Theoretical Analyses of Superconductivity in Iron Based Superconductor Ba1−xKxFe2As2

This paper focuses on the theoretical analysis of superconductivity in iron based superconductor Bax-1KxFe2As2. After reviewing the current findings on this system, we suggest that phononexciton combined mechanism gives a right order of superconducting transition temperature (TC) for 2 2 1 As Fe K Ba x x  . By developing a model Hamiltonian for the system under consideration, using double time temperature dependent Green’s function formalism and a suitable decoupling approximation technique, we have analyzed theoretically the superconductivity of iron-based superconductor Ba1−xKxFe2As2. Furthermore, by using the experimental and plausible theoretical values of the parameters in the obtained expressions, phase diagrams of superconducting transition temperature (TC) versus electron coupling constant ( ) e  and superconducting transition temperature (TC) versus superconducting gap parameter ) ( have been plotted. Our findings demonstrate that, as the electron coupling constant ( ) e  increases the superconducting transition temperature (TC) increases and vice versa and also as the temperature increases the superconducting order parameter decreases and vanishes at the transition temperature (TC) for Ba1−xKxFe2As2. Our findings are in a broad agreement with the experimental observations.

. By developing a model Hamiltonian for the system under consideration, using double time temperature dependent Green's function formalism and a suitable decoupling approximation technique, we have analyzed theoretically the superconductivity of iron-based superconductor 1− 2 2 . Furthermore, by using the experimental and plausible theoretical values of the parameters in the obtained expressions, phase diagrams of superconducting transition temperature (TC) versus electron coupling constant ( ) e  and superconducting transition temperature (TC) versus superconducting gap parameter ) ( have been plotted. Our findings demonstrate that, as the electron coupling constant ( ) e  increases the superconducting transition temperature (TC) increases and vice versa and also as the temperature increases the superconducting order parameter decreases and vanishes at the transition temperature (TC) for 1−

INTRODUCTION
Superconductivity is a remarkable macroscopic quantum phenomenon, which was discovered by Onnes (1911) while investigating the electrical resistivity of mercury at low temperature. Onnes (1911) found that, when mercury was cooled below TC =4.2K, the resistivity abruptly dropped to a value close to zero and the material became a superconductor. The discovery of the phenomenon of zero electric resistivity in mercury was soon followed by the observation of the superconducting state in other materials. The microscopic understanding of the phenomenon, however, was not given due attention until the formulation of the superconductivity theory by Bardeen, Cooper and Schrieffer in 1957(BCS) who provided a microscopic theoretical analysis suitable for describing the observed superconducting state (Bardeen et al., 1957). As is well known, the core point of the BCS theory is the formation of Cooper pairs. Superconducting materials are characterized by the presence of mainly two characteristic effects, the effect of zero electrical resistivity and the diamagnetism or the Meissner effect, which is the complete expulsion of magnetic field from the interior of a given superconducting material for temperatures below the critical temperature (TC). Furthermore, there was a great excitement after the discovery of high temperature superconductor cuprate by Bednorz and Müller (1986).
The recent discovery of superconductivity in iron-pnictides with transition temperature, TC = 26K (Kamihara et al., 2008) has generated enormous interest in the study of these superconductors. This discovery has provided an excellent opportunity to understand the behavior of unconventional or low temperature superconductors. After the discovery of iron based superconductors by Kamihara et al. (2008) different compounds were discovered by replacing lanthanum by magnetic rare earth elements such as Ce, Sm, Nd or Pr and the critical temperature could be easily raised up to 56K (Wang et al., 2008a). In iron based superconductors (IBSC), superconductivity usually occurs by doping either electrons or holes in the FeAs layers. The iron, Fe 2+ forms tetrahedron within the layers. This means that, ironpnictide Fermi level is formed by 3dxy, 3dyz, or 3dzx orbitals. Very intriguing property of the IBSC is rather high flexibility concerning elemental substitution, leading to the formation of several families of iron based superconductors. Based on their chemical composition, iron based superconductors are categorized as 1111 family (Kamihara et al., 2008), 122 family (Rotter et al., 2008), 111 family (Wang et al., 2008b) and 11 family (Hsu et al., 2008). These newly discovered iron based layered pnictides have triggered challenges towards understanding of their pairing mechanisms. Superconductivity in iron-pnictides appears by doping the parent compounds with charge carriers. Furthermore, there is a resistivity anomaly in the parent compounds of LaFeAsO at around 150K which disappears as superconductivity emerges in addition to a small anomaly in the dc magnetic susceptibility. Optical conductivity and theoretical calculations suggest that, LaFeAsO exhibits a spin density wave (SDW) instability, that is, suppressed by doping with electrons to induce superconductivity (Kamihara et al., 2008).

MODEL HAMILTONIAN
In order to obtain an expression for superconducting transition temperature (TC) and superconducting order parameter (Δ) from exclusively phonon-exciton combined mechanism, we calculated TC, reduced gap ( ) 2 C B T k  and compared the values with the experimental one for the Iron based superconductor Ba 1−x K x Fe 2 As 2 .
The Hamiltonian of a system of conduction electrons interacting with phonons (phonon-exciton) can be described by the two level systems as follows (Singh and Sinha, 1990); and (Allender et al., 1973).
Ĥ ph is the standard phonon Hamiltonian, which can be written as, Ĥ el−ph is the electron-phonon interaction Hamiltonian and is expressed as, and is the screened Coulomb interaction between conduction electrons.

Temperature dependent superconducting energy gap
To obtain the temperature dependent superconducting energy gap, we use the same techniques as above to solve the integral, given by, After a couple of steps we obtain, C T = ln 1.14 ℏ − ℎ − ∆ 2 ( 1 ) 2 (1.05) + ⋯ But from the BCS model, Thus, equation (25) can be simplified and obtain, and rearranging, we obtain, Equation (26) demonstrates how the superconducting order parameter, ∆(T) varies with temperature and is analogous to the BCS model.

RESULTS AND DISCUSSION
In this section, we analyzed the results which are obtained using the model Hamiltonian developed. We obtained the expressions for the superconducting order parameter (∆) and superconducting transition temperature (TC). Now, by using equation (24)

CONCLUSION
In this investigation, by considering the phonon-exciton combined mechanism and by using the experimental and plausible theoretical values of the parameters in the obtained expressions, we plotted phase diagrams of superconducting transition temperature (TC) versus electron coupling