Superconductivity was discovered in 1911 by Kamerlingh Onnes [1] when a so-called “Blue Boy” noticed that the resistivity of Hg metal vanished abruptly at a temperature of about 4.2 K. Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization and is one of the strongest forms of magnetism. It is responsible for most of the magnetic behaviors encountered in everyday life and is the basis for all permanent magnets (as well as the metals that are noticeably attracted to them). In particular, a material is ferromagnetic in narrower sense only if all of its magneticions add a positive contribution to the net magnetization. If some of the magneticions subtract from the net magnetization, that is, if they are partially anti-aligned, then the material is ferrimagnetic. If the ions anti-align completely so as to have zero netmagnetization, despite the magnetic ordering, then it is an antiferromagnet. Thus, ferromagnetic materials exhibit parallel alignment of moments resulting in large net magnetization even in the absence of a magnetic field.

Superconductivity in Ferromagnetic must result from a different type of electronpairing mechanisms. In these materials, electrons with spins pointing in the same direction team up with each other to form Cooper pairs with one unit of spin resulting in a triplet superconductivity. In contrast, conventional superconductivity also known as s-wave singlet superconductivity occurs when electrons with oppositespins bind together to form Cooper pairs with zero momentum and spin.

The coexistence of superconductivity and ferromagnetism has been studied theoretically and experimentally. The coexistence of ferromagnetism and superconductivity was first addressed theoretically by Ginzburg in 1957 [2] and experimental investigation was made by Matthias et al. [3] . The interplay between superconducting and ferromagnetic long range order has been recently attracting new interest due to the discovery of superconductivity in ferromagnetic compounds such as UGe₂ [4] , URhGe [5] , ZrZn₂ [6] , and in RuSr₂RECu₂O₈ compounds (with RE = Eu or Gd) [7] . The relationship between magnetism and superconductivity has received renewed attention since the discovery of ternary superconducting materials which also achieved long-range magnetic ordering at low temperatures. Ferromagnetic alignment can be expected to be strongly opposed by superconductivity. Such a long-period magnetic ordering was actually found in HoMo₆S₈ and in ErRh₄B₄. In ErRh₄B₄, Sinha et al. [8] carried out a detailed study on a single crystal in order to characterize this phase. For HoMo₆S₈, the study was done by Lynn et al. [9] only on polycrystalline samples.

In HoMo₆S₈, the ferromagnetic state destroys the superconductivity at sufficiently low temperatures. Recently, an experiment on HoMo₆S₈ [10] has shown that, the superconducting ordering parameter has a distinct maximum between some critical temperatures T_c1 and T_c2 (lower and upper superconducting critical temperatures), respectively and vanishes for T < T_c1 and T > T_c2.

Among the “Chevrel phases”, HoMo₆S₈ has been extensively studied in recent years [11] . HoMo₆S₈ becomes superconducting at T_c1 ≌ 1.82 K, but at a lower temperature T_c2 ≌ 0.64 K, it re-enters the normal state at the onset of long range ferromagnetic order. In a narrow temperature range T_c2 < T < T_m, superconductivity coexists with a modulated magnetic structure [12] .

In order to study the coexistence of ferromagnetism and superconductivity in superconducting HoMo₆S₈ theoretically in general and to find the expressions for transition temperature and order parameters in particular, a system of conduction and localized electrons have been considered. The exchange interaction acts between the conduction and the localized electrons. Thus, within the frame work of the BCS model [13] , the model system Hamiltonian can be formulated as follows.

where

and is the Hamiltonian or energy of mobile (conduction) electrons and localized electrons respectively.

Here, the operators and are the creation (annihilation) operators for conduction

and localized electrons respectively with the wave vector and the spin projection on z-axis σ = ↑ or ↓. is the one electron kinetic energy measured relative to the chemical potential. is the interaction (electron- electron) through boson (phonon) exchange and is given by,

where defines the matrix element of the interaction potential. is the interaction term between con-

duction electrons and localized electrons due to some unspecified mechanism with some coupling constant (α) and is expressed as,

Now, let us evaluate the following commutation relation,

From which we obtain,

Following similar procedure as above, we get,

and

Substituting Equations (6), (7) and (8) into the equation of motion given by,

we obtain,

In general, we have to write the higher order Green’s function into lower order Green’s function by using Wick’s theorem. Thus, we have,

Now, substituting Equation (11) into Equation (10), we get,

where and

One can also obtain the equation of motion for the expression and obtain,

For and, we obtain,

Now, using Equations (12) and (14), the equation of motion becomes,

From which we obtain,

Using the relation for ∆, given by,

and by changing the summation into integration and by introducing the density of states at the Fermi level, (N(0)), we get,

Now, changing, we use the Matsubara frequency [14] ,

Now, using Equation (18) in Equation (17), we get,

where Since attraction is effective in the region and taking the density of state to be constant in this region and using the relation,

We can write Equation (19) as,

Let

In this section, we describe the results which are obtained using the model Hamiltonian developed. We obtain the expressions for the superconducting ordering parameter (∆) and magnetic order parameter (η) with respect to

superconducting transition temperature (T_c) and magnetic order temperature (T_m) respectively. First, using Equation (27) and the experimental value of T_c for the superconducting HoMo₆S₈ and plausible approximations for other parameters, we plotted the transition temperature (T_c) versus magnetic order parameter (η) as shown in Figure 1. As can be seen from the figure, when the magnetic order parameter increases the superconducting transition temperature decreases. Second, by employing Equation (42) and the experimental value, T_m ≈ 0.67 K for HoMo₆S₈ and some suitable approximations for the other parameters in the equation, we plotted the magnetic order temperature versus magnetic order parameter as demonstrated in Figure 2. From the figure, it is vivid that, as the magnetic order parameter increases the magnetic order temperature also increases. Furthermore, the superconducting order parameter (∆) is expressed as a function of the transition temperature (T_c) and is plotted in Figure 3. The expression we obtained for the pure superconductor in Equation (47) is in agreement with the BCS model for η = 0. It is clear that from Figure 3, the superconducting order parameter, which is the measure of pairing energy decreases with increasing temperature and vanishes at the transition temperature (T_c). From Figure 4, we observe that, T_c decreases with increasing η, whereas T_m increases with increasing η and there is a small region of temperature where both superconductivity and ferromagnetism coexist in HoMo₆S₈. Our finding is in agreement with the experimental observation [15] .

In the present work, we have demonstrated the basic concepts of superconductivity with special emphasis on the BCS model and Cooper pair focusing on the interaction between superconductivity and ferromagnetism which are closely connected to the particular crystal of superconducting HoMo₆S₈. Employing the double time temperature dependent retarded Green’s functions formalism, we developed the model Hamiltonian for the system and derived equations of motion for conduction electrons, localized electrons and for pure superconducting system and carried out various correlations by using suitable decoupling procedures. In developing the model Hamiltonian, we considered spin triplet pairing mechanism and obtained expressions for superconducting order parameter, magnetic order parameter, superconducting transition temperature and magnetic order temperature. By using appropriate experimental values and considering suitable approximations, we plotted figures using the equations developed. As is well-known, superconductivity and ferromagnetism are two cooperative phenomena which are mutually antagonistic since superconductivity is associated with the pairing of electron states related to time reversal while in the magnetic states the time reversal symmetry is lost. Because of this, there is a strong competition between the two phases. This competition between superconductivity and magnetism made coexistence unlikely to occur. However, the model we employed in this work, shows that, there is a small region of temperature where both superconductivity and ferromagnetism can coexist in superconducting HoMo₆S₈.

TadesseDesta,GebregziabherKahsay, (2015) Coexistence of Superconductivity and Ferromagnetism in Superconducting HoMo₆S₈. World Journal of Condensed Matter Physics,05,27-36. doi: 10.4236/wjcmp.2015.51004