## Abstract

In recent years the intermetallic ternary RE_{2}MgGe_{2} (RE = rare earth) compounds attract interest in a variety of technological areas. We therefore investigate in the present work the structural, electronic, magnetic, and thermodynamic properties of Nd_{2}MgGe_{2} and Gd_{2}MgGe_{2}. Spin–orbit coupling is found to play an essential role in realizing the antiferromagnetic ground state observed in experiments. Both materials show metallicity and application of a Debye-Slater model demonstrates low thermal conductivity and little effects of the RE atom on the thermodynamic behavior.

## Introduction

The examination of intermetallic compounds with Mo_{2}FeB_{2} structure type (space group *P4/mbm*), as ordered variant of the U_{3}Si_{2} structure type, has been the subject of many studies in recent years due to a variety of intriguing physical properties^{1}. Examples reach from superconductivity to heavy fermion behavior, Kondo lattices, and magnetocalorics^{2}. Well-known members of the material class are U_{2}SnCo_{2} and U_{2}InPt_{2}, non-Fermi liquid systems on the verge of long-range magnetic ordering, and the hybridization-gap semiconductor U_{2}SnRu_{2}^{3}. Y_{1.6}Ce_{0.4}InPd_{2} and Lu_{1.6}Ce_{0.4}InPd_{2} show heavy fermion behavior and in Ce_{2}InRh_{2} the Ce atoms realize a mixed valent state with high spin fluctuation temperature^{4}. Yb_{2}AlSi_{2}, on the other hand, shows an intermediate valent state^{5}. More than 300 intermetallic compounds M_{2}XT_{2} (M = rare earth or actinoid metal; X = Mg, In, Sn, Cd; T = transition metal) are known. The X and T atoms constitute planar [XT_{2}] networks such that each X atom is coordinated to four T_{2} dumb-bells^{6,7}. The crystal structure can be understood as packing of distorted fragments of AlB_{2} and CsCl structure types, which form a network of octahedra and trigonal bipyramids with interstitial sites favorable for H allocation. The M atoms form a planar structure with triangular motif, which, depending on the exchange interaction, can lead to magnetic frustration, as described by the two-dimensional Shastry-Sutherland Hamiltonian \(H = J\sum\nolimits_{NN} {S_{i} S_{j} } + J^{\prime } \sum\nolimits_{NNN} {S_{i} S_{j} }\), where *S*_{i} represents the magnetic moment at site *i*^{8}*.* Magnetic frustration exists when the nearest neighbor interaction *J* and next nearest neighbor interaction *J′* are both antiferromagnetic (AF)^{9}. Since *J* > *J*′, nearest neighbor atoms form a network of *J-*coupled dimers that are coupled by *J′*. The ground state is a disordered spin liquid with energy gap between the singlet and triplet states or an antiferromagnet with gapless magnetic excitations. Transition is predicted for *J*′/*J* ≈ 0.6–0.7 at 0 K^{10}, although symmetry arguments suggest that an intermediate state is required, such as a helical magnet or a spin density wave^{11}.

Cermets with Mo_{2}FeB_{2} structure type show excellent wear resistance, low friction to non-ferrous metals, and thermal expansion coefficients close to those of steel. While they are not as strong as the commonly used hard metals^{12}, the mechanical properties can be improved by Cr and Ni additions. B, V, and Mn additions reduce the grain size and remarkably increase the transverse rupture strength^{13}. RE_{2}MgT_{2} (RE = rare earth) compounds are used for lightweight construction in the automobile and aerospace industries^{14}. They show high corrosion resistance and thus have been investigated intensively with respect to their microstructure and mechanical properties^{15}. The compounds play an essential role as bio-compatible materials for healing or replacing natural bone^{16}. They are also able to absorb large amounts of H, up to 8 atoms per formula unit, which weakens the magnetism, as the RE–RE exchange interaction and magnetic coupling via the conduction electrons are suppressed (while for U_{2}MgT_{2} the upper limit of H absorption is 2 atoms per formula unit and hydrogenation enhances the magnetism)^{17,18}. Interstitial doping with H atoms, which induces internal pressure and/or modifies the bonding between the other atoms, can be used to tune the crystal and electronic structures, particularly the magnetism, which depends critically on details of the hybridization and charge localization^{19}.

M_{2}XT_{2} compounds can be magnetocaloric with field-induced magnetic and/or structural transitions, where for M = RE the magnetic structure is determined mainly by the (i) RKKY (Ruderman, Kittel, Kasuya, and Yosida) interaction and (ii) crystalline electric field (which is responsible for magnetic phase transitions, for example, for RE = Nd)^{20}. In magnetocaloric ferromagnets/antiferromagnets the entropy change during isothermal magnetization is negative/positive (termed positive/negative magnetocaloric effect). A positive magnetocaloric effect is interesting for magnetic refrigerators and a negative magnetocaloric effect for heat pumps^{21}.

In U_{2}SnT_{2} compounds (non-magnetic or magnetic depending on T) the interatomic distance between U and T increases from U_{2}SnFe_{2} to U_{2}SnCo_{2} and U_{2}SnNi_{2}, even though the atomic radius decreases from Fe (1.27 Å) to Co (1.25 Å) and Ni (1.24 Å), and the spin–orbit coupling plays an important role for the electronic states^{22}. While in U_{2}SnNi_{2} the *a* lattice parameter increases with the temperature and the *c* lattice parameter decreases, in the cases of U_{2}SnCo_{2} and U_{2}SnPd_{2} both lattice parameters increase^{22}. U_{2}SnNi_{2} shows AF ordering below 25 K with the U moments aligned parallel to the *c-*axis^{23}. The band width of the U 5f. states decreases for heavier T = Co, Ni, Rh, Pd, Ir, and Pt^{24}. Giant magnetoresistance is predicted for U_{2}SnPd_{2} and U_{2}InPd_{2}^{25}, and specific heat data classify single crystalline U_{2}SnCo_{2} and U_{2}InPt_{2} as non-Fermi-liquid materials^{26}. Enhanced hybridization between the U 5f. and Ni 3*d* states leads in U_{2}Sn(Ni_{1-x}Co_{x})_{2} solid solutions for increasing *x* to a loss of AF ordering at *x* = 0.3 and in U_{2}Sn(Ni_{1-x}Pd_{x})_{2} solid solutions to a decrease of the Neel temperature (*T*_{N}) up to *x* = 0.3, which is reproduced by the RKKY model^{22}. At 1.5 K the U magnetic moments switch from in-plane ordering in U_{2}SnPd_{2} (2 μ_{B}) to out-of-plane ordering in U_{2}Sn_{0.65}Pd_{2.35} (0.9 μ_{B}) due to enhanced hybridization between the U 5f. and Pd 4*d* states (reduced U-Pd distance)^{22}.

The magnetic behavior of the Ce_{2}InT_{2} compounds is determined by the filling of the T *d* bands, with Ce being trivalent in Ce_{2}InCu_{2} and Ce_{2}InAu_{2} but not in Ce_{2}InPd_{2}^{27}. First principles calculations demonstrate that in Ce_{2}SnPd_{2} the hybridization between the Ce 4f. and Pd 4*d* states is weak (strong localization of the Ce 4f. states and large Ce magnetic moments)^{26}. Ce_{2}(In/Sn)Pd_{2} alloys display transitions between AF and ferromagnetic (FM) ground states as a function of the Sn/In ratio^{28,29}. Ce_{2}PbPt_{2} realizes AF ordering below 3.4 K^{30}. Tb_{2}InCu_{2} is FM up to 81 K^{29}, and Nd_{2}InAu_{2} and Tb_{2}InAu_{2} are ferrimagnetic and FM up to 36 and 73 K with magnetic moments of 3.5 and 9.31 μ_{B}, respectively^{30}. Pr_{2}InNi_{2} and Nd_{2}InNi_{2} undergo second order FM to paramagnetic transitions at 7.5 and 10.5 K, respectively, according to Ref. 31, whereas Ref. 32 reports Nd_{2}InNi_{2} to be AF with *T*_{N} = 8 K. Nd_{2}InNi_{2} can absorb up to 7 H atoms per formula unit at room temperature, which leads to expansion of the unit cell along the *a*- and *c*-axes and compression along the *b*-axis (orthorhombic structure with space group *Pbam*)^{33}. The Nd magnetic moment of 3.55 μ_{B} resembles that of a free Nd^{3+} ion (3.62 μ_{B})^{34,35}. Nd_{2}InNi_{2} is an Ising antiferromagnet^{36}.

Tb_{2}InPd_{2} shows a significantly higher *T*_{N} = 32 K than Pr_{2}InPd_{2} (5 K) and Nd_{2}InPd_{2} (8 K)^{37}, with a spin-reorientation transition at μ_{0}H_{c} = 4 T. Ref. 38 confirms *T*_{N} = 29.4 K for Tb_{2}InPd_{2}. Based on neutron powder diffraction, the Tb magnetic moments of μ_{eff} = 10.54 μ_{B} (aligned along the *c*-axis) ecxeed the theoretically predicted value of 9.72 μ_{B}^{35,37}. Substitution of In by Pd in Ce_{2}InPd_{2} results in a transition from FM to AF ordering^{39}. The fact that the specific heat of Yb_{2}InPd_{2} is one order of magnitude larger than that of Yb_{2}InAu_{2} is due to a high Yb 4f. density of states at the Fermi level, the intermediate valence of Yb being explained by hybridization with the Pd 4*d* states^{40}. RE_{2}PbPd_{2} compounds with RE = Ce, Nd, Sm, Gd, Tb, Dy, Ho, Er, and Tm show AF ordering at low temperatures (*T*_{N} in the range of 3.6–35 K), Pr_{2}PbPd_{2} is a Curie–Weiss paramagnet down to 1.72 K, and Y_{2}PbPd_{2} and La_{2}PbPd_{2} are weak Pauli paramagnets^{41}. Ce_{2}PbPd_{2} is subject to a weak Kondo effect with a Ce magnetic moment of 2.61 μ_{B}, and RE magnetic moments of 4.05 and 9.85 μ_{B} are found for Nd_{2}PbPd_{2} and Tb_{2}PbPd_{2}, respectively^{41}. Ce_{2}MgSi_{2} shows helical AF ordering at *T*_{N} = 12 K with magnetocrystalline anisotropy and a Ce magnetic moment of 2.47 μ_{B}^{42}. An anomaly in the electrical resistivity of Nd_{2}InGe_{2} at 9 K points to AF ordering with a small FM component^{43}.

In the case of the RE_{2}SnNi_{2} compounds the smaller RE elements Ho, Er, Tm, Lu, and Sc result in a Mo_{2}FeB_{2} structure, whereas the larger RE elements Ce, Pr, Nd, Sm, Gd, Tb, Dy, and Y result in an orthorhombic W_{2}CoB_{2} structure (space group *Immm*)^{44,45}. Nd_{2}SnNi_{2} exhibits AF ordering below *T*_{N} = 21 K (which can be turned into FM ordering by a moderate magnetic field of 0.25 T) and two further magnetic transitions at 17.7 and 14–15 K^{20}. Ce_{2}SnNi_{2} is a Kondo system with *T*_{N} = 4.7 K and *T*_{K} ≈ 8 K, and Gd_{2}SnNi_{2} and Tb_{2}SnNi_{2} show oscillatory magnetocaloric effects^{20}. Tb_{2}SnNi_{2} transforms under high pressure (8 GPa) and high temperature (1470 K) to the Mo_{2}FeB_{2} structure^{44}. In the W_{2}CoB_{2} structure it shows AF ordering below *T*_{N} = 66 K (Tb magnetic moment of 8.7 μ_{B}; very close to a free Tb^{3+} ion) with magnetic transitions at 42 and 8 K (probably coexistence of FM and AF ordering), and a Curie–Weiss behavior above 80 K (Tb magnetic moment of 7.7 μ_{B})^{20}. In the temperature range of 5–220 K, Nd_{2}SnNi_{2} and Tb_{2}SnNi_{2} do not reach their theoretical saturation magnetization in a magnetic field of 100 kOe, which may be attributed to a canted magnetic structure^{20}. For the isothermal magnetic entropy change in a magnetic field of 50 kOe values of 7.2, 0.1, 4.6, and 2.8 J/kg K are reported for Nd_{2}SnNi_{2}, Sm_{2}SnNi_{2}, Gd_{2}SnNi_{2}, and Tb_{2}SnNi_{2}, respectively^{20}. Tb_{2}SnNi_{2} shows striking similarities to (Pr,Ca)MnO_{3}, since both these compounds are subject to coexistence of AF and FM ordering at low temperature with a magnetocaloric effect that switches from negative to positive when the temperature increases^{46}.

Replacing the transition metal with a main group element has a drastic effect on the RE element and thus on the magnetic properties. To give an example, *T*_{N} grows from 49 K in Gd_{2}MgNi_{2} (additional magnetic transitions at 20*.*7 and 4*.*5 K^{47}) to 150 K in Gd_{2}MgGe_{2}^{48}. The RE_{2}MgGe_{2} compounds with RE = Nd, Gd, Tb, Dy, Ho, Er, and Tm show Curie–Weiss paramagnetism at high temperature, while Y_{2}MgGe_{2} and Lu_{2}MgGe_{2} display Pauli-like temperature independent paramagnetism^{49}. AF ordering at 14, 13, 32, 55, 24, and 14 K is found for Nd_{2}MgGe_{2}, Sm_{2}MgGe_{2}, Gd_{2}MgGe_{2}, Tb_{2}MgGe_{2}, Dy_{2}MgGe_{2}, and Ho_{2}MgGe_{2}, respectively, while Er_{2}MgGe_{2} and Tm_{2}MgGe_{2} do not undergo magnetic ordering at least down to 5 K^{49}. Since the type of magnetic ordering and ordering temperature are determined by the atomic interactions^{50} and there is a lack of detailed understanding, the present work addresses the class of RE_{2}MgGe_{2} compounds from a first principles perspective.

## Results and discussion

We use the full potential linear augmented plane wave plus local orbitals software WIEN2k^{51}, which provides highly accurate results due to the fact that it is an all-electron implementation of density functional theory. The electronic wave functions are expanded in spherical harmonics (up to *l*_{max} = 10) within non-overlapping muffin-tin spheres centered at the nuclear sites and plane waves (limited by *K*_{max} = 7/*R*_{MT,min}) in the remaining space (interstitial region). The generalized gradient approximation is chosen for the exchange–correlation functional^{52}. We employ 6 × 6 × 11 and 6 × 6 × 5 Monkhorst–Pack k-grids in the structural optimizations of unit cells and 1 × 1 × 2 supercells, respectively. The unit cells are non-primitive with two formula units (Fig. 1a). In the supercell calculations spin-polarization is taken into account with the RE atoms coupled ferromagnetically within the *ab*-plane and antiferromagnetically along the *c*-axis (Fig. 1b). Densities of states are calculated using a refined 10 × 10 × 9 Monkhorst–Pack k-grid and adding spin–orbit coupling by the polarized orbital method to obtain correct magnetic moments^{53,54}.

In the tetragonal unit cell of RE_{2}MgGe_{2} (Fig. 1a) the atoms occupy the Wykhoff positions RE 4 h (*x*_{RE}, *x*_{RE} + 1/2, 1/2), Ge 4 g (*x*_{Ge}, *x*_{Ge} + 1/2, 0), and Mg 2a (0, 0, 0). Both for non-magnetic and AF configurations, we relax *x*_{RE} and *x*_{Ge} at different volumes and fit the total energy to the Murnaghan equation of state^{55} in order to obtain the equilibrium volume (Fig. 2a,b). Then we relax *x*_{RE} and *x*_{Ge} at different *c/a* ratios and again fit the total energy to the Murnaghan equation of state in order to obtain the equilibrium *c/a* ratio (Fig. 2c,d) as well as the bulk modulus and its pressure derivative. Table 1 shows that AF ordering significantly modifies the unit cell volume for Nd_{2}MgGe_{2} but not for Gd_{2}MgGe_{2}. It turns out that AF ordering results in energy gain of 0.5 eV per unit cell for Nd_{2}MgGe_{2} and 2.3 eV per unit cell for Gd_{2}MgGe_{2} with respect to the non-magnetic solution. We do not further investigate Sm_{2}MgGe_{2}, Tb_{2}MgGe_{2}, Dy_{2}MgGe_{2}, and Ho_{2}MgGe_{2}, as no qualitative difference can be expected. Y_{2}MgGe_{2} and Lu_{2}MgGe_{2} are not of interest, because no magnetic ordering is obtained, in agreement with Ref. 49. For Er_{2}MgGe_{2} and Tm_{2}MgGe_{2} our calculations predict magnetic ground states. However, absence of magnetic ordering above 5 K implies that the compounds undergo low temperature phase transitions, i.e., our results are not of experimental relevance and therefore excluded from the following discussions.

The obtained AF lattice constants in Table 1 deviate from the experimental values by less than 0.4% (*a*) and 2.3% (*c*)^{48,49}. They turn out to be smaller for Gd_{2}MgGe_{2} than Nd_{2}MgGe_{2} though Gd is heavier than Nd (lanthanide contraction) and the shortest RE–RE distances within the *ab*-plane (3.72 Å for Nd_{2}MgGe_{2}, 3.66 Å for Gd_{2}MgGe_{2}) are significantly smaller than those along the *c*-axis (4.46 Å for Nd_{2}MgGe_{2}, 4.30 Å for Gd_{2}MgGe_{2}). This confirms the magnetic structure model of Ref. 49 that the localized RE 4f. spins realize FM ordering within the *ab*-plane and AF ordering mediated by RKKY interaction along the *c*-axis. As expected, we find *B* ∝ *V*_{0}^{‒1}, where *V*_{0} is the equilibrium unit cell volume. Rather small values of *B* reflect softness of the compounds under study. The magnetic moment \(M = g \sqrt {J\left( { J + 1 } \right)}\) depends on the Landé factor \(g = 1 + \frac{{J\left( {J + 1} \right) + S\left( {S + 1} \right) - L\left( {L + 1} \right)}}{{2J\left( {J + 1} \right)}}\) with *J* = *L* ± *S* for a less/more than half-filled shell^{56}. Comparison of the obtained magnetic moments with experiment in Table 2 confirms the validity of our theoretical approach. The fact that they are almost purely of 4f. origin and close to those of free atoms demonstrates localized magnetism.

To study the nature of the chemical bonding, we evaluate the electronic density of states (DOS) in Figs. 3 and 4. We note that the local density approximation yields qualitatively the same behavior (Figure S1 in the Supporting Information) and that the wave function expansions are well converged (Figure S2 in the Supporting Information). For both the non-magnetic and AF configurations of the two compounds we obtain similar results, in particular a finite DOS at the Fermi level, i.e., a metallic state. The high non-magnetic DOS at the Fermi level (mainly Nd/Gd 4f. states) points to magnetism according to the Stoner criterion^{57}. The symmetric shape of the AF DOS is due to the absence of a total magnetic moment. The AF DOS shows strong contributions of the Ge 4* s* states at low energy, the Ge 4*p* states below the Fermi level, and the Nd/Gd 4f. states above the Fermi level. The Nd 4f. states are found from − 2 eV to 4 eV, whereas the Gd 4f. states give rise to pronounced peaks around − 4 eV and just above the Fermi level. As the compounds are isostructural, the nature of the chemical bonding is similar (except for the splitting of the Nd/Gd 4f. states) and the physical properties thus are expected to be comparable.

To better understand the magnetism, we evaluate as dominant perturbation the crystal field splitting of the localized and correlated 4f. states. We follow the methodology of Refs.^{58,59,60,61,62} to extract the crystal field parameters from our first principles calculations and obtain the ground state multiplet energies 0, 2, 13, 15, and 16 meV (first excited state: 245 meV) for the Nd^{3+} ions and 0, 0.01, 0.03, and 0.05 meV (first excited state: 3980 meV) for the Gd^{3+} ions. Nd^{3+} (4*f*^{3} electronic configuration) and Gd^{3+} (4*f*^{7} electronic configuration) are Kramer ions. While the ground state of Nd^{3+} (^{4}*I*_{9/2}, *S* = 3/2, *L* = 6, *J* = 9/2) is split by the crystal field into five Kramer doublets (in agreement with Ref. 63), Gd^{3+} has zero orbital moment in the ground state (^{8}*S*_{7/2}, *S* = 7/2, *L* = 0, *J* = 7/2) and, consequently, is not influenced by the crystal field in first approximation. The angular dependence of the ground state multiplet energies in a magnetic field aligned within the *ab*-plane (Fig. 5a,b) shows that *a* is the easy magnetic axis and *b* is the hard magnetic axis. The obtained magnetic susceptibilities approach the experimental curve for increasing temperature (Fig. 5c,d).

We analyze the electron density in terms of the quantum theory of atoms in molecules^{64} (CRITIC code^{65}) by means of the critical points (CPs), which are categorized into nuclear, bond, ring, and cage CPs according to the Hessian matrix. Table 3 lists the locations and characteristics of all the symmetry-inequivalent CPs. As expected, the CPs are very similar for the two compounds. In Fig. 6 we represent the bond CPs (same order as in Table 3) by red spheres to identify the atoms linked by them. For Nd_{2}MgGe_{2} the Ge-Mg bond CP is closer to Mg (2.03 Å) than Ge (3.41 Å), while the Ge–Nd bond CP is more centered. The electron density at the bond CPs is small and the Laplacian is close to zero (positive in the cases of the Ge-Mg and Ge–Nd bonds, negative in the cases of the Mg-Mg and Ge–Ge bonds; Table 3). For Gd_{2}MgGe_{2} the Ge-Gd bond CP is closer to Gd (2.86 Å) than Ge (3.00 Å). Again the electron density at the CPs is small and the Laplacian is close to zero (positive in the case of the Ge-Gd bonds, negative in the cases of the Ge–Ge and Mg-Mg bonds; Table 3).

Table 4 lists the atomic volumes and corresponding integrated charges. 48% and 51% of the unit cell volume is filled by Ge for Nd_{2}MgGe_{2} and Gd_{2}MgGe_{2}, respectively, 40% and 38% by RE, and 11% by Mg. We find a net transfer of charge from Nd/Gd and Mg to Ge, in agreement with the Pauling electronegativity. The obtained oxidation states deviate significantly from the nominal values of + 3 (Nd/Gd), + 2 (Mg), and ‒4 (Ge), confirming a metallic nature^{48}, in contrast to the semiconducting charge distribution with neutral Mg proposed in Ref. 66. The global charge transfer is described by the ionicity \(c = \frac{1}{N}\sum\nolimits_{i = 1}^{N} {\frac{{\Delta Q_{i} }}{{OS_{i} }}}\)^{67}, where *N* is the number of atomic sites *i* and \(\Delta Q_{i}\)/*OS*_{i} is the ratio of the actual and nominal oxidation states. The values of 37% and 39% for Nd_{2}MgGe_{2} and Gd_{2}MgGe_{2}, respectively, are indicative of dominant covalent bonding. Indeed, the electron localization function demonstrates strong covalent Mg-Mg bonds (Fig. 7a,c) and the electronic charges in Table 4 in conjunction with Fig. 7b,d point to polarized covalent Mg-Ge bonds. We estimate the degree of metallicity in terms of the electron density flatness *f* = ρ_{min}/ρ_{max}, where ρ_{min} is the minimum electron density (at the cage1 CP) and ρ_{max} is the maximum electron density (at the bond5 CP)^{65}. As *f* = 1 represents metallic bonding and *f* = 0 represents localized bonding, the obtained values of 20% and 18% for Nd_{2}MgGe_{2} and Gd_{2}MgGe_{2}, respectively, show that metallic bonding plays a limited role.

We use a Debye-Slater model to drive the thermodynamic properties^{68}. Table 5 indicates that the two compounds behave similarly although Nd_{2}MgGe_{2} shows slightly larger bulk modulus and Debye temperature. The heat capacity at constant pressure (*C*_{p}) turns out to be larger than the heat capacity at constant volume (*C*_{V}), consistent with the relation *C*_{p} − *C*_{V} = (*α*_{V})^{2}*BV*_{p}*T*, where *α*_{V}, *B*, *V*_{p}, and *T* are the thermal expansion coefficient at constant volume, bulk modulus, volume of the primitive unit cell (Nd_{2}MgGe_{2}: ½ × 1592.4 Bohr^{3}; Gd_{2}MgGe_{2}: ½ × 1590.2 Bohr^{3}), and absolute temperature, respectively. According to Fig. 8a, *C*_{V} ∝ *T*^{3} at low temperature and the Dulong-Petit limit is approached at high temperature. Low Debye temperatures reflect low thermal conductivities and melting temperatures. According to Fig. 8b, the volume expansion starts to become linear in *T* at about 150 K.

## Conclusions

The structural, electronic, magnetic, and thermodynamic properties of the intermetallic compounds Nd_{2}MgGe_{2} and Gd_{2}MgGe_{2} have been investigated by full potential linearized augmented plane wave plus local orbitals calculations, employing the generalized gradient approximation for the exchange–correlation potential. The calculated lattice constants agree well with the available experimental data. Accounting for the spin–orbit coupling turns out to be mandatory to obtain correct magnetic moments and evaluate the electronic properties. Both compounds are found to combine metallicity with an AF ground state with localized magnetic moments. The chemical bonding turns out to be predominantly covalent. According to a Debye-Slater model, the thermal conductivity is low and the choice of the RE atom hardly affects the thermodynamic behavior.

## References

Villars, P. & Calvert, L. D. (eds.), 2nd edn. https://searchworks.stanford.edu/view/2016804 (American Society for Metals, 1991).

Szytula, A. & Leciejewicz, J.

*Handbook of Crystal Structures and Magnetic Properties of Rare Earth Intermetallics*(CRC Press, 1994).Kaczorowski, D. & Gulay, L. D. Magnetic and electrical properties of a novel compound U

_{2}Pd_{2}Pb.*J. Alloys Compd.***419**, 11–14 (2006).Mallik, R., Sampathkumaran, E. V., Dumschat, J. & Wortmann, G. Magnetic ordering and spin fluctuation behavior in compounds of the type Ce

_{2}(Pd, Rh)_{2}In.*Solid State Commun.***102**, 59–64 (1997).Gannon, W. J.

*et al.*Intermediate valence in single crystalline Yb_{2}Si_{2}Al.*Phys. Rev. B***98**, 075101 (2018).Zaremba, V. I., Johrendt, D., Rodewald, U., Nychyporuk, P. & Pöttgen, R. Structure and chemical bonding of Ce

_{2}Ge_{2}In and Ce_{2}Pt_{2}In.*Solid State Sci.***7**, 998–1002 (2005).Wang, B., Liu, Y., Ye, J. & Wang, J. Electronic, magnetic and elastic properties of Mo

_{2}FeB_{2}: first-principles calculations.*Comput. Mater. Sci.***70**, 133–139 (2013).Shastry, B. S. & Sutherland, B. Exact ground state of a quantum mechanical antiferromagnet.

*Phys. B***108**, 1069–1070 (1981).Miyahara, S. & Ueda, K. Theory of the orthogonal dimer Heisenberg spin model for SrCu

_{2}(BO_{3})_{2}.*J. Phys. Condens. Matter***15**, R327–R366 (2003).Havela, L., Mašková, S., Daniš, S., Stelmakhovych, O. & Miliyanchuk, K. Large H absorption in Nd

_{2}Ni_{2}In; magnetism in a new structure type.*Mater. Res. Soc. Symp. Proc.***1216**, W03-W12 (2010).Carpentier, D. & Balents, L. Field theory for generalized Shastry–Sutherland models.

*Phys. Rev. B***65**, 024427 (2002).Sun, C., Yu, H. & Liu, W. Microstructure, mechanical properties and first-principles calculations of Mo

_{2}FeB_{2}-based cermets with Mn addition.*J. Ceram. Soc. Jpn.***125**, 677–680 (2017).Xuming, P., Yong, Z., Shaogang, W. & Qiuhong, W. Effect of Mn on valence-electron structure and properties of hard phase in Mo

_{2}FeB_{2}-based cermets.*Int. J. Refract. Met. Hard Mater.***27**, 777–780 (2009).Aghion, E., Bronfin, B. & Eliezer, D. The role of the magnesium industry in protecting the environment.

*J. Mater. Process. Technol.***117**, 381–385 (2001).Kraft, R., Fickenscher, T., Kotzyba, G., Hoffmann, R. & Pottgen, R. Intermetallic rare earth (RE) magnesium compounds REPdMg and RE

_{2}Pd_{2}Mg.*Intermetallics***11**, 111–118 (2003).Staiger, M., Pietak, A., Huadmai, J. & Dias, G. Magnesium and its alloys as orthopedic biomaterials: a review.

*Biomaterials***27**, 1728–1734 (2006).Ourane, B. Recherche Exploratoire de Nouveaux Intermétalliques Ternaires à Base de Magnésium, Application au Stockage d’Hydrogène, Doctoral Thesis, Bordeaux University (2014).

Miliyanchuk, K., Maskova, S., Havela, L. & Gladyshevskii, R. Influence of hydrogenation on the magnetic properties of Er

_{2}Ni_{2}Al.*Chem. Met. Alloys.***9**, 169–173 (2016).Mašková, S. Structure and Magnetic Properties of F-Electron Compounds and Their Hydrides, Doctoral Thesis, Charles University in Prague, Faculty of Mathematics and Physics (2013).

Kumar, P., Singh, N. K., Suresh, K. G. & Nigam, A. K. Magnetocaloric and magnetotransport properties of R

_{2}Ni_{2}Sn compounds (RE = Ce, Nd, Sm, Gd, and Tb).*Phys. Rev. B***77**, 184411 (2008).Proceedings of the First International Conference on Magnetic Refrigeration at Room Temperature, edited by P. W. Egolf, International Institute of Refrigeration, Paris, France (2005).

Lafargue, D. Structures Magnétiques de Nouveaux Stannures Ternaires a Base d'Uranium ou de Terres Rares T

_{2}M_{2}Sn (T = RE, U et M = Ni, Pd), Doctoral Thesis, Sciences et Technologies University, Bordeaux I (1997).Mašková, S.

*et al.*U_{2}Ni_{2}Sn and the origin of magnetic anisotropy in uranium compounds.*Phys. Rev. B***99**, 064415 (2019).Prokeg, K., Brfick, E., Nakotte, H., de Chfitel, P. F. & de Boer, F. R. Simple calculation of hybridization effects in UTX and U

_{2}T_{2}X compounds.*Phys. B***206**(207), 8–10 (1995).Richter, M., Zahn, P., Divis, M. & Mertig, I. Giant magnetoresistance in uranium intermetallics: ab initio calculations for U

_{2}Pd_{2}In and U_{2}Pd_{2}Sn.*Phys. Rev. B***54**, 11985–11988 (1996).Lukachuk, M. & Pöttgen, R. Intermetallic compounds with ordered U

_{3}Si_{2}or Zr_{3}Al_{2}type structure-crystal chemistry, chemical bonding and physical properties.*Z. Kristallogr.***218**, 767–787 (2003).Hulliger, F. On tetragonal M

_{2}Au_{2}In and related compounds.*J. Alloys Compd.***232**, 160–164 (1996).Sereni, J. G., Roberts, J., Gastaldo, F. & Giovannini, M. Suppression of the Shastry–Sutherland phase driven by electronic concentration reduction in magnetically frustrated Ce

_{2.15}Pd_{1.95}(Sn_{1−y}In_{y})_{0.9}alloys.*Phys. Rev. B***100**, 054421 (2019).Sereni, J. G., Roberts, J., G-Bastaldo, F., Gerisso, M. & Giovannini, M. Shastry–Sutherland phase formation in magnetically frustrated Ce

_{2}Pd_{2}In_{1−x}Sn_{x}alloys.*Mater. Today Proc.***14**, 80–83 (2019).Kabeya, N.

*et al.*Antiferromagnetic ground state and heavy-fermion behavior in Ce_{2}Pt_{2}Pb.*Phys. Rev. B***98**, 035131 (2018).Zhang, Z., Wang, P., Ronga, H. & Li, L. Structural and cryogenic magnetic properties of RE

_{2}Ni_{2}In (RE = Pr, Nd, Dy and Ho) compounds.*Dalton Trans.***48**, 17792–17799 (2019).Maskova, S., Danis, S., Llobet, A., Nakotte, H. & Havela, L. Large magnetocaloric effect in Nd

_{2}Ni_{2}In.*Acta Phys. Pol. A***126**, 282–283 (2014).Mašková, S.

*et al.*Impact of hydrogen absorption on crystal structure and magnetic properties of geometrically frustrated Nd_{2}Ni_{2}In.*J. Alloys Compd.***566**, 22–30 (2016).Mašková, S., Havela, L. & Daniš, S. Enormous Hydrogen Absorption in Nd

_{2}Ni_{2}In, in*WDS'09 Proceedings of Contributed Papers, Part III*109–112 (2009).Maškova, S.

*et al.*Magnetic properties of Tb_{2}Pd_{2}In.*Single Cryst. Study Solid State Phenom.***194**, 58–61 (2013).Sala, G., Mašková, S. & Stone, M. B. Frustrated ground state in the metallic ising antiferromagnet Nd

_{2}Ni_{2}In.*Phys. Rev. Mater.***1**, 054404 (2017).Fischer, P.

*et al.*Antiferromagnetic rare-earth ordering in the intermetallic compounds R_{2}Pd_{2}In (R = Pr, Nd).*J. Phys. Condens. Matter***12**, 7089–7098 (2000).Maskova-Cerna, S.

*et al.*New type of magnetic structure in the R_{2}T_{2}X group: Tb_{2}Pd_{2}In.*J. Phys. Condens. Matter***32**, 345801 (2020).Giovannini, M.

*et al.*Effect of nonstoichiometry on the transition from ferromagnetism to antiferromagnetism in the ternary indides Ce_{1.95}Pd_{2+2x}In_{1−x}and Ce_{2+x}Pd_{1.85}In_{1−x}.*Phys. Rev. B***61**, 4044–4053 (2000).Giovannini, M.

*et al.*Characterization and physical properties of the indides Yb_{2}T_{2}In (T = Cu, Pd, Au).*Intermetallics***9**, 481–485 (2001).Kaczorowski, D. & Gulay, L. D. Magnetic and electrical properties of RE

_{2}Pd_{2}Pb (RE = Y, La-Sm, Gd-Tm) compounds.*J. Alloys Compd.***442**, 169–171 (2007).Dhar, S. K., Manfrinetti, P. & Palenzona, A. Magnetic ordering in CeMg

_{2}Si_{2}and Ce_{2}MgSi_{2}.*J. Alloys Compd.***252**, 24–27 (1997).Zaremba, V. I., Kaczorowski, D., Nychyporuk, G. P., Rodewald, U. C. & Pöttgen, R. Structure and physical properties of RE

_{2}Ge_{2}In (RE = La, Ce, Pr, Nd).*Solid State Sci.***6**, 1301–1306 (2004).Heymann, G.

*et al.*High-pressure phases of Tb_{2}Ni_{2}Sn and Dy_{2}Ni_{2}Sn.*Monatsh. Chem.***145**, 863–867 (2014).Heying, B., Rodewald, U., Chevalier, B. & Pottgen, R. The stannides RE

_{2}Ni_{2}Sn (RE = Pr, Ho, Er, Tm)—structural transition from the W_{2}B_{2}Co to the Mo_{2}B_{2}Fe type as a function of the rare earth size.*Z. Naturforsch. B***68**, 10–16 (2014).Gomes, A. M., Garcia, F., Guimarães, A. P., Reis, M. S. & Amaral, V. S. Field-tuned magnetocaloric effect in metamagnetic manganite system.

*Appl. Phys. Lett.***85**, 4974–4976 (2004).Łatka, K., Kmiec, R., Pacyna, A. W., Mishra, R. & Pöttgen, R. Magnetism and hyperfine interactions in Gd

_{2}Ni_{2}Mg.*Solid State Sci.***3**, 545–558 (2001).Choe, W., Mille, G. J. & Levin, E. M. Crystal structure and magnetism of Gd

_{2}MgGe_{2}.*J. Alloys Compd.***329**, 121–130 (2001).Suen, N., Tobash, P. H. & Bobev, S. Synthesis, structural characterization and magnetic properties of RE

_{2}MgGe_{2}(RE = Rare-Earth Metal).*J. Solid State Chem.***184**, 2941–2947 (2011).Kraft, R. & Pottgen, R. Ternary germanides RE

_{2}Ge_{2}Mg (RE = Y, La-Nd, Sm, Gd, Tb).*Monatsh. Chem.***135**, 1327–1334 (2004).Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, D. & Luitz, J.

*WIEN2k: An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties*(Technische Universität Wien, 2001).Perdew, J., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple.

*Phys. Rev. Lett.***77**, 3865–3868 (1996).Brooks, M. S. S. Calculated ground state properties of light actinide metals and their compounds.

*Phys. B***130**, 6–12 (1985).Eriksson, O., Johansson, B. & Brooks, M. S. S. Meta-magnetism in UCoAl.

*J. Phys. C***1**, 4005–4011 (1989).Murnaghan, F. D. The compressibility of media under extreme pressures.

*Proc. Natl. Acad. Sci. USA***30**, 244–247 (1944).Buschow, K. H. J. & de Boer, F. R.

*Physics of Magnetism and Magnetic Materials*(Kluwer Academic Publishers, 2003).Stoner, E. C. Collective electron specific heat and spin paramagnetism in metals.

*Proc. R. Soc. A***154**, 656–678 (1936).Novák, P., Knížek, K. & Kuneš, J. Crystal field parameters with wannier functions: application to rare-earth aluminates.

*Phys. Rev. B***87**, 205139 (2013).Mihóková, E., Novák, P. & Laguta, V. Crystal field and magnetism with Wannier functions: rare-earth doped aluminum garnets.

*J. Rare Earth***33**, 1316–1323 (2015).Novák, P., Kuneš, J. & Knížek, K. Crystal field of rare earth impurities in LaF

_{3}.*Opt. Mater.***37**, 414–418 (2014).Novák, P., Knížek, K., Maryško, M., Jirák, Z. & Kuneš, J. Crystal field and magnetism of Pr

^{3+}and Nd^{3+}ions in orthorhombic perovskites.*J. Phys. Condens. Matter***25**, 446001 (2013).Novák, P., Nekvasil, V. & Knížek, K. Crystal field and magnetism with Wannier functions: orthorhombic rare-earth manganites.

*J. Magn. Magn. Mater.***358–359**, 228–232 (2014).Popova, M. N.

*et al.*Optical spectra, crystal-field parameters, and magnetic susceptibility of multiferroic NdFe_{3}(BO_{3})_{4}.*Phys. Rev. B***75**, 224435 (2007).Bader, R. F. W.

*Atoms in Molecules*(Oxford University Press, 1990).Otero-de-la-Roza, A., Blanco, M. A., Martin Pendas, A. & Luana, V. Critic: a new program for the topological analysis of solid-state electron densities.

*Comput. Phys. Comm.***180**, 157–166 (2009).Whangbo, M.-H., Lee, C. & Köhler, J. Transition-metal anions in solids and their implications on bonding.

*Angew. Chem.***45**, 7465–7469 (2006).Sanchez, P. M., Pendas, A. M. & Luana, V. A classification of covalent, ionic, and metallic solids based on the electron density.

*J. Am. Chem. Soc.***124**, 14721–14723 (2002).Otero-de-la-Roza, A., Abbasi-Pérez, D. & Luana, V. Gibbs

_{2}: a new version of the quasiharmonic model code. II. Models for solid-state thermodynamics, features and implementation.*Comput. Phys. Commun.***182**, 2232–2248 (2011).

## Acknowledgements

The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST).

## Author information

### Authors and Affiliations

### Contributions

S.M. and O.M.A. conducted the calculations. U.S. contributed to the analysis of the results and writing of the manuscript. All authors reviewed the manuscript.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary Information

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Menouer, S., Abid, O.M., Benzair, A. *et al.* First principles calculations of the structural, electronic, magnetic, and thermodynamic properties of the Nd_{2}MgGe_{2} and Gd_{2}MgGe_{2} intermetallic compounds.
*Sci Rep* **11**, 10870 (2021). https://doi.org/10.1038/s41598-021-89042-5

Received:

Accepted:

Published:

DOI: https://doi.org/10.1038/s41598-021-89042-5

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.