## Abstract

A well-known result in unconventional superconductivity is the fragility of nodal superconductors against nonmagnetic impurities. Despite this common wisdom, Bi_{2}Se_{3}-based topological superconductors have recently displayed unusual robustness against disorder. Here, we provide a theoretical framework that naturally explains what protects Cooper pairs from strong scattering in complex superconductors. Our analysis is based on the concept of superconducting fitness and generalizes the famous Anderson’s theorem into superconductors having multiple internal degrees of freedom with simple assumptions such as the Born approximation. For concreteness, we report on the extreme example of the Cu* _{x}*(PbSe)

_{5}(BiSe

_{3})

_{6}superconductor. Thermal conductivity measurements down to 50 mK not only give unambiguous evidence for the existence of nodes but also reveal that the energy scale corresponding to the scattering rate is orders of magnitude larger than the superconducting energy gap. This provides the most spectacular case of the generalized Anderson’s theorem protecting a nodal superconductor.

## INTRODUCTION

Unconventional superconductors distinguish themselves from conventional ones by breaking not only *U*(*1*) gauge but also additional symmetries, usually reducing the point group associated with the normal-state electronic fluid. This extra symmetry reduction stems from the development of order parameters with nontrivial form factors, typically introducing point or line nodes in the excitation spectra. Nodal gap structures are especially known to give rise to power-law behavior in transport and thermodynamic quantities, which can be clearly detected in experiments, and are established as a key signature of unconventional superconductivity (*1*, *2*). However, nodal structures are also known to make superconductivity fragile in the presence of impurities, and many unconventional superconductors have actually been shown to be extremely sensitive to disorder (*3*, *4*).

Against all odds, the superconductivity in Bi_{2}Se_{3}-based materials was recently reported to present unusual robustness against disorder (*5*, *6*), despite showing nematic properties that point to unconventional topological superconductivity (*7*). Here, we report a notable observation that the Cu* _{x}*(PbSe)

_{5}(Bi

_{2}Se

_{3})

_{6}(CPSBS) superconductor (

*8*), which also shows nematic properties (

*9*), gives unambiguous evidence for the existence of gap nodes, while the scattering rate is more than an order of magnitude larger than the gap, a circumstance where nodal superconductivity is completely suppressed according to common wisdom. To understand this apparent puzzle, we generalize Anderson’s theorem (

*10*,

*11*) to complex superconducting (SC) materials encoding extra internal degrees of freedom (DOF), such as orbitals, sublattices, or valleys. It turns out that as long as the pairing interaction is isotropic, superconductors having a momentum-dependent gap structure (which manifests itself in the band basis) are generically protected from nonmagnetic scattering that does not mix the internal DOF. Our analysis is performed in the Born approximation and is based on the concept of SC fitness (

*12*,

*13*), a useful tool for understanding the robustness of SC states involving multiple DOF.

## RESULTS

### 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 Hamiltonian*c*_{mkσ}) creates (annihilates) an electron in the internal DOF *m* with momentum **k**, and spin σ = { ↑ , ↓ }. *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 Bi

_{2}Se

_{3}-based superconductors), then

*i*= {1,2,3}) are the Pauli matrices to encode the orbital and spin DOFs, respectively, and

*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

*h*

_{00}(

**k**), with the associated matrices

In Eq. 1,

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*becomes independent of

_{ab}**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 Bi

_{2}Se

_{3}-based superconductors (

*14*,

*15*). In the following, we focus on

**k**-independent

*d*and show that this assumption allows for a consistent description of the phenomenology of this family of materials.

_{ab}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 basis*x*) is the digamma function, and ħΓ_{Eff} encodes all pair-breaking mechanisms through*12*, *13*)

This expression is valid for **k**-independent _{0} 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 *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

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 Bi_{2}Se_{3}-based materials

We can now use the fitness function to discuss the robustness of the SC state observed in the Bi_{2}Se_{3}-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

*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.

The gap matrix can be parametrized as in Eq. 3 in the orbital basis. As already noted, we focus on **k**-independent _{2}Se_{3}-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:

**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).

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 *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

Note that the scattering associated with nonmagnetic impurities is rather trivial in its matrix form, *E _{u}* channel is zero, even though the gap changes sign across the nodes, which are present in this

*E*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 Bi

_{u}_{2}Se

_{3}-based materials because the identity matrix

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 Bi_{2}Se_{3}, 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

*7*).

### The case of CPSBS

CPSBS is a superconductor obtained by intercalating Cu into its parent compound (PbSe)_{5}(Bi_{2}Se_{3})_{6}, which is a member of the (PbSe)_{5}(Bi_{2}Se_{3})_{3m} homologous series realizing a natural heterostructure formed by a stack of the trivial insulator PbSe and the topological insulator Bi_{2}Se_{3} (*8*). It was recently elucidated (*9*) that CPSBS belongs to the class of unconventional superconductors derived from Bi_{2}Se_{3}, including Cu* _{x}*Bi

_{2}Se

_{3}(

*7*), Sr

*Bi*

_{x}_{2}Se

_{3}(

*24*,

*25*), and Nb

*Bi*

_{x}_{2}Se

_{3}(

*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

*Bi*

_{x}_{2}Se

_{3}(

*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_{el}*T* + β_{ph}*T*^{3} (*9*), and we extract the phononic specific-heat coefficient β_{ph} = 5.1 (5.2) mJ/molK^{4} and the electronic specific-heat coefficient γ_{el} = 5.8 (6.9) mJ/molK^{2} for sample I (II). The κ/*T* data present no anomaly at *T*_{c} (Fig. 3B), suggesting that electron-electron scattering is not dominant.

In the κ(*T*) data, one can separate the phononic and the electronic contributions to the heat transport when the κ/*T* versus *T*^{2} 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*_{ph}*T*^{3} (see the Supplementary Materials). A finite intercept of the linear behavior in this plot means that there is a residual electronic thermal conductivity κ_{0} that originated from residual quasiparticles, whose contribution increases linearly with *T*, i.e., κ_{0} = *a*_{e}*T*. 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 *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.

## DISCUSSION

To put the observed magnitude of κ into context, the Wiedemann-Franz law κ_{0}/*T* = *L*_{0}/ρ_{res} is useful (_{res} is the residual resistivity). Using this formula and the observed κ_{0}/*T* values in the normal state, we obtain ρ_{res} of 4.6 and 6.3 μΩm for samples I and II, respectively, which compares well to the direct measurements of ρ_{res} (*8*). We now make an order-of-magnitude estimate of the scattering time τ from ρ_{res} using the simple Drude model ρ_{res} = *m**/(*ne*^{2}τ) and the relation between the effective mass *m** and γ_{el} for a two-dimensional free electron gas. With γ_{el} = 6.9 mJ/molK^{2} of sample II, one obtains *V*_{mol} = 115.8 cm^{3}/mol is the Bi_{2}Se_{3} molar volume used for the normalization of *c*_{p}, and *c*_{0} = 1.27 nm is the height of the corresponding unit cell. With the typical carrier density 1.2 × 10^{21} cm^{−3} in CPSBS (*8*), one obtains τ = 2.2 × 10^{−14} s for sample II. Since *m** = 4.7*m*_{e} obtained from γ_{el} is likely an overestimate of the transport effective mass, it only gives an upper bound for τ. Hence, we obtain a lower bound of the scattering rate ħΓ = ħ/τ = 30 meV, which is already more than an order of magnitude larger than the SC gap Δ_{0} ≃ 0.5 meV. We note that the mobility in CPSBS is only ~10 cm^{2}/Vs, which precludes the determination of *m** from quantum oscillations, although *m** ≃ 0.2*m*_{e} has been estimated from quantum oscillations in Cu* _{x}*Bi

_{2}Se

_{3}(

*33*) and Nb

*Bi*

_{x}_{2}Se

_{3}(

*34*). Note that, if the actual effective mass is lighter than 4.7

*m*

_{e}in CPSBS, then ħΓ becomes larger, and the conclusion about the robustness becomes even stronger. The estimates of ħΓ for other Bi

_{2}Se

_{3}-based superconductors from the same Drude analyses unanimously give values larger than Δ

_{0}[see section S5 and (

*24*,

*35*–

*37*)], indicating the universal nature of the robustness in this family of unconventional superconductors.

It is crucial to notice that the universal thermal conductivity (*38*–*40*), which is expected only in clean superconductors satisfying ħΓ ≪ Δ_{0}, is not observed here. A simple estimate of the expected magnitude of the universal thermal conductivity *39*) makes this situation clear: By using the Fermi velocity *v*_{F} = 4.8 × 10^{5} m/s obtained from the angle-resolved photoemission experiments on CPSBS (*41*), one finds _{0}/*T* in CPSBS in 0 T, indicating that the κ_{0}/*T* value is significantly reduced from its clean-limit value due to strong impurity scattering. This gives convincing evidence that the strong scattering corresponding to ħΓ ≫ Δ_{0} is at work not only in the normal state but also in the SC state. Note that in high-*T*_{c} cuprates, the strong scattering leading to the “bad metal” behavior in the normal state is suppressed in the SC state, leading to the universal thermal conductivity to be observed in the mK region; clearly, this is not the case here.

Hence, one can safely conclude that in CPSBS, the energy scale of the scattering rate is much larger than the SC gap, which would normally preclude the realization of unconventional superconductivity with a nodal gap. This provides a spectacular proof of the generalized Anderson’s theorem in a multi-DOF superconductor. It is useful to note that the unusual robustness in *T*_{c} against disorder was already noted for Cu* _{x}*Bi

_{2}Se

_{3}(

*5*) and Nb

*Bi*

_{x}_{2}Se

_{3}(

*6*), and the penetration-depth measurements of Nb

*Bi*

_{x}_{2}Se

_{3}also found evidence for nodes (

*42*), but the origin of the robustness remained a mystery. This mystery has actually been a reason for hindering part of the community from accepting Bi

_{2}Se

_{3}-based materials as well-established unconventional superconductors. The present work finally solved this mystery, and it further provides a new paradigm for understanding the robustness of unconventional superconductivity. The new framework presented here will form the foundation for understanding the superconductivity in novel quantum materials where extra internal DOF such as orbitals, sublattices, or valleys govern the electronic properties.

## MATERIALS AND METHODS

High-quality CPSBS single crystals were grown by a modified Bridgman method as described before (*9*). Two samples from the same growth batch were measured. The dimensions of samples I and II were 2.8 mm by 2.5 mm by 0.35 mm and 5.6 mm by 2.0 mm by 0.20 mm, respectively. The exact *x* values of samples I and II were 1.47 and 1.29, respectively. The specific heat *c*_{p} was measured with a relaxation method in a Quantum Design PPMS down to 300 mK. Following previous works on CPSBS (*8*, *9*), the SC volume fraction was estimated from the *c*_{p} data by subtracting the phononic contribution *c*_{ph} and fitting the electronic contribution *c*_{el} with a line-nodal gap theory (*30*), yielding 85 and 100% for samples I and II, respectively. The shielding fraction at 1.8 K measured with a SQUID magnetometer in 0.2 mT applied parallel to the *ab* plane was 75 and 88% in samples I and II, respectively. The thermal conductivity κ was measured on the same samples in a dilution refrigerator (Oxford Instruments Kelvinox 400) with the standard steady-state method depicted in Fig. 3C in the main text using RuO_{2} thermometers. The temperature gradient ∇*T* was applied parallel to the *b* axis, and the magnetic field was applied along the *c** axis; note that CPSBS belongs to the *C*2/*m* space group, where

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/9/eaay6502/DC1

Section S1. The concept of SC fitness and the effective scattering rate

Section S2. The normal-state Hamiltonian for materials in the family of Bi_{2}Se_{3}

Section S3. The order parameters for materials in the family of Bi_{2}Se_{3}

Section S4. Analysis for *C*_{2h} symmetry

Section S5. Drude analysis of the scattering rates in Bi_{2}Se_{3}-based superconductors

Section S6. Phononic contribution to the thermal conductivity

Table S1. Parametrization of the normal-state Hamiltonian.

Table S2. Superconducting order parameters for the materials in the family of Bi_{2}Se_{3}.

Table S3. Analysis of the gap structure for the materials in the family of Bi_{2}Se_{3}.

Table S4. Estimates of the scattering rates in Bi_{2}Se_{3}-based superconductors from the simple

Drude analysis as was done for CPSBS in the main text.

Fig. S1. Behavior of phonons.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**A.R. would like to thank F. Herman for the valuable discussions.

**Funding:**This work was funded by the Deutsche Forschungsgemeinschaft (DFG; German Research Foundation) under CRC 1238-277146847 (A04 and B01) and under Germany’s Excellence Strategy–Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769. A.R. acknowledges the support of Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) project 2018/18287-8, Fundação para o Desenvolvimento da UNESP (FUNDUNESP) process 2338/2014-CCP, and ICTP-SAIFR FAPESP grant 2016/01343-7.

**Author contributions:**Y.A. conceived the experiment and A.R. constructed the theory. L.A. and Z.W. performed sample growth and characterizations. L.A. measured the specific heat and thermal conductivity with help from T.L. and L.A. T.L. and Y.A. analyzed the data. Y.A., A.R., and L.A. wrote the manuscript with inputs from all authors.

**Competing interests:**The authors declare that they have no competing interests.

**Data availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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