The compound Sr₂YRuO₆ belongs to the family of double perovskites with general formula O₆, where the site occupied by a large cation A = Ca, Sr, Ba is capable of 12-fold coordination by oxygen, while the B and sites represent smaller cations suitable for octahedral coordination, generally being lanthanides, yttrium, or 4d/5d transition metals.^[1,2] The compound Sr₂YRuO₆ is really interesting, with a frail equilibrium between ferromagnetic (FM) and antiferromagnetic (AFM) interactions engendering a complex magnetic response initially described as “anomalous”.^[1] The compound Sr₂YRuO₆ crystallizes in the ordered double perovskite structure with Ru⁵⁺ ions described by an FCC magnetic system, which align in an AFM structure at low temperature.^[3] Specific heat and magnetic susceptibility measurements reveal two phase transitions at ∼30 K and ∼26 K respectively,^[1] while the physical state of Sr₂YRuO₆ between these two transition temperatures continues to be elucidated. Additional interest in this material is originated from the occurrence of superconductivity when Ru is partially (up to 15%) replaced by Cu in both powders^[3–7] and single crystals,^[8–10] with superconducting volume fractions still under 10%. A broad knowledge of the magnetic state of Sr₂YRuO₆ is fundamental to understand how the superconducting and ferromagnetic order parameters can adjust themselves in the crystal lattice.^[11] However, since only one magnetic sublattice is present in Sr₂YRuO₆, this knowledge will contribute to the characterization of other double perovskites with two magnetic sublattices, like Sr₂FeReO₆ and Sr₂FeMoO₆ which are interesting for their potential application in spin electronics at room temperature.^[12] Hence, even though Sr₂YRuO₆ was synthesized 30 years ago as reported in Ref. ^[13], crucial and precise analysis of the experimental results with different techniques is still needed. The anomalies possess unusual transport and magnetic properties that emphasize the strong interplay of spin, charge, and orbital degrees of freedom. In some of our recent works,^{[14–17,32,32,32,32,32,32]} we have provided the magnetic behavior and phase diagrams using Monte Carlo simulations for the perovskite BiFeO₃, also we have presented and discussed the corresponding hysteresis cycles.

Within this context, the goal of this paper is to make a theoretical investigation of the magnetic properties of the Sr₂YRuO₆ compound. To reach this goal, a thoughtful analysis of the ground states, magnetizations, and susceptibilities as a function of the temperature and the effect of the crystal field along with the behavior of the hysteresis loop are discussed.

This paper is organized as follow. In section 2, we present the model and the Hamiltonian describing the compound Sr₂YRuO₆. Section 3 is devoted to the interpretation of the obtained results from the ground state phase diagrams as well as from the Monte Carlo simulations. In section 4, we provide our conclusions.

The theoretical model used to simulate the magnetic properties of the double perovskite Sr₂YRuO₆ using Monte Carlo simulation is presented in this section. The structure of the studied compound corresponds to the space group as presented in Fig. 1.^[32] The unit cell crystal structure of Sr₂YRuO₆ is illustrated in Fig. 1(a), whereas the octahedrons of RuO₆ and YO₆ are presented in Fig. 1(b). The structural parameters of Sr₂YRuO₆ are listed in table 1 according to Ref. ^[32]. The coordinates of the atoms Sr, Y, Ru, O1, O2, and O3 are given in table 2.^[32]

Fig. 1.

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	Fig. 1. (a) The unit cell crystal structure of Sr2YRuO6, and (b) RuO6, YO6 octahedrons.

Table 1.

Table 1.

Table 1. Structure parameters of Sr2YRuO6.[32] . Structure parameters Values Space group Cell parameters a=5.7761 Å, b=5.7840 Å, c=8.1689 Å, ,

Table 1. Structure parameters of Sr2YRuO6.[32] .

Table 2.

Table 2.

Table 2. Coordinates of Sr2YRuO6.[32] . Site X Y Z Magnetic moment/ Sr 0.0053 0.0267 0.2379 0 Y 1/2 0 0 0 Ru 1/2 0 1/2 3/2 O1 0.2648 0.2954 0.0338 0 O2 0.1988 0.7710 0.0345 0 O3 0.9331 0.4865 0.2352 0

Table 2. Coordinates of Sr2YRuO6.[32] .

The magnetic ordering of this compound can occur only by the Ru moments as illustrated in Fig. 2. The Hamiltonian controlling the system reads 1 In the above equation, the notation corresponds to the nearest-neighbors’ spins. The spins S_i of the Ru atom take the values ±1/2, ±3/2. The exchange coupling constant is donated by J. The last two terms are the Zeeman energy and the crystal field energy, where H and are the external magnetic field and the crystal field, respectively. The notations and stand for the perpendicular and parallel directions. The external magnetic field H and the crystal field are acting on all the system spins. The crystal field is originated from the competition between Ru–O interactions in this compound. In order to simulate this system’s magnetic behavior, with the above Hamiltonian, Monte Carlo simulations are performed under the Metropolis algorithm.

Fig. 2.

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	Fig. 2. The unit cell of Sr2YRuO6, showing the only magnetic Ru ions.

In this work, we give all physical parameters in the MKSA units. In fact, the temperature is in Kelvin, the energies are in Joule, and the external magnetic and crystal fields are in Tesla. For simplicity, we have fixed the Boltzmann constant at its unit value K_B =1 (dimensionless). This is the case for the earlier references.^{[15,16,32,32,32,32,32]}

The magnetic properties of the double perovskite Sr₂YRuO₆ are investigated using Monte Carlo simulations. The calculations are performed to simulate the Hamiltonian given in Eq. (1), considering the cyclic boundary conditions. For each spin configuration, we perform 10⁵ Monte Carlo steps eliminating the first 10⁴ generated configurations. While skimming all the system sites, single-flip attempts are made, and they either are accepted or rejected according to the Metropolis algorithm. In fact, the average of each parameter has been obtained for each Monte Carlo simulation, when averaging over several initial conditions, under the well-known Jackknife method.^[32] For every iteration, the following parameters are calculated.

The internal energy per site 2 where N_T is the total number of atoms consisting the unit cell.

The magnetization of the system 3 The susceptibility χ is given by 4 where K_B is the Boltzmann constant fixed at its unit value (K_B=1), and T is the absolute temperature. In this study, we will limit to the results of a fixed system size N_T=5×5 ×5, using the free boundary conditions.

We have studied the magnetic properties of the double perovskite Sr₂YRuO₆ using Monte Carlo simulations. We presented and discussed the ground state phase diagrams at T = 0 K. We showed the effect of the competition between the crystal field and the exchange coupling and . The four possible phases are found to be stable in the plane ( , for H = 0 and . Also we found that for and , the only stable phases are ±1/2. While for and , the stable phases are ±3/2. These phases are separated by a linear curve with negative slope in the plane ( , . The hysteresis loops of the Sr₂YRuO₆ compound as a function of the external magnetic field were presented and discussed. For , , , and the fixed temperature T = 21 K, the saturation is reached at H = 10. The delayed magnetization saturation is caused by the competition between the negative value of the exchange coupling and the increasing effect of both temperature and crystal field. We have also obtained the Néel temperature of Sr₂YRuO₆ ( K). The obtained results in this work are in good agreement with the earlier experimental results.^[1]