# The Study on Hybridized Two-Band Superconductor

Authors: T. Chanpoom; J. Seechumsang; S. Chantrapakajee; P. Udomsamuthirun
Journal: Advances in Condensed Matter Physics
 (2013)

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## 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-Tc 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-Tc ratio, and higher value of isotope effect coefficient are found.


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## Body

## 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 Ek=εf+Unσ, where εf was bare f-electron energy and Unσ 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-Tc 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
(1)H=∑k,σεkCkσ+Ckσ+εf∑k,σfkσ+fkσ+γ0∑k,σ(fkσ+Ckσ+Ckσ+fkσ)+U2∑i,σniσfni,-σf-Δ∑k(Ck↑+C-k↓++C-k↓Ck↑),
where Ckσ+(Ckσ) and fkσ+(fkσ) 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. nif=fiσ+fiσ is the intraatomic Coulomb interaction between f-electron. They [26] linearised the (U/2)∑i,σniσfni,-σf term by Hartree-Fock approximation that (U/2)∑i,σniσfni,-σf≈U∑i,σn-σfiσ+fiσ; then the Hamiltonian became
(2)H=∑k,σεkCkσ+Ckσ+∑k,σE0fkσ+fkσ+γ0∑k,σ(fkσ+Ckσ+Ckσ+fkσ)-Δ∑k(Ck↑+C-k↓++C-k↓Ck↑),
where E0=εf+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(3)H=∑k,σεkCkσ+Ckσ+∑k,σE0fkσ+fkσ+γ0∑k,σ(fkσ+Ckσ+Ckσ+fkσ)+U2∑i,σniσfni,-σf-Δ∑k(Ck↑+C-k↓++C-k↓Ck↑)-Δ∑k(fk↑+f-k↓++f-k↓fk↑),
where Ckσ+(Ckσ) and fkσ+(fkσ) 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. nif=fiσ+fiσ 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
(4)H=H1+H2+H12,
where   (5a)H1=∑kσεkCkσ+Ckσ-Δ∑k(Ck↑+C-k↓++C-k↓Ck↑),(5b)H2=∑kσEkfkσ+fkσ-Δ∑k(fk↑+f-k↓++f-k↓fk↑),(5c)H12=γ0∑kσ(fkσ+Ckσ+Ckσ+fkσ). 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 εk and Ek are the band energies of the conduction electron and other-electron band measured from the Fermi energy.E
k
=
E
0
+
U
n
-
σ is the energy collected the non-interaction with a modified other-electron band. Ckσ+(Ckσ) and fkσ+(fkσ) 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
(6)Δ=V2∑k(〈Ck↑+C-k↓+〉+〈fk↑+f-k↓+〉),
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:(7)G(k,τ)=-〈Tτψk(τ)ψk+(0)〉,
where ψk+=(Ck↑+,C-k↓,fk↑+,f-k↓) and Tτ is the ordering operator for imaginary time, τ=it.After some calculations, the Green function in Nambu representation is obtained:(8)G(ωn,k)=(iωn-(εk-Ek2)ρ3σ3-(εk+Ek2)σ3+Δσ1-γ0ρ1σ3εk-Ek2)-1,
where ωn=πT(2n+1), T is temperature, n is an integer, and ρi and σi(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(ωn,k)=iωn-γkρ3σ3-δkρ0σ3+Δρ0σ1+wρ1σ3. All parameters detailed can be found in [25]. We find that (εk-Ek)/2≡γk and (εk+Ek)/2≡δk, where γk and δk 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
(9)1V=14∑k(tanh(Δ2+ε-2/2T)Δ2+ε-2+tanh(Δ2+ε+2/2T)Δ2+ε+2),
where ε+=(εk+Ek)/2+((εk-Ek)/2)2+γ02 and  ε-=(εk+Ek)/2-((εk-Ek)/2)2+γ02. 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 Tc by putting Δ→0; then
(10)1V=14∑k(tanh(ε-/2Tc)ε-+tanh(ε+/2Tc)ε+).
In the absence of the hybridization interaction, γ0=0; that is,
(11)1V=12∑k(tanh(ε/2Tc0)ε)or1λ=ln(2γωDπTc0),
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 ωD is the Debye cutoff energy. Tc0 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 (εk≈Ek) and the superconductor with conduction electron band coexisting with lower-energy other-electron band (εk≫Ek,  Ek≈0).Case 1.
The superconductor that the conduction electron band having the same energy as other-electron band.In this case, the approximation isεk≈Ek. Then, we get ε-≈εk-γ0 and ε+≈εk+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
(12)1λ=14(∫-ωDωDtanh(ε-/2Tc)ε-dεk+∫-ωDωDtanh(ε+/2Tc)ε+dεk)≈ln(2γωD2-γ02πTc).
The critical temperature is
(13)Tc=1.13ωD2-γ02e-1/λ.
And the zero-temperature energy gap can be found as
(14)1λ=14∫-ωDωD(1Δ2(0)+(εk-γ0)2+1Δ2(0)+(εk+γ0)2)dεk=12(sinh-1(ωD-γ0Δ(0))+sinh-1(ωD+γ0Δ(0))).
For ωD≫Δ(0), we can get
(15)Δ(0)=2ωD2-γ02e-1/λ.
From (13) and (15), the gap-to-Tc ratio is obtained:
(16)R=2Δ(0)Tc=3.53.
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-Tc ratio.To investigate the effect of hybridization and the Debye cutoff on gap-to-Tc ratio, we rewrite the gap equation at critical temperature and at zero-temperature into the form [29]
(17)∫(-ωD+γ0)/(2Tc)(ωD+γ0)/(2Tc)tanhxxdx=∫-2(ωD-γ0)/Tc2(ωD+γ0)/Tc1R2+x2dx.
The numerical calculation of (17) is shown in Figure 2.Within the definition of isotope effect coefficient in harmonic approximation;α=(1/2)(ωD/Tc)(dTc/dωD) and equation (9), we
(18)α=ωD2(tanh(ωD-/2Tc)/ωD-+tanh(ωD+/2Tc)/ωD+tanh(ωD-/2Tc)+tanh(ωD+/2Tc)),
where ωD+=ωD+γ0 and ωD-=ωD-γ0.Consider the limiting cases that the hybridization is so small with respect to Debye cutoff energy,ωD≫γ0; for ωD>2Tc, we can get that α≈(ωD/4)(1/ωD-+1/ωD+)≈1/2, and for ωD<2Tc, we get α≈(ωD/2)((1/Tc)/(ωD/Tc))≈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. Thef-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 Ek≈0; then
(19)ε-≈εk2-γ0,ε+≈εk2+γ0,forεk2<γ0,ε-≈0,ε+≈εk,forεk2>γ0.Substituting above approximation into (10), we can get
(20)1λ=14(∫-ωDωDtanh(ε-/2Tc)ε-dεk+∫-ωDωDtanh(ε+/2Tc)ε+dεk)≈ln(2γπTc2ωDγ0)+12(ωD2Tc-γ0Tc),
where,
(21)∫-ωDωDtanh(ε-/2Tc)ε-dεk≈2(ωD2Tc-2γ02Tc)+2ln(4γγ0πTc),∫-ωDωDtanh(ε+/2Tc)ε+dεk≈2ln(2γωDπTc).
The critical temperature is
(22)Tc=1.132γ0ωDe-1/λ+(1/2)(ωD/2Tc-γ0/Tc).
The gap equation as zero-temperature is
(23)1λ=14∫-ωDωD(1Δ2(0)+(ε-)2+1Δ2(0)+(ε+)2)dεk,
where,
(24)∫-ωDωD1Δ2(0)+(ε-)2dεk≈(2Δ(0))(ωD-2γ0)+2sinh-1(2γ0Δ(0)),∫-ωDωD1Δ2(0)+(ε+)2dεk≈2sinh-1(ωDΔ(0)).
Then, we get
(25)Δ(0)=22γ0ωDe-1/λ+(1/2)((ωD-2γ0)/Δ(0)).
According to (22) and (25), the gap-to-Tc ratio is
(26)R=2Δ(0)Tc=3.53e((ωD-2γ0)/Tc)(1/R-1/4).To investigate the effect of hybridization and the Debye cutoff on gap-to-Tc ratio, we rewrite the gap equation at critical temperature and at zero-temperature into the form [29]
(27)∫-ωD/TcωD/Tctanh(y/4-(y/4))xdx=∫-2(ωD-γ0)/Tc2(ωD+γ0)/Tc1R2+x2dx.
The numerical calculation of this equation is shown in Figure 2.Within the definition of isotope effect coefficient in harmonic approximation, (10) and Ek≈0, we can get
(28)α=(ωD2)×(1+(2Tc/ωD)tanh(ωD/2Tc)ωD-2γ0+2Tc(tanh(ωD/2Tc)+tanh(γ0/Tc))).


## 3. Results and Discussions

We use the hybridized two-band Hamiltonian to investigate the critical temperature, zero-temperature order parameter, gap-to-Tc 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 (εk≈Ek) and the superconductor that conduction electron band coexists with lower-energy other-electron band (εk≫Ek,  Ek≈0). The critical temperature, zero-temperature order parameter, gap-to-Tc 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 ofTc, (Figure 1), we find that the weak-coupling limit (λ<0.4) can be found in range of ωD/Tc>10. Then, we investigate the effect of hybridization in weak-coupling limit with ωD/Tc>10 and γ0/Tc=0.2,4.0. We find that Tc is decreased when the hybridization coefficient increased in Case 2 and no effect in Case 1. In Figure 2, the gap-to-Tc 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 ωD/Tc→∞. Case 2, R≈3.3–3.9 can be found to depend strongly on the value of γ0/Tc. The higher value of γ0/Tc 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
TheTc 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 Figure3. 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 ωD/Tc→∞.Figure 3
The isotope effect coefficient of Cases1 and 2 with varied hybridization coefficients.The Fe-based superconductors are mutiband system which comprises at least two bands which proposes-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 αFe=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-Tc 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. Tc 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.



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