Unconventional disorder effects in correlated superconductors
UUnconventional disorder effects in correlated superconductors
Maria N. Gastiasoro , Fabio Bernardini , and Brian M. Andersen ∗ Niels Bohr Institute, University of Copenhagen,Juliane Maries Vej 30, 2100 Copenhagen, Denmark CNR-IOM-Cagliari and Dipartimento di Fisica, Universit`a di Cagliari, 09042 Monserrato, Italy (Dated: October 14, 2018)
The understanding of disorder has profoundlyinfluenced the development of condensed matterphysics, explaining such fundamental effects as,for example, the transition from ballistic to diffu-sive propagation, and the presence of quantizedsteps in the quantum Hall effect. For supercon-ductors, the response to disorder reveals crucialinformation about the internal gap symmetries ofthe condensate, and thereby the pairing mech-anism itself. The destruction of superconduc-tivity by disorder is traditionally described byAbrikosov-Gorkov (AG) theory,[1, 2] which how-ever ignores spatial modulations and ceases to bevalid when impurities interfere, and interactionsbecome important. Here we study the effectsof disorder on unconventional superconductors inthe presence of correlations, and explore a com-pletely different disorder paradigm dominated bystrong deviations from standard AG theory dueto generation of local bound states and cooper-ative impurity behavior driven by Coulomb in-teractions. Specifically we explain under whichcircumstances magnetic disorder acts as a strongpoison destroying high-T c superconductivity atthe sub-1% level, and when non-magnetic disor-der, counter-intuitively, hardly affects the uncon-ventional superconducting state while concomi-tantly inducing an inhomogeneous full-volumemagnetic phase. Recent experimental studies ofFe-based superconductors (FeSC) have discoveredthat such unusual disorder behavior seem to beindeed present in those systems. For cuprates, heavy-fermions, and FeSC the study ofdisorder currently constitutes a very active line of re-search, motivated largely by the fact that these sys-tems are made superconducting by ”chemical disorder-ing” (charge doping), but also boosted by controversies ofthe correct microscopic model, and a rapid developmentof local experimental probes. [3–5] Focusing on multi-band FeSC, disorder studies have proven exceptionallyrich and strongly material dependent. [6] Scanning tun-neling spectroscopy found a plethora of exotic atomic-sized impurity-generated states, [7–10] NMR and neu-trons observed clear evidence of glassy magnetic behav-ior, [11, 12] and µ SR discovered magnetic phases gen-erated by non-magnetic disorder. [13, 14] The resultingcomplex inhomogeneous phases and their properties in terms of thermodynamics and transport constitute animportant open problem in the field.Here, we present a theoretical study of correlation-driven emergent impurity behavior of both magnetic andnonmagnetic disorder in unconventional s ± multi-bandsuperconductors. For the case of magnetic disorder,we find that correlations anti-screen the local moment,and significantly enhance the inter-impurity Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interactions byinducing non-local long-range magnetic order which op-erates as an additional competitor to superconductivity.This results in an aggressive T c suppression rate wheresuperconductivity is wiped out by sub-1% concentrationsof disorder. This mechanism explains the ”poisoning ef-fect” discovered in Mn-substituted optimally doped (OD)LaFeAsO − x F x (La-1111) pnictide where less than 0.2%Mn is enough to suppress the optimal T c ∼
30K tozero, well beyond standard AG behavior. [15, 16] Bycontrast, for non-magnetic disorder the s ± supercon-ducting state is largely immune to disorder, in agree-ment with earlier one-band studies, finding that corre-lations enhance the screening of disorder potentials andthereby reduce pair-breaking and scattering rates com-pared to the non-interacting case. [17–21] In the cur-rent multi-orbital case, however, additional impurity-generated bound states play an important unexpectedrole in supporting T c . This resilience to non-magneticdisorder is remarkable since favorable clusters of impu-rities locally pin magnetic order, eventually causing avolume-full inhomogeneous magnetic state which coex-ists with superconductivity. These latter results are inagreement with extensive systematic experimental stud-ies of Ru-substituted 1111 superconducting materials. Model.
Interactions are included in the model by thestandard multi-orbital Hubbard term H int = U (cid:88) i ,µ n i µ ↑ n i µ ↓ + ( U (cid:48) − J (cid:88) i ,µ<ν,σσ (cid:48) n i µσ n i νσ (cid:48) (1) − J (cid:88) i ,µ<ν (cid:126)S i µ · (cid:126)S i ν + J (cid:48) (cid:88) i ,µ<ν,σ c † i µσ c † i µ ¯ σ c i ν ¯ σ c i νσ , where µ, ν are orbital indices, i denotes lattice sites, and σ is the spin. The interaction includes intraorbital (in-terorbital) repulsion U ( U (cid:48) ), the Hund’s coupling J , andthe pair hopping energy J (cid:48) . We assume U (cid:48) = U − J and J (cid:48) = J and fix J = U/
4. Non-magnetic and mag-netic disorder give rise to terms of the form H imp = a r X i v : . [ c ond - m a t . s up r- c on ] J un (cid:80) µ { i ∗ } V µ n i ∗ µ and H imp = I (cid:80) { i ∗ } µσ σS µ c † i ∗ µσ c i ∗ µσ , re-spectively. Here V µ ( S µ ) denotes the impurity potential(magnetic moment) in orbital µ at the disorder sites givenby the set { i ∗ } coupled to the charge (spin) density of theitinerant electrons. For concreteness we focus on FeSCand hence use a five-band model H = (cid:88) ij ,µν,σ t µν ij c † i µσ c j νσ − µ (cid:88) i µσ n i µσ , (2)with tight-binding parameters appropriate for 1111 pnic-tides [22]. The model H + H int exhibits a transitionto a bulk SDW phase at a critical repulsive interac-tion U c , and we parametrize the interactions in termsof u = U/U c . Superconductivity is included by H BCS = − (cid:88) i (cid:54) = j ,µν [∆ µν ij c † i µ ↑ c † j ν ↓ + H.c.] , (3)with ∆ µν ij = (cid:80) αβ Γ βνµα ( r ij ) (cid:104) ˆ c j β ↓ ˆ c i α ↑ (cid:105) being the supercon-ducting order parameter, and Γ βνµα ( r ij ) denoting the ef-fective pairing strength between sites (orbitals) i and j ( µ , ν , α and β ). In agreement with a general s ± pairingstate in FeSC, we include next-nearest neighbor (NNN)intra-orbital pairing. For further computational detailsand parameter dependence, we refer to the Supplemen-tary Material (SM). Magnetic disorder.
The study of magnetic disorderis motivated largely by the following remarkable experi-mental facts shown in Figs. 1(a,b): in OD La-1111 with ( a ) La - ( OD ) ( b ) Sm - ( OD ) ■■ ■■ ■▼ ▼ ▼ ▼■ T c ▼ T m [%] T [ K ] ■ ■ ■ ■ ■■ T c [%] T [ K ] ( c ) ( d ) ■ ■ ■■■ ■ ■▼▼ ▼ ■ T c ▼ T m
0. 10 20 30 40 50 6001020304050 Ru content [%] T [ K ] ■ ■ ■ ■ ■ ■■■ ■ ■▼ ▼ ▼ ▼ ▼ ▼▼▼ ▼ ▼■ T c ▼ T m
0. 10 20 30 40 50 6001020304050 Ru content [%] T [ K ] FIG. 1. Experimentally obtained superconducting T c andmagnetic T m transition temperatures in OD La-1111 (a,c)and Sm-1111 (b,d) versus magnetic disorder (a,b) and non-magnetic disorder (c,d). The data was adapted from Refs.13–16, and 23. correlations ■ ■ ■ ■■■■■ ■■ ✶ ✶ ✶ ✶ ✶ ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● u = ■ u = ✶ AG [%] k T c [ m e V ] FIG. 2.
Superconducting critical temperature T c as afunction of magnetic impurity concentration. Impu-rity moments are destructive for superconductivity, and the T c suppression rate is strongly modified by electronic corre-lations as seen by comparing the two T c curves at u = 0 and u = 0 .
97. The u = 0 curve follows the behavior described bystandard AG-theory. All impurity moments are modelled byorbitally independent exchange with IS µ = 0 .
38 eV. As seen,correlations act to poison the superconducting state suppress-ing it entirely (at all sites) after only ∼ .
5% disorder as seenby the orange curve. Larger u exhibits even more severe sup-pression rates. T c ∼
30K a mere ∼ .
2% magnetic Mn ions is enoughto destroy the superconducting state.[15] This extremedestruction rate of bulk superconductivity has been re-cently dubbed ”the poisoning effect” [16]. Interestingly,immediately beyond ∼ . T m . Recently it was found that this magneticphase is ( π, − x F x (Sm-1111) near optimaldoping, the corresponding T c suppression rate is muchslower with ∼
8% of Mn being required to wipe outsuperconductivity[23] [Fig. 1(b)].Figure 2 shows the suppression of T c as a functionof magnetic impurity concentration obtained within ourmodel. Without correlations ( u = 0) the T c suppressionfollows the curve expected from AG theory. The main re-sult of Fig. 2 is the much faster T c -suppression rate whenincluding Coulomb interactions as seen by comparisonof the two curves in Fig. 2. How may one understandthis result which appears at odds with the expectationthat correlations screen disorder and limit their damagingeffects?[17–21] The answer to this question necessitatesa deeper understanding of correlation effects at both the local scale (immediate vicinity of the impurity sites) and non-local scale (inter-impurity regions). Both effects areintimately tied to the fact that magnetic impurity mo-ments induce spin polarizations of the surrounding itin-erant electrons m i µ , which renormalize the exchange cou-pling such that ˜ H imp = ˜ I (cid:80) i µσ σ ˜ S i µ c † i µσ c i µσ , where˜ I ˜ S i µ = IS µ δ ii ∗ − U m i µ + J (cid:88) ν (cid:54) = µ m i ν (4) ≡ [ IS µ δ ii ∗ + I ind s i µ ] , is the emergent interaction-generated extended magneticimpurity potential (see SM for more details) generatedby the induced part, I ind s i µ . Focussing first on the local part of the effective potential, a line-cut of the inducedmagnetic potential I ind s i µ through a single impurity asa function of u is shown in Fig 3(a). As seen, the extentand amplitude of the resulting magnetic puddle growssignificantly with u , and results in a real-space structureillustrated in Fig. 3(b). The renormalized moment is sig-nificantly enhanced at the impurity site, even exceedingthe bare moment at large u , and exhibits sizable anti-parallel neighboring spins. Superconductivity is stronglyaffected by the additional pair-breaking caused by the en-hanced local moments, and therefore the suppression ofthe order parameter ∆ i increases accordingly, as shownin Fig. 3(c). In this way, approaching the magnetic in-stability has conspicuous local damaging effects on super-conductivity. This enhanced local pair-breaking is not,however, the sole reason for the enhanced T c suppres-sion rate, which also includes a cooperative (non-local)multi-impurity effect.Indeed, when multiple impurities are included, the cor-relations among their moments become crucial for lower-ing the free energy. Specifically, the spin polarized cloudsaround the impurities prefer to constructively interfere,thereby generating a quasi-long-range ordered magneticstate. [25] The inter-impurity regions acquire a result-ing finite magnetization due to this enhanced RKKY-like interaction between the impurities. Figures 3(e) and3(f) compare directly the case in point with 0 .
5% un-correlated disorder ( u = 0) versus the correlated situa-tion ( u = 0 . π,
0) LROmagnetization (see also SM) in agreement with exper-iments [24], constituting the additional non-local com-petitor to superconductivity. We show in Fig. 3(d) a plotof these two separate (local vs. non-local) effects on thesuperconducting order parameter suppression. The bluesurface is the self-consistent solution of ∆ i of the u = 0system shown in Fig. 3(e). The superconducting orderparameter is hardly affected by the bare magnetic poten-tials, and this is reflected in the correspondingly low T c suppression of Fig. 2. The green surface of Fig. 3(d) isthe resulting substantially reduced inhomogeneous ∆ i so-lution of the gap equation due to the renormalized local ( a ) I ind s i μ ( b )( c ) Δ i μ / Δ ( d ) IS μ local full ( e ) IS μ ( f ) I ind s i μ - x y - - x y FIG. 3.
Local and non-local effect of correlations onthe impurity response from magnetic disorder. (a) In-duced magnetic potential, I ind s i µ , along a cut through the im-purity and as a function of u . Correlations strongly renormal-ize the local moments and lead to typical real-space extendedmagnetic puddles similar to the one shown in (b) where theorange (blue) arrows show the induced (bare) moments for u = 0 .
97. The resulting disorder potential significantly mod-ifies the suppression of the superconducting order parameteras seen in (c). (d) Real-space map of the superconducting or-der in the presence of 0 .
5% magnetic disorder. The blue sur-face shows the suppression from only the bare moments, i.e. u = 0, but self-consistently obtained gaps beyond AG theory.Including the local correlation-enhanced magnetic momentsleads to the green surface, and only by including both localand non-local effects is superconductivity fully destroyed (or-ange). (e,f) Real-space maps of the (e) bare moments IS µ and (f) induced magnetic potential I ind s i µ for u = 0 .
97. Thecorrelations strongly enhance the inter-impurity coupling byinducing a LRO magnetic phase in-between the disorder sites,which further competes with superconductivity and efficientlysuppresses T c . For all results in this figure, the bare momentsare the ones used in Fig. 2 and µ = d xz orbital. potentials, cf. Fig. 3(b). Only when the second non-local magnetic order is also included, superconductivityis completely wiped out as illustrated by the orange sur-face in Fig. 3(d), explaining the physics of the aggressivesub-1 % T c suppression rate shown in Fig. 2.Within the above scenario, why does it require an orderof magnitude more magnetic disorder to suppress T c tozero in, for example, Sm-1111 compared to La-1111? Wepoint out two main reasons: 1) OD Sm-1111 exhibits alarger T c (compared to OD La-1111) (see SM for details),and 2) Consistent with transport studies, [26] Sm-1111is less correlated than La-1111, and hence described byeffective interactions further away from the quantum crit-ical point (QCP) at u = 1 than La-1111. In the SM weshow that indeed 8% critical amounts of magnetic dis-order in OD Sm-1111 is consistent with our modelling.Recently it was shown that Y-substitution for La can sim-ilarly shift OD La-1111 away from the QCP and removethe poisoning effect. [27] Non-magnetic disorder.
We now turn to the discussionof non-magnetic disorder, and again motivate the studyby a set of puzzling experimental findings from FeSCssummarized in Figs. 1(c,d), which compare the effect on T c and T m of Ru ions substituting for Fe in OD La-1111and Sm-1111[13, 14, 28]. Ru is isovalent to Fe, and there-fore expected to be a source of weak disorder, consistentwith the huge amount of ∼
60% of Ru required to sup-press T c , as seen in Fig. 1(c). An unexpected magneticphase is induced at intermediate values of Ru content x ,centered roughly around x = 0 .
25, and existing only ata finite span ∆ x of disorder as seen in Fig. 1(d). Themagnetic phase is most pronounced with largest ∆ x andhighest T m in Sm-1111 and only marginally present in La-1111 even though the latter system displays the poisoningeffect and hypothetized to be more correlated (than Sm-1111) in the above discussion. Finally we point out theremarkable counterintuitive levelling-off of the T c sup-pression rate concomitant with the value of Ru content x c where magnetic order sets in, as seen most clearly inthe case of Sm-1111 in Fig. 1(d).In order to capture correctly the effects of large (com-positional changing) amounts of Ru substitution, it isimperative to include the effect of Ru on the band-structure itself. Our first-principles calculations showthat the bandwidth roughly doubles with Ru contentgoing from x = 0 to x = 1 in both LaFe − x Ru x AsOand SmFe − x Ru x AsO (see SM for details). This band-widening effect is accounted for by an overall renormal-ization of the hopping amplitudes t µν ij → (1 + x ) t µν ij in Eq. (2). For concreteness, we focus initially on acase with correlations of intermediate strength, u = 0 . V µ = 0 .
03 eV on all orbitals but allow for a phe-nomenological tuning of the potential on the d z − r or-bital ( V d z − r = 0 . T c and T m as a function of x . As seen, in addition to amuch slower T c suppression rate as compared to Fig. 2,a magnetic phase centered around x ∼
25% is gener- T c ■ ■■ ■■ ■■ ■■ ■■▲ ▲▲ ▲▲▲ ▲▲▲ ▲▲ ▲▲■ T c ▲ T m [%] k T c [ m e V ] k T m [ m e V ] FIG. 4.
Critical temperatures T c and T m as a func-tion of non-magnetic impurity concentration x . Thecritical temperature T c versus disorder concentration (redsquares). The dashed curve shows the T c for the clean system T c where only the band-widening effect has been included(i.e. no disorder), effectively reducing the pairing strengthby ˜Γ ≡ Γ / (1 + x ). As seen there is a region ∆ x of disorderconcentration (10% (cid:46) x (cid:46) T c for the disordered case ascompared to the clean system. This regime is characterizedby the existence of a bulk magnetic phase (green triangles)induced by the disorder and is seeded by favorable local im-purity structures as explained in Fig. 5. Note the kink in dT c /dx and a reduced T c suppression rate around x ∼ ated above a certain concentration x c of Ru ions. Asa function of x , T c exhibits an initial drop, but, inter-estingly, the induction of the magnetic phase does notenhance the T c suppression rate as expected from naivecompetitive considerations, but rather seems to furtherstabilize superconductivity. The origin for these uncon-ventional disorder effects is explained in Fig. 5 and the as-sociated caption. In essence, the emergence of favorableimpurity clusters (dimers and trimers, Fig. 5(a)) lead tosubstantial LDOS enhancements of the d z − r orbital(Fig. 5(b)), which drive both 1) induced magnetization(Fig. 5(d)) through local crossings of the Stoner insta-bility, [30, 31] and 2) an associated enhancement of thesuperconducting order parameter ∆ i d z − r (Fig. 5(f)).Through inter-orbital couplings the boost of ∆ i d z − r near the dimers is enough to cause the support for theentire superconducting condensate evident in Fig. 4 atintermediate disorder content ∆ x , where the enhancedpairing overcompensates the pair-breaking effect of boththe disorder and the induced magnetic phase.The dimer-induced LDOS enhancement mechanismnaturally explains the increase of T m starting at interme-diate values of impurity concentration x c ∼ x c . Asthe concentration of disorder increases, more dimer-like ( a ) Impurity positions ( b ) LDOS i μ xy xy ( c ) B ( r ) ( d ) m i μ xy - - xy - - ( e ) ( f ) Δ i μ / Δ ● ● ● ● ● ● ● ● ● ● ● ● ● ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● Dimer [%] ○ /( + x ) [%] D i m e r c on c en t r a t i on [ % ] B and - w i den i nge ff e c t xy FIG. 5.
Effect of impurity dimers on T m and T c . (a)Black and red tiles both indicate the positions of a random setof 15% disorder. The red tiles highlight favorable dimer-likearrangements, defined by all the impurity sites with an occu-pied NNN site but not more than one occupied NN site. (b)Real-space map of the LDOS of the d z − r orbital at T > T c at the Fermi level, revealing explicitly the correlation betweenthe brightest sites (largest LDOS) with the red dimer sites in(a). (c) Local dipolar field B ( r ) = (cid:80) i m i | r i | ( r i is the distancebetween the muon site r and the moment position m i of theitinerant electrons) felt by muons with orange (blue) colorindicating regions with field strength larger (smaller) than0 . − . d z − r orbital which dominates the to-tal magnetization. From a comparison to (b) it is evident thatthe LDOS enhancement near the dimers freeze magnetic orderin their vicinity. (e) Dimer density (blue dots) and the band-width renormalization parameter 1 / (1 + x ) (orange circles) asa function of disorder concentration x . (f) Superconductingorder parameter of the d z − r orbital ∆ i d z − r / ∆ d z − r relative to its value in the clean system, revealing remarkableorder-of-magnitude local enhancements in the vicinity of theimpurity dimers. structures with high LDOS form, and the system eventu-ally acquires a large enough magnetic volume fraction tosupport a non-zero T m . Specifically, T m is defined iden-tically to the experimental µ SR definition by the highest T exhibiting a 50% magnetic volume fraction. A site isdefined to contribute to the volume fraction if its internaldipolar local field exceeds | . | mT [13, 14]. In the caseof 15% disorder discussed in Fig. 5 we find a nearly sat-urated volume fraction as shown in panel 5(c), in agree-ment with experiments [13, 14]. For more details on thedefinition of T m and the resulting short-range magneticstructure induced by the dimers, we refer to the SM.From Fig. 5(e), showing the dimer concentration as afunction of x one expects a max T m near x ∼ W → (1 + x ) W resultsin lowered effective Coulomb correlations, and pushesthe magnetic dome to lower x . Thus, the position ofthe induced magnetic dome is a compromise betweenthe dimer-enhanced LDOS and the weakening of correla-tions due to band-widening. The resulting x -dependenceof both T c and T m seen in Fig. 4 appear in excellentoverall agreement with the experimental results shownin Fig. 1(d).Returning to the discussion of the distinction betweenthe two 1111 materials shown in Fig. 1(c,d), a remainingquestion is the origin of the significantly smaller inducedmagnetic phase in La-1111 compared to Sm-1111. La-1111 exhibits the poisoning effect explained above by alarger u (compared to Sm-1111) in this material, and ac-cordingly one may naively expect the induced magneticphase to be even more pronounced for La-1111 than forSm-1111. However, the larger correlations also act toscreen the non-magnetic disorder which results in lowereffective potentials (see SM for details) on the e g orbitalswhich then become unable to cause the LDOS enhance-ments shown in Fig. 5. The modified potentials simplyshift the bound-state structure away from the Fermi level,locally weakening the Stoner condition and thereby over-compensating the larger u as explained in the SM.We end by pointing out that the main effects dis-cussed in this work, i.e. the poisoning effect by mag-netic disorder and the resilience of superconductivity tononmagnetic disorder and its induced magnetization, arenot a pecularity of iron-based systems, but rather quitegeneral effects expected to exist in multi-orbital corre-lated superconducting systems. By tuning other mate-rials close to a magnetic instability, for example, mag-netic disorder should exhibit a similar aggressive T c -suppression rate. Likewise, when nonmagnetic disorderleads to large enough LDOS enhancements of orbitalsthat do not dominate the spectral weight in the cleansystem near the Fermi level, a disorder-induced coexis-tence phase of magnetism and superconductivity is ex-pected to occur. Our findings also serve as a warningto draw strong conclusions about the pairing symmetrybased on T c -suppression rates of unconventional corre-lated systems without detailed theoretical modelling be-yond standard AG-theory. ACKNOWLEDGEMENTS
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Here, we provide computational details of the results presented in the main part of the paper. We elaborate bothon the applied band structure, and the mean-field decoupled Hamiltonian in real space. We outline the details of ourAbrikosov-Gor’kov calculations in orbital space. Finally, we discuss the role of magnetic (non-magnetic) disorder inSm-1111 (La-1111), and show how the relevant experimental findings for those cases may also be naturally reconciledwithin the correlated disorder scenario presented in the main part of the paper.
MODEL
The starting Hamiltonian defined on a two-dimensional lattice is given by H = H + H int + H BCS + H imp , (S1)describes a superconducting system in the presence of correlations and disorder.We use a five-orbital tight-binding band relevant to the 1111 pnictides [S1] H = (cid:88) ij ,µν,σ t µν ij c † i µσ c j νσ − µ (cid:88) i µσ n i µσ . (S2)We stress that for the 1111 systems a two-dimensional model should be appropriate since the dispersion along the k z direction is essentially absent.[S1] Figure S1 shows the Fermi surface and its main orbital character. The presenceof the γ pocket at ( π, π ) was found to depend on the pnictogen height [S2], which can be controlled by the nearest-neighbor (NN) hopping parameter t xy . Here, we use this result to model the difference between La-1111 (withoutthe γ pocket) and Sm-1111 (with the γ pocket) by setting t Laxy = 1 . t Smxy . The resulting Fermi surfaces are shown inFig. S1.After a mean-field decoupling, the interacting multi-orbital Hubbard interaction becomes H MFint = (cid:88) i ,µ (cid:54) = ν,σ [ U n i µσ + U (cid:48) n i νσ + ( U (cid:48) − J ) n i νσ ]ˆ c † i µσ ˆ c i µσ , (S3)where n i µσ ≡ (cid:104) ˆ c † i µσ ˆ c i µσ (cid:105) . We apply the spin and orbital rotational invariance relations U (cid:48) = U − J and J (cid:48) = J throughout this work, and additionally set J = U/ H BCS = − (cid:88) i (cid:54) = j ,µν [∆ µν ij c † i µ ↑ c † j ν ↓ + H.c.] , (S4)with ∆ µν ij = (cid:80) αβ Γ βνµα ( r ij ) (cid:104) ˆ c j β ↓ ˆ c i α ↑ (cid:105) being the superconducting order parameter, and Γ βνµα ( r ij ) denoting the effectivepairing strength between sites (orbitals) i and j ( µ , ν , α and β ). In agreement with a general s ± pairing state, we ( a ) ( b ) XY Γ XY Γ FIG. S1. Fermi surface used to model (a) La-1111 and (b) Sm-1111. Different colors represent the main orbital character ofthe bands (purple: d xz ; green: d yz ; orange: d xy ). As the pnictogen height decreases (Sm → La) the d xy γ pocket at ( π, π )disappears from the Fermi surface. In our modelling, the presence of this pocket is controlled by the NN d xy hopping parameter, t Laxy = 1 . t Smxy . include next-nearest neighbor (NNN) intra-orbital pairing, Γ µ ≡ Γ µµµµ ( r nnn ). Additionally, the standardly obtainedreduced pairing vertex of the e g orbitals (see for example Ref. S3) is accounted for by reducing them by roughly afactor of two: Γ t g = 0 .
293 eV and Γ e g = 0 . t g . This reduction limits the T c enhancement found in the nonmagneticdisorder case (see Fig.4 in main text), where the e g orbitals play an important role. By contrast, in the case ofmagnetic disorder the reduction in the e g orbital pairing does not influence the results since both superconductivityand induced long-range polarization are mainly determined by the t g orbitals.The last term in the Hamiltonian introduces disorder in the system. Non-magnetic and magnetic disorder give riseto terms of the form H imp = (cid:80) µ { i ∗ } V µ n i ∗ µ and H imp = I (cid:80) { i ∗ } µσ σS µ c † i ∗ µσ c i ∗ µσ , respectively. Here V µ ( S µ ) denotesthe impurity potential (magnetic moment) in orbital µ at the disorder sites given by the set { i ∗ } coupled to the charge(spin) density of the itinerant electrons.By using the spin-generalized Bogoliubov transformation,ˆ c i µσ = (cid:88) n ( u n i µσ ˆ γ nσ + v n ∗ i µσ ˆ γ † nσ ) , (S5)we arrive to the Bogoliubov-de Gennes (BdG) equations (cid:32) ˆ ξ ↑ ˆ∆ ij ˆ∆ ∗ ji − ˆ ξ ∗↓ (cid:33) (cid:18) u n v n (cid:19) = E n (cid:18) u n v n (cid:19) . (S6)The transformation (cid:0) u n ↑ v n ↓ E n ↑ (cid:1) → (cid:0) v n ∗↑ u n ∗↓ − E n ↓ (cid:1) maps two of the equations onto the other two and thuswe drop the spin index from the eigenvectors and eigenstates of the BdG equations. The matrix operators are definedas: ˆ ξ σ u i µ = (cid:88) jν t µν ij u j ν + (cid:88) µ (cid:54) = ν [ − µ + Ω δ ii ∗ δ µν + U n i µσ + U (cid:48) n i νσ + ( U (cid:48) − J ) n i νσ ] u i µ , (S7)ˆ∆ µν ij u i µ = − (cid:88) j ν ∆ µν ij u j ν , where Ω = V µ for nonmagnetic disorder and Ω = σIS µ for magnetic disorder. The five-orbital BdG equations aresolved on 30 ×
30 lattices with stable solutions found through iterations of the following self-consistency equations n i µ ↑ = (cid:88) n | u n i µ | f ( E n ) , (S8) n i µ ↓ = (cid:88) n | v n i µ | (1 − f ( E n )) , ∆ µ ij = Γ µ (cid:88) n u n i µ v n ∗ j ν f ( E n ) , where (cid:80) n denotes summation over all eigenstates n . We stress that the solutions are fully unrestricted and allowedto vary on all lattice sites and orbitals. The superconducting order parameter shown in the main manuscript is thebond averaged singlet component: ∆ i µ = 14 (cid:88) j
12 (∆ µ ij + ∆ µ ji ) (S9)where j are four nearest neighbors. The inclusion of several impurities leads to a spatially varying order parameter∆ i µ , and lowers the transition temperature T c at which a non-zero solution of the gap equation exists. Eventually,for a sufficiently high critical concentration of impurities superconductivity is destroyed at all sites, i.e. ∆ i µ = 0 , and T c = 0. INTERACTION-GENERATED NONMAGNETIC AND MAGNETIC POTENTIALS AROUNDDISORDER
In order to shed light on the interaction-generated extended potentials we rewrite Eq. (S3) in terms of the chargedensity n i µ and spin density m i µ fields, by using n i µσ = ( n i µ + σm i µ ) / H MFint = 12 (cid:88) i µ (cid:54) = νσ [ U n i µ + (2 U − J ) n i ν ] ˆ c † i µσ ˆ c i µσ − (cid:88) i µ (cid:54) = νσ σ [ U m i µ + Jm i ν ] ˆ c † i µσ ˆ c i µσ . (S10) ( a ) IS μ ( b ) I ind s i μ ( c ) V ind,i μ ( d ) | m ( q )| xy - - - xy - - xy - - - FIG. S2. (a) Bare magnetic impurities IS µ = 0 .
38 eV and the correlation-induced (b) magnetic potential I ind s i µ and (c)nonmagnetic potential V ind, i µ for the d xz orbital. (d) Fourier transform of the total magnetization. For all panels in this figure u = 0 . The effect of impurities on both charge and spin densities is given by n i µ = n i µ + ∆ n i µ and m i µ = m i µ + ∆ m i µ ,where n i µ and m i µ are the fields of the disorder-free system ( m i µ = 0 throughout this study) and ∆ n i µ and ∆ m i µ thedisorder-induced changes. Introducing these expressions in Eq. (S10) we obtain extended impurity-like terms V ind, i µ and I ind s i µ that result in the following effective disorder potentials˜ V i µ = V µ δ ii ∗ + 12 U ∆ n i µ + (2 U − J ) (cid:88) ν (cid:54) = µ ∆ n i ν ≡ [ V µ δ ii ∗ + V ind, i µ ] , (S11)˜ I ˜ S i µ = IS µ δ ii ∗ − U ∆ m i µ + J (cid:88) ν (cid:54) = µ ∆ m i ν ≡ [ IS µ δ ii ∗ + I ind s i µ ] , where V µ and IS µ are the bare nonmagnetic and magnetic potentials, respectively. Thus, in the presence of interactions( u (cid:54) = 0), the field modulations ∆ n i µ and ∆ m i µ induced by disorder give rise to effective nonmagnetic and magneticpotentials, respectively. An example relevant for bare magnetic disorder is shown in Fig. S2. As seen, besides theinduced magnetic potential shown in Fig. S2(b), a concomitant induced nonmagnetic potential is generated fromcharge density modulations, shown in Fig. S2(c). For the superconducting state studied in the main part of thepaper, the pair-breaking effect of the charge potential is weak and thus most of the correlation effects arise from theinduced magnetic local and non-local components. A Fourier transform of the total magnetization is displayed inFig. S2(d), clearly showing the dominant (0 , π ) LRO of the induced magnetization arising from the regions in-betweenthe impurity sites.We point out that early local paramagnon theories relevant e.g. to disorder in metallic Pd have also discussedeffects of Stoner enhanced susceptibilities.[S4] BANDWIDTH INCREASE THROUGH RU SUBSTITUTION
In Fig. S3(a), we compare the band structures of LaFeAsO and LaRuAsO to understand the effect of Ru substitu-tion. The first-principles calculations of the electronic structure were performed within the density functional theoryusing the full-potential linearized augmented plane-wave method with the addition of local-orbital basis functionsas implemented in the WIEN2K code. [S5–S7] For the exchange and correlation functional we use the generalizedgradient approximation (GGA) of Perdew, Burke, and Ernzerhof in its revised form. [S8] We used muffin-tin radii of2.40 a for Sm and La, 2.20 a for Fe and Ru, 2.0 a for As, and 1.90 a for O.To help the comparison of the results, we rescaled the abscissas in Fig. S3(a) to fit the band structure of LaRuAsOwith the Brillouin zone for the LaFeAsO system. The band structure of LaFeAsO (solid lines) is characterized by avalence band originating from Fe-3 d orbitals. We see that the band ranges from 0.15 to -2.15 eV and is separated fromthe As-4 p band by a small pseudo gap. The width of the Fe-3 d band is 2.3 eV. The short (black) arrow in Fig. S3(a)shows the estimated width of the band from its topmost d xz /d yz state to the lowermost d x − y dominated band. TheLaRuAsO and LaFeAsO band structures clearly differ in the dispersion of the Ru-4 d orbitals related band. To helpthe readability of the band structure in Fig. S3(a), we used the so-called fat-bands representation, where the size ofthe dots is proportional to the weight of the Ru-4 d orbitals. We see that Ru-4 d states span over a range of ∼ FIG. S3. (a) Band structures for LaFeAsO (lines) and LaRuAsO (dots). The size of the dots is proportional to the weightof the Ru-4 d orbitals. The LaRuAsO band structure is rescaled to fit the first Brillouin zone of LaFeAsO (see text). (b) Bandparameters for transition metal-related d -bands. Solid lines and filled symbols refer to LaFe − x Ru x AsO, while dashed linesand open symbols refer to SmFe − x Ru x AsO.
In Fig. S3(a), it is still possible to identify the states with the d xz /d yz and d x − y characters at Γ to define the widthof the valence band in LaRuFeAs. The long (red) arrow shows the estimated width of the Ru-4 d band.In order to provide a quantitative estimate of the band width as a function of Ru content x (in e.g. LaFe − x Ru x AsO),we introduce a criterion based on the density of states projected onto the Fe and Ru d -orbitals to define the d bandwidth. We define the function P ( E ) by P ( E ) = (cid:90) E −∞ [(1 − x ) D Fe ( ε ) + xD Ru ( ε )] dε, (S12)where D Fe ( ε ) and D Ru ( ε ) are the values of the projected density of states (PDOS) onto the Fe-3 d and Ru-4 d orbitals,respectively. The physical meaning of P ( E ) is the amount of electron density for energies below E coming from d states of Fe and Ru. We use P ( E ) to define unambiguously the band width in R Fe − x Ru x AsO. We find that P ( E ) inLaFeAsO is 1.515 for E = − .
15 eV and 6.3 for E = 0 .
15 eV. We define the minimum of the transition metal-related d -bands, E L , as the value that fulfills the relation P ( E L ) = 1 . E U , as the value for which P ( E U ) = 6 .
3. With this criterion, we computed the lower and upper limits of the d related band in R Fe − x Ru x AsO.In Fig. S3(b), we show the energies of the upper and lower edges of the transition metal-related band; the bandcenter defined as the average E av = ( E U + E L ); the bandwidth E W = E U − E L . Fig. S3(b) shows that the La vs.Sm substitution does not influence the position and the width of the transition metal d band. The band center energy E av is weakly dependent of Ru concentration. The bandwidth increases from 2.3 to 4.8 with Ru content. This isthe most relevant effect of Ru substitution on the band structure of R Fe − x Ru x AsO. The change in the bandwidthis due, in equal amount, to an increase of the band maximum and a decrease of the band minimum, with respect tothe Fermi energy. The band minimum goes from -2.15 to -3.2 eV, increasing the hybridization of Ru-4 d orbitals withthe As-4 p . The band maximum increases from 0.15 to 1.6 eV, showing that Ru related bands extend far beyond theFermi level into the unoccupied states. ABRIKOSOV-GOR’KOV THEORY
In the main text we compare the T c -suppression rates with standard Abrikosov-Gor’kov theory. In this section webriefly outline the procedure used to obtain those results.After averaging over random distributions of the impurities, the Green’s function describing the electron recoverstranslational symmetry. Thus, the full Green’s function is generally given by( G ( k , iω n )) − = ( G ( k , iω n )) − − Σ( k , iω n ) , (S13) ( a ) ( b ) ( c ) ( d ) Sm Nd La ✶✶ ✶✶ ✶✶ D O S ( ω = ) x y - - x y - - ■ ■ ■ ■ ■ ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● La ( u = ) ■ Sm ( u = ) Sm ( IS ˜= * IS ) [%] k T c [ m e V ] FIG. S4. (a) Schematic phase diagram of the L-1111 (L=Sm, Nd, La) members as a function of correlations u and DOS atthe Fermi level. Correlation-induced magnetic potential I ind s i µ of the d yz orbital for (b) La-1111 ( u = 0 .
97) and (c) Sm-1111( u = 0 .
70) for a 0 .
55% impurity configuration. (d) T c as a function of magnetic impurity concentration for La- and Sm-1111systems. with G ( k , iω n ) = ( iω n − H ( k ) ρ − ∆( k ) ρ σ ) − , (S14)the Green’s function in the impurity-free system, where ω n = 2 πT ( n + ) and T is the temperature. ρ i and σ i denote Pauli matrices operating on the electron and hole states and ordinary spin states, respectively. In the finalspin-orbital-nambu space all quantities ( G ( k , iω n ), Σ( k , iω n ), and G ( k , iω n )) are 10 ×
10 matrices. The SC orderparameter is obtained from the gap equation∆( k ) = T V (cid:88) ω n , k (cid:48) Γ( k (cid:48) ) Tr [ ρ σ G ( k (cid:48) , iω n )] , (S15)with Γ( k (cid:48) ) = 4Γ µ cos k (cid:48) x cos k (cid:48) y and in Born approximation the self-energy is given byΣ( k , iω n ) = n V (cid:88) k (cid:48) Ω G ( k (cid:48) , iω n )Ω , (S16)where Ω = V µ ρ for nonmagnetic impurities and Ω = σ IS µ ρ for magnetic impurities. The calculated Abrikosov-Gor’kov curves are obtained through iterative convergence of Eqs. (S13)-(S16) using 600 Matsubara frequencies whichwas checked to be enough in the cases presented.We stress that interference between the impurities, and all the interaction effects discussed in the main text areneglected within Abrikosov-Gor’kov theory. MAGNETIC DISORDER IN SM-1111
In this section we show the T c -suppression as a function of magnetic impurity concentration for the Sm-1111 case, tobe contrasted with the T c -suppression of La-1111 (poisoning effect) presented in the manuscript. Figure S4(a) showsa schematic phase diagram of the L-1111 (L=Sm, Nd, La) members within the present scenario. The main proposeddifferences between the compounds are: 1) Sm-1111 exhibits a larger DOS (than Nd- and La-11111) at the Fermilevel (see γ pocket evolution in Fig. S1) and hence a larger T c as explicitly shown in Fig. S4(d), and 2) Sm-1111 is lesscorrelated (than Nd- and La-11111), and hence described by a u parameter further away from the critical value u c .The second assumption implies weaker correlation effects, i.e. the additional interaction-induced pair-breaking effectis diminished, as evident from comparison of Figs. S4(b) ( u = 0 .
97 for La-1111) and S4(c) ( u = 0 . T c -suppression rate withan 8% critical impurity concentration similar to that found experimentally in optimally doped Sm-1111 [S9]. ( a ) % ( b ) % ( c ) % x x - - x - - ( d ) ( e ) ( f ) x x - - x - - FIG. S5. Induced magnetization m i with (a) 5%, (b) 25% and (c) 50% non-magnetic disorder content. Panels (d)-(f) displaythe respective fields B ( r ) associated with (a)-(c). COMPARISON WITH µ SR The magnetic ordering temperature T m determined from µ SR experiments is extracted from the T evolution ofthe magnetic volume fraction defined by the fraction of muons that detect a local field exceeding ∼ . T m is defined as the highest T where the volume fraction is 50%. The local field is proportional tothe staggered moment mainly through the dipolar coupling. Here we implement the map from a given staggeredmagnetization field at the muon sites r : B ( r ) = (cid:88) i µ m i µ | r i | , (S17)where r i = ( ax i − ( a/ ax ) , ay i − ( a/ ay ) , c ) denotes the relative distance between the muon site r and themoment position at site i . We have used the symmetric position of the main muon site ( a/ , a/ , c ), with a = 2 . c = 0 . B ( r ) in Fig. S5.Next, we turn to a brief discussion of the momentum structure of the magnetic phase induced by Ru substitution.The Fourier transformed magnetic structure of a single dimer shown Fig. S6(a) can be seen in Fig. S6(b). It consistsof broad peaks at low wave-vectors and near ( π, π ). In general, adding Ru to the system will result in a roughly equaloccupation of oppositely oriented dimer structures, as illustrated in Fig. S6(c). Thus the overall structure factor inFig. S6(d) respects tetragonal symmetry. We verified that the magnetic structure of the disordered-induced magneticphase remains dominated by the ”single-dimer” results of Figs. S6(a-d) by calculating the average of the magneticstructure factor for twelve different configurations with 15% nonmagnetic disorder, shown in Fig. S6(f). The inducedmagnetic (short-range) order in Figs. S6(e-f) is clearly very different from the stripe-like long-range magnetic orderinduced by magnetic impurities (see for example Fig. S2(b)), with sharp ( π, / (0 , π ) peaks. NONMAGNETIC DISORDER IN LA-1111
Finally, we return briefly to the discussion of the distinction between the three different 1111 materials shown inFig. S4(a). A necessity for La-1111 to exhibit the poisoning effect in the case of magnetic disorder is the closeness to u c for this material as illustrated in Fig. S4(a). For the case of nonmagnetic disorder, La-1111 will generally exhibita pronounced magnetic phase, and in particular, exhibit a larger magnetic phase than e.g. Sm-1111 contrary toexperimental findings. This, however, is only true for identical local Ru potentials in La-1111 and Sm-1111. Differentmaterial parameters will lead to differences in the relevant extracted Ru potentials. There is another effect, however, ( a ) ( c ) ( e ) y y y - ( b ) ( d ) ( f ) lowhigh FIG. S6. Dimer-induced magnetic structure. (a) Single dimer, (b) two dimers with opposite orientations, and (c) 15% non-magnetic disorder concentration. (d)-(f) The Fourier transformed | m ( q ) | of the cases (a)-(c), respectively. The 15% case hasbeen averaged over twelve different random configurations. which becomes important for correlated systems, which is the additional screening caused by the Hubbard cost ofcharge modulations. [S12]Figure S7(a) shows the electron density along a line cut through a nonmagnetic potential. As seen, the correlationseffectively screens the bare site potential. Such effects are typically not included in DFT-extracted impurity potentials,and could be an important source of discrepancy between bare and dressed potentials in correlated systems. In thecurrent case, such renormalized potentials mainly relevant to La-1111 will shift the bound-state structure away fromthe Fermi level, locally weakening the Stoner condition and thereby compensating the larger u . We show in Fig. S7(b)the magnetic dome for a renormalized V = 0 . u = 0 .
97 La-1111 system, much smaller than thedome of the Sm-1111 system despite the weaker correlations in the latter compound. ( a ) ( b ) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● u = ▲ u = n ■ ■■ ■■■ ■■■ ■■ ■■ La ( u = ) ■ Sm ( u = ) [%] k T m [ m e V ] FIG. S7. (a) Total electron density as a function of position along a cut through a single non-magnetic impurity. As seen bycomparison of the black ( u = 0 .
0) and red curves ( u = 0 .
97) correlations weaken the charge modulations. (b) Comparison ofInduced magnetic order for Sm-1111 (blue curve) and La-1111 (orange curve) showing a less pronounced magnetic phase forLa-1111 despite its larger correlations.[S1] H. Ikeda, R. Arita, and J. Kunes, Phys. Rev. B , 054502 (2010). [S2] K. Kuroki, H. Usui, S. Onari, R. Arita, and H. Aoki, Phys. Rev. B , 224511 (2009).[S3] M. N. Gastiasoro, P. J. Hirschfeld, and B. M. Andersen, Phys. Rev. B , 220509 (2013).[S4] P. Lederer and D. L. Mills, Phys. Rev. , 837 (1968).[S5] D. Singh, Phys. Rev. B , 6388 (1991).[S6] E. Sj¨ostedt, E, L Nordstr¨om, and D. Singh, Solid State Communications , 15 (2000).[S7] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k , An Augmented Plane Wave + Local Or-bitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universit¨at Wien, Austria), 2001. ISBN3-9501031-1-2, [S8] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,Phys. Rev. Lett. , 136406 (2008).[S9] S. J. Singh, J. Shimoyama, A. Yamamoto, H. Ogino, and K. Kishio, Physica C (Amsterdam) , 57 (2011).[S10] S. Sanna, R. De Renzi, T. Shiroka, G. Lamura, G. Prando, P. Carretta, M. Putti, A. Martinelli, M. R. Cimberle, M.Tropeano, and A. Palenzona, Phys. Rev. B , 060508 (2010).[S11] B. P. P. Mallett, Yu. G. Pashkevic, A. Gusev, T. Wolf, C. Bernhard, Europhys. Lett. , 57001 (2015).[S12] B. M. Andersen and P. J. Hirschfeld, Phys. Rev. Lett.100