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Figure 12. Atom-resolved DOS for the experimental lattice parameter for NiMnSb and CoMnSb, compared with the once compressed or expanded by 2%. With the small arrow we denote the edges of the minority gap.

Figure 12 Atom-resolved DOS for the experimental lattice parameter for NiMnSb and CoMnSb, compared with the once compressed or expanded by 2%. With the small arrow we denote the edges of the minority gap.

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Figure 1. Schematic representation of the density of states for a half-metal with respect to normal metals and semiconductors.
Figure 1. Schematic representation of the density of states for a half-metal with respect to normal metals and semiconductors.
Introduction to half-metallic Heusler alloys Figure 2. Cl, and £2, structures adapted by the half- and full-Heusler alloys. The lattice is consisted of 4 interpenetratating fcc sublattices. The unit cell is that of a fec lattice with four atoms as basis, e.g. CoMnSb: Co at (000), Mn at (4 4 4), a vacant site at (5 54) and Sb at (3 $3) in Wyckoff coordinates. In the case of the full Heusler alloys also the vacant site is occupied by a Co atom. Note also that if all atoms were identical, the lattice would be simply the bcc.
Introduction to half-metallic Heusler alloys Figure 2. Cl, and £2, structures adapted by the half- and full-Heusler alloys. The lattice is consisted of 4 interpenetratating fcc sublattices. The unit cell is that of a fec lattice with four atoms as basis, e.g. CoMnSb: Co at (000), Mn at (4 4 4), a vacant site at (5 54) and Sb at (3 $3) in Wyckoff coordinates. In the case of the full Heusler alloys also the vacant site is occupied by a Co atom. Note also that if all atoms were identical, the lattice would be simply the bcc.
Figure 3. Atom-resolved density of states (DOS) of NiMnSb for a paramagnetic (upper left) and ferromagnetic (right) calculation. In the left bottom panel the DOS for the semiconductor CoTiSb. The zero energy value corresponds to the Fermi level Er.
Figure 3. Atom-resolved density of states (DOS) of NiMnSb for a paramagnetic (upper left) and ferromagnetic (right) calculation. In the left bottom panel the DOS for the semiconductor CoTiSb. The zero energy value corresponds to the Fermi level Er.
Figure 4. DOS of XMnSb compounds for X= Ni, Pd, Pt and Co, Rh, Pd
Figure 4. DOS of XMnSb compounds for X= Ni, Pd, Pt and Co, Rh, Pd
Table 1. Calculated spin magnetic moments in zg for the XMnSb compounds. (Th experimental lattice constants [4] have been used.) The deviations from integral tota moment for the half-metallic compounds are due to numerical angular momentun cutoff (see text). small contributions of Fe and V states. Analogously the Fe and V bands have smal contributions of Sb p states (and even smaller contributions from the Sb s states). The V to, and e, states exhibit a small crystal field splitting of 0.7 eV, with the e, level beins lower due to their weaker hybridisation with the Fe e, states in tetrahedral symmetry while for the Fe states this splitting is negligible. Most important is, that the gap arise from the hybridisation of the Fe and the V states and that the Sb p states are very deey and do not affect the gap.
Table 1. Calculated spin magnetic moments in zg for the XMnSb compounds. (Th experimental lattice constants [4] have been used.) The deviations from integral tota moment for the half-metallic compounds are due to numerical angular momentun cutoff (see text). small contributions of Fe and V states. Analogously the Fe and V bands have smal contributions of Sb p states (and even smaller contributions from the Sb s states). The V to, and e, states exhibit a small crystal field splitting of 0.7 eV, with the e, level beins lower due to their weaker hybridisation with the Fe e, states in tetrahedral symmetry while for the Fe states this splitting is negligible. Most important is, that the gap arise from the hybridisation of the Fe and the V states and that the Sb p states are very deey and do not affect the gap.
Figure 5. Schematic illustration of the origin of the gap in the minority band in half- Heusler alloys and semiconductors: The energy levels FE, of the energetically lower lying bonding hybrids are separated from the levels Ey of the antibonding hybrids by a gap, such that only the bonding states are occupied. Due to legibility reasons, we use dl, d2 and d3 to denote the dzy, dyz and dz tzg-orbitals, respectively, and d4, d5 for the d,2, d,2_y2 eg-orbitals. Figure 5. Schematic illustration of the origin of the gap in the minority band in half zinc-blende NiMn and PtMn, with the same lattice constant as NiMnSb. Indeed a gay is formed in the minority band. In the hypothetical NiMn the Fermi energy is slighth above the gap, however the isoelectronic PtMn compound shows nearly half-metallicity In this case the occupied minority states consist of six bands, a low-lying s-band anc five bonding d-bands, both of mostly Pt character. Since the total number of valenc electrons is 17, the majority bands contain 11 electrons, so that the total moment pe unit cell is 11 — 6 5{4B, which is indeed obtained in the calculations. This is th largest possible moment for this compound, since in the minority band all 5 Mn d-state: are empty while all majority d-states are occupied. The same limit of 5p is also th maximal possible moment of the half-metallic C1, Heusler alloys. Tha: wave a tfha half swatallieae “4° 1s eAmnaAannaAd Jaman alleetnaAs wart <=th thw waaay
Figure 5. Schematic illustration of the origin of the gap in the minority band in half- Heusler alloys and semiconductors: The energy levels FE, of the energetically lower lying bonding hybrids are separated from the levels Ey of the antibonding hybrids by a gap, such that only the bonding states are occupied. Due to legibility reasons, we use dl, d2 and d3 to denote the dzy, dyz and dz tzg-orbitals, respectively, and d4, d5 for the d,2, d,2_y2 eg-orbitals. Figure 5. Schematic illustration of the origin of the gap in the minority band in half zinc-blende NiMn and PtMn, with the same lattice constant as NiMnSb. Indeed a gay is formed in the minority band. In the hypothetical NiMn the Fermi energy is slighth above the gap, however the isoelectronic PtMn compound shows nearly half-metallicity In this case the occupied minority states consist of six bands, a low-lying s-band anc five bonding d-bands, both of mostly Pt character. Since the total number of valenc electrons is 17, the majority bands contain 11 electrons, so that the total moment pe unit cell is 11 — 6 5{4B, which is indeed obtained in the calculations. This is th largest possible moment for this compound, since in the minority band all 5 Mn d-state: are empty while all majority d-states are occupied. The same limit of 5p is also th maximal possible moment of the half-metallic C1, Heusler alloys. Tha: wave a tfha half swatallieae “4° 1s eAmnaAannaAd Jaman alleetnaAs wart <=th thw waaay
Figure 7. Calculated total spin moment per unit cell as a function of the total number Z; of valence electrons per unit cell for all the studied half Heusler alloys. The dashed line represents the Slater-Pauling behavior. With open circles we present the compounds deviating from the SP curve. Some experimental values for bulk systems near the SP curve from reference [4]: NiMnSb 3.85 4p, PdMnSb 3.95 ep, PtMnSb 4.14 wp and finally CoTiSb non-magnetic.
Figure 7. Calculated total spin moment per unit cell as a function of the total number Z; of valence electrons per unit cell for all the studied half Heusler alloys. The dashed line represents the Slater-Pauling behavior. With open circles we present the compounds deviating from the SP curve. Some experimental values for bulk systems near the SP curve from reference [4]: NiMnSb 3.85 4p, PdMnSb 3.95 ep, PtMnSb 4.14 wp and finally CoTiSb non-magnetic.
Figure 8. Atom-resolved DOS for the CogMnZ compounds with Z= Al, Si, Ge, Sn compounds
Figure 8. Atom-resolved DOS for the CogMnZ compounds with Z= Al, Si, Ge, Sn compounds
Table 2. Calculated spin magnetic moments per unit cell in wp for the CogMnZ compounds, where Z stands for the sp atom. (The experimental lattice constants [4] have been used.) The deviations from integral total moment for the half-metallic compounds are due to numerical angular momentum cutoff (see text). All compounds are half-metallic. LOLA Opt vuIts tO POULT LILAL *~“€g — *~t2g° In a second step we consider the hybridization of these Co-Co orbitals with the Mn l-orbitals. As we show in the right-hand part of figure [9] the doubly degenerated e, orbitals hybridize with the d,2 and d,2_, same representation. They create a doubly degenerate bonding e, state that is very 2 of the Mn that transform according to the low in energy and an antibonding one that is unoccupied and above the Fermi level. The 3 X ta, Co orbitals couple to the dry 2,22 of the Mn and create 6 new orbitals, 3 of which are bonding and occupied and the other three are antibonding and high in energy. Since each Co atom sits in the center of a Mn tetrahedron, the crystal field splitting
Table 2. Calculated spin magnetic moments per unit cell in wp for the CogMnZ compounds, where Z stands for the sp atom. (The experimental lattice constants [4] have been used.) The deviations from integral total moment for the half-metallic compounds are due to numerical angular momentum cutoff (see text). All compounds are half-metallic. LOLA Opt vuIts tO POULT LILAL *~“€g — *~t2g° In a second step we consider the hybridization of these Co-Co orbitals with the Mn l-orbitals. As we show in the right-hand part of figure [9] the doubly degenerated e, orbitals hybridize with the d,2 and d,2_, same representation. They create a doubly degenerate bonding e, state that is very 2 of the Mn that transform according to the low in energy and an antibonding one that is unoccupied and above the Fermi level. The 3 X ta, Co orbitals couple to the dry 2,22 of the Mn and create 6 new orbitals, 3 of which are bonding and occupied and the other three are antibonding and high in energy. Since each Co atom sits in the center of a Mn tetrahedron, the crystal field splitting
Figure 9. Schematic illustration of the origin of the gap in the minority band in full-Heusler alloys. Due to legibility reasons, we use d1, d2 and d3 to denote the dzy, dy, and dz, orbitals, respectively, and d4, d5 for the d,2, d,2_,2 orbitals.
Figure 9. Schematic illustration of the origin of the gap in the minority band in full-Heusler alloys. Due to legibility reasons, we use d1, d2 and d3 to denote the dzy, dy, and dz, orbitals, respectively, and d4, d5 for the d,2, d,2_,2 orbitals.
Figure 10. Atom- and symmetry-resolved DOS for the CogMnGe compound.
Figure 10. Atom- and symmetry-resolved DOS for the CogMnGe compound.
Figure 11. Calculated total spin moments for all the studied full Heusler alloys. The dashed line represents the Slater-Pauling behavior. With open circles we present the compounds deviating from the SP curve. Some experimental values for bulk systems near the SP curve from reference fd: CogMnAl 4.01 we, Cog2MnSi 5.07 wp, CogMnGa 4.05 up, CogMnGe 5.11 wg, CogMnSn 5.08 wg, Co2FeSi 5.9 wp, Mn2VAI -1.82 wp and finally Fez VAI non-magnetic.
Figure 11. Calculated total spin moments for all the studied full Heusler alloys. The dashed line represents the Slater-Pauling behavior. With open circles we present the compounds deviating from the SP curve. Some experimental values for bulk systems near the SP curve from reference fd: CogMnAl 4.01 we, Cog2MnSi 5.07 wp, CogMnGa 4.05 up, CogMnGe 5.11 wg, CogMnSn 5.08 wg, Co2FeSi 5.9 wp, Mn2VAI -1.82 wp and finally Fez VAI non-magnetic.
Figure 13. Position of the conduction band maximum (CBM) and valence band minimum (VBM) relative to the Fermi level for CogMnSi, CogMnGe, and Co2MnSn, calculated for a range of lattice parameters within the LSDA and the GGA. The gap width is also shown. The calculated equilibrium lattice parameter is indicated by deg. (Figure redrawn after Picozzi et al. [43].)
Figure 13. Position of the conduction band maximum (CBM) and valence band minimum (VBM) relative to the Fermi level for CogMnSi, CogMnGe, and Co2MnSn, calculated for a range of lattice parameters within the LSDA and the GGA. The gap width is also shown. The calculated equilibrium lattice parameter is indicated by deg. (Figure redrawn after Picozzi et al. [43].)
Table 3. Calculated spin polarisation P at the Fermi level (Er) and in the middle of the spin-down gap (£,), and ratio of spin-down/spin-up DOS in the middle of the gap (n,/n+)(Em), for various half-Heusler alloys. The alloys PdMnSb and PtMnSb present a spin-down gap. but are not half-metallic. as Ep is slightly below the gap.
Table 3. Calculated spin polarisation P at the Fermi level (Er) and in the middle of the spin-down gap (£,), and ratio of spin-down/spin-up DOS in the middle of the gap (n,/n+)(Em), for various half-Heusler alloys. The alloys PdMnSb and PtMnSb present a spin-down gap. but are not half-metallic. as Ep is slightly below the gap.
Table 4. Spin (mspin) and orbital (morpit) Magnetic moments in j»p for the XMnSb half-Heusler compounds. The last three columns are the total spin and orbital magnetic moment and their sum, respectively are considered reliable. The highest orbital moment, almost 0.1 jug, appears at the Ir and Mn atoms in IrMnSb, but with opposite signs for the two atoms, so that the total Morb is close to zero (note that also the Ir spin moment is opposite to the Mn spin moment; for Ir in this compound we obtain a ratio of Mop/Mspin & 1/2).
Table 4. Spin (mspin) and orbital (morpit) Magnetic moments in j»p for the XMnSb half-Heusler compounds. The last three columns are the total spin and orbital magnetic moment and their sum, respectively are considered reliable. The highest orbital moment, almost 0.1 jug, appears at the Ir and Mn atoms in IrMnSb, but with opposite signs for the two atoms, so that the total Morb is close to zero (note that also the Ir spin moment is opposite to the Mn spin moment; for Ir in this compound we obtain a ratio of Mop/Mspin & 1/2).
Table 5. Spin (mspin) and orbital (morpit) magnetic moments in yg for the X2YZ full-Heusler compounds. The last three colums are the total spin and orbital magnetic moments and their sum, respectively NiMnSb) the gap is created by the hybridization and bonding-antibonding splitting between the Mn d and the Ni d states. In the case of ull-Heusler alloys (e.g. Co2MnSi) the gap originates hybridization of the d states of the two Co atoms anc the subsequent interaction of these hybrids with the Mn d sta atom is a spectator; nevertheless, due to its electronegativity i contributing to a shift of the spin-u p bands and therefore to t tes. At this stage, the sj t acts as a hole reservoir he value of the total spir moment. Due to the different exchange splitting of the different atoms in the unit cell the hybridization of spin-up states d oes not show the same pattern as for the spin-dowr states. Thus, in the end, a gap appears only among the spin-down bands, while fo: spin-up a metallic character arises. Spiga + . 1 wa
Table 5. Spin (mspin) and orbital (morpit) magnetic moments in yg for the X2YZ full-Heusler compounds. The last three colums are the total spin and orbital magnetic moments and their sum, respectively NiMnSb) the gap is created by the hybridization and bonding-antibonding splitting between the Mn d and the Ni d states. In the case of ull-Heusler alloys (e.g. Co2MnSi) the gap originates hybridization of the d states of the two Co atoms anc the subsequent interaction of these hybrids with the Mn d sta atom is a spectator; nevertheless, due to its electronegativity i contributing to a shift of the spin-u p bands and therefore to t tes. At this stage, the sj t acts as a hole reservoir he value of the total spir moment. Due to the different exchange splitting of the different atoms in the unit cell the hybridization of spin-up states d oes not show the same pattern as for the spin-dowr states. Thus, in the end, a gap appears only among the spin-down bands, while fo: spin-up a metallic character arises. Spiga + . 1 wa
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