The Study on Hybridized Two-Band Superconductor

Abstract

The two-band hybridized superconductor which the pairing occurred by conduction electron band and other-electron band are considered within a mean-field approximation. The critical temperature, zero-temperature order parameter, gap-to-𝑇_𝑐 ratio, and isotope effect coefficient are derived. We find that the hybridization coefficient shows a little effect on the superconductor that conduction electron band has the same energy as other-electron band but shows more effect on the superconductor that conduction electron band coexists with lower-energy other-electron band. The critical temperature is decreased as the hybridization coefficient increases. The higher value of hybridization coefficient, lower value of gap-to-𝑇_𝑐 ratio, and higher value of isotope effect coefficient are found.

1. Introduction

Since Moskalenko [1, 2] Suhl et al. [3] introduced the two-band model that accounts for multiple energy bands in the vicinity of the Fermi energy contributing electron pairing in superconductor, the two-band model has been applied to high temperature superconductor in copper oxides [4–10], MgB₂ superconductor [11–13], and heavy Fermion superconductor [14, 15]. Dolgov et al. [16] studied the thermodynamic properties of the two-band superconductor: MgB₂. The superconducting energy gap, free energy, the entropy, and heat capacity were calculated within the framework of two-band Eliashberg theory. Mazin et al. [17] studied the effect of interband impurity scattering on the critical temperature of two-band superconductor in MgB₂. Askerzade and Tanatar [18] and Changjan and Udomsamuthirun [19] calculated the critical field of the two-band superconductor by Ginzburg-Landau approach and applied it to Fe-based superconductors. Golubov and Koshelev [20] investigated the two-band superconductor with strong intraband and weak interband electronic scattering rates in the framework of coupled Usadel equation.

The interplay of superconductivity and magnetism is one of the most interesting phenomena of superconductor. The cuprate superconductor exhibits the phase diagram having the magnetic ordered states in the vicinity of the superconducting phase. The antiferromagnetic and ferromagnetic phases are also found in the heavy Fermion superconductor. In ErRh₄B₆ [21] and HoMo₆S₈ [22] system, 𝑠-wave superconductivity shows ferromagnetism in the ground state at intermediate temperature.

The nesting properties of Fermi surface in low dimensional system arise the spin density wave (SDW) state and charge density wave (CDW) state in the interplay of superconductivity and magnetism system. The SDW and the CDW states occurred by Coulomb interaction between electron, and electron-phonon interaction, respectively. Nass et al. [23] used the BCS-type pairing to explain the antiferromagnetic superconductor. Suzumura and Nagi [24] investigated some properties of antiferromagnetic superconductor. They proposed the Hamiltonian of the superconductivity associated with conduction 𝑑-electron and the antiferromagnetism associated with 𝑓-electron of rare earth atoms that formed the BCS-type pairing. Ichimura et al. [25] investigated the effect of the CDW on BCS superconductor within a mean-field approximation. In this model, the two order parameters of CDW and superconductor were introduced. The tight-binding band in 2D square lattice and nesting vector 𝑄=(𝜋,𝜋) was used in their calculations. Rout and Das [26] applied the periodic Anderson model (PAM) for calculating the nonmagnetic ground state of heavy Fermion superconductor. The Hamiltonian of the heavy Fermion systems composed of conduction electron band, 𝑓-electron band, the hybridization of conduction electron and 𝑓-electron band, BCS-like pairing band, and the intra-atomic Coulomb interaction of 𝑓-electron. The Hamiltonian was simplified by linearising the intra-atomic Coulomb interaction with the Hartree-Fock approximation then the 𝑓-electron band energy was 𝐸_𝑘=𝜀_𝑓+𝑈𝑛_𝜎, where 𝜀_𝑓 was bare 𝑓-electron energy and 𝑈𝑛_𝜎 was the Coulomb energy of 𝑓-electron. Finally, their Hamiltonian were consisted of conduction and 𝑓-electron band including BCS-like pairing band and hybridization term. Panda and Rout [27] studied the interplay of CDW, SDW, and superconductivity in high temperature superconductor in low doping phase. The model of mean-field Hamiltonian including the CDW, SDW and superconductivity was introduced.

In this paper, we modified the hybridized Hamiltonian of Rout and Das [26] to be the two-band superconductor with hybridization. Some physical properties of two-band hybridized superconductor, that is, critical temperature, zero-temperature order parameter, gap-to-𝑇_𝑐 ratio, and isotope effect coefficient, were investigated.

2. Model and Calculation

According to the hybridized Hamiltonian [26] that consisted of conduction electron and 𝑓-electron band, BCS-like pairing band, and hybridization term, we have 𝐻=_𝑘,𝜎𝜀_𝑘𝐶⁺_𝑘𝜎𝐶_𝑘𝜎+𝜀_{𝑓𝑘,𝜎}𝑓⁺_𝑘𝜎𝑓_𝑘𝜎+𝛾_0𝑘,𝜎𝑓⁺_𝑘𝜎𝐶_𝑘𝜎+𝐶⁺_𝑘𝜎𝑓_𝑘𝜎+𝑈2_𝑖,𝜎𝑛^𝑓_𝑖𝜎𝑛^𝑓_𝑖,−𝜎−Δ_𝑘𝐶⁺_𝑘↑𝐶⁺_−𝑘↓+𝐶_−𝑘↓𝐶_𝑘↑, (1) where 𝐶⁺_𝑘𝜎(𝐶_𝑘𝜎) and 𝑓⁺_𝑘𝜎(𝑓_𝑘𝜎) are the creation (annihilation) operator of conduction electron and 𝑓-electron. Δ is the superconducting order parameter that Cooper pairs involve only the conduction electron. 𝛾₀ is the hybridization interaction coefficient of 𝑓-electron band and conduction band. 𝑛^𝑓_𝑖=𝑓⁺_𝑖𝜎𝑓_𝑖𝜎 is the intraatomic Coulomb interaction between 𝑓-electron. They [26] linearised the (𝑈/2)∑_𝑖,𝜎𝑛^𝑓_𝑖𝜎𝑛^𝑓_𝑖,−𝜎 term by Hartree-Fock approximation that (𝑈/2)∑_𝑖,𝜎𝑛^𝑓_𝑖𝜎𝑛^𝑓_𝑖,−𝜎≈𝑈∑_𝑖,𝜎𝑛_−𝜎𝑓⁺_𝑖𝜎𝑓_𝑖𝜎; then the Hamiltonian became 𝐻=_𝑘,𝜎𝜀_𝑘𝐶⁺_𝑘𝜎𝐶_𝑘𝜎+_𝑘,𝜎𝐸₀𝑓⁺_𝑘𝜎𝑓_𝑘𝜎+𝛾_0𝑘,𝜎𝑓⁺_𝑘𝜎𝐶_𝑘𝜎+𝐶⁺_𝑘𝜎𝑓_𝑘𝜎−Δ_𝑘𝐶⁺_𝑘↑𝐶⁺_−𝑘↓+𝐶_−𝑘↓𝐶_𝑘↑, (2) where 𝐸₀=𝜀_𝑓+𝑈𝑛_−𝜎 that is the energy collected the non-interaction with a modified 𝑓-level.

In our model, the two-band superconductor comprise of conduction electron and other-electron band. The superconducting order parameters can occurr by conduction electron and other-electron band. The conduction band makes the intra-atomic Coulomb interaction with other-electron band. We set 𝐻=_𝑘,𝜎𝜀_𝑘𝐶⁺_𝑘𝜎𝐶_𝑘𝜎+_𝑘,𝜎𝐸₀𝑓⁺_𝑘𝜎𝑓_𝑘𝜎+𝛾_0𝑘,𝜎𝑓⁺_𝑘𝜎𝐶_𝑘𝜎+𝐶⁺_𝑘𝜎𝑓_𝑘𝜎+𝑈2_𝑖,𝜎𝑛^𝑓_𝑖𝜎𝑛^𝑓_𝑖,−𝜎−Δ_𝑘𝐶⁺_𝑘↑𝐶⁺_−𝑘↓+𝐶_−𝑘↓𝐶_𝑘↑−Δ_𝑘𝑓⁺_𝑘↑𝑓⁺_−𝑘↓+𝑓_−𝑘↓𝑓_𝑘↑, (3) where 𝐶⁺_𝑘𝜎(𝐶_𝑘𝜎) and 𝑓⁺_𝑘𝜎(𝑓_𝑘𝜎) are the creation (annihilation) operators of conduction electron and other-electron band. Δ is the superconducting order parameter. 𝛾₀ is the hybridization interaction coefficient of other-electron band and conduction band. 𝑛^𝑓_𝑖=𝑓⁺_𝑖𝜎𝑓_𝑖𝜎 is the intra-atomic Coulomb interaction between other-electron. The Hamiltonian is linearised by [26]’s technique; then we get the simplified two-band-hybridized Hamiltonian. We can write the Hamiltonian as 𝐻=𝐻₁+𝐻₂+𝐻₁₂, (4) where 𝐻₁=_𝑘𝜎𝜀_𝑘𝐶⁺_𝑘𝜎𝐶_𝑘𝜎−Δ_𝑘𝐶⁺_𝑘↑𝐶⁺_−𝑘↓+𝐶_−𝑘↓𝐶_𝑘↑,𝐻₂=_𝑘𝜎𝐸_𝑘𝑓⁺_𝑘𝜎𝑓_𝑘𝜎−Δ_𝑘𝑓⁺_𝑘↑𝑓⁺_−𝑘↓+𝑓_−𝑘↓𝑓_𝑘↑,𝐻₁₂=𝛾_0𝑘𝜎𝑓⁺_𝑘𝜎𝐶_𝑘𝜎+𝐶⁺_𝑘𝜎𝑓_𝑘𝜎. (5a)(5b)(5c) The first Hamiltonian describes the conduction electron Hamiltonian, the second Hamiltonian describes the Hamiltonian of other-electron band, and the third Hamiltonian describes the interact Hamiltonian. Where 𝜀_𝑘 and 𝐸_𝑘 are the band energies of the conduction electron and other-electron band measured from the Fermi energy.

𝐸_𝑘=𝐸₀+𝑈𝑛_−𝜎 is the energy collected the non-interaction with a modified other-electron band. 𝐶⁺_𝑘𝜎(𝐶_𝑘𝜎) and 𝑓⁺_𝑘𝜎(𝑓_𝑘𝜎) are the creation (annihilation) operator of conduction electron and other-electron. 𝛾₀ is the hybridization interaction coefficient of other-electron band and conduction band. Δ is the effective superconducting order parameter occurred by conduction electron and other-electron and assumed to be homogeneous in space. The effective superconducting order parameters is 𝑉Δ=2_𝑘𝐶⁺_𝑘↑𝐶⁺_−𝑘↓+𝑓⁺_𝑘↑𝑓⁺_−𝑘↓, (6) where the 𝑠-wave like BCS pairing interaction having the same coupling interaction potential is assumed. The effective superconducting order parameters are the coupling equation of conduction electron and other-electron that can occurr in magnetic superconductor [23–25]. However for simplicity of calculation, the same pairing strength is taken [28].

We introduce the finite-temperature Green function: 𝐺(𝑘,𝜏)=−𝑇_𝜏𝜓_𝑘(𝜏)𝜓⁺_𝑘(0), (7) where 𝜓⁺_𝑘=(𝐶⁺_𝑘↑,𝐶_−𝑘↓,𝑓⁺_𝑘↑,𝑓_−𝑘↓) and 𝑇_𝜏 is the ordering operator for imaginary time, 𝜏=𝑖𝑡.

After some calculations, the Green function in Nambu representation is obtained: 𝐺𝜔_𝑛,𝑘=𝑖𝜔_𝑛−𝜀_𝑘−𝐸_𝑘2𝜌₃𝜎₃−𝜀_𝑘+𝐸_𝑘2𝜎₃+Δ𝜎₁−𝛾₀𝜌₁𝜎₃⁻¹, (8) where 𝜔_𝑛=𝜋𝑇(2𝑛+1), 𝑇 is temperature, 𝑛 is an integer, and 𝜌_𝑖 and 𝜎_𝑖(𝑖=1,2,3) are the Pauli matrices. Our Green function obtained shows the same form as [25] that investigated the effect of the CDW on BCS superconductor within a mean-field approximation, 𝐺⁻¹(𝜔_𝑛,𝑘)=𝑖𝜔_𝑛−𝛾_𝑘𝜌₃𝜎₃−𝛿_𝑘𝜌₀𝜎₃+Δ𝜌₀𝜎₁+𝑤𝜌₁𝜎₃. All parameters detailed can be found in [25]. We find that (𝜀_𝑘−𝐸_𝑘)/2≡𝛾_𝑘 and (𝜀_𝑘+𝐸_𝑘)/2≡𝛿_𝑘, where 𝛾_𝑘 and 𝛿_𝑘 are the band structure energies in 2D square lattice of nearest-neighbor and next-nearest-neighbor transfer, respectively. And −𝛾₀≡𝑤 which 𝑤 is the order parameter of CDW. This result means that the CDW consideration gives the same result as the hybridization consideration within a mean-field approximation.

From (6) and (8), the superconducting gap equation is 1𝑉=14_𝑘tanhΔ²+𝜀²₋/2𝑇Δ²+𝜀²₋+tanhΔ²+𝜀²₊/2𝑇Δ²+𝜀²₊, (9) where 𝜀₊=(𝜀_𝑘+𝐸_𝑘)/2+((𝜀_𝑘−𝐸_𝑘)/2)²+𝛾²₀ and 𝜀₋=(𝜀_𝑘+𝐸_𝑘)/2−((𝜀_𝑘−𝐸_𝑘)/2)²+𝛾²₀. The 𝜀₊ and 𝜀₋ represent the upper and lower bands of quasiparticle energy spectra of the hybridization system. We can determine the superconducting critical temperature 𝑇_𝑐 by putting Δ→0; then 1𝑉=14_𝑘tanh𝜀₋/2𝑇_𝑐𝜀₋+tanh𝜀₊/2𝑇_𝑐𝜀₊. (10) In the absence of the hybridization interaction, 𝛾₀=0; that is, 1𝑉=12_𝑘tanh𝜀/2𝑇_𝑐0𝜀1or𝜆=ln2𝛾𝜔_𝐷𝜋𝑇_𝑐0, (11) where 𝛾=1.78. 𝜆=𝑁(0)𝑉, 𝜆 is the coupling constant, and 𝑁(0) is the constant density of state at the Fermi surface, and 𝜔_𝐷 is the Debye cutoff energy. 𝑇_𝑐0 is the critical temperature of superconductor without hybridization that the BCS’s result.

Because of the complicated quasi-particle energy spectra obtained, we introduce the two approximated conditions to calculate analytically; the superconductor with conduction electron band having the same energy as other-electron band (𝜀_𝑘≈𝐸_𝑘) and the superconductor with conduction electron band coexisting with lower-energy other-electron band (𝜀_𝑘≫𝐸_𝑘,  𝐸_𝑘≈0).

Case 1. The superconductor that the conduction electron band having the same energy as other-electron band.

In this case, the approximation is 𝜀_𝑘≈𝐸_𝑘. Then, we get 𝜀₋≈𝜀_𝑘−𝛾₀ and 𝜀₊≈𝜀_𝑘+𝛾₀. The difference of the lower and upper energy spectra is equal to 2𝛾₀. If 𝛾₀=0, the BCS’s superconductor is obtained. In this case, the effect of the hybridization interaction on the BCS superconductor is considered.

We substitute above approximations into (10); then the gap equation becomes 1𝜆=14^𝜔𝐷_−𝜔𝐷tanh𝜀₋/2𝑇_𝑐𝜀₋𝑑𝜀_𝑘+^𝜔𝐷_−𝜔𝐷tanh𝜀₊/2𝑇_𝑐𝜀₊𝑑𝜀_𝑘≈ln2𝛾𝜔²_𝐷−𝛾²₀𝜋𝑇_𝑐. (12) The critical temperature is 𝑇_𝑐=1.13𝜔²_𝐷−𝛾²₀𝑒^−1/𝜆. (13) And the zero-temperature energy gap can be found as 1𝜆=14^𝜔𝐷_−𝜔𝐷1Δ²(0)+𝜀_𝑘−𝛾₀²+1Δ²(0)+𝜀_𝑘+𝛾₀²𝑑𝜀_𝑘=12sinh⁻¹𝜔_𝐷−𝛾₀Δ(0)+sinh⁻¹𝜔_𝐷+𝛾₀Δ(0). (14) For 𝜔_𝐷≫Δ(0), we can get Δ(0)=2𝜔²_𝐷−𝛾²₀𝑒^−1/𝜆. (15) From (13) and (15), the gap-to-𝑇_𝑐 ratio is obtained: (0)𝑅=2Δ𝑇_𝑐=3.53. (16) In this case, we find that the hybridization interaction coefficient decreases the critical temperature and zero-temperature energy gap but has no effect on gap-to-𝑇_𝑐 ratio.

To investigate the effect of hybridization and the Debye cutoff on gap-to-𝑇_𝑐 ratio, we rewrite the gap equation at critical temperature and at zero-temperature into the form [29] ^{(𝜔𝐷+𝛾0)/(2𝑇𝑐)}_{(−𝜔𝐷+𝛾0)/(2𝑇𝑐)}tanh𝑥𝑥𝑑𝑥=^{2(𝜔𝐷+𝛾0)/𝑇𝑐}_{−2(𝜔𝐷−𝛾0)/𝑇𝑐}1𝑅²+𝑥²𝑑𝑥. (17) The numerical calculation of (17) is shown in Figure 2.

Within the definition of isotope effect coefficient in harmonic approximation; 𝛼=(1/2)(𝜔_𝐷/𝑇_𝑐)(𝑑𝑇_𝑐/𝑑𝜔_𝐷) and equation (9), we 𝜔𝛼=^𝐷2tanh𝜔⁻_𝐷/2𝑇_𝑐/𝜔⁻_𝐷+tanh𝜔⁺_𝐷/2𝑇_𝑐/𝜔⁺_𝐷tanh𝜔⁻_𝐷/2𝑇_𝑐+tanh𝜔⁺_𝐷/2𝑇_𝑐, (18) where 𝜔⁺_𝐷=𝜔_𝐷+𝛾₀ and 𝜔⁻_𝐷=𝜔_𝐷−𝛾₀.

Consider the limiting cases that the hybridization is so small with respect to Debye cutoff energy, 𝜔_𝐷≫𝛾₀; for 𝜔_𝐷>2𝑇_𝑐, we can get that 𝛼≈(𝜔_𝐷/4)(1/𝜔⁻_𝐷+1/𝜔⁺_𝐷)≈1/2, and for 𝜔_𝐷<2𝑇_𝑐, we get 𝛼≈(𝜔_𝐷/2)((1/𝑇_𝑐)/(𝜔_𝐷/𝑇_𝑐))≈1/2 that the BCS’ result.

Case 2. The superconductor that the conduction electron band coexisting with lower-energy other-electron band.

Because of the hybridization Hamiltonian having the same Green’s function as the charge density wave model, we can apply this model to the superconducting state found in heavy Fermion superconductor. The heavy Fermion superconductor has its origin in the interplay of strong Coulomb repulsion in 4f- and 5f-shells and their hybridizations with the conduction band. The 𝑓-electron is associated with the magnetic ordering having lower energy than conduction electron. We can make the assumption that the 𝑓-electron band is at the Fermi level which can be taken as 𝐸_𝑘≈0; then 𝜀₋≈𝜀_𝑘2−𝛾₀,𝜀₊≈𝜀_𝑘2+𝛾₀𝜀,for^𝑘2<𝛾₀,𝜀₋≈0,𝜀₊≈𝜀_𝑘𝜀,for^𝑘2>𝛾₀. (19)

Substituting above approximation into (10), we can get 1𝜆=14^𝜔𝐷_−𝜔𝐷tanh𝜀₋/2𝑇_𝑐𝜀₋𝑑𝜀_𝑘+^𝜔𝐷_−𝜔𝐷tanh𝜀₊/2𝑇_𝑐𝜀₊𝑑𝜀_𝑘≈ln2𝛾𝜋𝑇_𝑐2𝜔_𝐷𝛾₀+12𝜔_𝐷2𝑇_𝑐−𝛾₀𝑇_𝑐, (20) where, tanh𝜀^𝜔𝐷_{−𝜔𝐷−}/2𝑇_𝑐𝜀₋𝑑𝜀_𝑘𝜔≈2^𝐷2𝑇_𝑐−2𝛾₀2𝑇_𝑐+2ln4𝛾𝛾₀𝜋𝑇_{𝑐𝜔𝐷−𝜔𝐷},tanh𝜀₊/2𝑇_𝑐𝜀₊𝑑𝜀_𝑘≈2ln2𝛾𝜔_𝐷𝜋𝑇_𝑐. (21) The critical temperature is 𝑇_𝑐=1.132𝛾₀𝜔_𝐷𝑒^{−1/𝜆+(1/2)(𝜔𝐷/2𝑇𝑐−𝛾0/𝑇𝑐)}. (22) The gap equation as zero-temperature is 1𝜆=14^𝜔𝐷_−𝜔𝐷1Δ²(0)+𝜀₋²+1Δ²(0)+𝜀₊²𝑑𝜀_𝑘, (23) where, ^𝜔𝐷_−𝜔𝐷1Δ²(0)+𝜀₋²𝑑𝜀_𝑘≈2Δ(0)𝜔_𝐷−2𝛾₀+2sinh⁻¹2𝛾₀Δ(0)_{𝜔𝐷−𝜔𝐷},1Δ²(0)+𝜀₊²𝑑𝜀_𝑘≈2sinh⁻¹𝜔_𝐷Δ(0). (24) Then, we get Δ(0)=22𝛾₀𝜔_𝐷𝑒^{−1/𝜆+(1/2)((𝜔𝐷−2𝛾0)/Δ(0))}. (25) According to (22) and (25), the gap-to-𝑇_𝑐 ratio is (0)𝑅=2Δ𝑇_𝑐=3.53𝑒^{((𝜔𝐷−2𝛾0)/𝑇𝑐)(1/𝑅−1/4)}. (26)

To investigate the effect of hybridization and the Debye cutoff on gap-to-𝑇_𝑐 ratio, we rewrite the gap equation at critical temperature and at zero-temperature into the form [29] _{𝜔𝐷/𝑇𝑐−𝜔𝐷/𝑇𝑐}tanh𝑦/4−𝑦/4𝑥𝑑𝑥=^{2(𝜔𝐷+𝛾0)/𝑇𝑐}_{−2(𝜔𝐷−𝛾0)/𝑇𝑐}1𝑅²+𝑥²𝑑𝑥. (27) The numerical calculation of this equation is shown in Figure 2.

Within the definition of isotope effect coefficient in harmonic approximation, (10) and 𝐸_𝑘≈0, we can get 𝜔𝛼=^𝐷2×1+2𝑇_𝑐/𝜔_𝐷tanh𝜔_𝐷/2𝑇_𝑐𝜔_𝐷−2𝛾₀+2𝑇_𝑐tanh𝜔_𝐷/2𝑇_𝑐+tanh𝛾₀/𝑇_𝑐. (28)

3. Results and Discussions

We use the hybridized two-band Hamiltonian to investigate the critical temperature, zero-temperature order parameter, gap-to-𝑇_𝑐 ratio, and isotope effect coefficient of superconductor. The Green function and gap equation are derived analytically. However, the quasi-particle energy spectra obtained are complicated; then we introduce two approximated cases as, the superconductor that conduction electron band has the same energy as other-electron band (𝜀_𝑘≈𝐸_𝑘) and the superconductor that conduction electron band coexists with lower-energy other-electron band (𝜀_𝑘≫𝐸_𝑘,  𝐸_𝑘≈0). The critical temperature, zero-temperature order parameter, gap-to-𝑇_𝑐 ratio, and isotope effect coefficient are shown in the exact forms.

We use the integration forms of involved equations for more accuracy in numerical calculation. After the numerical calculations of 𝑇_𝑐, (Figure 1), we find that the weak-coupling limit (𝜆<0.4) can be found in range of 𝜔_𝐷/𝑇_𝑐>10. Then, we investigate the effect of hybridization in weak-coupling limit with 𝜔_𝐷/𝑇_𝑐>10 and 𝛾₀/𝑇_𝑐=0.2,4.0. We find that 𝑇_𝑐 is decreased when the hybridization coefficient increased in Case 2 and no effect in Case 1. In Figure 2, the gap-to-𝑇_𝑐 ratios (𝑅) of Cases 1 and 2 with varied hybridization coefficients are shown. In Case 1, the 𝑅 is tended to BCS value, 3.53, as 𝜔_𝐷/𝑇_𝑐→∞. Case 2, 𝑅≈3.3--3.9 can be found to depend strongly on the value of 𝛾₀/𝑇_𝑐. The higher value of 𝛾₀/𝑇_𝑐 and the lower value of 𝑅 are found to agree with superconducting gap’s behavior of HF system of Rout et al. [30].

Figure 1 

The 𝑇𝑐 versus hybridization coefficients of Cases 1 and 2.

Figure 2 

The gap-to-𝑇𝑐 ratio of Cases 1 and 2 with varied hybridization coefficients.

The isotope effect coefficient is also investigated and shown in Figure 3. In Case 2, we find that the isotope effect coefficient can be more than the BCS (𝛼>0.5) and less than BCS value (𝛼<0.5), but in Case 1 we can find only for 𝛼>0.5. However, 𝛼 in both cases is converse to 0.5 as 𝜔_𝐷/𝑇_𝑐→∞.

Figure 3 

The isotope effect coefficient of Cases 1 and 2 with varied hybridization coefficients.

The Fe-based superconductors are mutiband system which comprises at least two bands which propose 𝑠-wave paring state. The 𝑅=3.68 [31] which shows a consistent manner with the BCS prediction is found. The value of isotope effect coefficient were found to be 𝛼≈0.35--0.4  [32] and 𝛼_Fe=0.81 [33]. These results indicate that electron-phonon interaction plays some role in the superconducting mechanism by affecting the magnetic properties [31]. According to our model, the effective superconducting order parameters are the coupling equation of conduction electron and other-electron and the magnetic order are included in our calculation as in Case 2. We can get the experimental data of Fe-based superconductor from Figures 2 and 3.

4. Conclusions

The two-band hybridized superconductor that the pairing occurred by conduction electron band and other-electron band is studied in weak-coupling limit. The formula of critical temperature, zero-temperature order parameter, gap-to-𝑇_𝑐 ratio and isotope effect coefficient are calculated. For the superconductor that conduction electron band has the same energy as other-electron band, the hybridization coefficient shows a little effect. The numerical results do not differ much from the BCS’s results. For the superconductor that conduction electron band coexists with lower-energy other-electron band, the hybridization coefficient show more effect. 𝑇_𝑐 is decreased when the hybridization coefficient increases. We can get higher and lower value of 𝑅 and 𝛼 than BCS’s results depending on the hybridization coefficient. Higher value of hybridization coefficient, lower value of 𝑅, and higher value of 𝛼 are found.

Acknowledgments

The financial support of the Office of the Higher Education Commission, Srinakhariwirout University, and ThEP Center is acknowledged.