The structural and electronic properties of hypothetical Ru$_x$Fe$_{1-x}$Se and Ru$_x$Fe$_{1-x}$Te systems have been investigated from first principles within the density functional theory (DFT). Reasonable values of lattice parameters and chalcogen atomic positions in the tetragonal unit cell of iron chalcogenides have been obtained with the use of norm-conserving pseudopotentials. The well known discrepancies between experimental data and DFT-calculated results for structural parameters of iron chalcogenides are related to the semicore atomic states which were frozen in the used here approach. Such an approach yields valid results of the electronic structures of the investigated compounds. The Ru-based chalcogenides exhibit the same topology of the Fermi surface (FS) as that of FeSe, differing only in subtle FS nesting features. Our calculations predict that the ground states of RuSe and RuTe are nonmagnetic, whereas those of the solid solutions Ru$_x$Fe$_{1-x}$Se and Ru$_x$Fe$_{1-x}$Te become the single- and double-stripe antiferromagnetic, respectively. However, the calculated stabilization energy values are comparable for each system. The phase transitions between these magnetic arrangements may be induced by slight changes of the chalcogen atom positions and the lattice parameters $a$ in the unit cell of iron selenides and tellurides. Since the superconductivity in iron chalcogenides is believed to be mediated by the spin fluctuations in single-stripe magnetic phase, the Ru$_x$Fe$_{1-x}$Se and Ru$_x$Fe$_{1-x}$Te systems are good candidates for new superconducting iron-based materials.
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