Ideal nodal net phonons in Pn3¯m type Ag2O

The topological phonon state is a new field that has sparked much interest. Weyl phonons in FeSi, for example, have been theoretically proposed and experimentally identified. In this work, via the first-principle calculation, we prove the ideal nodal net phonons exist in a realistic material Ag₂O with $Pn\overline{3}m$ type structure. With the help of the Berry phase calculations, we find that the nodal net phonons in Ag₂O are topologically non-trivial. The phononic surface states are visible, which benefits the experimental detections. The results in this work contribute to the material realization of topological nodal net phonons. The author hopes the experimental verification of the novel topological phonons can be performed in the following investigations.

Introduction

Topological semimetals [1–10] with symmetry-protected bands crossing around the Fermi level have inspired enormous interest in condensed matter physics. As a typical family of topological semimetals, node-line semimetals [11–20] have high band degeneracy along a certain line in the Brillouin zone (BZ), and the resultant drumhead surface states at the boundary. Suppose more than one nodal line/ring appears in momentum space. In that case, these nodal lines/rings may form complex topological nodal structures in the three-dimensional (3D) Brillouin zone (BZ), such as nodal-chain [21–28], nodal-box [29], nodal-link [30–35], and nodal-net [36, 37] structures.

Topological state research has recently been extended to bosonic systems, such as photons in photonic crystals, phonon systems in 3D solids, and classical elastic waves in macroscopic artificial phononic crystals. In realistic materials, topological phonons [38–45] could play an essential role in thermal transports, electron-phonon coupling, and other phonon-related processes. Thus far, researchers have predicted some realistic materials [46–65] to host nodal net phonons. For example, using first-principles calculations, [66] discovered a nodal net state in the semiconductor copper chloride (CuCl). CuCl has a cubic crystal structure with the space group Pa3 (No. 205). In momentum space, the nodal net has a hexahedral shape and is made up of interconnected quadruple degenerate straight nodal lines. Moreover, with the help of first-principle calculations and symmetry analysis, [67] proposed the coexistence of the three-nodal surface and nodal net phonons in space groups with numbers 61 and 205. Some realistic materials, such as ZnSb with SG No. 61 and RuS₂, P₂Pt, and OsS₂ with SG No. 205, hosting three-nodal surface and nodal net phonons have also been identified by [67].

In this paper, based on first-principles calculations, we contribute to one more realistic material with ideal nodal net phonons. Ag₂O is Cuprite structured and crystallizes in the cubic $Pn\overline{3}m$ space group. Ag¹⁺ is bonded in a linear geometry to two equivalent O²⁻ atoms. O²⁻ is bonded to four equivalent Ag¹⁺ atoms to form corner-sharing OAg₄ tetrahedra. The lattice constants for the cubic Ag₂O are optimized via first-principle calculations. The obtained results from the calculations for the lattice constants are a = b = c = 4.81 Å, which are in good agreement with the experimental data [68], i.e., a = b = c = 4.73 Å. The crystal structure of the relaxed Ag₂O is shown in Figure 1A, where the Ag atoms and O atoms occupy the 2a, and 4c Wyckoff positions, respectively.

FIGURE 1

FIGURE 1. (A) The crystal structure of Ag₂O, (B) 3D bulk Brillouin zone (BZ), and the two-dimensional (2D) surface BZ.

Methods

The calculations for the realistic material Ag₂O were performed using the Vienna ab initio Simulation Package [69] and the framework of density functional theory. The calculation’s energy and force convergence criteria were set to 10⁻⁶ eV and −0.01 eV/A, respectively. The plane-wave expansion was truncated at 500 eV, and the entire BZ was sampled by a 7 × 7 × 7 Monkhorst-Pack grid. We used the PHONOPY code to generate the symmetry information and construct the constant force matrices for phonon spectra calculations. To calculate the phonon surface states, we used the WannierTools package [70] in conjunction with the iterative Green function method to construct the tight-binding model Hamiltonian.

Weyl nodal net phonons

Based on the determined lattice constants, the phonon dispersion for a 2 × 2 × 2 supercell of Ag₂O is shown in Figure 2A. Note that the phonon-related properties in this work are calculated based on the density functional perturbation theory (DFPT). According to Figure 2A, there is no imaginary frequency in the phonon spectrum, indicating the dynamical stability of cubic Ag₂O. Around the frequency of five THz, two phonon bands along X-R-Γ-M and Γ-X merged into one twofold degenerate phonon bands along the X-M path. We want to point out that a similar case can also be found around the frequency of 14 THz. Here, we only focus on the twofold degenerate phonon bands along the X-M path. The three-dimensional plot of the twofold degenerate phonon bands along the X-M is shown in Figure 2B, and one can see a cross-shaped nodal structure appears (see the dotted lines) in Figure 2B.

FIGURE 2

FIGURE 2. (A) Phonon dispersion of Ag₂O along the Γ-X-M-Γ-R-X|R-M paths. (B) The 3D plot of the cross-shaped nodal structure.

To better view the cross-shaped nodal structure in the phonon dispersion of Ag₂O, we plot the phonon band structure of the crossing branches in ky = π plane (See Figure 3A). The cross-shaped nodal structure in the k_y = π plane is formed from crossing two straight Weyl nodal lines. We select some symmetry points along the X-M path and show the phonon dispersions for Ag₂O along the a-b-a’, c-d-c’, e-f-e’, and g-h-g’ in Figure 3C. All the crossing points are doubly degenerate points with a linear phonon band dispersion (see Figure 3C). Note that the cross-shaped nodal structures can also exist in k_y = -π, k_x = ± π, k_z = ± π plane. That is, these straight nodal lines are perpendicular to each other in different planes, forming a Weyl nodal net in the 3D BZ, as illustrated in Figure 3B.

FIGURE 3

FIGURE 3. (A) A map of energy gaps between the two crossing branches around 5 THz in the ky = π plane. (B) A schematic diagram of Weyl nodal net in 3D BZ. (C) Phonon dispersions of Ag₂O along the a-b-a’, c-d-c’, e-f-e’, and g-h-g’ paths. (D) Phononic drumhead-like surface states in (010) surface.

Phononic surface states

To examine the topological nature of the Weyl nodal net phonons in Ag₂O, we calculate its Berry phase using the following formula: ${\gamma }_n={\oint }_C{A}_n\left(\mathbf{k}\right)\cdot d\mathbf{l}$, where $A_n\left(\mathbf{k}\right)=-i{{\displaystyle \sum }}_n〈u_n\left(\mathbf{k}\right)\left|{\nabla }_{\mathbf{k}}\right|u_n\left(\mathbf{k}\right)〉$ is the Berry connection. Calculated results show that the Weyl nodal net phonons host a π Berry phase, indicating topologically non-trivial behaviors. The non-trivial nodal net phonons will lead to phononic drumhead-like surface states. As shown in Figure 3D, we project the two Weyl points (belong to the Weyl net) into $\stackrel{\sim }{R}$ and $\stackrel{\sim }{X}$ surface points to the (010) surface (see Figure 2B). The phonon LDOS projected on the (010) surface BZ along $\stackrel{\sim }{R}$-$\stackrel{\sim }{X}$. Obviously, a clear drumhead-like surface state, connected by the projections of the Weyl points, is visible and marked by the black arrow. The bulk stats do not cover such a phononic surface state and would benefit the experimental detections shortly.

Summary and remarks

We show that nodal net phonons exist in Ag₂O using first-principles calculations. Straight lines constrained in the high-symmetry line X-M at the BZ boundary represent the nodal net. Because there is no spin in phononic systems, the nodal net phonons in Ag₂O are resistant to time-reversal symmetry breaking. Before closing the paper, we would like to point out that our results can also guide the investigations of the Weyl nodal net in spinless electronic systems (such as topological semimetals without the consideration of spin-orbital coupling). We present a spinless lattice model to demonstrate the existence of Weyl nodal net states in spinless materials with SG 224. A unit cell with one site (0,0,0) was considered in this spinless lattice model, and s orbital was placed on this site. The four-band tight-binding (TB) Hamiltonian is shown as follows: $\mathcal{H}=\left(\begin{array}{cccc}\mathrm{e}& \mathrm{A}& \mathrm{B}& \mathrm{C}\\ \mathrm{A}& \mathrm{e}& \mathrm{D}& \mathrm{E}\\ \mathrm{B}& \mathrm{D}& \mathrm{e}& \mathrm{F}\\ \mathrm{C}& \mathrm{E}& \mathrm{F}& \mathrm{e}\end{array}\right)$, where $A=4s\mathrm{Cos}\left(k_x\right)\mathrm{Cos}\left[\left(k_y+k_z\right)/2\right]$, $B=4s\cos \left(k_y\right)\mathrm{Cos}\left[\left(k_x+k_z\right)/2\right]$, $C=4s\mathrm{Cos}\left[\left(k_x+k_y\right)/2\right]\mathrm{Cos}\left(k_z\right)$, $D=4s\mathrm{Cos}\left[\left(k_x-k_y\right)/2\right]\mathrm{Cos}\left(k_z\right)$, $E=4s\mathrm{Cos}\left(k_y\right)\mathrm{Cos}\left[\left(k_x-k_z\right)/2\right]$, $\mathrm{F}=4\mathrm{s}\mathrm{C}\mathrm{o}\mathrm{s}\left({\mathrm{k}}_{\mathrm{x}}\right)\mathrm{Cos}\left[\left({\mathrm{k}}_{\mathrm{y}}-{\mathrm{k}}_{\mathrm{z}}\right)/2\right]$. When we set e = 0.4 and s = 0.1, the band structure of this spinless lattice model is shown in Figure 4. From Figure 4, one finds that the appearance of the Weyl nodal line along the X-M path further reflects the occurrence of the Weyl nodal net in 3D BZ.

FIGURE 4

FIGURE 4. Band structure of the spinless lattice model with SG 224.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

Calculating, writing, and researching are done by YL.

Funding

This work is supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-K202104101 and No. KJQN202204104) and the school-level Scientific Research Project of Chongqing Youth Vocational and Technical College (Grant No. CQY2021KYZ03).

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The reviewer HZ declared a shared affiliation with the author(s) add initials here unless all authors are concerned to the handling editor at the time of review.

Publisher’s note

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