Dirty two-band superconductivity with interband pairing order

We study theoretically the effects of random nonmagnetic impurities on the superconducting transition temperature $T_c$ in a two-band superconductor characterized by an equal-time s-wave interband pairing order parameter. The Fermi-Dirac statistics of electrons allows a spin-triplet s-wave pairing order as well as a spin-singlet s-wave order parameter due to the two-band degree of freedom. In a spin-singlet superconductor, $T_c$ is insensitive to the impurity concentration when we estimate the self-energy due to the random impurity potential within the Born approximation. On the other hand in a spin-triplet superconductor, $T_c$ decreases with the increase of the impurity concentration. We conclude that Cooper pairs belonging to odd-band-parity symmetry class are fragile under the random impurity potential even though they have s-wave pairing symmetry.


Introduction
Conventional wisdom suggests that the dependence of superconducting transition temperature T c on the concentration of nonmagnetic impurities is closely related to the momentum-symmetry of the pair potential. It is well known that T c of an s-wave superconductor is insensitive to the impurity concentration. [1,2,3] On the other hand, unconventional superconductivity such as p-and d-wave symmetry is fragile in the presence of impurities. The robustness of an s-wave Cooper pair under potential disorder, however, may be weakened in a two-band superconductor as discussed in previous literature [4,5,6,7,8,9,10]. In these papers, the intraband pairing order is assumed in each conduction band. Namely, two electrons at the first (second) band form the pair potential ∆ 1 (∆ 2 ). Such theoretical model would describe the superconducting states in MgB 2 [11,12] and iron pnictides [13,14]. The suppression of T c by the interband impurity scatterings is a common conclusion of all the theoretical studies.
In addition to the intraband pair potentials, the interband (or interorbital) Cooper pairing order has been discussed in a topological superconductor Cu x Bi 2 Se 3 [15,16,17]. Various types of multiband superconductivity would be expected in topological-material based superconductors because the band-crossing plays an essential role in realizing the topologically nontrivial states. Moreover, a possibility of interband/interorbital Cooper pairing is pointed out also in a heavy fermionic superconductor UPt 3 [18,19] and an antiperovskite superconductor Sr 3−x SnO. [20]. In addition to the spin-singlet order parameter, the spin-orbit coupling may make the spin-triplet order parameter possible. Thus a superconductor with the interband pairing order can be a superconductor of a novel class. So far, however, little attention has been paid to physical phenomena unique to an interband superconductor.
In this paper, we theoretically study the effects of nonmagnetic random impurities on T c in a two-band superconductor characterized by an equal-time s-wave interband pairing order. The pair potential is defined by the product of two annihilation operators of an electron. Therefore, the pair potential must be antisymmetric under the commutation of the two annihilation operators, which is the requirement from the Fermi-Dirac statistics of electrons. Due to the two-band degree of freedom, a spintriplet s-wave pair potential is allowed as well as a spin-singlet s-wave one. The latter is symmetric under the permutation of the two band indices (even-band-parity), whereas the former is antisymmetric (odd-band-parity). The effects of impurity potential are considered through the self-energy estimated within the Born approximation. The transition temperature is calculated from the linearized gap equation. We find that T c is insensitive to the impurity concentration in a spin-singlet s-wave interband superconductor. However, T c in a spin-triplet s-wave case decreases with the increase of the impurity concentration. We conclude that odd-band parity Cooper pairs are fragile under the potential disorder even though they belong to s-wave symmetry class. This paper is organized as follows. In Sec. 2, we explain the normal state that makes possible spatially uniform interband Cooper pairing orders. The gap equation in the clean limit is derived for both a spin-singlet superconductor and a spin-triplet superconductor. The effects of random impurities on the superconducting transition temperature are studied in Sec. 3. The conclusion is given in Sec. 4. Throughout this paper, we use the units of k B = c = = 1, where k B is the Boltzmann constant and c is the speed of light.

Interband pairing order
The interband s-wave pair potential is defined by where ψ † λ,σ (r) (ψ λ,σ (r)) is the creation (annihilation) operator of an electron with spin σ (=↑ or ↓) at the λ th conduction band and g > 0 represents the interband attractive interaction. By applying the Fourier transformation, the pair potential becomes In the second line, we assume the spatially uniform order parameter which is realized at k + k ′ = 0. To apply the weak coupling mean-field theory, the state at k with spin σ in the first band and the state at −k with spin σ ′ in the second band must be degenerate at the Fermi level. Otherwise interband Cooper pairs have the center-of-mass momenta and their order parameter oscillates in real space [21,22,23]. Thus the interband pair potential requires a characteristic band structure. In this paper, we consider a normal state described by the Hamiltonian, where m is the mass of an electron, µ is the chemical potential, and v represents the hybridization between the two conduction bands. Generally speaking, the hybridization potential is a complex number characterized by a phase θ. We will show that observable values in a superconductor are independent of θ although the expression of the Green function depends on it. Throughout this paper, Pauli matrices in spin, two-band, particle-hole spaces are denoted byσ j ,ρ j , andτ j for j = 1 − 3, respectively. In addition, σ 0 ,ρ 0 , andτ 0 are the unit matrices in these spaces. Since the two bands are identical to each other, the Hamiltonian preserves the symmetry described by where T is the time-reversal operator, K means the complex conjugation. Thus Γ represents the combined operation of the time-reversal and the exchange between the two bands. The normal state Hamiltonian in Eq. (6) is simplest model which satisfies Eq. (8). The conclusions of this paper are insensitive to the normal state Hamiltonian. We will explain the reasons after reaching the main results. The electronic structure given in Eq. (6) may poses both the interband and the intraband s-wave order parameters in its superconducting phase. The effects of potential disorder on T c for intraband superconductivity have been already studied theoretically in previous papers [5,6,7,8,9,10]. In our model, the amplitudes of two intraband pair potentials are expected be equal to each other because of the symmetry in the two conduction bands. It has been well established that T c of intraband superconductivity in such symmetric case is insensitive to the impurity scatterings. [5,7,10] Thus we focus only on interband superconductivity in this paper. According to Eq. (4), we define the spatially uniform superconducting order parameter explicitly as In the two-band model, it is possible to define two types of interband pairing order: spin-singlet and spin-triplet. In spin-singlet symmetry, the pair potential in Eq. (10) is symmetric (antisymmetric) under the permutation of band (spin) indices On the other hand in spin-triplet symmetry, the pair potential in Eq. (10) is antisymmetric (symmetric) under the permutation of band (spin) indices In what follows, we consider opposite-spin-triplet pairing order. The Bogoliubov-de Gennes (BdG) Hamiltonian in momentum space is represented bȳ where∆ S and∆ T represent the spin-singlet pair potential and the spin-triplet one, respectively. Hereafter we fix the superconducting phase at zero for simplicity. The BdG Hamiltonian can be described in reduced 4 × 4 matrix form by choosing spin of an electron as ↑ and that of a hole as ↓, where s s = 1 for a spintriplet superconductor and s s = −1 for a spin-singlet superconductor. We note in the normal state that ξ * (−k) = ξ(k) holds true in the presence of time-reversal symmetry. The Green function is obtained by solving the Gor'kov equation, where ω n = (2n + 1)πT is a fermionic Matsubara frequency with T being a temperature. The solution of the normal Green function within the first order of ∆ is represented aŝ where we omit k from ξ(k) for simplicity. The results are common in both spin-singlet and spin-triplet cases because the normal Green function does not include the pair potential in the lowest order of ∆. The anomalous Green functions for a spin-singlet superconductor within the first order of ∆ is calculated aŝ Theρ 1 component in Eq. (20) is linked to the pair potential through the gap equation where N 0 is the density of states at the Fermi level per spin. We have used the relation where a and b are constants. The last equation in Eq. (22) is identical to the gap equation in the BCS theory. The hybridization generates theρ 0 andρ 3 components in Eq. (20) which belong to even-frequency spin-singlet even-momentum-parity even-band-parity (ESEE) symmetry class.

Effects of impurities
Let us consider the nonmagnetic random impurities described by A =τ 3ρ1 cos θ −ρ 2 sin θ.
The first and the second terms in Eq. (27) cause the intraband and the interband scatterings, respectively. We assume that the impurity potential satisfies the following properties, where · · · means the ensemble average, n imp is the impurity concentration, and v imp represents the strength of the impurity potential. We also assume that the attractive electron-electron interactions are insensitive to the impurity potentials [3]. To discuss the effects of impurities with Eqs. (29) and (30), Hamiltonian in real space is necessary. The impurity Hamiltonian in Eq. (27) is described in real space as well as the kinetic part and the hybridization in Eqs. (5) and (6). In the real space representation with the basis shown in Eq. (5), the random potential V imp (r) should be independent of band indices. The phase of random potential generating the interband scattering must be equal to that of the hybridization. Otherwise, time-reversal symmetry is broken. The effects of the impurity scatterings are taken into account through the self-energy estimated within the Born approximation. The Green function in the presence of the impurity potential is calculated within the second order perturbation expansion with respect to the impurity potential, where 0 in the subscript indicates unperturbed Green function. By considering Eqs. (29) and (30), we obtaiň The second and the third terms are derived from the intraband impurity scatterings and the interband impurity scatterings, respectively. By applying the Fourier transformation, the Green function becomeš whereΣ intra andΣ inter are the self-energy due to the intraband impurity scatterings and that of interband impurity scatterings, respectively. The details of the derivation are given in Appendix. In the Born approximation, the self-energies are represented aš The total self-energy is calculated aš Here the Green function after carrying out the summation of k is indicated by · · · as, whereρ ν with ν = 0 − 3 are the Pauli matrices in band space. The Gor'kov equation in the presence of impurities is expressed by Eq. (37) with Eqs. (38)-(43) give the general expression self-energy due to impurity scattering within the Born approximation. The properties in the normal state and those in the superconducting state are mainly embedded in the normal Green function in Eq. (42) and in the anomalous Green function in Eq. (43), respectively. Therefore the results can be applied to various two-band superconductors. Here we briefly mention a general feature of the self-energy. In Eq. (39), f 1 ρ 1 is present but f 2 ρ 2 is absent inΣ F because of the anticommutation relations amongρ ν . This feature is independent of the normal state Hamiltonian. As shown in the remaining part of this section, the effects of random nonmagnetic impurity scatterings on the transition temperature T c depends on spin symmetry of the pair potential. The difference comes from such general property ofΣ F . We will explain details of the difference in the following subsections.

spin-singlet
The normal part of the self-energy is calculated aŝ where τ imp represents the life time due to impurity scatterings. The factor 2 in Eq. (47) stems from the two contributions of different scattering processes: the intraband impurity scatterings and the interband impurity scatterings. In a spin-singlet superconductor, the self-energy of the anomalous part results in because Eq. (39) includes f 1 ρ 1 . As a consequence, the Gor'kov equation in the presence of impurities becomes, ω n = ω n η n ,∆ = ∆ η n , η n = 1 + 1 2τ imp |ω n | .
The self-energy renormalizes the frequency and the pair potential exactly in the same manner as ω n →ω n and ∆ →∆. As a consequence, the anomalous Green function can be calculated aŝ whereF 0 on the right hand side is shown in Eq. (20). The gap equation in the presence of impurities is given by Eq. (21) withF 0 (k, ω n ) →F (k, ω n ). The resulting gap equation remains unchanged from that in the clean limit. Thus the impurity scatterings do not change T c in a spin-singlet superconductor. The argument here is exactly the same as that in Ref. [1] for a single-band spin-singlet s-wave superconductor and is consistent with the Anderson's theorem [3].

spin-triplet
In a spin-triplet superconductor, the Green function in Eq. (43) with Eq. (24) is calculated as By substituting the results into Eq. (39), we find because Eq. (39) does not include f 2 ρ 2 . The resulting Gor'kov equation becomes, The impurity self-energy renormalizes the frequency as ω n →ω n but leaves the pair potential as it is. Thus the anomalous Green function in the presence of impurities becomesF whereF 0 on the right hand side is given in Eq. (24). The gap equation Eq. (21) witĥ The results suggest that the impurity scatterings decrease T c for a spin-triplet superconductor.
In Fig. 1, we show T c of a spin-triplet interband superconductor as a function of ξ 0 /ℓ, where T 0 is the transition temperature in the clean limit in the absence of the hybridization (i.e., v = 0), ξ 0 = v F /2πT 0 is the coherence length, v F = k F /m is the Fermi velocity, and ℓ = v F τ imp is the mean free path due to the impurity scatterings. We numerically solve Eq. (58) with ω c /2πT 0 = 10 3 . The results show that T c decreases with the increase of ξ 0 /ℓ. In the clean limit, T c decreases with the increase of the hybridization v as indicated in Eq. (25). The superconducting phase vanishes when the amplitude of hybridization goes over its critical value of v c ≈ 2πT 0 /C, where C = 4 e γ E and γ E = 0.577 is the Euler's constant. In the presence of impurities, the interband spin-triplet superconductivity vanishes at ξ 0 /ℓ ≈ 2/C = 0.281 at v = 0, ξ 0 /ℓ ≈ 0.244 at v = 0.5 v c , and ξ 0 /ℓ ≈ 0.168 at v = 0.8 v s .
The suppression of T c by impurities in a spin-triplet case can be interpreted as follows. The interband impurity scatterings hybridize the electronic states in the two bands and average the pair potential over the two-band degree of freedom. As shown in Eq. (15), the sign of pair potential in one sector is opposite to that in the other where we set s s = 1 for a triplet superconductor. Thus the pair potentials in the two sectors cancel each other when the interband impurity potential hybridizes the two sectors. As a result, the anomalous part of the self-energy vanishes as shown in Eq. (55). Namely, the impurity self-energy does not renormalize the pair potential, which leads to the  Figure 1. The transition temperature T c versus ξ 0 /ℓ. The impurity concentration is proportional to ξ 0 /ℓ, where ξ 0 is the coherence length and ℓ is the elastic mean free path. In a spin-singlet case, T c is independent of ξ 0 /ℓ within the Born approximation as shown with a broken line, which is consistent with the Anderson's theorem. The results for a spin-triplet interband superconductor at v = 0 are identical to those for a single-band unconventional superconductor characterized such symmetry as spinsinglet d-wave or spin-triplet p-wave.
suppression of T c . The absence of f 2 ρ 2 in Eq. (39) can be understood by such physical interpretation. It would be worth mentioning that the gap equation in Eq. (58) with v = 0 is identical to that for a single-band unconventional superconductor under the potential disorder. In a p-wave or d-wave superconductor, the anomalous Green function F 0 (ω n ) vanishes due to their unconventional pairing symmetries, which leads to Σ F = 0 and the suppression of T c . We conclude that the odd-band-parity pairing correlation is fragile under impurity potential even though it belongs to s-wave momentum parity symmetry class. Therefore, a clean enough sample is necessary to observe spin-triplet interband superconductivity in experiments.
Mathematically, the robustness of a spin-singlet s-wave interband superconducting state is described by the anomalous part of the self-energyΣ F =∆/2τ imp |ω n | in Eq. (48). The suppression of T c in a spin-triplet superconductor is described byΣ F = 0 in Eq. (55). As we already explained below Eq. (44), these features are derived from the general expression of the self-energy in Eq. (39) and are independent of the normal state Hamiltonian. Therefore, our conclusions are valid for various interband superconductors.

Conclusion
We studied the effects of random nonmagnetic impurities on the superconducting transition temperature T c in a two-band superconductor characterized by an equal-time s-wave interband pair potential. Due to the two-band degree of freedom, both spinsinglet and spin-triplet pairing order parameters satisfy the requirement from the Fermi-Dirac statistics of electrons. The effects of impurity potential is considered through the self-energy obtained within the Born approximation. The transition temperature is calculated from the linearized gap equation. In a spin-singlet superconductor, the random potential does not change T c . On the other hand in a spin-triplet superconductor, T c decreases with the increase of the impurity concentration. We conclude that Cooper pairs belonging to odd-band-parity symmetry class are fragile under the random impurity potential even though they belong to s-wave momentum symmetry.
A ± =ρ 1 cos θ ±ρ 2 sin θ. (62) Here we focus on the anomalous part of the self-energy because its general expression is important to justify the main conclusion. We find the relation, The most important feature is that f 2 ρ 2 component changes its sign due to the anticomutation relations amongρ j . Together with the intraband contribution ν f ν ρ ν , we obtain the general expression of the anomalous part in Eq. (39).