Critical Temperature of Site-Diluted Spin-1/2 Systems with Long-Range Ferromagnetic Interactions

Abstract

In the paper the Pair Approximation (PA) method for studies of the site-diluted spin-1/2 systems of arbitrary dimensionality with the long-range ferromagnetic interactions is adopted.The method allows to take into account arbitrary anisotropy of the interactions in the spin space, so it is not limited to purely Ising couplings. Within this approach, the Gibbs free energy is obtained, which allows to derive all the further interesting thermodynamic properties.In particular, we obtain an equation for the critical temperature of the second-order phase transitions for the model in question. In the study we focus our attention on the systems with ferromagnetic interactions decaying with the distance according to the power law J(r) ∝ r −n .We discuss the dependence of the critical temperature on the concentration of magnetic component and the index n for selected one-, two-and three-dimensional lattices. We confirm the absence of the critical concentration for a diluted magnet with infinite interaction range. In the regime of the low concentrations of magnetic component, we find a non-linear increase of the critical temperature with the concentration in the form of T c ∝ p n/d , depending on the system dimensionality d and the index n, whereas n > d.

Dilute magnetism in graphene

“…In full, the isotropic case, J = J z = J, with JS = t [74], shows a linear behavior for low impurity concentrations, with no signs of a critical concentration below which the ferromagnetic order is lost. This agrees with approximate results for a system of diluted spins interacting through a ferromagnetic, long ranged RKKY-like interaction [58]. For x < 15%, the temperature dependence on x is well described by the linear function T C ≈ 1400x (K).…”

“…It couples magnetic moments on the same sublattice ferromagnetically, and antiferromagnetically on different sublattices. In two-dimensions, particularly in graphene, the ordering of a diluted set of magnetic atoms coupled via RKKY interaction has been shown to be possible [56][57][58][59]. However, the use of an effective RKKY-type interaction, often derived perturbatively, carries the disadvantage of losing information about the underlying electronic system.…”

The Pair Approximation method for the ferromagnetic Heisenberg model with spin $S=1$ and arbitrary range of interactions. Application for the magnetic semiconductor CrIAs

“…In the progress of further studies, the CVM, and at the same time PA, have been mastered with a tendency to clarify and simplify the formalism [4,5,[7][8][9][10][11][12][13][14][15][16][17]. This enabled application of the method to many physical problems, especially in the field of magnetism [18][19][20][21][22][23][24][25][26][27][28][29][30]. For instance, the PA method has been first applied to the Heisenberg [4,5,[8][9][10]24] and classical Ising [11,13,17,22] models.…”

“…For instance, the PA method has been first applied to the Heisenberg [4,5,[8][9][10]24] and classical Ising [11,13,17,22] models. The studies included also dilute systems, for instance, involving the Ising model with long-range interactions [29], as well as the Heisenberg anisotropic model [25,31]. The Ising-type models with higher spins have also been considered [11].…”

“…It has also been shown that for the Blume-Emery-Griffiths model with z = 5 the PA method gives exact results on a Bethe lattice [19]. Moreover, regarding dilute systems, the PA method yields reasonable values of the critical concentrations [25,29]. An essential advantage of the PA method is that the Gibbs energy can be obtained in an analytical form, from which all thermodynamic properties can be calculated in a self-consistent way.…”