ORIGINAL RESEARCH article
Sec. Quantum Materials
Volume 9 - 2022 | https://doi.org/10.3389/fmats.2022.977595
Topological states in boron phosphide with zinc-blende structure
- 1Aviation and Automobile School, Chongqing Youth Vocational and Technical College, Chongqing, China
- 2College of Physics, Chongqing University, Chongqing, China
The field of topological states in phonon of solids have been rapidly developing in recent years. This work examined the phonon dispersion of a compound Boron Phosphide (BP) with a Zinc-Blende structure via first-principle calculation. The results show that BP is a stable compound in theory and hosts rich topological signatures in its phonon dispersion. Specifically, Weyl and quadratic nodal line states can be found in the acoustic branches, and triple point and quadratic contact triple point can be found in the optical branches. It is hoped that the rich topological states in BP can be imaged by inelastic x-ray scattering or neutron scattering in the near future.
Topological electronic materials have been one of the most active research fields in the past decade. To this date, many types of topological electronic materials (Wang, 2008; Li et al., 2009; Hasan and Kane, 2010; Moore, 2010; Qi and Zhang, 2011; Gao et al., 2016; Tokura et al., 2019; Wang et al., 2020a; Yue et al., 2020), including topological insulators (Hasan and Kane, 2010; Moore, 2010; Qi and Zhang, 2011; Tokura et al., 2019), spin-gapless semiconductors (Wang, 2008; Li et al., 2009; Gao et al., 2016; He et al., 2017; Wang et al., 2020a; Yue et al., 2020; Yang et al., 2021), topological semimetals (Vazifeh and Franz, 2013; Yang et al., 2015; Chang et al., 2016; Liu et al., 2019a; Wang et al., 2020b; Li and Xia, 2020; Lim et al., 2020; Liu et al., 2022a), (Li et al., 2020b; Li et al., 2020c; Li et al., 2020d), and topological superconducting (Chung et al., 2011; Stoudenmire et al., 2011; Sato and Ando, 2017; Zhang et al., 2018; Frolov et al., 2020), have been predicted in theory; some of them are also confirmed in experiments. For example, some three-dimensional (3D) material databases documented tens of thousands of candidates as topological semimetals. In 2019, Zhang et al. Zhang et al. (2019a) swept through 39,519 materials in the crystal database and proposed that 8,056 of 39,519 materials have topologically nontrivial properties. In the same year, Vergniory et al. Vergniory et al. (2019) performed a high-throughput search of high-quality materials in the Inorganic Crystal Structure Database (ICSD) database and found 4,078 topological semimetals. Tang et al. Tang et al. (2019) listed 692 topological semimetals with symmetry-dominated band crossing points around the Fermi level using symmetry indicators. In 2022, Yu et al. Yu et al. (2022) exhibited an encyclopedia of emergent particles in three-dimensional crystals. They completed a complete list of all possible particles in time-reversal-invariant systems using systematic symmetry analysis.
Very recently, the studies of topological states are intensively generalized to phonons in crystalline materials (Liu et al., 2020a; Chen et al., 2021a; Li et al., 2021). Many realistic solids with diverse topological signatures (Jin et al., 2018a; Jin et al., 2018b; Liu et al., 2019b; Xia et al., 2019; Li et al., 2020a; Liu et al., 2020b; Wang et al., 2020c; Peng et al., 2020; Zheng et al., 2020; Liu et al., 2021a; Wang et al., 2021a; Xie et al., 2021a; Zhou et al., 2021a; Chen et al., 2021b; Liu et al., 2021b; Wang et al., 2021b; Xie et al., 2021b; Zhou et al., 2021b; Liu et al., 2021c; Wang et al., 2021c; Liu et al., 2021d; Wang et al., 2021d; Zheng et al., 2021; Zhong et al., 2021; Ding et al., 2022a; Wang et al., 2022a; Ding et al., 2022b; Liu et al., 2022b; Wang et al., 2022b; Liu et al., 2022c; Yang et al., 2022) are reported via first-principle calculation and symmetry analysis. For example, in theory, solids with Weyl point phonons (Xia et al., 2019; Liu et al., 2020b; Wang et al., 2020c; Liu et al., 2021b), Dirac point phonons (Chen et al., 2021b), nodal chain phonons (Zhou et al., 2021b), nodal box phonons (Zhou et al., 2021b), nodal cage phonons (Ding et al., 2022a; Wang et al., 2022b), and nodal surface phonons (Xie et al., 2021b; Wang et al., 2021d) have been reported. More interestingly, some topological signatures in the phonon spectrum have been confirmed in an experiment with the help of meV-resolution inelastic X-ray scattering. In two consecutive works (Miao et al., 2018; Zhang et al., 2019b), the double Weyl points and the parity-time reversal symmetry-dominated helical nodal lines are realized in the phonon spectrum of FeSi and MoB2, respectively.
In particular, the coexistence of topological phonons in realistic materials has attracted much attention, such as the coexistence of zero-, one-, and two-dimensional degeneracy phonons (Wang et al., 2021a), hybrid-type nodal-ring and quadratic nodal-line phonons and nodal-net and nodal-link phonons (Zhou et al., 2021b), Dirac nodal line and Weyl nodal line phonons (Wang et al., 2022a). In this work, we focus on a realistic material (Popper and Ingles, 1957), Boron Phosphide (BP), with a Zinc-Blende structure, and study the topological signatures in the phonon dispersion of BP. BP has already been prepared before by the reaction of boron and red phosphorus in an evacuated, sealed silica tube at 1,100°C. BP is with space group F-43m (space group number 216) and contains two atoms, B and P, located at the (0,0,0) and (0.75,0.75,0.75) Wyckoff sites, respectively (see Figure 1A,B). The lattice structure of BP is totally relaxed via first-principle calculation, and the obtained lattice constant is a = b = c = 3.217 Å. To the best of our knowledge, the prosperous topological state has not even been previously reported in the phonon dispersion of BP, and this is the first time to report the topological phonons in the acoustic and optical branches of BP.
The theoretical calculation is performed in the framework of density functional theory (Cohen et al., 2012), which is implemented in the Vienna ab initio Simulation Package (Hafner, 2008). The generalized gradient approximation with Perdew-Burke-Ernzerhof formalism (Perdew et al., 1996) is used for the exchange-correlation energy, and the projector augmented wave method (Blöchl, 1994) is applied for the interactions between the ions and valence electrons. Moreover, a cutoff energy of 500 eV is chosen for the plane wave set, and a Γ-centered k grid of 5 × 5 × 5 is sampled for the Brillouin zone. Based on this optimized lattice of BP, we have computed the phonon dispersion spectrum of 2 × 2 × 2 supercell by the density functional perturbation theory, as implemented within the PHONOPY package (Togo and Tanaka, 2015).
Topological states in optical branches
The phonon dispersion of BP along the
For the three optical branches, one finds two crossing points, one is at
As shown in Figure 4A, the crossing point on
Topological states in acoustic branches
Then, we come to study the topological states in acoustic branches in BP. As shown in Figure 1C, one finds two doubly degenerate acoustic phonon bands appear along the
The crossing bands along the
FIGURE 5. (A) Selected symmetry points a1 (0.4, 0.0, 0.4), a2 (0.3, 0.0, 0.3), a3 (0.2, 0.0, 0.2), a4 (0.1, 0.0, 0.1), b1 (0.9, 0.5, 0.4), b2 (0.8, 0.5, 0.3), b3 (0.7, 0.5, 0.2), and b4 (0.6, 0.5, 0.1). (B) Enlarged phonon dispersions of BP along the bn-an-bn (n = 1–4) paths.
Actually, the crossing bands along the
FIGURE 6. (A) Selected symmetry points c1 (0.4, 0.4, 0.4), c2 (0.3, 0.3, 0.3), c3 (0.2, 0.2, 0.2), c4 (0.1, 0.1, 0.1), d1 (0.4, 0.5, 0.4), d2 (0.3, 0.4, 0.3), d3 (0.2, 0.3, 0.2), and d4 (0.1, 0.2, 0.1). (B) Enlarged phonon dispersions of BP along the dn-cn-dn (n = 1–4) paths.
In this work, based on theoretical calculations, we studied the WNL, QNL, TP, and QTCP in the phononic system BP. The acoustic branches form the WNL and QNL and the optical branches form the TP and QCTP. There is a frequency gap between the acoustic and the optical branches. Furthermore, the rich nodal points and rich nodal lines are distributed at different frequency ranges. The WNL phonons along the
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
The author confirms being the sole contributor of this work and has approved it for publication.
This work is supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-K202104101), the Open Project Funding of Theoretical Physics Academic Exchange Platform of Chongqing University (Grant No. 2, in the year of 2021), 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.
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Keywords: topological state, topological phonons, density functional theory, phonon dispersion, optical branch
Citation: Li Y (2022) Topological states in boron phosphide with zinc-blende structure. Front. Mater. 9:977595. doi: 10.3389/fmats.2022.977595
Received: 24 June 2022; Accepted: 17 August 2022;
Published: 20 September 2022.
Edited by:Xiaotian Wang, Southwest University, China
Reviewed by:Guangqian Ding, Chongqing University of Posts and Telecommunications, China, Gokhan Surucu, Gazi University, Turkey, Rabah Khenata, University of Mascara, Algeria
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*Correspondence: Yang Li, firstname.lastname@example.org