Pair breaking in multi-orbital superconductors: an application to oxide interfaces
We investigate the impact of impurity scattering on superconductivity in an anisotropic multi-orbital model with spin-orbit coupling which describes the electron fluid at two-dimensional oxide interfaces. As the pairing mechanism is under debate, both conventional and unconventional superconducting states are analyzed. We consider magnetic and nonmagnetic spin-dependent intra- and interorbital scattering and discuss possible microscopic realizations leading to these processes. It is found that, for magnetic disorder, the unconventional superconductor is protected against interband scattering and, thus, more robust than the conventional condensate. In case of nonmagnetic impurities, the conventional superconductor is protected as expected from the Anderson theorem and the critical scattering rate of the unconventional state is enhanced by a factor of four due to the spin-orbit coupling and anisotropic masses in oxide interfaces.
To understand the properties of many of the recently discovered superconductorsKamihara et al. (2006); Reyren et al. (2007), a multi-band description, which is often tied to the existence of multiple orbital degrees of freedom, is essential. Taking into account several Fermi surfaces does not only yield quantitative corrections, but also makes qualitatively different concepts possible such as pairing states that have no analogue in a single-band picture. One famous example is the state that has been proposed, e.g., in the iron-based superconductorsMazin et al. (2008); Kuroki et al. (2008) and earlier in other systemsAgterberg et al. (1999). While the pairing field is constant on a given Fermi surface, it changes sign between the electron and hole band.
The identification of a pairing state and mechanism constitutes a challenging task, in particular, in a multi-band system. The behavior of superconductivity under the influence of different types of disorder can give important hints about the order parameter and pairing glue. In case of conventional -wave superconductivity, the transition temperature is only very weakly affected by the presence of nonmagnetic impurities except for very strong disorder where the mean-free path is comparable to or smaller than the Fermi wavelength . This is usually referred to as the Anderson theoremAnderson (1959); Abrikosov and Gorkov (1958, 1959). When magnetic impurities are considered, the transition temperature is suppressedAbrikosov and Gorkov (1961) and vanishes when becomes of the order of the superconducting coherence length even when . Since various unconventional pairing states have been shownBalian and Werthamer (1963); Annett et al. (1991); Golubov and Mazin (1997); Mackenzie and Maeno (2003); Balatsky et al. (2006); Alloul et al. (2009) to be strongly suppressed already by nonmagnetic impurities, the sensitivity of the transition temperature to nonmagnetic disorder is commonly used as an indication of unconventional pairing. Unconventional superconductivity is thus expected to occur only in very clean systems.
However, several studiesGolubov and Mazin (1997); Michaeli and Fu (2012); Korshunov et al. (2014); Hoyer et al. (2015) of both magnetic and nonmagnetic disorder have revealed that, in some cases, unconventional superconductors can be much more robust than naively expected or even enjoy an analogue of the Anderson theorem. Refs. Golubov and Mazin, 1997 and Michaeli and Fu, 2012; Korshunov et al., 2014; Hoyer et al., 2015 indicate that it is essential to take into account all characteristic details of the system, such as multiple orbitals and spin-orbit coupling, in order to judge reliably the robustness of unconventional superconductors in the presence of disorder. From a theoretical point of view, it then becomes more difficult to find an approach that captures the essential features but leads to results that are universal in the sense that they do not depend on irrelevant microscopic details.
One example of a recently discovered superconductingReyren et al. (2007) system where the multi-orbital characterJoshua et al. (2012); Fête et al. (2012), spin-orbit couplingCaviglia et al. (2010); Ben Shalom et al. (2010) as well as strongly anisotropic effective massesSantander-Syro et al. (2011) seem to be crucial for understanding its properties is provided by the two-dimensional (2D) electron liquid that formsOhtomo and Hwang (2004) at the interface between the insulators LaAlO (LAO) and SrTiO (STO). In Ref. Scheurer and Schmalian, 2015a, two possible pairing states have been proposed for this system. If superconductivity arises as a consequence of conventional electron-phonon coupling, the pairing field will have the same sign on both spin-orbit split Fermi surfaces. In case of an unconventional, i.e., purely electronic, pairing mechanism, the order parameter has opposite signs on the two Fermi surfaces. Despite the similarities to the state known from the iron-based superconductors, there is one main crucial difference. The Fermi surfaces in LAO/STO interfaces are singly degenerate due to the combination of broken inversion symmetry and spin-orbit coupling which has important consequences for superconductivity (see also, e.g., Refs. Gor’kov and Rashba, 2001; Scheurer and Schmalian, 2015b). For example, the unconventional pairing state proposedScheurer and Schmalian (2015a) for oxide interfaces has been shown to be a time-reversal invariant topological superconductor which is intimately related to the Fermi surfaces being split by spin-orbit coupling.
Consequently, it constitutes an important open question whether such an exciting new state of matter is indeed realized in oxide interfaces. An analysis of the impact of disorder on the superconducting states proposed in Ref. Scheurer and Schmalian, 2015a is thus required. It is particularly important to perform a quantitative analysis beyond a simple one-orbital toy model calculation since experimental estimates of the coherence lengthReyren et al. (2007) and the mean-free pathBen Shalom et al. (2009); Chang et al. (2013) are of the same order.
In this paper, we consider impurity scattering in superconducting LAO/STO interfaces using a two-band model that captures the multi-orbital character, the spin-orbit splitting as well as the anisotropy of the masses. The resulting highly non-circular form of the Fermi surfaces and anisotropic textures of the wavefunctions are conveniently taken into account by introducing patches in the nested regions of the Fermi surfaces. Due to the strong anisotropy, the wavefunctions are approximately constant within each patch yielding results that only depend on a small set of parameters. We find that the spin and orbital polarization of the states at the Fermi surfaces enhances the stability of both the conventional and unconventional superconducting phase against impurity scattering. Furthermore, contrary to common wisdom, the unconventional superconductor is found to be, by a factor of two, more robust against magnetic scattering than the conventional condensate.
The remainder of the paper is organized as follows. In Sec. II, we introduce the clean model of superconductivity. The impurity configurations consistent with the symmetries of the system and possible microscopic realizations thereof are discussed in Sec. III. In Sec. IV, we present the approximations used to calculate the transition temperature in the disordered system. Finally, the results can be found in Sec. V.
Ii Superconductivity in the clean system
To describe superconductivity in oxide heterostructures, we use an effective Hamiltonian of the Ti and orbitals. To justify why these are the essential degrees of freedom, let us recall that the orbitals relevant for the electronic low-energy physics are the Ti states of the manifold, i.e. the , and orbitals. The states of the interface electron system derived from the orbital mainly localize and possibly order magneticallyPentcheva and Pickett (2007, 2008); Michaeli et al. (2012); Santander-Syro et al. (2014). This explains the discrepancy between the carrier concentration in the two-dimensional electron liquid expected from the polar catastrophe mechanism and that seen in transport measurements. The electronic states that lead to superconductivity are therefore most likely due to the and orbitals which is supported by recent experimentsJoshua et al. (2012); Fête et al. (2012).
Let us first focus on the non-interacting part of the effective Hamiltonian of these states,
where and describe the annihilation and creation of quasi-particles of crystal momentum in state , with labeling the four distinct combinations of the two orbitals (, ) and the spin orientations. The Hamiltonian is characterized by two ingredients, .
Firstly, the different overlap of the two orbitals along the - and -direction leads to anisotropic masses described by
where the upper (lower) component refers to the () orbital and is the unit matrix in spin space. Experimentally, large anisotropy ratios have been reportedSantander-Syro et al. (2011). The orbital mixing term describes interorbital second-nearest neighbor hopping.
Secondly, the properties of the electron liquid are strongly affected by spin-orbit coupling Caviglia et al. (2010); Ben Shalom et al. (2010). As shown in Ref. Scheurer and Schmalian, 2015a, this leads to the contribution
to the Hamiltonian. Here and are real constants and () denote Pauli matrices in spin (orbital) space. We see that besides the usual Rashba coupling (), the multi-orbital character of the model also allows for the atomic spin-orbit term proportional to and two Dresselhaus couplings (, ). Eq. (3) is the most general spin-orbit coupling Hamiltonian (up to linear order in ) consistent with time-reversal and the point symmetry of the system. Starting from a three-orbital model that also includes the band and integrating outScheurer and Schmalian (2015a) the latter reproduces Eq. (3) with the additional constraint .
Note that we do not include any coupling to the magnetic moments that possibly develop in the bands. The reason is that no significant correlation between the inhomogeneous magnetization of the orbitals and the superconducting response has been observedBert et al. (2011) and, hence, the effective exchange coupling between the localized magnetic states and the superconducting electrons is expected to be negligible for describing superconductivity. This can be understood as a consequence of the spatial separation of the localized magnetic and the itinerant superconducting bands and the small orbital admixture of in the states that host superconductivity.
The spin-orbit coupling (3) breaks inversion symmetry and removes the spin-degeneracy of the Fermi surfaces. For chemical potentials close (deviation ) to the bottom of the spectrum of , only two out of the four singly degenerate bands of Eq. (1) are relevant. As carrier concentration dependent studiesJoshua et al. (2012); Fête et al. (2012) show that the sweet spot of superconductivity is associated with the Fermi level entering the - and -bands, we will focus on the two bands of with lower energy. The corresponding form, spin and orbital polarization of the Fermi surfaces are shown in Fig. 1. In the patches, highlighted in green, the Fermi surfaces are nearly straight lines which are, in groups of four, approximately parallel (nesting). Within each patch, the wavefunctions are almost constant. The large mass anisotropy in leads to strong orbital polarization: The patches in the lobes along () are mainly -polarized (-polarized). Similarly, the spin-orbit coupling largely aligns the spin parallel or antiparallel to one of the in-plane coordinate axes. The strong spin and orbital polarization of the patch wavefunctions is of crucial relevance for impurity scattering as it can lead to the suppression of certain processes. This will be discussed in detail in Sec. V.
In Ref. Scheurer and Schmalian, 2015a, it has been shown that the system can have two distinct superconducting instabilities: In case of a conventional, electron-phonon induced, pairing mechanism, the pairing field has the same sign on both Fermi surfaces, which will be referred to as in the following. If the superconductor is due to the exchange of particle-hole fluctuations, the sign of the order parameter will change between the Fermi surfaces. Correspondingly, this unconventional superconducting state will be denoted by .
At low energies, the relevant interaction channel that distinguishes between these two superconductors is pair hopping, i.e., the annihilation of a Kramers pair of electrons from one band and the creation of a Kramers pair in the other. Depending on the sign of the associated coupling constant, either the or superconductor will be favored. As the sign conventions for this interaction term sensitively depends on the phase convention for the eigenstates of , the mathematical formulation of the pair hopping interaction will be postponed to Sec. IV.2.
Iii Disorder configurations
Here we discuss the possible impurity configurations that can be present in oxide interfaces. The most general (quadratic) Hamiltonian of a given disorder realization reads
where, for convenience, we have switched to a real-space representation with and denoting the Fourier-conjugates of and , respectively.
In what follows, we not only allow for disorder configurations that are consistent with the point group of the system but also take into account impurities that locally break any subset of the point symmetries. We only require these symmetries to be preserved on average. For example, oxygen vacancies that locally break the fourfold rotation symmetry occur with equal probability on bonds along the two perpendicular directions parallel to the interface. This effect is most easily discussed on the level of correlation functions. Focusing on Gaussian-distributed disorder, the entire information about the statistics is contained in the correlator
Here represents the disorder average only taking into account Hermitian disorder configurations, . The latter constraint makes the averaging procedure equivalent to a Gaussian integration over a real-valued field.
Within the replica approachEdwards and Anderson (1975), averaging over leads to an effective impurity-induced electron-electron interaction that reads in the action description as
Here, , are the Grassmann analogues of , with the additional replica index , where denotes the number of replicas. At the end of the calculation of physical quantities, the limit has to be taken. We see that the correlator defined in Eq. (5) plays the role of the bare disorder vertex function. The same holds in the diagrammatic disorder-averaging technique of Ref. Abrikosov et al., 1975.
For simplicity, we will restrict our analysis to spatially local disorder, i.e. in Eq. (6), and -correlated statistics which, in addition, forces . Furthermore, let us assume the disorder distribution to be homogeneous. Taken together, the impurity vertex becomes spatially local and translation invariant,
iii.1 Symmetry properties of the impurity vertex
To derive and classify the disorder configurations relevant for oxide interfaces, we have to analyze how the vertex transforms under the point group operations and time-reversal.
To begin with the former, consider an arbitrary operation of the point group. The field operators transform according to
where with and denoting the representation of on vectors and pseudovectors, respectively. Here represents the total angular momentum operator. From Eq. (4) readily follows that
which immediately yields the transformation behavior of the correlator (5). As mentioned, while a generic given disorder realization will break all point symmetries of the system, on average the full point symmetry of the normal state has to be restored. In other words, the correlator (5) must transform trivially which leads, within the approximation (7), to the constraint
for all operations .
To distinguish between nonmagnetic and magnetic disorder, we have to take into account the properties under time reversal. Let us denote the representation of time reversal on wavefunctions by with unitary and representing complex conjugation. Within the basis used in Eq. (3), it holds . According to Eq. (4), the disorder Hamiltonian transforms under time-reversal as
Here we have already taken advantage of the Hermiticity of which allows for a representation that does not explicitly involve complex conjugation. This is convenient as the restriction on time-reversal symmetric (TRS), , or time-reversal antisymmetric (TRA), , disorder realizations can then be reformulated in terms of the constraints
on the disorder vertex. Note that it is sufficient to impose the condition in the first line as the second line automatically follows from the property
which is a direct consequence of the definition (5).
Because of the very different behavior of superconductivity in the two cases, we will discuss TRS and TRA disorder separately, which will be referred to as “nonmagnetic” and “magnetic” in the following.
iii.2 Nonmagnetic disorder
Let us represent the basis matrices as the possible tensor products of the Pauli matrices and , , referring to the spin and orbital degrees of freedom as in Eq. (3). This constitutes a convenient choice since these basis matrices both transform under the irreducible representations of the point group and are either symmetric or antisymmetric under time-reversal as summarized in Table 1. Recalling Eq. (15), one can then readily construct the most general vertex in case of nonmagnetic disorder,
characterized by six independent real coupling constants. To gain physical insight into Eq. (16), let us first discuss the meaning of the different terms and then analyze how they can emerge microscopically.
The contributions proportional to , and correspond to spin-trivial disorder with orbital wavefunctions of different point symmetry: The different terms transform as , and under which corresponds to -, -, and -wave. Noting that is the projection of the -component of the orbital angular momentum operator on the subspace spanned by the and orbitals, the terms with and represent (atomic) spin-orbit disorder with out-of-plane/longitudinal and in-plane/transversal orientation of the spin, respectively. The prefactors of the two terms in the last line of Eq. (16) must be identical in order to ensure -rotation symmetry of the disorder distribution. Finally, the contribution proportional to describes the fact that the point group allows for an admixture of -wave disorder and longitudinal spin-orbit disorder on the same site.
To understand how the different terms can be realized microscopically let us investigate the schematic illustration of the crystal structure in Fig. 2. We begin with oxygen vacancies, which are frequently discussed in the context of LAO/STO interfaces (see, e.g., Refs. Huijben et al., 2013; Liu et al., 2013). Assume that one oxygen at position , right below the central Ti at position , is vacant. In this case, the defect preserves the full point symmetry of the crystal such that only the first terms in Eq. (16) can be present. The explicit values of , and depend on microscopic details of the system, however, as will be seen in Sec. V, only the sum of these scattering rates enters the effective pair-breaking strength. Here we mention that all coupling constants in the impurity vertex are proportional to the concentration of the defects considered. If instead an oxygen adjacent to the central Ti, e.g., at position , is vacant, all point symmetries of the lattice are locally broken except for the mirror symmetry with respect to the -plane. This symmetry constraint rules out the contribution proportional to and the first term in the last line of Eq. (16) as both and are odd under . Correspondingly, if we consider oxygen vacancies at position , the second term in the last line of Eq. (16) and again the contribution with will be forbidden. Recalling that the prefactors of the two terms in the last line of Eq. (16) have to be equal by rotation symmetry, we see that, as expected, vacancies at position and have to be included with equal probability to restore the full point symmetry after averaging. In summary, single oxygen vacancies allow for all contributions in Eq. (16) except for the - term.
To see how the latter can occur, let us now assume that a Sr atom is vacant, e.g., at position in Fig. 2. Similarly as above, all point symmetries are broken save for the mirror reflection at the plane. This now renders the -term possible which is even under the reflection at . Assuming that, with the same probability, a Sr atom is missing at position , one restores rotation symmetry. In this case, all terms in Eq. (16) are allowed except for the one proportional to . Finally, we note that the same is true for a local oxygen double vacancy where oxygen atoms are missing at position and at the same time. We have thus provided examples of how all the terms in the most general non-magnetic impurity vertex (16) can be realized.
iii.3 Magnetic disorder
In case of magnetic disorder, i.e. in Eq. (12), we proceed in the same way as in Sec. III.2. From Table 1, one finds that there are three independent terms in the impurity vertex that correspond to in-plane magnetic moments,
|and five terms,|
where the magnetic moment is oriented perpendicular to the plane. As before, all coupling constants must be real due to Eq. (15).
Physically, the three terms , and in correspond to impurities with in-plane spin-magnetization and distinct orbital symmetry (-, - and -wave). Note that no orbital-magnetic disorder with in-plane orientation can occur in the two-orbital model as the projection of the - and -component of the orbital angular momentum operator onto the and subspace vanishes identically. The analogous terms in with the same orbital symmetry but spin polarization along the -direction are proportional to , and , respectively. Finally, corresponds to purely orbital magnetism along the -direction and shows that an arbitrary admixture of spin and orbital magnetism along the -direction is allowed by symmetry.
Similarly to the nonmagnetic case, let us discuss some examples of how the terms in Eq. (17) can emerge microscopically. One can imagine that locally one Ti atom orders magnetically due to the proximity of the system to a competing magnetic instabilityScheurer and Schmalian (2015a). Numerical calculations indicatePavlenko et al. (2012a, b) that magnetic moments on Ti atoms can be induced by the presence of oxygen vacancies. Alternatively, we may think of Ti being replaced by a different magnetic atom. In all cases, the form of the resulting impurity potential crucially depends on the orbitals that host the magnetic moment and whether it is orbital or spin magnetism. Thus, in general, all terms in Eq. (17) are feasible. To be more specific, recalling that the observed magnetism at the interface is mainly due to the bandPentcheva and Pickett (2007, 2008); Michaeli et al. (2012); Santander-Syro et al. (2014), one may expect that, most likely, the spin degree of freedom of the Ti orbital orders locally. In that case, the impurity vertex would be described by the terms proportional and as the associated orbital structure, , transforms as under .
If, instead, a magnetic moment develops at a site that does not lie at a high symmetry point with respect to the Ti atoms, the symmetry constraints will be less restrictive. Assuming, e.g., a spin-magnetic moment along the -axis with -wave orbital symmetry (i.e. ) at the oxygen site () in Fig. 2, the disorder potential is only restricted to be odd both under time-reversal and mirror reflection symmetry with respect to the -plane (-plane). The reduction of symmetry constraints simply follows from the fact that, irrespective of the symmetries of the local impurity, the presence of a perturbation at () already breaks all symmetries except for time-reversal and one mirror reflection. Under these two remaining symmetry operations, the resulting impurity potential in the effective model for the conducting Ti orbitals must have the same behavior as (in both cases being odd). From these criteria one finds that only the term proportional to can be excluded in Eq. (17).
Iv Critical temperature of the disordered superconductor
In this section, we provide details about how the critical temperature of the superconductors is calculated in the presence of disorder. Readers that are mainly interested in the results of this paper and the physical interpretation thereof can skip this part and directly proceed with Sec. V.
iv.1 Patch approximation
For our calculation, it will be convenient to work in the eigenbasis of the normal state Hamiltonian in Eq. (1). For this purpose, we introduce new operators and via
where are the four eigenstates of . By design, the normal state Hamiltonian (1) becomes diagonal,
In the new basis, the momentum space representation of the impurity vertex in Eq. (7) reads
which depends on all four external momenta. To simplify the analysis, we take advantage of the fact that, within each of the patches of the Fermi surfaces in Fig. 1, the wavefunctions are approximately constant. Together with the fact that this set of patches covers the main part of both Fermi surfaces, it is justified to replace
We only keep the fermions from the patches as dynamical degrees of freedom characterized by a constant spin and orbital polarization in a given path. This is what we refer to as the “patch approximation”. In Eq. (21) and in the following, denotes the two Fermi surfaces and , () represent the four distinct patches on each Fermi surface in the half-space with () as illustrated in Fig. 3. The additional index will be convenient for describing superconductivity as, for fixed and , the states with and are related by time reversal. We use to denote the deviation of the crystal momentum from the centers of the different patches. We always assume that these momenta are cut off in a way that does not allow for patches to overlap. Note that applying the patch approximation automatically takes advantage of the fact that, as discussed in Sec. II, two out of the four bands of the normal state Hamiltonian (1) can be neglected for energetic reasons.
Using the compact multi-index notation , the disorder induced interaction (6) finally becomes within the patch approximation
where, using the expansion (14),
In the second line of Eq. (23), we have introduced the diagrammatic symbol for the impurity vertex that will be used in Sec. IV.3. Due to the translation invariance of the disorder distribution (see Eq. (7)), the impurity induced interaction (22) must conserve momentum, which is accounted for by . Here we will not specify the explicit form of . Let us only note that it crucially depends on details of the band structure. To see this, consider, e.g., the scattering process of an electron from to with an electron that starts and does not leave the patch in Fig. 3. Obviously, the available phase space of this process depends both on the nesting vector between and as well as on the geometry of the patches. Fortunately, as we will explain in Sec. IV.3, the transition temperature of the superconducting states will not depend on these details of the band structure.
We emphasize that the vertex contains the full information about disorder scattering within the patch approximation, combining knowledge about the microscopic structure (see Sec. III) of the impurities involved and the form of the eigenstates (see Fig. 1) between which electrons are scattered. Motivated by the strong orbital and spin polarization of the patch wavefunctions, we approximate the states to be perfectly polarized. More precisely, within each patch, we take the spin to be fully aligned parallel to one of the coordinate axes and the orbital weight to be either fully or corresponding to the approximate polarization of the wavefunctions illustrated in Fig. 1. This simplification allows us to obtain analytic expressions for the scattering vertex . In Sec. V, we will also comment on the impact of deviations from perfect polarization.
iv.2 Phase convention for the eigenstates
To have a well-defined representation of the superconducting order parameter in the eigenbasis of the normal-state Hamiltonian, it is essential to constrain the phases of the eigenstates in Eq. (18). To see this, consider the microscopic order parameter defined by the mean-field Hamiltonian
which reads in the eigenbasis (18) as
Here it has already been taken into account that we focus on order parameters that have no Fermi-surface off-diagonal elementsScheurer and Schmalian (2015b). Clearly, Eq. (25) is not invariant under a general -space local -gauge transformation of the wavefunctions.
To remove this ambiguity, we take advantage of the fact that the normal state Hamiltonian is even under time reversal. Since the Fermi surfaces are singly degenerate, one can always adjust the phases of the eigenstates to satisfy
where, for convenience, we have switched to the patch notation. The phase factor is actually arbitrary but has been chosen to reproduce the conventions of Ref. Scheurer and Schmalian, 2015a. Note that the sign factor is necessary due to the property of the time-reversal operator. Imposing Eq. (26) fixes the relative phase between and , thus, rendering Eq. (25) gauge invariant.
To classify the possible superconducting phases, it is essential to know how the order parameter transforms under the elements of the point group of the system. Applying Eq. (8) in Eq. (24) one readily finds that
Taking into account the phase convention (26) and using one more time that the Fermi surfaces are singly degenerate, it follows that the representation (25) of the order parameter transforms according to (again switching to the patch notation)
where denotes the patch obtained when applying the point group operation on the subset of momentum space belonging to patch .
General symmetry and energetic argumentsScheurer and Schmalian (2015b) indicate that the superconducting order parameter of oxide interfaces transforms according to a one-dimensional irreducible representation of the point group . ExperimentsRichter et al. (2013) indicate that the superconductor is fully gapped. Together with the result that time-reversal symmetry is not brokenScheurer and Schmalian (2015b), the order parameter must transform under the trivial representation of (-wave) and, hence,
The microscopic calculation of Ref. Scheurer and Schmalian, 2015a yields and corresponding to the and superconductors as possible candidates. As already discussed in Sec. II, the crucial interaction leading to these two instabilities is pair-hopping, which is given by
Within the phase convention (26), taking () leads to the conventional (unconventional ) superconductor.
iv.3 Ginzburg-Landau expansion in the presence of disorder
To determine the transition temperature of the superconductor, we first decouple the pair-hopping interaction (30) by a mean-field approximation in the Cooper channel assuming -wave symmetry as defined in Eq. (29). This yields
Let us denote the free energy for a given disorder realization in Eq. (4) by . Assuming that the system is self-averaging, the free energy of the superconductor is given by the disorder average that only depends on the disorder correlator (5) and not on details of the microscopic . As we are only interested in the transition temperature of superconductivity, it is sufficient to expand the free energy up to second order in the order parameters. One finds
as , where the coefficients are given by
in the path integral representation. Here denotes the expectation value with respect to the action associated with the normal state Hamiltonian (19). The transition temperature and the order parameter then simply follow from diagonalizing the quadratic form in Eq. (32). These two quantities are determined by the eigenvalue that first changes sign and the associated eigenvector, respectively.
The disorder averaging in Eq. (33) leads to the effective interaction (22) with the impurity line as defined in Eq. (23). The exact expression for the coefficient is represented diagrammatically in Fig. 4(a) and requires knowing the full self energy and vertex correction. In this paper, we will assume that the disorder scattering is sufficiently weak in the sense that the mean-free path is much larger than the inverse Fermi momentum . This assumption is justified by the experimental observation that the typical normal state sheet resistance is by more than one order of magnitude smaller than , where and denote the electron charge and Planck’s constant.Reyren et al. (2007); Ben Shalom et al. (2009); Chang et al. (2013) Being suppressed by a factor , we neglect all diagrams with crossed impurity lines in the self-energy as well as in the vertex correction. Consequently, only the “rainbow diagrams” and “Cooperon ladder”, which are shown in Fig. 4(b) and (c), have to be taken into account.
From the diagrammatic representation, one observes that the impurity vertex does not enter as a full tensor but only in form of the index combinations
in the self-energy and vertex correction, respectively. Here the patch momenta and have been added for later convenience and denotes the Kramers partner of patch . In Eq. (35), we have taken into account that the superconducting vertex only couples Kramers partners, , characterized by zero total momentum. As the patches are assumed to be disjoint in momentum space, momentum conservation allows for Cooper pairs only being scattered into pairs of states of the form .
Note that this crucially reduces the complexity of impurity scattering as the tensor with possible index combinations is replaced by two scattering matrices and with only entries each. More importantly, we directly see that the momenta and can freely run over all values within the associated patches since momentum conservation is fulfilled in the diagrams of Eqs. (34) and (35) for arbitrary and independent values of and . We emphasize that this crucial simplification is a consequence of the superconducting state being characterized by Cooper pairs with vanishing center of mass momentum and the absence of diagrams with crossed impurity lines in the diagrammatic expansion of Fig. 4. Had we included diagrams with crossed lines, the impurity vertex would have entered in a more general form with three arbitrary external momenta (one momentum is always fixed by momentum conservation). In this case, also scattering processes of an electron that changes its patch with an electron that stays in the same patch contribute. As discussed in Sec. IV.1, the phase space of these processes strongly depends on quantitative details of the band structure and of the patch geometry. From these considerations, we see that that focusing on the limit not only allows for a controlled summation of a minimal set of diagrams but also renders the final result independent of the aforementioned microscopic details.
In the explicit summation and evaluation of the diagrams, which can be found in the Appendix, the component of the patch momentum perpendicular to the Fermi surface is constrained by an energetic cutoff . The longitudinal limits of the patches is absorbed by introducing the density of states per patch. For simplicity, we assume that the density of states does not depend on and introduce , the density of states per Fermi surface. In Sec. V, we will also comment on the modifications in case of an imbalance in the density of states on the different Fermi surfaces.
Finally, we note that the two scattering matrices in Eqs. (34) and (35) that determine the behavior of the superconductor under the influence of impurity scattering are made unique by the phase convention (26): Obviously, is already invariant under arbitrary -gauge transformations of the wavefunctions, whereas is fixed due to Eq. (26) since states with the same and but opposite always come in pairs.
Let us begin our discussion with the clean case. Calculating the Ginzburg-Landau expansion in the absence of any disorder reproduces the results of Ref. Scheurer and Schmalian, 2015a: For attractive (repulsive) pair hopping, i.e., () in Eq. (30), the () superconductor is realized. In both cases, the transition temperature assumes the usual BCS form
with denoting Euler’s constant.
To proceed with the impact of disorder on superconductivity, we first focus on the case of nonmagnetic disorder which is characterized by the vertex in Eq. (16). For attractive pair hopping, we again obtain the state. Within the approximations of Sec. IV.3, the associated transition temperature is unaffected by disorder irrespective of the values of the different coupling constants in the impurity vertex (16). This result is a manifestation of the well-known Anderson theoremAnderson (1959); Abrikosov and Gorkov (1958, 1959).
In case of repulsive pair hopping, , we still find that, as in the clean case, the superconductor condenses. In accordance with the usual expectation that unconventional superconductors are less robust against impurity scattering, the transition temperature is now reduced by disorder. Despite the complicated form of the vertex (16), it turns out that the impurity distribution only enters the Ginzburg-Landau expansion and, thus, the transition temperature, in form of the combined scattering rate
Note that , which describes the admixture of -wave and spin-orbit disorder on the same site, does not enter at all. This follows from the strong orbital polarization of the wavefunctions in the patches (see Fig. 1), which we have idealized to be perfect in our calculation. It is easily seen upon inserting the term with in Eq. (16) into the two relevant scattering matrices defined in Eqs. (34) and (35). Since the two scattered electrons either both switch the orbital or both keep their orbital character, either or in Eq. (16) yields zero in and