Theoretical modeling of the electronic structure and Fermi surfaces of Gd4Sb3 and GdSb2 compounds

Our theoretical modeling of the electronic structure in the intermetallic Gd4Sb3 and GdSb2 compounds has been done in the framework of density functional theory accounting for spin-orbit coupling. It revealed the metallic character of the summed total density of electronic states for both materials. The complicated Fermi surfaces were found in the half-metallic Gd4Sb3 compound which corresponds to the band structure. The GdSb2 material is obtained to be a bad metal with the low density of states near the Fermi energy, the Fermi surfaces of GdSb2 are found to have the cylindrical shape.


Introduction
There are many classes of materials which are now classified as quantum materials [1] because of quantum effects governing their outstanding physical properties.One of the most exciting class of materials among them is the class of topological materials [2] with exotic Dirac or Weyl-featured physical properties [3].These features come from the band structure of these compounds being a result of quantum effects on related electronic states in reciprocal space.Such features may appear in spectral properties of insulators, superconductors, metals, or semimetals, e.g., LaBi with Dirac-conelike features [4].However, the other experimental manifestations can be revealed in extreme magnetoresistance (XMR), e.g., in HoBi [5].Recently, a number of RSb based on rare-earths R and antimony Sb have been found to contain topological features in the band structure: LaSb [6], NdSb [7], etc.The GdSb monoantimonide with the equiatomic composition was investigated both experimentally and theoretically [8,9,10,11].Experimental angle-resolved photoemission spectroscopy discovered pockets at Г and X in spectra of RSb for R = Y, Ce, Gd, Dy, Ho, Tm, Lu [12].
There are non-equiatomic Gd-Sb compounds which have been reported without further study.In particular, GdSb [13], GdSb2, Gd4Sb3, and Gd5Sb3 [14] intermetallic compounds were studied using high-temperature differential thermal analysis in [15].Among these synthesized compounds, the rareearth diantimonides, i.e., RSb2, were reported in a LaSb2-structural type and high-pressure orthorhombic type of structure [16].In [11], we found that Gd4Sb3 is a half-metallic ferromagnet (HMF) with the gap of 0.67 eV only in the minority spin projection, while the majority spin projection is characterized by intensive metallic density of states.In the band structure of GdSb2, a Dirac-conelike feature near the Г, S points was theoretically predicted [11].
In this manuscript, the results of the theoretical modeling for Gd4Sb3 and GdSb2 are reported.The electronic structure and Fermi surface were obtained taking into account spin-orbit coupling.We relate the features of the Fermi surfaces to the previous band structure calculations.

Crystal structure and computational details
The intermetallic Gd4Sb3 and GdSb2 materials are crystallized in the and Th3P4-and SmSb2-types, respectively, i.e., space groups I-43d, # 220, and Cmce, # 64, for parameters see [11].The theoretical modeling was carried in Quantum ESPRESSO set of computer programs [17,18].The generalized gradient approximation with the Perdew-Burke-Ernzerhof (abbreviated as PBE) exchange-correlation functional was chosen for the electronic structure computations.Spin-orbit coupling was previously found to be necessary for correct band structure and band gap values.Therefore, in the present study, spin-orbit coupling was taken into account in all calculations via full relativistic ultrasoft pseudopotentials (PPs) as set in the standard Quantum Espresso PPs library [19].In these PPs, the Gd 4f states are not taken as the valence electronic states.It was demonstrated in [11,20] that a ferromagnetic ordering of the Gd 4f moments provides magnetic moment of about 7 μB/Gd in the Gdbased intermetallic materials.

Electronic structure and Fermi surfaces of Gd4Sb3
In figure 1, the theoretically modelled total, Gd 5d and Sb 5p partial densities of states (DOS) for Gd4Sb3 are shown.The Fermi energy is equal to zero, it is plotted as a vertical orange dashed line.From figure 1a one can conclude that Gd4Sb3 has a lot of DOS at the Fermi energy.This result is in agreement with the previous study [11], whereas the spin-resolved DOS allowed us to specify that the minority spin projection has the gap of 0.67 eV.Thus, from our study, Gd4Sb3 is a half-metal which sums up to a metallic DOS in figure 1a.In the current calculation Sb 5p states fully lay below the Fermi level within the range of (-4.6, -1.0) eV, figure 1c.Calculated Gd 5d states are found to be mostly unoccupied contributing to the density of states on the Fermi energy and above it, figure 1b.One can also notice the dip in the total density of states, figure 1a, around the -0.7 eV where density of states is almost zero, which means that by tuning in Fermi level it is possible to get the case of a semiconductor with a small gap or semimetal.
The Fermi surfaces of Gd4Sb3 with distinct shapes are plotted in figure 2. The first shown Fermi surface figure 2a has a shape of a small cone and is located around high symmetry point N which correspond to the small hole pockets around same symmetry point in the band structure of the intermetallic Gd4Sb3 compound [11] along direction H-N.The second Fermi surface figure 2b has a more complex shape which could be described as connected paraboloids with a dent around high symmetry point P (direction N-P-Г), the same band also creates a sphere-like Fermi surface around Г. Another surface in figure 2c also represents a paraboloid with four bulges around P in the band structure [11].And the last shown Fermi surface figure 2d has a simple parabolic shape without bulges or dents around the same high symmetry point P.

Electronic structure and Fermi surfaces of GdSb2
In figure 3, the theoretically modelled total, Gd 5d and Sb1, Sb2 5p partial DOS for GdSb2 are shown.From figure 3a one can conclude that the GdSb2 is a bad metal with pretty low density of states at the Fermi level and around it, that corresponds to the previous study of this compound [11].All of the three shown states Gd 5d, Sb1 5p and Sb2 5p are hybridized and contribute equally to the valence band, see the panels b-c of figure 3, and the conduction band is mostly made of Gd 5d states, figure 3b.From this study and [11], one can conclude that spin-orbit coupling is negligible for GdSb2.

Conclusions
In this study, the intermetallic Gd4Sb3 and GdSb2 materials were investigated theoretically in terms of their electronic structure.The calculations were performed in the framework of quantum-mechanical approach in density functional theory accounting for spin-orbit coupling.The former compound (Gd4Sb3) is found to be metallic in nature which agrees with the previous study [11] considering that the summation of densities from both spin projections is required.Our modeling of the Fermi surfaces for Gd4Sb3 compound has showed that most of them are located near to the high symmetry points N and P and that they are parabolic in shape.The investigations show that the latter considered compound (GdSb2) is also a metal but with the low density of states at the Fermi energy, which is usually classified as a bad metal.The overall effect of spin-orbit coupling in the GdSb2 compound is found to be almost unnoticeable.The calculated Fermi surfaces for the second compound have cylinder-like shapes and are located near to the Г and S high symmetry points.

Figure 1 .
Figure 1.Calculated total (a), partial densities of states (b,c) for the Gd4Sb3 intermetallic material.