Origin of electronic dimers in the spin-density wave phase of Fe-based superconductors
OOrigin of electronic dimers in the spin-density wave phase of Fe-based superconductors
Maria N. Gastiasoro , P. J. Hirschfeld , and Brian M. Andersen Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen, Denmark Department of Physics, University of Florida, Gainesville, Florida 32611, USA (Dated: November 27, 2018)We investigate the emergent impurity-induced states arising from point-like scatterers in thespin-density wave phase of iron-based superconductors within a microscopic five-band model. Inde-pendent of the details of the band-structure and disorder potential, it is shown how stable magnetic( π, π ) unidirectional nematogens are formed locally by the impurities. Interestingly, these nemato-gens exhibit a dimer structure in the electronic density, are directed along the antiferromagnetic a -axis, and have typical lengths of ∼
10 lattice constants in excellent agreement with recent scanningtunnelling experiments. These electronic dimers provide a natural explanation of the dopant-inducedtransport anisotropy found e.g. in the 122 iron pnictides.
PACS numbers: 74.20.-z, 74.70.Xa, 74.62.En, 74.81.-g
Hints of electronic nematicity, i.e. spontaneous break-ing of discrete rotational symmetry while preservingtranslational symmetry, appear in many cases amongthe Fe-based materials[1, 2]. These anisotropies, de-tected by STM[3], transport (on detwinned samples)[4–6], ARPES[7–10], neutron scattering[11], optical[12] andRaman[13] spectroscopy, and torque magnetometry[14],have largely been interpreted in terms of an intrinsic ten-dency of these systems to break C symmetry globallydue to nematic correlations, due either to spin nematiceffects[1, 2] or orbital ordering.[15–18] However, thereare also indications of local defect states which break C symmetry[19–24]. Since impurities are known to nucleatemagnetic order locally[25–27], it is reasonable to assumethat incipient nematic order in a fluctuating state canbe similarly condensed around a defect, to create a localnematic electronic state. Once present, such anisotropicdefect structures can influence macroscopic anisotropy aswell, and it has been suggested[3, 23, 28, 29] that theyare responsible for the resistivity anisotropy observed indetwinned Ba-122 crystals[5, 6, 29].Because impurities represent a well-defined perturba-tion which can be examined locally, studying the elec-tronic states they create can yield important informa-tion on the background correlations present in the puresystem[25]. In YBCO, for example, magnetic dropletsformed around Zn impurities are known to have a sizeconsistent with the pure antiferromagnetic (AF) correla-tion length.[25] At present, the microscopic mechanismresponsible for the creation of local defect states in Fe-based materials and for breaking of rotational symmetryis unclear. Some clues are offered by STM experiments ondefects in the underdoped, magnetically ordered phase,where the symmetry is already broken by the ( π,
0) spin-density wave (SDW) order (the wave vector is given in theeffective 1-Fe Brillouin zone). Fourier transform scanningtunneling spectroscopy (STS) deduced the existence ofelectronic defects with C symmetry[3] nucleated by the Co dopants in Ca(Fe − x Co x ) As , while a more detailedanalysis reported a dimer structure.[23] These dimers areapproximately eight lattice constants ( a ) long and ori-ented along the AF a -axis, consistent with the larger re-sistivity found along the ferromagnetic b -axis.As a first step in understanding the origin of local C symmetry breaking observed in several experiments onvarious materials, it seems useful to study a situationwhere a known chemical impurity substitutes at a knownposition, and ask why such a dimer-like structure (withcharge or local density of states (LDOS) peaks locatedsuch a great distance from the impurity site) should beinduced. As in the cuprates, this problem should be ac-cessible to weak-coupling theories of these systems, pro-vided they account for the electronic states of the sys-tem to which the impurity couples and treat interactionson the average. Hints that an unusual electronic statemight be induced were found already in first principlescalculations of a Co dopant in Ba-122, where the mag-netic potential due to the Co was found to be oscillatoryand exceed several unit cells[30]. Impurity-induced C structures have been previously studied within a strong-coupling model[31], and a scenario based on a competingpocket density wave order which has, however, not beenobserved[32]. Finally, in Ref. 33, impurities were shownto pin fluctuating orbital order and create local stateswith broken C symmetry but no dimer-like character.Here, we present a general study the origin of elec-tronic dimer states by an explicit, unbiased microscopicfive-band calculation of the local electronic densities neara point-like impurity potential in the SDW phase of theiron pnictides. The impurity causes a local ( π, π ) mag-netic instability, which combined with the ( π,
0) orderof the bulk SDW phase, results in unidirectional mag-netic defects oriented along the AF a -axis, and associ-ated electronic density dimers. These dimers are causedby local density modulations of the SDW phase, and notdependent on the details of the band-structure, disorder a r X i v : . [ c ond - m a t . s up r- c on ] O c t potential, or local impurity-modified interaction parame-ters. This is a concrete example of a phenomenon whichis largely unexplored, the local nucleation of a particu-lar magnetic order in a bulk state with different mag-netic order. We show how such emergent impurity statesevolve from droplets at high temperatures T to nemato-gens in the low- T SDW phase. The final size of the low- T dimers is consistent with recent STM measurements[23],but depends within our theory on the ”cooling rate”, andwe show how dimers of other lengths may also be ob-tained. Finally, we compute the LDOS characteristics ofthe dimer states to compare with STM experiments, anddiscuss how our model can be used to perform realisticcalculations of effective defect potentials for applicationto transport experiments.The five-orbital Hamiltonian consists of three terms H = H + H int + H imp , (1)with H the kinetic part arising from a tight-binding fitto the density functional theory (DFT) band-structure ofRef. 34 (similar results arise by using e.g. the band ofIkeda et al. [38]) H = (cid:88) ij ,µν,σ t µν ij c † i µσ c j νσ − µ (cid:88) i µσ n i µ.σ . (2)The operators c † i µσ ( c i µσ ) create (annihilate) an electronat site i in orbital state µ with spin σ , and µ is thechemical potential fixed such that the doping δ = (cid:104) n (cid:105) − . µ and ν denote the five ironorbitals d xy , d xz , d yz , d x − y , and d z − r . The secondterm in Eq.(1) 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 intraorbital (interorbital) interaction U ( U (cid:48) ), the Hund’s rule coupling J and the pair hoppingenergy J (cid:48) . We assume spin rotation invariance U (cid:48) = U − J and J (cid:48) = J . In this work we additionally take J = U/
4. The last term in Eq.(1) describes the impurity H imp = V imp (cid:88) µσ c † i ∗ µσ c i ∗ µσ , (4)which adds a local potential V imp at the impurity site i ∗ .In the present work, we neglect the orbital dependenceof the impurity potential for simplicity.After a mean-field decoupling of Eq.(3) we solve the fol-lowing eigenvalue problem (cid:80) j ν H µν ij σ u n j ν = E n u n i µ , where H µν ij σ = t µν ij + δ ij δ µν [ − µ + δ ii ∗ V imp + U (cid:104) n i µ ¯ σ (cid:105) (5)+ (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) )] , (cid:64) eV (cid:68) V i m p (cid:64) e V (cid:68) (cid:72) a (cid:76) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:230) Impurity (cid:224)
Bulk (cid:144) T N m (cid:144) Μ B (cid:72) b (cid:76) FIG. 1. (Color online) (a) Single impurity phase diagram dis-playing the impurity-induced magnetization at the impuritysite at T = 0 vs. U and V imp . Inset: spatial magnetizationof the induced impurity state at T = 0 in the paramagneticstate. (b) Magnetization vs. T (normalized to the bulk N´eeltemperature T N ) at the impurity site (red curve) and in thebulk (black) with U = 1 . T N = 0 . U ), and V imp = 0 . on N x × N y lattices with self-consistently obtained (cid:104) n i µσ (cid:105) = (cid:80) n | u n i µσ | f ( E nσ ), with f the Fermi function.We note that while several calculations of inhomogeneousstates have been performed using realistic, five-orbitalmodels[35–41], none have discussed the role of residualelectronic correlations and induced local SDW order.In the absence of disorder, the ground state of the un-doped system is a normal metal or a metallic ( π,
0) SDWdepending on the value of U . However, even withoutSDW order, impurities can pin magnetic order locally[25–27] similar to what has been discussed extensively forcuprate superconductors.[42–48] In Fig. 1(a) we show thesingle impurity phase diagram at low T for impurity-induced magnetic order vs. U and V imp (below the bulkSDW phase transition at U c (cid:39) . T = 0). Asseen, the potential generates local commensurate ( π, π )magnetic order in a regime of intermediate strength re-pulsive V imp for sufficiently large U .When U exceeds U c , similar impurity-induced ordertakes place at high T above T N as shown in Figs. 2(a,e).The local magnetic moment at the potential site is dis-played in Fig. 1(b) where one clearly sees the extendedmagnetic impurity phase above T N , and the enhancedimpurity moments at low T over the bulk SDW magneti-zation. Upon lowering T , the C symmetric high- T mag-netic ( π, π ) droplets shown in Fig. 2(a,e) have to competewith the surrounding bulk ( π,
0) SDW order. As shownin Fig. 2, the C structure of the SDW phase leads toa magnetic cigar-like impurity structure, a nematogenaligned along the AF a -axis, which still exhibits the in-ternal ( π, π ) magnetic structure of the high- T phase, butinherits an overall C symmetry from the SDW back-ground. While these low- T nematogens shown in Fig. 2are stable and fully converged, their final length dependson the path of convergence, i.e. the number of stepstaken in the cooling process, as seen explicitly by com-paring the right and left columns in Fig. 2. The origin (cid:72) a (cid:76) (cid:72) e (cid:76) (cid:45) (cid:72) b (cid:76) (cid:72) f (cid:76) (cid:45) (cid:72) c (cid:76) (cid:72) g (cid:76) (cid:45) (cid:72) d (cid:76) (cid:72) h (cid:76) (cid:45) FIG. 2. (Color online) Magnetization in real-space upon low-ering T with U = 1 . V imp = 0 . T /T N = 0 . , . , . , .
25. At each T , the systemis iterated until the bulk magnetization has converged. Thefinal low- T nematogen is stable and fully converged, but itslength depends on the ”cooling rate”, as seen by comparisonof the left and right columns distinguished by different cool-ing steps ∆ T /T N =0.21 (left), 0.105 (right). Note that onlyhalf of the steps are shown for the slow cooling case. of this ”cooling rate” dependence is simply a competi-tion between the impurity-induced ( π, π ) moments andthe SDW long-range order. It is natural to speculatethat this numerical cooling-rate dependence is relatedto the significant annealing dependence of the resistiv-ity anisotropy found recently.[12, 29] We note that theexistence of the nematogens is robust, and not depen-dent on the cooling rate, band-structure, or correlationstrength U (as long as U > U c ), or strength of the im-purity potential V imp , provided the latter two are able tocreate a high- T impurity bound state [Fig. 2(a,e)].Figure 3 summarizes the magnetic and electronic prop-erties of the unidirectional low- T nematogens. As seenfrom Fig. 3(a,d), the impurity-induced magnetization ne-matogen consists of an odd number of sites (even numberof lattice spacings) in length and width, and is always ori-ented along the AF a -axis. This is clearly because sucha structure can lower its energy with a long unfrustratedboundary with the ( π,
0) order. In addition, in Fig. 3(b,e)we see that the nematogen exhibits peaks at both endsresulting in an electronic dimer-like structure of the totalcharge density. This appears to be a general character-istic of these emergent impurity states, and also followsfrom the same energetic considerations, since the mag-netic energy can be lowered by creating a charge state as homogeneous as possible; thus the excess charge fromthe impurity site is moved to the ends of the nemato-gen. Previous STM work has discovered the existenceof electronic dimers in the LDOS,[19–21, 24] and recentpartially integrated LDOS within the SDW phase foundstrong evidence for electronic dimers near Co dopantsin CaFe As .[23] Within the present scenario, the exis-tence of density dimers naturally explains the presenceof LDOS dimers. However, as opposed to the robustexistence of the nematogens, the detailed structure ofthe LDOS near the impurity sites is more sensitive toparameters. In Fig. 3 we show cases where the onsiteimpurity potential is weak, V imp = 0 . ∼ a in both cases.Figure 4 displays in greater detail the LDOS proper-ties of the nematogens. The first point we wish to il-lustrate is that the size of the dimers deduced from thelow-energy integrated LDOS is not necessarily the sameas that in the (fully integrated) charge distribution. Thelow-energy integrated LDOS shown in Figs. 3(c,f) ex-hibits the dimer structure because of a peak in the LDOSat a particular site within the low-energy integration win-dow. Figures 4(a,d) show the LDOS at an energy cor-responding to the peak in the LDOS at this site, yield-ing essentially the same result as in Figs. 3(c,f). Forthe current parameters, the LDOS at the (distinct) siteswhere the charge distribution is maximal [Fig. 3 (b,e)]exhibits a peak at higher binding energy, as shown inFigs. 4(b,e). The structure of the low-energy LDOS canbe more clearly inferred from Figs. 4(c,f) which show theenergy dependence of the LDOS at several relevant sitesin the nematogen. From the LDOS on the site corre- (cid:72) a (cid:76) (cid:72) d (cid:76) (cid:45) (cid:72) b (cid:76) (cid:72) e (cid:76) (cid:72) c (cid:76) (cid:72) f (cid:76) FIG. 3. (Color online) 2D real-space maps of (a,d) the mag-netization, (b,e) the total electron charge density, and (c,f)the low-energy integrated LDOS for the same two low- T ne-matogens shown in Fig. 2(d,h). sponding to the low-energy dimer peak, for example, itis evident why the low-energy integrated LDOS exhibitsa peak at roughly 4 a from the impurity site in this case.For the LDOS results presented here we have useda potential strength of V imp = 0 . d z − r component is important for inducing the im-purity bound state and magnetic structure at both low- T [Fig. 1(a)] and high- T [Fig. 2(a,e)]. The LDOS curves inFigs. 4(a-d) are also mainly dominated by the d z − r or-bital. This is somewhat unusual since the Fermi surfaceand the bulk SDW magnetization is dominated by the t g orbitals. We believe that the importance of the d z − r orbital arises from the fact that the impurity potential al-lows for mixing with the d z − r -dominated band whichexhibits a sharp peak in the DOS about 0.25 eV below theFermi level for the bandstructure used in this work.[34]Local C symmetry breaking defect signatures, includ-ing vortex states[19], have been observed in many differ-ent Fe based superconductors. We have not exhibited amechanism for explaining all of them, but have begunthe process of realistic modeling of those which can beexpected to be driven by bulk magnetic order. The re-markable existence of large scale charge dimers inducedaround the impurities in our calculations is consistentwith STM measurements on Ca-122[23], but it is intrigu-ing that even larger electronic dimers have also been im-aged by STM in FeSe samples which are not thoughtto be magnetic[19, 24]. This may imply that the thinfilm FeSe samples used in STM in fact display at leastintermediate range magnetic order which is difficult todetect[50], or possibly that dimers can be created in astrongly fluctuating paramagnetic phase if the C sym-metry is already broken. The latter possibility is an im-portant question to address in the context of anisotropictransport measurements on Ba-122[5, 6, 29], where sig-nificant anisotropy enhancement is observed below thestructural phase transition T s but above T N . While the (cid:72) a (cid:76) (cid:72) d (cid:76) (cid:72) b (cid:76) (cid:72) e (cid:76) (cid:45) Ω (cid:64) eV (cid:68) L DO S (cid:64) a . u . (cid:68) (cid:72) c (cid:76) (cid:45) Ω (cid:64) eV (cid:68) (cid:72) f (cid:76) FIG. 4. (Color online) (a,b,d,e) 2D real-space maps of theLDOS at the characteristic low-energy dimer energy ω = − ω = − ω = − ω for different sites in the sys-tem: bulk (black), impurity site (green), at the low-energydimer peak (red), and at the charge density peak (blue). simulations we have presented here are consistent withthe sign and magnitude of the transport anisotropy inthe ordered phase T < T N , with the current tetragonalbandstructure we find that impurity-induced nematogensdo not form above T N . Whether they are stable in thestructural symmetry broken phase T N < T < T s will bethe subject of a future study.In summary, we have discovered the emergence of uni-directional electronic dimer states induced by point-likedefects in the SDW phase of iron pnictides. 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