Tunable valley filtering in graphene with intervalley coupling

Charge carriers in honeycomb structures, such as graphene, silicene and transition metal dichalcogenides, etc., have recently attracted great interest for the great potential of the basic and applied researches [1–16]. Among 2D materials, particularly graphene has shown various interesting properties such as the Klein paradox [17] and the Zitterbewegung [18,19]. Graphene is also a suitable material for observing spintronics effects due to its weak SO coupling. [20–22]. Meanwhile, there are two inequivalent points at the K and $K'$ points in the Brillouin zone of graphene which are linked by the spatial inversion operation [23]. Moreover, the intervalley scattering in graphene is severely suppressed due to the large valley separation in momentum space [24,25].

Valleytronics, in analogy with spintronics [26], has received a boost which aims to control the valley degree of freedom [27–33]. To control the number of electrons in each valley, a number of proposals have been made. For example, it is possible to induce the valley polarization through zigzag edges [28], lattice strain [34–37], magnetic field [30,38–42], line defects [43,44], optical field [45–47] or a one-dimensional valley junction model [33]. Recently, the investigation of the intervalley coupling in monolayer graphene has attracted attention theoretically [48–51]. Specially, Yafei Ren et al. [50] has considered a tight-binding model of a graphene sheet with periodically top-adsorbed atoms. As the periodicity can cause that the inequivalent $KK'$ valleys are mapped into the same Γ-point, the low-energy effective Hamiltonian for such a system can be described by intervalley coupled Dirac fermions. This allows one to use the intervalley coupling to realize valley-polarized electronic transport in valleytronics devices.

In this article, we propose a structure to realize valley filtering electronic transport, which consists mainly of a valley coupled graphene as the scatter. We find that the valley-dependent transmission probability in this junction is sensitive to the incident angle. We also find that the differential valley transmission P_T is an odd function of the incident angle while the total transmission T is an even function of the incident angle. In addition, we also note that the sign of the valley polarization component P_x of the outgoing current can switch from positive to negative and its magnitude can be changed by manipulating the Fermi energy E or the electronic potential V in the valley-coupled graphene region.

This paper is organized as follows. In the next section we introduce the model and the mode-matching approach. In the third section we present our numerical results and discussions. The final section is devoted to a brief summary.

In this article, we consider a ballistic normal/valley-coupled/normal graphene (NG/VCG/NG) junction occupying the $(x,~y)$-plane, as illustrated in fig. 1(a). The low-energy effective Hamiltonian for an electron in this structure is given by [50]

$$\begin{eqnarray} H&= \hbar {v_F}\left( {{k_x}{\sigma _x}{\tau _z} + {k_y}{\sigma _y}} \right)+\biggl[ {\Delta _1}{\sigma _z} + \frac{{{\Delta _2}}}{2}\left( {1 + {\sigma _z}} \right){\tau _x} \nonumber\\ &\quad +\, \left( {{U_0}I + V} \right)\hat I\biggr]\Theta \left( x \right)\Theta \left( {L - x} \right), \end{eqnarray} \tag{ 1 }$$

where ${v_F} \approx {0.86\times10^6}\ \text{m/s}$ is the Fermi velocity of the structure, $p_x~(p_y)$ is the x component (y component) of the electron momentum. The matrices $\sigma_i$ and $\tau_i$, with $i=x,y,z$, characterize the Pauli matrices which operate on the sublattice (A and B sites) and valley (K and $K'$ points) spaces, respectively. $\Theta(x)$ is the step function. $\Delta _1$ is the on-site potential difference between the A and B sublattices. $\Delta _2$ is the strength of the intervalley coupling which leads to a cross-correlation between K and $K'$ valleys. Such an intervalley coupling can be used to control the valley-polarized electronic transport. ${U_0} = {\Delta _2} - {\Delta _1}$ stands for the on-site energy. L is the width of the scattering region. V is the value of the electric potential which can be tuned by the gate voltage. In the following we set $\hbar = {v_F} = 1$.

Zoom In Zoom Out Reset image size

The eigenstates and eigenvalues in the graphene with the homogeneous intervalley coupling are characterized by

$$\begin{eqnarray} &&\psi _{\beta\varepsilon }^ \pm \left( {{k_x},{k_y}} \right) = \frac{1}{{{N_{\beta \varepsilon}}}}\left({{\varsigma _{\beta \varepsilon }}, \pm {k_ \pm },\varepsilon {\varsigma _{\beta \varepsilon }},} \right.{\left. { \mp \varepsilon {k_\mp}} \right)^T}, \end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray} &&E_{\beta \varepsilon}=\beta \sqrt{k^2+ {{\left({\Delta _1} + \varepsilon \frac{{{\Delta _2}}}{2}\right)}^2}} + \varepsilon \frac{{{\Delta _2}}}{2} + \left( {{U_0} + V} \right), \end{eqnarray} \tag{ 3 }$$

Here, ${k_ \pm} = {k_x} \pm {k_y}$, $k = \sqrt {k_x^2 + k_y^2}$, ${\varsigma _{\beta \varepsilon}}=E_{\beta \varepsilon } - {U_0} - V + {\Delta _1}$ and ${N_{\beta \varepsilon }} = 2\sqrt {{\varsigma _{\beta \varepsilon }}\left( {{\varsigma _{\beta \varepsilon }} - {\Delta _1} - \varepsilon \frac{{{\Delta _2}}}{2}} \right)}$. $\beta=+1~(-1)$ specifies the conduction (valence) band. $\varepsilon=\pm1$ denotes the two valley-split subbands of the conduction and valence bands.

We assume that the electron in the conductance band of the NG region $(x<0)$ has positive energy $(E>0)$ with the incidence angle φ and the valley τ and moves to the right, as shown in fig. 1(b). Then, the wave functions in each region are given by

$$\begin{equation} \begin{array}{rcl} {\Psi _I}\left( x \right)&=& \psi _{N\tau }^ + + {r_{\tau \tau }}\psi _{N\tau }^ - + {r_{\tau '\tau }}\psi _{N\tau '}^ -, \\ {\Psi _{II}}\left( x \right) &=& {p_1}\psi _1^ + + {p_2}\psi _1^ - + {p_3}\psi _2^ + + {p_4}\psi _2^ -, \\ {\Psi _{III}}\left( x \right) &=& {t_{\tau \tau }}\psi _{N\tau }^ + + {t_{\tau '\tau }}\psi _{N\tau '}^ +, \end{array} \end{equation} \tag{ 4 }$$

with

$$\begin{equation} \begin{array}{rcl} \psi _{NK}^ \pm \left( {x,y} \right) &=& \displaystyle\frac{1}{{\sqrt 2 }}{\left( { \pm {e^{ \mp i\varphi }},1,0,0} \right)^T}{e^{ \pm i{q_x}x + i{q_y}y}},\\ \psi _{NK'}^ \pm \left( {x,y} \right) &=& \displaystyle\frac{1}{{\sqrt 2 }}{\left( {0,0, \mp {e^{ \pm i\varphi }},1} \right)^T}{e^{ \pm i{q_x}x + i{q_y}y}},\\ \psi _1^ \pm \left( {x,y} \right) &=& \displaystyle\left( {{E_N}, \pm {k_{1 \pm }},{E_N},}{ \mp {k_{1 \mp }}} \right)^T{e^{ \pm i{k_{1x}}x + i{k_y}y}},\\ \psi _2^ \pm \left( {x,y} \right)&=& \displaystyle{\left( {{E_N}, \pm {k_{2 \pm }}, - {E_N}, \pm {k_{2 \mp }}} \right)^T}{e^{ \pm i{k_{2x}}x + i{k_y}y}}, \end{array} \end{equation} \tag{ 5 }$$

Here, ${q_x} = E\cos\left(\varphi \right)$, ${q_y} = E\sin \left( \varphi\right)$, $E_{N}= E-{U_0} - V + {\Delta _1}$, $\text{k}_{1\left( 2 \right)x} = \sqrt {{E_N}\left( {{E_N} - 2{\Delta _1} \mp {\Delta _2}} \right) - k_y^2}$ and ${k_{1\left(2\right) \pm }} = {k_{1\left( 2 \right)x}} \pm i{k_y}$. Moreover, the translational invariance along the interface leads to the conservation of the momentum in the y-axis direction, namely, $q_y=k_y$. ${r_{\tau'\tau}}$ and ${t_{\tau'\tau}}$ are the scattering amplitudes for an electron with valley τ being scattered into valley $\tau'$. The same analysis is applicable to the electron coming from the valance band.

By applying the boundary conditions ${\Psi _I}\left( 0 \right) = {\Psi _{II}}\left( 0 \right)$ and ${\Psi _{II}}\left( L \right) = {\Psi _{III}}\left( L \right)$, we can provide the detailed expressions for the valley-dependent scattering amplitudes ${t_{\tau'\tau}}$, which can be given by

$$\begin{equation} \hspace{-10pt}\begin{array}{rcl} &&{t_{KK(K'K')}}= \\ &&-\,2i{E_N}\cos \left( \varphi \right){e^{ - i{q_y}L}}\left\{ 2i{E_N}{k_{1x}}{k_{2x}} \cos \left( \varphi \right)\left( {\cos \left( {{k_{1x}}L} \right)}\right.\right. \\ &&+\,\left. {\cos \left( {{k_{2x}}L} \right)} \right) + {k_{2x}}\sin \left( {{k_{1x}}L} \right)\left[ {k_{1x}^2 + {E}_N^2 + E\left( {E - 2{E_N}} \right)} \right]\\ &&\times\,{\sin ^2}\left( \varphi \right) + {k_{1x}}\sin \left( {{k_{2x}}L} \right)\left[ {k_{2x}^2 + {E}_N^2 + E\left( {E - 2{E_N}} \right)}\right.\\ &&\times\,\left.\left.{ {{\sin }^2}\left( \varphi \right)} \right] \right\}/A,\\ &&t_{K'K\left( {KK'} \right)} =\\ &&2{E_N}\cos \left( \varphi \right){e^{ - i\left( {{q_y}L \pm 2\varphi } \right)}}\left\{ -{k_{1x}}{k_{2x}}\left( \cos \left( {{k_{1x}}L} \right) \right.\right.\\ &&-\,\left.\cos \left( {{k_{2x}}L} \right) \right) \left[ {E{e^{ \mp i\varphi }} + \left( {2{E_N} - E} \right){e^{ \pm i\varphi }}} \right] \\ && +\, i\left[ {k_{1x}}\sin \left( {{k_{2x}}L} \right)\left( {e^{ \pm 2i\varphi }}\left( E- {E_N} \right){E_N} -\left( E{E_N} + k_{2x}^2 \right.\right.\right.\\ &&-\, \left.\left.{E^2}{{\sin }^2}\left( \varphi \right) \right) \right) - {k_{2x}}\sin \left( {{k_{1x}}L} \right)\left( {{e^{ \pm 2i\varphi }}}\left( {E - {E_N}} \right){E_N}\right.\\ &&\left. {\left. {\left. { -\, \left( {E{E_N} + k_{1x}^2 - {E^2}{{\sin }^2}\left( \varphi \right)} \right)} \right)} \right]} \right\}/A, \end{array} \end{equation} \tag{ 6 }$$

for the electron coming from the conductance band,

$$\begin{equation} \hspace{-10pt}\begin{array}{rcl} &&{t_{KK(K'K')}} =\\ &&-\,2i{E_N}\cos \left( \varphi \right){e^{i{q_y}L}}\left\{ 2i{E_N}{k_{1x}}{k_{2x}}\cos \left( \varphi \right)\left( {\cos \left( {{k_{1x}}L} \right)}\right.\right.\\ &&+\,\left. {\cos \left( {{k_{2x}}L} \right)} \right) + {k_{2x}}\sin \left( {{k_{1x}}L} \right)\left[ k_{1x}^2 + {E}_N^2 \right.\\ &&+\,\left. E\left( {E - 2{E_N}} \right) \right]{\sin ^2}\left( \varphi \right) + {k_{1x}}\sin \left( {{k_{2x}}L} \right)\left[ k_{2x}^2 + {E}_N^2 \right.\\ &&+\, \left.\left.E\left( {E - 2{E_N}} \right)~ {{{\sin }^2}\left( \varphi \right)} \right] \right\}/A,\\ &&{t_{K'K\left( {KK'} \right)}} =\\ &&2{E_N}\cos \left( \varphi \right){e^{i\left( {{q_y}L \pm 2\varphi } \right)}}\left\{ - {2}{k_{1x}}{k_{2x}}\left( \cos \left( {{k_{1x}}L} \right)\right.\right.\\ &&-\, \left.\cos \left( {{k_{2x}}L} \right) \right) \left[ {{E_N}{e^{ \mp i\varphi }} \pm iE\sin \left( \varphi \right)} \right] + i\left[ - {k_{1x}}\sin \left( {{k_{2x}}L} \right)\right.\\ &&\times\, \left.\left( {k_{2x}^2 + \left( {{e^{ \mp i\varphi }}{E_N}} \right.} \right. {{{\left. { \pm\; iE\sin \left( \varphi \right)} \right)}^2}} \right)+ {k_{2x}}\sin \left( {{k_{1x}}L} \right)\\ &&\times\, \left.\left.\left( {k_{1x}^2{+}\left( {{e^{ \mp i\varphi }}{E_N} \pm iE} \right.} \left. {{{\left. { \times\, \sin \left( \varphi \right)} \right)}^2}} \right)\right.\right] \right\}/A, \end{array} \end{equation} \tag{ 7 }$$

for the electron coming from the valance band.

Here

$$\begin{equation} \hspace{-10pt}\begin{array}{rcl} &&A=\\ &&\left\{{\! - 4{{\left( {E - {E_N}} \right)}^2}{k_{1x}}{k_{2x}}{{\sin }^2}\!\left( \varphi \right){- 4}i{E_N}\cos \left( \varphi \right)} \right.{k_{2x}}\cos \left( {{k_{2x}}L} \right)\\ &&\times\, \sin \left( {{k_{1x}}L} \right)\left[ {k_{1x}^2 + E_N^2 + E\left( {E - 2{E_N}} \right){{\sin }^2}\left( \varphi \right)} \right]\\ &&-\, \sin \left( {{k_{1x}}L} \right)\sin \left( {{k_{2x}}L} \right)\left[ {2{E}_N^4 + 2k_{1x}^2k_{2x}^2 + E_N^2\left( {k_{1x}^2 + k_{2x}^2} \right)} \right.\\ &&+\, E_N^2\cos \left( {2\varphi } \right)\left( {k_{1x}^2 + k_{2x}^2 + 2{{E}^2}\cos \left( {2\varphi } \right) - 2{{E}^2}} \right) + 8EE_N^2\\ &&\left. { \times\, \left( {E - {E_N}} \right){{\sin }^2}\left( \varphi \right) + 2{E^3}\left( {E - 4{E_N}} \right){{\sin }^4}\left( \varphi \right)} \right] + 2{k_{1x}}\\ &&\times\, \cos \left( {{k_{1x}}L} \right)\left[ {{k_{2x}}\cos \left( {{k_{2x}}L} \right)\left( {E_N^2\left( {3 + \cos \left( {2\varphi } \right)} \right) + 2E} \right.} \right.\\ &&\left. { \times\, \left( {E - 2{E_N}} \right){{\sin }^2}\left( \varphi \right)} \right) - 2i{E_N}\cos \left( \varphi \right)\sin \left( {{k_{2x}}L} \right)\left( {E_N^2} \right.\\ &&\left. {\left. {\left. { +\, k_{2x}^2 + E\left( {E - 2{E_N}} \right){{\sin }^2}\left( \varphi \right)} \right)} \right]} \right\}. \end{array} \end{equation} \tag{ 8 }$$

Notice that the valley-conserved transmission ${t_{KK(K'K')}}$ and valley-flip transmission ${t_{K'K\left( {KK'} \right)}}$ are completely suppressed if $E_{N}=0$. This is because the electron wave vectors ${k_{1(2)x}}$ are purely imaginary when $E_{N}=0$, there are no propagating modes in the intervalley coupled graphene region.

Meanwhile, according to the Landauer-Büttiker formula, it is straightforward to obtain the zero-temperature ballistic conductance G through the junction as ${G = G_{kk} + {G_{kk'}} + {G_{k'k}} + {G_{k'k'}}}$. Furthermore, due to the presence of the intervalley coupling in the system, the electron valley is not conserved. Therefore, we need to know all three components of the outgoing polarization vector P to understand the different transformations that the device can perform on the valley of an incoming electron. In analogy to the spin polarization of the outgoing current [52–55], we introduce the valley polarization of the outgoing current which can be evaluated by the following expressions:

$$\begin{equation} \begin{array}{rcl} {P_x}&=&\displaystyle \frac{{2\text{Re} \left[ {{G_{xy}}} \right]}}{G},\\ {P_y}&=& \displaystyle\frac{2\text{Im}\left[ {{G_{xy}}} \right]}{G},\\ {P_z}&=& \displaystyle\frac{{\left( G_{KK} + {G_{KK'}} \right) - \left( {{G_{K'K}} + {G_{K'K'}}} \right)}}{G}, \end{array} \end{equation} \tag{ 9 }$$

with

$$\begin{equation} \begin{array}{rcl} {G_{\tau '\tau }}&=& \displaystyle\frac{{{G_0}}}{2}\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {{T_{\tau '\tau }}\cos \left( \varphi \right)\text{d}\varphi },\\ {G_{xy}}&=& \displaystyle\frac{{{G_0}}}{2}\int_{ - \frac{\pi }{2}}^{\frac{\pi }{2}} {\left( {{{\text{t}}_{K'K}}{\text{t}}_{KK}^ * + {{\text{t}}_{{\text{K}'\text{K}'}}} {\text{t}}_{{\text{KK}'}}^ * } \right)\cos \left( \varphi \right)\text{d}\varphi }. \end{array} \end{equation} \tag{ 10 }$$

Here, ${G_0} = \frac{{2{e^2}{L_y}{k_F}}}{{{\pi }h }}$ and ${T_{\tau \tau '}} = {\left| {{t_{\tau ' \tau}}} \right|^2}$. L_y is the length of the sample in the y-axis direction, k_F is the Fermi wave vector. P_x, P_y and P_z represent the valley polarization components of the outgoing current propagating through the right NG attached to the VOG.

In order to better understand the graphene with homogeneous intervalley coupling $\Delta_{2}$ and on-site potential difference $\Delta_{1}$, first let us briefly investigate the energy dispersions corresponding to eq. (3) for $k_y=0$, V = 0, $\Delta_1=1$ and $\Delta_2=1$, as shown in fig. 2(a). For $\varepsilon =-1$, there is a crossing point (at $k = 0$) between conduction and valence bands. For $\varepsilon =+1$, however, the conduction and valence bands are split by a local band gap. Thus, the introduction of $\Delta_{1}$ and $\Delta_{2}$ in the system has a strong influence on the dispersion relation which completely changes the low-energy behavior of the electrons. In addition, we also study the orientations of valley ${\hat M}$ in the states $\psi _{\beta \varepsilon }^ \pm$ which are given by $\langle {\psi _{\beta \varepsilon }^ \pm }|{{\hat M}_i}| {\psi _{\beta \varepsilon }^ \pm } \rangle$ with ${\hat{M}_i} ={\scriptsize{\left({\begin{array}{c@{\enskip}c} {{\tau _i}}&0\\ 0&{{\tau _i}^ * } \end{array}}\right)}}$. After calculation, we find that the orientations of the valley are given by

$$\begin{equation} M = \frac{1}{{2\left( {{E_{F}} - V - {U_0} - \varepsilon \frac{{{\Delta _2}}}{2}} \right)}}\left( {{k_x}, - {k_y},{\Delta _1} + \varepsilon \frac{{{\Delta _2}}}{2}} \right). \end{equation} \tag{ 11 }$$

In fig. 2(b), we demonstrate the orientations of the valley as a function of the in-plane electron wave vector $\left( {{k_x},{k_y}} \right)$ for $E_{F}=4$, V = 0, $\Delta_{1}=1$ and $\Delta_2=1$. We find that the expectation values of the in-plane valley orientations are completely aligned to the direction of propagation. Moreover, according to eq. (11), it can be concluded that the out-of plane valley direction T_z does not depend on the wave vector. Here it is also interesting to note that the three components of the valley T are dependent on the electronic potential V and the Fermi energy E_F. Therefore, we can use V and E_F to generate and manipulate valley-polarized current.

Zoom In Zoom Out Reset image size

We now turn to studying the general behavior of the valley-dependent transmission probability $T_{\tau'\tau}$ which is calculated numerically based on eq. (6) and eq. (7), as shown in fig. 3. In this figure, we set L = 3, $\Delta_1=1$ and $\Delta_2=1$. One can clearly see that the valley-dependent transmission probability exhibits the oscillatory behaviors in all cases which can be understood by the self-interference of the waves reflected inside the intervalley coupled graphene region. This can also be explained by the fact that $\sin \left( {{k_{1(2)x}}L} \right)$ and $\cos \left( {{k_{1(2)x}}L} \right)$ in eq. (6) and eq. (7) are the periodic functions when k_1x and k_2x are real. We also find that the valley-conserved transmission ${T_{KK }}~({T_{K'K'}})$ is invariant under the change $\varphi \to - \varphi$ since the valley-conserved scattering amplitudes in eq. (6) and eq. (7) are even functions of $\varphi~[{t_{KK\left( {K'K'} \right)}}\left( \varphi \right) = {t_{KK\left( {K'K'} \right)}}\left( { - \varphi } \right)]$.

Zoom In Zoom Out Reset image size

Furthermore, it is worth noting that the valley-flipped transmission ${T_{K'K }}~({T_{K K'}})$ is an even function of φ when $V = 1$ (fig. 3(a) and fig. 3(c)). For ${V \ne 1}$, we notice that ${T_{K'K }}~({T_{K K'}})$ becomes asymmetric with respect to the angle of incidence, as shown in fig. 3(b) and fig. 3(d). These can be explained by the fact that when $V= 1$, ${U_0} + V - {\Delta _1} = 0$, namely, E_N in eq. (6) and eq. (7) is equal to E, the numerators of ${t_{K'K\left( {KK'} \right)}}$ in eq. (6) and eq. (7) are reduced to ${e^{ - i\left( {{q_y}L \pm 2\varphi } \right)}} \cdot D$ and ${e^{ i\left( {{q_y}L \pm 2\varphi } \right)}} \cdot D$, respectively. Here

$$\begin{equation} \begin{array}{rcl} &&D = 2E\cos \left( \varphi \right)\left\{ { - 2E{k_{1x}}{k_{2x}}\cos \left( \varphi \right)} \right.\\ &&\times \,\left( {\cos \left( {{k_{1x}}L} \right) - \cos \left( {{k_{2x}}L} \right)} \right) - i\left[ {{k_{1x}}\sin \left( {{k_{2x}}L} \right)} \right.\\ &&\left. { \times\, \left( {k_{2x}^2} \right. + {E^2}{{\cos }^2}\left( \varphi \right)} \right) - {k_{2x}}\sin \left( {{k_{1x}}L} \right)\\ &&\left. {\left. { \times \,\left( {k_{1x}^2+{E^2}{{\cos }^2}\left( \varphi \right)} \right)} \right]} \right\}, \end{array} \end{equation} \tag{ 12 }$$

Based on eq. (12), we know that ${T_{K'K\left( {KK'} \right)}} = {\left| {{t_{K'K\left( {KK'} \right)}}} \right|^2}$ is invariant with respect to φ. When ${V\ne 1}$ is present, however, the symmetry of ${t_{K'K }}~({t_{K K'}})$ about φ is broken due to $\sin \left( \varphi \right)$ in eq. (6) and eq. (7). This is one of our main results. In fig. 3, we also note that ${T_{K K }}\left(\varphi \right) = {T_{K'K'}}\left( { \varphi } \right)$ and ${T_{K' K }}\left( \varphi \right) = {T_{K K'}}\left( { - \varphi } \right)$ which arise from $t_{kk}\left( \varphi \right) = {{t}_{{K'K'}}} \left( \varphi \right)$ and ${{t}_{{K'K}}}\left(\varphi \right) = {{t}_{{KK'}}}\left( { - \varphi } \right)$ in eq. (6) and eq. (7).

In analogy with the differential spin transmission for spintronics [56–60], we define the differential valley transmission ${P_T} = \frac{{\left( {{T_{KK}} + {T_{KK'}}} \right) - \left( {{T_{K'K}} + {T_{K'K'}}} \right)}}{T}$ that describes the difference of transmission for the K and $K'$ states through the scattering region with the total transmission $T = {T_{KK}} + {T_{KK'}} + {T_{K'K}} + {T_{K'K'}}$. In fig. 4, we show the contour plot of the differential valley transmission P_T and the total transmission T in the $(E,k_y)$-plane for V = 2 (fig. 4(a) and (b)) and in the $(V,k_y)$-plane for E = 3 (fig. 4(c) and fig. 4(d)). Here we set L = 3, $\Delta_1=1$ and $\Delta_2=1$. In fig. 4(a) and fig. 4(c), we note that the differential valley transmission P_T is quite large and can change between −1 and 1 when adjusting the Fermi energy E as well as the electronic potential V. Moreover, we also note that when the incident angle is moved from $-\varphi$ to $+\varphi$, the positive and negative peaks of P_T are present. This means that the sighs of P_T can be switched by varying the incident angle φ. However, P_T is odd with respect to φ, namely, ${P_T}\left( \varphi \right) = -{P_T}\left( { - \varphi }\right)$. Therefore, we can use this result to choose preferential directions along which the valley polarization of an initially unpolarized electron is strongly enhanced. On the other hand, the total transmission T shows the symmetry of $T\left( \varphi \right) = T\left( { - \varphi } \right)$ and oscillates with the Fermi energy E as well as the electronic potential V, as shown in fig. 4(b) and fig. 4(d).

Zoom In Zoom Out Reset image size

Finally, we discuss the variation of the valley polarization $\textbf{P}= \left( {{P_x},{P_y},{P_z}} \right)$ of the outgoing current and the charge conductance G as a function of the Fermi energy E for V = 2 (fig. 5(a) and fig. 5(b)) and electronic potential V for $E'=3$ (fig. 5(c) and fig. 5(d)) under several values of the thickness of the VOG region $L=1,2,3$. Note that only the valley polarization along the x-direction P_x is shown in fig. 5. This is because the y and z components of the valley polarization P are zero. These characteristics can be understood by using the eq. (6), eq. (7) and eq. (9). After carrying out a little tedious algebra, we find that ${\text{Im}}\left({{{t}_{K'K}}{t}_{KK}^ * +{{t}_{{K'K'}}} {t}_ {{KK'}}^ * } \right)$ is an odd function of the injection angle φ, thus P_y is zero. As shown in fig. 4(a) and fig. 4(c), ${P_T} = \frac{{\left( {{T_{KK}} + {T_{KK'}}} \right) - \left( {{T_{K'K}} + {T_{K'K'}}} \right)}}{T}$ is odd with respect to φ, so ${P_z} = \frac{{\int_{ - \frac{\pi }{2}}^{ - \frac{\pi }{2}} {\left[ {\left( {{T_{KK}} + {T_{KK'}}} \right) - \left( {{T_{K'K}} + {T_{K'K'}}} \right)} \right]\cos \left( \varphi \right)}\text{d}\varphi }}{{\int_{ - \frac{\pi }{2}}^{ - \frac{\pi }{2}} {T\cos \left( \varphi \right)} \text{d}\varphi }}$ is also zero. ${\text{Re}} \left( {{{t}_{K'K}}{t}_{KK}^ * +{{t}_{{K'K'}}}{t}_{{KK'}}^ * } \right)$, however, is an even function of the incident ϕ. Therefore, the non-zero P_x can be generated. This is another of our important results. Moreover, a new feature may be noted that the sign of the valley polarization component P_x can switch from positive to negative and its magnitude can be tuned with the Fermi energy E or the electronic potential V induced by a gate voltage in the VOG region, as shown in fig. 5(a) and fig. 5(c). From fig. 5(b) and fig. 5(d), we can see that the conductance spectrum G can form the obvious resonance peaks because the total transmission T oscillates with the fermi energy E or the electronic potential V, as displayed in the bottom panels of fig. 4.

Zoom In Zoom Out Reset image size

In summary, we have investigated valley transports in the NG/VCG/NG junction. We have shown that the valley-dependent transmission probability in this junction exhibits considerable angle-dependent features. We also find that the differential valley transmission P_T is an odd function of the incident angle while the total transmission probability is an even function of the incident angle. Moreover, we also obtain that the magnitude and sign of the valley polarization component P_x of the outgoing current can be effectively controlled by slight changes in the Fermi energy E or the electronic potential V in the VOG region. These results offer some helpful strategies to design the graphene valleytronics devices.

This work is supported by the State Key Program for Basic Research of China under Grants No. 2011CB922103 and No. 2010CB923400, and the National Natural Science Foundation of China under Grants No. 11174125, No. 11074109, and No. 11447112, the Scientific Research Program Funded by Shaanxi Provincial Education Department Grant No. 15JK1132, the Opening Project of Shanghai Key Laboratory of High Temperature Superconductors Grant No. 14DZ2260700, and the Scientific Research Foundation of Shaanxi University of Technology Grant No. SLG-KYQD2-01.