Valley filtering effect of phonons in graphene with a grain boundary

Due to their possibility to encode information and realize low-energy-consumption quantum devices, control and manipulation of the valley degree of freedom have been widely studied in electronic systems. In contrast, the phononic counterpart—valley phononics—has been largely unexplored, despite the importance in both fundamental science and practical applications. In this work, we demonstrate that the control of “valleys” is also applicable for phonons in graphene by using a grain boundary. In particular, perfect valley filtering effect is observed at certain energy windows for flexural modes and found to be closely related to the anisotropy of phonon valley pockets. Moreover, valley filtering may be further improved using Fano-like resonance. Our findings reveal the possibility of valley phononics, paving the road towards purposeful phonon engineering and future valley phononics.

Figure 1

(a) The atomic configuration used for phonon transport calculation. Three horizontal long blocks represent three unit cells along the $y$ direction. The red-shaded and blue-shaded areas on two sides mark the left (L) and right (R) thermal leads, which are semi-infinite crystalline graphene sheets. As divided by vertical dotted lines, each principal layer of both thermal leads consists of eight atoms in a unit cell. (b) Phonon dispersion of graphene calculated using RESCU [27]. (c) ${\mathbf{a}}_{1,2}$ are primitive lattice vectors of graphene and ${\mathbf{A}}_{1,2}$ are lattice vectors of an eight-atom unit cell of graphene as shown in the L/R thermal leads in (a). ${\mathbf{b}}_{1,2}$ and ${\mathbf{B}}_{1,2}$ are the reciprocal lattice vectors of graphene and the eight-atom unit cell, with the first Brillouin zones (BZs) shaded in blue and grey, respectively. [(d)–(e)] $\mathbf{k}$-resolved transmission at $\omega =200$ and $400{\mathrm{cm}}^{-1}$ shown in the folded $1^{\mathrm{st}}$ BZ (left) and unfolded BZ (middle). In the middle panel, the folded BZs are also indicated in dotted boxes, covering the hexagonal unfolded BZ. Contour plots showing the branch information [ZA (red), TA (green), LA (blue), ZO (cyan), and LO (magenta)] are shown in the right panel.

Figure 2

$\mathbf{k}$-resolved transmission function. (a) Schematic plots of $K$ valleys in the ZA/ZO (left panel) and TA/LO (right panel) branches of graphene. The hexagonal dashed lines indicate the first BZ of graphene. (b) $\mathbf{k}$-resolved transmission at $\omega =500,600,800$, and $1300{\mathrm{cm}}^{-1}$. Transmission of modes that have negative group velocities (left-going) is also drawn by time-reversal symmetry $[\mathrm{\Xi }(k_x,k_y)=\mathrm{\Xi }(-k_x,-k_y)]$ for clarity. Boundaries of the first BZ are marked by black solid lines.

Figure 3

Angular dependence of (a) transmission of valley $K$ (thin solid line) and $K^{\prime }$ (dotted thick line) phonons and (b) the corresponding valley polarization at 500 (ZA), 600 (ZO), 800 (TA), and 1300 (LO) ${\mathrm{cm}}^{-1}$. (c) Transmission plots with velocity vectors shown for valley $K$ and $K^{\prime }$ phonons at $500{\mathrm{cm}}^{-1}$. (d) Dependence of valley polarization on phonon wave number of the ZA and ZO branches for an injection angle of ${30}^{\circ }$ (red solid line) and ${40}^{\circ }$ (black dot solid line).

Figure 4

Schematic plot of the anisotropy-induced valley-polarized transmission. (Top) Under the mirror symmetry about the $x$ axis, transmission of one phonon valley can be inferred from that of the other valley. Anisotropic transmission naturally leads to valley-polarized transmission. (Bottom) When transmission of one valley also mirror-symmetric about the $x$ axis, transmission has no valley polarization.

Figure 5

(a) Illustration of the scattering wave functions of most-transmitted modes [indicated by arrows in Fig. 3] at $\omega =600,800,$ and $1300{\mathrm{cm}}^{-1}$. Effective oscillation amplitudes, which are defined as $u_{\mathrm{eff}}=\sqrt{u_x^2+u_y^2+u_z^2}$ are strongest in red and weakest in blue. (b) Transmission spectrum around $800{\mathrm{cm}}^{-1}$ at $K_y=-0.4008(2\pi /A_y)$, which leads to the transmission peak indicated by an arrow in Fig. 3 at $\omega =800{\mathrm{cm}}^{-1}$. (c) Fano-like resonance model and (d) a numerical result using $\gamma \left(\omega \right)=\omega {\gamma }_0,{\omega }_c=0.84{\gamma }_0,{\omega }_{\mathrm{defect}}=0.7{\gamma }_0,\text{and}t=0.05{\gamma }_0^2$.