In this work, we review the progress on the Heusler-based materials with spin-gapless semiconducting behaviour, including half-Heusler compounds, full-Heusler compounds, DO₃-type compounds, and LiMgPdSn-type quaternary Heusler compounds.

Magnetic Heusler compounds (MHCs) have recently attracted great attention since these types of material provide novel functionalities in spintronic and magneto-electronic devices. Among the MHCs, some compounds have been predicted to be spin-filter semiconductors [also called magnetic semiconductors (MSs)], spin-gapless semiconductors (SGSs) or half-metals (HMs). In this work, by means of first-principles calculations, it is demonstrated that rare earth-based equiatomic quaternary Heusler (EQH) compounds with the formulaMCoVZ(M= Lu, Y;Z= Si, Ge) are new spin-filter semiconductors with total magnetic moments of 3 µ_B. Furthermore, under uniform strain, there are physical transitions from spin-filter semiconductor (MS) → SGS → HM for EQH compounds with the formula LuCoVZ, and from HM → SGS → MS → SGS → HM for EQH compounds with the formula YCoVZ. Remarkably, for YCoVZEQH compounds there are not only diverse physical transitions, but also different types of spin-gapless feature that can be observed with changing lattice constants. The structural stability of these four EQH compounds is also examined from the points of view of formation energy, cohesive energy and mechanical behaviour. This work is likely to inspire consideration of rare earth-based EQH compounds for application in future spintronic and magneto-electronic devices.

Nodal-line semimetals in two-dimensional (2D) materials have attracted intense attention recently. From fundamental physics and spintronic application points of view, high Curie temperature ferromagnetic (FM) ones with nodal lines robust against spin-orbit coupling (SOC) are extremely desirable. Here, we propose that the FM ${\mathrm{K}}_2\mathrm{N}$ monolayer is such a Weyl nodal-line semimetal. We show that the ${\mathrm{K}}_2\mathrm{N}$ monolayer is dynamically stable and has a FM ground magnetic state with out-of-plane [001] magnetization. It shows two nodal lines in the low-energy band structures. Both nodal lines are robust against SOC, under the protection of mirror symmetry. We construct an effective Hamiltonian, which can well characterize the nodal lines in the system. Remarkably, the nodal-line semimetal proposed here is distinct from the previously studied ones in that the ${\mathrm{K}}_2\mathrm{N}$ monolayer is a 2D $d^0$-type ferromagnet with the magnetism arising from the partially filled the $p$ orbitals of the N atom, which can bring special advantages in spintronic applications. In addition, the Curie temperature in the ${\mathrm{K}}_2\mathrm{N}$ monolayer is estimated to be 942 K, which is significantly higher than previous FM nodal-line materials. We also find that specific tensile strains can transform the nodal line from type-I to type-II, making its nodal-line characteristics even more interesting.

As a novel type of fermionic state, the hybrid nodal loop with the coexistence of both type-I and type-II band crossings has attracted intense research interest. However, it remains a challenge to realize the hybrid nodal loop in both two-dimensional (2D) materials and in ferromagnetic (FM) materials. Here, we propose a FM hybrid nodal loop in a 2D CrN monolayer. We show that the material has a high Curie temperature ($>600$ K) FM ground state, with the out-of-plane [001] magnetization. It shows a half-metallic band structure with two bands in the spin-up channel crossing each other near the Fermi level. These bands produce both type-I and type-II band crossings, which form a fully spin-polarized hybrid nodal loop. We find the nodal loop is protected by the mirror symmetry and robust against spin-orbit coupling. An effective Hamiltonian characterizing the hybrid nodal loop is established. We further find the configuration of the nodal loop can be shifted under external perturbations such as strain. Most remarkably, we demonstrate that both type-I and type-II Weyl nodes can be realized from such FM hybrid nodal loop by simply shifting the magnetization from out of plane to in plane. Our work provides an excellent candidate to realize a FM hybrid nodal loop and Weyl fermions in 2D material, and is also promising for related topological applications with their intriguing properties.