Nondegenerate chiral phonons in the Brillouin-zone center of $\sqrt{3}\times \sqrt{3}$ honeycomb superlattices

The theoretical finding on chiral phonons at Brillouin-zone corners (valleys) of two-dimensional honeycomb lattices and its experimental verification in monolayer tungsten diselenide [H. Zhu, J. Yi, M. Li, J. Xiao, L. Zhang, C. Yang, R. A. Kaindl, L. Li, Y. Wang, and X. Zhang, Science 359, 579 (2018)], have attracted wide attention in the study of phonon chirality very recently. In this paper, to make chiral phonons more measurable, the valley phonons were folded to Brillouin-zone center in $\sqrt{3}\times \sqrt{3}$ honeycomb superlattices, and near the center topological chiral phonons can be observed. The chiral phonons which are not the superposition of linear modes are nondegenerate and can be optically excited in helicity-resolved Raman scattering. Moreover, by adjusting the doping mass, the topological chiral phonons can be engineered in specific branches. Finally, in deformed $\sqrt{3}\times \sqrt{3}$ honeycomb superlattices, topological chiral phonons can also be found near the Brillouin-zone center. We believe that the findings of nondegenerate topological chiral phonons in the Brillouin-zone center help to enrich our understanding of chiral phonons and promote future applications in phononics.

Figure 1

(a) Schematic representation of the orthohexagonal $\sqrt{3}\times \sqrt{3}$ superlattices. The blue balls, red balls, and cyan balls represent sublattices 1, 3, and 5 with $m_a$, sublattices 2 and 6 with $m_b$, and sublattice 4 with $m_c$, respectively. The magenta lines represent lattice primitive cells basic vectors ${\mathbf{\alpha }}_1$ and ${\mathbf{\alpha }}_2$. (b), (c) Phonon dispersion and phonon polarization along the $k_x$-directions. The red solid, blue dash, olive dash dot, and orange short dash lines correspond to branches 5, 6, 7, and 8, respectively. (d) Contour plot denotes value of phonon polarization from $-1$ to $+1$ for branch 7. Here, $m_a=1, m_b=1.2$, and $m_c=1.5$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 2

At a fixed point ($k_x=\frac{1}{10a}, k_y=0)$ near the zone center $\mathbf{\Gamma }$. (a) Phonon vibration modes of branches 5, 6, 7 and 8 in $\sqrt{3}\times \sqrt{3}$ honeycomb superlattices. (b) Phonon vibration modes of branches 2 and 3 in honeycomb AB lattices. The gray balls represent original location of all sublattices in a unit cell, the arrow denotes rotation direction. Here, $m_a=1, m_b=1.2$ and $m_c=1.5$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 3

The phonon Berry curvature of branches 7 and 8 along the $k_x$-directions. Here, $m_a=1, m_b=1.2$ and $m_c=1.5$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 4

(a) At the $\mathbf{\Gamma }$ point, phonon dispersion of branches 5, 6, 7, and 8 versus $m_c$. (b) The phonon polarization of the four branches when $m_c=1.2$. (c)–(f) The phonon dispersion relation of the four branches at $m_c=2.0$, 1.0, 0.6, and 0.1, respectively. (g)–(j) The phonon polarization of the four branches at $m_c=2.0$, 1.0, 0.6, and 0.1, respectively. Here, $m_a=1$ and $m_b=1.2$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 5

(a) The phonon frequency difference between branches 5 and 6, branches 7 and 8 changes versus $m_c$ at a fixed point ($k_x=\frac{1}{100a}, k_y=0)$ which is near zone center $\mathbf{\Gamma }$. (b) The phonon polarization of the four branches changes versus $m_c$ at a fixed point ($k_x=\frac{1}{100a}, k_y=0)$. The inset shows a part of the phonon polarization in the $k_x$ direction (from $k_x=-\frac{1}{2a}$ to $k_x=\frac{1}{2a}$). Here, $m_a=1$ and $m_b=1.2$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 6

(a) Schematic representation of the deformed $\sqrt{3}\times \sqrt{3}$ superlattices. (b), (c) Phonon dispersion and phonon polarization along the $k_x$ directions. (d) Contour plot of the phonon polarization for branch 5. Here, $m_a=1, m_b=1.2$ and $m_c=1.5$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 7

(a) Schematic representation of the honeycomb AB superlattices. The blue balls and red balls represent sublattices 1, 3, and 5 with $m_a$, sublattices 2, 4, and 6 with $m_b$, respectively. The magenta lines represent lattice primitive cells basic vectors ${\mathbf{\alpha }}_1$ and ${\mathbf{\alpha }}_2$. (b), (c) Phonon dispersion and phonon polarization along the $k_x$ directions. The red solid, blue dash, olive dash dot, and orange short dash lines correspond to branches 5, 6, 7, and 8, respectively. (d) Contour plot of the phonon polarization for branch 7. Here, $m_a=1$ and $m_b=1.2$; the spring constants are $K_L=1$ and $K_T=0.25$.

Figure 8

The structure of one unit cell of hexagonal boron nitride structure with doping of one silicon atom.