Highlights•Magnetism of Sr2XMoO6 (X = Ge,Si) is studied by LSDA and LSDA+U.•Mo-t2gt2g states and O-2p states make the most contribution to magnetic moment.•Thermodynamic properties are studied using a quasi-harmonic Debye model.We present a study of the electronic and thermodynamic properties of Sr2XMoO6 (X=Ge and Si) based on the density-functional theory first-principles calculations. It is found that Sr2GeMoO6 is a half-metallic ferromagnet and Sr2SiMoO6 is a metallic ferromagnet within the local spin density approximation (LSDA). But using a correlated band theory LSDA+U, the results indicate that both compounds are half-metallic ferromagnets with a total spin magnetic moment per formula unit of 2.00 μB. Through the analysis of crystal-field-splitting and spin-splitting, we find that the main magnetic moment contribution comes from the Mo-t2gt2g states and O-2p2p states for both compounds. Finally, by using a quasi-harmonic Debye model, the bulk modulus, heat capacity, Debye temperature and the coefficient of thermal expansion have also been obtained in the present work. The present calculations show that Sr2GeMoO6 and Sr2SiMoO6 may be promising materials for future spintronics applications.Graphical abstract

•Six Zr-based quaternary Heusler spin-gapless semiconductors (SGS) are studied.•The Slater–Pauling behaviours of the SGSs are discussed.•The alloys are stable in the aspects of cohesion energy and formation energy.•This is the first paper commented on 4d transition element based SGSs.Employing first-principle calculations, we have investigated electronic and magnetic properties of the Zr-based quaternary Heusler alloys: ZrCoVIn, ZrFeVGe, ZrCoFeP, ZrCoCrBe and ZrFeCrZ (Z = In and Ga). Our calculation results show that all the alloys are (or nearly) spin-gapless semiconductors. The Slater–Pauling behaviours of these alloys are discussed as well. The cohesion energy and formation energy of these alloys have also been discussed, and the results indicate the studied alloys are stable.

Using the full-potential local orbital minimum-basis method, the Ti2-based full-Heusler alloys are studied. The results show that these compounds exhibit a half-metallic behavior, however, in contrast to the conventional full-Heusler alloys, the full-Heusler alloys show a Slater–Pauling rule Mt = Zt − 18 between the total spin magnetic moment (Mt) and valence electron concentration (Zt) per unit cell. Low formation enthalpy implies these Heusler alloys can be fabricated experimentally. The origin of the gap in these half-metallic alloys are well understood. It is found that the half-metallic properties of Ti2-based compounds are insensitive to the lattice distortion and a fully spin polarization can be obtained within a wide range of lattice parameters. This is favorable in practical application.Highlights► State-of-the-art Ti2-based Heusler alloys are predicted half-metallic materials. ► Calculated total spin magnetic agree with the Slater–Pauling rule: Mt = Zt − 18. ► The bulk modulus at equilibrium state and the range of pressure exhibiting half-metallic properties are listed. ► The influence of sp atom on structural stability and half-metallic properties also is illustrated quite well.

•Six Zr-based quaternary Heusler spin-gapless semiconductors (SGS) are studied.•The Slater–Pauling behaviours of the SGSs are discussed.•The alloys are stable in the aspects of cohesion energy and formation energy.•This is the first paper commented on 4d transition element based SGSs.Employing first-principle calculations, we have investigated electronic and magnetic properties of the Zr-based quaternary Heusler alloys: ZrCoVIn, ZrFeVGe, ZrCoFeP, ZrCoCrBe and ZrFeCrZ (Z = In and Ga). Our calculation results show that all the alloys are (or nearly) spin-gapless semiconductors. The Slater–Pauling behaviours of these alloys are discussed as well. The cohesion energy and formation energy of these alloys have also been discussed, and the results indicate the studied alloys are stable.