3.2. Density Functional Theory Calculations
As has been mentioned above, a crystal structure prediction was previously made for EuScCuSe
3, and the space group
Pnma was supposed [
3]. Due to the fact that the sample experimentally synthesized in our work was solved in space group
Cmcm, we did a comprehensive investigation of the EuScCuSe
3 crystal structure stability in both space groups,
Pnma and
Cmcm.
At the first step of density functional theory calculations, crystal structures of EuScCuSe
3 in
Pnma and
Cmcm space groups were totally optimized, and the obtained lattice parameters are presented in
Table 3. The simulated structural data get close to the experiments in both cases. It should be noted that the energy per formula unit is almost the same for both structure types and differs only in the fifth decimal place: −9634.327521165 at. un (
Pnma), −9634.327541515 at. un (
Cmcm).
The next mandatory part of the crystal structure stability investigation is the simulation of elastic properties [
33]. Calculations of the elastic constants were performed using the built-in functionality of CRYSTAL17 code. The obtained data for EuScCuSe
3 in
Cmcm and
Pnma structures are presented in
Table 4. The necessary and sufficient
Born criteria [
34] for the orthorhombic crystal-system stability are
C11 > 0,
C11C22 >
C12,
C11C22C33 + 2
C12C13C23 − C
11C
232 − C
22C
132 − C
33C
122 > 0, C
44 > 0, C
55 > 0, C
66 > 0. All the above conditions are satisfied both for the real
Cmcm and predicted
Pnma-structure of EuScCuSe3 previously.
As any data on elastic properties of EuScCuSe
3 are absent at this time in databases or articles, we present calculation of the bulk modulus,
Young’s modulus, and shear modulus in the
Voigt,
Reuss, and
Hill approximations (
Table 5). The dependence of
Young’s modulus on the crystal directions demonstrates a significant anisotropy of the elastic properties in both the
Cmcm and the
Pnma structure (
Figure S2).
The calculated values of the shear modulus and bulk modulus make it possible to estimate the
Vickers hardness for EuScCuSe
3 (
Table 5). To estimate the
Vickers hardness, the empirical formula (3.3.1) from work [
1] was used.
This formula well describes the hardness of a row of compounds with an ionic and covalent type of chemical bond (about 40 compounds were considered in work [
1]). In Formula (1),
G and
B are the shear modulus, and bulk modulus by
Hill is estimated. The experimental values of hardness are absent from research papers. According to calculations, the elastic constants and hardness of EuScCuSe
3 differ significantly for the
Cmcm and
Pnma structures (
Table 4).
As vibrational spectroscopy is a powerful tool for the determination of crystal structure details, simulation of Raman and infrared spectra for the experimentally obtained data in this work (
Cmcm structure) and possibly earlier predicted
Pnma structure [
1] were done. The results for the infrared-active modes, Raman modes, and “silent” modes at the Г point are given in
Tables S4 and S5 of the SM. The degree of participation of each ion in a particular mode is estimated from the analysis of displacement vectors obtained from these ab-initio calculations. The ions that are shifted significantly in the mode are listed in the column “participants” (
Tables S4 and S5). The values of ion displacements for vibrational modes are shown in
Figure S3.
The number of formula units in the
Pnma structure is equal to 4 (Z = 4), and this value is the same for the
Cmcm structure, see
Table 1. However, the primitive cell of the
Cmcm structure contains only two formula units (
Figure S4). Thus, the number of vibrational modes should be larger in the
Pnma case. The Raman-active modes for
Pnma and
Cmcm structures should be listed as 12
Ag + 6
B1g +12
B2g + 6
B3g and 5
Ag + 4
B1g +
B2g + 5
B3g, correspondingly [
35]. The result of Raman and infrared spectra simulations for both structures are presented in
Figure 3. Despite the fact that the number of vibrational modes is different for the structures in
Cmcm and
Pnma, the simulated Raman and infrared spectra are quite similar. Thus, we suppose that the definition of the correct space group (
Cmcm or
Pnma) using experimental vibrational spectroscopy is almost impossible in this case.
The only possible indicator for the
Pnma structure is the low-lying weak band in the Raman spectrum (
Figure 3a) which is associated with very strong movements of all ions except for Cu
+ (
Figure S3). However, the wavenumber value of this vibrational mode is the lowest in both structures. In this regard, the calculation of phonon dispersion curves was done for the
Pnma structure, and the results of the simulation in Γ–X direction are shown in
Figure 4a. The key factor of the dynamical stability of crystal lattice is the absence of imaginary (unstable) phonon modes and this approach works in for the case of experimentally observed crystal structures [
36] as for crystal structure stability prediction [
37,
38]. According to the obtained data (
Figure 4a), we can say that the crystal structure of EuScCuSe
3 in the previously supposed space group
Pnma should be unstable. This fact, among other things, is consistent with the experimentally obtained space group
Cmcm obtained for the real EuScCuSe
3 in this work. At the same time, simulated phonon dispersion for the
Cmcm structure do not contain unstable phonon modes over all of the high-symmetric
Brillouin zone points (
Cmcm).
The band structure and the density of states for EuScCuSe
3 calculated using hybrid PBE0 functional are shown in
Figure 5. The path in the
Brillouin zone is plotted through the most highly symmetric points. For the space group
Cmcm, the path is made along Г–Y–T–Z–S–R–Г. The coordinates of the points are (0,0,0,), (
1/
2,
1/
2,0), (
1/
2,
1/
2,
1/
2), (0,0,
1/
2), (0,
1/
2,0), (0,
1/
2,
1/
2), (0,0,0) respectively. The Bilbao crystallographic server was used [
35]. Since for europium pseudopotential that replaced their core shells, the 4
f inclusive was used, the band structure does not include 4
f states. For the Eu
2+ cations, only outer shells (5
s25
p6) were taken into account by means of valence basis sets [
39]. The projected DOS onto the whole set of atomic orbitals of Eu, Sc, Cu, and Se atoms was calculated near the band gap. According to these calculations, the DOS of copper and selenium are located near the top of the valence band. The DOS of scandium and europium are located near the bottom of the conduction band. The band gap value is defined as the difference in energy between the top of the valence band and the bottom of the conduction band. Calculations predict for EuScCuSe
3 the indirect electronic transition with a band gap value of 3.27 eV. It should be noted, that in the case of the dynamically unstable
Pnma structure, the band gap value is the same, but the calculated electronic transition is direct (
Figure S5).
3.3. Magnetic Properties
The temperature dependence of the specific magnetization was measured in the temperature range from 2 to 300 K (
Figure 6). Based on it, the temperature dependences of the direct and reciprocal values of the molar magnetic susceptibility are calculated.
The main contribution to the magnetic properties of EuScCuSe
3 is made by the Eu
2+ cations with unfilled
f-shells. There is no significant effect of the crystal field on the magnetic moment since, in the ground state (
8S
7/2), this cation has a zero-orbital momentum. Its temperature dependence of magnetic susceptibility in the paramagnetic region should be well described by the
Curie-Weiss law:
considering the temperature-independent term χ
TIP. Approximation of the experimental dependence by this formula gives the following values: χ
TIP = 1.04·10
−5 m
3 kmol
−1,
C = 0.0977 K m
3 kmol
−1,
θW = 6.0 K. The deviations of the experimental points from the approximating curve in the temperature ranging from 40 to 300 K are no more than 1%, and from 10 to 40 K about 2.5%. A comparison of the characteristics obtained with those calculated for non-interacting Eu
2+ cations is given in
Table 6.
There is a sharp deviation from the Curie-Weiss law at temperatures below 5 K. This deviation is obviously due to the ferromagnetic transition, although there is no noticeable discrepancy in the data for the FC and ZFC
The experimental curve of magnetization at a temperature of 2 K (
Figure 7b) has the form characteristic of magnetically soft ferromagnets. The coercive force is less than 2 kA m
−1, and saturation occurs in a field of about 500 kA m
−1. The magnetization in a field of 4 MA m
−1 per formula unit is 6.5
μB, which is close to the theoretical value of about 7
μB for a free Eu
2+ cation.