Phononic metagrating for lattice wave manipulation

The control of the phonon-wave behavior is of prominent interest for applications in thermal management and the processing of quantum information. In this work, we design and characterize a phonon-based grating nanostructure in a suspended single-layer graphene. The anomalous transmission of lattice waves and the controllable angle of the transmitted wave can be achieved via a skillful design of the underlying nanostructure and the mode conversion can be monitored in situ. An interesting nonlocal effect emerges due to the inherently strong covalent bondings. These features distinctly differ from former scenarios with acoustic waves. This work paves the way for the regulation of phonon transport by designing phonon-based metagrating structures.

Figure 1

(a) Schematic design of the phononic grating structure embedded in a single-layer graphene. A normal incident lattice wave ($k_y^0=0$) is scattered into different diffraction orders after passing through the grating structure. The grating structure has a thickness of $D_x$, and a period length of $D_y$ along the $y$ direction. (b) Lattice structure of graphene in the real space. ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$ are the primitive-cell vectors, while A (blue) and B (red) represent two atoms ($j=1$ and $j=2$) in the primitive cell. The C–C bond length is denoted as $l_0$. (c) Phase changes of lattice waves passing through ${}^{14}\mathrm{C}$ layers with different lengths $L$. The inset shows the atomic structure in one segment, where the blue (red) atom represents ${}^{14}\mathrm{C}$ (${}^{12}\mathrm{C}$) atom. (d) Schematic of the functional unit for phase control. Top panel shows the discrete design by vertically stacking eight ${}^{14}\mathrm{C}$ layers with different lengths together, while the bottom panel shows the continuous design via a triangular ${}^{14}\mathrm{C}$ layer. All simulations in this work are carried out based on the continuous design.

Figure 2

The 2D FFT of different atoms (a) $j=1$ and (b) $j=2$ in the primitive cell for propagation of lattice wave that evolves a long time in pure suspended single-layer graphene. The top and bottom panel show the 2D FFT of the atomic displacement $u_y$ and $u_x$, respectively. The side-bar color codings represent the amplitude of the FFT results $B_{j\alpha }\left(\mathbf{k}\right)$ for the transmitted modes in the unit of $\AA$. Here, the TA mode with $k_0=$ 0.0373 ($2\pi /\AA$) is used as the normally incident wave packet.

Figure 3

(a) The distribution of the atomic displacement $u_y$ for the transmitted wave in real space. Here, $D_y=124.6\AA$ and $D_x=550$ $\AA$ are used in the simulations. (b) The 2D FFT of the atomic displacement $u_y$ (left panel) and $u_x$ (right panel) in the reciprocal space. The dashed (solid) line box represents the TA (LA) mode. Notice there exists weak $n=-3$ LA mode and $n=-4$ TA mode. (c)–(f) The inverse 2D FFT for different diffraction orders in real space. The arrow points to the propagation direction, and the angle denotes the diffraction angle. The side-bar color codings in (a) and (c)–(f) represent the displacement amplitude in real space, while that in (b) represents the amplitude of the FFT results in the reciprocal space. All color codings carry a dimension of length.

Figure 4

(a) The phonon dispersion of graphene at low frequency in ΓM direction from the lattice dynamics calculation. $k_0$ represents the wave-vector amplitude of the initially launched TA mode along $x$ direction, and $k_0^{{}^{\prime }}$ denotes the wave-vector amplitude of the corresponding LA mode with the same frequency. The solid horizontal line is plotted as a guide to the eye. The low-lying curve represents the ZA branch with quadratic dispersion. (b) and (c) show the dependence of transmission angle on the structure period $D_y$ for TA and LA modes, respectively. (d) The transmission angle vs. wavelength of the TA mode for a fixed $D_y=99.7 \AA$. The solid lines in (b) and (c) represent the calculation results of Eq. (1), while the symbols denote the simulation results.

Figure 5

The energy distribution of transmitted lattice wave. The side-bar color coding represents the amplitude of mode energy in the unit of ${\AA }^2$. Here, $D_y=124.6\AA$ and $D_x=550 \AA$ are used in the simulations, and the TA mode with $k_0=$ 0.0373 ($2\pi /\AA$) is used as the normally incident wave packet.

Figure 6

The effect of different atoms (a) $j=1$ and (b) $j=2$ in the primitive cell for the propagation of lattice wave in the suspended single-layer graphene with the metagrating structure. The top, middle, and bottom panels show, respectively, the spatial distribution of $u_y$, the corresponding 2D FFT, and the inverse 2D FFT of the $n=1$ mode. The side-bar color codings of the top and bottom panels represent the amplitude of transmitted modes in real space. The color codings for the middle panel represent the amplitude of the FFT results $B_{j\alpha }\left(\mathbf{k}\right)$ for the transmitted modes. All the color codings have the length dimension. Here, $D_y=124.6\AA$ and $D_x=550 \AA$ are used in the simulations, and the TA mode with $k_0=$ 0.0373 ($2\pi /\AA$) is used as the normally incident wave packet.

Figure 7

The 2D FFT of $u_y$ after the phonon wave passes through the metagrating structure with (a) $D_x=200 \AA$ and (b) $D_x=700 \AA$. The side-bar color coding represents the amplitude of the FFT results $B_{j\alpha }\left(\mathbf{k}\right)$ for the different transmitted modes in the unit of $\AA$. The angle and number of transmission modes remain the same, showing only minor variations in the intensity, as shown with the side bar. Here, $D_y=124.6\AA$ is used in the simulations, and the TA mode with $k_0=$ 0.0373 ($2\pi /\AA$) is used as the normally incident wave packet.

Figure 8

The 2D FFT of $u_y$ after the phonon wave passes through the metagrating structure with different widths $D_y$. (a) $D_y=49.8\AA$; (b) $D_y=62.3\AA$; (c) $D_y=74.8\AA$; (d) $D_y=87.2\AA$; (e) $D_y=99.7\AA$; and (f) $D_y=112.1\AA$. The side-bar color coding represents the amplitude of the FFT results $B_{j\alpha }\left(\mathbf{k}\right)$ for the transmitted modes in the unit of $\AA$. Here, $D_x=550 \AA$ is used in the simulations, and the TA mode with $k_0=0.0373$ ($2\pi /\AA$) is used as the normally incident wave packet.