Point Phonon Research Articles

Development of a genetic algorithm based interatomic potential and application in thermal conductivity study of ThO2 grain boundaries

ThO2 is a promising fuel for next-generation reactors due to its superior thermo-physical properties. In this study, we present a Coulomb-Buckingham-Morse interatomic potential for ThO2, developed using a homemade genetic algorithm (GA) method. The potential accurately captures fundamental properties, such as lattice constants, elastic constants, point defect energies, phonon spectra, and thermal conductivity. We apply this potential to perform a systematic study of thermal transport of twenty-one tilt grain boundaries (GBs) in ThO2, which have a significant effect on the macroscopic properties of polycrystalline materials. We demonstrate that the microscopic structure, GB excess energy, and GB disorder index are well correlated with the Kapitza conductance for a wide variety of GBs. Additionally, we discuss the effect of high temperature on the Kapitza conductance, where the GB structure undergoes disordering when approaching the melting temperature, resulting in an increase of thermal resistance. Finally, a detailed analysis of the phonon density of states is presented to reveal the mechanisms of thermal conductance in ThO2.

2D ZrS2: An experimentally realizable monolayer with multiple linear Weyl point phonons

In recent years, there has been a great deal of interest in predicting Weyl physics in phonons due to the unique properties of topological phonons. In this study, we postulate that ZrS2 is a candidate material with multiple Linear Weyl point (LWP) phonons at the K (or K') high-symmetry point in its phonon curves. The existence of an opposite quantized Berry phase at the K and K' valleys determines the topologically nontrivial nature of these LWPs. Fascinatingly, the nonzero quantized Berry phase at the two valleys produced phononic edge states on the (100) surface. On the one hand, our research emphasizes for the first time the topological phononic properties of 2D ZrS2. On the other hand, note that 2D ZrS2 is an experimentally feasible material, making it a concrete material platform for investigating and confirming the predicted topological physics.

Complete list of valley linear Weyl point phonons in two dimensions

The discovery of topological quantum states in two-dimensional (2D) systems is one of the most promising advancements in condensed matter physics. Linear Weyl point (LWP) phonons have been theoretically investigated in some 2D materials. Especially, Jin, Wang, and Xu [Nano Lett. 18, 7755 (2018)] proposed in 2018 that the candidates with threefold rotational symmetry at the corners of the hexagonal Brillouin zone can host LWP phonons with a quantized valley Berry phase. Note that all the candidates with hexagonal lattices may not host LWP phonons at $K$ (${K}^{\ensuremath{'}}$) high-symmetry points (HSPs). Hence a strategy for narrowing the search range for LWP phonons in 2D is highly required. This work provides an exhaustive list of valley LWP phonons at HSPs in 2D by searching the entire 80 layer groups (LGs). We found that the valley LWP phonons can be obtained at HSPs in 11 of the 80 LGs. Guided by the symmetry analysis, we also contributed to realizing the ideal 2D material with valley LWP phonons. We identified the existence of the valley LWP phonons in 11 2D material candidates with 11 LGs. This work offers a method to search for valley LWPs in 2D phononic systems and proposes 2D material candidates to obtain the valley LWP phonons.

Lattice dielectric properties of rutile TiO2 : First-principles anharmonic self-consistent phonon study

We calculate the lattice dielectric function of strongly anharmonic rutile ${\mathrm{TiO}}_{2}$ from ab initio anharmonic lattice dynamics methods. Since an accurate calculation of the $\mathrm{\ensuremath{\Gamma}}$ point phonons is essential for determining optical properties, we employ the modified self-consistent approach, including third-order anharmonicity as well as fourth-order anharmonicity. The resulting optical phonon frequencies and linewidths at the $\mathrm{\ensuremath{\Gamma}}$ point agree much better with experimental measurements than those from a perturbative approach. We show that the four-phonon scattering process contributes as much as the third-order anharmonic term to phonon linewidths of some phonon modes. Furthermore, incorporating the frequency dependence of phonon linewidth reveals that experimentally known but unidentified peaks of the dielectric function are due to the two-phonon process. This work emphasizes the importance of the self-consistent approach in predicting the optical properties of highly anharmonic materials.

Maximally charged single-pair multi-Weyl point phonons in P23 -type BeH2

The realization of multi-Weyl systems with the minimum nonzero number of Weyl points and the maximum charge number remains challenging in topology physics. In this work, based on first-principles calculations, we propose that ${\mathrm{BeH}}_{2}$ is thermodynamically, mechanically, and dynamically stable in a cubic crystal structure with the $P23$ space group. Importantly, this is the first work to report the appearance of single-pair multi-Weyl point phonons with the maximum charge number in $P23$-type ${\mathrm{BeH}}_{2}$. Furthermore, the number and charge of the Weyl points and the phononic surface modes can be tuned by applying $1%$ and $2%$ uniaxial strains along the [100] and [111] directions, respectively. Finally, we report that clean charge-two single-pair triple-point phonons appear in $P23$-type ${\mathrm{BeH}}_{2}$ and investigate the related chiral phonon transition under the uniaxial strains.

Topological phonons in Cs-Te binary systems

In this paper, we used symmetry analyses and first-principles calculations to discover seven Cs-Te binary systems that have different crystal structures and can host symmetry-enforced topologically nontrivial phonons: $P{2}_{1}{2}_{1}{2}_{1}\text{\ensuremath{-}}{\mathrm{Cs}}_{2}\mathrm{Te}, Pbam$-CsTe, $Pnma\text{\ensuremath{-}}{\mathrm{Cs}}_{2}\mathrm{Te}, Pm\overline{3}m$-CsTe, $Cmcm\text{\ensuremath{-}}{\mathrm{Cs}}_{2}{\mathrm{Te}}_{5}, Cmc{2}_{1}\text{\ensuremath{-}}{\mathrm{Cs}}_{2}{\mathrm{Te}}_{3}$, and $P{2}_{1}/c\text{\ensuremath{-}}{\mathrm{CsTe}}_{4}$. These phonons include charge-two Dirac point phonons, charge-one Weyl point phonons, quadratic contact triple point phonons, triple point phonons, butterflylike Weyl nodal line phonons, Dirac nodal line phonons, Weyl nodal loop phonons, multiple straight Weyl nodal line phonons, nodal cage phonons, one-nodal surface phonons, two-nodal surface phonons, and three-nodal surface phonons. Furthermore, the relationship between the crystal structure and the topological properties was thoroughly investigated in this study. More importantly, the Cs-Te binary systems clearly exhibit phononic arc-shaped, phononic nodal-line-shaped, phononic drumheadlike, and phononic torus surface states, which benefit experimental detections. Our theoretical results not only propose various experimentally prepared Cs-Te binary systems with exotic topological phonons and phononic surface states but also explain the structure-property relationship for Cs-Te binary systems based on symmetry analyses.

Dirac phonons in two-dimensional materials

Phonons are an ideal platform for realizing stable spinless two-dimensional (2D) Dirac points because they have a bosonic nature and hard-to-break time-reversal symmetry. It should be noted that the twofold degenerate nodal points in the phonon dispersions of almost all reported 2D materials are misclassified as ``Dirac points'' owing to a historical issue. The correct name for these twofold degenerate nodal points should be ``Weyl'' because 2D phononic systems are essentially spinless and because each twofold degenerate point is described by a Weyl model in two dimensions. To date, reports of fourfold degenerate Dirac point phonons in 2D materials are lacking. In this paper, we searched through all 80 layer groups (LGs) and discovered that Dirac phonons can be realized in 7 of the 80 LGs. Moreover, the Dirac points in the phonon dispersions of 2D materials can be divided into essential and accidental degenerate points, which appear at high-symmetry points and on high-symmetry lines, respectively. Guided by symmetry analysis, we identified the presence of Dirac phonons in several 2D material candidates with six LGs. This work offers a method for identifying Dirac phonons in 2D and proposes 2D material candidates for realizing Dirac phonons.

Dirac point phonons at high-symmetry points: Towards materials realization

Dirac point semimetals and related Fermi arc surface states are extremely important in topological electronic systems. In 2021, Chen et al. [Phy. Rev. Lett. 126, 185301 (2021)] proposed that Dirac points can appear on high-symmetry lines (HSLs) or at high-symmetry points (HSPs) in three-dimensional (3D) phonon systems as an extension from Dirac points in electronic systems. Inspired by this work, we present an exhaustive list of Dirac point phonons (DPPs) at HSPs in 230 space groups (SGs) by checking the encyclopedia of emergent particles in 3D crystals. The DPPs are divided into four categories: charge-zero (C-0) DPPs, charge-two (C-2) DPPs, quadratic DPPs (QDPPs), and cubic crossing DPPs (CCDPPs). 29 SGs, 6 SGs, 19 SGs, and 5 SGs are identified as candidate SGs for obtaining C-0 DPPs, C-2 DPPs, QDPPs, and CCDPPs at HSPs, respectively. Importantly, herein we contribute to the realization of materials with DPPs at HSPs and through the following: (i) we propose five realistic materials, namely, $Pna{2}_{1}$-type ${\mathrm{Li}}_{3}{\mathrm{AsS}}_{3}, P{2}_{1}$-type NiSbS, $P\overline{4}{2}_{1}c$-type ${\mathrm{NaBH}}_{4}$, and $Pm\overline{3}n$-type ${\mathrm{Ti}}_{3}\mathrm{Au}$ and ${\mathrm{Ta}}_{3}\mathrm{Sn}$ as materials with C-0 DPPs, C-2 DPPs, QDPPs, and CCDPPs at HSPs, respectively, based on the first-principles calculations. The surface states for these five materials are also examined in this work. (ii) We present 626 candidates with C-0 DPPs, 183 candidates with C-2 DPPs, 433 candidates with QDPPs, and 102 candidates with CCDPPs at HSPs, which were discovered by checking the phonon dispersions of 10 037 materials listed in the phonon database at Kyoto University. (iii) Due to the materials with SGs 103, 104, 106, 158, 184, 222, 223, 226, and 228 are not included in the phonon database at Kyoto University, we propose 18 candidates with C-0 DPPs, 5 candidates with QDPPs, and 8 candidates with CCDPPs at HSPs with above-mentioned SGs by screening the Inorganic Crystal Structure Database (ICSD). Therefore, our findings can be used to guide research into Dirac points at HSPs in 3D phonon systems.

Cubic Ca3I3P with ideal charge-two triple point

A recent study reported the enhancement of thermoelectric response due to the presence of triple point phonons [Phys. Rev. Mater 2, 114204 (2018)]. Because the topological triple point phonons field is still at its early stage, it is necessary to search for realistic materials with ideal triple point phonons. In this work, we show that cubic Ca3I3P is an excellent material to host the ideal charge-two triple points at high-symmetry points Γ and H. Because the charge-two triple points have a topological charge of |C| = 2, two arc-shaped surface states emanating from the projections of each charge-two triple nodal point can be found in cubic Ca3I3P. It is hoped that the ideal charge-two triple point phonons and the clean phononic arc-shaped surface states in Ca3I3P can be experimentally confirmed shortly.

Ideal phononic charge-two nodal point and long nontrivial surface arcs in Na2Zn2O3

Recently, there has been significant interest in exploring the chiral quasiparticles in phonons, which describe the atomic lattice vibrations in solids. In this work, using first-principle calculation, we select a realistic material Na2Zn2O3 as an example to demonstrate that it is an ideal candidate with charge-two Dirac point phonons and charge-two Weyl point phonons at high-symmetry points A and Γ, respectively. The phononic charge-two nodal points in Na2Zn2O3 are visible and almost ideal. That is, there are no other phonon bands nearby. Moreover, nontrivial phononic surface arcs span the whole surface Brillouin zone. Such clean and long nontrivial arc-shaped phononic surface states benefit the experimental detection. The current work is hoped to guide the investigations of chiral nodal points in phononic systems.

Phononic nodal point in two dimensions: A mini-review

In recent decades, nodal point states in electronic systems have attracted significant interest in current research. Recently, the conceptual framework of nodal point states has been extended to bosonic systems, especially the phononic one. It is well known that the nodal point states may exist much more universally in materials other than topological electronic systems. Fortunately, a series of nodal point phonons are reported in three-dimensional realistic materials, and some are certified in experiments. However, to our knowledge, the study of phononic 2D nodal points is still relatively primitive. Hence, a highlight of research in the emerging area covering approximately the last two-three years is necessary. This mini-review will summary the recent advances in the phononic nodal point in two dimensions. Some typical examples, including graphene, CrI3 monolayer, YGaI monolayer, TiB4 monolayer, Ti2P monolayer, and Cu2Si monolayer, are concluded in this mini-review. The topological properties and possible applications of these material candidates are also summarized.

Polarized Raman spectroscopy on topological semimetal Co3Sn2S2

AbstractWe present polarized Raman spectroscopy of the topological semimetal Co3Sn2S2, which was recently shown to host a Weyl semimetal phase. Stokes Raman spectra were obtained with the incident light parallel to the c‐axis of Co3Sn2S2. Two major phonon Raman peaks were observed at 289 and 386 cm−1 over continuous background emission signals. The intensity of the low‐wavenumber (289 cm−1) peak showed no polarization dependence. The high‐wavenumber (386 cm−1) peak and the continuous background signal were strongly polarized in the incident light polarization direction. These responses were almost independent of the in‐plane crystal orientation to the incident polarization, as is the manifestation of the point group symmetry of the unit cell of Co3Sn2S2. According to the group theory and Raman tensor analyses, the low‐ and high‐wavenumber Raman signals are attributed to Γ point phonon modes with and symmetries, respectively. Furthermore, line shape analyses revealed that the high‐wavenumber mode exhibited asymmetric peak feature well described by the Breit–Wigner–Fano function. These results suggest the Fano resonance between the phonon scattering with the continuous electronic background associated with low energy excitations near the Fermi energy. The clarified phonon energies and symmetries, as well as the electronic contribution to the Raman scattering, will not only be useful as a fingerprint to readily verify the experimentally grown or theoretically calculated crystal structure but also suggest importance of Raman spectroscopy as an effective tool to study low energy excitations and their interactions in Co3Sn2S2.

Material realization of double-Weyl phonons and phononic double-helicoid surface arcs with P213 space group

Unconventional chiral quasiparticles with twofold, threefold, and fourfold degeneracies and Chern number ($C$) = $\ifmmode\pm\else\textpm\fi{}2$ have recently sparked interest because of their topological chirality in momentum space. Herein, based on first-principles calculations and symmetry analysis, we propose 14 synthesized topological phonon systems with space group $P2{}_{1}3$ (No. 198) that host two types of almost-ideal double-Weyl points, which are the fourfold degenerate charge-2 Dirac point and the threefold degenerate spin-1 Weyl point. Based on the projected local density of states on the [100] and [101] surfaces of these selected realistic materials, we discovered two visible winding around the projected double-Weyl points, demonstrating the appearance of phononic double-helical surface arcs. Furthermore, we constructed a $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model to determine how the two types of double-Weyl point phonons coexist in the system with the $P2{}_{1}3$ space group. We believe that this work provides directions for investigating double-Weyl point phonons and an ideal platform for investigating phononic double-helical surface states in realistic materials.

The in-plane and cross-plane thermal conductivity of RBa 2 Cu 3 O 7−x (R = Eu, Gd, Dy, Er, Y) superconductors

We present an analysis of thermal conductivity of a series of rare earth substituted high temperature superconductors (HTS) of the form RBa 2 Cu 3 O 7−x (R = Eu, Gd, Dy, Er, Y). Interaction of lattice vibrations with structural defects, surface boundaries, dislocations or point defects, phonons and electrons can lead to substantial thermal resistance. This work deals with the combined impact of all these scattering processes on the thermal conductivity of these HTSs and thus, presents a complete picture. The role of various scattering processes in thermal transport has been explored based on the many-body quantum dynamics for the bulk material. We have tried to interpret which scattering processes are important in different temperature ranges. We note that the boundary scattering is the sole contributor to thermal resistance at very low temperatures, in excellent agreement with experimental observations. Further, we also studied the anisotropic nature of thermal conductivity in single-crystal YBCO. Our results bear excellent agreement with the available experimental data for both the in-plane and cross-plane thermal conductivity. It is observed that the in-plane thermal conductivity is much higher than the cross-plane thermal conductivity imparting a 2 − D nature to the thermal transport in such materials. We conclude that of the various scattering processes, anisotropy in the thermal transport is largely owing to the defect scattering, phonon-phonon scattering and interference scattering processes. Such an analysis addresses the possibility of limiting the heat flow in a particular direction while allowing it in the other; a feature that can be exploited for the development of exotic technological high-temperature superconductor crystals for industrial use.

Role of the M point phonons for the dynamical stability of B2 compounds

Although many binary compounds have the B2 (CsCl-type) structure in the thermodynamic phase diagram, an origin of the dynamical stability is not understood well. Here, we focus on 416 compounds in the B2 structure extracted from the Materials Project, and study the dynamical stability of those compounds from first principles. We demonstrate that the dynamical stability of the B2 compounds lies in whether the lowest frequency phonons around the M point in the Brillouin zone are endowed with a positive frequency, except for VRu. We show that the interatomic interactions up to the fourth nearest neighbor atoms are necessary for stabilizing such phonon modes, which should determine the minimum cutoff radius for constructing the interatomic potentials of binary compounds with guaranteed accuracy.

Thermodynamic stability and optical properties of C-doping-induced defects in hexagonal boron nitride as potential single-photon emitters

Quantum information technologies have triggered enormous interest in single-photon emitters (SPEs), such as local defects in wide-band-gap semiconductors. Recently, visible single-photon emission was observed in carbon-doped hexagonal boron nitride (h-BN), yet the structure giving rise to the emission has not been unambiguously identified. Further progress in the rational design of SPEs requires a deep insight into all the ground-state properties, the electronic structure of the defects in question, and their relationship with the optical properties. To this end, we apply state-of-the-art computational methods to evaluate the stability and optical properties of defects that can be responsible for the above-mentioned experimental findings. For our analysis, we select seven possible C-based defects: (i) C substitutes B $({\mathrm{C}}_{\mathrm{B}})$, (ii) C substitutes N $({\mathrm{C}}_{\mathrm{N}})$, (iii) C substitutes N and nearest B and N are interchanged $({\mathrm{C}}_{\mathrm{N}}{\mathrm{N}}_{\mathrm{B}})$, (iv) C substitutes N and there is a nearby B vacancy $({V}_{\mathrm{B}}{\mathrm{C}}_{\mathrm{N}})$, and (v) three different configurations of ${V}_{\mathrm{N}}{\mathrm{C}}_{\mathrm{B}}$ (C substitutes B and a nearby N vacancy). We perform calculations of the electronic structure using the linear response approach ($GW$ method), followed by finding the solution to the Bethe-Salpeter equation to obtain the excitation spectra of dynamically stable defects. Our density-functional-theory-based calculations of the optimized geometries and their \ensuremath{\Gamma}-point phonon frequencies reveal that ${V}_{\mathrm{B}}{\mathrm{C}}_{\mathrm{N}}$ is dynamically unstable and undergoes a transformation to one corrugated ${V}_{\mathrm{N}}{\mathrm{C}}_{\mathrm{B}}$ structure. Likewise, the flat ${V}_{\mathrm{N}}{\mathrm{C}}_{\mathrm{B}}$ (with ${C}_{2\mathrm{v}}$ symmetry) lies on a very shallow and anharmonic minimum that would also relax toward the corrugated ${V}_{\mathrm{N}}{\mathrm{C}}_{\mathrm{B}}$ structure. Clearly, our results demonstrate how important is to attest to the dynamical stability of any proposed structure. Upon comparing our results of the considered defects with the reported experimental SPE spectra, we show that only the ${\mathrm{C}}_{\mathrm{N}}$ defect has an excitation spectrum fitting the conditions for the observed single-photon emission. Importantly, this defect is found to be the most thermodynamically stable among those under consideration.

First-principles insights into mechanical, optoelectronic, and thermo-physical properties of transition metal dichalcogenides ZrX2 (X = S, Se, and Te)

Transition metal dichalcogenides (TMDCs) belong to technologically important compounds. We have explored the structural, elastic, bonding, optoelectronic, and some thermo-physical properties of ZrX2 (X = S, Se, and Te) TMDCs in detail via the ab initio technique in this work. Elastic anisotropy indices, atomic bonding character, optoelectronic properties, and thermo-physical parameters, including melting temperature and minimum phonon thermal conductivity, are investigated for the first time. All the TMDCs under investigation possess significant elastic anisotropy and layered structural features. ZrX2 (X = S, Se, and Te) compounds are fairly machinable, and ZrS2 and ZrSe2 are moderately hard. ZrTe2, on the other hand, is significantly softer. Both covalent and ionic bondings contribute in the crystals. Electronic band structure calculations display semiconducting behavior for ZrS2 and ZrSe2 and metallic behavior for ZrTe2. Energy dependent optoelectronic parameters exhibit good correspondence with the underlying electronic energy density of state features. ZrX2 (X = S, Se, and Te) compounds absorb ultraviolet radiation effectively. The reflectivity spectrum, R(ω), remains over 50% in the energy range from 0 to ∼20 eV for ZrTe2. Therefore, this TMDC has a wide band and nonselective high reflectivity and can be used as an efficient reflector to reduce solar heating. The Debye temperature, melting point, and minimum phonon thermal conductivity of the compounds under study are low and show excellent correspondence with each other and also with the theoretically predicted elastic and bonding characteristics.

Three-nodal surface phonons in solid-state materials: Theory and material realization

This year, Liu \textit{et al}. [Phys. Rev. B \textbf{104}, L041405 (2021)] proposed a new class of topological phonons (TPs; i.e., one-nodal surface (NS) phonons), which provides an effective route for realizing one-NSs in phonon systems. In this work, based on first-principles calculations and symmetry analysis, we extended the types of NS phonons from one- to three-NS phonons. The existence of three-NS phonons (with NS states on the $k_{i}$ = $\pi$ ($i$ = $x$, $y$, $z$) planes in the three-dimensional Brillouin zone (BZ)) is enforced by the combination of two-fold screw symmetry and time reversal symmetry. We screened all 230 space groups (SGs) and found nine candidate groups (with the SG numbers (Nos.) 19, 61, 62, 92, 96, 198, 205, 212, and 213) hosting three-NS phonons. Interestingly, with the help of first-principles calculations, we identified $P2_{1}$2$_{1}$2$_{1}$-type YCuS$_{2}$ (SG No. 19), $Pbca$-type NiAs$_{2}$ (SG No. 61), $Pnma$-type SrZrO$_{2}$ (SG No. 62), $P4_{1}$2$_{1}$2-type LiAlO$_{2}$ (SG No. 92), $P4_{3}$2$_{1}$2-type ZnP$_{2}$ (SG No. 96), $P2_{1}$3-type NiSbSe (SG No. 198), $Pa\bar{3}$-type As$_{2}$Pt (SG No. 205), $P4_{3}$32-type BaSi$_{2}$ (SG No. 212), and $P4_{1}$32-type CsBe$_{2}$F$_{5}$ (SG No. 213) as realistic materials hosting three-NS phonons. The results of our presented study enrich the class of NS states in phonon systems and provide concrete guidance for searching for three-NS phonons and singular Weyl point phonons in realistic materials.

Coexistence of phononic sixfold, fourfold, and threefold excitations in the ternary antimonide Zr3Ni3Sb4

Three-, four-, and sixfold excitations have significantly extended the subjects of condensed-matter physics. There is an urgent need for a realistic material that can have coexisting three-, four-, and sixfold excitations. However, these materials are uncommon because these excitations in electronic systems are usually broken by spin-orbit coupling (SOC) and are normally far from the Fermi level. Unlike the case in electronic systems, phonon systems with a negligible SOC effect, not constrained by the Pauli exclusion principle, provide a feasible platform to realize these excitations in a wide frequency range. Hence, in this work, we demonstrate by first-principles calculations and symmetry analysis that perfect three-, four-, sixfold excitations appear in the phonon dispersion rather than the band structures of ${\mathrm{Zr}}_{3}{\mathrm{Ni}}_{3}{\mathrm{Sb}}_{4}$, which is a well-known indirect-gap semiconductor with an ${\mathrm{Y}}_{3}{\mathrm{Au}}_{3}{\mathrm{Sb}}_{4}$-type structure. This material features a threefold-degenerate quadratic contact triple-point phonon, fourfold-degenerate Dirac point phonon, and sixfold degenerate point phonon. Moreover, these nodal point phonons are very robust to uniform strain. Two obvious phonon surface arcs of the (001) plane are extended in the whole Brillouin zone, which will facilitate their detection in future experimental studies. The current work provides an ideal model to investigate rich excitations in a single material.

Coexistence of zero-, one-, and two-dimensional degeneracy in tetragonal SnO2 phonons

Based on the dimension of degeneracy, topological electronic systems can roughly be divided into three parts: nodal point, line and surface materials corresponding to zero-, one- and two-dimensional degeneracy, respectively. In parallel to electronic systems, the concept of topology was extended to phonons, promoting the birth of topological phonons. Till date, few nodal point, line and surface phonons candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, for the first time, we prove that zero-, one- and two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material SnO$_2$ with \textit{P}4$_2$/\textit{mnm} structure. In contrast to the previously reported electronic systems, the topological phonons observed in SnO$_2$ are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, SnO$_2$ with multiple dimensions of degeneracy phonons is a good platform for studying the entanglement among nodal point, line and surface phonons. Moreover, obvious phonon surface states are visible, which is beneficial for experimental detection.