Valley Filtering and Electronic Optics Using Polycrystalline Graphene

In this Letter, both the manipulation of valley-polarized currents and the optical-like behaviors of Dirac fermions are theoretically explored in polycrystalline graphene. When strain is applied, the misorientation between two graphene domains separated by a grain boundary can result in a mismatch of their electronic structures. Such a discrepancy manifests itself in a strong breaking of the inversion symmetry, leading to perfect valley polarization in a wide range of transmission directions. In addition, these graphene domains act as different media for electron waves, offering the possibility to modulate and obtain negative refraction indexes.

Figure 1

Atomistic structure of graphene grain boundary systems (a) $(2,1)|(1,2)$ and (b) $(0,7)|(3,5)$. Charge carriers transmitted across the boundary under a uniaxial strain of magnitude $\sigma$ and direction $\theta$ are considered. (c) Diagrams illustrating the strain effects on the band structure of graphene domains (left) and the momentum conservation rule (right).

Figure 2

(a),(b) Transmission probability in the two valleys and (c),(d) valley polarization as a function of incident angle for energy $ϵ=0.3\mathrm{eV}$ and different strains. (a),(c) and (b),(d) present the data obtained in the systems GB1 (for strain direction $\theta =45°$) and GB2 (for $\theta =22°$) in Figs. 1 and 1, respectively.

Figure 3

Strain-induced modulation of refraction index in the (a) GB1 and (b) GB2 systems extracted from the data in Fig. 2. (c) Diagrams illustrating the properties of electron beams with negative refraction [particularly, for $\sigma =3\%$ in (a)] and color maps showing the transmission function for different outgoing angles.

Figure 4

(a) Conductance ${\mathcal{G}}_{D,D^{\prime }}$ (left axis) in unit of $(2e^{2}W/3\pi h{a}_0)$ ($a_0=1.42\AA$) and valley polarization $P_{\text{val}}$ (right axis) as a function of outgoing angle ${\varphi }_2$ for Fermi level $E_F=0.3\mathrm{eV}$ and at different temperatures. ($E_F$, ${\varphi }_2$) maps of (b),(c) ${\mathcal{G}}_{D,D^{\prime }}$ and (d) $P_{\text{val}}$ at 300 K. All data were obtained for $\sigma =3\%$ in the GB1 system. (e) Simple schematic of multiple electrode devices for measuring the directional currents.