Theoretical Study Of Topological Phonon Coexistence States In Solid Materials

In recent years,topological phonon materials have been a hot topic in condensed matter physics,and many types of topological phonon materials have been discovered in theory and experiments.Each type of topological element has different physical properties.The novel physical properties of solid materials with topological phonon coexistence state are worth further study.Unlike electronic systems,topological phonons are not restricted by the Pauli exclusion principle.Therefore,the properties of topological phonons can be studied in the whole frequency range of materials,which provides a basis for the study of topological phonon coexistence states.However,the current research focuses on single topological types of materials,and there are few reports on phonon coexistence states in multi-types of topologies.The reason for this situation is that the coexistence of multiple topological phonon types tends to lead to a large number of phonon bands with high overlap,and the surface states of the materials are covered.The Fermi arc（drumhead-like state）of nodes（nodal lines）cannot be observed,which will bring trouble to the experimental confirmation work.Therefore,to study the topological phonon coexistence states of materials,it is necessary to meet the requirements that the materials are easy to prepare,the phonon band is"clean,"and the surface state projected onto the plane is transparent to ensure the experimental verification of the topological properties of materials.Based on density functional theory,four kinds of topological phonon coexistence states are studied by first principles and surface Green’s function.First,taking KCuS,a single material with space group number 62,as an example,two coexisting states are proposed:the coexistence of the Weyl nodal line and Dirac nodal line;drumhead-like surface states and torus surface states coexist.By first-principles calculation,it is found that there are Dirac nodal lines along the S-R path in the phonon spectrum of KCu S,and the₌0 plane has a Weyl nodal line.The phonon bands of Dirac and Weyl nodal line are almost flat and well separated from other phonons,and the Weyl nodal lines hinge in momentum space.Based on the symmetry analysis,the Dirac nodal line and the Weyl nodal line are symmetrically related,and the coexisting states of the two nodal lines are found in the phonon dispersion of the9)-type real materials Ba Si₂,Ti B,and Zr Si.In addition,based on the calculation of surface Green’s function,clear torus surface states and drumhead-like surface states appear on the surfaces[100]and[001],respectively.The reason for different surface states is explained by calculating the Zak phase of each surface.This work proposed for the first time the coexistence of nodal and surface states of different degenerates in a single material.Therefore,9)-type KCu S is a good platform for studying the entanglement of Dirac nodal and Weyl nodal lines and realizing torus and drumhead-like surface states.Second,the coexistence of zero-dimensional,one-dimensional,and two-dimensional degeneracy of phonon bands was first discovered in the phonon dispersion of a single material structure,4₂/-type Sn O₂.Based on the calculation of first principles,it is found that in the high symmetric paths A-M andΓ-Z,the Dirac point of zero degeneracy and the triple degeneracy point appear respectively,the one-dimensional linear Weyl nodal of double degeneracy appears in the Z-R path,and the nodal surface of two-dimensional degeneracy appears in the_/=plane.By calculating the Berry phase of a linear nodal line,the nodal line is a notrivial nodal line with Berry phase valueπ.By phonon band,irreducible representation,nodal point,nodal lines,and nodal planes are protected by symmetry.In addition,there are no surface states except the nodal surface,and both nodal lines have obvious Fermi arc and drumhead-like surface states.It is worth mentioning that there have been more than ten kinds of high-purity preparation methods for4₂/-type Sn O₂in the experiment.It is found that phonon bands forming nodal point,nodal lines,and nodal surface are well separated from other phonons,which is conducive to the observation of bulk phonons by inelastic X-ray scattering or neutron scattering in the experiment.The surface states of nodal points and nodal line can be verified by high-resolution electron energy loss spectroscopy,helium atom scattering,or terahertz spectroscopy.Therefore,4₂/-type Sn O₂is an excellent vector for studying zero-dimensional,one-dimensional,and two-dimensional subband degenerate coexistence.Finally,the coexistence of three-nodal surfaces and Dirac nodal-line net are found in the phonon dispersion of the materials with spatial groups 61 and 205.The phonon dispersion of9)-type Zn Sb and9)3-type Ru S₂,P₂Pt,and Os S₂is calculated,and it is found that the phonon dispersion of Zn Sb will form nodal surface in the_//=plane,and Dirac nodal line will appear on S-R,U-R,and T-R paths.These nodal lines will converge at R,forming a net of nodal lines in momentum space.The phonon spectrum of Zn Sb is still stable under the uniform stress of 5%and-5%,while the Dirac nodal-line net and the three-nodal surfaces still exist.Three distinct drumhead-like surface states of Dirac nodal lines can be observed at the[100]and[001]planes.Similarly,coexisting states of three-nodal surfaces and Dirac nodal lines can also be found in phonon dispersion of Ru S₂,P₂Pt,and Os S₂,and clear drumhead-like surface states of Dirac nodal lines can be observed in the plane[001].Four realistic materials have obvious surface states,which will be conducive to experimental detection.Based on the irreducible phonon representation and symmetry analysis,it is found that the three-nodal surface and Dirac nodal-line net in phonon dispersion of the above four materials are protected by symmetry.Therefore,the materials proposed above are ideal for studying solid materials’coexistence states of the three-nodal surface phonons and the phononic Dirac nodal-line net.