Resistive anisotropy due to spin-fluctuation scattering in the nematic phase of iron pnictides
RResistive anisotropy due to spin-fluctuation scatteringin the nematic phase of iron pnictides
Maxim Breitkreiz, ∗ P. M. R. Brydon, and Carsten Timm † Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, USA 20742 (Dated: August 29, 2014)The large in-plane anisotropy of the resistivity is a hallmark of the nematic state of the iron pnic-tides. Solving the Boltzmann transport equation, we show that the prominent doping dependenceas well as the large values of the anisotropy can be well explained by momentum-dependent spin-fluctuation scattering without assuming anisotropic impurity states. Due to the forward-scatteringcorrections, the hot spots contribute to the resistive anisotropy even in the case of strong spin fluc-tuations, which makes large values of the anisotropy possible. The ellipticity of the electron pocketsplays an important role in explaining the dominance of positive values of the anisotropy, i.e., largerresistivity in the direction with weaker spin fluctuations, throughout the doping range.
PACS numbers: 72.10.Di, 72.15.Lh, 74.70.Xa
Introduction.
Currently, one of the most intensivelydiscussed topics in the field of high- T c superconductivityis the origin of the nematic phase of the iron pnictides[2, 3]. The nematic phase transition occurs at temper-atures T s above or coinciding with the magnetic order-ing temperature T N , at which a stripe antiferromagneticstate with ordering vector Q X = ( π, (defining the x -direction in this work) is established. The nematic phasefound for T N < T < T s is characterized by a brokenrotational symmetry between the x and y directions inthe absence of magnetic order. Although one of its mostobvious manifestations is the orthorhombic distortion ofthe lattice, it is generally considered that the nematicstate arises from electronic correlations [3]. However, theprecise mechanism is still under debate [4–8].Another key experimental signature of the nematicphase is the pronounced difference between the resistivi-ties along the x and y directions, ∆ ρ ≡ ( ρ y − ρ x ) /ρ x [3, 9–12]. Understanding the origin of the resistive anisotropyshould offer crucial insights into the origin of the nematic-ity. Two scenarios are debated: (i) the scattering offanisotropic impurity states [9, 13–16] and (ii) the scat-tering off fluctuating collective excitations with spectrumreflecting the underlying nematicity [11, 17].The existing description of the resistive anisotropy dueto spin fluctuations [17], i.e., within scenario (ii), is re-stricted to the limit of weak spin-fluctuation scatteringcompared to isotropic impurity scattering, although theformer is likely stronger than the latter, except at verylow temperatures when the spin fluctuations are frozenout [18–22]. Naturally, this limit is only compatible withsmall values of ∆ ρ , since the dominant impurity partleads to isotropic resistivity. Though in disagreementwith the huge positive anisotropy up to ∆ ρ ≈ . ob-served in experiments on electron-doped samples [3, 9], ∗ [email protected] † [email protected] the theory correctly predicts negative ∆ ρ for hole-dopedsamples [11].Within scenario (i), the much larger ∆ ρ in electron-doped Ba(Fe − x Co x ) As [3] compared to hole-dopedBa − x K x Fe As [11, 12] is explained as a consequence ofthe stronger scattering off Co-dopands placed within theiron plane [9, 13, 16]. The observed anisotropic impuritystates are all elongated in the x -direction, hence givinga larger scattering cross-section in the y -direction [13].The negative ∆ ρ measured for hole-doped samples thenarises due to details of the band structure [16]. The de-pendence of ∆ ρ on the degree of disorder is controversial:some experiments show, in agreement with scenario (i),a reduction of ∆ ρ upon sample annealing, which is sup-posed to lower the degree of disorder [9], while othersreport a much weaker disorder dependence [10].In this work, we consider scenario (ii) with spin-fluctuation scattering of arbitrary strength. For spin-fluctuation and isotropic impurity scattering of compara-ble strength, we reproduce both the small negative ∆ ρ forhole-doped samples and the large positive ∆ ρ in electron-doped samples. We also show that the reduction of ∆ ρ in electron-doped samples upon annealing is consistentwith the spin-fluctuation scenario. In a nutshell, our re-sults follow from the role of the spin-fluctuation scatter-ing strength in controlling the size of the Fermi-surfaceareas that contribute to the resistive anisotropy. Model and Method.
We describe the band structureby an effective two-dimensional model [5, 11, 17, 22, 23]with a nearly circular hole Fermi pocket at the centerof the Brillouin zone and two elliptical electron pockets eX and eY displaced by Q X = ( π, and Q Y = (0 , π ) ,respectively, where length is measured in units of theiron-iron separation. We use the same dispersions as inRef. [23] and fix the ellipticity of the electron pockets bychoosing ξ e = 2 . The Fermi pockets are sketched in Fig.1. The sizes of the pockets depend on the doping level,which is controlled by the electron filling n [22]. Thevalidity of the minimal model for the case of 122 pnictides a r X i v : . [ c ond - m a t . s t r- e l ] S e p Figure 1. (Color online) Hole ( h ) and electron ( eX and eY )Fermi pockets of the two-band model. In the nematic phase,scattering between h and eX is stronger than between h and eY , as indicated by the arrows marked W sf , giving rise tothe resistive anisotropy. As discussed in the main text, theelectron pockets can be divided into regions that contributepositively (red) or negatively (blue) to the anisotropy, de-pending on the direction of the Fermi velocity. States on eachFermi surface are parametrized by the angle θ to the x -axiswith respect to the center of the pocket. has been discussed in the supplementary information forRef. [11].To focus on the impact of the spin-fluctuation scat-tering, in the following we neglect the distortion of theFermi pockets due to the splitting of the iron d yz and d xz orbital levels [1, 2]. In the Supplemental Material [26] weshow that this splitting gives rise to an additional re-sistive anisotropy. By itself, this shows poor agreementwith experiment, however, and the effect of nematicityin the spin-fluctuation scattering is the dominant mech-anism over a large parameter range.We assume transport to be dominated by scatteringoff spin fluctuations and isotropic impurities. The spin-fluctuation scattering amplitude is determined by theimaginary part of the spin susceptibility. We use a phe-nomenological model for the susceptibility in the nematicphase that has been employed for calculations in theimpurity-dominated regime [17, 27, 28]. Following Ref.[22], we introduce a total elastic scattering rate betweenstates | s, θ (cid:105) on the Fermi pockets, parametrized by thepocket index s and the angle θ (cf. Fig. 1), W s (cid:48) θ (cid:48) sθ ≡ (1 − δ bb (cid:48) ) W sf α × (cid:90) dε (cid:48) ε (cid:48) coth ε (cid:48) k B T − tanh ε (cid:48) k B T ε (cid:48) + ω q + W imp , (1)where ω q = Γ (cid:0) ξ − ∓ φ + q x (1 ± η ) + q y (1 ∓ η ) (cid:1) with q = k ( s, θ, ε F ) − k ( s (cid:48) , θ (cid:48) , ε (cid:48) ) , where the wave vectors aremeasured from the center of the corresponding Fermipocket. Further, b ( b (cid:48) ) is the band giving rise to the Fermipocket s ( s (cid:48) ), φ is the nematic order parameter, ξ is thecorrelation length in the isotropic phase, Γ is the Lan-dau damping parameter, and η is the in-plane anisotropy of the correlation length. The upper (lower) sign corre-sponds to the scattering between the hole pocket and theelectron pocket eX ( eY ). W sf and W imp represent theoverall strengths of the scattering off spin fluctuationsand impurities, respectively, and the numerical factor α = 10 ensures that at the highest considered temper-ature (see below) W sf /W imp is of the same order as theinverse ratio of average lifetimes due to scattering off spinfluctuations and impurities only, W sf /W imp ∼ τ imp /τ sf .The susceptibility entering Eq. (1) is peaked at thenesting vectors Q X and Q Y for all dopings, consistentwith the observed stability of commensurate antiferro-magnetic order against doping. The resulting scatteringrate is therefore larger for scattering wave vectors close to Q X or Q Y . The strongest scattering is found at the “hotspots,” i.e., the points on the Fermi pockets connectedby the nesting vectors. The position of the hot spots de-pends on the doping level [22, 23]. In the nematic phasea finite order parameter φ > breaks the C symmetry.This enhances the peak at Q X in the susceptibility, lead-ing to stronger scattering between the hole pocket h andthe electron pocket eX than between the pockets h and eY , as indicated in Fig. 1.We focus on the dependence of the resistive anisotropyon doping (electron filling n ) and on the relative strengthsof spin-fluctuation and impurity scattering (controlled by W sf /W imp ). The explicit temperature T in Eq. (1) con-trols the energy available for spin excitations and thusadditionally affects the strength of spin-fluctuation scat-tering. In the relevant limit k B T (cid:28) ω q , this leads to thefamiliar T dependence. Since the nematic phase appearsin a narrow temperature interval above the Néel tempera-ture T N ( n ) , we choose the temperature T ( n ) = T N ( n ) = T (1 − [( n − . / . ) with T = max[ T N ( n )] = 137 K .This mimics the situation in 122 pnictides, where themagnetic order is suppressed upon doping the parentcompound, here taken to correspond to n = 2 . [23].Our results are qualitatively insensitive to the specificform of T ( n ) . Since the temperature tracks T N ( n ) , itis reasonable to keep the parameters ξ , φ , and Γ fixed;we have checked that the qualitative behavior does notdepend on their precise values.We employ the non-equilibrium Green-function for-malism [29] in the Boltzmann approximation, where thelinear-response distribution function at the Fermi energyis determined by the vector mean free paths Λ sθ [30, 31]of the states | s, θ (cid:105) . The vector mean free path obeys thekinetic equation [22] Λ sθ = τ sθ v sθ + τ sθ (cid:88) s (cid:48) (cid:90) dθ (cid:48) π N s (cid:48) θ (cid:48) W s (cid:48) θ (cid:48) sθ Λ s (cid:48) θ (cid:48) , (2)where v sθ ≡ (cid:126) − ∇ k ε b k | s,θ is the velocity, N sθ = | d k sθ /dθ | /π (cid:126) | v sθ | is the density of states, and τ sθ = (cid:18) π (cid:88) s (cid:48) (cid:90) dθ (cid:48) N s (cid:48) θ (cid:48) W s (cid:48) θ (cid:48) sθ (cid:19) − , (3)is the lifetime of the state | s, θ (cid:105) . The first term on ∆ ρ n (a) (b)(c) (d)-2-10123 0 0.1 0.2 0.3 0.4 0.5 ∆ ρ e θ θ/π n = 2 . e-dopingh-doping -2-10123 0 0.1 0.2 0.3 0.4 0.5 θ/π n = 2 . -2-10123 0 0.1 0.2 0.3 0.4 0.5 θ/π n = 2 . W sf /W imp (a)1510 1.9 2 2.1 2.2 ρ ( T ( n )) / ρ ( ) n n W s f / W i m p -0.100.10.20.30.4 ∆ ρ Figure 2. (Color online) (a) Resistive anisotropy as a function of doping (parametrized by n ) and the relative strengths ofspin-fluctuation and impurity scattering. (b) Resistive anisotropy as a function of doping for W sf /W imp = 0 . , , and . (c)Angle-resolved contributions of the electron pockets to the resistive anisotropy as defined in Eq. (7). While for W sf /W imp = 0 . only regions close to the hot spots (indicated by arrows) contribute, for increasing W sf /W imp the contributing regions grow.(d) Ratio of averaged resistivities at the temperatures T ( n ) considered in (a)–(c) and at T = 0 K . We choose the parameters η = 0 . , Γ = 350 meV, ξ − = 0 . , and φ = 0 . . the right-hand side of Eq. (2) represents the relaxation-time approximation, while the second incorporates theforward-scattering corrections.The resistivity ρ i in the direction i = x, y is determinedby the vector mean free path, ρ i = (cid:16) e (cid:88) s (cid:90) dθ π N sθ v isθ Λ isθ (cid:17) − ≡ (cid:16) (cid:88) s (cid:90) dθ π σ isθ (cid:17) − , (4)where σ isθ is the contribution of the state | s, θ (cid:105) to the totalconductivity σ i = (cid:80) s (cid:82) dθ π σ isθ . It is useful to resolvethe resistive anisotropy in terms of band and angularcontributions, ∆ ρ = (cid:90) dθ π (cid:0) ∆ ρ hθ + ∆ ρ eθ (cid:1) , (5)where the contributions from hole and electron pocketsread, respectively, ∆ ρ hθ ≡ σ y (cid:0) σ xh,θ − σ yh,θ + σ xh,θ + π/ − σ yh,θ + π/ (cid:1) , (6) ∆ ρ eθ ≡ σ y (cid:0) σ xeY,θ − σ yeY,θ + σ xeX,θ + π/ − σ yeX,θ + π/ (cid:1) . (7)In Eq. (6), we consider the contributions from the hole-pocket states | h, θ (cid:105) and | h, θ + π/ (cid:105) together, since only the joint contribution vanishes in the normal, C -sym-metric phase. For the same reason, the states | eY, θ (cid:105) and | eX, θ + π/ (cid:105) are considered together in Eq. (7). Ac-cording to the definition of ∆ ρ eθ , the contributions fromstates close to the minor axis of the elliptical electronpockets are found at θ ≈ , while the contributions fromstates close to the major axis are found at θ ≈ π/ . Results.
Figure 2 summarizes the results for the resis-tive anisotropy obtained by solving Eq. (2) numerically[22]. In Fig. 2(a) the resistive anisotropy is plotted asa function of doping and the ratio W sf /W imp , while inFig. 2(b) the doping dependence is illustrated for threecharacteristic values of W sf /W imp . The contributions ∆ ρ eθ from the electron pockets are found to dominatethe anisotropy, for which reason only these contributionsare shown in Fig. 2(c). As evident from Fig. 2(c) andillustrated in Fig. 1, the electron pockets can be dividedinto positively and negatively contributing parts, withthe crossover located roughly where the Fermi velocitypoints in the diagonal direction; the parts close to the mi-nor axis of the electron pockets contribute with positivesign, while the parts close to the major axis contributewith negative sign. This is because the conductivity ofthe electron pocket eY is larger than that of eX due tothe stronger scattering for the latter.The total resistive anisotropy in Figs. 2(a) and (b)shows a strong doping dependence, which changes quali-tatively with W sf /W imp . The angle-resolved plots in Fig.2(c) show that for increasing W sf /W imp the contributingregions of the electron pockets expand. This is schemati-cally illustrated in Fig. 3. For small W sf /W imp , the resis-tive anisotropy is dominated by regions close to the hotspots, whereas the “cold” regions, where spin-fluctuationscattering is weaker, give small contributions. Since theelectron pockets have negatively and positively contribut-ing parts, the position of the hot spots determines thesign of the resistive anisotropy. The negative (positive)extremum is found for the filling n ≈ . ( n ≈ . ),for which the hot spots lie on the major (minor) axis ofthe electron pockets. The difference between the posi-tive and negative extrema is due to different velocitiesand densities of states at the major and minor axes.In the impurity-dominated limit, W sf /W imp (cid:28) ,the anisotropy is very small as impurity scattering isisotropic. With increasing W sf /W imp , the contributingregions of the electron pockets expand and the extremaof ∆ ρ grow, until the active region starts to include partscontributing with the opposite sign. Upon further ex-pansion, the positive and negative contributions beginto partially compensate each other. Since the negativelycontributing regions are smaller, the negative extremumof ∆ ρ is suppressed at a smaller ratio W sf /W imp thanthe positive extremum. At W sf /W imp ≈ this results ina strong doping asymmetry with small negative valueson the hole-doped side and large positive values on theelectron-doped side.We emphasize that the result that the hot spots con-tribute to ∆ ρ even for dominant spin-fluctuation scat-tering, as sketched in Fig. 3, is not obvious. Since inthis limit the scattering at the hot spots is much strongerthan in the cold regions, one would naively expect the hotspots to be short circuited by the cold regions [32], i.e., tobe irrelevant for the transport, in which case ∆ ρ wouldbe significantly smaller [11, 17]. However, as we haveshown for the C -symmetric state of the pnicitides [22],the short-circuiting is compensated by enhanced forward-scattering corrections.To compare the results to measurements, we have toidentify the relevant range of W sf /W imp . In Fig. 2(d),we plot the calculated ratio of the averaged resistivity ρ ( T ) ≡ ( ρ x + ρ y ) / at T = T ( n ) and at T = 0 K , wherethe spin excitations are frozen out and the resistivity isdue to impurity scattering alone, which we assume to betemperature independent. Ignoring for the moment thatthe system is antiferromagnetic at T = 0 K , we observethat for W sf /W imp = 1 and W sf /W imp = 10 the resistivityratios are comparable to those measured for as-grown andannealed samples, respectively [9]. The reduction of thedensity of states in the antiferromagnetic phase shouldincrease the T = 0 K resistivity, however, and so ourargument likely underestimates W sf /W imp .For W sf /W imp = 1 , Figs. 2(a) and (b) show a large pos-itive peak with ∆ ρ ≈ . in electron-doped samples and asmall negative peak with ∆ ρ ≈ − . in hole-doped sam- Figure 3. (Color online) Increasing strength of spin-fluctuation scattering extends the contributing regions of theelectron pockets. Two characteristic filling levels are consid-ered, n ≈ . and n ≈ . , with hot spots at the major andthe minor axis of the electron pockets, respectively. ples. This is in good agreement with experimental obser-vations [3, 9, 11]. The results also show that in electron-doped samples an increase of W sf /W imp beyond about leads to a reduction of the peak value of ∆ ρ . A reductionof ∆ ρ upon annealing was indeed observed in electron-doped Ba(Fe − x Co x ) As [9], where this effect has beentaken as strong evidence that the resistive anisotropymainly stems from scattering at anisotropic impuritystates. Our results show, however, that such a reduc-tion is also consistent with anisotropic spin-fluctuationscattering. For the hole-doped samples, we predict an increase in ∆ ρ with annealing if W sf /W imp (cid:38) , see Figs.2(a) and (b), which to our knowledge has not been mea-sured so far.In the Supplemental Material [26], we show thatanisotropy due to orbital splitting adds nearly additivelyto ∆ ρ , indicating the robustness of the results againstband details. This is in line with the fact that the mainfeatures of ∆ ρ are explained by a mechanism that doesnot rely on the details of the model. Summary.
We have studied the resistive anisotropyin the nematic state of iron pnictides. We have con-sidered a two-band model and assumed scattering to bedominated by spin fluctuations and isotropic impurities.The inclusion of forward-scattering corrections is crucialfor the correct description [22]. The obtained resistiveanisotropy ∆ ρ shows good agreement with experimentalresults for annealed and as-grown samples. In particular,we have shown that the twin puzzles of the doping asym-metry of ∆ ρ and the reduction of ∆ ρ upon annealing canbe explained within the spin-fluctuation scenario. Thequalitative behavior is governed by the contributing re-gions on the elliptical electron pockets, in particular theirgrowth with increasing spin-fluctuation strength. Impor-tantly, the hot spots contribute to ∆ ρ even for strongspin-fluctuation scattering, contrary to what was thoughtpreviously. Since spin fluctuations are particularly strongat the hot spots, this naturally leads to large anisotropies. Acknowledgments.
Financial support by the DeutscheForschungsgemeinschaft through Research TrainingGroup GRK 1621 is gratefully acknowledged. The au-thors thank B. M. Andersen, E. Babaev, M. N. Gasti- asoro, P. J. Hirschfeld, D. Inosov, and J. Schmiedt foruseful discussions. [1] R. M. Fernandes, A. V. Chubukov, and J. Schmalian, Na-ture Phys. , 97 (2014).[2] J. C. Davis and P. J. Hirschfeld, Nature Phys. , 184(2014).[3] J.-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon,Z. Islam, Y. Yamamoto, and I. R. Fisher, Science , 824(2010).[4] W. Lv and P. Phillips, Phys. Rev. B , 174512 (2011).[5] R. M. Fernandes, A. V. Chubukov, J. Knolle, I. Eremin,and J. Schmalian, Phys. Rev. B , 024534 (2012).[6] S. Onari and H. Kontani, Phys. Rev. Lett. , 137001(2012).[7] V. Stanev and P. B. Littlewood, Phys. Rev. B , 161122(2013).[8] S. Liang, A. Moreo, and E. Dagotto, Phys. Rev. Lett. ,047004 (2013).[9] S. Ishida, M. Nakajima, T. Liang, K. Kihou, C. H. Lee,A. Iyo, H. Eisaki, T. Kakeshita, Y. Tomioka, T. Ito, andS. Uchida, Phys. Rev. Lett. , 207001 (2013).[10] H.-H. Kuo and I. R. Fisher, Phys. Rev. Lett. , 227001(2014).[11] E. C. Blomberg, M. A. Tanatar, R. M. Fernandes,I. I. Mazin, B. Shen, H.-H. Wen, M. D. Johannes, J.Schmalian, and R. Prozorov, Nature Commun. , 1914(2013).[12] J. J. Ying, X. F. Wang, T. Wu, Z. J. Xiang, R. H. Liu,Y. J. Yan, A. F. Wang, M. Zhang, G. J. Ye, P. Cheng,J. P. Hu, and X. H. Chen, Phys. Rev. Lett. , 067001(2011).[13] M. P. Allan, T-M. Chuang, F. Massee, Yang Xie, Ni Ni,S. L. Bud’ko, G. S. Boebinger, Q. Wang, D. S. Dessau, P.C. Canfield, M. S. Golden, and J. C. Davis, Nature Phys. , 220 (2013).[14] Y. Inoue, Y. Yamakawa, and H. Kontani, Phys. Rev. B , 224506 (2012).[15] M. N. Gastiasoro, P. J. Hirschfeld, and B. M. Andersen,Phys. Rev. B , 100502(R) (2014).[16] M. N. Gastiasoro, I. Paul, Y. Wang, P. J. Hirschfeld, andB. M. Andersen, arXiv:1407.0117.[17] R. M. Fernandes, E. Abrahams, and J. Schmalian, Phys.Rev. Lett. , 217002 (2011).[18] L. Fang, H. Luo, P. Cheng, Z. Wang, Y. Jia, G. Mu, B.Shen, I. I. Mazin, L. Shan, C. Ren, and H.-H. Wen, Phys.Rev. B , 140508(R) (2009).[19] S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S.Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya,K. Hirata, T. Terashima, and Y. Matsuda, Phys. Rev. B , 184519, (2010).[20] L. Fanfarillo, E. Cappelluti, C. Castellani, and L. Ben-fatto, Phys. Rev. Lett. , 096402 (2012).[21] M. Breitkreiz, P. M. R. Brydon, and C. Timm, Phys.Rev. B , 085103 (2013).[22] M. Breitkreiz, P. M. R. Brydon, and C. Timm, Phys.Rev. B , 245106 (2014).[23] P. M. R. Brydon, J. Schmiedt, and C. Timm, Phys. Rev.B , 214510 (2011).[24] M. Yi, D. Lu, J.-H. Chu, J. G. Analytis, A. P. Sorini, A. F. Kemper, B. Moritz, S.-K. Mo, M. G. Moore, M.Hashimoto, W.-S. Lee, Z. Hussain, T. P. Devereaux, I. R.Fisher, and Z.-X. Shen, Proc. Natl. Acad. Sciences ,6878 (2011).[25] K. Nakayama, Y. Miyata, G. N. Phan, T. Sato, Y.Tanabe, T. Urata, K. Tanigaki, and T. Takahashi,arXiv:1404.0857.[26] See Supplemental Material at http://XXX for the dis-cussion of the effect of splitting of d yz and d xz orbital levelson the resistive anisotropy within the present model.[27] D. S. Inosov, J. T. Park, P. Bourges, D. L. Sun, Y.Sidis, A. Schneidewind, K. Hradil, D. Haug, C. T. Lin,B. Keimer, and V. Hinkov, Nature Phys. , 178 (2010).[28] S. O. Diallo, D. K. Pratt, R. M. Fernandes, W. Tian, J.L. Zarestky, M. Lumsden, T. G. Perring, C. L. Broholm,N. Ni, S. L. Bud’ko, P. C. Canfield, H.-F. Li, D. Vaknin,A. Kreyssig, A. I. Goldman, and R. J. McQueeney, Phys.Rev. B , 214407 (2010).[29] J. Rammer and H. Smith, Rev. Mod. Phys. , 323(1986).[30] G. D. Mahan, Many-Particle Physics , 3rd edition,Plenum, New York (2000).[31] E. H. Sondheimer, Proc. R. Soc. London, Ser. A , 100(1962); P. L. Taylor, Proc. R. Soc. London, Ser. A ,200 (1963).[32] R. Hlubina and T. M. Rice, Phys. Rev. B , 9253 (1995). SUPPLEMENTAL MATERIAL
In the main text we calculate the resistive anisotropy due to scattering off nematic spin fluctuations for a C -symmetric band structure. The degeneracy of the iron d yz and d xz orbitals is lifted in the nematic phase [1, 2],however, lowering the symmetry of the band structure to C . In this supplemental material we consider the effect ofthis orthorhombic distortion in the band structure on the resistive anisotropy.The increased (decreased) iron-iron separation along the x ( y ) axis in the orthorhombic state decreases (increases)the onsite energy of the iron d xz ( d yz ) orbital. To model the resulting changes in our band structure, we follow Ref.[3] and decrease the size of the eX pocket, increase the size of the eY pocket, and elongate the hole pocket alongthe x direction, see Fig. S1(a). This distortion is motivated by the orbital composition of the Fermi pockets [4]. Weimplement the distortion by introducing a parameter δ > in the dispersion relations for the two bands h and e : ε h k = ε h − µ + 2 t h (cid:2) (1 − δ ) cos k x + (1 + δ ) cos k y (cid:3) , (1) ε e k = ε e − µ + t e, cos k x cos k y − t e, ξ (cid:2) (1 + δ ) cos k x + (1 − δ ) cos k y (cid:3) , (2)where length is measured in units of the iron-iron separation. We choose a relatively large orthorhombic distortionof the band structure with δ = 0 . , for which the relative difference of the electron-pocket areas is about . Allother band parameters are as in the main text.For a nonzero orthorhombic distortion, the model displays a resistive anisotropy ∆ ρ even when the nematic param-eter in the susceptibility vanishes, φ = 0 . We present results for this case in Fig. S1. The calculated ∆ ρ is in ratherpoor agreement with experimental findings: neither the minimum near optimal doping nor the significant extent ofnegative values is observed. Note that while the magnitude of ∆ ρ scales with δ , its qualitative behavior does notchange significantly.Figure S2 shows the result for the combined effect of orbital splitting ( δ = 0 . ) and the nematicity in the spinsusceptibility ( φ = 0 . ). The effect of the two sources of anisotropy appear to be additive and the characteristicsignatures of the nematic spin fluctuations are still conspicuous. In particular, the large positive anisotropy in electron-doped samples and the much smaller anisotropy in hole-doped samples for W sf /W imp (cid:46) is still present, as is thereduction of the anisotropy in electron-doped samples for W sf /W imp (cid:38) . On the other hand, for W sf /W imp (cid:29) ,the weak contribution of the spin fluctuations in the case of electron doping means that the resistive anisotropy iscontrolled by the distortion of the band structure and becomes negative, as in Fig. S1.In summary, the effect of orbital splitting alone cannot account for the observed resistive anisotropy. Betteragreement might be achieved for a more sophisticated model of the band structure, although this would be at theexpense of fine tuning. In contrast, including the nematicity in the spin fluctuation spectrum gives much betteragreement with experimental results, is robust against the distortion of the band structure, and dominates thecontribution of the distorted band structure to the resistive anisotropy over a large parameter range. [1] M. Yi, D. Lu, J.-H. Chu, J. G. Analytis, A. P. Sorini, A. F. Kemper, B. Moritz, S.-K. Mo, M. G. Moore, M. Hashimoto,W.-S. Lee, Z. Hussain, T. P. Devereaux, I. R. Fisher, and Z.-X. Shen, Proc. Natl. Acad. Sciences , 6878 (2011).[2] K. Nakayama, Y. Miyata, G. N. Phan, T. Sato, Y. Tanabe, T. Urata, K. Tanigaki, and T. Takahashi, arXiv:1404.0857.[3] R. M. Fernandes, A. V. Chubukov, and J. Schmalian, Nature Phys. , 97 (2014).[4] S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, New J. Phys. , 025016 (2009). e-dopingh-doping 00.10.20.30.40.51.9 2 2.1 2.2 ∆ ρ n (a)(a) (b) (c)(b) (c) W sf /W imp e-dopingh-doping -0.100.10.21.9 2 2.1 2.2 ∆ ρ n n W s f / W i m p -0.100.10.20.30.40.5 ∆ ρ n W s f / W i m p -0.2-0.100.10.20.3 ∆ ρ Figure S1. (Color online) (a) Sketch of the Fermi pocket distortion and the scattering strength between the hole and the electronpockets. (b), (c) Resistive anisotropy in the presence of orbital splitting ( δ = 0 . ) and a paramagnetic spin susceptibility( φ = 0 ). e-dopingh-doping 00.10.20.30.40.51.9 2 2.1 2.2 ∆ ρ n (a)(a) (b) (c)(b) (c) W sf /W imp e-dopingh-doping -0.100.10.21.9 2 2.1 2.2 ∆ ρ n n W s f / W i m p -0.100.10.20.30.40.5 ∆ ρ n W s f / W i m p -0.2-0.100.10.20.3 ∆ ρ Figure S2. (Color online) (a) Sketch of the Fermi pocket distortion and the scattering strength between the hole and theelectron pockets. (b), (c) Resistive anisotropy in the presence of orbital splitting ( δ = 0 . ) and nematic spin susceptibility( φ = 0 .017