Theoretical Investigation of the Interplay of Superconductivity and Magnetism in Ba 1 − x K x Fe 2 As 2 Superconductor in a Two-Band Model by Using the Bogoliubov Transformation Formalism

. The main focus of this article is to investigate the theoretical interplay of magnetism and superconductivity in a two-band model for the iron-based superconductor Ba 1 − x K x Fe 2 As 2 . On the basis of experimental results, the two-band model Hamiltonian was considered. We obtained mathematical statements for the superconductor Ba 1 − x K x Fe 2 As 2 superconducting (SC) transition temperature, spin-density-wave (SDW), transition temperature, superconductivity order parameter, and SDW order parameter from the Bogoliubov transformation formalism and the model Hamiltonian. Furthermore, an expression for the dependence of the SDW transition temperature on the SDW order parameter and the dependence of the SC transition temperature on the SDW order parameter was obtained for Ba 1 − x K x Fe 2 As 2 . By substituting the experimental and theoretical values of the parameters in the derived statements, phase diagrams of the SC transition temperature versus the SDW order parameter and the SDW transition temperature versus the SDW order parameters have been plotted to demonstrate the dependence of the SDW order parameter on transition temperatures. By combining the two-phase diagrams, we depicted the possible coexistence of superconductivity and magnetism for the Ba 1 − x K x Fe 2 As 2 superconductor. The phase diagrams of temperature versus SC order parameter and temperature versus SDW order parameter were also plotted to show the dependence of order parameters on temperature for the Ba 1 − x K x Fe 2 As 2 superconductor.


Introduction
Fe-based superconductors (FebSc), which greatly build up the family of unconventional superconductors, have many different systems. The major FebSc systems are 1111, 122, 111, 11, 245, 42622, 1144, and 12442 (e.g., LaFeAsO, Ba 1−x K x Fe 2 As 2 , LiFeAs, FeSe 1−x Te x , Rb 2 Fe 4 Se 5 , Sr 4 V 2 O 6 Fe 2 As 2 , RbEuFe 4 As 4 , and KCa 2-Fe 4 As 4 F 2 ) [1][2][3][4][5]. Iron-based square-planar sheets, which are essential for the unconventional superconductivity of all FebSc systems, represent their shared structural component [6]. The parent compounds of FebSc systems are antiferromagnets, spindensity-wave (SDW) metals, and they are magnetic bad metals. Increasing doping in 122 systems can destroy antiferromagnetism and lead to superconductivity [7]. Superconductivity occurs in specific planes, Fe-As plane [8]. The iron-containing plane of the superconductor, in which pnictogen or chalcogen atoms protrude above and below the plane, is not flat because the pnictogen and chalcogen atoms are much larger than iron atoms [9]. The pnictogen and chalcogen atoms are tightly grouped in a tetrahedral structure [10]. Due to the close packing of the Fe atoms, the five Fe 3d orbitals serve as charge carriers in superconductivity [11]. The sign-changing s-wave symmetry characterizes the FebSc symmetry [12].
Most of the iron-pnictides become superconductors only when doped (added an impurity) with holes or electrons. T c depends on doping concentration [13]. For the 122 ironbased superconductors family, AFe 2 An 2 , where (A indicates alkaline earth metals such as Ba, Sr or Ca and Eu) and An is a pnictide (As, P), superconductivity can be achieved with the introduction of dopants [14]. There are several ways to introduce dopants [15]. These are (1) hole doping is achieved with substituting A for monovalent B + (B ¼ Cs, K, Na) atoms partially in the blocking layer, and this substitution should add an excess hole into the system, for example, Ba 1−x K x Fe 2 As 2 [16,17] (2) partially substitute Fe for transition metals (Co, Ni, Pd, Rh) into FeAs layers and yields electrons into the system. In this method, dopants are directly doped into the Fe layer, which can stabilize the system; for example, Co (A(Fe 1−x Co x ) 2 As 2 ), Rh (A(Fe 2−x Rh x ) As 2 ), Ni (A(Fe 1−x Ni x ) 2 As 2 ) [18][19][20] and we get electrondoped pnictides that form an abundant phase diagram where superconductivity and magnetism exist together, and (3) replacing arsenic partially with phosphorus, and phosphorus generates a chemical pressure effect that suppress SDW and emerges superconductivity at the corresponding unit cell volume [21,22].
The parent compound BaFe 2 As 2 is an iron arsenide compound and crystallizes with the tetragonal ThCr 2 Si 2 type structure, space group I4/mmm [23]. Its crystal structure and SDW transition are strongly affected by hole doping with potassium [24]. The compound BaFe 2 As 2 changes structural and magnetic transitions simultaneously at T s ¼ T N ≈ 140 K [25]. Magnetic transition is strongly coupled with structural distortion. Potassium substitutions with hole doping suppress the SDW state of BaFe 2 As 2 , resulting in the befall of superconductivity [26]. The suppression of longrange magnetic ordering and the emergence of the superconducting (SC) state at the same time are correlated, suggesting that the spin fluctuation of the Fe moments is crucial in creating the superconductive state [27]. The coupling between lattice and spins results from the proximity of the structural and magnetic transitions [28]. The crystal distortion is driven by magnetisms because lower symmetry allows the spins to order and thus relieve magnetic frustrations [6]. Superconductivity, which is closely related to electron pair instability on the Fermi surface, SDW, which has been deduced from the nesting Fermi surface and observed directly in neutron diffraction experiments in the BaFe 2 As 2 (122) class of materials. The same electronic state would participate in both the SC and SDW electron-pairs instability of the Fermi surface, and indeed the hole and electron-like Fermi sheets in Ba 1−x K x Fe 2 As 2 have been observed [29].
Experiment has been revealed that in potassium substitution barium iron arsenide superconductor (Ba 1−x K x Fe 2 As 2 ), superconductivity appears for 0:1 ≤ x ≤ 1 in both the orthorhombic and tetragonal phases and for 0:1 ≤ x <0:4, the SDW and SC orders interplay at low temperature [30]. The coexistence of magnetism and superconductivity strongly emphasizes that magnetic spin fluctuations are involved in the Cooper pairing in Fe-based superconductors [24]. As the doping level (x in Ba 1−x K x Fe 2 As 2 ) increases, the electron Fermi pocket continuously shrinks, whereas the hole pocket becomes larger. SDW ordering coexisting with superconductivity at ¼ 0:2 T c ¼ ð 24 KÞ. When the doping concentration reaches 0.3, the SDW transition is completely suppressed, and T c increases to 33 K and at x ¼ 0:4, T c ¼ 38 K. The electron pocket finally vanishes beyond x ¼ 1, where superconductivity also disappears. In this article, we study the interaction (coexistence) between magnetism and superconductivity in a twoband model for the iron-based superconductor Ba 1−x K x Fe 2 As 2 by using the Bogoliubov transformation formalism.
Based on the concepts of electronic structure, this article theoretically investigates the coexistence of magnetism and superconductivity with potassium substitution (or hole doping) in the blocking layer of barium iron arsenide (Ba 1−x K x Fe 2 As 2 ) superconductor in the two-band model. By considering a two-band model of Hamiltonian and using the Bogoliubov transformation formalism methodology, It tried to find the mathematical expression for the SC critical temperature T c ð Þ, SC order parameter Δ s ð Þ, the SDW order parameter M ð Þ and SDW transition temperature (T M ).

Mathematical Formulation of the Problem
A system of Hamiltonian conduction electron interacting with phonons for the interplay of superconductivity and magnetism in our compound (Ba 1−x K x Fe 2 As 2 ) in a twoband model, band a and band b, can be expressed as follows [31][32][33][34][35]: Here, the first term is the kinetic energy. The second term is the kinetic energy of holes. The third term is the pair interaction of the SC order parameter, and the fourth term is the pair hopping between the fermion SC channel due to magnetic interactions with some coupling constant (U m kσ ). k and σ are the wave vector and the spin projection on the z-axis. V k is the interaction potential of fermions, ε s and ε b energies of free electrons measured with respect to the chemical potential. â † k↑ â k↓ and b † k↑ b k↓ are creation (annihilation) operators of the electrons and holes, respectively.
To decouple the model Hamiltonian, one uses the meanfield approximation, and thus the mean-field Hamiltonian of both SC state and SDW state, expressed as follows: where f is the Fermi-Dirac statistics, E g is the Energy at Fermi level.
Applying the properties of Fermi-Dirac statistics, the term 1− j 2f E g k ð Þ j becomes the following: The coherence factors u k and v k related as follows: Inserting Equations (13) and (14) into Equation (12), one can get the following: Substituting Equation (15) into Equation (3), the gap parameter Δ s k ð Þ becomes the following: where . Equation (16) is the SC order parameter in the s band.
Similarly, the b-band mean field value is found as follows: where f is the Fermi-Dirac statistics, E c is the energy at the Fermi level.

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Applying the properties of Fermi-Dirac statistics, the term 1− j 2f E c k ð Þ ð Þjalso becomes the following: The coherence factors x k and y k related as follows: Inserting Equations (18) and (19) into Equation (17), one gets the following: Substituting Equation (13) into Equation (4), the gap parameter Δ b k ð Þ becomes the following: where . Equation (21) is the SC order parameter in the b band. (2) can yield an SDW phase that is a pure magnetic phase for zero SC interaction. The mean-field Hamiltonian for the pure magnetic phase is as follows:

Pure SDW State. The Hamiltonian Equation
The mean-field Hamiltonian in Equation (22) can be diagonalized using the relevant Bogoliubov transformation or quasiparticle transformation, and its diagonalized form is as follows: The Bogoliubov quasi-particle matrix for our transformation is as follows: Operators in the Hamiltonian are expressed as follows: The coherency factors are given by the following: Therefore, the diagonalized Hamiltonian will have the form as follows: where E h and E e are the eigenvalues and given by an expression Using the Bogoliubov transformation, we defined the SDW state in Equation (23); we can now calculate the expectation value of the operators in Equation (22) to find the SDW gap equation.
2.3. SC Order Parameter Δ s ð Þ. There is clear information that show the coexistence of SDWs and superconductivity in both hole-and electron-doped families in most FebSc [22]. In our compound, Ba 1−x K x Fe 2 As 2 , Superconductivity can be obtained by (1) chemical doping [36][37][38] and/or (2) the application of pressure [39]. The addition of a chemical dopant and/or the application of pressure on BaFe 2 As 2 results in the eradication of the SDW and, therefore, in the reduction of magnetic order.
Superconductivity is observed when the magnetic order is sufficiently weak; thus, superconductivity and weak magnetic order can still coexist [37,38,40]. The exact mechanism for the emergence of superconductivity is, however, still unknown. Therefore, we need to study the coupled system below the SC transition temperature. The mean-field Hamiltonian without the interaction part for both the SC state and the magnetic state is stated in Equation (2). Applying the second quasi-particle transformation to Equation (2), it gives the following: Using Nambu notion where the Nambu spinors are as follows: The energy matrix Diagonalizing Equation (31) for each band gives the following: where ŷ † gk and ŷ † hk are the Bogoliubov quasi-particle matrix which is defined as the following transformations: Their relation to fermion operators is defined as follows: where U k , V k , X k and Y k are coherence factors that satisfy the following relation: and where E A and E B are the eigenvalues or excitations and are given by the following: Therefore, the diagonalized Hamiltonian will have the form as follows: Following the same procedure as for the SC state and the magnetic state, the gap equations for both the magnetic and SC states, the gap equations or order parameters are given by the following: where . Near the edge of the Fermi-surface, there is homogeneity, and the two order parameters can be taken as equal and opposite; with this assumption, we can take Δ b k ð Þ ¼ Δ s k ð Þ, Advances in Condensed Matter Physics and ϵ s ¼ ϵ b . With this in mind, the two SC gap equations of Equation (40) are simplified to the following: where For simplicity, the interaction potential V sc k ¼ V and Equation (41) is simplified to the following: where Changing summation to integration in the region − ℏω F <ϵ s k ð Þ<ℏω F and by introducing the density of state at the Fermi level of the interband interaction, N 0 ð Þ that is as follows: The density of state N 0 ð Þ is equal to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N s 0 ð ÞN b 0 ð Þ p and Equation (42), becomes the following: Let where If T À! T c; Δ s k ð Þ À! 0, and evaluating the integrals, , then Equation (56) written as follows: From Laplacian transform with the Matsuber relation result, we can write Equation (58) as follows: Apply the following equality. where From the above equality relations z ¼ β ϵ s k ð Þ 2 and dz ¼ β dϵ s k ð Þ 2 For low-temperature tanh β 2 ℏω F À Á À! 1 and γ denotes the Euler's constant and its value is given by γ ¼ 1:78.
ð Þ ¼ 1:202.so after some steps Equation (66) can be written as follows: The second integral I 2 can be calculated using L'Hopitals Rule as follows: Advances in Condensed Matter Physics After some steps, Equation (72) becomes the following: Now Equation (54) yields, Applying the same procedure h 2 can be written as follows: Substituting Equations (74) and (75) into Equation (51), it gives the following: For small M, one can ignore the M 2 term. Thus, Equation (76) reduces to the following: The expression for dependence of T c on M in the inter band interaction becomes the following: Equation (81) clearly shows the dependence of the SC transition temperature on the SDW order parameter. In the pure diamagnetism region, M ¼ 0, and Equation (81) becomes the following: This is like the well-known BCS expression for the SC transition temperature T c ð Þ. From Equation (82), the SC coupling parameter can be found as follows: After some steps, Equation (83) becomes the following: From the BCS theory, For small values of M, we ignore M 2 term. Thus, Equation (107) reduces to the following: where ψ is the SDW coupling parameter. Equation (110) shows that the SDW order parameter increases as the SDW transition temperature increases. For M ¼ 0, Equation (110) is simplified to the following: Let say From Laplacian transform with the Matsuber relation result, we can write the equation as follows: Now let us apply the following equality: where From the above equality relations z ¼ β ϵ s k ð Þ For low-temperature tanh β 2 ℏω F À Á À! 1 and γ denotes the Euler's constant and its value is given by γ ¼ 1:78. (123) can be written as follows: Therefore, it gives the following: For a very small value of M, M 2 goes to zero and T À! T M . The magnetic coupling parameter becomes as follows: Substituting Equation (127) into Equation (126), we get the following:

Conclusions
In this work, the possible coexistence of superconductivity and magnetism has been studied in the two-band model for the iron-based superconductor Ba 1−x K x Fe 2 As 2 . The superconductivity order parameter is weakened when the temperature increases and gets lost at the SC critical temperature. The SC transition temperature is weakened as the magnitude of the SDW order parameter increases. The SDW transition temperature increases with increasing SDW order parameter for Ba 1−x K x Fe 2 As 2 . This finding also proofs the dependence of the SDW order parameter on the temperature in the pure magnetic region of the iron-based superconductor Ba 1−x K x Fe 2 As 2 . The SDW order parameters is weakened when the temperature  increases and vanishes at the SDW transition temperature. We have depicted the possible interplay of superconductivity and magnetism of the Ba 1−x K x Fe 2 As 2 superconductor. The possible interplay of superconductivity and magnetism in Ba 1−x K x Fe 2 As 2 initiates further study in the field of condensed matter physics for further studies. The results obtained in this article work are in agreement with previous studies [25,30] that the coexistence implies that there is a weak exchange coupling between the itinerant SC electrons and the localized ordered spins in the Ba 1−x K x Fe 2 As 2 superconductor.

Data Availability
The data used to support the findings of this study are included in the article.

Ethical Approval
The conducted research is not related to either human or animal use. Advances in Condensed Matter Physics 13