_{1}

^{*}

With the use of a model Hamiltonian and retarded double time green’s function formalism, we obtain mathematical expressions for spin density wave and superconductivity parameters. The model reveals a distinct possibility of the coexistence of magnetic phase and superconductivity, which are two usually irreconcilable cooperative phenomena. The work is motivated by the recent experimental evidences of coexistence of spin density wave and superconductivity in a number of FeAs-based superconductors. The theoretical results are then applied to show the coexistence of spin density wave and superconductivity in iron pnictide compound Ba
_{1-x}K
_{x}Fe
_{2}As
_{2} (0.2 ≤ x < 0.4).

The interplay between superconductivity and magnetism has been an interesting topic in condensed mater physics which has been considered until very recently hostile and incompatible. Since the discovery of superconductivity in quaternary pnictide-oxides with critical temperatures (T_{C}) up to 55 K a lot of tremendous interest has been generated in the study of coexistence of these two cooperative phenomena of superconductivity and magnetism. After first reports on superconductivity in undoped LaNiPO [_{C} of 26 K in the F-doped arsenide LaO_{1}_{−}_{x}F_{x}FeAs system [

In addition to this several groups reported an increase of T_{C} values by replacing La with smaller-size rare- earth ions like CeO_{1}_{−x}F_{x}FeAs [_{1−x}F_{x})FeAs with a critical temperature T_{C} of 55 K [_{C} [

The FeSC is quite promising for applications. Having much higher Hc than cuprates and high isotropic critical currents [

By combining transport, X-ray and neutron diffraction experiment studies, the first member of a new family of iron pnictide superconductors (Ba_{1}_{−}_{x}K_{x})Fe_{2}As_{2} with the ThCr_{2}Si_{2}-type structure was discovered a bulk superconductor with T_{C} = 38 K and both the SDW and the superconducting orders coexist in the (Ba_{1}_{−}_{x}K_{x})Fe_{2}As_{2} (0.2 ≤ x < 0.4). The structural and electronic properties of the parent compound BaFe_{2}As_{2} are closely related to LaFeAsO. The induced superconductivity by hole doping is found to have a significantly higher TC in comparison with hole doped LaFeAsO. In contrast to previously stated opinions, the results prove that hole doping is definitely a possible pathway to induce high-T_{C} superconductivity, at least in the oxygen-free compounds [

The above exciting discovery stimulated a lot of interest in the study of coexistence of superconductivity and magnetism. The proximity of the magnetic and superconducting (SC) phases suggests a close relationship between the two phenomena. It is generally believed that the magnetic couplings between the itinerant electrons and/or between the itinerant electron and local spin are essential to both spin density wave instability and superconductivity. Besides, other experimental and theoretical findings, especially the antiferromagnetic ground state and the SDW anomaly of LaFeAsO strongly suggest that the pairing mechanism of the electrons is likely to be connected with spin fluctuations, as it has been assumed for the cuprates [

Triplet superconductivity appears provided that we have coexistence of singlet superconductivity and SDW. In many high

Research on superconducting iron arsenides has largely focused on ternary compounds with the ThCr_{2}Si_{2}-type structure, rather than arsenide oxides (LaFeAsO derivatives) [_{0.6}K_{0.4})Fe_{2}As_{2}, [

The relation between the spin-density-wave (SDW) and superconducting order is a central topic in current research on the superconducting iron pnictide based high TC superconductors. So, in this paper, we start with a model Hamiltonian which incorporates not only terms of the BCS but also by assuming the pairing interaction is due to spin fluctuations for iron pnictide superconductors Ba_{1−x}K_{x}Fe_{2}As_{2}, to examine the coexistence of spin density wave and superconductivity.

The purpose of this work is to study theoretically the co-existence of spin density wave and superconductivity properties in the compound Ba_{1−x}K_{x}Fe_{2}As_{2} in general and to find expression for transition temperature and order parameter in particular. For this purpose, we tried to find the mathematical expression for the superconducting critical temperature (T_{C}), superconducting order parameter (∆_{sc}) the magnetic order parameter (M) and SDW transition temperature (T_{SDW}). Within the framework of the BCS model, the model of the Hamiltonian for coexistence SDW and superconductivity in the compound can be express as:

where

Whereas (

The Double time dependent Green’s function equal to the change of the average value of some dynamic quantity by the time t and useful because they can be used to describe the effect of retarded interactions and all quantities of physical interest can be derived from them. To get the equation of motion we use the double-time temperature dependent retarded Green function is given by Zubarev [

where

The Fourier transformation

Taking the Fourier transform we get:

From (4), it follows that

where the anti-commutation relation,

has been used. To derive an expression for

(1) and using the identities and

Solving the commutator in Equation (5) by using the Hamiltonian in e Equation (1), we get

After some lengthy but straightforward calculations; we arrive at the following results:

Substituting (8) in to (5), we get

The equation of motion for the correlation

Evaluating the commutator in Equation (10) using Hamiltonian:

After some lengthy but straightforward calculations; we arrive at the following results:

Substituting (11) in to (10), we get

Similarly as we did in the above the equation of motion for the correlation

is given by:

and

From Equation (12), we obtain:

And from Equation (14):

Plugging Equations (15) and (16) in (9), yields:

And insert Equations (15) and (16) in (13), we have:

Applying nesting condition

and

Let

Then Equations (19) and (20) respectively becomes:

Finally we can express:

Using the expression

To take into account the temperature dependence of order parameters, we shall write as:

where

Using Equation (24) into Equation (25), we obtain

Let us use

and

Plugging Equation (28) and Equation (29) in Equation (27), we get:

For mathematical convenience, we replace the summation in (27) by integration. Thus

where

The density of state

Assume

where

Finally we can write Equation (31) as:

From (32), it clearly follows that the order parameters

We now consider the equations of motion for SDW, we can write,

Doing a lot as we did in the above, we finally get:

and

Doing a lot as we did in the previous, we finally get:

Using Equation (38) in to Equation (26), the SDW order Parameter M is given by:

or

So, finally we get:

From (41), it is again evident that the order parameters

It is, therefore, possible that in some temperature interval, SDW and superconductivity can co-exist, although one phase has a tendency to suppress the critical temperature and the order parameter of the other phase.

To study Equation (32), we consider the case, when

We can then replace

In (32) and get,

Using the integral relation,

the above equation reduces to,

from the BCS theory, the order parameter_{C} is given by

using this result in (43), we obtain

To solve (45) numerically we use Debay temperature and the interband BCS coupling constant.

To estimate α, I consider the case

From (32), we then have

Putting

Using Laplacian’s transformation with Matsuber relation result we can write,

where

Using the fact that, for low temperature,

We can write (48) as,

Using L’Hospital’s rule, it is easy to show that

which can be neglected since

Substituting (49) in (46), we then obtain

This implies,

which can be used to estimate _{1−x}K_{x}Fe_{2}As_{2}, using the experimental value

energy.

To study how M depends on the magnetic transition temperature

proceeding as before, it is easy to show that,

Neglecting

This gives;

we can use (52) to draw the phase diagram for M and

In this section we want to drive an expressions for the order parameters of SDW, M, and triplet superconductivity,

where the superconducting order parameter is given by:

We now consider the equation of motion:

Doing a lot as we did in the above for the commutation and using the assumption

The nesting property of the Fermi surface that expected for low dimensional band structure and attributed to the SDW ordering gives as an expression

Finally:

Since we are dealing with only the triplet pair; we can ignore the singlet correlation.

The equation of motion for correlation in RHS of (57) is written as:

and

which can be rewritten, after solving the commutation relation and removing the singlet pair.

From Equations (58) and (59), we will get;

With help of Equation (60) and Equation (57):

which in turn can be written as:

Applying nesting condition

Using the expression

where

and

By taking an approximation over the superconducting order parameter, such that it is independent of wave vector, finally we get:

We now consider the equations of motion for SDW, we can write,

Doing a lot as we did in the above, we finally get:

So,

Starting with a model Hamiltonian for the system and using Green’s function formalism, we obtained expressions for superconducting transition temperature (T_{C}), magnetic order parameter (M) and spin density wave transition temperature (T_{SDW}). Based on these result we found two very vital equations ((45) and (52)). Moreover, we scrutinized the effect of magnetic order parameter (M) on superconducting transition temperature (T_{C}) and spin density wave transition temperature (T_{SDW}) in Ba_{1−x}K_{x}Fe_{2}As_{2} by using the relevant parameters. For this purpose, we have used (45) which have been numerically solved using the relevant parameters to plot the phase diagram for magnetic order parameter (M) versus superconducting transition temperature (T_{C}). In the same figure, we have also plotted the phase diagram of magnetic ordering (M) versus spin density wave transition temperature (T_{SDW}), using (52). From the graph we observe T_{C} decreases with increase in M, whereas T_{SDW} increases with increase in M. The phase diagrams of M versus T_{C} and M versus T_{SDW}, were merged to obtain the region where both spin density wave and superconductivity co-exist. The regions of intersection of the two merged graphs showed in _{1−x}K_{x}Fe_{2}As_{2}.

Using a model Hamiltonian consisting of spin density wave and superconducting part and applying Green’s function formalism we have got an expression which shows the relation of the two order parameters and their variation with temperature. From _{C} decreases with increase in M, whereas T_{SDW} increases with increase in M. The spin density wave and superconducting phases, therefore, resist each other. However, the present work shows that there is a small region of temperature, where both the phases may be in existence together, which is indicated by (SC + SDW) in the figure. In the absence of spin density wave the expression for both singlet and triplet cases reduces to the well known BCS result. My study explicitly shows that

spin density wave and superconductivity truly coexist in Ba_{1−x}K_{x}Fe_{2}As_{2} (0.2 ≤ x < 0.4) in some range of magnetic order.

I thank Prof. Amarendra Rajput for providing me constructive comments, suggestions and valuable support.

HaftuBrhane, (2015) Coexistence of Spin Density Wave (SDW) and Superconductivity in Ba_{1-x}K_{x}Fe_{2}As_{2}. World Journal of Condensed Matter Physics,05,319-331. doi: 10.4236/wjcmp.2015.54032