$T_{3}$Pb$_{2}Ch_{2}$ ($T$=Pd,Pt and $Ch$=S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

Shandite with Ni $_{3}$ Pb $_{2}$ S $_{2}$ chemical formula and R $\overset{ˉ}{3}$ m symmetry, contains the kagome sublattice formed by the transition metal atoms. Recent experimental results confirmed the possibility of successfully synthesizing Pd $_{3}$ Pb $_{2} C h_{2}$ ( $C h$ =S,Se) with the same structure. In this paper, we theoretically investigate the dynamical properties of such compounds. Furthermore, we study the possibility of realizing Pt $_{3}$ Pb $_{2} C h_{2}$ with the shandite structure. We show that the Pd $_{3}$ Pb $_{2}$ {\it Ch} $_{2}$ and Pt $_{3}$ Pb $_{2}$ S $_{2}$ are stable with R $\overset{ˉ}{3}$ m symmetry. In the case of Pt $_{3}$ Pb $_{2}$ S $_{2}$ , there is a soft mode, which is the source of the structural phase transition from R $\overset{ˉ}{3}$ m to R $\overset{ˉ}{3}$ c symmetry, related to the distortion within the kagome sublattice. We discuss realized phonon nodal lines in the bulk phonon dispersions in upper frequency modes. We show that the shandite structure can host the phonon surface states, with strong dependence by the surface kind. Additionally, chiral phonons with circular motion of the Pb atoms around the equilibrium position are realized.

$T_{3}$ Pb $_{2} C h_{2}$ ( $T$ =Pd,Pt and $C h$ =S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

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$T_{3}$ Pb $_{2} C h_{2}$ ( $T$ =Pd,Pt and $C h$ =S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

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T3T3​Pb2Ch22​Ch2​ (TT=Pd,Pt and ChCh=S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

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T3T3​Pb2Ch22​Ch2​ (TT=Pd,Pt and ChCh=S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

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$T_{3}$ Pb $_{2} C h_{2}$ ( $T$ =Pd,Pt and $C h$ =S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

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$T_{3}$ Pb $_{2} C h_{2}$ ( $T$ =Pd,Pt and $C h$ =S,Se) with transition metal kagome net: Dynamical properties, phonon nodal line, phonon surface states, and chiral phonons

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