Fully Valley-Polarized Electron Beams in Graphene

We propose a device to break the valley degeneracy in graphene and produce fully valley-polarized currents that can be either split or collimated to a high degree in a experimentally controllable way. The proposal combines two recent seminal ideas: negative refraction and the concept of valleytronics in graphene. The key new ingredient lies in the use of the specular shape of the Fermi surface of the two valleys when a high electronic density is induced by a gate voltage (trigonal warping). By changing the gate voltage in a $n\mathrm{\text{−}}p\mathrm{\text{−}}n$ junction of a graphene transistor, the device can be used as a valley beam splitter, where each of the beams belong to a different valley, or as a collimator. The result is demonstrated through an optical analogy with two-dimensional photonic crystals.

Figure 1

(a),(b): The two limiting cases for the deposition of the gate over the graphene sample. The green area denotes the region doped with holes at energies where the TW becomes relevant. (a) Interface of the gate parallel to the zigzag orientation and (b) the armchair interface. (c),(d) represent the Fermi surface of the two valleys and the conservation of the parallel wave vector for the zigzag and armchair interface, respectively. For the zigzag junction, the refractions in the two valleys have different orientations. For the armchair interface, the transmission in one of the valleys is in the forward direction, and there is very little dispersion in the second valley.

Figure 2

(a) Potential profile of a $n\mathrm{\text{−}}p\mathrm{\text{−}}{n}^{-}$ junction and schematic band structure of one valley. (b) Theoretical ray tracing of the beams coming from a point source in a junction with the interface parallel to the zigzag orientation. $E_F=0.05\mathrm{eV}$, $V_1=0.10\mathrm{eV}$, and $V_2=-1.5\mathrm{eV}$. The beam splitting is noticeable. (c) Armchair junction. We now obtain a collimation perpendicular to the interface.

Figure 3

(a) Current versus voltage for two transmission angle ${\theta }_t$ of observation. (b) Transmission probability for the $K$ (blue lines) and $K^{\prime }$ (red lines) as a function of the output angle ${\theta }_t$ for different negative potentials $V_2$. The parameters are as follows: barrier width, $d=100\mathrm{nm}$; initial energy, $E_F=0.05\mathrm{eV}$; and initial potential, $V_1=0.14\mathrm{eV}$.

Figure 4

(a) Geometry of the trigonal PC of air holes and optical analogy of a $p\mathrm{\text{−}}n$ junction in graphene. (b) Brillouin zone and isofrequency lines for the two PCs with $n_1$ (red circles) and with $n_2$ (blue triangles). (c) Band diagram for a trigonal PC described in text with refractive index $n_1=3.04$ (red bands) and $n_2=2.65$ (blue bands). The two insets show details of the zone around the $K$ point where we observe that, for the wavelength $\lambda =4.08a_{\mathrm{PC}}$, $n_1$ corresponds to the conduction band, whereas $n_2$ is located at the TW valence band.

Figure 5

Maps of the electric field modulus of a wave originated at a point source located inside a PC with matrix refractive index $n_1$ and propagating through another PC of dielectric matrix refractive index $n_2$. (a) Armchair orientation, illustrating the beam splitting of the two different valleys $K$ and $K^{\prime }$. (b) Zigzag orientation. We can see a collimation of the beams since the curvature in the trigonal distorted band is smaller than the one of the circular isoline in the proximity of the Dirac point. The angular dispersion is low.