Impurity bound states and disorder-induced orbital and magnetic order in the s+- state of Fe-based superconductors
IImpurity bound states and disorder-induced orbital and magneticorder in the s ± state of Fe-based superconductors Maria N. Gastiasoro and Brian M. Andersen
Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen, Denmark (Dated: April 2, 2019)We study the presence of impurity bound states within a five-band Hubbard modelrelevant to iron-based superconductors. In agreement with earlier studies, we findthat in the absence of Coulomb correlations there exists a range of repulsive impuritypotentials where in-gap states are generated. In the presence of weak correlations,these states are generally pushed to the edges of the gap, whereas for larger correla-tions the onsite impurity potential induces a local magnetic region which reintroducesthe low-energy bound states into the gap. a r X i v : . [ c ond - m a t . s up r- c on ] O c t I. INTRODUCTION
There are at least two reasons why the study of disorder effects in the high-T c super-conductors remain an important topic. First, the superconducting state itself is generatedby chemical doping which inevitably disorders the samples, and second, local probes of thequasi-particle states near the impurity sites can provide important information on the under-lying system. In the case of the cuprates, for example, it was shown how disorder acts topin competing correlations, providing a natural explanation for the so-called spin-glass phasein the underdoped regime, and STM measurements near isolated Ni impurities showedclear evidence for d -wave pairing symmetry of the superconducting order parameter. In the iron-pnictides, several experimental scanning tunneling studies have been per-formed to investigate the modulations in the electronic spectrum caused by various defects.
In the case of LiFeAs, for example, it has been recently shown how several kinds of defectsexist on the surface, with distinct local structures in the local density of states (LDOS). Atpresent there is no theoretical model capturing the details of the LDOS near these differentimpurities.Theoretically, it is well-known that both potential and magnetic impurities can giverise to in-gap states in d -wave and multi-band s ± superconductors. In the latter case,several theory studies of the single-impurity problem have been reported both with simplifiedtwo-band models, and within a five-band approach. Here, we extend the study ofthe single-impurity problem within a five-band model including interactions treated in anunrestricted Hartree-Fock approximation. It is shown how an impurity locally induces orbitaland magnetic order which can strongly modify the positions of the spectral in-gap boundstates detectable by STM.
II. MODEL
The five-orbital model Hamiltonian is given by H = H + H int + H BCS + H imp . (1)The first term is a tight-binding 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)Here the operators c † i µσ ( c i µσ ) create (annihilate) an electron at the i -th site in the orbital µ and with spin projection σ , and µ is the chemical potential. The indices µ and ν runthrough 1 to 5 corresponding to the five d xz , d yz , d x − y , d xy and d z iron orbitals. Thehopping integrals t µν ij are the same as those in Graser et al. , included up to fifth nearestneighbors. Here, as elsewhere in this paper, the energy units are in electron volt (eV).The second term describes the onsite Coulomb interaction, 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) (3) − 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 , which includes the intra-orbital (inter-orbital) interaction U ( U (cid:48) ), the Hund’s rule coupling J and the pair hopping energy J (cid:48) . We will assume orbital and spin rotational invariancewhere the relations U (cid:48) = U − J and J (cid:48) = J hold.The third term is a phenomenological BCS pairing term H BCS = − (cid:88) i (cid:54) = j ,µν [∆ µν ij c † i µ ↑ c † j ν ↓ + H.c. ] , (4)with the superconducting (SC) order parameter ∆ µν ij = V ij (cid:104) c j ν ↓ c i µ ↑ (cid:105) where V ij denotes thestrength of the effective attraction. The pairing is chosen as next-nearest-neighbor intra-orbital pairing, which reproduces the fully gapped s ± state.The last term in the Hamiltonian is a nonmagnetic impurity term H imp = V imp (cid:88) i ∗ µσ c † i ∗ µσ c i ∗ µσ , (5)which adds a local potential V imp at a single site i ∗ in all five orbitals, neglecting the orbitaldependence for simplicity.After a mean-field decoupling of the onsite interaction term (3) in both the “density” andthe “Cooper” channels, a Bogoliubov transformation results in the following multi-orbitalBogoliubov de-Gennes equations (cid:88) j ν H i µ j νσ ∆ i µ j ν ∆ ∗ i µ j ν − H ∗ i µ j ν ¯ σ u n j ν v n j ν = E n u n i µ v n i µ , (6)where H i µ j νσ = t µν ij + δ ij δ µν [ − µ + δ ii ∗ V imp + U (cid:104) n i µ ¯ σ (cid:105) (7)+ (cid:88) µ (cid:48) (cid:54) = µ ( U (cid:48) (cid:104) n i µ (cid:48) ¯ σ (cid:105) + ( U (cid:48) − J ) (cid:104) n i µ (cid:48) σ (cid:105) )] , and ∆ i µ j ν = δ ij δ µν [∆ µµ ( U ) ii + 2 (cid:88) µ (cid:48) (cid:54) = µ ∆ µ (cid:48) µ (cid:48) ( J (cid:48) ) ii ] (8)+ 2 δ ij (cid:88) µ (cid:48) (cid:54) = µ [∆ µµ (cid:48) ( U (cid:48) ) ii + ∆ µ (cid:48) µ ( J ) ii ] − ∆ µν ij . The local densities and the SC order parameters are obtained through the following self-consistency equations (cid:104) n i µ ↑ (cid:105) = (cid:88) n | u n i µ | f ( E n ) , (9) (cid:104) n i µ ↓ (cid:105) = (cid:88) n | v n i µ | (1 − f ( E n )) , ∆ µν ( X ) ii = X (cid:88) n u n i µ v n ∗ i ν f ( E n ) , (10)∆ µν ij = V ij (cid:88) n u n i µ v n ∗ j ν f ( E n ) , (11)where X = U , U (cid:48) , J or J (cid:48) . III. RESULTS AND DISCUSSION
We focus on signatures associated with the s ± state, generated by an intra-orbital su-perconducting pairing V ij = 0 .
65 between next-nearest neighbor sites. The chemical po-tential µ is fixed so that the total density is n = 6 . x imp , y imp ) = (14 ,
14) in a 28 ×
28 lattice.For repulsive potentials ( V imp > d xz and d yz at each Fe site, theirhopping amplitudes t µν ij along x and y directions are the same but rotated by π/ d yz with a π/ d xz , and a local twofold orbitalordering is induced around the impurity. Figure 1(a) shows the orbital ordering around arepulsive impurity. For attractive potentials ( V imp < π/ U c (with J = U/ m i = (cid:80) µ ( n i µ ↑ − n i µ ↓ ) µ B is plotted in figure 1(b). For attractive potentialson the contrary, no induced magnetization is found.Let us analyze why this effect depends on the type of impurity. The local density ofstates (LDOS) at site i is given by N i ( ω ) = − π Im (cid:88) nµ ( | u n i µ | ω − E n + iη + | v n i µ | ω + E n + iη ) , (12)and calculated using the “supercell” method, with a 25 ×
25 copies of the original 28 × η = 0 . U = J = 0 uncorrelated case. Around therepulsive impurity, states are generated inside the SC gap; on the contrary, the gap is almostunchanged and clean around the attractive impurity. These results agree with a previousfive-orbital single-impurity study. In the static long wavelength limit ( ω = 0, k → χ ( k → ,
0) is proportional to the DOS at the Fermilevel, and the Stoner criterion becomes
U N ( E F ) →
1. Because of the presence of bound (a) x y n xz (cid:45) n yz (cid:45) (b) x y m (cid:72) Μ B (cid:76) (cid:45) (c) x y (cid:68) xz S FIG. 1: Real-space distribution of the self-consistent mean fields for a repulsive impurity( V imp = 1). The interacting parameters have been chosen as U = 1 .
35 and J = U/ d xz orbital in meV.states in the case of repulsive impurities, the DOS is generally higher at the Fermi level,allowing the Stoner instability to be crossed locally around the impurity site. Figure 1(c) shows the obtained SC order parameter and its modulation around the im-purity site. A representative singlet component for each orbital µ is given by,∆ S i µ = 12 (cid:88) j (∆ µµ ij − ∆ µµ ji ) . (13)The spatial modulation of this orbitally resolved SC order parameter around the impurityfollow the symmetry of their corresponding orbitals; twofold symmetry for ∆ Sxz and ∆
Syz ,and fourfold for the rest of the orbitals.Finally, we analyze the role of the strength of the correlations on the final LDOS. We focuson repulsive potentials, where local in-gap bound states are generated and magnetizationcan be induced around the impurity. The correlation strength dependence of the LDOSis summarized in figure 3. The uncorrelated U = J = 0 case can be seen in figure 3(a).There are four in-gap bound states at the nearest-neighbor and impurity sites. When thecorrelations are slightly increased, these states are pushed away from the Fermi level (panels3(b)-(c)). At values of U below the critical Stoner U c , the impurity-state formation happensat the edges or outside of the gap. For higher strength of correlations, U starts getting closeto the critical value ( U → U c ), and in-gap states are formed again. It is then possible for (a) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . (b) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . FIG. 2: LDOS for a (a) repulsive ( V imp = 1) and (b) attractive ( V imp = −
1) impurity. Theinteraction parameters are chosen to be U = J = 0. The black curve is the DOS far awayfrom the impurity site and the red and green curves the DOS at the nearest neighbor andimpurity site, respectively.the system to locally cross the Stoner instability, and magnetization is induced. Finally, thehigh correlation case U > U c , is shown in figure 3(f). Strong magnetization sets in in thevicinity of the potential (see figure 1(b)), and new local in-gap magnetic features appearin the LDOS around the Fermi level associated with the effective cluster of ”magnetic”impurities surrounding the non-magnetic potential. (a) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . (b) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . (c) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . (d) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . (e) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . (f ) (cid:45) (cid:45) Ω (cid:72) eV (cid:76) N (cid:72) Ω (cid:76) bulkonsiten .n . FIG. 3: LDOS around a repulsive impurity ( V imp = 1), for various correlation strengths U and J = U/
4. (a) U = 0, (b) U = 0 .
2, (c) U = 0 .
4, (d) U = 1 . < ∼ U c = 1 .
1, (e) U = 1 . > ∼ U c and (f) U = 1 . > U c . The black curve is the DOS far away from theimpurity site and the red and green curves the DOS at the nearest neighbor and impuritysite, respectively. IV. CONCLUSIONS
In summary, we have studied the single-impurity problem in an effective five-orbitalHubbard model with interactions included at the mean field level. Superconductivity isincluded by a phenomenological BCS term where pairing between next-nearest neighborsgenerate a fully gapped s ± state.Local properties such as orbital ordering and magnetization are induced around the im-purity potentials. The orbital ordering appears around repulsive and attractive potentials,because of an effective hopping asymmetry of the ordered orbitals. Magnetization is inducedaround repulsive scatterers. These kind of impurities are pair-breaking and develop localin-gap bound states, enhancing the LDOS around the Fermi level. The Stoner condition canthen be locally satisfied for strong enough correlations. By contrast, attractive impuritieshave an almost uniform clean gap, and do not induce local magnetization.Finally, we discuss the role of correlations on the impurity bound states. At low correla-tion strengths, the bound states tend to be pushed out of the SC gap. However, when thecorrelation strength approach a critical value U c , the in-gap bound states are pushed backinto the gap, and finally when local magnetization is induced around the potential additionalsub-gap peaks appear in the LDOS resulting from the magnetization. V. ACKNOWLEDGEMENT
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