Generalized Anderson’s theorem for superconductors derived from topological insulators

Generalizing Anderson’s theorem

We start generalizing Anderson’s theorem to superconductors having extra internal DOF. To address the effects of impurities in such superconductors, it is useful to consider a Bogoliubov–de Gennes HamiltonianĤBdG(k)=Ψk†(Ĥ0(k)Δ̂(k)Δ̂†(k)−Ĥ0*(−k))Ψk(1)written in terms of a multi-DOF Nambu spinor Ψk†=(Φk†,Φ−kT), encoding several DOF within Φk†=(c1k↑†,c1k↓†,…,cnk↑†,cnk↓†). Here, cmkσ† (c_mkσ) creates (annihilates) an electron in the internal DOF m with momentum k, and spin σ = { ↑ , ↓ }. Ĥ0(k) is the normal-state Hamiltonian in this multi-DOF basis, which can be parametrized asĤ0(k)=∑a,bhab(k)[τ̂a⊗σ̂b](2)where h_ab(k) are momentum-dependent real functions with subscripts a and b corresponding to the extra internal DOF and the spin DOF, respectively. If we focus on the case of two orbitals as the extra internal DOF (as in the Bi₂Se₃-based superconductors), then τ̂i and σ̂i (i = {1,2,3}) are the Pauli matrices to encode the orbital and spin DOFs, respectively, and τ̂0 and σ̂0 are identity matrices. In the most general case, there are 16 parameters h_ab(k). However, in the presence of time-reversal and inversion symmetries, the number of allowed h_ab(k) terms is reduced to only five plus h₀₀(k), with the associated matrices τ̂a⊗σ̂b forming a set of totally anticommuting matrices.

In Eq. 1, Δ̂(k) is the gap matrix, which can be parametrized in a similar formΔ̂(k)=∑a,bdab(k)[τ̂a⊗σ̂b(iσ̂2)](3)

Here, d_ab(k) denote form factors, which, in general, can have a k dependence determined by the pairing mechanism. However, when superconductivity is driven by phonons or by local interactions, the pairing force is isotropic and d_ab becomes independent of k. Previous works have shown that conventional electron-phonon coupling in the presence of Coulomb repulsion is a possible mechanism to drive odd-parity superconductivity in Bi₂Se₃-based superconductors (14, 15). In the following, we focus on k-independent d_ab and show that this assumption allows for a consistent description of the phenomenology of this family of materials.

The effects of impurities in multi-DOF superconductors can be understood by calculations similar in spirit as the standard calculations for simple metals within the Born approximation (11) (see details in section S1), from which we can infer the behavior of the critical temperature, T_c, as a function of the effective scattering rate ħΓ_Eff associated with depairing mechanisms. The calculations yield a familiar result, which is now generalized to encode the complexity of the normal and SC states in the multi-DOF basislog (TcTc0)=Ψ(12)−Ψ(12+ħΓEff2πkBTc)(4)where Tc0 is the critical temperature of the clean system, Ψ(x) is the digamma function, and ħΓ_Eff encodes all pair-breaking mechanisms throughħΓEff=14〈Tr[F˜C†(Ωk)F˜C(Ωk)]¯〉k(5)which is determined solely by the SC fitness function (12, 13)F̂C(k−k′)=V̂(k−k′)Δ̂−Δ̂V̂*(k−k′)(6)

This expression is valid for k-independent Δ̂ matrices of arbitrary dimension. Here, V̂(k−k′) is the matrix impurity scattering potential encoding all DOFs, F∼C(Ωk)=F̂C(Ωk)/Δ0, Δ₀ is the magnitude of the gap, Ω_k is the solid angle at the Fermi surface, the horizontal bar indicates impurity averaging, and the brackets indicate the average over the Fermi surface. This form of effective scattering rate accounts for the potentially nontrivial dependences of the pair wave functions and different scattering processes among the multiple DOFs through the SC fitness function F̂C(k). It is prudent to mention that the SC fitness concept was originally introduced as a measure of the incompatibility of the normal-state electronic structure with the gap matrix (12, 13). The effects of disorder on the SC state can also be inferred directly from the SC fitness function, if one introduces an impurity scattering potential to the normal-state Hamiltonian.

On the basis of Eqs. 4 to 6, we can now formulate the generalized Anderson’s theorem as follows: For a momentum-independent SC order parameter in the orbital basis encoded by the matrix Δ̂, the SC state is robust against impurities if the associated scattering potential V̂(k−k′) in the orbital basis satisfiesF̂C(k−k′)=0(7)

This condition recovers Anderson’s original result in the one band limit (in which case a nonmagnetic scattering potential is proportional to the identity) but now allows us to address more complex materials with extra internal DOF. As will be shown in the next section, for a scenario in which inter-DOF scattering is not allowed, Anderson’s theorem can still hold even if the gap has nodes when projected to the Fermi surface.

Robust superconductivity in the Bi₂Se₃-based materials

We can now use the fitness function to discuss the robustness of the SC state observed in the Bi₂Se₃-based materials. The normal state can be described by focusing on the quintuple-layer (QL) units, as schematically depicted in Fig. 1. The QL has D_3d point group symmetry, and the low-energy electronic structure can be described by an effective two-orbital model (16). The orbitals stem from Bi and Se atoms and have p_z character. By a combination of hybridization, crystal field effects, and spin-orbit coupling, one can identify two effective orbitals with opposite parity, labeled P_1z+ and P_2z−, with the ± sign indicating the parity (17). A schematic representation of the orbitals is given in Fig. 1B. With the definition Φk†=(c1↑†,c1↓†,c2↑†,c2↓†)k, the normal-state Hamiltonian can be parametrized as Eq. 2. In the presence of time-reversal and inversion symmetries, only the terms with (a, b) = {(0,0), (2,0), (3,0), (1,1), (1,2), (1,3)} are allowed in the Hamiltonian. The properties of the respective matrices under the point group operations allow us to associate each of these terms to a given irreducible representation of D_3d, therefore constraining the momentum dependence of the form factors h_ab(k). More details on the parametrization of the Hamiltonian are given in section S2.

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The gap matrix can be parametrized as in Eq. 3 in the orbital basis. As already noted, we focus on k-independent Δ̂ matrices in this basis because the pairing force is considered to be isotropic in Bi₂Se₃-based superconductors (14, 15, 18). Within the D_3d point group symmetry, the allowed order parameters are summarized in table S2. In Fig. 2A, we provide a schematic representation of pairing in the orbital basis, in which one can distinguish intraorbital pairing in the even A_1g channel from interorbital pairing in the odd channels (19). Given the experimental evidence for nodes along the y direction in CPSBS (9), here we focus on the following E_u order parameter:Δ̂=Δ0[iτ̂2⊗σ̂1(iσ̂2)]=Δ0(00−10000110000−100)(8)which is spin triplet and orbital singlet. Under the action of the parity operator P̂=τ̂3⊗σ̂0, one can infer that this is an odd-parity state, even though the gap matrix is momentum independent. The oddness of this order parameter stems from the different parity of the two underlying orbitals. This k-independent order parameter in the orbital basis acquires nodes along the y axis once projected to the Fermi surface in the band basis (as shown schematically in Fig. 2B; see section S3 for explicit calculations).

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Given the order parameter from Eq. 8, we can now use the SC fitness function to understand the robustness of the SC state in CPSBS. We write the explicit form of the matrix impurity scattering potential V̂(k−k′) asV̂(k−k′)=[V0(k−k′)τ̂0⊗σ̂0+Vs(k−k′)S⋅(τ̂0⊗σ̂)](9)where V_a(k − k′) is the Fourier transform of the scattering potential introduced by a localized impurity in real space. Here, a = {0, s} indicates nonmagnetic and magnetic impurity scattering, respectively; S signifies the spin of the magnetic impurities, and σ̂ the spin of the scattered electrons. The key aspect of the scattering potential for this material is the absence of orbital mixing, namely, no terms with τ̂1 or τ̂2 contributions. This is guaranteed by the opposite parity of the effective orbitals within the assumption of an inversion symmetric impurity potential. Another key aspect is the fact that the two orbitals are symmetric and antisymmetric combinations of the same two orbitals in the top/bottom layers; therefore, there is also no τ̂3 term in the scattering potential (since it would imply that the scattering amplitude for the two effective orbitals is different). Under these circumstances, the scattering potential can only have the simple form given above.

Note that the scattering associated with nonmagnetic impurities is rather trivial in its matrix form, ∼ τ̂0⊗σ̂0, which always commute with the gap matrices Δ̂. As a consequence, the generalized Anderson’s theorem, as formulated in Eq. 7, is satisfied, and the effective scattering rate for nonmagnetic impurities in CPSBS with the SC order parameter in the E_u channel is zero, even though the gap changes sign across the nodes, which are present in this E_u channel. Note that in this theoretical framework, the gap nodes are induced by the normal-state band structure once one translates the problem from the orbital basis to the band basis, as schematically shown in Fig. 2B and discussed in detail in section S3. The conclusion of zero scattering rate is valid for any momentum-independent SC order parameter possible for the Bi₂Se₃-based materials because the identity matrix τ̂0⊗σ̂0 commutes with any Δ̂ of the form τ̂a⊗σ̂b.

Previous theoretical works have discussed unexpected effects of impurities in multiband and multiorbital superconductors, mostly focusing on Fe-based superconductors with s₊₊ or s₊₋ SC states, which ultimately belong to the trivial irreducible representation of the respective point group (20, 21). Here, we proposed a novel scenario, in which order parameters in a nontrivial irreducible representation displaying symmetry-protected nodes are shown to be robust against disorder. In the context of superconductors derived from Bi₂Se₃, Michaeli and Fu discussed how spin-orbit locking could parametrically protect unconventional SC states, but their results are valid only for states with pairs of electrons of the same chirality, restricting the analysis to order parameters in the A_1g and A_1u representations (22). More recently, Nagai proposed that the interorbital spin-triplet state with E_u symmetry can be mapped to an intraorbital spin-singlet s-wave pairing if the roles of spin and orbital are exchanged in the Hamiltonian, and he argued that this mapping could provide a mechanism for Anderson’s theorem to remain valid in the presence of strong spin-orbit coupling (23). Both works rely on assumptions that are not valid for all SC symmetry channels and depend on strong spin-orbit coupling. These restrictions are not required for the above generalization of Anderson’s theorem for multi-DOF superconductors, which shows that the robustness of the SC state against impurities is guaranteed by the isotropic nature of the pairing interaction written in the local orbital basis (leading to a momentum-independent order parameter in this microscopic basis), under the requirement that impurity scattering is not allowed between orbitals with opposite parity. These considerations are concisely captured by the SC fitness function F̂C(k). We emphasize that this framework has a particular importance in the context of topological superconductors because the topological nature is often endowed by the extra DOF (7).

The case of CPSBS

CPSBS is a superconductor obtained by intercalating Cu into its parent compound (PbSe)₅(Bi₂Se₃)₆, which is a member of the (PbSe)₅(Bi₂Se₃)_3m homologous series realizing a natural heterostructure formed by a stack of the trivial insulator PbSe and the topological insulator Bi₂Se₃ (8). It was recently elucidated (9) that CPSBS belongs to the class of unconventional superconductors derived from Bi₂Se₃, including Cu_xBi₂Se₃ (7), Sr_xBi₂Se₃ (24, 25), and Nb_xBi₂Se₃ (26, 27), that have a topological odd-parity SC state, which spontaneously breaks rotation symmetry (28). In contrast to the fully opened gap in Cu_xBi₂Se₃ (29), the gap in CPSBS appears to have symmetry-protected nodes (9).

Figure 3A shows the temperature dependence of the electronic specific heat c_el, which is obtained from the total specific heat c_p by subtracting the phononic contribution c_ph (9), for the two samples studied in this work. The line-nodal gap theory (30) describes the c_el(T) data well, and the fits using this theory allow us to estimate the SC volume fraction, which is 85 and 100% for samples I and II, respectively. The thermal conductivity κ was measured on the same samples down to 50 mK (Figs. 3B and 4) with the configuration depicted in Fig. 3C. Note that our previous study of c_p in CPSBS in rotating magnetic field has revealed that line nodes are located in the a direction (9). The c_p(T) data in the normal state obey c_p = γ_elT + β_phT³ (9), and we extract the phononic specific-heat coefficient β_ph = 5.1 (5.2) mJ/molK⁴ and the electronic specific-heat coefficient γ_el = 5.8 (6.9) mJ/molK² for sample I (II). The κ/T data present no anomaly at T_c (Fig. 3B), suggesting that electron-electron scattering is not dominant.

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In the κ(T) data, one can separate the phononic and the electronic contributions to the heat transport when the κ/T versus T² plot shows a linear behavior at low enough temperature. In our samples, this happens for T ≲ 100 mK (Fig. 4, A and B), where phonons enter the boundary scattering regime and the phononic thermal conductivity κ_ph changes as b_phT³ (see the Supplementary Materials). A finite intercept of the linear behavior in this plot means that there is a residual electronic thermal conductivity κ₀ that originated from residual quasiparticles, whose contribution increases linearly with T, i.e., κ₀ = a_eT. In nodal superconductors, it has been established (1) that impurity scattering gives rise to a finite density of residual quasiparticles even at zero temperature, which is responsible for the finite a_e. Upon application of a magnetic field H, vortices create additional quasiparticles that affect κ. In both samples, the magnetic-field dependence of a_e is sublinear (see Fig. 4, C and D), and this is most likely due to the Doppler shift of the superfluid around vortices, which leads to a ∼ H increase in c_el in a nodal superconductor (31). Note that the exact H dependence of a_e would not be simple because vortices enhance both the quasiparticle density and their scattering rate (32). In 2.5 T, the superconductivity is fully suppressed, and the κ/T data are those of the normal state.

At this point, it is important to notice that these κ/T data unambiguously show the presence of residual mobile quasiparticles down to 50 mK, which gives convincing evidence for the existence of gap nodes. In particular, sample II is essentially 100% SC as indicated by the c_p data, and yet, this sample in 0 T shows significant electronic heat conduction in the zero-temperature limit, which accounts for ∼24% of the normal-state heat conduction (see Fig. 4D). This is impossible for a fully gapped superconductor. The case for sample I is similar: Although the SC volume fraction of this sample is ∼85% and hence one would expect some residual heat conduction at the level of 15% of the normal-state value (shown by the hatch at the bottom of Fig. 4C) due to the non-SC portion of the sample, the actual residual heat conduction in 0 T accounts for ∼45% of the normal-state value, which strongly points to the contribution of residual nodal quasiparticles.