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ORIGINAL RESEARCH article

Front. Phys., 07 December 2023
Sec. Condensed Matter Physics

Theoretical study of the interplay of spin density wave and superconductivity in nickel substitution of the strontium–iron–arsenide (SrFe2−xNixAs2) superconductor in a two-band model

Gedefaw Mebratie Bogale
&#x;Gedefaw Mebratie Bogale1*Dagne Atnafu Shiferaw&#x;Dagne Atnafu Shiferaw2
  • 1Department of Physics, Mekdela Amba University, Dessie, Ethiopia
  • 2Department of Physics, Dilla University, Dilla, Ethiopia

The main objective of this manuscript is to focus on the computational study of the interplay of spin density wave (SDW) and superconductivity using a two-band model for SrFe2−xNixAs2. We derived mathematical statements for the superconducting critical temperature, SDW critical temperature, superconductivity order parameter, and the SDW order parameter using the Hamiltonian model and Green’s function formalism for the SrFe2−xNixAs2 superconductor. A mathematical expression for the dependence of transition temperatures on the SDW order parameter was obtained for SrFe2−xNixAs2. Using these mathematical statements, transition temperatures versus the SDW order parameter phase diagrams were plotted to show the dependence of the SDW order parameter on transition temperatures. By merging these diagrams, we have depicted the intriguing possibility of the interplay of superconductivity and magnetism for the SrFe2−xNixAs2 superconductor. Phase diagrams of temperature versus superconducting order parameters and the SDW order parameter were also plotted to show the dependence of order parameters on temperature for the SrFe2−xNixAs2 superconductor.

1 Introduction

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 Tc superconductors marks a turning point. In 1979, the discovery of superconductivity in the heavy-fermion compound, CeCu2Si2 [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 La2−xBaxCuO4 with Tc = 35K [8]. Iron-based superconductors (FebSc), with Tc = 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, Rb2Fe4Se5, CsEuFe4As4, Sr4V2O6Fe2As2, RbCa2Fe4As4F2, SmO1−xFxFeAs, NaFeAs, Fe1+yTe1−xSex, and SrFe2−xNixAs2) [1216]. In the 245 FebSc system, Rb2Fe4Se5, 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 (dxy, dyz, and dxz) 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, (dx2y2, dyz, and dxy), 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 dyz, whereas the hole Fermi pocket around the (0, 0) point is made up of a combination of dxz and dyz orbitals. The inter-band spin fluctuation resulting from this nesting is actually the particle hole scattering that occurs between the dxz and dyz orbitals since the component of the hole Fermi pocket coupled by the nesting wave vector (π, 0) is primarily of the dxz 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 kz = π. For electron-doped 122 systems such as SrFe2−xNixAs2, 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 SrFe2As2 [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 SrFe2−xNixAs2 occur simultaneously at TS = TN ≈ 205K [25, 32, 33]. The transition can be tracked to the concentration x = 0.15, where TS (TN) ≈ 40K [34], but TS (TN) disappears for x > 0.15 or it is not observed in x = 0.16, which leads to maximum Tc [35]. In the concentration range 0.10 ≤ x ≤ 0.22, superconductivity is observed for SrFe2−xNixAs2. As the doping level [x in SrFe2−xNixAs2] 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 SrFe2As2 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 SrFe2−xNixAs2 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 (SrFe2−xNixAs2) 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 (Tc and TM) and order parameters (ΔSc and M).

2 Mathematical formulation of the problem

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 (dyz, say s and dxz, say d) can be expressed as [3941]

Ĥ=k,σεskŝk,σŝk,σ+k,σεskd̂k,σd̂k,σΔScskŝkŝk+ŝkŝkΔScdkd̂kd̂k+d̂kd̂kΔScsdkŝkŝk+d̂kd̂kΔScsdkd̂kd̂k+ŝkŝkMkŝk+p,d̂k+d̂k+p,ŝk,(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. ɛs(k) and ɛd(k) are energies of an electron measured with respect to the Fermi energy in s and d bands, respectively. ŝk(ŝk) are the creation (annihilation) operators in the s-intra-band interaction having the wave number k (−k) and spin (), respectively. d̂k(d̂k)are the creation (annihilation) operators in the d-intra-band interaction having the wave number k (−k) and spin (), respectively. ŝ(k+p)(d̂(k+p)) are the operators which create fermions with momentum (k + p) in the s(d) band interaction, respectively.

ΔScs is the superconducting order parameter (mean field) due to the intra-band interactions within the s band and is given by

ΔScs=kUsŝk,ŝk(2)

where Us is the intra-band interaction potential in the s-band.

ΔScd is the superconducting order parameter (mean-field) due to the intra-band interactions within the d-band and is given by

ΔScd=kUdd̂k,d̂k(3)

where Ud is the intra-band interaction potential in the d-band.

ΔScsd is the superconducting order parameter (mean-field) due to inter-band interactions between the s- band and d-bands, which is given by

ΔScsd=kUsd2ŝk,ŝk+d̂k,d̂k(4)

where Usd 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

Rrtt=D̂t,Êt=iθttD̂t,Êt.(5)

Rr (tt′) is boson operators’ retarded response function. and ≪…≫ denote the thermodynamic average and abbreviated notation for the Green’s function in the system, respectively. D̂(t) and Ê(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(tt′) represents the Heaviside step function and defined as θ(tt′) = 1 if t > t′ and 0 if t < t′. D̂(t) can be written as D̂(t)=eiĤtD(o)eiĤt. D̂(t),Ê(t) is the commutator or anti-commutator. This is described as D̂(t),Ê(t)=D̂(t)Ê(t)ςÊ(t)D̂(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

ddtRr(tt)=ddtiθ(tt)D̂(t),Ê(t)

ddtRr(tt)=iddtθ(tt)D̂(t),Ê(t)iθ(tt)ddtD̂(t),Ê(t), and this is simplified to

iṘrtt=ddtθttD̂t,Êt+θttddtD̂t,Êt.(6)

The Dirac’s delta function δ(tt′) is related to the Heaviside step function as

θtt=tδtt.(7)

This implies the following:

ddtθtt=δtt.(8)

Now let us introduce the Heisenberg equations of motion as

ddtD̂t=D̂t,Ĥ.(9)

Substituting Eq. 8 and Eq. 9 into Eq. 6, we have the following:

iṘrtt=δttD̂t,Êt+D̂t,Ĥ,Êt.(10)

Let Rr(ω) be the Fourier transform of Rr (tt′), which will be given by

Rrtt=Rrωeiωttdω.(11)

The inverse Fourier transform is

Rrω=12πRrtteiωttdtt.(12)

The first-order derivative of Eq. 11 with time will be

iṘrtt=iωRrωeiωttdω.(13)

The Dirac delta function will be defined as

δtt=12πeiωttdω.(14)

Substituting Eqs 13, 14 into Eq. 10, we have

ωRrω=D̂t,Êt+D̂t,Ĥ,Êt.(15)

Last but not the least, the formalism for the double-time temperature-dependent Green’s function is

ωD̂t,Êt=D̂t,Êt+D̂t,Ĥ,Êt.(16)

Here, D̂(t),Ê(t) denotes the Fourier transform of Green’s function involving the operators D̂(t) and Ê(t). We will apply the anti-commutation relation ŝkσ,ŝkσ=ŝkσ,ŝkσ=0 and ŝkσ,ŝkσ=δkkδσσ to solve the equation of motion, where δkk=1,ifk=k0,otherwise and δσσ=1,σ=σ0,otherwise

For simplification, Eq. 1 can be written as

Ĥ=Ĥs+Ĥd+Ĥsd+ĤM,(17)

where

Ĥs=k,σεskŝk,σŝk,σΔScskŝkŝk+ŝkŝk.

This is s intra-band interaction’s mean-field Hamiltonian.

Ĥd=k,σεskd̂k,σd̂k,σΔScdkd̂kd̂k+d̂kd̂k.

This is d intra-band interaction’s mean-field Hamiltonian.

Ĥsd=ΔScsdkŝkŝk+d̂kd̂kΔScsdkd̂kd̂k+ŝkŝk.

This is the mean-field Hamiltonian in the inter-band interaction

ĤM=Mkŝk+p,d̂k+d̂k+p,ŝk.

This is the mean-field Hamiltonian due to the magnetic interaction of conduction electrons.

2.1 Superconducting order parameters in the pure superconducting region

2.1.1 ΔScs due to intra-band interactions within the s-band

To describe the superconducting order parameter due to the intra-band interaction within the s-band in the pure superconducting region, one can use the equation of motion for the correlation ŝk,ŝk, and the following equation can be written:

ωŝk,ŝk=ŝk,ŝk+ŝk,Ĥ,ŝk,(18)
ωŝk,ŝk=0+ŝk,Ĥs+Ĥd+Ĥsd+ĤM,ŝk.(19)

The commutation relations in Eq. 19 can be solved as follows:ŝk,Ĥs=ŝk,k,σεs(k)ŝk,σŝk,σΔScsk(ŝkŝk+ŝkŝk)ŝk,Ĥs=k,σεs(k)ŝk,ŝk,σŝk,σΔScsk(ŝk,ŝkŝk+ŝk,ŝkŝk)ŝk,Ĥs=k,σεs(k)(ŝk,ŝk,σŝk,σŝk,σŝk,ŝk,σ)ΔScskŝk,ŝkŝkŝkŝk,ŝk+ŝk,ŝkŝkŝkŝk,ŝk

ŝk,Ĥs=εskŝk+ΔScsŝk.(20)

ŝk,Ĥd=ŝk,k,σεs(k)d̂k,σd̂k,σΔScdkd̂kd̂k+d̂kd̂kŝk,Ĥd=k,σεs(k)(ŝk,d̂k,σd̂k,σ)ΔScdk(ŝk,d̂kd̂k+ŝk,d̂kd̂k)ŝk,Ĥd=k,σεs(k)(ŝk,d̂k,σd̂k,σd̂k,σŝk,d̂k,σ)ΔScdk(ŝk,d̂kd̂kd̂kŝk,d̂k+ŝk,d̂kd̂kd̂kŝk,d̂k)

ŝk,Ĥd=0.(21)

ŝk,Ĥsd=ŝk,ΔScsdk(ŝkŝk+d̂kd̂k)ΔScsdk(d̂kd̂k+ŝkŝk)ŝk,Ĥsd=ΔScsdk(ŝk,d̂kd̂k+ŝk,d̂kd̂k)ΔScsdk(ŝk,d̂kd̂k+ŝk,ŝkŝk)ŝk,Ĥsd=ΔScsdk(ŝk,d̂kd̂kd̂kŝk,d̂k+ŝk,d̂kd̂kd̂kŝk,d̂k)ΔScsdk(ŝk,d̂kd̂kd̂kŝk,d̂k)+ŝk,d̂kd̂kd̂kŝk,d̂k

ŝk,Ĥsd=ΔScsdŝk.(22)

ŝk,ĤM=ŝk,Mkŝ(k+p)d̂k+d̂(k+p)ŝkŝk,ĤM=Mk(ŝk,ŝ(k+p)d̂k+ŝk,d̂(k+p)ŝk)ŝk,ĤM=Mk(ŝk,ŝ(k+p)d̂kŝ(k+p)ŝk,d̂k+ŝk,d̂(k+p)ŝkd̂(k+p)ŝk,ŝk)

ŝk,ĤM=Md̂k+p.(23)

Ignoring Eq. 23 and inserting Eq. 20 to Eq. 22 into Eq. 19, we get

ωŝk,ŝk=εskŝk+ΔScsŝk+ΔScsdŝk,ŝk,(24)
ωŝk,ŝk=εskŝk,ŝk+ΔScsŝk,ŝk+ΔScsdŝk,ŝk,(25)
ω+εskŝk,ŝk=ΔScs+ΔScsdŝk,ŝk,(26)
ŝk,ŝk=ΔScs+ΔScsdω+εskŝk,ŝk.(27)

The equation of motion for the homologous ŝk,ŝk in Eq. 27 is described as

ωŝk,ŝk=ŝk,ŝk+ŝk,Ĥ,ŝk,(28)
ωŝk,ŝk=1+ŝk,Ĥs+Ĥd+Ĥsd,ŝk.(29)

Evaluating the commutation relations in Eq. 29 which yields,ŝk,Ĥs=ŝk,k,σεs(k)ŝk,σŝk,σΔScsk(ŝkŝk+ŝkŝk)ŝk,Ĥs=k,σεs(k)ŝk,ŝk,σŝk,σΔScsk(ŝk,ŝkŝk+ŝk,ŝkŝk)ŝk,Ĥs=k,σεs(k)(ŝk,ŝk,σŝk,σŝk,σŝk,ŝk,σ)ΔScsk(ŝk,ŝkŝkŝkŝk,ŝk+ŝk,ŝkŝkŝkŝk,ŝk)

ŝk,Ĥs=εskŝk+ΔScsŝk.(30)

ŝk,Ĥd=ŝk,k,σεs(k)d̂k,σd̂k,σΔScdkd̂kd̂k+d̂kd̂kŝk,,Ĥd=k,σεs(k)(ŝk,d̂k,σd̂k,σ)ΔScdk(ŝk,d̂kd̂k+ŝk,d̂kd̂k)ŝk,Ĥd=k,σεs(k)(ŝk,d̂k,σd̂k,σd̂k,σŝk,d̂k,σ)ΔScdk(ŝk,d̂kd̂kd̂kŝk,d̂k+ŝk,d̂kd̂kd̂kŝk,d̂k)

ŝk,Ĥd=0.(31)

ŝk,Ĥsd=ŝk,ΔScsdk(ŝkŝk+d̂kd̂k)ΔScsdk(d̂kd̂k+ŝkŝk)ŝk,Ĥsd=ΔScsdk(ŝk,d̂kd̂k+ŝk,d̂kd̂k)ΔScsdkŝk,d̂kd̂k+ŝk,ŝkŝkŝk,Ĥsd=ΔScsdk(ŝk,d̂kd̂kd̂kŝk,d̂k+ŝk,d̂kd̂kd̂kŝk,d̂k)ΔScsdk(ŝk,d̂kd̂kd̂kŝk,d̂k)+ŝk,d̂kd̂kd̂kŝk,d̂k

ŝk,Ĥsd=ΔScsdŝk.(32)

Inserting Eq. 30 to Eq. 32 in Eq. 29, we get

ωŝk,ŝk=1+εskŝk+ΔScsŝk+ΔScsdŝk,ŝk,(33)
ωŝk,ŝk=1+εskŝk,ŝk+ΔScs+ΔScsdŝk,ŝk,(34)
ωεskŝk,ŝk=1+ΔScs+ΔScsdŝk,ŝk,(35)
ŝk,ŝk=1ωεsk+ΔScs+ΔScsdωεskŝk,ŝk.(36)

Substituting Eq. 36 in Eq. 27, we get

ŝk,ŝk=ΔScs+ΔScsdω+εsk1ωεsk+ΔScs+ΔScsdωεskŝk,ŝk,(37)
ŝk,ŝk=ΔScs+ΔScsdω+εskωεsk+ΔScs+ΔScsdω+εskΔScs+ΔScsdωεskŝk,ŝk,(38)
ŝk,ŝk=ΔScs+ΔScsdω+εskωεsk+ΔScs+ΔScsd2ω+εskωεskŝk,ŝk,(39)
1ΔScs+ΔScsd2ω+εskωεskŝk,ŝk=ΔScs+ΔScsdω+εskωεsk,(40)
ω+εskωεskΔScs+ΔScsd2ω+εskωεskŝk,ŝk=ΔScs+ΔScsdω+εskωεsk,(41)
ŝk,ŝk=ΔScs+ΔScsdω+εskωεskΔScs+ΔScsd2.(42)

By ignoring ΔScsd, Eq. 42 reduces to the following.

ŝk,ŝk=ΔScsω+εskωεskΔScs2.(43)

By decoupling Eq. 43 using the partial fraction decomposition method, we have

ŝk,ŝk=ΔScsω+εskωεskΔScs2,(44)
ŝk,ŝk=ΔScsω2εs2kΔScs2.(45)

Using the expression ωn, where ωn is the Matsubara frequency and written as

ωn=2n+1πβ,(46)
ŝk,ŝk=ΔScsωn2εs2kΔScs2,(47)
ŝk,ŝk=ΔScsωn2+εs2k+ΔScs2,(48)
ŝk,ŝk=ΔScs2n+1πβ2+εs2k+ΔScs2,(49)
ŝk,ŝk=β2ΔScs2n+1π2+β2εs2k+ΔScs2.(50)

ΔScs is written as

ΔScs=Usβk,nŝk,ŝk.(51)

Substituting Eq. 50 in Eq. 51, we have

ΔScs=Usk,nβΔScs2n+1π2+β2εs2k+ΔScs2.(52)

Let ξ=βεs2(k)+ΔScs2 and 1((2n+1)π)2+ξ2=tanh(ξ2)2ξ, then Eq. 52 can be written as

ΔScs=Us2k,nΔScs2βtanhξ2ξ.(53)

Let us change summation into integration in the region −ℏωF < ɛs(k) < ℏωF, and at the Fermi level, the density of state, Ns(o), is

kωFωFNsodεsk,(54)
ΔScs=Us2ωFωFNsoΔScs2βtanhξ2ξdεsk,(55)
1UsNso=0ωF2βtanhξ2ξdεsk.(56)

Let α = UsNs(o), which is called the superconducting coupling constant in the s-band intra-band interaction.

1α=0ωFtanhβ2εs2k+ΔScs12εs2k+ΔScs12dεsk.(57)

Case I: If T → 0, β this implies that tanhβ2εs2(k)+ΔScs121

1α=0ωF1εs2k+ΔScs12dεsk.(58)

By applying the integral relation 1x+y2dx=lnx+x+y2, Eq. 58 gives

1α=lnωFΔScs+ωFΔScs2+112,(59)
1αln2ωFΔScs,(60)
ΔScs=2ωFexp1α.(61)

From the concept of BCS theory, the superconducting order parameterΔScs at T = 0 for a given superconductor with critical temperature Tc is written as 2ΔScs(o)=3.53KBTc.

2ΔScso=3.53KBTc=4ωFexp1α,(62)

This gives

Tc=1.14ωFkBexp1UsNso,(63)

where Us = 0.323meV, Ns(o) = 1.5 (meV)−1, kB = 0.086 meV/K and ℏωF = 6 meV [29]. Substituting these experimental values, we get Tc = 10.09K that agrees with the experiment.

Case 2: If 0 < T < Tc, then Eq. 57 is simplified as follows:

1α=2β0ωF12n+1π2+γ2dεsk,(64)
1α=2β0ωF1ωn2+εs2k+ΔScs2dεsk.(65)

From the Laplacian transform with the Matsubara relation result, we can write Eq. 65 as

1α=2β0ωF1ωn2+εs2kdεskΔScs22β0ωF1ωn2+εs2kdεsk,(66)
1α=2β0ωF12n+12π2+βεsk2dεskΔScs22β0ωF12n+12πβ2+εs2kdεsk,(67)
1α=0ωFtanhβ2εskεs2kdεsk2ΔScs22β0ωF012n+12πβ2+εs2kdεsk.(68)

Applying the following equality 01(y2+εs2(k))2dεs(k)=201y4(1+x2)2dεs(k)where y2=(2n+1)2(πβ)2 and x2=εs2(k)y2.

1α=0ωFtanhβ2εskεs2kdεskΔScs24β0ωF01y41+x22dεsk.(69)

From the above equality relations x=βεs(k)2 and dx=βdεs(k)2.

1α=0βωF2tanhxxdx4β00ΔScs2y31+x22dx,(70)
=lnβωF2tanhβωF20βωF2lnxcosh2xdx4ΔScs2βy30011+x22dx,(71)
=lnβωF2tanhβωF2lnπ4γ4β2ΔScs2π3012n+13011+x22dx,(72)
=lnβωF2tanhβωF2lnπ4γ4β2ΔScs2π378ξ3π4.(73)

For low temperature, tanh(βωF2)1 and γ denotes the Euler’s constant, and its value is given by γ = 1.78. 01(1+x2)2dx=π4 and 01(2n+1)p=(12p)ξ(p); this means ξ(3) = 1.202. So, after some steps, Eq. 73 can be written as

1α=ln1.14ωFkBTc1.052ΔScsπkBTc2.(74)

From Eq. 74, we have

1α=ln1.14ωFkBTc,(75)
ln1.14ωFkBTc=ln1.14ωFkBTc1.052ΔScsπkBTc2,(76)
ln1.14ωFkBTcln1.14ωFkBT=1.052ΔScsπkBTc2,(77)
ln1.14ωFkBTc1.14ωFkBT=1.052ΔScsπkBTc2,(78)
lnTTc=1.052ΔScsπkBTc2.(79)

Using logarithmic series

ln(±k)=±k12k2±13k3

ln11TTc=1TTc121TTc2131TTc3.(80)

Leaving the high-order terms, we get the following.

ln11TTc1TTc,(81)
1TTc=1.052ΔScsπkBTc2,(82)
1.052ΔScsπkBTc=1TTc12,(83)
ΔScs=π1.025kBTc1TTc12,(84)
ΔScsT=3.063kBTc1TTc12.(85)

This equation tells us the superconducting order parameter ΔScs as a function of temperature. As temperature increases, the order parameter decrease and vanishes at Tc (10.09K). If T is zero, ΔScs=2.658meV.

2.1.2 ΔScd due to intra-band interactions within the d-band

ΔScd due to the intra-band interactions within the d band provided by

ΔScd=Usβk,nd̂k,d̂k.(86)

By applying the same procedure as above, the superconducting transition temperature Tc due to the intra-band interactions within the d-band is written as

Tc=1.14ωFkBexp1UdNdo,(87)

where Ud = 0.267meV, Nd(o) = 1.80 (meV)−1, kB = 0.086 meV/K and ℏωF = 6 meV [29]. Substituting these experimental values in this equation, we get Tc = 9.92K that agrees with the experiment. The dependence of the superconducting order parameter ΔScd on temperature due to the intra-band interactions within the d-band is also given by

ΔScdT=3.063kBTc1TTc12.(88)

Eq. 88 indicates the dependence of the superconducting order parameter on temperature in the d-band intra-band interaction in the pure superconducting region. The superconducting order parameter decreases as the temperature increases. It vanishes at Tc (9.92K). At T = 0, ΔScd=2.163meV.

2.1.3 ΔScsd due to the inter-band interaction between the s- and d-bands

The inter-band interaction between the s- and d-bands causes the superconducting order parameter, which can be connected to Green’s function as

ΔScsd=Usd2kŝk,ŝk+d̂k,d̂k(89)

Applying the same procedure, the equation of motion for the correlations ŝk,ŝk and d̂k,d̂k to describe the superconducting order parameter by using Green function formalism similar to the previous procedures due to the inter-band interaction is given by

ŝk,ŝk=ΔScs+ΔScsd(ω+εs(k)M)(ωεs(k)+M)ΔScs+ΔScsd2 andd̂k,d̂k=ΔScd+ΔScsd(ω+εs(k)M)(ωεs(k)+M)ΔScd+ΔScsd2.

Ignoring the intra-band superconducting order parameter terms and by decoupling these equations using the partial fraction decomposition method, we will have ŝk,ŝk=12i=12β2Δi(k)((2n+1)π)2+β2(εs2(k)+Δi2(k)) andd̂k,d̂k=12i=12β2Δi(k)((2n+1)π)2+β2(εd2(k)+Δi2(k))

where Δi(k)=ΔScsd1(1)iM, which is called the effective order parameter. Substituting these equations into Eq. 89, we get

ΔScsd=Usd2k12i=12β2Δik2n+1π2+β2εs2k+Δi2k+Usd2k12i=12β2Δik2n+1π2+β2εd2k+Δi2k.(90)

Let ɛs(k) = ɛd(k) in the inter-band interaction between the two bands, and Eq. 90 becomes

ΔScsd=Usd2k12i=1,22β2Δik2n+1π2+β2εs2k+Δi2k.(91)

Let μ=βεs2(k)+Δi2(k) and 1((2n+1)π)2+μ2=tanh(μ2)2μ, then Eq. 91 is written as

ΔScsd=Usd4k,i=1,2Δiktanhβ2εs2k+Δi2kεs2k+Δi2k.(92)

Changing summation into integration in the region −F < ɛd(k) < ℏωF, at the Fermi level, the density of state in the inter-band interaction is Nsd(o), that is, kωFωFNsd(o)dεs(k) The density of state Nsd(o) is equal to Ns(o)Nd(o), then Eq. 92 becomes

ΔScsd=Usd4ωFωFNsdoΔiktanhβ2εs2k+Δi2kεs2k+Δi2kdεsk,(93)
2UsdNsNdo=0ωFΔikΔScsdtanhβ2εs2k+Δi2ksqrtεs2k+Δi2kdεsk,(94)
2UsdNsNdo=0ωFΔScsdMΔScsdtanhβ2εs2k+ΔScsdM2εs2k+ΔScsdM2dεsk+0ωFΔScsdMΔScsdtanhβ2εs2k+ΔScsdM2εs2k+ΔScsdM2dεsk.(95)

After a couple of steps, the superconducting transition temperature Tc is given by

Tc=1.14ωFkBexp1UsdNsNdoM4KBTclnωF+MωFM,(96)

where Usd = 0.297meV, Nd(o) = 1.80 (meV)−1, Ns(o) = 1.5 (meV)−1, kB = 0.086 meV/K, and ℏωF = 6 meV [29]. Eq. 96 clearly shows that the superconducting transition temperature depends on the SDW order parameter.

In the pure diamagnetism region M = 0, and Eq. 91 becomesTc=1.14ωFkBexp1UsdNsNd(o) with Tc = 10.20K, which agrees with the experiment.For perfect diamagnetism, M = 0, and after some steps, the superconducting order parameter will be

ΔScsdT=3.063kBTc1TTc12.(97)

This equation shows that the dependence of the superconducting order parameter on the temperature in the inter-band interactions between the s- and d-bands. The superconducting order parameter suppresses as the temperature increases. It vanishes at the superconducting transition temperature (Tc = 10.20K). If T=0,ΔScsd(0)=2.687meV.

2.2 SDW order parameter (M)

Using the double-time temperature-dependent Green’s function to the equation of motion for the correlation ŝ(k+p),d̂k, we will find the magnetic order parameter. To solve this problem, we start from the equation of motion for the correlation ŝ(k+p),d̂k, and from double-time temperature-dependent Green’s function, it can be described as

ωŝk+p,d̂k=ŝk+p,d̂k+ŝk+p,Ĥ,d̂k,(98)
ωŝk+p,d̂k=ŝk+p,d̂k+ŝk+p,Ĥs+Ĥd+Ĥsd+ĤM,d̂k.(99)

The commutation relations in Eq. 99 can be solved as follows:

ŝk+p,Ĥs=εsk+pŝk+p+ΔScsŝk+p,(100)
ŝk+p,Ĥd=0,(101)
ŝk+p,Ĥsd=ΔScsdŝk+p,(102)
ŝk+p,ĤM=Mŝk+p.(103)

Substituting Eq. 100 to Eq. 103 in Eq. 99, it gives

ωŝk+p,d̂k=ΔScs+ΔScsdω+ϵsk+pMŝk+p,d̂k.(104)

Now, let us find the equation of motion for the correlation ŝ(k+p),d̂k in Eq. 104.

ωŝk+p,d̂k=1+ŝk+p,Ĥs+Ĥd+Ĥsd+ĤM,d̂k.(105)

Solving the commutation relations in Eq. 105, we get

ŝk+p,Ĥs=εsk+pŝk+p+ΔScsŝk+p,(106)
ŝk+p,Ĥd=0,(107)
ŝk+p,Ĥd=ΔScsdŝk+p,(108)
ŝk+p,Ĥd=Mŝk+p.(109)

Substituting Eq. 106 to Eq. 109 in Eq. 105, we get

ωŝk+p,d̂k=1ωεk+p+M+ΔScs+ΔScsdωεk+p+Mŝk+p,d̂k.(110)

Substituting Eq. 110 into Eq. 104 and after some mathematics, we get

ŝk+p,d̂k=12ΔScs+Mω2+ϵs2k+p+ΔScs+M2+12ΔScsMω2+ϵs2k+p+ΔScsM2,(111)

where Δj(k)=ΔScsd1(1)jM, which is called the effective magnetic order parameter. Using the expression ωn and applying the nesting condition, ϵs2(k+p)=ϵs2(k). Eq. 111 gives

ŝk+p,d̂k=12j=1,21jΔjkωn2+ϵs2k+Δj2k.(112)

The magnetic order parameter is given by

M=Uβkŝk+p,d̂k.(113)

Substituting Eq. 112 into Eq. 113; transforming summation into integration in the boundary −ℏωF < ɛs(k) < ℏωF and by presenting the density of state (DOS) at the Fermi level is N(o), that is, kωFωFN(o)dεs(k), and after a couple of steps, we get

2UNo=1jΔjM0ωFtanhβ2εs2k+Δi2kεs2k+Δi2kdεsk,(114)
2UNo=ΔSsc+MM0ωFtanhβ2εs2k+ΔSsc+M2εs2k+ΔSsc+M2dεskΔSscMM0ωFtanhβ2εs2k+ΔSscM2εs2k+ΔSscM2dεsk.(115)

After some steps, we get

1UNo=ln1.14ωFkBTM1.052MπkBTM2+βM4lnωF+MωFM.(116)

For small values of M, we ignore the M2 term. Thus, Eq. 116 reduces to

TM=1.14ωFkBexp1UNo+βM4lnωF+MωFM,(117)

where UN(o) = 1.68 [29] and it is the SDW coupling parameter. Eq. 117 shows that the SDW order parameter increases as the SDW transition temperature increases.

For the pure magnetic region ΔScs=0 and Eq. 115 becomes

1UNo=0ωFtanhβ2εs2k+M2εs2k+M2dεsk.(118)

This is simplified to

1UNo=ln1.14ωFkBT1.052MπkBT2.(119)

For a little value of M, M2 → 0 and TTM. Eq. 119 reduces

1UNo=ln1.14ωFkBTM.(120)

This implies that

ln1.14ωFkBTM=ln1.14ωFkBT1.052MπkBTM2.(121)

Eq. 121 simplifies to

M=πkBTM1.0521TTM12,(122)
MT=3.063kBTM1TTM12.(123)

Eq. 123 indicates that if temperature rises, the SDW order parameter suppresses.

3 Results and discussion

In this part, we discussed how temperature (T) affects superconducting order parameters (ΔSc) and the SDW order parameter (M), and M affects on both the SDW transition temperature (TM) and superconducting transition temperature (Tc). We expand on the analysis. In a two-band model for SrFe2−xNixAs2, we created the theoretical examination of the coexistence of superconductivity and SDW. With the aid of a two-band Hamiltonian model and the double-time temperature-dependent Green’s function formal consideration, we were able to derive the mathematical expressions for the superconducting transition temperature (Tc), superconducting order parameters for each intra- and inter-band interactions, SDW order parameter (M), and SDW transition temperature (TM).

From Eqs 63, 87, 96, we obtain the superconducting transition (critical) temperatures for each intra- and inter-band interactions of SrFe2−xNixAs2. Using these (Tc) values and Eqs 85, 88, 97, respectively, we plotted the phase diagram of ΔSc versus T within each intra-band and inter-band interactions, Figure 1A.

FIGURE 1
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FIGURE 1. Superconducting order parameters vs. temperature for each interactions of the SrFe2−xNixAs2 superconductor (A) and SDW order parameter (M) vs. temperature in the pure magnetic region of the SrFe2−xNixAs2 superconductor (B).

As illustrated in Figure 1A, the superconducting order parameter decreases as the temperature increases. It vanishes as the temperature is equal to the critical temperature. For the s intra-band interaction, the maximum value of the superconducting order parameter, (ΔScs)=2.658meV, occurs at T = 0, and it vanishes at the superconducting transition temperature Tc = 10.09K. For the d intra-band interaction, the maximum value of the superconducting order parameter,ΔScd=2.613meV, occurs at T = 0, and ΔScd=0 at the superconducting transition temperature Tc = 9.92K. Moreover, ΔScsd=2.687meV at T = 0 and ΔScsd=0 at the superconducting transition temperature Tc = 10.20K for the inter-band interaction. For each intra-band and inter-band interactions of the SrFe2−xNixAs2 superconductor, the theoretical values of superconducting transition temperatures agree with the experimental value, which is around Tc = 10K, as discussed in Table 1 [34].

TABLE 1
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TABLE 1. Superconducting transition temperature Tc values and the superconducting order parameter at T = 0 in different interactions for our compound SrFe2−xNixAs2. The mean value of Tc is nearly 10K.

Based on Eq. 123, we plotted the phase illustration for the dependence of the magnetic order parameter (M) on temperature in the pure magnetic region illustrated in Figure 1B. As illustrates from this figure, magnetism decreases as the temperature enhances and vanishes at the SDW transition temperature TM = 205K. The maximum value of the SDW order parameter, M = 54 meV, occurs at T = 0. This finding is also in agreement with experimental observations [34, 37].

Based on Eq. 96, we plotted the phase diagrams of Tc versus M. As illustrated from Figure 2A, when the value of the SDW order parameter enhances, the superconducting transition temperature is suppressed for SrFe2−xNixAs2. From this figure, one can see the SDW order parameter promotes the magnetic nature and suppresses superconductivity in the system.

FIGURE 2
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FIGURE 2. Superconducting transition temperature (TC) versus the SDW order parameter(M) for the inter-band interaction of the SrFe2−xNixAs2 superconductor (A) and SDW transition temperature (TM) versus SDW order parameter(M) of SrFe2−xNixAs2 (B).

Using Eq. 117, we also plotted the phase plotting of the SDW transition temperature (TM) versus the SDW order parameter (M), as seen in Figure 2B. As demonstrated from this figure, (TM) progressively gets bigger with the SDW order parameter of the SrFe2−xNixAs2 superconductor.

Finally, by combining Figures 2A, B, this article depicted a region where both SDW and superconductivity coexist, as shown in Figure 3. Because of their coexistence, the iterating superconducting electrons and spins are thought to have a weak exchange coupling for SrFe2−xNixAs2. This figure shows that the possible interplay of superconductivity and SDW for SrFe2−xNixAs2. As indicated in this figure, our finding is in agreement with experimental observations [34, 37]. This figure also depicts there are regions that show the superconducting and anti-ferromagnetic states segregate, which indicates that there are regions where magnetic and superconducting phases are not mixed.

FIGURE 3
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FIGURE 3. Superconducting critical temperature and magnetic transition temperature vs. the SDW order parameter for the SrFe2−xNixAs2 superconductor.

4 Conclusion

In this work, we have studied the possibility of coexisting superconductivity and magnetism for the iron-based superconductor SrFe2−xNixAs2. The superconductivity order parameter for FebSc SrFe2−xNixAs2 is suppressed as the temperature raises and vanishes at the superconducting critical temperature. The magnitude of the SDW order parameter for SrFe2−xNixAs2 is suppressed as the superconducting critical temperature increases and increases with increasing the SDW transition temperature. We depicted the possibility of coexisting superconductivity and magnetism in the SrFe2−xNixAs2 superconductor. For the iron-based superconductor SrFe2−xNixAs2, we further studied the reliance of the SDW order parameter M on temperature in the pure magnetic region. When temperature increases, the SDW order parameter is suppressed and is zero at the SDW transition temperature of the SrFe2−xNixAs2 superconductor.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: superconductivity, superconducting transition temperature, superconducting order parameters, SDW transition temperature, SDW order parameter, magnetism

Citation: Bogale GM and Shiferaw DA (2023) Theoretical study of the interplay of spin density wave and superconductivity in nickel substitution of the strontium–iron–arsenide (SrFe2−xNixAs2) superconductor in a two-band model. Front. Phys. 11:1235105. doi: 10.3389/fphy.2023.1235105

Received: 05 June 2023; Accepted: 20 November 2023;
Published: 07 December 2023.

Edited by:

Jeffrey W. Lynn, National Institute of Standards and Technology (NIST), United States

Reviewed by:

Jianbao Zhao, Canadian Light Source, Canada
Minghu Fang, Zhejiang University, China

Copyright © 2023 Bogale and Shiferaw. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Gedefaw Mebratie Bogale, gedefawmebratie22@gmail.com

These authors have contributed equally to this work

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