Insight into the Topological Nodal Line Metal YB2 with Large Linear Energy Range: A First-Principles Study

The presence of one-dimensional (1D) nodal lines, which are formed by band crossing points along a line in the momentum space of materials, is accompanied by several interesting features. However, in order to facilitate experimental detection of the band crossing point signatures, the materials must possess a large linear energy range around the band crossing points. In this work, we focused on a topological metal, YB2, with phase stability and a P6/mmm space group, and studied the phonon dispersion, electronic structure, and topological nodal line signatures via first principles. The computed results show that YB2 is a metallic material with one pair of closed nodal lines in the kz = 0 plane. Importantly, around the band crossing points, a large linear energy range in excess of 2 eV was observed, which was rarely reported in previous reports that focus on linear-crossing materials. Furthermore, YB2 has the following advantages: (1) An absence of a virtual frequency for phonon dispersion, (2) an obvious nontrivial surface state around the band crossing point, and (3) small spin–orbit coupling-induced gaps for the band crossing points.

However, following the discovery of new topological semimetals/metals, the detection of materials suitable for experimental operation represents a significant challenge. To be suitable for experimental conjoin, new types of topological materials are formed, such as nodal chain materials [36][37][38], nodal net materials [39], and nodal link materials [40][41][42].
However, following the discovery of new topological semimetals/metals, the detection of materials suitable for experimental operation represents a significant challenge. To be suitable for experimental operation, materials should have the following basic properties: (1) The target materials are already prepared or at the least are, in theory, phase stable, (2) a large linear energy range should be present around the band crossing points, and (3) the influence of spin-orbit coupling effect on the electronic structure should be relatively small [29].
In this study, we found that P6/mmm-type YB2 possesses the above-mentioned advantages. We would like to point out that P6/mmm-type YB2 is an existing material and its experimental lattice constants are a = b = 3.3042 Å, c = 3.8465 Å [43]. Note that Song et al. [44] revealed that high-purity YB2 powders can be obtained by vacuum solid-state reaction at 1800 °C for 10 h. We studied the structural stability, electronic structure, and topological signatures of this material via first principles. Our calculations showed that YB2 is a topological semimetal with one pair of nodal lines in the kz = 0 plane and a large linear band dispersion around the band crossings, which should facilitate further experimental investigation. To determine the structural stability of this system, the phonon dispersion was also examined. When the effect of spin-orbit coupling is considered, small gaps (up to 40 meV) are found at the band crossing points. Therefore, YB2 is a good candidate for the investigation of the physical properties of nodal line fermions. We used first principles to calculate the electronic structure of the entire YB2 material. For the exchange-correlation potential, the generalized gradient approximation (GGA) [46] of the Perdew-Burke-Ernzerhof (PBE) [47] function was adopted. In this study, the cutoff energy was set at 600 eV and the Brillouin zone was sampled using the Monkhorst-Pack k-point mesh with a size of 11 × 11 × 8. To determine the nontrivial surface states in YB2, the WANNIERTOOLS package (version 2.5.0) [48] was selected.
The crystal structure of YB2 was fully relaxed and the optimized lattice parameters were found to be a = b = 3.29 Å and c = 3.85 Å. These values are in a good agreement with the experimental ones [43]. Also, these values match well those in the Materials Project Database (a = b= 3.30 Å, c = 3.85 Å). We used first principles to calculate the electronic structure of the entire YB 2 material. For the exchange-correlation potential, the generalized gradient approximation (GGA) [46] of the Perdew-Burke-Ernzerhof (PBE) [47] function was adopted. In this study, the cutoff energy was set at 600 eV and the Brillouin zone was sampled using the Monkhorst-Pack k-point mesh with a size of 11 × 11 × 8. To determine the nontrivial surface states in YB 2 , the WANNIERTOOLS package (version 2.5.0) [48] was selected.
The crystal structure of YB 2 was fully relaxed and the optimized lattice parameters were found to be a = b = 3.29 Å and c = 3.85 Å. These values are in a good agreement with the experimental ones [43]. Also, these values match well those in the Materials Project Database (a = b = 3.30 Å, c = 3.85 Å).
The Phonopy code was used to plot the phonon dispersion curve using the supercell-based approach. A 2 × 2 × 2 supercell was built and the phonon dispersion of the YB 2 supercell along the Γ-M-K-Γ-A-L-H-A paths (see Figure 1B) is shown in Figure 2. The absence of imaginary frequencies in the first Brillouin zone confirms its dynamic stability.
Materials 2020, 10, x FOR PEER REVIEW 3 of 9 The Phonopy code was used to plot the phonon dispersion curve using the supercell-based approach. A 2 × 2 × 2 supercell was built and the phonon dispersion of the YB2 supercell along the -M-K--A-L-H-A paths (see Figure 1B) is shown in Figure 2. The absence of imaginary frequencies in the first Brillouin zone confirms its dynamic stability.

Results and Discussion
Using the GGA-PBE method, the density of states of YB2 was produced and can be seen in Figure 3. We can see that this material exhibited metallic properties; however, the total density of states value was low, at around 2 eV. The total density of states in the range 0-2 eV was mainly dominated by Y-d orbitals (see the yellow area in Figure 3). To fully understand the nontrivial band crossing points and the corresponding topological signatures, we also determined the band structure of YB2, as shown in Figure 4. From Figure 4, we can see two obvious band crossing points, named A1 and A2, near the K high symmetry points. Before discussing the topological signatures of both band crossing points, we should point out that the energy range of the linear band dispersion around the A1 and A2 band crossing points was more than 2 eV (see the yellow and green areas in Figure 4). Such a large linear energy range is

Results and Discussion
Using the GGA-PBE method, the density of states of YB 2 was produced and can be seen in Figure 3. We can see that this material exhibited metallic properties; however, the total density of states value was low, at around 2 eV. The total density of states in the range 0-2 eV was mainly dominated by Y-d orbitals (see the yellow area in Figure 3).
Materials 2020, 13, 3841 3 of 9 The Phonopy code was used to plot the phonon dispersion curve using the supercell-based approach. A 2 × 2 × 2 supercell was built and the phonon dispersion of the YB2 supercell along the -M-K--A-L-H-A paths (see Figure 1B) is shown in Figure 2. The absence of imaginary frequencies in the first Brillouin zone confirms its dynamic stability.

Results and Discussion
Using the GGA-PBE method, the density of states of YB2 was produced and can be seen in Figure 3. We can see that this material exhibited metallic properties; however, the total density of states value was low, at around 2 eV. The total density of states in the range 0-2 eV was mainly dominated by Y-d orbitals (see the yellow area in Figure 3). To fully understand the nontrivial band crossing points and the corresponding topological signatures, we also determined the band structure of YB2, as shown in Figure 4. From Figure 4, we can see two obvious band crossing points, named A1 and A2, near the K high symmetry points. Before discussing the topological signatures of both band crossing points, we should point out that the energy range of the linear band dispersion around the A1 and A2 band crossing points was more than 2 eV (see the yellow and green areas in Figure 4). Such a large linear energy range is To fully understand the nontrivial band crossing points and the corresponding topological signatures, we also determined the band structure of YB 2 , as shown in Figure 4. From Figure 4, we can see two obvious band crossing points, named A1 and A2, near the K high symmetry points. Before discussing the topological signatures of both band crossing points, we should point out that the energy range of the linear band dispersion around the A1 and A2 band crossing points was more than 2 eV (see the yellow and green areas in Figure 4). Such a large linear energy range is substantially larger than most other proposed linear-type band dispersion materials [49]. Moreover, such a large linear energy range makes YB 2 a promising candidate to study the physics related to experimental band crossings. To obtain an accurate range of the large linear band dispersion and band crossing points of YB 2 , the hybrid functional [50] was used to calculate the electronic structure along the M-K-Γ paths, and the results are shown in Figure 5. It is clear that the two band crossing points, A1 and A2, and the large range (larger than 2 eV) of the linear band dispersion were retained. such a large linear energy range makes YB2 a promising candidate to study the physics related to experimental band crossings. To obtain an accurate range of the large linear band dispersion and band crossing points of YB2, the hybrid functional [50] was used to calculate the electronic structure along the M-K- paths, and the results are shown in Figure 5. It is clear that the two band crossing points, A1 and A2, and the large range (larger than 2 eV) of the linear band dispersion were retained.  Next, we discuss the topological signatures of these two band crossing points, A1 and A2, based on the arguments presented by Weng et al. [51]. These double-degenerated crossings should be assigned to a line, and the band crossing points should not be seen as isolated points. To further prove that A1 and A2 reside on a nodal line, the K-centered three-dimensional (3D) plot of the two bands in the kz = 0 plane as well as the K-centered 2D plot of the two bands in the kz = 0 plane are given in Figure 6A and Figure 6B, respectively. The white lines in Figure 6 show the intersections between the two bands, namely, an obviously closed line. As shown in Figure 6A, we can see that the band crossing points belong to a nodal line in the kz = 0 plane, and this nodal line has a slight energy variation. The 2D plane figure of the K-centered nodal line is shown in Figure 6B. such a large linear energy range makes YB2 a promising candidate to study the physics related to experimental band crossings. To obtain an accurate range of the large linear band dispersion and band crossing points of YB2, the hybrid functional [50] was used to calculate the electronic structure along the M-K- paths, and the results are shown in Figure 5. It is clear that the two band crossing points, A1 and A2, and the large range (larger than 2 eV) of the linear band dispersion were retained.  Next, we discuss the topological signatures of these two band crossing points, A1 and A2, based on the arguments presented by Weng et al. [51]. These double-degenerated crossings should be assigned to a line, and the band crossing points should not be seen as isolated points. To further prove that A1 and A2 reside on a nodal line, the K-centered three-dimensional (3D) plot of the two bands in the kz = 0 plane as well as the K-centered 2D plot of the two bands in the kz = 0 plane are given in Figure 6A and Figure 6B, respectively. The white lines in Figure 6 show the intersections between the two bands, namely, an obviously closed line. As shown in Figure 6A, we can see that the band crossing points belong to a nodal line in the kz = 0 plane, and this nodal line has a slight energy variation. The 2D plane figure of the K-centered nodal line is shown in Figure 6B. Next, we discuss the topological signatures of these two band crossing points, A1 and A2, based on the arguments presented by Weng et al. [51]. These double-degenerated crossings should be assigned to a line, and the band crossing points should not be seen as isolated points. To further prove that A1 and A2 reside on a nodal line, the K-centered three-dimensional (3D) plot of the two bands in the k z = 0 plane as well as the K-centered 2D plot of the two bands in the k z = 0 plane are given in Figure 6A,B, respectively. The white lines in Figure 6 show the intersections between the two bands, namely, an obviously closed line. As shown in Figure 6A, we can see that the band crossing points belong to a nodal line in the k z = 0 plane, and this nodal line has a slight energy variation. The 2D plane figure of the K-centered nodal line is shown in Figure 6B.
A YB 2 crystal structure has two mechanisms to protect the nodal line: (1) A horizontal mirror plane with the nodal line located in the mirror-invariant k z = 0 plane and (2) inversion symmetry and time-reversal symmetry. It should be noted that the system has time-reversal symmetry and, thus, there should be the same nodal line centered at the K' point, as shown in Figure 7. Note that the band structures of ScB 2 , VB 2 , ZrB 2 , NbB 2 , HfB 2 , and TaB 2 with P6/mmm structure were calculated by Zhang et al. [52]. Based on their work, one can see that these above-mentioned materials also have large linear energy range. A YB2 crystal structure has two mechanisms to protect the nodal line: (1) A horizontal mirror plane with the nodal line located in the mirror-invariant kz = 0 plane and (2) inversion symmetry and time-reversal symmetry. It should be noted that the system has time-reversal symmetry and, thus, there should be the same nodal line centered at the K' point, as shown in Figure 7. Note that the band structures of ScB2, VB2, ZrB2, NbB2, HfB2, and TaB2 with P6/mmm structure were calculated by Zhang et al. [52]. Based on their work, one can see that these above-mentioned materials also have large linear energy range.   A YB2 crystal structure has two mechanisms to protect the nodal line: (1) A horizontal mirror plane with the nodal line located in the mirror-invariant kz = 0 plane and (2) inversion symmetry and time-reversal symmetry. It should be noted that the system has time-reversal symmetry and, thus, there should be the same nodal line centered at the K' point, as shown in Figure 7. Note that the band structures of ScB2, VB2, ZrB2, NbB2, HfB2, and TaB2 with P6/mmm structure were calculated by Zhang et al. [52]. Based on their work, one can see that these above-mentioned materials also have large linear energy range.  The surface states along the (001) direction for the nodal line are given in Figure 8. In Figure 8B, the band crossing points are shown using yellow balls and the nontrivial surface states using red lines. From Figure 8B, one can see that the surface states along the K − Γ path are merged in the bulk state; however, the surface states along the M − K path are clearly defined. Note that the investigation of nodal line material is at initial stage, and some of them [53][54][55] have been confirmed in experiment. For example, the nodal line fermions of ZrSiSe [55] were proven in de Haas-van Alphen (dHvA) quantum oscillations. 8B, the band crossing points are shown using yellow balls and the nontrivial surface states using red lines. From Figure 8B, one can see that the surface states along the   K path are merged in the bulk state; however, the surface states along the K -M path are clearly defined. Note that the investigation of nodal line material is at initial stage, and some of them [53][54][55] have been confirmed in experiment. For example, the nodal line fermions of ZrSiSe [55] were proven in de Haas-van Alphen (dHvA) quantum oscillations. The calculated electronic structures displayed above did not consider the spin-orbit coupling effect. Therefore, as a final consideration, we will discuss the effect of spin-orbit coupling on the band crossing points in the YB2 system. The results are shown in Figure 9; one can see that the spin-orbit coupling-induced gaps for points A1 and A2 were 31 meV and 40 meV, respectively. For almost all topological materials with nodal line states, gaps can be formed between their nodal lines via spin-orbit coupling effects. However, the spin-orbit coupling gaps in the YB2 system were relatively small in comparison to some well-known nodal line semimetals/metals, such as CaTe (~50 meV) [56], BaSn2 (>50 meV) [57], CaAgBi (>80 meV) [58], and TiOs (>100 meV) [59].  The calculated electronic structures displayed above did not consider the spin-orbit coupling effect. Therefore, as a final consideration, we will discuss the effect of spin-orbit coupling on the band crossing points in the YB 2 system. The results are shown in Figure 9; one can see that the spin-orbit coupling-induced gaps for points A1 and A2 were 31 meV and 40 meV, respectively. For almost all topological materials with nodal line states, gaps can be formed between their nodal lines via spin-orbit coupling effects. However, the spin-orbit coupling gaps in the YB 2 system were relatively small in comparison to some well-known nodal line semimetals/metals, such as CaTe (~50 meV) [56], BaSn 2 (>50 meV) [57], CaAgBi (>80 meV) [58], and TiOs (>100 meV) [59].

Conclusions
In conclusion, we have reported a perfect topological metal, P6/mmm-type YB 2 , using first principles. The P6/mmm-type YB 2 has high structural stability, one pair of nodal lines in the k z = 0 plane, and a large linear energy range near the band crossings. The single pair of nodal lines are protected by two independent mechanisms: (1) Mirror symmetry and (2) inversion and time-reversal symmetries. We observed the nontrivial surface states in the (001) plane. Under the effect of spin-orbit coupling, gaps were present between the nodal lines with values of up to 40 meV. It should be noted that the spin-orbit coupling-induced gaps were smaller than some predicted nodal line state topological materials.

Conflicts of Interest:
The authors declare no conflict of interest.