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This work focuses on the theoretical investigation of the coexistence of superconductivity and ferromagnetism in the superconducting HoMo
_{6}S
_{8}. By developing a model Hamiltonian for the system and using the Green’s function formalism and equation of motion method, we have obtained expressions for superconducting transition temperature (
T_{c}), magnetic order temperature (
T_{m}), superconductivity order parameter (
D
) and magnetic order parameter (η). By employing the experimental and theoretical values of the parameters in the obtained expressions, phase diagrams of energy gap parameter versus transition temperature, superconducting transition temperature versus magnetic order parameter and magnetic order temperature versus magnetic order parameter are plotted separately. By combining the phase diagrams of superconducting transition temperature versus magnetic order parameter and magnetic order temperature versus magnetic order parameter, we have demonstrated the possible coexistence of superconductivity and ferromagnetism in superconducting HoMo_{6}S_{8}.<

Superconductivity was discovered in 1911 by Kamerlingh Onnes [

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} [_{2} [_{2}RECu_{2}O_{8} compounds (with RE = Eu or Gd) [_{6}S_{8} and in ErRh_{4}B_{4}. In ErRh_{4}B_{4}, Sinha et al. [_{6}S_{8}, the study was done by Lynn et al. [

In HoMo_{6}S_{8}, the ferromagnetic state destroys the superconductivity at sufficiently low temperatures. Recently, an experiment on HoMo_{6}S_{8} [_{c}_{1} and T_{c}_{2} (lower and upper superconducting critical temperatures), respectively and vanishes for T < T_{c}_{1} and T > T_{c}_{2}.

Among the “Chevrel phases”, HoMo_{6}S_{8} has been extensively studied in recent years [_{6}S_{8} becomes superconducting at T_{c}_{1} ≌ 1.82 K, but at a lower temperature T_{c}_{2} ≌ 0.64 K, it re-enters the normal state at the onset of long range ferromagnetic order. In a narrow temperature range T_{c}_{2} < T < T_{m}, superconductivity coexists with a modulated magnetic structure [

In order to study the coexistence of ferromagnetism and superconductivity in superconducting HoMo_{6}S_{8} 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 [

where

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

Here, the operators

and localized electrons respectively with the wave vector

where

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

One can also obtain the equation of motion for the expression

For

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

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

where

We can write Equation (19) as,

Let

Now, let us study equation (21) by considering different cases.

Case (I): As

Hence, Equation (21) becomes,

Using the integral

For

This implies that,

For

The experimental value of HoMo_{6}S_{8} is, T_{c} ≈ 1.82 K.

Thus,

Case (II): At_{ }

Now, employing Equation (27) and the experimental value of T_{c} for the superconducting HoMo_{6}S_{8} and plausible approximations for other parameters, we plotted the transition temperature (T_{c}) versus magnetic ordering parameter (η) as shown in

For η = 0, we get the expression for T_{c} to be,

Using Green’s function formalism, the equation of motion for the localized electrons is obtained to be,

Now, using the Hamiltonian given in Equation (1), we evaluated the commutation

where

Applying similar procedure as above and assuming

Now, from Equations (31) and (32), we get,

From which we get,

The equation of motion that shows the correlation between the conduction and localized electrons can be demonstrated. Using similar definition as for ∆, we can write the magnetic ordering parameter, η as,

Changing the summation into integration and by introducing the density of states, N(0), we get,

Using the Matsubara frequency,

Equation (36) becomes,

where

Now, let us first solve the following expression.

Using Laplace’s transform and Matsubara frequency, Equation (38) becomes,

where

and

Then,

Since ∆_{l} is very small,

From which we get,

Using Equation (42) and the experimental value, T_{m} ≈ 0.67 K for HoMo_{6}S_{8} and some plausible approximations for other parameters in the equation, we plot the magnetic order temperature (T_{m}) versus magnetic order parameter as shown in

For pure superconducting system, that is, when magnetic order cannot appear or magnetic effect is zero, we can ignore η and our previous calculation gives the following results which is similar to the well-known BCS model.

As T → 0, η → 0 and tanh(βE/2) → 1, Equation (21) reduces to,

From which we obtain,

Furthermore, for T → T_{c}, η = 0 and for low temperature, i.e.

From which we get,

To obtain the temperature dependency of energy gap in Equation (21), we used the same techniques to solve the integral,

But from the BCS model

For

Using the relation

From which we can get,

Equation (47) shows how the superconducting order parameter, ∆(T) varies with temperature when η = 0 and is similar to the BCS model.

Using the experimental value, T_{c} ≈ 1.82 K for HoMo_{6}S_{8} and some plausible approximations, we plot ∆ versus T_{c} as shown in

Now, by combining _{6}S_{8} as shown in

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 HoMo_{6}S_{8} and plausible approximations for other parameters, we plotted the transition temperature (T_{c}) versus magnetic order parameter (η) as shown in _{m} ≈ 0.67 K for HoMo_{6}S_{8} and some suitable approximations for the other parameters in the equation, we plotted the magnetic order temperature versus magnetic order parameter as demonstrated in _{c}) and is plotted in _{c}). From _{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 HoMo_{6}S_{8}. Our finding is in agreement with the experimental observation [

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 HoMo_{6}S_{8}. 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 HoMo_{6}S_{8}.

TadesseDesta,GebregziabherKahsay, (2015) Coexistence of Superconductivity and Ferromagnetism in Superconducting HoMo_{6}S_{8}. World Journal of Condensed Matter Physics,05,27-36. doi: 10.4236/wjcmp.2015.51004