Theoretical study of phase stability, crystal and electronic structure of MeMgN2 (Me = Ti, Zr, Hf) compounds

Scandium nitride has recently gained interest as a prospective compound for thermoelectric applications due to its high Seebeck coefficient. However, ScN also has a relatively high thermal conductivity, which limits its thermoelectric efficiency and figure of merit (zT). These properties motivate a search for other semiconductor materials that share the electronic structure features of ScN, but which have a lower thermal conductivity. Thus, the focus of our study is to predict the existence and stability of such materials among inherently layered equivalent ternaries that incorporate heavier atoms for enhanced phonon scattering and to calculate their thermoelectric properties. Using density functional theory calculations, the phase stability of TiMgN2, ZrMgN2 and HfMgN2 compounds has been calculated. From the computationally predicted phase diagrams for these materials, we conclude that all three compounds are stable in these stoichiometries. The stable compounds may have one of two competing crystal structures: a monoclinic structure (LiUN2 prototype) or a trigonal superstructure (NaCrS2 prototype; R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar{3} $$\end{document}3¯mH). The band structure for the two competing structures for each ternary is also calculated and predicts semiconducting behavior for all three compounds in the NaCrS2 crystal structure with an indirect band gap and semiconducting behavior for ZrMgN2 and HfMgN2 in the monoclinic crystal structure with a direct band gap. Seebeck coefficient and power factors are also predicted, showing that all three compounds in both the NaCrS2 and the LiUN2 structures have large Seebeck coefficients. The predicted stability of these compounds suggests that they can be synthesized by, e.g., physical vapor deposition.


Introduction
Thermoelectric materials and devices, which directly convert a thermal gradient into an external voltage, are reliable and low-maintenance power-generating materials used for niche applications such as solidstate cooling or electric power supplying units in deep-space exploration. However, the use of thermoelectrics is presently limited [1] by their low efficiency and high cost. For example, the crustal abundance and global production of tellurium is low [2,3]. This limits widespread use of the benchmark thermoelectric materials (Bi 2 Te 3 and PbTe). Thus, there is a need for replacement materials.
The thermoelectric efficiency is directly connected to the dimensionless figure of merit: where S is the Seebeck coefficient, r is the electrical conductivity, j is the thermal conductivity, and T is the absolute temperature [4]. The product S 2 r is known as the power factor. In the limit of zT ! 1, the Carnot engine efficiency (i.e., the maximum efficiency achievable in a heat engine) is obtained. However, designing materials with higher zT values is a difficult challenge, as all three terms are interrelated in a way that typically limits zT to below unity in commonly available materials. In order to overcome this barrier, Slack proposed the phonon glass-electron crystal (PGEC) approach for thermoelectric material design [5][6][7]: one should seek a material with a high Seebeck coefficient value and engineer it in such a way that it will behave like a crystal for electrons, but scatter phonons similarly to glass. As a result, added material optimization processes are required to increase the zT of any given material.
As a starting point for this approach of engineering a high zT material, prior works have suggested cubic scandium nitride (ScN) [8]. The Seebeck coefficient of ScN is relatively large (reaching -180 lV=K at 800 K) and because of its low electrical resistivity, large power factors between 2.5 and 3.5 9 10 -3 Wm À1 K À2 have been reported [9,10]. Doping and alloying ScN with heavy elements [11,12] and/or creating artificial layer interfaces such as metal/semiconductor superlattices [13][14][15][16] can alter properties and decrease the thermal conductivity, resulting in an enhanced zT.
Furthermore, ScN can also become p-type by Sc-site doping [17,18]. Although the direction of research is promising, ScN does have a relatively large thermal conductivity [19][20][21][22] of approximately 8-12 Wm À1 K À1 . Scandium and nitrogen are both light atoms compared to their heavier counterparts such as lead, bismuth and tellurium which effectively scatter phonons [23], and artificial interfaces seen in superlattices are synthesized at a sub micrometer scale, while thermoelectric power generation requires millimeter-sized bulk samples [24]. Also, scandium does not have phonon isotope scattering as it is an isotopically pure element.
In a recent paper, Alling [25] addressed these issues by proposing an equivalent ternary based on ScN. Scandium (which is a group-3 element) can be replaced with one group-2 and one group-4 element in a 50/50 proportion to cover the same electron valence. The final compound should then have a MeAEN 2 stoichiometry, with Me representing a transition metal from the group-4 elements and AE belonging to the group-2 (alkaline earth) elements, such as magnesium. TiMgN 2 was predicted to be stable using density functional theory (DFT). Band structure calculations predicted stoichiometric TiMgN 2 to have a 1.11 eV band gap using the HSE06 [25,26] hybrid functional. This methodology has also been used by Tholander et al. [27] to predict zincbased group-4 transition metal nitride stability and crystal structure. While much research has been done regarding Ti-Si-N [28][29][30] and Ti-Al-N [31][32][33][34] which show superior hardness and/or oxidization resistance compared to TiN, there are much fewer studies reported for Ti-Mg-N [35][36][37][38][39]. TiMgN 2 may crystallize in the B1-L1 1 superstructure [25], which could open a new opportunity for hard coating research by inter-layer dissipation of heat or research for hard coatings with better mechanical properties.
In this paper, we continue the work in investigating ternary structures based on ScN. We also computationally study the phase stability, band structure, Seebeck coefficient and power factor of two more candidate compounds potentially useful in thermoelectric applications, ZrMgN 2 and HfMgN 2 . As Ti, Zr and Hf belong to group 4 of the periodic table, all three share similar physical and chemical properties, and it can be assumed that any stable Ti-based ternary may also exist for Zr and Hf.

Computational details
Over 60 different and chemically stoichiometric crystal structures registered in the Inorganic Crystal Structure Database (ICSD) [40] were studied in order to calculate the formation enthalpy of Ti-Mg-N, Zr-Mg-N and Hf-Mg-N and prepare the necessary phase diagrams. Although the binary nitrides are well known, TiMgN 2 , ZrMgN 2 and HfMgN 2 are not present in either the Materials Project database [41] or the ICSD. Half of these crystal structures follow the MeMgN 2 stoichiometry, while the remaining crystal structures belong to various Mg-, Ti-, Zr-and Hfbased ternaries. In addition, the opposite sequence, MgMeN 2 , was also studied in case some structures would show a different phase when switching the positions of the metal atoms in their respective sublattice.
First-principles calculations were employed using DFT [42,43] with the projector augmented wave method (PAW) [44] implemented in the Vienna ab initio simulation package (VASP) [45][46][47] version 5.2. Electronic exchange correlation effects and the electronic band structure were modeled with the generalized gradient approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) functional [48]. It should be noted that the Kohn-Sham gaps of standard GGA calculations are systematically smaller than experimental band gaps, but for the present work this is not an issue since we are mostly concerned with dismissing metallic compositions. To the extent that we identify relevant compounds, they can be further investigated by in-depth theoretical work and/or by laboratory synthesis of the three ternary nitrides. The plane wave energy cutoff was set at 400 eV. The required structure files for the crystal structures were obtained from the ICSD and converted to VASP input files using cif2cell [49]. Phase diagrams were prepared using the software package Pymatgen (Python Materials Genomics) [50], the band structure illustrations by the high-throughput toolkit (httk) [51] and the crystal structures by VESTA [52]. For the phase diagrams, the formation energy per atom was calculated for each ternary compound and related to competing ternary stoichiometries and neighboring binary compounds. The Materials Project database provided the formation enthalpies of all of the binaries (TiN, ZrN, HfN, Mg 3 N 2 , etc.).
The present work uses the same correction of the N 2 energy as used in the Materials Project, based on work by Wang et al. [53] as standard GGA exchangecorrelation functionals in DFT are known to, in general, have systematic errors in the prediction of energy differences between solid and gas phase systems [54]. Hence, to accurately reproduce the formation energy of a system relative to a gas end point, it is common to adjust the gas phase energy.
The calculations used an 11 9 11 9 11 k-point mesh for Brillouin zone sampling and were executed with the Monkhorst-Pack scheme [55]. For band structure calculations, the tetrahedron method was used in order to obtain band gap values with spin polarization included [56].
Finally, the Seebeck coefficient S and power factor S 2 rs À1 (being the charge carrier relaxation time) of the predicted semiconductors is calculated at room temperature and 600 K as functions of the chemical potential using Boltzmann transport theory with the constant relaxation time approximation. We use the software BoltzTraP [57] on DFT calculations with a 40 9 40 9 40 k-point mesh for Brillouin zone sampling. Figure 1a shows the phase diagram for Ti-Mg-N. Although 28 different crystal structures other than those that follow the MeMgN 2 formula (such as Ca 4 TiN 4 [58], perovskite CaTiO 3 [59], Ti 2 AlN and Ti 4 AlN 3 MAX-phases [60]) were tested, only the ordered TiMgN 2 stoichiometry is found to be thermodynamically stable relative to known and investigated phases with the other ordered stoichiometries being either unstable or metastable. Random Ti 1-x Mg x N solid solutions with the rocksalt structure have, however, been found to be thermodynamically stable for a range of compositions [25]. This precise stoichiometry occurred in 29 of the investigated crystal structures. These include the trigonal NaCrS 2 (R 3mH) superstructure [61], the tetragonal BaNiS 2 (P4=n m m Z) superstructure [62], tetragonal LiUN 2 [63], ZnGeN 2 [64] (based on the NaFeO 2 -beta structure) and the inverse-MAX BaCeN 2 [65].

TiMgN 2
In order to differentiate between these structures, Table 1 lists a selected group of examples with their respective formation enthalpies. These results show that crystallization into the NaCrS 2 is the most likely outcome with a -1.299 eV formation enthalpy and a 0.04 eV difference compared to the LiUN 2 structure which agrees with the findings mentioned in Ref. [25]. It should be noted that the difference between the formation enthalpies of these two crystal structures would most likely mean that NaCrS 2 is the preferred structure, but LiUN 2 is also studied for any comparison needed between TiMgN 2 , ZrMgN 2 and HfMgN 2 .
Both crystal structures are shown in Fig. 2. The results suggest that TiMgN 2 will crystallize into the NaCrS 2 superstructure (also viewed as a NaCl-B1 superstructure that includes three alternating layers of Ti and Mg) which could cause phonon scattering at the interface of each layer as mentioned in the introduction. Figure 3a, d shows the band structures for TiMgN 2 in the NaCrS 2 and LiUN 2 structures. According to these results, TiMgN 2 is a semiconductor with a Kohn-Sham PBE band gap of 0.26 eV in the NaCrS 2 structure (Fig. 3a). However, the case for LiUN 2 (Fig. 3d) is different, as band structure calculations show no band gap, i.e., predicting metallic properties. It is possible that TiMgN 2 could crystallize in the LiUN 2 structure as a metastable phase. Table 2 shows the lattice parameters and the band gap energy in both crystal structures. These results show that although the trigonal NaCrS 2 crystal structure remains with only the lattice parameters changing, the LiUN 2 structure relaxes from tetragonal to monoclinic according to the calculated unit cell lattice parameters. Figure 4a, b shows the Seebeck coefficient of TiMgN 2 versus the chemical potential at room temperature and 600 K, respectively. Only the NaCrS 2 structure was studied as the LiUN 2 structure was predicted with no band gap. These results show relatively high Seebeck coefficient values at the Fermi level. Figure 5a, b shows S 2 r À Á =s versus the chemical potential at room temperature and 600 K, respectively. Depending on the assumed relaxation time, predicted power factor values could exceed those of ScN (Fig. 5k, l). Figure 1b shows the phase diagram for ZrMgN 2 . Also here the only stable ternary has the MeMgN 2 stoichiometry. As for the preferred crystal structure, formation enthalpies for the selected crystal structures are shown in Table 3. In contrast to TiMgN 2 , the LiUN 2 structure competes with the NaCrS 2 structure with less than 0.01 eV formation enthalpy difference. The predicted band structures are shown in Fig. 3b, e. In both cases, ZrMgN 2 is a semiconductor regardless of crystal structure. However, for the NaCrS 2 crystal structure we find an indirect Kohn-Sham PBE band gap of 0.89 eV and for the LiUN 2 structure, a direct band gap of 0.46 eV. The respective lattice parameters and band gap energy are shown in Table 4. ZrMgN 2 relaxes in a similar way as TiMgN 2 with the NaCrS 2 structure remaining the same while the tetragonal LiUN 2 structure relaxes into a monoclinic structure according to the calculated unit cell lattice parameters. Figure 4b, g (room-temperature calculations) and Fig. 4d, h (600 K calculations) shows the Seebeck coefficient of ZrMgN 2 versus the chemical potential in the NaCrS 2 and the LiUN 2 structures. These results show an increase in the Seebeck coefficient values and a slight shift in the chemical potential compared to TiMgN 2 with higher values seen in the NaCrS 2 structure. Figure 5b, g and d, h shows the S 2 r À Á =s versus chemical potential at room temperature and 600 K, respectively. These results predict power factor values close to the Fermi level which are larger than those of ScN (Fig. 5k, l). Figure 1c shows the phase diagram for Hf-Mg-N. Similar to both TiMgN 2 and ZrHfN 2 , the HfMgN 2 stoichiometry is predicted to be stable. Table 5 compares a selected group of crystal structures and shows the NaCrS 2 and LiUN 2 structures with similar formation enthalpies (less than 0.01 eV difference), thus predicting a competition between the two structures. Figure 3c, f shows the predicted band structures for both NaCrS 2 and LiUN 2 . Similar to ZrMgN 2 , an indirect band gap of 1.19 eV is predicted for the NaCrS 2 structure, while a 0.77 eV direct band gap is predicted for the LiUN 2 structure. The respective lattice parameters and band gap energies are shown in Table 6. Similar to TiMgN 2 and ZrMgN 2 , HfMgN 2  preserves the trigonal NaCrS 2 structure but relaxes from tetragonal LiUN 2 into a monoclinic structure.  to both TiMgN 2 and ZrMgN 2 with higher values seen in the NaCrS 2 structure. Figure 5e, i and f, j shows the S 2 r À Á =s versus chemical potential at room temperature and 600 K, respectively. These results predict power factor values almost equal to those of ZrMgN 2 and larger than that of ScN close to the Fermi level (Fig. 5k, l).

Discussion
For ZrMgN 2 and HfMgN 2 , the formation enthalpies of the NaCrS 2 and the LiUN 2 structure are close, within the accuracy of our approach. This suggests that both of these structures may be possible to synthesize, i.e., with the one higher in energy as a longlasting metastable state. The shifting between the NaCrS 2 and the LiUN 2 structures could be done by   choosing suitable substrates for epitaxial stabilization during the synthesis process. Despite that we cannot with certainty determine which of the structures for ZrMgN 2 and HfMgN 2 are thermodynamically stable, both are semiconductors. This motivates future studies on synthesis for thermoelectrics and other applications. It should be noted that the NaCrS 2 structures show indirect band gaps with larger values and large slopes for the density of states at the Fermi level compared to their direct band gap counterparts in the LiUN 2 structure. Another feature seen in all three compounds is the relation between band gap values and lattice parameters with the transition metal, Me. As the smaller Ti atom is replaced with the larger Zr atom, the lattice parameters, cell volume and band gap value increase, which is expected. However, only the band gap value increases when Zr is replaced with Hf as the f orbital electrons are not effective at screening the increasing charge, resulting in similar atomic size (lanthanide contraction [66]) and similar lattice parameters.
Band gap (eV)   Although the present results are promising, actual attempts to synthesize these prospective compounds would be important. Similar to the synthesis of MAXphase [67] thin films, it should be possible to synthesize ordered TiMgN 2 , ZrMgN 2 and HfMgN 2 outside thermodynamic equilibrium in a magnetron sputtering system. All of the mentioned elements are vacuum compatible, and one could use the deposition parameters needed for stoichiometric TiN, ZrN, HfN and Mg 3 N 2 to reach a Me=Mg ¼ 1 ratio and fine-tune the MeMgN 2 stoichiometry. References [36,38] note the deposition temperature for rocksalt (Ti, Mg)N alloys to be between 200 and 300°C with oxidization resistance close to 700°C (suitable for mid-temperature thermoelectric applications). If the layered NaCrS 2 superstructure is preferred, it would be advisable to use either high-temperature direct growth or low-temperature deposition, followed by high-temperature annealing [68] (in ammonia or nitrogen). In this case, GaN or SiC [69] substrates could be considered for their suitable lattice constant and thermal stability.
As for the thermoelectric properties, the calculated Seebeck coefficient values show that in the range of a moderate change in the Fermi level, high room-temperature Seebeck coefficient values can be achieved (Fig. 4), although it seems that HfMgN 2 is either an insulator or would require elemental doping due to the larger shift in the chemical potential.
Note that what we have calculated is the power factor divided by the relaxation time. The results (Fig. 5) can be used as an estimate of the difference in thermoelectric performance at various doping levels between the studied compounds and known materials, e.g., ScN, as shown for comparison in Figs. 4k, l and 5k, l. However, such a comparison is made under the assumptions that the constant relaxation time approximation holds sufficiently well and that the relaxation time for the compounds is similar. For more precise predictions, the relaxation time value needs to be obtained from experimental data, as it can for example for common thermoelectric materials such as Bi 2 Te 3 [70,71].
As ordered TiMgN 2 , ZrMgN 2 and HfMgN 2 have not yet been studied experimentally, such data do not exist, and obtaining meaningful numbers for the electrical conductivity is difficult. However, using experimental data from Burmistrova et al. [19] and the classical equation for conductivity (r ¼ ne 2 sm À1 ), the constant relaxation time s for ScN (which the ternaries were modeled after) is estimated to be equal to 6:5 Â 10 À14 s.

Conclusions
Theoretical methods were used to study the phase stability and band structure of TiMgN 2 , ZrMgN 2 and HfMgN 2 . In all three cases, only MeMgN 2 is predicted to be the stable stoichiometry. It is shown that stoichiometric TiMgN 2 crystallizes into the hexagonal NaCrS 2 superstructure with a 0.26 eV indirect Kohn-Sham PBE band gap. ZrMgN 2 and HfMgN 2 were also studied, which shows tendency to crystallize in both the NaCrS 2 superstructure and the LiUN 2 prototype monoclinic structure. Both show semiconducting properties regardless of the crystal structure. ZrMgN 2 shows a 0.89 eV indirect band gap when crystallizing into the NaCrS 2 structure, while as crystallization into the LiUN 2 structure results in a 0.46 eV direct band gap. As for HfMgN 2 , the band gap increases as crystallization into NaCrS 2 results in a 1.19 eV indirect band gap and crystallization into LiUN 2 results in a 0.77 eV direct band gap. Lattice parameters and cell volumes increase with the substitution of Ti with Zr, but slightly decrease when Zr is substituted with Hf.
Finally, the Seebeck coefficient and power factor was calculated for all of the semiconducting compounds. The results show that in the range of a moderate change in the Fermi level, high room-temperature Seebeck coefficient values can be achieved.
Thus, the predicted stability and semiconducting properties of these compounds can be further studied both theoretically and experimentally for any prospective thermoelectric properties. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.