First-principles prediction of half-Heusler half-metals above room temperature

TL;DR

This study uses first-principles calculations to identify new half-Heusler alloys that are ferromagnetic, half-metallic, and stable above room temperature, advancing spintronic material development.

Contribution

The paper reports the discovery of 28 new ferromagnetic, half-metallic, vibrationally stable half-Heusler compounds with high Curie temperatures, including five promising candidates for room-temperature applications.

Findings

28 new ferromagnetic half-Heusler compounds identified

17 compounds remain magnetically ordered above 300 K

5 compounds are promising for room-temperature spintronics

Abstract

Half-metallicity (HM) offers great potential for engineering spintronic applications, yet only few magnetic materials present metallicity in just one spin channel. In addition, most HM systems become magnetically disordered at temperatures well below ambient conditions, which further hinders the development of spin-based electronic devices. Here, we use first-principles methods based on density functional theory (DFT) to investigate the electronic, magnetic, structural, mixing, and vibrational properties of $90$ $XYZ$ half-Heusler (HH) alloys ($X =$ Li, Na, K, Rb, Cs; $Y =$ V,Nb, Ta; $Z =$ Si, Ge, Sn, S, Se, Te). We disclose a total of $28$ new HH compounds that are ferromagnetic, vibrationally stable, and HM, with semiconductor band gaps in the range of $1$-$4$ eV and HM band gaps of $0.2$-$0.8$ eV. By performing Monte Carlo simulations of a spin Heisenberg model fitted to DFT energies, we estimate the Curie temperature, $T_{\rm C}$, of each HM compound. We find that $17$ HH HM remain magnetically ordered at and above room temperature, namely, $300 \le T_{\rm C} \le 450$ K, with total magnetic moments of $2$ and $4$ $\mu_{\rm B}$. A further materials sieve based on zero-temperature mixing energies let us to conclude $5$ overall promising ferromagnetic HH HM at and above room temperature: NaVSi, RbVTe, CsVS, CsVSe, and RbNbTe. We also predict $2$ ferromagnetic materials that are semiconductor and magnetically ordered at ambient conditions: LiVSi and LiVGe.

Figures

Tables

Table 1. Table 1: Properties summary of the most relevant HH alloys predicted in this study considering their lowest-energy configuration. The 7 7 7 overall most promising HH compounds are highlighted in bold. All systems are vibrationally stable and display FM spin ordering. The electronic properties are calculated with the Tran-Blaha modified Becke-Johnson (TB-mBJ) meta-GGA exchange-correlation functional tran09 . The numerical uncertainty in our T C subscript 𝑇 C T_{\rm C} results is ± 25 plus-or-minus 25 \pm 25  K.

$\mathrm{Compound}$	$\mathrm{Structure}$	$a_0$	$\mathrm{Electronic}$	$M$	$E_{\mathrm{BG}}$	$E_{\mathrm{HM}}$	$T_C$	$E_{\mathrm{mix}}$
		$\left(\AA \right)$	$\mathrm{behavior}$	$\left({\mu }_B\right)$	$\left(\mathrm{eV}\right)$	$\left(\mathrm{eV}\right)$	$\left(K\right)$	$(\mathrm{eV}/f.u.)$
$\mathrm{𝐋𝐢𝐕𝐒𝐢}$	$\mathrm{T1}$	$5.85$	$\mathrm{𝐒𝐂}$	$2.0$	$1.43$	$-$	$\mathrm{𝟑𝟎𝟎}$	$-0.11$
$\mathrm{𝐋𝐢𝐕𝐆𝐞}$	$\mathrm{T1}$	$5.93$	$\mathrm{𝐒𝐂}$	$2.0$	$1.06$	$-$	$\mathrm{𝟑𝟕𝟓}$	$-0.11$
$\mathrm{𝐍𝐚𝐕𝐒𝐢}$	$\mathrm{T1}$	$6.25$	$\mathrm{𝐇𝐌}$	$2.0$	$1.11$	$0.52$	$\mathrm{𝟑𝟎𝟎}$	$+0.21$
$\mathrm{NaVTe}$	$\mathrm{T1}$	$6.84$	$\mathrm{HM}$	$4.0$	$4.18$	$0.54$	$150$	$-0.12$
$\mathrm{KVSi}$	$\mathrm{T3}$	$6.55$	$\mathrm{HM}$	$2.0$	$1.49$	$0.21$	$375$	$+0.56$
$\mathrm{𝐑𝐛𝐕𝐓𝐞}$	$\mathrm{T3}$	$7.27$	$\mathrm{𝐇𝐌}$	$4.0$	$3.73$	$0.83$	$\mathrm{𝟑𝟕𝟓}$	$+0.14$
$\mathrm{𝐂𝐬𝐕𝐒}$	$\mathrm{T3}$	$6.93$	$\mathrm{𝐇𝐌}$	$4.0$	$3.92$	$0.68$	$\mathrm{𝟑𝟎𝟎}$	$+0.01$
$\mathrm{𝐂𝐬𝐕𝐒𝐞}$	$\mathrm{T3}$	$7.13$	$\mathrm{𝐇𝐌}$	$4.0$	$3.66$	$0.66$	$\mathrm{𝟑𝟕𝟓}$	$+0.11$
$\mathrm{RbNbSi}$	$\mathrm{T3}$	$6.77$	$\mathrm{HM}$	$2.0$	$1.24$	$0.57$	$450$	$+0.83$
$\mathrm{RbNbSn}$	$\mathrm{T3}$	$7.19$	$\mathrm{HM}$	$2.0$	$1.19$	$0.33$	$450$	$+0.83$
$\mathrm{𝐑𝐛𝐍𝐛𝐓𝐞}$	$\mathrm{T3}$	$7.41$	$\mathrm{𝐇𝐌}$	$4.0$	$3.82$	$0.75$	$\mathrm{𝟒𝟓𝟎}$	$+0.40$
$\mathrm{CsNbSi}$	$\mathrm{T3}$	$6.87$	$\mathrm{HM}$	$2.0$	$0.47$	$0.11$	$375$	$+0.91$
$\mathrm{CsNbGe}$	$\mathrm{T3}$	$6.96$	$\mathrm{HM}$	$2.0$	$0.62$	$0.14$	$375$	$+0.91$
$\mathrm{CsNbSn}$	$\mathrm{T3}$	$7.28$	$\mathrm{HM}$	$2.0$	$0.69$	$0.16$	$375$	$+0.94$
$\mathrm{CsNbTe}$	$\mathrm{T3}$	$7.55$	$\mathrm{HM}$	$4.0$	$3.23$	$0.39$	$450$	$+0.49$
$\mathrm{KTaSn}$	$\mathrm{T1}$	$7.17$	$\mathrm{HM}$	$2.0$	$1.31$	$0.54$	$300$	$+0.88$
$\mathrm{RbTaSi}$	$\mathrm{T1}$	$6.89$	$\mathrm{HM}$	$2.0$	$1.36$	$0.66$	$300$	$+0.92$
$\mathrm{RbTaGe}$	$\mathrm{T1}$	$6.99$	$\mathrm{HM}$	$2.0$	$1.39$	$0.55$	$300$	$+0.97$
$\mathrm{RbTaSn}$	$\mathrm{T3}$	$7.16$	$\mathrm{HM}$	$2.0$	$1.05$	$0.35$	$375$	$+1.03$
$\mathrm{RbTaTe}$	$\mathrm{T3}$	$7.37$	$\mathrm{HM}$	$4.0$	$3.70$	$0.24$	$375$	$+0.75$

Equations

$$E_{\mathrm{spin}}=E_0+\frac{1}{2}\sum _{ij}{J}_{ij}{S}_i{S}_j,$$

$$E_0=\frac{1}{2}\left(E_{\mathrm{FM}}+E_{\mathrm{AFM}}\right)$$

$$J_{ij}=\frac{1}{8|S{|}^2}\left(E_{\mathrm{FM}}-E_{\mathrm{AFM}}\right),$$

$$M=N_t-8,$$

$$E_{\mathrm{mix}}=E_{XYZ}^{\mathrm{HH}}-\left(E_X^{\mathrm{fcc}}+E_Y^{\mathrm{fcc}}+E_Z^{\mathrm{fcc}}\right),$$

Full text

I Introduction

Half-metals (HM) with full spin polarization at the Fermi level are of great potential for spintronic applications felser07 ; zutic03 . In particular, HM are considered to be ideal for injecting spin-polarized currents into semiconductors bhat16 ; dash09 and for manufacturing electrodes for magnetic tunnel junctions and giant magnetoresistance devices tanaka99 ; hordequin98 . Half-Heusler (HH) alloys comprise a relatively large family of multifunctional materials with chemical formula $XYZ$ and cubic symmetry (space group $F\overline{4}3m$); the archetypal HH compound NiMnSb was the first half-metal material to be ever reported groot83 . The structural compatibility of HH HM with typical cubic semiconductors, the potential huge number of current HH HM compounds casper12 , and the possibility of combining different HH to form HM layered structures azadani16 , open great prospects in the field of spin-based electronics.

In recent years, HH alloys have been studied extensively with computational first-principles methods martin12 ; cazorla17 . The structural simplicity, rich variety, and predictable electronic behaviour (e.g., modified Slater-Pauling rule damewood15 ) of HH alloys convert these materials into a perfect target for automated computational searches of HM ma17 ; legrain17 ; sattar18 . The structural stability and electronic and magnetic properties of HH calculated at zero temperature are the usual materials descriptors employed to guide the theoretical searches of candidate HM. However, analysis of the corresponding magnetic properties at room temperature, which are crucial for the engineering of practical applications tu16 ; cao17 , usually are neglected due to the large computational load associated with first-principles simulation of thermal effects curtarolo13 . Furthermore, ferromagnetic (FM) spin ordering is widely assumed in such computational investigations galanakis06 ; wei12 regardless of the fact that anti-ferromagnetic (AFM) spin ordering is also possible in HM materials damewood15 ; leuken94 ; luo08 ; hu11 . In order to provide improved guidance to future experiments, it is convenient then to examine the magnetic properties of HH HM at $T\neq 0$ K conditions, along with their vibrational and mixing stabilities.

In this article, we investigate the electronic, magnetic, structural, mixing, and vibrational properties of $90$ $XYZ$ HH alloys ($X=$ Li, Na, K, Rb, Cs; $Y=$ V, Nb, Ta; $Z=$ Si, Ge, Sn, S, Se, Te) with first-principles methods based on density functional theory (DFT). Our selection of materials has been motivated by recent encouraging results reported by other authors for some similar systems damewood15 ; hussain18 ; wang17 . We introduce a simple HH spin Heisenberg model fitted to FM and AFM DFT energies that allows for fast and systematic monitoring of the magnetization as a function of temperature. We predict that a total of $17$ HH HM are vibrationally stable and remain magnetically ordered at and above room temperature, with total magnetic moments of $2$ and $4$ $\mu_{\rm B}$ and semiconductor (half-metal) band gaps in the range of $1$–$4$ ($0.2$–$0.8$) eV. A further materials sieve based on zero-temperature mixing energies allows us to identify $5$ HH HM that are most promising for electronic applications. Meanwhile, a total of $21$ HH alloys are found to exhibit an anti-ferromagnetic ground state but all of them turn out to be metallic. We also predict $2$ new semiconductor FM materials that possess high thermodynamic stability and Curie temperatures. General structural, thermodynamic, and functional trends are identified across the $X$, $Y$, and $Z$ series. Hence, our computational study discloses a number of electronic materials and design strategies that should be useful for spintronic applications.

II Methods

In what follows, we explain the technical details of our first-principles calculations. We also describe the simple spin Heisenberg model that we have devised to analyse the magnetic properties of HH alloys at finite temperatures, along with the technical details of the accompanying Monte Carlo simulations.

II.1 First-principles calculations

Density functional theory (DFT) calculations have been performed with the self-consistent full potential linearized augmented plane wave (FP-LAPW) method, as implemented in the WIEN2K code wien2k . The generalised gradient approximation to the exchange-correlation energy due to Perdew-Burke-Ernzerhof (PBE) pbe has been employed in this study to estimate energies and determine equilibrium geometries. The value of the R${}_{\rm mt}\times$Kmax product, where Rmt is the value of the muffin-tin sphere radii and Kmax of the plane wave cut-off energy, was fixed to $9$ in order to provide highly converged results. Likewise, a dense k-point mesh of $21\times 21\times 21$ was used for integrations within the first Brillouin zone (IBZ). The self-consistent threshold values adopted for the calculation of energies and equilibrium geometries were $10^{-5}$ eV and $10^{-4}$ eV/Å, respectively. In order to estimate accurate electronic and magnetic properties (e.g., band gaps and magnetic moments) at reasonable computational cost, we employed the Tran-Blaha modified Becke-Johnson (TB-mBJ) meta-GGA exchange-correlation functional tran09 over the equilibrium structures generated with PBE.

The pseudopotential plane-wave DFT code VASP vasp has been also used for determining the energy difference between FM and AFM spin orderings, and for estimating vibrational lattice phonons. We used the projector-augmented wave method to represent the ionic cores bloch94 , considering the following electrons as valence states: $X$ $s$, $p$, and $d$; $Y$ $s$, $p$, and $d$; and $Z$ $s$ and $p$. Wave functions were represented in a plane-wave basis truncated at $650$ eV and for integrations within the IBZ we employed a dense $\Gamma$-centered k-point mesh of $16\times 16\times 16$. The calculation of phonon frequencies was performed with the small displacement method alfe09 and the PHONOPY code phonopy . The following parameters provided sufficiently well converged phonon frequencies: 190–atom supercells (i.e., $4\times 4\times 4$ replication of the conventional HH unit cell) and atomic displacements of $0.02$ Å.

II.2 Spin Heisenberg model and Monte Carlo simulations

To analyse the effects of temperature on the magnetic properties of HH alloys, we define the following spin Heisenberg Hamiltonian:

[TABLE]

where $E_{0}$ is a reference energy, $S_{i}$ represent the magnetic moment of ion $i$, and $J_{ij}$ the exchange interactions of ion $i$ with the rest. In our model, we only consider spin couplings between nearest magnetic ions (which add up to $12$, given that the symmetry of the magnetic sublattice is face centered cubic) and assume that all exchange interactions are equal. Based on such simplifications, the value of the model parameters can be calculated straightforwardly with DFT methods as:

[TABLE]

where $E_{\rm FM}$ and $E_{\rm AFM}$ are the energy of the crystal considering perfect ferromagnetic and anti-ferromagnetic spin arrangements, respectively. (We recall that in the AFM spin configuration each magnetic ion sees $8$ out of its $12$ nearest neighbors with opposite spin orientation singh17 .) It is worth noting that in spite of the simplicity of the adopted spin Heisenberg Hamiltonian, similar models have been able to provide accurate results for the $T$-dependence of the magnetization of complex magnetic materials (e.g., multiferroic oxide perovskites cazorla13 ; cazorla17b ; cazorla18 ).

Classical Monte Carlo (MC) simulations of the spin Heisenberg Hamiltonian just explained were performed to estimate the magnetization of HH alloys (that is, equal to the sum of all invidual spins) as a function of temperature. We used a periodically-repeated simulation box containing $20\times 20\times 20$ spins. Thermal averages were computed from runs of $50,000$ MC sweeps performed after equilibration (which consisted of the same number of MC steps). A small symmetry-breaking magnetic anisotropy was introduced in the MC simulations to facilitate the accompanying numerical analysis escorihuela12 . By using this setup and monitoring the evolution of the magnetization as a function of $T$, we were able to estimate magnetic transition temperatures, $T_{\rm C}$, with a numerical accuracy of $25$ K.

III Results and Discussion

We start this section by providing an overview of the computational strategy that we have followed to determine the ground-state configurations of HH alloys, and the general classification that results from our calculations based on their stuctural, electronic, and magnetic properties. Then, we explain in detail the electronic and vibrational properties of the disclosed HH HM, followed by a discussion on their magnetic phase transition temperatures and zero-temperature mixing energies. Finally, we highlight the HH alloys that according to our calculations are most promising for spintronics applications.

III.1 Overview of the $90$ $XYZ$ HH alloys

The $XYZ$ HH alloys considered in this study feature an alkali metal in the $X$ position (Li, Na, K, Rb, and Cs), a transition metal in the $Y$ position (V, Nb, and Ta), and a non-metal $sp$-element in the $Z$ position (Si, Ge, Sn, S, Se, and Te). The cubic HH unit cell consists of three interpenetrating face centered cubic (fcc) sublattices that together render a crystal structure with $F\overline{4}3m$ symmetry. There exist three possible atomic arrangements compatible with this structure, namely, T1, T2, and T3, which are generated by exchanging the Wyckoff positions of the $X$, $Y$, and $Z$ ions (see Fig. 1). The physical properties of HH alloys may be greatly influenced by their specific atomic arrangement sattar18 , hence configurations T1, T2, and T3 need to be all considered when performing thorough functionality searches within this family of materials.

Systematic fixed-volume geometry optimizations have been carried out for each HH compound to determine the energetically most favorable atomic arrangement, equilibrium volume, and magnetic ordering. First, the equilibrium volume of each possible structure considering FM spin ordering is obtained by fitting the series of calculated energy points to a Birch-Murnaghan equation of state cazorla15 ; subsequently, the energy associated with AFM spin ordering is analyzed for the lowest-energy atomic arrangement obtained in the FM case. Figures 2a-b illustrate such a computational procedure for the particular case of NaVTe, which turns out to exhibit a T1–FM ground state and an equilibrium volume of $320$ Å3. Once the optimal atomic arrangement, equilibrium volume, and magnetic ordering have been determined at the PBE level, we accurately calculate the corresponding electronic properties (e.g., energy band gaps and magnetic moments) with the TB-mBJ functional tran09 .

Figure 3 shows a general classification of the $90$ HH compounds analyzed in this study made on basis to their structural, magnetic, and electronic properties. A total of $56$ materials present lowest energy on the T1 arrangement, $32$ on the T3, and only $2$ on the T2. Compounds with a light-weight alkali metal in the $X$ position (e.g., Li, Na, and K) tend to be more stable in the T1 arrangement, whereas those with heavy-weight alkali metals (e.g., Rb and Cs) in the T3. On the other hand, compounds with a heavy-weight transition metal in the $Y$ position (e.g., Nb and Ta) mostly are stabilized in the T1 configuration, made the exception of the HH alloys containing Cs which always adopt the T3 arrangement. Meanwhile, the role of the non-metal element occupying the $Z$ position on choosing the most favorable structure appears to be negligible. We note that the two compounds that adopt the T2 configuration turn out to be metallic (i.e., LiTaSn and LiNbSn); thereby, the T2 arrangement will be ignored for the remainder of the article.

At zero-temperature conditions, we find that $55$ HH alloys are FM, $21$ AFM, and $14$ non-magnetic (see Fig.3). A total of $39$ FM compounds are found to be half-metallic, $2$ FM semiconductor (SC), and the rest metallic. In what follows, we describe with detail the electronic, vibrational, magnetic, and mixing properties of the $41$ new HM and SC compounds that have been determined in our investigation, all of which present FM spin ordering.

III.2 Electronic and vibrational properties

Figure 4 shows the energy and half-metallic band gaps, $E_{\rm BG}$ and $E_{\rm HM}$, estimated for the $90$ $XYZ$ HH alloys considered in this study. $E_{\rm BG}$ is the energy difference between the top of the valence band and bottom of the conduction band in the semiconductor spin channel, and $E_{\rm HM}$ the minimum energy that an electron requires to surpass the spin gap (see Fig.5, where we represent those quantities for CsVSe). One can observe a certain correlation between these two quantities, namely, large $E_{\rm BG}$’s are accompanied by large $E_{\rm HM}$’s. For instance, RbNbS (T1 arrangement) has an energy band gap of $5.08$ eV and HM band gap of $1.46$ eV, while CsVSn (T3 arrangement) has $1.30$ eV and $0.18$ eV, respectively. Nevertheless, there are few compounds that in spite of having a relatively small energy band gap exhibit a relatively large HM band gap (e.g., RbNbSi in the T3 configuration for which we estimate $E_{\rm BG}=1.24$ eV and $E_{\rm HM}=0.57$ eV).

The color code employed in Fig.4 shows that the HH alloys consisting of heavy-weight $X$-ions (K, Rb, and Cs), light-weight $Y$-ions (V and Nb), and chalcogen elements (S, Se, and Te) in the $Z$ position, generally possess the largest $E_{\rm BG}$ and $E_{\rm HM}$. By contrast, HH compounds with a light-weight (heavy-weight) alkali (transition) metal element like Li (Ta) in the $X$ ($Y$) position mostly are metallic or present medium band gaps.

Figure 6 shows the spin-projected density of electronic states calculated for the ground state of RbVTe (T3 configuration). This compound is half-metallic with a large $E_{\rm HM}$ of $0.83$ eV and large energy band gap of $3.73$ eV, hence is an interesting case for which to analyze the electronic features. In the majority band (spin up), is appreciated that the V $d$–$t_{2g}$ and Rb $d$ orbitals are highly hybridized in the region surrounding the Fermi level, which leads to the appearance of metallicity in that channel. In the minority band (spin down), however, the V and Rb $d$ orbitals are shifted to higher energies and remain unoccupied; consequently, a HM band gap appears in that channel between the occupied $d$ bonding and unoccupied $d$ antibonding states. Meanwhile, the majority of states forming the top of the valence band in the minority band (spin down) correspond to Te $p$ orbitals, which are shifted to energies well below those of the V and Rb $d$ orbitals.

We note that when the chalcogen element (S, Se, and Te) in the $Z$ position is substituted by a group–XIV element (Si, Ge, Sn), $E_{\rm BG}$ and $E_{\rm HM}$ generally undergo a significant reduction (see Fig.4). For instance, RbVSn has an energy band gap of $1.13$ eV and a HM band gap of $0.20$ eV. Such a band gap cloisure is consequence of a decrease in the energy separation between the $Z$ $p$ orbitals and $X$–$Y$ $d$ bonding and antibonding states (not shown here), and it occurs when the number of valence electrons in the system changes from $12$ (S, Se, Te) to $10$ (Si, Ge, Sn). Interestingly, the energy band gap changes as induced by $Z$–element substitutions correlate directly with changes in the magnetic moment of the ions; we will discuss in detail this and other magnetic effects in the next subsection.

Out of the $90$ HH alloys investigated in this study, we have found that $39$ ($2$) are ferromagnetic and half-metallic (magnetic semiconductor) at zero-temperature conditions. In order to provide useful guides for the experiments, is necessary also to assess the vibrational stability of the candidate materials proposed by theory. Here, we have calculated the lattice phonons for each HH alloy at the high-symmetry reciprocal point $\Gamma$, and concluded that $28$ FM HM along with the $2$ FM SC are vibrationally well behaved (that is, do not present imaginary phonon frequencies at the IBZ center). For some selected cases (namely, the overall most promising materials that will be discussed in Sec.III.4), we have calculated also the full phonon spectrum over the entire IBZ and found that most of them are vibrationally stable (see Fig.7, where we enclose a couple of relevant examples). We have not been able to draw any robust correlation between vibrational instability and chemical structure for the HH alloys analyzed in this study. For the remainder of the article, we will focus on describing the magnetic and mixing properties of the $28$ ($2$) FM, HM (SC), and vibrational stable compounds that we have predicted.

III.3 Magnetism and mixing stability

Recently, Damewood et al. damewood15 have proposed a modified Slater-Pauling (mSP) rule for describing the magnetic moment of the half-Heusler alloys LiMn$Z$ ($Z=$ N, P, Si). Specifically, the total magnetic moment per formula unit (expressed in units of $\mu_{\rm B}$), $M$, has been parametrized as:

[TABLE]

where $N_{t}$ is the total number of valence electrons in the unit cell. In the present case, $N_{t}$ is equal to $12$ for alloys displaying S, Se, or Te in the $Z$ position, and to $10$ for Si, Ge, and Sn (we recall that $N_{t}=1$ for the $X$ elements considered in this study, $N_{t}=5$ for the $Y$’s, and $N_{t}=4$ or $6$ for the $Z$’s). We have found that all non-metallic and magnetic $XYZ$ alloys disclosed in this study (i.e., either HM or SC) in fact follow the mSP rule proposed by Damewood et al. damewood15 (some examples are provided in Table I). Interestingly, in those cases we also observe a clear correlation between the total magnetization and size of the energy band gap: large $M$’s and large $E_{\rm BG}$’s occur simultaneously in the same compounds. From an applied point of view, this seems to be a positive finding since wide band gap HM alloys then are likely to be also magnetically robust. Nevertheless, the size of the magnetic moments does not provide any information on the variation of the magnetization at $T\neq 0$ conditions, which is related to the exchange interactions between the magnetic moments.

We have estimated the magnetic transition temperature, $T_{\rm C}$, for the $28$ ($2$) FM, HM (SC), and vibrational stable compounds determined in this study, by using the computational methods explained in Sec. II.2. Our $T_{\rm C}$ results are shown in Fig.8. It is appreciated that compounds with a heavy-weight alkali metal in the $X$ position (Rb and Cs) concentrate the highest magnetic transition temperatures (see also Table I). For instance, RbNbTe displays a Curie temperature of $450\pm 25$ K and CsVSe of $375\pm 25$ K, while NaVTe becomes paramagnetic at $150\pm 25$ K and KTaGe at $225\pm 25$ K. We ascertain the lack of any correlation between the size of the ionic magnetic moments, $M$, and the Curie temperature of the system; for example, RbNbSn displays $M=2~{}\mu_{\rm B}$ and $T_{\rm C}=450\pm 25$ K while for RbNbTe we estimate $M=4~{}\mu_{\rm B}$ and the same magnetic transition temperature. Interestingly, we find that a total of $17$ ($2$) FM, HM (SC), and vibrational stable compounds remain magnetically ordered at or above room temperature. All of them are listed in Table I and will be discussed in the next subsection.

Finally, we calculated the zero-temperature mixing energy, $E_{\rm mix}$, of the $28$ ($2$) FM, HM (SC), and vibrational stable compounds disclosed in this work, by using the formula:

[TABLE]

where $E_{XYZ}^{\rm HH}$ represents the ground-state energy of the HH alloy, and $E_{A}^{\rm fcc}$ the energy of the bulk $A$ crystal considering an equilibrium fcc structure. $E_{\rm mix}$ provides a quantitative estimation of how stable the $XYZ$ system is against decomposition into $X$-, $Y$-, and $Z$-rich regions. In particular, negative (positive) values of the mixing energy indicate high (low) stability of the compound against phase decomposition. We should note, however, that the configurational entropy of the HH alloy, which is totally neglected in Eq.(4), will always contribute favourably to the free energy and enhance the chemical stability of the $XYZ$ compound at finite temperatures page16 ; rost15 . Consequently, a positive but small value of $E_{\rm mix}$ does not necessarily imply phase separation under realistic $T\neq 0$ K conditions. Here, we (somewhat arbitrarily) consider that a $E_{\rm mix}$ threshold value of $0.2$ eV per formula unit can be used to sieve materials with reasonably good mixing stability from those with tendency for phase separation shenoy19 .

Figure 9 shows our $E_{\rm mix}$ results; only $7$ compounds out of $30$ (i.e., $28$ HM and $2$ SC) display zero-temperature mixing energies below $0.2$ eV per formula unit. We note that all of those low mixing energy compounds contain V in the $Y$ position, namely, LiVSi (SC), LiVGe (SC), NaVTe (HM), NaVSi (HM), RbVTe (HM), CsVS (HM), and CsVSe (HM) (see Table I). Specifically, only the first three alloys listed above present negative $E_{\rm mix}$ values, while for CsVS we obtain a practically null mixing energy. On the other hand, KVSi (HM), RbNbTe (HM), and CsNbTe (HM) exhibit mixing energies close to $0.50$ eV/f.u., and for the rest of compounds we estimate $E_{\rm mix}$’s that are close to $1.00$ eV per formula unit. These results indicate that most of the HH HM reported in this work, in spite of possessing relatively high Curie temperatures, are likely to present phase separation issues, which is not desirable for practical applications.

Meanwhile, we observe that when the element occupying the $Z$ position in the HH alloy is a chalcogen (S, Se, and Te) the resulting mixing energy is noticeably smaller than when the element belongs to group–XIV of the periodic table (Si, Ge, Sn). For instance, we find that $E_{\rm mix}$ amounts to $-0.12$ eV/f.u. for NaVTe and to $0.21$ eV/f.u. for NaVSi (see Table I). The same behaviour is observed also for other compounds presenting either Nb or Ta in the $Y$ position (e.g., for RbTaSn we estimate $1.03$ eV/f.u. and for RbTaTe $0.75$ eV/f.u.). Therefore, a possible strategy for improving the mixing stability of some of the new HH HM compounds reported in this study (e.g., KVSi with $E_{\rm mix}=0.56$ eV/f.u. and RbNbSi with $E_{\rm mix}=0.83$ eV/f.u.) may consist in doping with light-weight alkali metals (Li and Na) in the $X$ position (although this may also lead to some unwanted decrease in the energy band gap and Curie temperature of the alloy, see present and previous sections) and with chalcogen species in the $Z$ position.

III.4 Most promising magnetic HH alloys

Table I shows the $19$ vibrationally stable and ferromagnetic compounds predicted in this study that possess a Curie temperature equal or above room temperature. Two out of those $19$ alloys are semiconductor while the rest are half-metallic. We have also included NaVTe in the table, in spite of presenting a relatively low Curie temperature of $150$ K, owing to its good mixing stability properties ($E_{\rm mix}=-0.12$ eV/f.u.). From an applied perspective, an overall promising HM (or magnetic semiconductor) material should present the following qualities: (1) being vibrationally and chemically stable, (2) high Curie temperature, (3) large energy band gaps, (4) large magnetic moment, and (5) being structurally compatible with other semiconductor materials typically employed in electronic devices (e.g., silicon and GaAs with respective lattice parameters of $5.4$ and $5.6$ Å).

Except NaVTe and CsNb$Z$ with $Z=$ Si, Ge, and Sn, all the compounds reported in Table I fulfill conditions (2), (3), and (4) above. Besides, compounds LiVSi (SC), LiVGe (SC), and NaVSi (HM) also fulfill (1) and (5), which indicates that these materials are in fact very promising for spintronics applications tu16 ; cao17 . Meanwhile, RbVTe (HM), CsVS (HM), and CsVSe (HM) fulfill (1)–(4) and only partially (5); however, the energy band gaps estimated for these compounds are so large that they also deserve to be highlighted. Finally, we mention RbNbTe as the most encouraging case of a HH HM not containing vanadium in the $Y$ position; this alloy fulfills conditions (2)–(4), as many other compounds, but the corresponding $E_{\rm BG}$, $E_{\rm HM}$, and $T_{\rm C}$ values are exceedingly large.

Overall, the most promising HH alloys for use in spintronics applications predicted by our computational research are LiVSi (SC), LiVGe (SC), NaVSi (HM), RbVTe (HM), CsVS (HM), CsVSe (HM) and RbNbTe (HM), all of which possess magnetic transition temperatures at or above room temperature. The only apparent disadvantage of some of these compounds (i.e., NaVSi and RbNbTe) are their positive and large mixing energies (see Table I), which suggests the likely existence of phase separation issues in practice. Nevertheless, as we have mentioned earlier, a likely stategy for solving this problem may consist in doping to some extent with light-weight alkali metals (Li and Na) in the $X$ position, which in turn would improve their structural compatibility with typical semiconductor materials, and/or with chalcogen atoms in the $Z$ position.

IV Conclusions

We have performed a comprehensive first-principles study of the structural, electronic, structural, vibrational, and mixing properties of $90$ $XYZ$ half-Heusler alloys ($X=$ Li, Na, K, Rb, Cs; $Y=$ V, Nb, Ta; $Z=$ Si, Ge, Sn, S, Se, Te). In contrast to previous computational studies dealing with a large number of candidate materials, we have analyzed the magnetic features of most HH alloys at finite temperatures since this piece of information is crucial for guiding the experimental searches of technologically relevant materials. A total of $17$ alloys are predicted to be vibrationally stable, half-metallic, and magnetically ordered at room temperature, with total magnetic moments of $2$ and $4$ $\mu_{\rm B}$ and semiconductor band gaps in the range of $1$–$4$ eV. On the other hand, all the HH alloys that have been identified as anti-ferromagnetic, $21$ in total, turn out to be metallic. We have also found $2$ new magnetic semiconductors that exhibit high thermodynamic stability and Curie temperatures. After analyzing the mixing stability of the vibrationally well-behaved HH alloys, we have identified the following compounds as overall most promising for spintronics applications: LiVSi (SC), LiVGe (SC), NaVSi (HM), RbVTe (HM), CsVS (HM), CsVSe (HM) and RbNbTe (HM). On the other hand, we have argued that simple doping strategies may be used to improve the mixing stability of some of the discarded HH half-metals. Hence, we hope that our computational study will stimulate new experimental efforts leading to progress in the field of spin-based electronics.

Acknowledgments

This research was supported under the Australian Research Council’s Future Fellowship funding scheme (No. FT140100135). M. A. S. acknowledges financial support from the Higher Education Commission (HEC) of Pakistan under the IRSIP scholarship (PIN:IRSIP 35 PSc 11). Computational resources and technical assistance were provided by the Australian Government and the Government of Western Australia through the National Computational Infrastructure (NCI) and Magnus under the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre.