Theoretical Study On Phonon Nodal Lines In Solid Materials

The discovery of topological states in the electronic band structure of materials opens a new chapter in condensed matter physics and materials science.With the deepening of the research,the topological state has gradually expanded from electron to phonon.Phonon materials are highly sought after by researchers because their topological states are not affected by spin orbit coupling and can be observed in the full frequency range.The current topological phonon materials are mainly divided into three types:nodal point,nodal line and nodal plane.Phonon nodal line material has rich physical states in momentum space,such as closed nodal line,open nodal line,skeleton nodal line,chain nodal line,etc.This article,on the basis of first-principles calculation,combined with k·p model,the study of phonon section of line of solid materials,the concrete content is as follows:Firstly,based on the first-principles calculation,nodal-chain,nodal-net and nodal-cage formed by the intertwining of nodal lines are investigated.The study of the phonon spectrum of Li Al Se₂in the Pna2₁space group has revealed that it forms a network of nodal lines along both directions,U-R and U-Z,in the Brillouin zone in the vicinity studied.The phonon spectrum of Na Mg H₃in the Pnma space group is found to form nodal-chain in the planes k_x-k_z and k_x-k_y and Dirac nodal-line along the S-R path.The nodal-chain and Dirac nodal-line have clean surface states in the（001）and（010）planes.The phonon spectrum of Au Br in the P4₂/ncm space group is found to have nodal-cage on the k_z=0,k_z=π,k_x=0andk_y=0 planes.Further calculations show that nodal-net,nodal-chain and nodal-cage all have non-zero Berry phases and that these complex nodal states are topologically non-trivial.Surface state calculations of the above materials reveal that they all have clean surface states.Secondly,the coexistence of open and closed nodal lines in phonon materials is investigated.Based on the first-principles calculation,the coexistence of the two types of nodal lines is achieved for Co As S of space group Pca2₁.The existence of closed nodal lines in the k_y=0 and k_y=π planes is connected by the existence of open nodal lines in the k_y=0 plane with contact points at P1 onΓ-Z and P2 on X-U respectively,together they form a complex nodal lines.Coexistence of open and closed nodal lines of Na2Cu P of the cmcm type,with point P3 onΓ-Y belonging to the（110）plane and point P5 on the Z-B₀path belonging to the（110）plane.The open and closed nodal lines intersect at point P4 on theΓ-Z path,which belongs to both the（110）and（1 10）planes.As a result,the closed and open nodal lines in Na₂Cu P can form a complex nodal line structure.It is also noted that both the closed and open nodal lines present at this point have a non-zero Berry phase,and that both Co As S and Na₂Cu P,have clean surface states.Finally,the interconversion between open and closed nodal lines is investigated.Ba（Ag S）₂and Ca（Zn P）₂,two different materials in the same space group,are used as examples.Ba（Ag S）₂has a closed nodal line,formed by the points P1 on path K-Γand P2 on path H-A,respectively,and Ca（Zn P）₂has open nodal lines,consisting of a P3 point on path K-Γand a P4 point on path H-A together forming a serpentine open nodal line.It is found that both closed and open nodal lines have a non-zero Berry phase.Moreover,calculations of the surface Green’s functions revealed that Ba（Ag S）₂and Ca（Zn P）₂exhibit extremely clean surface states in both the（010）and（100）planes,respectively.Finally a simple tight-binding model was constructed by considering a_orbital at the Wyckoff position of 2d and reconstructing the Hamiltonian,where the Hamiltonian includes the hopping parameter for the interlayer interactions and characterises the intra-layer interactions.When there are no interlayer interactions,the nodal line will be an open nodal line with a topological classification ofZ³（0,0,1）Conversely,when the interlayer interactions are strong enough,the nodal line will become a closed nodal line.