Effect of disorder on the competition between nematic and superconducting order in FeSe

Crystalline FeSe is known to display strong nematic order below a weak tetragonal-orthorhombic structural transition around $T_s\sim 90$K, and a superconducting transition at $T_c\sim 9K$. Recently, it was shown that electron irradiation, which creates pointlike potential scattering defects, has the surprising effect of enhancing $T_c$ while suppressing $T_s$. Here we discuss a possible scenario for such an effect, which postulates a competition between $s_\pm$ superconductivity and nematic order. The transition to the nematic state is modeled by a mean field theory of a $d$-wave Pomeranchuk instability, together with a Cooper pairing interaction in both one- and multiband models. The effect of nonmagnetic impurities on both orders is treated on equal footing within the Born approximation. We find evidence that disorder can indeed enhance $T_c$ while suppressing the competition with nematic order, but only in a multiband situation. We discuss our results in the context of experimental data on FeSe crystals.

Crystalline FeSe is known to display strong nematic order below a weak tetragonal-orthorhombic structural transition around Ts ∼ 90K, and a superconducting transition at Tc ∼ 9K. Recently, it was shown that electron irradiation, which creates pointlike potential scattering defects, has the surprising effect of enhancing Tc while suppressing Ts. Here we discuss a possible scenario for such an effect, which postulates a competition between s± superconductivity and nematic order. The transition to the nematic state is modeled by a mean field theory of a d-wave Pomeranchuk instability, together with a Cooper pairing interaction in both one-and multiband models. The effect of nonmagnetic impurities on both orders is treated on equal footing within the Born approximation. We find evidence that disorder can indeed enhance Tc while suppressing the competing nematic order, but only in a multiband situation. We discuss our results in the context of experimental data on FeSe crystals.

I. INTRODUCTION.
Fe-based superconductors (FeSC) present a continuing challenge to the condensed matter physics community eight years after their discovery 1 . Pairing is thought to be electronic in nature, and originate from spin fluctuations, treated theoretically in various limits [2][3][4] . While many aspects of the superconducting state have been understood, the simplest material, FeSe, has proven to be one of the most elusive. It exhibits a tetragonal to orthorhombic structural phase transition at T s ∼ 90 K, and displays very strong electronic nematic behavior below this temperature (T ), but never orders magnetically as do the more familiar Fe pnictide systems. While its critical temperature T c ∼ 9 K is low for this class of materials, a sharp increase of T c to about 40 K is observed under a few GPa of pressure, which simultaneously suppresses T s 5, 6 . Various intercalates of FeSe have T c of roughly this magnitude at ambient pressure as well, and the highest T c of the entire class of Fe-based superconductors, 70-100 K, is achieved in monolayer films of FeSe grown on SrTiO 3 substrates. [7][8][9] Because of these intriguing hints of intrinsic higher-T c superconductivity, efforts to understand the properties of the basic bulk FeSe have accelerated recently.
The origin of the nematic order in FeSe has itself been the subject of considerable debate. In general the origin of strong nematic tendencies in the FeSC have been discussed in terms of a competition between fluctuations of structural, orbital, and spin degrees of freedom 10 . At first glance, the lack of long-range magnetic order in the ambient pressure phase diagram seems to suggest that the popular spin nematic scenario 11 might not be appropriate, and that orbital fluctuations might play a more important role 12,13 . However, the confirmation of a longrange magnetic state under a modest pressure 14,15 has lent support to other proposals that suggest that the ground state at ambient pressure may be a quantum paramagnetic state 16,17 or a state with long-range mag-netic order of "hidden" quadrupolar type 18,19 . The very small Fermi surfaces in this system may also be important to prevent long-range ordering 20 .
Recently, a new experiment by Teknowijoyo et al. 21 measuring T c and superfluid density ρ s in FeSe has appeared that may provide an important clue to this puzzle. The authors irradiated their sample with low-energy electrons, known to create homogeneously distributed Frenkel pairs of atomic vacancies and interstitials without doping the system 22 . This seems to be the best way to create pure potential disorder in these systems, where chemical substitutions have proven difficult to interpret 23 . In other Fe-based superconductors [24][25][26] , T c has been strongly suppressed by disorder of this type, as compared to chemical substitution suppression rates. In FeSe, on the other hand, Teknowijoyo et al. 21 found an increase of T c upon irradiation, and concomitant decrease of T s . While these trends are similar to the effect of hydrostatic pressure, the authors argued first that irradiation tended to increase rather than decrease the lattice volume, and that the magnitudes of the changes in lattice constants were in any case an order of magnitude smaller than in pressure experiments. They speculated that impurities might introduce local pair strengthening effects due to the proximity of FeSe to a magnetic transition 27 , or that the competition of superconductivity with nematic order might play a role.
An intriguing paradigm to enhance T c by disorder for an s ± state in the presence of competing stripelike "(π, 0)" magnetic order was offered by Fernandes et al. 28 , who pointed out that both interband and intraband impurity scattering processes would suppress magnetism, while only interband processes would break superconducting pairs. Whether a competition of superconductivity with nematic order can similarly lead to a T c enhancement is not a priori obvious, given that nematic order is a q = 0 distortion of the Fermi surface. In addition, since the gap function in FeSe is known to be highly anisotropic 2 , superconductivity will be suppressed by both types of scattering processes, independent of any sign change over the Fermi surface. From these perspectives, the T c enhancement mechanism of Ref. 28 should be irrelevant.
In this paper, we provide a concrete framework for the study of the effect of impurities on the competition between nematic and superconducting order. We first assume that that nematic order may be modeled by a dwave Pomeranchuk instability competing with superconductivity. Such an instability has been argued to qualitatively describe the nematic tendencies of both cuprate 29 and Fe-based superconductors 30,31 . We show here that the nematic order (Pomeranchuk instability) temperature, which we identify with T s , is suppressed naturally by disorder, as found by earlier authors 32 . In addition, however, we find that T c can actually be enhanced when the nematic order is weakened. Whether T c is enhanced or suppressed by disorder turns out to depend in a nontrivial way on the interplay of the anisotropy of the superconducting pairing interaction with that of the nematic distortion of the Fermi surface.
We now introduce the mean field treatment of a dwave Pomeranchuk distortion, first within a single band model for pedagogical purposes, and then generalized to multiband Fermi surfaces appropriate to the FeSC. Disorder is then treated in the Born approximation, and the instability temperatures of both nematic and superconducting transitions calculated, for the different model Fermi surfaces, and for different types of anisotropic swave superconductivity. We present full phase diagrams of superconductivity and nematic order as functions of disorder and temperature, and discuss the comparison with experiment.

A. Interplay of superconducting and nematic order
We wish to work with the simplest possible model capturing the competition of superconductivity and nematic order, and allowing for the introduction of the effects of nonmagnetic disorder. We begin therefore with a Hamiltonian describing fermions with an interaction which leads to a Pomeranchuk instability 29 where H 0 describes a noninteracting parabolic band of electrons, H 0 = kσ (k 2 /(2m) − µ)c † kσ c kσ , and where µ is the chemical potential, d k = cos 2φ, and φ is the angle around the Fermi surface. In the second line we have made a mean field approximation 29 and defined which leads to a self-consistency equation for the nematic order parameter The nematic transition temperature when Φ k → 0, T nem , is then given by where λ nem = mV nem /2π. It is worth noting that the nematic instability is of the Stoner type: unless V nem is larger than a threshold value, long range nematic order is not possible. Next, we study superconductivity in the nematic phase, and show that the interplay of nematicity and superconductivity depends sensitively on the structure of the superconducting pairing. We consider a sim- To study the interplay of such states with nematic order, we introduce the anisotropic pair potential V kk ′ = V sc Y(φ)Y(φ ′ ) on the Fermi surface, and for the moment assume it to be independent of nematic changes in the Fermi surface itself. The linearized gap equation in the presence of nematic order, Φ 0 > 0 may then be expressed as Here the pair bubble P(φ, ω n ) makes the maximum contribution along the long direction (φ = π/2) of the deformed Fermi surface. If the minimum of the absolute value of gap function is also along this direction (i.e. r > 0), then the deformed Fermi surface will suppress such gap states. We can approximate the effect of nematic order on T c in the limit of weak deformation of the Fermi surface here T nem c0 (T c0 ) is the superconducting transition temperature in the nematic (tetragonal) phase with no disorder. As expected from the structure of P, the critical temperature T c depends on the structure of the pairing through the model anisotropy parameter r, and can apparently increase or decrease depending on how the anisotropy is oriented with respect to the distorted C 2 Fermi surface. When the Fermi surface gets stretched along the direction of gap minima (r > 0, Fig. 1) T c is suppressed. In contrast, gap minima oriented perpendicularly to the stretched Fermi surface (r < 0, Fig. 1) lead to an increase of T c , since P(φ) and ∆(φ) have maxima on the same regions of the Fermi surface. This is similar to superconductivity in the presence of spin-density wave order, where gap maxima away from the nesting hot spots is favorable 33 .
We note the important role of particle-hole asymmetry for this problem. As seen explicitly from Eq. (5), in the limit of perfect particle-hole symmetry (µ → ∞), the coupling between superconducting and nematic order vanishes. Furthermore, the sign of the nematic distortion depends on the sign of µ: thus if one replaces the electron band assumed above with a hole band, the effect will be identical, except that T c will now be suppressed for r < 0. The relative orientation of the gap states described above remains unchanged, however: T c is suppressed relative to T c0 if gap minima are along the stretched direction.
These discussions assume that the nematic/superconducting coexistence solutions discussed above are ground states for the given parameters. However, an examination of the free energy (see Appendix) shows that those superconducting states with gap minima along the nematic elongation of the Fermi surface are higher energy than the homogeneous nematic state, and are therefore unstable, as illustrated in Fig.  2.

B. Effect of disorder
We first determine the nematic order parameter in the presence of nonmagnetic disorder above T c by defining a self-energy within the Born approximation,  where n imp is the concentration of impurities with potential V imp , and G(k, ω) = (iω − ξ k − Φ k + i0 + ) −1 at T nem , and iω = iω − Σ nem (ω). This renormalizes the single-particle energy and influences the nematic order through its self-consistency equation. Due to the d-wave Pomeranchuk form, the Φ k itself is not renormalized in this approximation.
To study superconductivity within the same framework, we introduce the disorder self-energy in Nambu space in the same approximation, (8) whereĜ is the Nambu Green's function Here the renormalized order parameter has the form ∆ k ≡∆ iso + ∆ ani d k , and we have defined ∆ ani = ∆ 0 r/ 1 + r 2 /2.
Note that the nematic order parameter Φ k and anisotropic gap component ∆ ani are unrenormalized by disorder. At T c , the self-consistency expressions for the amplitude of the d-wave nematic order parameter Φ 0 and the superconducting order parameter ∆ 0 are given by Fig. 3 now shows the result of a numerical evaluation of Eqs. (13)- (14). As expected, disorder gradually suppresses the nematic order as also found in Ref. 32. From the inset in Fig. 2, we can anticipate that the r > 0 case, which shows no competition, should lead to a T c suppression with disorder, as indeed shown in Fig. 3. For a small suppression of T nem , T c appears to increase when r > 0, i.e. when the gap minima are along the elongated direction of Fermi surface. However, as shown in Fig. 2, such a pairing state is a metastable state. Thus disorderinduced T c enhancement appears to be disfavored in this one band model.

III. THREE-BAND MODEL
We now investigate whether the basic notions which govern the effect of disorder on T c , T nem in the one-band case continue to hold in a somewhat more realistic multiband framework appropriate for the FeSe system that motivated this study, and ask whether a disorder-driven T c enhancement is possible as a consequence of multiband physics. Unlike its monolayer and intercalate cousins, bulk FeSe has well-established Fermi surface hole pockets at Γ and electron pockets at X and Y , as modeled in Fig. 4 (a) in the 1-Fe zone. Although many phenomenological assumptions for the forms of the interactions on these three bands are possible, we assume for simplicity the same Pomeranchuk harmonic for the nematic instability on all bands, leading to the distorted Fermi surface pockets also plotted in the figure. We further adopt various simple forms of the BCS pairing interaction on all three bands, including pairing anisotropy on one band or the other, as also illustrated in Fig. 4 (b)-(f). Because we operate without a microscopic theory, we have considerable freedom to choose gap structures consistent with C 2 symmetry in the nematic phase. We therefore consider anisotropy on the hole and the electron pockets independently, although in reality all will distort simultaneously. Note that all order parameters on all bands are determined self-consistently. Our goal is simply to demonstrate that a T c enhancement with disorder due to competition with nematic order is possible, and deduce what qualitative conclusions we can from that novel situation.

A. Electronic structure and pairing interaction
The Hamiltonian for this three band model is where c † i,k,σ (c i,k,σ ) creates (annihilates) a fermion with momentum k and spin σ in the i th band. Ψ is an extended Nambu basis for three bands, . The fermionic dispersions in the C 4 symmetric phase are, We assume a circular hole pocket and slightly elliptic electron pockets with ε = 0.6. The nematic and superconducting order parameters are taken to be Φ ki = Φ i d ki ∀i, and ∆ ki = ∆ i Y i (φ i ), respectively. The multiband self-consistency equations for the nematic and SC orders analogous to Eqs. (13)- (14) are then obtained as Here ω n is the fermionic Matsubara frequency at temperature T , and Y i = (1 + r i cos 2φ i )/ 1 + r 2 i /2, where the parameter r i controls the anisotropy of the SC order.
Note that while φ is always measured with respect to the positive x axis at each pocket, r ei will now be assumed to have same sign on pockets e 1 and e 2 , such that an overall C 2 state is realized. Two possible such choices are illustrated in Fig. 4

(e)-(f).
We use a separable form of interactions for the ne- Note that the Nambu components of the disorder selfenergy involve both intra-and interband scattering processes via the impurity scattering potential matrix in the band basis u ij , which is taken to have only two elements: v for intra-and u for interband scattering, for all band components, with η ≡ u/v. Here G 0(1)i is the τ 0(1) component of the i th band's Nambu Green's function and ∆ iso i is the isotropic component of the gap function for i th band. Note that we ignore the τ 3 component of the impurity self-energy, which mainly renormalizes the chemical potential.
It is now relatively easy to arrange for nematic order and superconductivity to compete, without fine tuning of the interactions. Bulk FeSe itself is known to have an order parameter that is highly anisotropic, with nodes or near-nodes somewhere on the Fermi surface 2 . We therefore focus particularly on cases with large gap anisotropy, either on the hole pocket (Fig. 4(c)-(d)) or on the electron pockets (Fig. 4(e)-(f)).
Note we only consider s ± type pairing states, with overall sign change between electron and hole pockets. However, our conclusions are mostly qualitatively valid for s ++ states as well, since minimal pairbreaking is required to obtain a T c enhancement in an s ± state, and we therefore are forced to assume relatively weak interband scattering, such that the sign difference does not play an essential role.

B. Results
We first focus on the effect of disorder on the pure nematic state. Both interband and intraband scattering suppress the nematic order in the current model, as shown in the inset to Fig. 5, which displays the variation of T nem with increasing impurity concentration. As the interband scattering increases, nematicity goes down rapidly; since interband scattering connects Fermi surfaces with different signs of Φ k , this is equivalent to the effect seen for the one-band case in Ref. 32.
The relative effects of interband impurity scattering obviously depends now on the gap structure, i.e. whether a sign changing state is realized. Evidence for sign changing gap behavior in this system is limited, and reviewed in Ref. 2. The most significant is probably the recent experiment by Wang et. al. 34 , who observed a q = (π, 0) low-energy inelastic neutron scattering resonance very similar to that generally taken as strong evidence for the s ± state in other Fe-based superconductors 35 . Earlier, STM 36 and some thermal conductivity and penetration depth measurements 37 had reported a gap with nodes.  More recently, and possibly on slightly different samples, a gap with very deep minima has been suggested to be consistent with STM 38 , penetration depth 21,39 , and low temperature thermal conductivity measurements 40 as well. For our purposes, such tiny differences in gap structure are probably irrelevant.
The superconducting transition temperature T c in the presence of disorder and nematicity is now determined by solving the multiband linearized gap equation, Eq. ((21)), which contains the nematic order parameter Φ ki , itself determined by Eq. 20. We first examine the simplest case, r i = 0 on all bands, i.e. an isotropic gap in the presence of the nematically distorted Fermi surfaces. The isotropic state in the presence of a single nematic pocket was not a stable solution for the nematic distortion chosen. However, in the presence of the electron pockets and an interband interaction, an isotropic state in the presence of nematic order can be lower in energy than the pure nematic one, as seen in Fig. 6 (a). The competition of the two order parameters is shown explicitly in Fig. 6(b) and in fact is seen to lead to a modest T c enhancement with disorder as seen in Fig. 6 (c) and (d). This effect may now be enhanced somewhat by considering nonzero r h . Note that these effects, while roughly consistent in magnitude with the T c enhancement effect seen in experiment, will be suppressed in an s ± state by any additional source of pairbreaking, e.g. as when the interband scattering rate is increased (Fig. 6 (d)).
Note that within the current model the ability of the hole band gap anisotropy to enhance T c further is limited by the narrow range of stability of this state (see Fig. 6 (a)-(c)). It is interesting therefore to explore the role of gap anisotropy on the electron pockets, which we illustrate in Fig. 7. In panels (a) and (c), the range of stability of a state which enhances T c with disorder is found to be much wider than in the hole pocket case. In addition, this range is asymmetric with respect to the anisotropy parameter r e , indicating that it is more likely to observe  T c enhancement if r e > 0, i.e. if the gap minima on the most nematically distorted pockets are aligned with the pocket elongation axis (Fig. 4).

IV. DISCUSSION
The motivation for this study has been a recent lowenergy electron irradiation experiment by Teknowijoyo et. al. 21 , which found a surprising enhancement of T c with increasing disorder in FeSe, a system with no longrange magnetic order but strong nematic order below the structural transition. It is worth noting that until now, all other Fe-based superconductors similarly irradiated have had their critical temperatures strongly suppressed by this type of disorder, which should create nearly ideal pointlike potential scatterers. This striking result raises several questions about the interplay of superconductivity and nematicity, and may provide an important clue to the physics of the mysterious FeSe material.
The experimental situation with regard to competition of superconductivity and nematic order, required for our scenario, is currently unclear. In several situations, including hydrostatic pressure 5,6 and doping by sulfur or field gating 41  clear evidence that the orthorhombic distortion was enhanced by the occurrence of superconductivity in S-doped FeSe crystals. This suggests that the interplay between the two orders may be more subtle than assumed here. If our approach is qualitatively correct, however, it implies that disorder-induced T c enhancement of the type reported in Ref. 21 and discussed here should disappear in the S-doped systems.
Clearly one aspect of the simple model presented here is unphysical, namely that the pairing interaction assumed is "rigid', in the sense that as the Fermi surface deformed as T c is lowered, our model does not capture the concomitant evolution of the pairing interaction before T c is reached (it does treat the anisotropy of the gap in the presence of a distorted Fermi surface selfconsistently). It seems therefore possible that our model overestimates competition of superconductivity, and nematicity and therefore artificially favors T c enhancement when nematicity is suppressed. A more complete microscopic treatment of the effect of nematic order on pairing is outside the scope of present manuscript and will be treated elsewhere.

V. CONCLUSIONS
In summary, we have studied the interplay between nematicity and superconductivity for a system with one band and three bands, modeling the the nematic instability with a mean field treatment of a d-wave Pomeranchuk transition. We have shown that in several physically plausible circumstances, nematic order competes with superconductivity, and may allow T c to rise, as observed in Ref. 21, when disorder suppresses nematicity. Indeed, such an enhancement appears rather natural in the presence of sufficiently strong nematicity. We note that the effect is qualitatively different from a T c enhancement resulting from the competition between superconductivity and antiferromagnetism discussed by earlier authors.
We have further discussed how the T c enhancement effect is sensitive to the degree and orientation of the gap anisotropy with respect to the deformed Fermi surface. In particular, we showed that superconducting gaps with minima along the stretched axis of the deformed Fermi surface are easily suppressed by nematic order. Upon introduction of disorder, which rapidly destroys nematic order, such states show significant T c enhancement. In the one-band case, these states do not appear to be thermodynamically stable, but they are stabilized by the addition of additional pockets, as in the three-band model studied here. In contrast, SC states with gap maxima along the stretched direction of Fermi surface, do not compete strongly with nematic order, so T c of an s ± state is suppressed as usual by disorder. mean field Hamiltonian, which can be evaluated using coherence factors For the one band case, the expectation value of the kinetic energy may be written as, whereξ k = ξ k + Φ k . The potential energy term for the one band case is and for three bands, Here we have assumed only interband interactions for both orders. Note that expressions for F in both one-and threeband cases manifestly reduce to the correct expressions in either the pure nematic or superconducting states.