Multifold degenerated phonon points in the cubic Nb₃Bi compound Open Access

Over the past few decades, there has been a notable surge in research enthusiasm, particularly in condensed matter physics, driven by the exploration of topological states in materials.^1–3 The heightened focus of research on these materials is attributed to their unique electron behavior, distinct physical traits, and potential applications in the next generations of energy and electronic devices.^4–10 In parallel to electronic systems, the expansion of topological concepts into the realm of phonons has given rise to the field of phonon topology. In recent times, the researchers have pinpointed specific real materials that are anticipated to show topological phonon properties.^11,12 Some of these predictions have been validated through experiments.^13,14 When analyzing the electronic structure, the focus on topological properties revolves around the region near the Fermi level within band structures. But considering phonons, the Fermi level concept is not relevant, as there is no obstacle for phonons to follow the Pauli exclusion principle. Therefore, it is possible to analyze the complete frequency range. As a result, phonons offer a reliable foundation for investigating topological behaviors.¹⁵ 

Unusual electronic topological properties are being observed in various A15 structured inter-metallic compounds, such as Na $3$Bi,¹⁶ Cd $3$As $2$⁠,¹⁷ and SmMg $2$Bi $2$⁠,¹⁸ now extending to the phononic topological properties.¹⁹ The A15 or Cr $3$Si structured type compounds are inter-metallic compounds with the chemical formula A $3$B where A is any transition element and B is any transition or non-transition element and possessing a specific structure. These compounds with unique physical and chemical properties, such as high superconducting temperatures, have significant technological applications.^20,21 Among many of the A15 inter-metallic compounds, Nb-based compounds are playing a crucial role in the present-day research.^21–27 

Researchers have put forth a variety of phononic band crossings, through first-principles calculations, featuring two-, three-, four-, and sixfold degenerate phonon points across different materials.^{11,12,28–32} It has been found¹² that the sixfold degenerate point (SDP) phonons are present in five potential space groups by conducting a thorough symmetry analysis. Also, five space groups are identified as potential candidates for hosting cubic contact Dirac point (CCDP) phonons at High-Symmetry Points (HSPs).^11,33–35 In addition to these, a quadratic contact triple point (QCTP) is also being observed in Ta $3$Sn,¹⁹ SiO $2$⁠,³² Zr $3$Ni $3$Sb $4$⁠,³¹ etc. These multi-fold degenerate crossings provide a versatile foundation for attaining more unconventional properties necessary for further development. The presence of these higher-order phonon points can be affecting the thermal properties of the material by enhancing phonon scattering rates. This can be further supported theoretically and experimentally, as was the case with a gray arsenic crystal.³⁶ 

Nb $3$Bi belongs to the space group 223, which is one among the five space groups as mentioned earlier. The phonon topological analysis of Ta $3$Sn¹⁹ having topological states with sixfold degeneracy as well as quadratic dispersion suggests the possibility of observing some interesting phonon topological properties in the realistic material Nb $3$Bi, which is similar to Ta $3$Sn. This offers great opportunity to study unusual quantum states in systems with multiple degeneracy.^37–39 

In this article, employing first-principles calculations, the dynamical properties of Nb $3$Bi are examined. Moreover, there is a study on the total and partial phonon density of states (PDOS) and the determination of an irreducible representation at the $Γ$ point, contributing to a more thorough understanding. The phonon dispersion curve was computed along the high-symmetric path and identified the presence of SDP and CCDP at the R point and QCTP at the $Γ$ point. An in-depth investigation was conducted by analyzing the bulk as well as the phonon surface states in their respective planes.

This paper is organized as follows. In Sec. II, we present computational details. Our results are presented and discussed in Sec. III. Finally, we summarize and conclude our findings in Sec. IV.

Our calculations of the geometric, elastic, and phonon properties are performed by employing density functional theory (DFT)^40,41 with the plane wave projected augmented wave (PAW) method⁴² as implemented in the Vienna Ab Initio Simulation Package (VASP).⁴³ The exchange-correlation is approximated by the Generalized Gradient Approximations (GGAs) using PBE (Perdew–Burke–Ernzerhof)⁴⁴ and PBEsol (Perdew–Berke–Ernzerhof for solids).⁴⁵ The electronic configurations of Nb and Bi are 4p $6$4d $4$5s $1$ and 6s $2$6p $3$⁠, respectively. The energy-cutoff of 350 eV was chosen, and the Brillouin zone was sampled with a $12×12$ $×12$ Monkhorst–Pack grid of $k$-points.⁴⁶ As a convergence condition of the optimization loop, we took the energy change below $10 − 6$ eV and $10 − 8$ eV for the ionic and electronic degrees of freedom, respectively.

The dynamical properties we investigated using a Parlinski–Li–Kawazoe method⁴⁸ are implemented in the phonopy⁴⁹ software. Within this method, the interatomic force constants (IFCs) are calculated from the Hellmann–Feynman forces generated by displacements of individual atoms inside the supercell. In such a case, the $2×2×2$ supercell was used with a reduced $4×4×4$ $k$-points grid. Finally, the obtained IFCs were used to generate the phononic Wannier tight-binding model and calculations of the surface states within WannierTools.⁵⁰ 

The Cr $3$Si type compound or A15 structured Nb $3$Bi possesses a cubic crystal structure as shown in Fig. 1(a). Nb $3$Bi belonging to symmetry Pm $3 ¯$n (space group No. 223) has eight atoms per unit cell of which two are Bi atoms and the remaining six are Nb atoms. In such a structure, Bi atoms are forming a bcc-lattice and each cube face contains two of the Nb atoms forming mutually orthogonal one-dimensional chains along edges. This is the main characteristic of the A15 structure. The Bi atoms occupy the $2a$ (0,0,0) Wyckoff site, and Nb atoms occupy the $6d$ (1/4,1/2,0) Wyckoff site. Furthermore, Bi and Nb atoms show $+1$ and $−3$ oxidation states, respectively. From the charge density distribution around atoms (not presented), which form a spherical distribution without overlapping, the bounding between Nb and Bi can be characterized as ionic.

To obtain the equilibrium structural parameters, the volume and the ion positions of the crystal are fully relaxed using the PBE and PBEsol approximations as tabulated in Table I. The computed lattice constant 5.329 Å and the cell volume $151.4 Å 3$ are in excellent agreement with the experimental values.⁴⁷ The parameters obtained from PBEsol are preferred as these are closer to the experimental values.

The phonon dispersion curve calculated and the irreducible representation at high symmetric points are depicted in Fig. 1(b). Initially, it can be noticeable that the acoustic and the optical branches exhibit coupling at low frequencies. Deeper into the analysis, it is observed that at high frequencies, there are more than twofold degeneracies present at the $Γ$ and R points. This is attributed to the heavy mass atoms. The frequencies of the Nb $3$Bi compound are all positive throughout the Brillouin zone, indicating that it is dynamically stable. The calculation of elastic constants was performed to confirm this stability. Since Nb $3$Bi has a cubic crystal structure, there are three independent ground state elastic constant tensor components, namely, C $11$⁠, C $12$⁠, and C $44$⁠, that are to be known. The values of these elastic constants are obtained as $C 11=313.521$ GPa, $C 12=104.786$ GPa, and $C 44=62.482$ GPa, respectively. The material satisfies the mechanical stability requirements for cubic structures. We analyze parameters, such as the bulk modulus (⁠ $B=174.371$ GPa), the shear modulus (⁠ $S=76.816$ GPa), and Poisson’s ratio (⁠ $n=0.308$⁠), which exhibit comparable values to those observed in previously studied Nb $3$X compounds.²⁵ All the values are positive and large in magnitude, which confirms that Nb $3$Bi is indeed stable. The computation of the phonon partial density of states (DOS) is conducted to analyze the individual contributions from each atom. As observed in Fig. 1(c), within the 0–4 THz range, the contribution from the Bi atom is predominant. While at higher frequencies (from 4 to 8 THz), the Nb atom becomes more influential due to its lighter nature compared to Bi in the compound, which is similar as Ti $3$Sb.⁵¹ 

The space group Pm $3 ¯$n symmetry is characterized by basic symmetry operations, which involves rotational symmetry ${ 2 001|0,0,0}$⁠, ${ 2 010|0,0,0}$⁠, ${ 2 100|0,0,0}$⁠, threefold rotation ${ 3 111 +|0,0,0}$⁠, ${ 3 1 ¯ 1 1 ¯ +|0,0,0}$⁠, ${ 3 1 1 ¯ 1 ¯ +|0,0,0}$⁠, etc., and mirror reflection ${ m 010|0}$⁠, ${ m 001|0}$⁠, and ${ m 100|0}$⁠. Apart from these, there are non-symmorphic symmetry characteristics, which are the glide plane ${ m 110|1/2,1/2,1/2}$⁠, ${ m 1 1 ¯ 0|1/2,1/2,1/2}$ and screw axes ${ 2 1 ¯ 01|1/2,1/2,1/2}$⁠, ${ 2 101|1/2,1/2,1/2}$⁠, etc. These symmetry operations are giving the highly degenerated phonon points at the high-symmetry points of the Brillouin zone.

As a result of such symmetries, highly degenerate phonon modes are found at the high symmetry points, as illustrated in Table III. The focus of our attention lies on the R point, specifically where a sixfold degenerate point (SDP) and a cubic contact Dirac point (CCDP) are manifested, as shown in Fig. 3. In practical terms, there exist three frequencies at which the sixfold degenerated point can be observed. However, it is essential to note that only at the highest frequency of approximately 6.17 THz is the SDP subjected to analysis in this context. This analysis sheds light on the intricate nature of the highly degenerate phonon modes present at the R point, offering a deeper understanding of the symmetries at play. The realization of the SDP and CCDP at this specific point highlights the complexities inherent in the phonon modes of the system under consideration. By delving into the intricacies of these degenerate points, researchers can unravel the underlying principles governing the behavior of phonon modes in materials exhibiting such symmetries. The exploration of these phenomena contributes to the broader field of materials science, offering insights that pave the way for advancements in various technological applications. Furthermore, the detailed examination of the SDP at the highest frequency provides valuable information for further research and theoretical developments in the study of phonon modes and their implications in materials science. In addition, there exist a fourfold degenerate point at $∼$4.22 THz, which is referred to as CCDP. Similarly at the $Γ$ point, we can observe several quadratic contact triple points (QCTPs) realized by threefold degenerated modes (modes with T symmetry, see Table II).

In this section, we will discuss the possible phononic surface states of Nb $3$Bi in the case of (001) and (1 $1 ¯$0) surfaces (Figs. 4 and 5, respectively). A slab-like system can realize different terminations due to the absence of clear layers in the cubic Nb $3$Bi. Thus, we can find two types of termination of a realized surface. In the first case, the surface is terminated by Nb atoms [see top surfaces in Figs. 4(a) and 5(a)]. Contrary to this, second termination is realized by mixing of Bi and Nb atoms [see bottom surfaces in Figs. 4(a) and 5(a)]. Independently by the termination, a surface exhibits twofolded rotation symmetry.

The realized surface states can be recognized from comparison of the calculated surface Green function (for specific termination) with the spectral function of the bulk states [calculated for the central part of the slab-like systems, presented in Figs. 4(a) and 5(b)]. The surface spectrum can be understand as a projection of all bulk states from a three dimensional (bulk) Brillouin zone on the two dimensional (surface) Brillouin zone [see Fig. 1(d)]. From this, the bulk spectral function is relatively complex, which makes it difficult to recognize the surface states. Nevertheless, some of them are well visible and marked by yellow arrows. In practice, the surface states are realized in a whole range of frequencies.

Realization of the twofolded rotational symmetry for the discussed surfaces is well visible on highest surface states for the (001) surface with Nb termination [i.e., the top surface in Fig. 4(a)]. The surface states are realized along the $Γ ¯$– $Y ¯$– $Γ ¯$ path [around $6.4$ THz, in Fig. 4(c)], while it is invisible along the $Γ ¯$– $X ¯$– $Γ ¯$ path. Similar behavior is visible for another surface states.

Probably most predominant are surface states associated with the Nb terminated (top) surface. In such a case, the surface states are realized at a high frequency range [e.g. around 6.5 THz in Fig. 5(d)], which is in agreement with the phonon density of states [Fig. 1(c)]. Similarly, the surface states associated with the Bi phonon modes are expected in an intermediate frequency range [such as two surface states in Fig. 5(c)], where the phonon density of states is associated mostly with Bi atoms.

In brief, our examination of Nb $3$Bi has presented compelling findings that support its stable Pm $3 ¯$n symmetry and the existence of multi-fold degenerate phonon points, indicative of its intriguing topological properties. The detailed analysis of the crystal structure has once again confirmed the A15 structure of Nb $3$Bi, validating specific atomic arrangements and oxidation states that align with experimental findings. By employing first-principles calculations, we have substantiated the stability of the compound within the Pm $3 ¯$n symmetry and identified multi-fold degenerate phonon points, which include significant features, such as the sixfold degenerate point (SDP), the cubic contact Dirac point (CCDP), and the quadratic contact triple point (QCTP). Analysis of the surface has revealed predominant terminations by Nb atoms, along with a mix of Bi and Nb atoms, showcasing a distinct twofold rotation symmetry pattern. Through meticulous comparisons with bulk states and projection onto a 2D surface Brillouin zone, we have successfully distinguished surface states, significantly enhancing our comprehension of bulk phonon dispersion curves. This thorough investigation has provided valuable insights into the unique phonon characteristics and surface states of Nb $3$Bi, underscoring its potential importance in the realm of topological materials research.

Some figures in this work were rendered using Vesta.⁵³ The authors are grateful for the computing resources provided by CMSD and the University of Hyderabad. A.T. is grateful for the financial assistance by the Prime Minister’s Research Fellow scheme (PMRF). A.P. kindly acknowledges support by the National Science Centre (NCN, Poland) under Project No. 2021/43/B/ST3/02166. G.V. would like to acknowledge the Institute of Eminence, University of Hyderabad (No. UoH-IoE-RC3-21-046) for funding.

The authors have no conflicts to disclose.

Manasa MSL: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Aiswarya T.: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Andrzej Ptok: Methodology (equal); Project administration (equal); Validation (equal); Writing – review & editing (equal). G. Vaitheeswaran: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.