Open Access Article

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Shenda
He‡
^{ab},
Ruirong
Kang‡
^{ab},
Pan
Zhou
*^{a},
Zehou
Li
^{b},
Yi
Yang
*^{b} and
Lizhong
Sun
^{ab}
^{a}Hunan Provincial Key laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University, Xiangtan 411105, China. E-mail: zhoupan71234@126.com
^{b}School of Materials Science and Engineering, Xiangtan University, Xiangtan 411105, China. E-mail: yangyi@xtu.edu.cn

Received
31st January 2022
, Accepted 6th May 2022

First published on 18th May 2022

Materials with spin-polarized electronic states have attracted a huge amount of interest due to their potential applications in spintronics. Based on first-principles calculations, we study the electronic characteristics of a series of AB_{2}X_{4} chalcogenide spinel structures and propose two promising candidates, VZn_{2}O_{4} and VCd_{2}S_{4}, that are spin-polarized semimetal materials. Both of them have ferromagnetic ground states. Their bands near the Fermi level are completely spin-polarized and form two types of nodal rings in the spin-up channel, and the large gaps in the spin-down channel prevent the spin-flip. Further symmetry analysis reveals that the nodal rings are protected by the glide mirror or mirror symmetries. Significantly, these nodal rings connect with each other and form a nodal chain structure, which can be well described using a simple four-band tight-binding (TB) model. The two ternary chalcogenide spinel materials with a fully spin-polarized nodal chain can serve as a prominent platform in the future applications of spintronics.

Topological semimetals have already been identified in a series of materials, including IrF_{4},^{18} Ba_{3}Si_{4},^{19} carbon networks,^{14} and β-cristobalite BiO_{2}.^{20} The majority of them, however, are nonmagnetic, whereas magnetic semimetals appear to exhibit exotic features such as tunable nodal points and the anomalous Hall effect. Initially the pyrochlore iridate AIr_{2}O_{7} (A is Y or a rare-earth element) was proposed to be a magnetic topological semimetal,^{21} and later the ferromagnetic (FM) material HgCr_{2}Se_{4} was proposed to be a novel Weyl semimetal.^{22} Soon after, HgCr_{2}Se_{4} has been experimentally confirmed as a half-metal by Guan et al.^{23} Recently, more materials having magnetic topological properties have been proposed. For example, LiV_{2}F_{6} has been theoretically demonstrated to be a Weyl monoloop half-semimetal,^{24} and Li_{3} (FeO_{3})_{2}^{25} has been proposed as a half-semimetal material with two independent Weyl loops. The Dirac line-nodal half-semimetal penta-X_{2}Y (X = N, C and Y = B, C, P) was proposed,^{26} and CaFeO_{3} was theoretically predicted to have line-surface electronic states.^{9} Type-II spin-polarized nodal lines exist in quasi-two-dimensional compounds X_{2}YZ_{4} (X = K, Cs, Rb, Y = Cr, Cu, Z = Cl, F),^{27} and nodal points, lines, and surfaces coexist in quasi-one-dimensional compounds XYZ_{3} (X = Cs, Rb, Y = Cr, Cu, Z = Cl, I).^{28} The spin-polarized Dirac point, triply degenerate nodal point, nodal loops, and nodal surface coexist in the anti-ferromagnetic (AFM) electride Ba_{4}Al_{5}·e^{−}.^{29}

Due to its various applications, the ternary chalcogenide spinel family is well-known and has been extensively researched for decades.^{17} They have a face centered cubic structure and are represented as AB_{2}X_{4}, where A and B are metal atoms that center the X (chalcogens) tetrahedrons and octahedrons, respectively. Almost all main group and transition metal elements may be synthesized in a stable spinel form, resulting in a diverse range of elemental compositions, electronic configurations, and valence states.^{30,31} Due to these features, the spinel has a wide range of magnetic, electrical, optical, and catalytic properties.^{32–36} Until now, a series of spinel materials, including HgCr_{2}Se_{4}, VMg_{2}O_{4}, LiV_{2}O_{4}, FeAl_{2}O_{4}, and NiAl_{2}O_{4},^{22,37–39} have been theoretically predicted to have the fully spin polarized electronic states. Transition metal spinels always have various tunable magnetic properties, which is significant for spintronic applications. Their established synthesis technology lends themselves to spintronic studies as well. Therefore, exploring ideal spinel structures with fully spin-polarized electronic states is beneficial for spintronic experimental research and potential applications.

In this work, we find two spinel structures, VZn_{2}O_{4} and VCd_{2}S_{4}, which have completely spin-polarized electronic states near the Fermi level. In their spin-up channel, they have a series of band crossings that belong to two different types of nodal rings and are protected by mirror or gliding mirror operations. We discovered that these rings form a chain-like structure. The nontrivial topological properties of the nodal chains are confirmed by the surface states. Only small gaps are opened after considering spin–orbit coupling (SOC). Hence, our work reveals a promising material platform for studying the fundamental physics of the fully spin-polarized nodal-chain, which also possesses great potential for future spintronic applications.

Fig. 1 (a) Primitive cell and (b) conventional cell of VZn_{2}O_{4} and VCd_{2}S_{4}. (c) BZ and the corresponding high-symmetry paths in the first Brillouin zone. |

Despite the fact that a large number of similar materials with the same crystal structure have been synthesized, the two spinel materials have yet to be synthesized in experiments. We will investigate their stability with the experimentally synthesized spinel material in the following section. To demonstrate their stability, we firstly calculate the cohesive energies (E_{c}) of VZn_{2}O_{4} and VCd_{2}O_{4}. The results are 4.23 eV per atom and 2.86 eV per atom, respectively. In addition, we also calculate the E_{c} of experimentally synthesized ZnV_{2}O_{4}.^{51} The value is 3.72 eV per atom, which is comparable to those of our structures, indicating that they have the potential to be synthesized. The phonon dispersions are calculated as well, and the results are shown in Fig. 2(a) and (b). The absence of imaginary frequency reveals that they are kinetically stable. Their first-principles molecular dynamics (MD) simulations are performed by using a 2 × 2 × 2 supercell [see Fig. 2(c) and (d)]. The evolutions of total energies prove that the two materials are thermodynamically stable. Furthermore, as shown in Table S2, ESI,† the linear elastic constants and Young's moduli computed using the stress vs. energy method meet the Born–Huang criteria (C_{11} − C_{12} > 0, C_{11} + 2C_{12} > 0, C_{44} > 0),^{52} indicating that they are mechanically stable.

Fig. 2 Phonon dispersions (a) for VZn_{2}O_{4} and (b) for VCd_{2}S_{4}. Their energy evolutions obtained by MD at 300 K for 6 ps (1 fs per step) are shown in (c) and (d), respectively. |

The orbital-resolved band structures in Fig. 4(a) indicate that the V-d_{z2} and V-d_{x2−y2} orbitals contribute the most to the conduction-band minimum (CBM) and valence-band maximum (VBM) of VZn_{2}O_{4}, but there are more orbitals that contribute to the low-energy electronic states of VCd_{2}S_{4} [see Fig. S2(a), ESI†]. The local band gaps between the CBM and VBM are calculated (as shown in Fig. 4 for VZn_{2}O_{4} and Fig. S2, ESI,† for VCd_{2}S_{4}). It can be seen that the band crossings are not isolated, and they combine to form two different types of closed loops. We use the abbreviation NR_{1} to represent the nodal ring that includes the crossing points on the paths of Γ–K, W–Γ, and Γ–X and centers around the Γ point in the k_{z} = 0 plane. On the other hand, NR_{2} represents the nodal ring centered around the X point in the Γ–X–L plane. Although a band crossing appears in the L–K path, it seemingly belongs to a new nodal ring in the L–Γ–K plane, which also contains the crossing in the Γ–K path. After analyzing the symmetry property of the space group Fdm, we discover that the L–Γ–K and Γ–X–L planes are equivalent, and the new nodal ring is equivalent to NR_{2}.

In order to decide whether the nodal rings are accidental or symmetry-protected, we further analyze their symmetries by calculating the irreducible representations (irreps) of the Bloch states around these crossing points (as shown in Fig. S3, ESI†). As a result, we find that the NR_{1} in the k_{z} =0 plane is under the protection of the glide mirror operation G_{z}:(x, y, z) → (x + 1/4, y + 3/4, −z + 1/2), which can be verified by the corresponding opposite sign of glide eigenvalues. Similarly, the NR_{2} in the Γ–X–L plane is protected by the mirror operation G_{01}:(x, y, z) → (z, y, x) (Fig. S3, ESI†). The glide or mirror eigenvalues are used to illustrate the formation of hourglass dispersion in the planes of Γ–K–W–X and Γ–X–U–L [see Fig. S4, ESI†]. These two planes are invariant planes of G_{z}:(x, y, z) → (x + 1/4, y + 3/4, −z + 1/2) and G_{01}:(x, y, z) → (z, y, x), respectively. The eigenvalues of the corresponding operations can then be used to label the Bloch states in these two planes. Here we use the representative paths W–Γ and X–L to demonstrate them. As shown in Fig. S4(b), ESI,† the four bands around the Fermi level form an hourglass dispersion. According to the ‘Bilbao Crystallographic Server’^{53} website's little group representation of the space group Fdm, only two two-order single-values representations exist at W (W_{1} and W_{2}). However, there are four one-order (Γ^{+}_{1}, Γ^{−}_{1}, Γ^{+}_{2}, and Γ^{−}_{2}), two two-order (Γ^{+}_{3} and Γ^{−}_{3}), and four third-order (Γ^{+}_{4}, Γ^{−}_{4}, Γ^{+}_{5}, and Γ^{−}_{5}) single-valued representations at Γ. Using the wavefunctions from VASP, we find that the representations at W (Γ) are W_{1} and W_{2} (Γ^{+}_{3} and Γ^{−}_{3}) and all of them are double-degenerate. Because the W–Γ is located on the invariant plane, the glide eigenvalues must be [where (k_{x}, k_{y}, k_{z}) is the fractional coordination of the k-points on the Γ–K–W–X plane], which indicate that the sign of the eigenvalues would be changed when k moves from W to Γ. By replacing the coordinates with the above eigenvalues, we obtain the glide eigenvalues at W and Γ, as shown in Fig. S4(b), ESI.† Because the double-degenerate states have opposite signs at W, the degenerate states at Γ have the same glide eigenvalues. Because the band evolution along W–Γ must be smooth, the two eigenspaces must be exchanged, resulting in hourglass dispersion. A similar argument can be applied to X–L with the operator of G_{01}. However, because this operation does not involve fractional translation, the eigenvalues in the whole plane must be +1 or −1.

Due to the cubic symmetry of the two materials, other equivalent rings can appear on the equivalent planes, for example, similar nodal rings NR_{1} can appear in the planes of k_{x} = 0, k_{y} = 0. Surprisingly, we find that NR_{1} and NR_{2} are not isolated but share the same nodal points in the paths of Γ–K and Γ–X, and form a chain-like band crossing structure in BZ in the spin-up channel, as shown in Fig. 5(a).

To further illustrate the topological properties of the nodal chain, we calculated the Berry phase with the help of MLWFs. As shown in Fig. S5, ESI,† the Berry phase is π (0) if the k-loop passes through NR_{1} or NR_{2} (NR_{1} and NR_{2}), which are the typical characteristics of a nodal chain. Nodal chains that are topologically protected always lead to nontrivial surface states. The 2D projection of bulk BZ onto the (001) plane and the related high-symmetry points are shown in Fig. 5(a). Fig. 5(b) shows the projection of a portion of the nodal rings. We chose a path cut in the projected 2D BZ to investigate the nontrivial surface states, and the electronic local density states are shown in Fig. 5(c). The surface states demonstrate that the nodal chain's two types of nodal lines are topologically nontrivial.

Recently, several spinels with nodal net states have been proposed. Here we compare their nodal states with VZn_{2}O_{4} and VCd_{2}S_{4}, including LiV_{2}O_{4}, FeAl_{2}O_{4} and NiAl_{2}O_{4}, MgV_{2}O_{4} and Mg_{2}VO_{4}.^{37,39,54} They have a half-semimetal electronic state near E_{f}, which is the same as our two materials. When compared to the nodal lines in VZn_{2}O_{4} or VCd_{2}S_{4}, the size of NR_{2} for LiV_{2}O_{4} is clearly different, resulting in the band crossing of the Γ–K path disappearing and the band structure also being different. Despite the fact that FeAl_{2}O_{4} and NiAl_{2}O_{4} have been proposed as having nodal chain states, the energy ranges of their band crossings near E_{f} are too broad and are disadvantageous to analyze the topological characters and surface states. However, the dispersion of the crossing points for VZn_{2}O_{4} and VCd_{2}S_{4} is small, making experimental detection much easier. The magnetic ground state of MgV_{2}O_{4} is AFM, with spin-polarized flat band states under a ferromagnetic state, as opposed to our ferromagnetic ground states and half-metallic nodal chains. Because Mg_{2}VO_{4} has a similar atomic occupation to our two materials, its nodal chain is similar to our two materials.

(1) |

(2) |

Onsite energy | ε | 0.012 eV |

Nearest neighbor | ddπ^{1} |
−1.097 eV |

ddδ^{1} |
1.869 eV | |

ddσ^{2} |
−0.555 eV | |

Next nearest neighbor | ddπ^{2} |
−0.045 eV |

ddδ^{2} |
0.257 eV |

Finally, we examine how the external strain and SOC affect the energy band structures of these two materials. As shown in Fig. 6(c), the band gaps of the spin-down bands become smaller with increasing lattice constants, but they are still insulated and the semimetal states around the Fermi level are still well preserved (Fig. S7, ESI†). The band structures with SOC are presented in Fig. S8 (ESI†). We find that all opened gaps are below 10 meV, which means they can always be neglected at room temperature.

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## Footnotes |

† Electronic supplementary information (ESI) available: Information of all spinels considered. Magnetic configurations and elastic constants. Orbital resolved bands and local gap distribution of VCd_{2}S_{4}. Eigenvalues of symmetry operations. Berry phase of the TB model. Energy band structures of VZn_{2}O_{4} with different Hubbard U values, strains and SOC. See DOI: https://doi.org/10.1039/d2ma00107a |

‡ These two authors equally contributed to this work. |

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