Tetragonal Strain Research Articles

Effect of Crystal Symmetry on the Spin States of Fe3+ and Vibration Modes in Lead-free Double-Perovskite Cs2AgBi(Fe)Br6.

We show by electron spin resonance (ESR) and Raman spectroscopies that the crystal phase transition of the lead-free double-perovskite Cs2AgBiBr6 has a profound symmetry-breaking effect on the high spin states of, for example, a transition-metal ion Fe3+ and the vibrational modes. It lifts their degeneracy when the crystal undergoes the cubic-tetragonal phase transition, splitting the six-fold degenerate S = 5/2 state of Fe3+ to three Kramer doublets and the enharmonic breathing mode Tg of the MBr6 octahedra (M = Ag, Bi, Fe) into Eg + Ag. The magnitudes of both spin and Raman line splitting are shown to directly correlate with the strength of the tetragonal strain field. This work, in turn, demonstrates the power of the ESR and Raman spectroscopies in probing structural phase transitions and in providing in-depth information on the interplay between the structural, spin, and vibrational properties of lead-free double perovskites, a newly emerging and promising class of materials for low-cost and high-efficiency photovoltaics and optoelectronics.

Electron-lattice interplay in LaMnO3 from canonical Jahn-Teller distortion notations

LaMnO$_3$ is considered as a prototypical Jahn-Teller perovskite compound, exhibiting a metal to insulator transition at $T_{JT} = 750K$ related to the joint appearance of an electronic orbital ordering and a large lattice Jahn-Teller distortion. From first-principles, we revisit the behavior of LaMnO$_3$ and show that it is not only prone to orbital ordering but also to charge ordering. Both charge and orbital orderings appear to be enabled by rotations of the oxygen octahedra and the subtle competition between them is monitored by a large tetragonal compressive strain, that is itself a Jahn-Teller active distortion. Equally, the competition of ferromagnetic and antiferromagnetic orders is slave of the same tetragonal strain. Our results further indicate that the metal to insulator transition can be thought as a Peierls transition that is enabled by spin symmetry breaking. Therefore, dynamical spin fluctuations in the paramagnetic state stabilize the insulating phase by the instantaneous symmetry breaking they produce and which is properly captured from static DFT calculations. As a basis to our discussion, we introduce canonical notations for lattice distortions in perovskites that distort the oxygen octhedra and are connected to charge and orbital orderings.

Multiferroic and photovoltaic current properties of tetragonally strained BiFeO3 thin films

Mn (5%) doped BiFeO3 (MBFO) thin films were epitaxially grown on La2/3Sr1/3MnO3/LaAlO3 substrates via pulsed laser deposition. The MBFO thin film was confirmed to be an epitaxial thin film having a (001) orientation crystal structure by X-ray diffraction experiments. Compared to the BiFeO3 (BFO) thin film, the MBFO thin films have a shrunk a-axis lattice constant and a significantly elongated c-axis lattice constant, showing a c/a ratio of 1.15, and exhibiting high tetragonal strain. Due to the high tetragonal strain, the MBFO thin film exhibited higher polarization, magnetization, and d33 values than the BiFeO3 (BFO) thin film, resulting in enhanced multiferroic properties. From the results of the study on the relationship between the ferroelectric domain structure and the thickness observed by piezoresponse force microscope, it was confirmed that the domain wall energies of the MBFO thin films were larger than those of the BFO thin films. In addition, the I-V curves (photovoltaic current) showed that the MBFO thin films had the open-circuit voltages and photovoltaic currents higher than that of the BFO thin films due to its higher ferroelectric polarization and smaller band gap than those of the BFO thin films.

Complete spin gapless semiconductivity in equiatomic quarternary Heusler material TiZrMnAl

A new equiatomic quarternary Heusler compound TiZrMnAl has been theoretically predicted. Two possible structural configurations are considered under the site preference rule and their equilibrium lattices have been derived. Electronic band structures identify TiZrMnAl compound as the spin gapless semiconductors in both two structure types. The total magnetic moments of TiZrMnAl are both equal to zero with the partial moments of Ti and Zr antiparallelly aligned to that of Mn. This integral value of total moment follows the Slater-Pauling rule in the form of Mt = Zt − 18, where Mt represents the total magnetic moment and Zt is the total number of valence electrons. The mechanical and dynamic stabilities of TiZrMnAl have also been validated and strong elastic anisotropy is further revealed with the calculated directional dependent Young’s modulus and shear modulus for both structures. Besides, the effect of random swap disorder between Ti and Zr atoms has been examined and it is found that through the whole swap range, TiZrMnAl maintains its fully compensated spin gapless semiconductivity, which is a very good property for possible applications in spintronic devices. Lastly, the uniform and tetragonal strains have been accessed and their effects on the electronic and magnetic properties have been studied. This systematic study can provide a comprehensive reference for the further development of Ti and Zr based quarternay Heusler compounds and even inspire other relative studies for the exploration of new spin gapless semiconductors.

Large magnetic anisotropy in highly strained epitaxial MgFe2O4 thin films

In order to acquire a soft magnetic film with low conductivity and large magnetic anisotropy, the strain-modulated magnetic anisotropy is studied in epitaxial MgFe2O4 (MFO) films. The MFO films with thicknesses of 25, 48, 75, and 110 nm are grown on the MgAl2O4 (MAO) (100) substrate using a pulsed laser deposition technique. Due to the large lattice-mismatch (3.34%) between MFO and MAO substrates, the interface exhibits a large tetragonal compressive strain. The results of field-dependent magnetization suggest the soft magnetic nature of all the measured MFO films. The analysis of angular dependent ferromagnetic resonance reveals the large uniaxial magnetic anisotropy energy (Ku) of −1.62 × 106 erg/cm3 in the highly strained 25 nm film, and Ku decreases with the increasing thickness due to strain relaxation. The observed large magnetic anisotropy in these highly strained MFO thin films is larger than most of the soft ferrite thin films, which arise due to tetragonal distortion and inverse magnetostriction. The epitaxial MFO thin films with enhanced magnetic anisotropy could be one of the potential candidates for spin filters.

Revisit of the shape and orientation of precipitates with tetragonal transformation strains that minimise the elastic energy

ABSTRACT The selection of the shape and orientation of precipitates with tetragonal misfit (transformation) strains that minimise the elastic energy is revisited. The complex map of preferred shapes suggested by Kaganova and Roitburd for inhomogeneous isotropic materials is extended to materials with cubic symmetry and for systems that compromise with the surface energy of close-packed planes. Several shape transitions are explained from the kinematics of precipitates with tetragonal transformation strains in solid matrices. Plate shaped precipitates are associated with minimum elastic energy in homogenous systems and for soft precipitates, in accordance with the theorem of Khachaturyan. We explain how compact shapes like rods and spheres arise when the matrix is softer than the precipitate. Analytic estimations are derived to determine the preferred shapes based on the difference between the elastic stiffness of the matrix and the precipitate. When the transformation strains differ in sign, they give rise to a large lattice rotation. This lattice rotation is necessary for the formation of twins by equivalent variants of the transformation strain.

Theoretical Study of the Electronic, Magnetic, Mechanical and Thermodynamic Properties of the Spin Gapless Semiconductor CoFeMnSi

CoFeMnSi has been both experimentally and theoretically proven as a novel spin-gapless semiconductor and resulted in a new research direction in equiatomic full Heusler compounds. Using the first-principles calculation method, we investigated the electronic, magnetic and mechanical properties of CoFeMnSi material in this study. The obtained lattice constant under the LiMgPdSn-type Heusler structure is 5.611 Å and it is fairly consistent with previous experimental results and theoretical calculations. Furthermore, the achieved total magnetic moment of 4 μB follows the Slater–Pauling rule as Mtotal = Ztotal − 24, where Mtotal is the total magnetic moment per formula unit and Ztotal is the total valence electron number, i.e., 28 for CoFeMnSi material. We have also examined the mechanical properties of CoFeMnSi and computed its elastic constants and various moduli. Results show CoFeMnSi behaves in a ductile fashion and its strong elastic anisotropy is revealed with the help of the 3D-directional-dependent Young’s and shear moduli. Both mechanical and dynamic stabilities of CoFeMnSi are verified. In addition, strain effects on the electronic and magnetic properties of CoFeMnSi have been investigated, including both uniform and tetragonal strains, and we found that the spin-gapless feature is easily destroyed with both strain conditions, yet the total magnetic moment maintains a good stability. Furthermore, the specific behaviors under various temperatures and pressures have been accessed by the thermodynamic properties with a quasi-harmonic Debye model, including bulk modulus, thermal expansion coefficient, Grüneisen constant, heat capacity and Debye temperature. This comprehensive study can offer a very helpful and valuable reference for other relative research works.

The structural, electronic, magnetic and mechanical properties of d0 binary Heusler alloys XF3 (X = Be, Mg, Ca, Sr, Ba)

The first-principles calculations within generalized gradient approximation (GGA) method are employed to study the structural, electronic, magnetic and mechanical properties of d0 binary Heusler alloys XF3 (X = Be, Mg, Ca, Sr, Ba). They have integer magnetic moments per formula unit 1μB/f.u., and the most stable state is ferromagnetic (FM) state. They have thermodynamic stabilities because the formation energies are negative and the cohesion energies are positive. The elastic constants Cij satisfy the mechanical stability conditions. Their HM properties are robust under the effect of hydrostatic strain and tetragonal strain.

Computational study of the electronic, magnetic, mechanical and thermodynamic properties of new equiatomic quaternary heusler compounds TiZrRuZ (Z = Al, Ga, In)

The electronic, magnetic, and mechanical properties of the new equiatomic quarternary Heusler compounds TiZrRuZ (Z = Al, Ga, In) have been studied with the first-principles calculations. Three types of different atomic configurations have been examined and results show the atomic site occupation follows the general site preference rule. The equilibrium lattice constants have been determined and also the total and atom-resolved magnetic moments have been calculated. These new equiatomic quarternary Heusler compounds TiZrRuZ (Z = Al, Ga, In) are ferromagnetic half metals at the ground states. In particular for the total magnetic moment (MT), an integral number of 1 µB is obtained with the main contribution from Ti atoms and it follows the general Slater-Pauling rule, i.e. MT = ZT -18, where ZT is the total valence electron number, 19 for TiZrRuZ. Furthermore, the derived mechanical properties have shown that these materials behave in a strong ductile manner with a high elastic anisotropy, which is further confirmed by the calculated 3D Young's modulus and shear modulus. Their mechanical and dynamic stabilities have also been verified with the generalized elastic criteria and phonon dispersion spectrum. Afterwards, the effects of the uniform and tetragonal strains have been studied. The variation of the conduction band minimum and the valence band maximum with both strains have been revealed and the materials show good robustness of the half metallicity. Also, the total magnetic moment maintains constant of 1 µB against both strains while the atomic magnetic moments vary. In the last, the thermodynamic properties have also been analyzed with the quasi-harmonic Debye model under pressure 0 to 10 GPa and temperature from 0 to 500 K. These materials can be good candidates for the future spintronic application and can even inspire other relative work for material research in this field.

Robust Spin-Gapless Behavior in the Newly Discovered Half Heusler Compound MnPK.

Spin gapless semiconductors have aroused high research interest since their discovery and a lot of effort has been exerted on their exploration, in terms of both theoretical calculation and experimental verification. Among different spin gapless materials, Heusler compounds stand out thanks to their high Curie temperature and highly diverse compositions. Especially, both theoretical and experimental studies have reported the presence of spin gapless properties in this kind of material. Recently, a new class of d0 − d Dirac half Heusler compound was introduced by Davatolhagh et al. and Dirac, and spin gapless semiconductivity has been successfully predicted in MnPK. To further expand the research in this direction, we conducted a systematical investigation on the spin gapless behavior of MnPK with both generalized gradient approximation (GGA) and GGA + Hubbard U methods under both uniform and tetragonal strain conditions by first principles calculation. Results show the spin gapless behavior in this material as revealed previously. Different Hubbard U values have been considered and they mainly affect the band structure in the spin-down channel while the spin gapless feature in the spin-up direction is maintained. The obtained lattice constant is very well consistent with a previous study. More importantly, it is found that the spin gapless property of MnPK shows good resistance for both uniform and tetragonal strains, and this robustness is very rare in the reported studies and can be extremely interesting and practical for the final end application. This study elaborates the electronic and magnetic properties of the half Heusler compound MnPK under uniform and tetragonal strain conditions, and the obtained results can give a very valuable reference for related research works, or even further motivate the experimental synthesis of the relative material.

Concentration-tuned tetragonal strain in alloys: Application to magnetic anisotropy of FeNi1−xCox

We explore an opportunity to induce and control tetragonal distortion in materials. The idea involves formation of a binary alloy from parent compounds having body-centered and face-centered symmetries. The concept is illustrated in the case of FeNi$_{1-x}$Co$_x$ magnetic alloy formed by substitutional doping of the L1$_0$ FeNi magnet with Co. Using electronic structure calculations we demonstrate that the tetragonal strain in this system can be controlled by concentration and it reaches maximum for $x=0.5$. This finding is then applied to create a large magnetocrystalline anisotropy (MAE) in FeNi$_{1-x}$Co$_x$ system by considering an interplay of the tetragonal distortion with electronic concentration and chemical anisotropy. In particular, we identify a new ordered FeNi$_{0.5}$Co$_{0.5}$ system with MAE larger by a factor 4.5 from the L1$_0$ FeNi magnet.

Investigation of the structural competing and atomic ordering in Heusler compounds Fe2NiSi and Ni2FeSi under strain condition.

The structural competing and atomic ordering of the full Heusler compounds Fe2NiSi and Ni2FeSi under uniform and tetragonal strains have been systematically studied by the first-principles calculation. Both Fe2NiSi and Ni2FeSi have the XA structure in cubic phase and they show metallic band structures and large magnetic moments (greater than 3μB) at equilibrium condition. Tetragonal distortion can further decrease the total energy, leading to the possible phase transformation. Furthermore, different atom reordering behaviours have been observed: for Fe2NiSi, atoms reorder from cubic XA-type to tetragonal L10-type; for Ni2FeSi, there is only structural transformation without atom reordering. The total magnetic moments of Fe2NiSi and Ni2FeSi are mainly contributed by Fe atoms, and Si atom can strongly suppress the moments of Fe atoms when it is present in the nearest neighbours of Fe atoms. With the applied strain, the distance between Fe and Si atoms play an important role for the magnetic moment variation of Fe atom. Moreover, the metallic band nature is maintained for Fe2NiSi and Ni2FeSi under both uniform and tetragonal strains. This study provides a detailed theoretical analysis about the full Heusler compounds Fe2NiSi and Ni2FeSi under strain conditions.

Electronic structures, magnetic properties and strain effects of quaternary Heusler alloys FeMnCrZ (Z = P, As, Sb, Bi, Se, Te)

We have systematically investigated the electronic structures, magnetic properties and strain effects of quaternary Heusler compounds FeMnCrZ (Z = P, As, Sb, Bi, Se, Te) by using the self-consistent full-potential linearized-augmented plane-wave (FPLAPW) method with the generalized gradient approximation (GGA) and the local density approximation (LDA) + U methods based on the density functional theory (DFT). By analyzing the electronic structures of these six compounds within GGA, we have found that the FeMnCrAs, FeMnCrSb, and FeMnCrTe alloys show excellent half-metallic ferrimagnetism, and the FeMnCrP and FeMnCrBi compounds present near half-metallicity. By using the LDA + U method, we found that FeMnCrP, FeMnCrAs and FeMnCrTe are half-metals. The total spin magnetic moments for all the half-metals satisfy the Slater-Pauling 24 electron-rule, confirming the existence of the half-metallic gap. The estimated Curie temperatures of all compounds exceed the room temperature, making the half-metal compounds promising for spintronic applications. Finally, the effects of uniform and tetragonal strain have been investigated. The half-metallicity can be either enhanced, reduced or destroyed by a uniform strain. A ferrimagnetism to ferromagnetism phase transition can be induced by a uniform lattice expansion within the LDA + U method. The effects of the tetragonal distortion turn out to be negligible for these compounds.

The Structural, Electronic, and Magnetic Properties of Binary Heusler Alloys ZCl3 (Z = Li, Na, K, Rb) with DO3-Type Structure from First Principle Calculations

The structural, electronic, and magnetic properties of binary Heusler alloys ZCl3 (Z = Li, Na, K, Rb) with DO3-type structure have been studied by means of the first principle calculations. The calculated results reveal that all four alloys are half-metallic (HM) materials at the stable ferromagnetic ground state. The total magnetic moments (Mt) of LiCl3, NaCl3, KCl3, and RbCl3 alloys are all 2.000 μB/f.u., following the Slater-Pauling rule Mt = 24 − Zt for the former two alloys, while satisfying the Slater-Pauling rule Mt = 30 − Zt for the latter two alloys. The spin-up band gaps of the binary Heusler alloys ZCl3 (Z = Li, Na, K, Rb) are 5.598, 5.687, 6.744, and 6.745 eV at equilibrium lattice constants 7.076, 7.237, 7.588, and 7.819 A, respectively. The positive cohesion energy and negative formation energy indicate that they are thermodynamically stable. Moreover, wide HM regions for binary Heusler alloys ZCl3 (Z = Li, Na, K, Rb) show their HM characters are robust when the lattices are expanded or compressed under uniform and tetragonal strains. Finally, the total charge density, magnetic charge density, and charge density difference of the (110) plane for LiCl3 alloy are analyzed in detail.

Enhanced multiferroic properties of tetragonally strained epitaxial BiMnO3 thin films grown on single crystal Rh substrates

Epitaxial BiMnO3 (BMO) thin films with tetragonally strained structures were fabricated and their multiferroic properties were investigated in order to improve the leakage current characteristics of the BMO thin films and to secure the excellent multiferroic properties. The x-ray diffraction experiment confirmed that the BMO thin films grown on the Rh substrates had noticeably tetragonally strained structure compared to the BMO thin films on the Nb doped SrTiO3 substrates. As a result, the leakage current characteristics of the BMO thin films were improved, and the ferroelectric hysteresis loops were also excellent enough to be used for the nonvolatile memory devices. In addition, it was confirmed that the BMO thin films had a multiferroic property by measuring the ferromagnetic hysteresis loops of the BFM thin films, and it was confirmed that the ferromagnetic properties were improved by the tetragonal strain. Further, the band gaps of the BMO thin films were measured and it was confirmed that the band gap of the strained BFO thin film was increased.

Large negative uniaxial magnetic anisotropy in highly distorted Co-ferrite thin films

The strain induced magnetic anisotropy (MA) of epitaxial Co-ferrite (CFO) thin films grown on MgAl2O4(001) (MAO) by reactive sputtering was studied. These films underwent large tetragonal compressive strain due to the lattice mismatch between the substrates and films, resulting in tetragonalities of up to −0.04. Scanning transmission electron microscopy observation combined with fast Fourier transform analysis revealed that the lattice distortion monotonically relaxed with the increasing film thickness. Unlike the CFO(001) films on MgO(001) substrates, a magnetically enhanced layer exists at the interface between CFO and MAO. A large negative uniaxial MA energy of −5.9 MJ/m3 was confirmed for the thinnest film of 12.9 nm at 300 K by magneto-torque measurements. The induced uniaxial anisotropy decreased with the increasing film thickness owing to misfit relaxation. The magneto-elastic (ME) constant, which was determined by a fitting line for films with different thicknesses and therefore different tetragonalities, was 0.15±0.01 GJ/m3. This result is consistent with the value of 0.14 GJ/m3 for the bulk and suggests that the framework of the phenomenological ME theory for CFO is valid for a wide range of tetragonalities, at least up to −0.04. Our results also indicate that the misfit engineering of CFO thin films has great potential in inducing enhanced uniaxial MA.

Ab initio calculations of temperature-dependent magnetostriction of Fe and A2Fe1−xGax within the disordered local moment picture

The fully relativistic disordered local moment (DLM) theory is used to perform calculations of the magnetic torque of tetragonally distorted Fe and fully disordered (A2) Fe1−xGax (0≤x≤0.2) alloys to describe the temperature-dependence of their magnetoelasticity. The finite temperature magnetoelasticity, in particular the magnetoelastic constant B1, is obtained within DLM theory by studying the response of the magnetic torque generated by the magnetocrystalline anisotropy to the application of a tetragonal strain. Calculations of B1 have been performed on bcc Fe across its ferromagnetic temperature range. Our results show good qualitative agreement with experiment, in particular reproducing the anomalous, non-monotonic thermal behavior of bcc Fe’s magnetostriction, which has been largely unexplained for more than 50 years. The method has also been used to calculate the finite temperature magnetoelasticity of the A2 phase of Fe1−xGax alloys as a starting point for further investigations into the giant magnetostriction of Galfenol alloys. Our calculations show that homogeneously doping bcc Fe with Ga does not produce an enhancement in magnetostriction and that the non-monotonic temperature dependence and significant volume dependence are suppressed by increasing Ga content.

Elastic and Magnetoelastic Properties of Fe-Rich Tetragonal Y–Fe Compounds: Phenomenology and Electron Theory

To study the strain effects on the lattice stability and magnetic anisotropy (MA) of Fe-rich tetragonal Y-Fe compounds, elastic constants of Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">14</sub> B-and YFe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> -type compounds were calculated in the first principles, and MA of impurity-doped YFe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> with tetragonal lattice strain was calculated in a real-space tight-binding method. It has been found that the YFe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> -type compounds are elastically softer than Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Fe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">14</sub> B, and that the uniaxial MA of Co-doped YFe <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sub> may be increased by decreasing c/a, where c and a are the lattice constants along c-axis and within ab-plane, respectively.

Enhanced piezoelectric properties and temperature stability of Bi4Ti3O12-based Aurivillius ceramics via W/Nb substitution

Bi4Ti3O12 (BIT), a typical Aurivillius ceramics with high Curie temperature (Tc ˜ 675 °C), has great potential for high temperature applications. This work provides an effective method of inducing structure distortion, relieving the tetragonal strain of the TiO6 octahedron and decreasing the concentration of oxygen vacancies to improve the piezoelectricity and temperature stability of BIT ceramics. Bi4Ti2.98W0.01Nb0.01O12 possesses an optimum piezoelectric coefficient (d33) of 32 pC/N, a high Tc of 655 °C and a large resistivity of 3 × 106 Ω·cm at 500 °C. The maximum d33 reported here is approximately quadruple than that of pure BIT (˜7 pC/N). Moreover, the d33 of W/Nb co-doped BIT and the in-situ temperature stability of the compression-mode sensor present a highly stable characteristic in the range of 25–600 °C. These results imply that W/Nb-modified BIT ceramics is a promising candidate for application at high temperatures of up to 600 °C.

Robust half metallicity state with the hydrostatic and tetragonal distortion for a new quaternary Heusler ZrTiRhGa: FP-LAPW calculations

The structural, mechanical, electronic and magnetic properties of quaternary Heusler ZrTiRhGa alloy have been investigated by using the first-principles calculations within the generalized gradient approximation GGA. The results of our calculations show that the type (I) of ZrTiRhGa, with the ferromagnetic (FM) state, is the most stable configuration among three possible configurations. The GGA + U calculations, where U is on-site Coulomb interaction correction, are performed to investigate the relationship between band gap and Hubbard potential.The calculated results showed that ZrTiRhGa compound is half-metallic material (HMM) with complete spin polarization around the Fermi level. The total magnetic moment is 2 μB per formula unit and follows the Slater-Pauling rule μt = Zt−18. The calculated elastic constants, cohesive and formation energy, for ZrTiRhGa compound, were analyzed in detail and reveal the mechanical stability. In addition, we investigated the sensitivity of the half-metallicity with two structural-type changes: hydrostatic and tetragonal deformation. Our calculations show that the half-metallicity can be preserved with lattice constants from 5.74 Å to 7.12 Å, under the influence of hydrostatic strain and in the range of the c/a ratio from −8% to 12% with the effect of the tetragonal strain. For ZrTiRhGa, since there is no available experimental data, our calculations are considered as first predictions.