Computational investigation of half-Heusler compounds for spintronics applications

We present first-principles density functional calculations of the electronic structure, magnetism, and structural stability of 378 XYZ half-Heusler compounds (with $X=$ Cr, Mn, Fe, Co, Ni, Ru, Rh; $Y=$ Ti, V, Cr, Mn, Fe, Ni; $Z=$ Al, Ga, In, Si, Ge, Sn, P, As, Sb). We find that a “Slater-Pauling gap” in the density of states (i.e., a gap or pseudogap after nine states in the three atom primitive cell) in at least one spin channel is a common feature in half-Heusler compounds. We find that the presence of such a gap at the Fermi energy in one or both spin channels contributes significantly to the stability of a half-Heusler compound. We calculate the formation energy of each compound and systematically investigate its stability against all other phases in the open quantum materials database (OQMD). We represent the thermodynamic phase stability of each compound as its distance from the convex hull of stable phases in the respective chemical space and show that the hull distance of a compound is a good measure of the likelihood of its experimental synthesis. We find low formation energies and mostly correspondingly low hull distances for compounds with $X=$ Co, Rh, or Ni, $Y=$ Ti or V, and $Z=$ P, As, Sb, or Si. We identify 26 18-electron semiconductors, 45 half-metals, and 34 near half-metals with negative formation energy that follow the Slater-Pauling rule of three electrons per atom. Our calculations predict several new, as-yet unknown, thermodynamically stable phases, which merit further experimental exploration—RuVAs, CoVGe, FeVAs in the half-Heusler structure, and NiScAs, RuVP, RhTiP in the orthorhombic MgSrSi-type structure. Further, two interesting zero-moment half-metals, CrMnAs and MnCrAs, are calculated to have negative formation energy. In addition, our calculations predict a number of hitherto unreported semiconducting (e.g., CoVSn and RhVGe), half-metallic (e.g., RhVSb), and near half-metallic (e.g., CoFeSb and CoVP) half-Heusler compounds to lie close to the respective convex hull of stable phases, and thus may be experimentally realized under suitable synthesis conditions, resulting in potential candidates for various semiconducting and spintronics applications.

Figure 1

Schematic of the XYZ half-Heusler $C{1}_b$ structure. It consists of three interpenetrating fcc sublattices with atomic sites $X\left(\frac{1}{4},\frac{1}{4},\frac{1}{4}\right), Y\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$, and $Z(0,0,0)$. The $\left(\frac{3}{4},\frac{3}{4},\frac{3}{4}\right)$ site is vacant.

Figure 2

Calculated total energies of CoMnAl in the half-Heusler $C{1}_b$ structure as a function of the lattice constant $a$ in ferrimagnetic, ferromagnetic, and nonmagnetic states.

Figure 3

Calculated total energies of RhCrSn in the half-Heusler $C{1}_b$ structure as a function of the lattice constant $a$ in two different ferrimagnetic states and the nonmagnetic state.

Figure 4

Calculated total energies of tetragonal distorted CrTiAs as a function of two lattice constants $a$ and $c$. There are two half-metallic energy minima corresponding to two different tetragonal phases, labeled by dashed circles, at $(a,c)=(5.52,6.66)\AA$ and $(a,c)=(5.97,5.67)\AA$. They differ in energy by 0.051 eV/f.u. The pink dashed line corresponds to cubic structures.

Figure 5

A schematic convex hull in the A-B chemical space. Phases ${\mathrm{S}}_i$ lie $on$ the convex hull and are thermodynamically stable, i.e., for each phase ${\mathrm{S}}_i$, there is no other phase or combination of phases at its composition lower in energy. Phases ${\mathrm{U}}_i$ are off the convex hull and thus unstable. For example, the formation energy of phase U2 is higher than that of a linear combination of phases S2 and S3. The distance from the convex hull ($\mathrm{\Delta }{E}_{\mathrm{HD}}$) of phase U2 is given by the difference between its formation energy and the energy of the convex hull at its composition ($E_{\mathrm{hull}}$, represented by the crimson star).

Figure 6

DFT-calculated formation energy vs hull distance of all compounds reported in the ICSD in the XYZ compositions considered in this work. A hull distance $\mathrm{\Delta }{E}_{\mathrm{HD}}=0$ indicates a stable ground state compound on the convex hull. Blue circles indicate half-Heuslers in the ICSD that have been experimentally synthesized. Green diamonds indicate XYZ phases other than $\mathrm{C}{1}_b$ that have been experimentally synthesized. Red pentagons indicate XYZ phases experimentally reported to be stable at high temperature or pressure. Blue-green squares indicate reported XYZ phases with site occupations that differ from the OQMD calculation. Yellow circles indicate $\mathrm{C}{1}_b$ phases sourced into ICSD from electronic structure calculations rather than from experiment.

Figure 7

DFT-calculated formation energy vs hull distance of all the 378 XYZ half-Heusler compounds considered in this work. A hull distance $\mathrm{\Delta }{E}_{\mathrm{HD}}=0$ indicates a stable ground state compound on the convex hull. Almost all the experimentally reported half-Heusler compounds (green squares, “In ICSD [e]”) have a hull distance less than $\sim 0.1$ eV/atom (the window represented by the two horizontal dashed lines); half-Heusler compounds sourced into the ICSD from previous computational work are represented by red diamonds (labeled “In ICSD [c]”).

Figure 8

DFT formation energies per atom for the 378 half-Heusler compounds considered in this work (colors represent the formation energy; blue and yellow = increasingly negative and positive formation energies respectively). The three coordinates represent the $X, Y$, and $Z$ species of the corresponding XYZ compound.

Figure 9

DFT-calculated distances from the convex hull for the 378 half-Heusler compounds considered in this work (colors represent the hull distance; blue and yellow = increasingly close to and away from the convex hull, respectively. All shades of blue represent hull distances $\mathrm{\Delta }{E}_{\mathrm{HD}}\le 0.1$ eV/atom). The three coordinates represent the $X, Y$, and $Z$ species of the corresponding XYZ compound.

Figure 10

DFT formation energies and hull distances for potential half-Heusler compounds grouped by the element on the $X$ site. The numbers near the top (in blue) and center (in brown) of each column denote the number of compounds with negative formation energy $\mathrm{\Delta }{E}_f$ and hull distance $\mathrm{\Delta }{E}_{\mathrm{HD}}\le 0.1$ eV/atom, respectively, in the corresponding $Z$-element group. Within a given $X$-element column, the compounds are ordered first by the element on the $Y$ site (same order as in Fig. 11) and then by the element on the $Z$ site (same order as in Fig. 12), i.e., $Z$ varies more rapidly than $Y$.

Figure 11

DFT formation energies and hull distances for potential half-Heusler compounds grouped by the element on the $Y$ site. The numbers near the top (in blue) and center (in brown) of each column denote the number of compounds with negative formation energy $\mathrm{\Delta }{E}_f$ and hull distance $\mathrm{\Delta }{E}_{\mathrm{HD}}\le 0.1$ eV/atom, respectively, in the corresponding $Y$-element group. Within a given $Y$-element column, the compounds are ordered first by the element on the $X$ site (same order as in Fig. 10) and then by the element on the $Z$ site (same order as in Fig. 12), i.e., $Z$ varies more rapidly than $X$.

Figure 12

DFT formation energies and hull distances for potential half-Heusler compounds grouped by the element on the $Z$ site. The numbers near the top (in blue) and center (in brown) of each column denote the number of compounds with negative formation energy $\mathrm{\Delta }{E}_f$ and hull distance $\mathrm{\Delta }{E}_{\mathrm{HD}}\le 0.1$ eV/atom, respectively, in the corresponding $Z$-element group. Within a given $Z$-element column, the compounds are ordered first by the element on the $X$ site (same order as in Fig. 10) and then by the element on the $Y$ site (same order as in Fig. 11), i.e., $Y$ varies more rapidly than $X$.

Figure 13

The distribution of half-Heusler compounds with negative ($\mathrm{\Delta }{E}_f<0$ eV/atom) and positive ($\mathrm{\Delta }{E}_f>0$ eV/atom) formation energies as a function of spin polarization $\mathcal{P}\left(E_F\right)$ [given by Eq. (3)]. In the central region, we show the number of half-Heusler compounds grouped by 10 percentage points of spin polarization. In an additional region to the left, we show the 24 semiconductors, including 23 compounds with a negative formation energy, and 1 with a positive formation energy. In the additional region on the right, we show 72 half-metals, including 42 and 30 with negative and positive formation energies respectively.

Figure 14

The distribution of half-Heusler compounds that lie on or close to ($\mathrm{\Delta }{E}_{\mathrm{HD}}<0.1$ eV/atom) and far away ($\mathrm{\Delta }{E}_{\mathrm{HD}}>0.1$ eV/atom) from the convex hull, as a function of spin polarization $\mathcal{P}\left(E_F\right)$ [given by Eq. (3)]. In the central region, we show the number of half-Heusler compounds grouped by 10 percentage points of spin polarization. We show the 24 semiconductors (of which 19 compounds have $E_{\mathrm{HD}}<0.1$ eV/atom) in the additional region to the left, and show the 72 half-metals (of which 15 compounds have $E_{\mathrm{HD}}>0.1$ eV/atom). Clearly, the existence of a gap at the Fermi level in one or both spin channels contributes to the stability of a half-Heusler compound.

Figure 15

Density of electronic states (DOS) of the 18-electron half-Heusler semiconductor RhTiP. The upper panel presents the total DOS (black) and the DOS projected on the $p$ orbitals of Rh (red), Ti (blue), and P (green). The lower panel presents the contribution from the corresponding $d$ orbitals of each atom. Zero energy corresponds to the Fermi level.

Figure 16

Charge densities for (a) RhTiP, (b) CoTiP, (c) RhTiSb, (d) CoTiSb, (e) RuVSb, and (f) FeVSb, for an isovalue of 0.405 $e{\text{Å}}^{-3}$, respectively. The figures were generated using vesta [72].

Figure 17

Calculated total magnetic moment $M_{\mathrm{tot}}$ as a function of the total number of valence electrons $N_V$ per f.u. for the 203 half-Heusler compounds with negative formation energies. The dash-dot line represents the Slater-Pauling rule $M_{\mathrm{tot}}=N_V-18$, and all the 45 half-metals listed in the boxes follow this rule precisely. Different colors indicate different sets of half-Heusler compounds based on the element on the $X$ site. Diamond, square, circle, and triangle symbols indicate ferro/ferrimagnets, half-metals, metals, and semiconductors, respectively. To avoid confusion about the signs of magnetic moments, we uniformly use the absolute values of magnetic moments in this diagram.

Figure 18

Calculated total and atom-resolved densities of electronic states for MnVAs, MnCrAs, MnMnAs, FeMnAs, CoMnAs, and NiMnAs. In each subplot, the upper (lower) panel shows the majority (minority) spin channel. The number of valence electrons per f.u. $N_V$ is also indicated for each system. Zero energy corresponds to the Fermi level.

Figure 19

Calculated density of electronic states (DOS) of CoMnSb in the ${\mathrm{Co}}_2\mathrm{MnSb}$-MnSb superstructure reported in Ref. [106]. Zero energy corresponds to the Fermi level.