Coexistence of Spin Density Wave ( SDW ) and Superconductivity in Ba 1 − xKxFe 2 As 2

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 Ba1−xKxFe2As2 (0.2 ≤ x < 0.4).


Introduction
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 [1] [2] below 5 K, shortly after this discovery the breakthrough was a T C of 26 K in the F-doped arsenide LaO 1−x F x FeAs system [3].
In addition to this several groups reported an increase of T C values by replacing La with smaller-size rareearth ions like CeO 1−x F x FeAs [4], and samarium-arsenide oxides Sm(O 1−x F x )FeAs with a critical temperature T C of 55 K [5] [6].The iron based superconductors promise interesting applications.While the interplay of superconductivity and magnetism, as well as their mechanisms remains the issues of active debates and studies, one thing in FeSC riddle is clear that it is the complex multi-band electronic structure of these compounds that determine their rich and puzzling properties.What is important and captivating is that this complexity seems to play a positive role in the struggle for understanding the FeSC physics and also for search of the materials with higher T C [7].
The FeSC is quite promising for applications.Having much higher Hc than cuprates and high isotropic critical currents [8] they are attractive for electrical power and magnet applications, while the coexistence of magnetism and superconductivity makes them interesting for spintronics [9].All the compounds share similar electronic band structure in which the electronic states at the Fermi level are occupied predominantly by the Fe 3d electrons [7].
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 [10].
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 [10].
Triplet superconductivity appears provided that we have coexistence of singlet superconductivity and SDW.In many high sc T superconductors, superconducting mechanism is attributed to strong coulomb interactions of the electrons in the system, which can also be the cause for the appearance of SDW state.This suggests the existence of competition between the two states [11].The properties of the unconventional triplet superconductivity and SDW with an emphasis on the analysis of their order parameter were reviewed.
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) [12].This is because single-phase samples and also large single crystals of the ternary compounds are much easier to obtain.Partial replacement of barium for potassium (hole doping) induced superconductivity at 38 K in (Ba 0.6 K 0.4 )Fe 2 As 2 , [13].
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.

Model Hamiltonian of the System
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 ( ) † ˆp p a a σ σ are the creation (annihilation) operators of an electron having the wave number p and spin σ .
Whereas ( SC ∆ ) superconducting order parameter and (M) SDW order parameters.The Hamiltonian in (1) will be used to determine the equations of motion in terms of the Green function.

Coupling of SDW and Superconducting Order Parameters
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 [14]: where Â and B are Heisenberg operators and ( ) is the Heaviside step function.Now, using Dirac delta function and Heisenberg operators, we can write as: The Fourier transformation Taking the Fourier transform we get: where the anti-commutation relation, has been used.To derive an expression for † † ˆ, a a  , using (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: The equation of motion for the correlation † † , ˆq a a 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 † , ˆq a a 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 17) and (14)  becomes:  19) and (20) respectively becomes: Finally we can express: Using the expression is effective order parameter and the Matsubara's frequency, we can write Equation (23) as: To take into account the temperature dependence of order parameters, we shall write as H. Brhane and ( ) Plugging Equation (28) and Equation (29) in Equation ( 27), we get: For mathematical convenience, we replace the summation in (27) by integration.Thus ( ) is the density of states at the Fermi level.The density of state ( ) ( ) ( ) Finally we can write Equation (31) as: From (32), it clearly follows that the order parameters SC ∆ and M, for superconductivity and SDW are interdependent.
We now consider the equations of motion for SDW, we can write, † † ˆˆˆ, , , .
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: So, finally we get: 41), it is again evident that the order parameters SC ∆ and M, for superconductivity and SDW are interdependent, as was the case from (32).
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.

Dependence of the Magnetic Order Parameter on the Transition Temperature for Spin Density Wave and Superconductivity
To study Equation (32), we consider the case, when 0 K, .T β → → ∞ We can then replace ( ) ( ) In (32) and get, ( ) ( ) the above equation reduces to, from the BCS theory, the order parameter SC ∆ , at T = 0 for a given superconductor with transition temperature T C is given by To solve (45) numerically we use Debay temperature and the interband BCS coupling constant.
To estimate α, I consider the case c T T → which implies, SC 0. ∆ → From (32), we then have and for SC 0 ∆ = we can write ( ) ( ) Using Laplacian's transformation with Matsuber relation result we can write, ( ) ( ) ( ) Using the fact that, for low temperature, tanh


, where γ is the Euler constant having the value 1.78 γ = (Hsian) [15] and the last equation can be neglected since 2 M is very small.

H. Brhane
We can write (48) as, Using L'Hospital's rule, it is easy to show that ( ) ( ) is very small.Substituting (49) in (46), we then obtain which can be used to estimate 1 exp for Ba 1−x K x Fe 2 As 2 , using the experimental value C T and cut-off energy.
To study how M depends on the magnetic transition temperature SDW T , we consider ( we can use (52) to draw the phase diagram for M and SDW T .

Pairing of Spin Density Wave (SDW) and Triplet Superconductivity
In this section we want to drive an expressions for the order parameters of SDW, M, and triplet superconductivity, SC ∆ , as a function of both of them and temperature, and to compare the variation of each with temperature.Still we can use the Hamiltonian given by Equation ( 1), but in this case the superconducting order parameter depends on spin alignment [16] The nesting property of the Fermi surface that expected for low dimensional band structure and attributed to the SDW ordering gives as an expression 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: 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: ( ) where 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, † † † , ˆˆˆˆ., , Doing a lot as we did in the above, we finally get: † 2 2 ˆ, .

Results
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 Figure 1 indicate co-existence of spin density wave and superconductivity for Ba 1−x K x Fe 2 As 2 .

Conclusion
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 Figure 1 we observe that T 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.

Figure 1 .
Figure 1.Co-existence of spin density wave (SDW) and superconductivity in Ba 1−x K x Fe 2 As 2 .
and they can be expressed as;