Switchable valley injection into graphene

The generation and control of the valley-polarized current play the key roles in valleytronics applications, but still remain as tough challenges. We propose a general idea of switchable valley injection, and predict theoretically that a ferromagnet-covered graphene junction would take the role of a valley injector, whose on/off state can be easily switched only by changing the magnetic direction. This scheme can be extended to other two-dimensional crystals and may lead to an alternative path for valleytronic practical applications.

Figure 1

Model and reciprocal spaces. (a) Model of the nickel-sandwiched graphene junction. The current is presented by a red arrow. (b) The supercell of the left lead. The crystal constants are $a, b$, and $c$. The blue and gray balls denote nickel and carbon atoms, respectively. (c) Reciprocal space of (b), stacked by a series of 2D hexagonal BZs with $k_z$ varying from 0 to 1 (the unit is $2\pi /c$).

Figure 2

Valley-splitting band structures. (a–d) Valley-splitting band structures of the nickel-graphene hybridized system in the left lead: (a) without SOC; (b) $M\parallel X$ with SOC; (c) $M\parallel Z$ with SOC; (d) a comparison between (b) and (c). In (a–c), the red and blue lines represent the bands of the valleys $K$ and $K^{\prime }$, respectively. In (d), the blue lines denote the bands in (b), and the green lines denote the average band energy defined as $\left(E_K+E_{K^{\prime }}\right)/2$ in (c). (e) A schematic diagram showing the process of the valley splitting. The red (blue) cones represent the valley $K\left(K^{\prime }\right)$, on which the up (down) purple arrows represent the directions of $M_v$. The green balls represent electrons, and the yellow arrows on the green balls represent the directions of $M_s$.

Figure 3

Transmission coefficients. The $k$-resolved transmission coefficients in the transverse BZ ($k_x-k_z$) at the energy of 30 meV, when (a) $M\parallel X$; (b) $M\parallel Z$. $k_x$ and $k_z$ have the units of $2\pi /a$ and $2\pi /c$, respectively, where $a$ and $c$ are the crystal constants mentioned in Fig. 1. (c) The valley transmission ratio ($\mathbf{\gamma }$) when $M\parallel Z$. The values are painted with colors. (d) The band structure with $k_z=0.5$ when $M\parallel Z$. The bands close to valleys $K$ and $K^{\prime }$ are drawn by red and blue lines, respectively. The cyan-rectangle-marked band segment at $K$ contributes to the red area in the transmission ratio plot in (c).

Figure 4

Valley-polarized transport properties. The valley-polarized $I-V$ characters when (a) $M\left|\right|X$ and (b) $M\left|\right|Z$. (c) The valley-dependent AMR ($\mathbf{\delta }$) of the $K$ and $K^{\prime }$. In (a–c), the red downward triangles and blue diamonds denote the valleys $K$ and $K^{\prime }$, respectively. (d) The valley-injecting efficiency $\left(\eta \right)$, when $M\left|\right|X$ (blue diamonds) and $M\left|\right|Z$ (red downward triangles).

Figure 5

Valley on/off switch system. (a) At off and (b) at on states. The green arrows indicate the magnetic directions. The valley-polarized currents are represented by the red and the blue arrows, and the carriers with opposite valley DOF are represented by the red and blue cones, respectively. Voltmeters are utilized to detect and measure the valley Hall voltages.