(\begin{array}{@{}cccc@{}} \gamma_1+v_0 & v_{\rm F}\pi^{\dagger} & -\delta & 0 \nonumber \\ v_{\rm F}\pi & \gamma_1+v_0 & 0 & -\delta \nonumber \\ -\delta & 0 & -\gamma_1+v_0& v_{\rm F}\pi^{\dagger} \nonumber \\ 0 & -\delta& v_{\rm F}\pi & -\gamma_1+v_0 \nonumber \\ \end{array} \right). \nonumber \end{align} \tag{ 8 }$$

For $\delta=0$, this Hamiltonian is block-diagonal and represents two Dirac cones as shown in figure 1(c). The two cones correspond to modes with wave vector $k^{\pm}$ given by

$$\begin{align} \displaystyle k^{\pm}=\left[-k^{2}_{y}+\left(\epsilon\pm \sqrt{(1+ \delta^{2})} \right)^2 \right]^{1/2}. \label{k_vector_AA} \nonumber \end{align} \tag{ 9 }$$

In figure 1(c) the blue bands corresond to the odd $k^{+}$ modes and and red bands, denoting the even modes, are given by the $k^{-}$ wavevector. In these equations, v₀ denotes the energy shift of the whole spectrum. This shift can be chosen zero by assigning the same magnitude but different signs to the electrostatic potentials on both layers $V_1=-V_2$. Equation (8) shows that for zero electric field ($\delta=0$) both cones are decoupled and the scattering between them is strictly forbidden. This was used before in [52] to propose AA-BL as a potential candidate for ' cone-tronics' based devices. However, this protected cone transport is broken for finite bias ($\delta\neq0$) and hence scattering between the cones is allowed. Furthermore, one might wonder if the charge carriers stay within their cone transport through a domain consisting of two decoupled layers.

In order to calculate the scattering probability in the reflection and transmission channel, we use the transfer matrix method together with boundary conditions that require the eigenfunctions in each domain to be continious for each sublattice [53, 54]. To conserve probability current we normalize transmission probabilities T and reflection probabilities R such that

$$\begin{align} \displaystyle \sum_{i,j}\left(T_i^{\,j}+R_i^{\,j} \right)=1, \nonumber \end{align} \tag{ 10 }$$

where the index i refers to the incoming mode while the index j denotes the outgoing mode. For a coupled bilayer the different modes are labelled by '−' for the modes that are even under in-plane inversion and by '$+$' for odd modes. For a decoupled 2SL system, we employ the notation t for the top layer and b for the bottom layer. For example, for the system 2SL-AB-2SL and for an incident particle in the top layer of 2SL gives $T_t^t+T_t^b+R_t^t+R_t^b=1$. In figure 2 all possible transition probabilities are shown schematically.

To obtain measureable quantities, we finally calculate the zero temperature conductance that can be obtained from the Landauer-Büttiker formula [55] where we have to sum over all the transmission channels,

$$\begin{align} \displaystyle \label{eq08} {G_i^{\,j}}(E)=G_{0}\frac{L_y}{2 \pi}\int_{-\infty}^{+\infty}{\rm d}k_{y} T_{i}^{\,j}(E,k_y), \nonumber \end{align} \tag{ 11 }$$

with L_y the length of the sample in the y-direction and $G_0=4\ e^2/h$. The factor 4 comes from the valley and spin degeneracy in graphene. The total conductance of any configuration is the sum of all available channels $G_T=\sum_{i, j}G_i^{\,j}$.

We consider charge carriers tunnelling through 2SL-AA and AA-2SL systems. In figure 4(a) we show the transmission and reflection probabilities for charge carriers impinging on pristine AA-BL as a function of incident angle ϕ. As a result of the layer symmetry, charge carriers incident from bottom/top layer of 2SL and transmitted into the lower Dirac cone ($k^+$) in the AA-BL will have the same transmission probability $T_{b}^+=T_{t}^+$. Similarly, for those charge carriers transmitted into the upper cone, they will also have the same probability $T_{b}^-=T_{t}^-$ regardless which layer they are incident from.

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This symmetry stems from the fact that the wavefunction in the 2SL are a superposition of two spinors corresponding to the two sublattices while in AA-BL it is a superposition of four. For this reason, charge carriers incident from top or bottom layer of 2SL have the same dynamics and hence share their transmission probability. A partial reflection into the same layer, $R_{b}^{b}=R_{t}^{t}$ is shown in figure 4(a), which corresponds to evanescent modes associated with the upper Dirac cone ($k^-$). As in transmission, charge carriers can be back scattered between the layers. However, the absence of the electrostatic potential results in a small scattering current as depicted in figure 4(a). In addition, scattering back from top to bottom layer or vice versa occurs also with the same reflection probabilities $R_{b}^{t}=R_{t}^{b}$.

Because of chiral decoupling of oppositely propagating waves in AA-BL and in SL, back-scattering is forbidden for normal incidence ($\phi=0$) and thus the reflection probabilities for each channel are zero, i.e. $R_{b}^{b(t)}(0) = R_{t}^{t(b)}(0)=0$. This is associated with perfect tunnelling $T_{b}^+(0) + T_{b}^-(0) = T_{t}^+(0) + T_{t}^-(0) = 1$. The effect holds for all forthcoming structures composed of AA-BL and 2SL.

Figure 4(b) shows the numerical results of the same system, 2SL-AA, but now in the AA region, the potential is increased to $v_0=1.5 \gamma_1$. This shifts the two Dirac cones in energy to $v_0\pm \gamma_1$. As a result of the presence of the electrostatic potential, a strong scattered reflection $R_{b}^{t}/R_{t}^{b}$ takes place when there are no propagating modes in the AA section.

In figures 4(c) and (d), we show the reversed configuration, i.e. an AA-2SL system. The transmission and reflection probabilities for zero ($v_0=0$) and with nonzero ($v_0=1.5 \gamma_1$) electrostatic potentials applied to 2SL are reported in panels (c) and (d) respectively. Similar to the 2SL-AA system, we can note that layer symmetry still holds such that $T_{+}^b = T_{+}^t$ and $T_{-}^b = T_{-}^t$. Furthermore, we find strong non-scattered reflection in the $R_{+}^+$ and $R_{-}^-$ channels that is associated with evanescent modes on both sides of AA-BL and 2SL whereas the scattered reflection channels $R_{-}^+$ and $R_{+}^-$ are always zero due to the protected cone transport discussed earlier.

In this section, we show the results of transport across two domain walls forming a system with three regions; where AA-BL is sandwiched between two regions of 2SL, see figure 2(a). Such a system can exhibit a strong layer selectivity when current flows through the intermediate region, i.e. AA-BL. This behaviour has already been investigated in [58]. Here, however, we go in much more detail to show how the different transmission and reflection channels are affected by the electrostatic potential or finite bias applied to the intermediate region.

In figures 5 and 6 we show the scattered and non-scattered channels for transmission and reflection for pristine AA-BL and with electrostatic potential of strength $v_0=1.5 \ \gamma_1$, respectively. Layer symmetry is preserved in both reflection and transmission channels as clarified in figure 5. Figures 5 and 6 also show strong scattered transmission, especially for normal incidence which can be altered depending on the width of the AA-BL. When an electrostatic potential is applied to the middle domain, resonances appear in the transmission probabilities for $v_0+\gamma_{1}>E>v_0-\gamma_1$ as shown in figure 6. This is a consequence of the finite size of the AA-BL and the presence of charge carriers with different chirality in the mentioned range of energies [52]. Introducing a finite bias $\delta =0.6\gamma_1$ on AA-BL breaks the layer symmetry of the system. As a result, $T_{b}^b \neq T_{t}^t$ and $R_{b}^b \neq R_{t}^t$. However, it is still preserved in the scattered channels $T_{t}^b = T_{b}^t$ and $R_{t}^b = R_{b}^t$ (see figure 7).

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It is worth mentioning here that the finite bias does not break the angular symmetry with respect to normal incidence in the transmission and reflection probabilities as it does for normal AB-BL [49]. This is a manifestation of the symmetric inter-layer coupling in AA-BL.

In this system we interchange the AA-BL and 2SL as shown in figure 2(d). In this case, scattering is defined between the two cones in the AA-BL regions. In figures 8 and 9 we show the transmission and reflection probabilities between the two Dirac cones through the pristine 2SL and in the presence of an electrostatic potential, respectively. The first and the last rows of figures 8 and 9 show the non-scattered transmission and reflection probabilities corresponding to the lower and upper Dirac cones, respectively. We notice that Klein tunnelling is preserved at normal incidence. This shows that Klein tunnelling in AA-stacked bilayer graphene is a robust feature that is insensitive to local changes in the inter-layer coupling. On the other hand we see that scattering between two different Dirac cones remains strictly forbidden even with a local decoupling of the two layers. Therefore, these devices could be used for conetronics. As a result, in the second row of figures 8 and 9 the scattered transmission and reflection channels are zero $T_{+}^- = T_{-}^+=R_{+}^- = R_{-}^+=0$.

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In figure 10 we plot the transmission and reflection probabilities for a potential strength $v_0=1.5\ \gamma_1$ and inter-layer bias $\delta = 0.3\ \gamma_1$. The shift in the bands of the top (white) and bottom (red) layer of 2SL is due to the inter-layer bias which couples the two Dirac cones as shown in equation (8). Therefore, the suppression of the scattering transmission and reflection probabilities due to the protected cone transport does not hold anymore. It is, therefore, possible that scattering between different cones takes place as clarified in the second row of figure 10.

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The conductance of two and three-block systems is shown in figures 11 and 12, respectively. For the two systems 2SL-AA and AA-2SL with pristine AA-BL and 2SL, the conductance for different channels is shown in figures 11(a) and (b). It shows that the conductance of these two systems are identical. Referring to figures 4(a) and (c) we notice that the transmission probabilities for pristine 2SL-AA and AA-2SL are quite different. However, the corresponding conductances (see figure 11) exhibit time reversal symmetry in spite of the fact that the domain wall separates two different systems. This is a strong point which can be verified experimentally even in the case of zero electrostatic potential.

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Adding an electrostatic potential to one of the two sides leads to different behavior in the conductance of the above mentioned two systems as depicted in figures 11(c) and (d). In figure 11(c) the charge carriers incident from 2SL and impinging on AA-BL whose bands are shifted by v₀. Each conductance channel gives zero at $E=0$ due to the absence of propagating states in the 2SL at this energy, even though there are propagating states available in AA-BL corresponding to two cones. We note also that $G_{b}^\pm = G_{t}^\pm$ are almost zero at upper and lower cones $v_0\pm \gamma_1$ as a result of the absence of states at these points as seen in figure 11(c). In figure 11(d) we see that the conductance of different channels is not zero in contrast to the previous case because here at $E=0$ there are propagating states available in both AA-BL and 2SL. Furthermore, all channels have one minimum, due to the lack of states, at $E = v_0$ which corresponds to the Dirac cone in 2SL shifted by v₀ while $G_-^{t/b}$ has also another minimum at the upper cone $E=\gamma_1$ as shown in figure 11(d). Finally, for comparison we add in figures 11(e) and (f) the conductance that will be measured in the absence of a domain wall for 2SL-2SL and AA-AA junctions with $v_0=0$ (blue curves). Our results indicate that domain walls are experimentally identifiable channels even in the absence of a gate. As a reference we also calculate the total conductance in the presence of an electrostatic potential ($v_0=1.5\gamma_1$) as shown with black curves in figures 11(e) and (f) which corresponds, in this case, to the usual p-n junctions in single-layer graphene and AA-BL, respectively.

The conductance of three-block systems is shown in figure 12 where left and right panels correspond to AA-2SL-AA and 2SL-AA-2SL structure, respectively. Protected cone transport leads to zero conductance in the scattered channels $G_-^+ = G_+^-=0$ as shown in figure 12(a). A close inspection also reveals that $G_-^- = G_+^+$ at $E=0$ with finite and non-zero values, regardless of the fact that in the 2SL region there are no available propagating states. This is attributed to the evanescent modes in 2SL at $E=0$ which are responsible for ballistic transport in graphene [59]. We thus also expect that $G_-^-$ (red curve in figure 12(a)) should be exactly zero at the Dirac cone $E = \gamma_1$ as a result of the absence of propagating states in the leads at this energy.

By shifting the bands of 2SL using a local potential with strength $v_0 = 1.5\gamma_1$, a local minimum appears in the conductance G_T at $E = v_0$ which corresponds to the position of the charge-neutrality point in 2SL as shown in figure 12(c). This minimum can be obtained by aligning the upper cone in AA-BL and the Dirac cone in 2SL such that they are located at the same energy, this can be achieved by choosing $v_0=\gamma_1$. The main difference introduced by applying an inter-layer bias is the broken protected cone transport where now $G_+^-=G_-^+\neq0$ as depicted in figure 12(e). For completeness, we performed similar calculations but now with 2SL as the leads (2SL-AA-2SL) and the results for the conductance with pristine, gated and biased AA-BL are shown in figures 12(b), (d) and (f), respectively. Here, all conductance channels are zero at $E=0$ such that $G_t^t=G_b^b$ and $G_t^b=G_b^t$ as shown in figures 12(b) and (d). Similarly, the main features in figure 12(f) are in qualitative agreement with those shown in figures 12(b) and (d) but now the tunnelling equivalence through the same channel is broken so that $G_t^t\neq G_b^b$. This is a direct consequence of the perpendicular electric field which leads to the breaking of the inter-layer sublattice equivalence. The peaks appearing in the total conductance are due to the finite size of the AA-BL region.

In this section, we evaluate how the stacking of the connected region changes the transport properties across a domain wall. The angle-dependent transmission and reflection probabilities for pristine systems 2SL-AB are plotted in figure 13(a). The charge carriers can be incident from the two layers in the 2SL structure and impinge on AB-BL where, depending on their energy, they can access only one propagating mode $k^+$ or two $k^\pm$ if the energy is large enough. Scattering from the top or bottom layer of 2SL into one of these modes is equivalent $T_{t}^\pm=T_{b}^\pm$ as well as backscattering $R_{t}^{t(b)}=R_{b}^{b(t)}$ and hence, as before, layer symmetry is preserved (see figure 13(a)). In figure 13(b) we show results with the AB-BL region subjected to an electrostatic potential of strength $v_0=1.5\gamma_1$. Surprisingly, we see that the layer symmetry is broken and an asymmetric feature with respect to normal incidence shows up in the transmission and non-scattered reflection probabilities, see appendix, such that $[T/R](\phi)=[T/R](-\phi).$ For example, $T_{b}^\pm(\phi)=T_t^\pm(-\phi)$ as well as the non-scattered reflection channels $R_{b}^b(\phi)=R_{t}^t(-\phi)$ as discussed in section 3. This asymmetric feature can be understood by resorting to the bands on both sides of the junction, where due to the electrostatic potential the band alignment of 2SL and AB-BL is altered. In this case, the center of the AB-BL band is shifted upwards in energy with respect to the crossing of the 2SL [38] energy bands. The origin of such asymmetry is a direct consequence of the asymmetric coupling in AB-BL which leads to shifting of the bands by $\gamma_1$. Therefore, at low energy $\left\vert E-v_{0} \right\vert< \gamma_1$ there is only one propagating mode $k^{+}$ (i.e. one type of charge carrier) and consequently only $T_{b(t)}^+$ is available. For larger energy, on the other hand, there are two modes available giving rise to four channels $T_{b(t)}^\pm$.

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The angular asymmetry feature is present only in the region in the $(E, k_y)$-plane where there is only one propagating mode. This can be also understood as a manifestation of the asymmetric amplitude of the wave function in the AB-BL side due to the evanescent modes in this region [26]. The theory of tunnelling through an interface of monolayer and bilayer was presented earlier [26] and such asymmetry was noticed as well. Moreover, in our case there are two single layer graphene sheets connected to the bottom and top layers of the bilayer system but the asymmetric feature in [26] will be recovered when considering only one propagation channel. For instance, the transmission probabilities $T_{t}^\pm$ and $\ T_{b}^\pm$ presented in figure 13(b) show the same asymmetric features discussed in [26]. This asymmetry feature is reversed in the other valley, so that the total transmission or reflection averaged over both layers is symmetric as can be seen from figure 13(b). However, this valley-dependent angular asymmetry could also be used for the basis of a layer-dependent valley-filtering device as proposed in other works [4, 60].

The above analogy, which is discriminating between the presence of one or two modes, applies also to the non-scattered reflection probabilities $R_{b}^b$ and $\ R_{t}^t$. These non-scattered currents are carried by the states localized on the disconnected sublattices $\alpha_2$ and $\beta_1$, as seen in figure 1. In that case, there is one traveling mode [19] and thus, inherently, a layer asymmetric feature will be present. In contrast, for the scattered channels $R_{b}^t$ and $\ R_{t}^b$ the charge carriers must jump between the layers of AB-BL. This occurs through the localized states on the connected sublattices $\alpha_1$ and $\beta_2$ where there are two travelling modes and, hence, these probabilities exhibit layer symmetry as shown in figure 13(b). In the AB-2SL configuration, where charge carriers incident from the AB-BL impinge on the 2SL, we show the angle-dependent transmission and reflection probabilities in figure 13(c) for pristine 2SL and AB-BL.

Similar to the previous configuration 2SL-AB, the results are symmetric in this case because the Dirac cones of both systems (2SL and AB-BL) are aligned. Furthermore, there is an equivalence in the transmission channels such that $T_{\pm}^t=T_{\pm}^b$ with partial reflection associated with the non-scattered channels $R_{-}^-$ and $R_{+}^+$. While for the scattered channels $R_{+}^-$ and $R_{+}^-$ are almost zero. This is due to efficient transmission resulting from the absence of the electrostatic potential in the 2SL. An electrostatic potential of strength $v_0=1.5 \gamma_1$ induces a scattering between the two modes in the reflection channels so that now $R_{-}^+=R_{+}^-\neq0$ as depicted in figure 13(d). In addition, it breaks the band alignment and gives rise to the layer asymmetry feature in the transmission probabilities $T_+^{b(t)}$ where only one travelling mode exists i.e. $E<\gamma_1$. Thus, $T_-^{b(t)}$ always preserves layer symmetry in this case, see figure 13(d), because the mode $k^-$ exists for $E>\gamma_1$ where also the mode $k^+$ is available as discussed above. This is also the same reason that configurations consisting of AA-BL always preserve layer symmetry. Indeed, AA-BL does not have a region in the $(E, k_y)$-plane with only one propagating mode, and there are always two travelling modes for all energies.

Different configurations have been proposed to connect a single layer to the AB-stacked bilayer graphene [5, 27, 38, 61]. Now, two SL are connected to the AB-stacked bilayer, see figure 2(a). In figure 14 we show the dependence of the transmission and reflection probabilities on the transverse wave vector k_y and the Fermi energy. It appears that all channels are symmetric with respect to normal incidence since the Dirac cones of AB and 2SL are aligned. It also implies that scattered and non-scattered channels of the transmission and reflection are equivalent such that $(T/R)_{b}^t=(T/R)_{t}^b$ and $(T/R)_{t}^t=(T/R)_{b}^b$ (see figure 14).

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Another interesting feature of this configuration is that for $E<\gamma_1$ the scattered and non-scattered transmissions are equal $T_{i}^{\,j}=T_{i}^{i}$. In this energy regime such device can be used as an electronic beam splitter [61, 62].

Figure 15 displays the same plot as in figure 14 but with an electrostatic potential on the AB-BL region. There is an important difference as compared to the pristine AB-BL case, the layer symmetry is broken such that $T_{t}^{b}(k_{y})= T_{b}^{t}(-k_y)$ as clarified in figure 15. This can be also understood by pointing out that charge carriers scattered from top to bottom when moving from left to right in the K valley are equivalent to charge carriers scattering from bottom to top when moving oppositely in the second valley $K^\prime$.

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Introducing a finite bias ($\delta>0$) to the AB-BL region along with an electrostatic potential ($v_0>0$) will shift the bands and opens a gap in the spectrum. As a result of the presence of a strong electric field, the transmission channels are completely suppressed inside the gap due to the absence of traveling modes as seen in figure 16. Moreover, non-zero asymmetric reflection appears in the gap as well as a violation of the equivalence of non-scattered transmission channels. This is a result of the breaking of inter-layer sublattice equivalence [49]. In addition, some localized states appear inside the 'Mexican hat' of the low energy bands where they are pushed by the strong electric field ($\delta=0.8 \gamma_1$), see figure 16.

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There is a link between the transmission probabilities of our system 2SL-AB-2SL and those investigated by González et al [38]. The channels $T_b^b$ and $T_b^t$ are qualitatively equivalent to those obtained in [38]. For example, $T_b^t$ shows electron-hole ($e-h$) and $\delta\rightarrow-\delta$ symmetry whereas $T_b^b$ exhibits another symmetry which can be obtained under the exchange $(e, \delta)\leftrightarrow\ (h, -\delta)$. The results in figure 17 are in good agreement with those of [38] where we fix $v_0=0$ and $d=25$ nm.

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For leads composed of AB-BL where the intermediate region is pristine 2SL, we show the results in figure 18 for the transmission and reflection probabilities. Now charge carriers will scatter between the different modes of the AB-BL on the left and right leads as shown in figure 2(b). As expected, all channels are symmetric and as a result of the finite size of the 2SL region, resonances appear in T as shown in figure 18. These so-called Fabry–Pérot resonances appear at quantized energy levels [63]

$$\begin{align} \displaystyle \label{eq019} E^n_{\rm SL}(k_y)=\sqrt{k_y^{2}+\left(\frac{n \pi \ }{d}\right)^{2}}. \nonumber \end{align} \tag{ 21 }$$

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This is the dispersion relation for modes confined in the 2SL region with width d.

The results presented in figure 18 reveal no scattering between the two modes $k^+$ and $k^-$ and charge carriers are only transmitted or reflected through the same channel from which they come from. Unexpectedly, introducing an electrostatic potential induces a strong scattering in the reflection channels ($R_+^-=R_-^+\neq0$) and very weak scattering in the transmission channels ($T_+^-=T_-^+\neq0$), as seen in figure 19. When the 2SL are biased, the Dirac cones at bottom and top layers will be shifted up (white dashed lines) and down (red dashed lines) in energy, respectively (see figure 20). This bias will strengthen the coupling between the two modes resulting in a strong scattering between them. In addition, the inversion symmetry is broken due to the bias leading to an asymmetry with respect to normal incidence.

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The conductance of the two-block system consisting of 2SL and BA-BL is shown in figure 21 for different values of the applied gate voltage. Figures 21(a) and (b) reveal that the system where charge carriers are incident from the 2SL and impinge on AB-BL and vice versa are equivalent to the case when both 2SL and AB-BL are at the same potential. As seen in figures 21(a) and (b), $G_+^{t(b)}=G_{t(b)}^+$ are contributing to the total conductance G_T starting from $E=0$ where the $k^+$ mode exists. On the contrary, $G_-^{t(b)}=G_{t(b)}^-$ only contributes when $E>\gamma_1$ where $k^-$ states are available and this appears as a sharp increase in G_T at $E=\gamma_1$. On the other hand, considering an applied electrostatic potential on the right side of the two-block system will break this equivalence as seen in figures 21(c) and (d). In addition, as a result of the shift of the Dirac cone in AB-BL (see figure 21(c)) or 2SL (see figure 21(d)) due to the electrostatic potential, all conductance channels are zero at $E=v_0$. Similar to the AA-BL case, the conductances of the pristine systems 2SL-AB/AB-2SL (see figures 21(a) and (b)) clearly preserve the time reversal symmetry. Even though, both systems have different transmission probabilities as can be seen from figures 13(a) and (c). We also show in figures 21(e) and (f) the total conductance in the absence of domain wall in 2SL-SL and AB-AB systems, respectively, for $v_0=0$ (blue curves) and $v_0=1.5\gamma_1$ (black curves). This shows that transport channels in the presence of domain walls are experimentally recognisable.

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In figure 22 we show the conductance in a 2SL-AB system as a function of the bias for transport using a single figure 22(a) or a double figure 22(b) mode. The results show that the contribution from the top and bottom layers to the conductances have opposite behaviours as a function of the inter-layer bias. The total conductance $G_{T}$, however, has a convex form, increasing with the application of an inter-layer bias. From figure 22(b), on the other hand, we see that when a second mode is available, four channels contribute to the conductance and the total conductance assumes a concave form, i.e. decreasing with increasing inter-layer bias. This is a characteristic experimental feature that can signal the presence of a second mode of propagation.

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For the three-block system we show the conductance of the configuration AB-2SL-AB and 2SL-AB-2SL in the left and right columns of figure 23, respectively. The resulting conductance of the first configuration shows only two non-zero channels $G_+^+$ and $G_-^-$, while the scattered ones $G_+^{-}=G_-^{+} =0$ since $T_+^{-}=T_-^{+} =0$ (see figure 23(a)). Furthermore, for low energy $G_T=G_+^+$ since the mode $k^{-}$ is not available in this regime but it starts conducting when $E>\gamma_1$. The applied electrostatic potential on the 2SL keeps the scattered conductance channels at zero and a minimum in the conductance appears around the shifted Dirac cone $E=v_0$ of the 2SL as depicted in figure 23(c). As pointed out before, if the Fermi energy approaches the strength of the electrostatic potential, a non-zero minimum is present in the conductance because charge carriers can be transmitted through a width d of 2SL via evanescent modes [59]. In figure 23(f) this minimum disappears and the conductance dramatically increases at $E=\gamma_1$. This is because the bias will couple the two modes and two additional scattered channels $G_+^-$ and $G_-^+$ start conducting. The resonant peaks resulting in the conductance, see figures 23(a), (c) and (e), are due to the finite size of the intermediate region and hence strictly depend on its width d.

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On the other hand, the conductance of the configuration 2SL-AB-2SL has different features. In figure 23(b) the four channels, in contrast to the previous configuration, start conducting from $E=0$. This possess layer symmetry such that $G_t^t=G_b^b$ and $G_b^t=G_t^b$. Of particular importance is the equivalence of the four channels for $E<\gamma_1$ while for $E>\gamma_1$ charge carriers strongly scatter between the layers (i.e. $\ G_i^{\,j}>G_i^i$) as shown in figure 23(b). This equivalence of the four channels in the regime $E<\gamma_1$ vanishes when an electrostatic potential is applied ($v_0>0$) to the intermediate region as seen in figure 23(d). However, the scattered and non-scattered conducting channels are still equivalent in this case where $G_t^{t(b)}=G_b^{b(t)}$ with $G_i^{\,j}>G_i^i$ for all energy ranges, see figure 23(d).

As discussed before, the most characteristic feature of the inter-layer bias in the AB-BL is the opening of a gap in the energy spectrum between $v_0\pm\delta$ which is reflected in the conductance as seen in figure 23(f). The resonant sharp peaks in the conductance near the edges of the gap result from the localized states inside the Mexican hat of the low energy bands. Another consequence of the inter-layer bias is the breaking of the equivalence in the non-scattered conducting channels where now $G_t^{t}\neq G_b^{b}$ as seen in figure 23(f).