^{1}

^{*}

^{1}

^{*}

**The interaction of superconductivity and magnetism is studied in iron based superconductor using the Hamiltonian consisting of the itinerant electrons, localized electrons moment, and s-f interaction. Using Greens function technique and equation of motion method, we have obtained an expressions for superconducting order parameters (△( T), △(0))**

**and critical temperature T**

_{C}, which reduce to BCS result in the absence of magnetic interactions. The result of the calculations shows that superconductivity can coexist with magnetism in iron based superconductor below the critical temperature.The discovery of high T_{C} iron based superconductor in 2008 [

The interplay of superconductivity and magnetism has been studied in iron based superconductors theoretically and experimentally [

In this work, we are trying to predict the interplay of superconductivity and magnetism on iron based superconductors which can help in explaining experimental observations.

Our model Hamiltonian is composed of

where the first term,

In the above pairing Hamiltonian the term

σ = ↑ or ↓; _{i} of the itinerant electrons and the five 3d spin S_{i} local moment located at site i, where g is the corresponding exchange constant.

To get an effective interaction we change the momentum term in to boson operator. Diagonalizing the Hamiltonian (H_{l}) using Bogoliubov transformation, the canonical form of the Hamiltonian in terms of spin waves,

We obtained the itinerant electrons and localized electrons moment using relations in spin operators like,

The electrons in the valence band which are interacting with an anti ferromagnetically ordered, localized spin system can be described by

We get an effective Hamiltonian

In order to calculate the superconducting parameter, we first need to obtain equation of motion. In this work we used Greens function equation of motion method. Applying elementary commutation relation we found two equations:

From these we get,

where

The superconducting order parameter can be expressed as

The sum may be changed to integral by introducing the density of state

Attractive interaction is effective for the region

Applying Laplaces transform with replacement of ω by Matsubara frequency

The equation becomes

For low temperature the first integral becomes

The second integral becomes

Hence,

This expression can be rewritten as

From this equation we can get the following important relations.

1) Superconducting order parameter as a function of temperature

This quantity (∆) is zero at critical temperature T_{C}. Substituting ∆ = 0, we get

2) using

Or

3) Equation (9) can be written as

As T → 0, and β → 0, gives

Applying standard integrals and approximation for x =1, and

Equation (13) is clearly in agreement with the fact that as the net magnetization increases, the induction of superconductivity decreases. In addition to this, in the absence of magnetic term, Equation (13) reduces to the well known BCS expression.

The results clearly show that superconductivity can coexist with magnetism in iron based superconductor below the critical temperature. Experimental findings show the coexistence of superconductivity and magnetism in some range of doping in some compounds [