Superconductivity was observed for the first time by a Dutch physicist Heike Kamerlingh Onnes. He found in 1911 that the electrical resistivity of Hg abruptly dropped to zero [1]. After 20 years of his discovery, in 1933, W. Meissner and his student R. Ochsenfeld found that the magnetic field is repelled by superconducting materials, called the Meissner effect [2]. This effect suggests a fundamental property of the superconducting state called perfect diamagnetism. In 1957, L. N. Cooper, J. Bardeen, and J. R. Schrieffer, the three American physicists, developed a quantum theory, called BCS theory, to elucidate the behaviour of superconducting materials at the microscopic level [3]. This theory claims that the formation of quasiparticles termed Cooper pairs by the electrons in a material leads to superconductivity [4]. Cooper pairs are bosons and can aggregate in the same low-energy fundamental state (ground state energy), which is the superconducting state. It bases on a critical assumption that there is an attractive force between electrons. Electrons are fermions and are subject to the Pauli exclusion principle [5].

In the development of condensed matter physics, the discovery of high-transition temperature T _c superconductors marks a turning point. In 1979, the discovery of superconductivity in the heavy-fermion compound, CeCu ₂ Si ₂ [6], came as a surprise because magnetic spin–spin interactions bind the superconducting charge carriers in pair and is highly unlikely by BCS theory [7]. The heavy-fermion system is the first class of unconventional superconductors. K. A. Muller and J. G. Bednorz discovered cuprates, the second class of high-temperature superconductors, in La _2−x Ba _x CuO ₄ with T _c = 35K [8]. Iron-based superconductors (FebSc), with T _c = 26K in LaOFeAs, are the other family of high-temperature materials. They were identified by Hosono et al., in 2008 [9].

FebScs have various distinct systems that significantly enlarge the class of unconventional superconducting materials. Numerous FebSc systems with various compositions and crystal structure classes are found still now. Different systems are distinguished for convenience by the stoichiometric proportions of the chemical components of their parent molecules [10, 11]. The most common FebSc systems that use this nomenclature are 245, 1144, 42622, 12442, 1111, 111, 11, and 122 (for example, Rb ₂ Fe ₄ Se ₅, CsEuFe ₄ As ₄, Sr ₄ V ₂ O ₆ Fe ₂ As ₂, RbCa ₂ Fe ₄ As ₄ F ₂, SmO _1−x F _x FeAs, NaFeAs, Fe _1+y Te _1−x Se _x , and SrFe _2−x Ni _x As ₂) [12–16]. In the 245 FebSc system, Rb ₂ Fe ₄ Se ₅, there is a unique phase separation phenomenon, unlike most other FebSc systems, and each phase separation has arrived from the competition between magnetic and superconducting ordering in the material [17, 18]. Despite having various structural variations, all FebSc systems have one thing in common: iron-based square-planar sheets [19]. FebSc systems contain iron-based layers, which are essential for their superconductivity, similar to copper oxide-based superconductors, in which oxygen—together with copper—creates the superconductivity layer [20].

According to theories, the parent materials of FebSc are semi-metallic, and the density of state close to Fermi’s surface is primarily supplied by the iron 3d electrons (orbitals) and all five of the 3d orbitals cross Fermi’s surface [21]. Just from the most experimental results and the band structure calculations, the results reveal three iron 3d orbitals (d _xy , d _yz , and d _xz ) provide the primary contribution to the density of states near the Fermi level and they ruffle weakly in the z direction [22]. Two electron pockets are situated at the margin of the Brillouin zone (BZ) point, and two hole pockets are distributed evenly throughout the resultant Fermi surface. The weights of the three orbitals, ( d x 2 − y 2 , d _yz , and d _xy ), are almost equal when doping is near optimum [23]. The two-orbital model and this idea of density of state will be used in this article’s computations.

Within the two-orbital model, the electron Fermi pocket is close to the (π, 0) point and is made up of the orbital d _yz , whereas the hole Fermi pocket around the (0, 0) point is made up of a combination of d _xz and d _yz orbitals. The inter-band spin fluctuation resulting from this nesting is actually the particle hole scattering that occurs between the d _xz and d _yz orbitals since the component of the hole Fermi pocket coupled by the nesting wave vector (π, 0) is primarily of the d _xz orbital characteristic [24]. The amount of doping affects how the electrical band structure is shaped. In electron-doped materials, such as 122 Fe-based superconductor compounds, the Fermi surface has many quasi 2D warped cylinders centred Γ point around (k = 0, 0) and M (k = π, π), as well as a potential quasi 3D pocket close to k _z = π. For electron-doped 122 systems such as SrFe _2−x Ni _x As ₂, the electron Fermi pocket expands as the doping level rises, while the hole Fermi pocket contracts until at the strongly doped level, where the hole Fermi pocket eventually disappears. The opposite is true for hole-doped systems [20].

The spin density wave (SDW) state, first proposed by Overhauser in 1962, is where the electronic spin density forms a static wave and makes it a type of anti-ferromagnetic state [25]. It happens in anisotropic low-dimensional compounds at low temperatures. The spin and SDW are coupled. It describes the spin density periodic modulation specified by the Fermi wave number [26]. There is no net magnetization over the entire volume with the density varying perpendicularly as a function of position. When delocalized or itinerant electrons, rather than localized electrons, are responsible for the spatial spin density modulation and the SDW transition takes place. Its origin can be electron–hole pairing or finite wave vector singularities of the magnetic susceptibility [27]. It is Fermi surface nesting that is responsible for SDW stabilization. Nesting of the Fermi surface and SDW are observed in SrFe ₂ As ₂ [28]. Although the SDW transition leads to the establishment of AFM order, which causes the moment of the iron atoms to become still along the longer axis in the orthorhombic phase, and the orthorhombic deformation of the crystal lattice transforms the structure of the crystal from the tetragonal to the orthorhombic phase [29].

The FeAs tetrahedral layers are present in the parent compounds of the 122 iron pnictide superconductors, which also show structural transitions from tetragonal to orthorhombic and from paramagnetic (PM) to anti-ferromagnetic (AFM) structures [30]. It has been demonstrated that the AFM orthorhombic phase is the parent phase of the pnictide superconductor that produces the SDW state, which can be suppressed to produce superconductivity by applying pressure or through chemical doping. It has been demonstrated that the parent pnictide compound structural and magnetic phase transitions are crucial in causing superconductivity [31]. The structural and magnetic transitions in our compound SrFe _2−x Ni _x As ₂ occur simultaneously at T _S = T _N ≈ 205K [25, 32, 33]. The transition can be tracked to the concentration x = 0.15, where T _S (T _N ) ≈ 40K [34], but T _S (T _N ) disappears for x > 0.15 or it is not observed in x = 0.16, which leads to maximum T _c [35]. In the concentration range 0.10 ≤ x ≤ 0.22, superconductivity is observed for SrFe _2−x Ni _x As ₂. As the doping level [x in SrFe _2−x Ni _x As ₂] increases, the hole Fermi pocket continuously shrinks, whereas the electron pocket becomes larger, and the hole pocket finally vanishes in the heavily doped region, where superconductivity also disappears. Magnetic transition is strongly coupled with the structural distortion. Superconductivity has been demonstrated that the result from the suppression of the AFM state of SrFe ₂ As ₂ by nickel substitution. The suppression of the long-range magnetic ordering and the sudden appearance of the Sc state at the same time are correlated, which suggests that the spin fluctuation of the Fe moments is crucial in creating the superconductive state [36]. The closeness of structural and magnetic transitions suggests that spin and lattice coupling has occurred. Lower symmetry enables the spins to organise and, therefore, reduce magnetic frustrations, which is assumed to be the source of the crystal deformation [37].

FebSc are so interesting because they show the coexistence of superconductivity and magnetism. Therefore, they show a lot of promise for applications. They are desirable for electrical power stations and magnetic applications (Maglev trains) because they have a substantially larger critical magnetic field than cuprates or heavy fermions and high isotropic critical currents. Superconductivity shows potential to have a significant impact on our community and the realisation of a world with minimal carbon emissions [38]. In this article, the researchers study the coexistence of superconductivity and SDW in a two-band model for the iron-based superconductor SrFe _2−x Ni _x As ₂ by using Green’s function formalism.

Based on the concepts of electronic structure, this article is investigated theoretically on the coexistence of superconductivity and SDW in nickel substitution of the strontium–iron–arsenide (SrFe _2−x Ni _x As ₂) superconductor in a two-band model. By considering a two-band model of Hamiltonian and using the double-time temperature-dependent Green’s function formalism methodology, researchers tried to discover the mathematical statements for the critical temperatures (T _c and T _M ) and order parameters (Δ _Sc and M).

We explore a two-band model that encapsulates the fundamental physics of the multi-band unconventional superconducting state and SDW order and has a self-consistent solution in order to investigate the coexistence of superconductivity and SDW. The mean-field model Hamiltonian for the interplay of SDW and superconductivity in our compound in a two-band model (d _yz , say s and d _xz , say d) can be expressed as [39–41] H ̂ = ∑ k , σ ε s k s ̂ k , σ † s ̂ k , σ + ∑ k , σ ε s k d ̂ k , σ † d ̂ k , σ − Δ S c s ∑ k s ̂ k ↑ † s ̂ − k ↓ † + s ̂ − k ↓ s ̂ k ↑ − Δ S c d ∑ k d ̂ k ↑ † d ̂ − k ↓ † + d ̂ − k ↓ d ̂ k ↑ − Δ S c s d ∑ k s ̂ k ↑ † s ̂ − k ↓ † + d ̂ − k ↓ d ̂ k ↑ − Δ S c s d ∑ k d ̂ k ↑ † d ̂ − k ↓ † + s ̂ − k ↓ s ̂ k ↑ − M ∑ k s ̂ k + p ↑ † , d ̂ k ↑ + d ̂ k + p ↑ † , s ̂ k ↑ , (1)

where the first term indicates the energy of conduction electrons in the s-band. The second term indicates the energy of conduction electrons in the d-band. The third and fourth terms are the energies involving superconductivity due to the intra-band interactions at the s- and d-bands, respectively. The fifth and sixth terms are the energies involving superconductivity due to the inter-band interaction between bands s and d. The last term is the mean-field Hamiltonian which describes the magnetic interactions. ɛ _s (k) and ɛ _d (k) are energies of an electron measured with respect to the Fermi energy in s and d bands, respectively. s ̂ k ↑ † ( s ̂ − k ↓ † ) are the creation (annihilation) operators in the s-intra-band interaction having the wave number k (−k) and spin ↑ (↓), respectively. d ̂ k ↑ † ( d ̂ − k ↓ † ) are the creation (annihilation) operators in the d-intra-band interaction having the wave number k (−k) and spin ↑ (↓), respectively. s ̂ ( k + p ) ↑ † ( d ̂ ( k + p ) ↑ † ) are the operators which create fermions with momentum (k + p) in the s(d) band interaction, respectively.

Δ S c s is the superconducting order parameter (mean field) due to the intra-band interactions within the s band and is given by Δ S c s = ∑ k U s ≪ s ̂ k ↑ † , s ̂ − k ↓ † ≫ (2) where U _s is the intra-band interaction potential in the s-band.

Δ S c d is the superconducting order parameter (mean-field) due to the intra-band interactions within the d-band and is given by Δ S c d = ∑ k U d ≪ d ̂ k ↑ † , d ̂ − k ↓ † ≫ (3) where U _d is the intra-band interaction potential in the d-band.

Δ S c s d is the superconducting order parameter (mean-field) due to inter-band interactions between the s- band and d-bands, which is given by Δ S c s d = ∑ k U s d 2 ≪ s ̂ k ↑ † , s ̂ − k ↓ † ≫ + ≪ d ̂ k ↑ † , d ̂ − k ↓ † ≫ (4) where U _sd is the inter-band interaction potential between the two bands.

The double-time temperature-dependent Green’s function is used to find the equation of motion. This formalism is defined as R r t − t ′ = ≪ D ̂ t , E ̂ t ′ ≫ = − i θ t − t ′ D ̂ t , E ̂ t ′ . (5)

R _r (t − t′) is boson operators’ retarded response function. … and ≪…≫ denote the thermodynamic average and abbreviated notation for the Green’s function in the system, respectively. D ̂ ( t ) and E ̂ ( t ′ ) are the Heisenberg notations of the field operators. They are expressed in terms of particle creation and annihilation operators or the quantized field function product. θ(t − t′) represents the Heaviside step function and defined as θ(t − t′) = 1 if t > t′ and 0 if t < t′. D ̂ ( t ) can be written as D ̂ ( t ) = e i H ̂ t D ( o ) e − i H ̂ t . D ̂ ( t ) , E ̂ ( t ′ ) is the commutator or anti-commutator. This is described as D ̂ ( t ) , E ̂ ( t ′ ) = D ̂ ( t ) E ̂ ( t ′ ) − ς E ̂ ( t ′ ) D ̂ ( t ) , where ς = 1 for bosons and ς = −1 for fermions. To obtain the equations of motion, we differentiate Eq. 5 with time (t) as

d d t R r ( t − t ′ ) = − d d t i θ ( t − t ′ ) D ̂ ( t ) , E ̂ ( t ′ )

d d t R r ( t − t ′ ) = − i d d t θ ( t − t ′ ) D ̂ ( t ) , E ̂ ( t ′ ) − i θ ( t − t ′ ) d d t D ̂ ( t ) , E ̂ ( t ′ ) , and this is simplified to i R ̇ r t − t ′ = d d t θ t − t ′ D ̂ t , E ̂ t ′ + θ t − t ′ d d t D ̂ t , E ̂ t ′ . (6) The Dirac’s delta function δ(t − t′) is related to the Heaviside step function as θ t − t ′ = ∫ − ∞ t δ t − t ′ . (7) This implies the following: d d t θ t − t ′ = δ t − t ′ . (8) Now let us introduce the Heisenberg equations of motion as d d t D ̂ t = D ̂ t , H ̂ . (9) Substituting Eq. 8 and Eq. 9 into Eq. 6, we have the following: i R ̇ r t − t ′ = δ t − t ′ D ̂ t , E ̂ t ′ + ≪ D ̂ t , H ̂ , E ̂ t ′ ≫ . (10) Let R _r (ω) be the Fourier transform of R _r (t − t′), which will be given by R r t − t ′ = ∫ − ∞ ∞ R r ω e − i ω t − t ′ d ω . (11) The inverse Fourier transform is R r ω = 1 2 π ∫ − ∞ ∞ R r t − t ′ e − i ω t − t ′ d t − t ′ . (12) The first-order derivative of Eq. 11 with time will be i R ̇ r t − t ′ = − i ω ∫ − ∞ ∞ R r ω e − i ω t − t ′ d ω . (13) The Dirac delta function will be defined as δ t − t ′ = 1 2 π ∫ − ∞ ∞ e − i ω t − t ′ d ω . (14) Substituting Eqs 13, 14 into Eq. 10, we have ω R r ω = D ̂ t , E ̂ t ′ + ≪ D ̂ t , H ̂ , E ̂ t ′ ≫ . (15) Last but not the least, the formalism for the double-time temperature-dependent Green’s function is ω ≪ D ̂ t , E ̂ t ′ ≫ = D ̂ t , E ̂ t ′ + ≪ D ̂ t , H ̂ , E ̂ t ′ ≫ . (16) Here, ≪ D ̂ ( t ) , E ̂ ( t ′ ) ≫ denotes the Fourier transform of Green’s function involving the operators D ̂ ( t ) and E ̂ ( t ′ ) . We will apply the anti-commutation relation s ̂ k σ † , s ̂ k σ ′ † = s ̂ k σ , s ̂ k σ ′ = 0 and s ̂ k σ † , s ̂ k σ ′ = δ k k ′ δ σ σ ′ to solve the equation of motion, where δ k k ′ = 1 , i f k = k ′ 0 , otherwise and δ σ σ ′ = 1 , σ = σ ′ 0 , otherwise

For simplification, Eq. 1 can be written as H ̂ = H ̂ s + H ̂ d + H ̂ s d + H ̂ M , (17) where H ̂ s = ∑ k , σ ε s k s ̂ k , σ † s ̂ k , σ − Δ S c s ∑ k s ̂ k ↑ † s ̂ − k ↓ † + s ̂ − k ↓ s ̂ k ↑ . This is s intra-band interaction’s mean-field Hamiltonian. H ̂ d = ∑ k , σ ε s k d ̂ k , σ † d ̂ k , σ − Δ S c d ∑ k d ̂ k ↑ † d ̂ − k ↓ † + d ̂ − k ↓ d ̂ k ↑ . This is d intra-band interaction’s mean-field Hamiltonian. H ̂ s d = − Δ S c s d ∑ k s ̂ k ↑ † s ̂ − k ↓ † + d ̂ − k ↓ d ̂ k ↑ − Δ S c s d ∑ k d ̂ k ↑ † d ̂ − k ↓ † + s ̂ − k ↓ s ̂ k ↑ . This is the mean-field Hamiltonian in the inter-band interaction H ̂ M = − M ∑ k s ̂ k + p ↑ † , d ̂ k ↑ + d ̂ k + p ↑ † , s ̂ k ↑ . This is the mean-field Hamiltonian due to the magnetic interaction of conduction electrons.