Model for topological phononics and phonon diode

The quantum anomalous Hall effect, an exotic topological state first theoretically predicted by Haldane and recently experimentally observed, has attracted enormous interest for low-power-consumption electronics. In this work, we derived a Schrödinger-like equation of phonons, where topology-related quantities, time-reversal symmetry, and its breaking can be naturally introduced similar to the process for electrons. Furthermore, we proposed a phononic analog of the Haldane model, which makes the novel quantum (anomalous) Hall-like phonon states characterized by one-way gapless edge modes immune to scattering. The topologically nontrivial phonon states are useful not only for conducting phonons without dissipation but also for designing highly efficient phononic devices, like an ideal phonon diode, which could find important applications in future phononics.

Figure 1

(a) A two-dimensional honeycomb lattice whose inversion symmetry is broken by different atomic masses of the two sublattices (red and blue) and TRS is broken by Coriolis field $\mathrm{\Omega }$ or magnetic field $\mathbf{B}$. (b) The phonon dispersion for no symmetry breaking ($m_I=m_T=0$), where Dirac-like linear dispersions occur at the $K$ and $K^{\prime }$ points. TA, LA,TO, and LO denote transverse acoustic, longitude acoustic, transverse optical, and longitude optical modes, respectively. The phonon dispersions for (c) broken inversion symmetry ($m_I=0.1$) and (d) broken TRS ($m_T=0.1$), where a band gap is opened near the Dirac point. Chern numbers of separated bands are labeled. The same values of Chern numbers were obtained by a different computational method in a previous work [19]. (e) and (f) Distribution of the Berry curvature in the Brillouin zone for the two lowest bands displayed in (c) and (d), respectively.

Figure 2

(a) Schematic of a nanoribbon structure with one-way edge states. A nanoribbon including 20 zigzag chains is calculated with broken TRS ($m_T=0.1$). (b) The phonon dispersion (middle panel), and density of states projected onto the left and right edges (left and right panels, where large (small) values are colored blue and red (black). The backward-moving (blue) and forward-moving (red) edge modes appear only on the left and right, respectively. (c) Phonon transmissions as a function of frequency $\omega$ for varying strength of disorder $\mathrm{\Delta }$. The disorders are introduced into the center part of a length of 20 unit cells by selecting atomic masses randomly within $[1-\mathrm{\Delta },1+\mathrm{\Delta }]$. Results of $\mathrm{\Delta }=0,0.1,0.2$ are denoted by black, red, and blue lines, respectively. The phonon transmission decreases gradually with increasing $\mathrm{\Delta }$, except for the one-way edge states within the bulk gap (shaded yellow).

Figure 3

(a) A phase diagram showing that the Chern number $\mathcal{C}$ of the Dirac gap is tuned by the two mass terms $m_I$ and $m_T$. (b) Schematic of patterning one-way edge states (denoted by red lines) at the interface between regions of $\mathcal{C}=0$ and $\mathcal{C}\ne 0$. (c) The phonon dispersion of $m_I=m_T=0.05$, in which one Dirac point at the $K^{\prime }$ valley is preserved and the other one at the $K$ valley is gapped as illustrated in the inset. This is an example showing that the band degeneracy of the $K$ and $K^{\prime }$ valleys is split by breaking inversion symmetry and TRS simultaneously.

Figure 4

(a) and (b) Phonon transmission functions for transport from terminal A to terminal B (red solid line) and from B to A (blue dashed line) in two-terminal and three-terminal transport systems ($m_I=0,m_T=0.1$). Results are shown only for phonon frequencies within the Dirac gap, where the one-way edge states (denoted by red lines in the insets) exist. The former system gives zero diode effect, while the latter acts as an ideal phonon diode. (c) A schematic diagram illustrating the concept of selective scattering for enhancing the diode effect. The real-space distribution of the forward- and backward-moving states (${\psi }_{\mathbf{k}}$ and ${\psi }_{-\mathbf{k}}$, denoted by red and blue, respectively) could be separated in real space when breaking TRS. Then, scatters (denoted by black dots) can be selectively placed on the transport path of the forward-moving state, introducing significant scattering for ${\psi }_{\mathbf{k}}$ but not ${\psi }_{-\mathbf{k}}$.