In-plane anisotropic magnetoresistance in antiferromagnetic Ba(Fe 1−x Co x ) 2 As 2 , (Ba 1−x K x )Fe 2 As 2 and Ba(Fe 1−x Ru x ) 2 As 2
IIn-plane anisotropic magnetoresistance in antiferromagneticBa(Fe − x Co x ) As , (Ba − x K x )Fe As and Ba(Fe − x Ru x ) As Gerald Derondeau, ∗ J´an Min´ar,
1, 2
Sebastian Wimmer, and Hubert Ebert Department Chemie, Physikalische Chemie, Universit¨at M¨unchen, Butenandtstr. 5-13, 81377 M¨unchen, Germany NewTechnologies-Research Center, University of West Bohemia, Pilsen, Czech Republic (Dated: October 10, 2018)Using the Kubo-Greenwood formalism the resistivity anisotropy for electron dopedBa(Fe − x Co x ) As , hole doped (Ba − x K x )Fe As and isovalently doped Ba(Fe − x Ru x ) As in theirantiferromagnetic state has been calculated in order to clarify the origin of this important phe-nomenon. The results show good agreement with experiment for all cases without considering im-purity states extending over several unit cells or temperature induced spin fluctuations. From thisit is concluded that the resistivity anisotropy at low temperatures is primarily caused by an in-planeanisotropic magnetoresistance. Accounting for the band dispersion with respect to k z is howevermandatory to explain the results, showing the importance of the three-dimensional character of theelectronic structure for the iron pnictides. Furthermore, it is shown that the counterintuitive sign ofthe resistivity anisotropy is no fundamental property but just a peculiarity of the anisotropic bandstructure. The iron pnictide superconductors are one of most im-portant examples for unconventioanl superconductivity,with the physical interest of the last few years mainly fo-cused on their strong in-plane anisotropy which emergesfor various physical properties [1–6]. This anisotropyis prominent in electrical transport [1–3, 7–11] but alsoshows up in angle-resolved photoemission spectroscopy(ARPES) [4, 12], neutron diffraction [5] or optical spec-troscopy [6, 13]. Of special interest is the in-plane re-sistivity anisotropy (∆ ρ = ρ a − ρ b ) for the BaFe As family as it seems to involve several physical phenom-ena. Notably, the higher resistivity was observed alongthe shorter b axis ( ρ b ) which corresponds to ferromag-netically coupled chains, while the lower resistivity wasfound along the a axis ( ρ a ) along which the spins cou-ple antiferromagnetically [1, 2]. A quite different be-havior is reported for different kinds of substitutionin BaFe As [9, 10, 14, 15]. Compared to electrondoped Ba(Fe x Co x ) As [1, 3] a smaller resistivity andanisotropy is seen in isovalent Ba(Fe x Ru x ) As [15]and an almost negligible resistivity and possibly invertedanisotropy is reported for hole doped (Ba x K x )Fe As [7, 10].Obviously, it turned out to be non-trivial to find a the-oretical explanation which can account for all these find-ings. There is recent theoretical work by several groupsbased on model Hamiltonian approaches [16–22] and lit-tle numerical work based on Monte Carlo simulations[23]. However, the origin of ∆ ρ is still under discus-sion. For example Kuo and Fisher [11] ascribe ∆ ρ exclu-sively to contributions of the intrinsic anisotropic bandstructure. Another approach is to assume extrinsic de-fect states which extend over several unit cells [3, 15].These can be explained theoretically [17–19] and mon-itored by scanning tunneling microscope (STM) exper-iments which see local anisotropic defect states with asize of approximately 22 ˚A [24–26]. What is missing so far is theoretical work that is not based on a modelHamiltonian but on a parameter-free application of den-sity functional theory (DFT) or a comparable scheme,showing whether and how the intrinsic band structurecan explain the observed resistivity anisotropy ∆ ρ .With this work we present a first principles DFT-based study on ∆ ρ in three exemplary iron pnic-tides: electron doped Ba(Fe x Co x ) As (Co-122), holedoped (Ba x K x )Fe As (K-122) and isovalent dopedBa(Fe x Ru x ) As (Ru-122). In order to reliably ac-count for the disorder induced by substitution we usethe Korringa-Kohn-Rostoker-Green function (KKR-GF)approach together with the coherent potential approx-imation (CPA) alloy theory, which proved already tobe a very powerful and accurate tool to study the elec-tronic structure of the substitutional iron pnictides [27–29]. Access to the longitudinal resistivity is given bythe Kubo-Greenwood equation [30, 31], which allows fora direct comparison with experiment. All calculationshave been performed self-consistently and fully relativis-tically within the four component Dirac formalism basedon the local density approximation (LDA) [32, 33]. Theapplied theory thus accounts for all effects of the intrinsicband structure and of disorder induced impurity scatter-ing in terms of the CPA. On the other hand, it does notaccount for anisotropic impurity states extending overseveral unit cells or for temperature induced spin fluc-tuations. More details are found in the SupplementalMaterial.[34]The magnetic properties of the investigated com-pounds are summarized in Fig. 1 together with the or-thorhombic unit cell in its experimental magnetic con-figuration. The obtained magnetic moment of undopedBaFe As is 0 . µ B and thus in good agreement withexperimental data (0 . µ B [35]). Using the CPA allowsfor a site resolved investigation of the magnetic moments,thus, the solid lines correspond to the average magnetic a r X i v : . [ c ond - m a t . s up r- c on ] A ug c ba BaFeAs m [ µ B ] Dopant concentration x m Fe (Co) m Fe (Ru) m avg (Co) m avg (Ru) m avg (K) FIG. 1. (Left) Magnetic moments for Co-122 (red), K-122(blue) and Ru-122 (black) depending on the dopant concen-tration x . The solid lines correspond to m avg , meaning thesubstitutional dependent average, while the dashed lines cor-respond to the pure Fe moment m Fe . (Right) Orthorhombicunit cell with the experimental magnetic configuration. moment while the dashed lines give the Fe contributiononly. The collapse of long range antiferromagnetic or-der at the critical concentration x crit for Co-122 andK-122 is in reasonable agreement with experiment, al-though it is slightly shifted to higher x values when com-pared with experiment (Co-122: x expcrit ≈ .
07 [36]; K-122: x expcrit ≈ .
25 [37]). Yet, for isovalent doped Ru-122 the av-erage magnetic moments decrease due to the increasingRu concentration which has a low magnetic moment inthe order of 0 . µ B (not shown in Fig. 1), however, theindividual Fe magnetic moments surprisingly increase. Inthe literature magnetic dilution was already discussed asthe main driving force for the magnetic breakdown inRu-122 [38, 39], yet in the investigated regime of x themagnetic order did not disappear, i.e. x crit should behigher than 0.4 (see Fig. 1).Accounting for the magnetic configuration the in-planelongitudinal resistivity was calculated, with the main re-sults shown in Fig. 2. First consider Co-122 in (A), wherethe resistivity of the antiferromagnetic (AFM) configu-ration has a dome-like variation with increasing dopantconcentration x until the collapse of the AFM order takesplace at x crit . Note that the resistivity ρ b AFM along theferromagnetic b axis (red) is always larger than the resis-tivity ρ a AFM along the antiferromagnetic a axis (blue)with a maximum of the anisotropy roughly at x crit /2.This behavior is in full agreement with experiment [1, 3].In Fig. 2 (A) we also give the experimental data of Ishida et al. [3] for ρ a and ρ b using specifically adapted scalesto the right and top of the figure. A direct comparisonof the theoretical and experimental data seems never-theless justified for two reasons. First of all, the self-consistently calculated collapse of the AFM order takesplace at a higher x crit compared to experiment, thus oneshould account for that by adjusting the experimental and theoretical x axis in a way that the breakdown ofmagnetic order coincides. Secondly, it is known that theresistivity decreases significantly for annealed crystals,indicating a strong contribution of crystal defects [3, 9].Thus it is expected to find the calculated resistivity to belower than the experimental one, consequently the axisto the right for the experimental resistivity is adjustedby a factor of 2. Most important are the order of mag-nitude and the dependency of ∆ ρ on the doping, whichare rather well reproduced by the presented calculations.For comparison we show also the calculated resistivity ofthe same orthorhombic crystal but for the nonmagnetic(NM) state, using green ( a axis) and purple ( b axis) linesin Fig. 2. Notably, the resistivity of the orthorhombicNM Co-122 is almost by a factor of 10 reduced comparedto the AFM case and it shows neither a dome-like behav-ior nor a significant anisotropy due to the lattice distor-tion. This unambiguously shows that the lattice has anegligible contribution to the anisotropic behavior com-pared to the magnetic structure, as was stressed beforein Refs [27, 29].Next, consider K-122 in Fig. 2 (B), where we show noexperimental data because ∆ ρ is reported to be almostimmeasurably small with even possible sign changes [7,10]. Indeed, the calculations find an in-plane resistivityalways below 8 µ Ω cm which is reduced by a factor of15 compared to Co-122. Replacing Fe with Co or Ruaffects the d -electronic states which are dominating at theFermi level ( E F ) for the iron pnictides. Thus, disorderintroduced for sp -elements like Ba or K hardly affectsthe resistivity of the compound. The calculations furthersupport the possibility of sign changes in ∆ ρ with x .Finally, consider Ru-122 in Fig. 2 (C), with recent ex-perimental data taken from Liu et al. [15] shown as forthe Co-122 case with axes at the top and on the rightadjusted the same way. The sign of ∆ ρ is the same asfor Co-122, however, the anisotropy does not disappearas the magnetic order in Ru-122 (see Fig. 1) also doesnot disappear for the investigated regime of substitution.In addition, one has to note that for comparable dopantconcentrations the resistivity in Ru-122 is reduced com-pared to Co-122, as it was also observed in experiment[15]. Thus, again the trends and the order of magnitudefor the resistivity are in reasonable agreement with recentexperimental data.In summary, in all three representative iron pnictidesit was possible to reproduce the qualitative behaviorof ∆ ρ in the antiferromagnetic phase based on a LDAapproach without considering spatially extended impu-rity states or spin fluctuations. Thus, the resistivityanisotropy of the iron pnictides at low temperatures canbe well understood in terms of an in-plane anisotropicmagnetoresistance (AMR) [40]. This leads to the ques-tion how the anisotropic band structure can influencethis AMR. The idea of linking ∆ ρ to the band structureas seen e.g. by ARPES was already proposed by Yi et (A) ρ [ µ Ω c m ] ρ ( e x p ) [ µ Ω c m ] x of Ba(Fe x Co x ) As x (exp) ρ a (exp) ρ b (exp) ρ a AFM ρ b AFM ρ a NM ρ b NM (B) ρ [ µ Ω c m ] x of (Ba x K x )Fe As ρ a AFM ρ b AFM ρ a NM ρ b NM (C) ρ [ µ Ω c m ] ρ ( e x p ) [ µ Ω c m ] x of Ba(Fe x Ru x ) As x (exp) ρ a (exp) ρ b (exp) ρ a AFM ρ b AFM ρ a NM ρ b NM FIG. 2. In-plane resistivity calculated for (A) Co-122, for (B) K-122 and for (C) Ru-122 up to x crit as a function of theconcentration x . The blue and red curves correspond to the antiferromagnetic (AFM) state, while the green and purple linescorrespond to the same orthorhombic lattice but for the nonmagnetic (NM) case. The pluses and crosses are experimental data[3, 15] with the corresponding concentration x (exp) and resistivity ρ (exp) axes given at the top and right of the figure. al. [4], however, corresponding explanations were hardlysuccessful so far [15]. The main reason for this apparentincompatibility is due to the fact that ARPES is a sur-face sensitive method. For the iron pnictides, however,it is absolutely mandatory to account for the dispersionof bands with k z as was also recently stressed in the con-text of the effective mass enhancement seen in ARPES[41]. Accounting for the k z dispersion of the anisotropicbands will thus allow for more meaningful results and anat least qualitative understanding of the observed trends.Discussing the presented resistivity data, one shouldfirst note that the resistivity anisotropy is also observedfor the parent compound BaFe As , although it is re-ported to be small after annealing [2, 9]. Still, the Lif-shitz transition [42], i.e. the topological change of theFermi surface when going from the nonmagnetic to theantiferromagnetic state, should influence the resistivitybehavior. Indeed, the formerly almost isotropic bands inthe NM state undergo a significant band splitting result-ing in a considerable anisotropy after the Lifshitz transi-tion to the AFM state [27, 29, 42]. When accounting forthe k z dispersion, one can qualitatively understand thehigher resistivity along b in terms of a hybridization ofhole and electron pockets seen also in ARPES [42]. Thisleads to the formation of a band gap and of minima fromelectron pockets at the Fermi level only along the b axis.This Lifshitz transition of BaFe As and the implicationsfor ∆ ρ are discussed in more detail in the SupplementalMaterial.[34]A qualitative discussion of ∆ ρ for the investigatedcompounds will be exemplarily done for Co-122 with x = 0 . ρ for Co sub-stitution and for Ru-122 with x = 0 .
10 as an exampleof isovalent doping. The band structures are shown fordifferent k z values in Fig. 3 with the paths ΓY and ΓXcorresponding to directions in k -space parallel to the b and a axes, respectively. First consider the Co-122 case in panels (A - E), where a strong dependence of the bandson k z is obvious. Of most interest are the W- or S-shapedbands (W-band: two minima along both directions ΓYand ΓX; S-band: only one minimum along ΓY) which area result of the previously discussed hybridization withsignificant band broadening due to the disorder. Thehighest ∆ ρ is observed for x = 0 .
075 in Co-122 becausefor exactly this concentration most of these W- and S-bands have their minimum precisely at E F . Approachingthese extrema, the slope of the bands is decreasing, lead-ing to a decrease of the quasiparticle velocities and thusto an increase of the resistivity. Additionally, the appar-ently strong disorder-induced broadening of the W- andS-bands leads to a decreased quasiparticle lifetime whichagain increases the resistivity. It would be difficult to ex-plain ∆ ρ based on their contribution at E F in Co-122 forsome values of k z , e.g. for Fig. 3 (A) k z = 0 .
0, (B) 0.25and (D) 0.75. However, for (C) k z = 0 . E F is clearly anisotropic with a higherimpact along the ΓY path, leading to a higher resistivity ρ b compared to ρ a .Analogously, ∆ ρ can be explained for the Ru-122 com-pound with x = 0 .
1, but now the strong anisotropiccontribution can be seen for Fig. 3 (F) k z = 0 . E F . The Ru-122system is an interesting prototype system for isovalentdoping because the Ru substitution induces disorder but E F is only marginally changed. The crucial impact ofthe anisotropic band structure can be easily shown byartificially changing the Fermi energy by ∆ E F and recal-culating the resistivity ρ a ( b ) for Ba(Fe . Ru . ) As . Theresults are presented in Fig. 4 where some exemplary val-ues are highlighted by the cyan, black, green and purplelines. The corresponding energies are indicated in Fig. 3(F - J) for comparison. It becomes immediately obvious,that the magnitude and the sign of ∆ ρ = ρ a − ρ b depends Ba(Fe − x Co x ) As for x = 0 . k z = 0 . E n e r g y [ e V ] Y Γ X (B) k z = 0 . k || ΓY k || ΓX (C) k z = 0 . k || ΓY k || ΓX (D) k z = 0 . k || ΓY k || ΓX (E) k z = 1 . (cid:48) Z X (cid:48) maxmin
Ba(Fe − x Ru x ) As for x = 0 . k z = 0 . E n e r g y [ e V ] Y Γ X (G) k z = 0 . k || ΓY k || ΓX (H) k z = 0 . k || ΓY k || ΓX (I) k z = 0 . k || ΓY k || ΓX (J) k z = 1 . (cid:48) Z X (cid:48) maxmin
FIG. 3. Bloch spectral functions for different values of k z in units of πc , with ΓY and ΓX corresponding to paths parallel to the b and a axes, respectively. The panels (A - E) and (F - J) show results for Co-122 with x = 0 .
075 and Ru-122 with x = 0 . strongly on the chosen energy. For ∆ E F = − .
125 eV(cyan line) one arrives at the minimum of a band at X,whose impact is strong enough to change the sign of ∆ ρ ,meaning a higher resistivity ρ a compared to ρ b . Thisis comparable to K-122, where the hole doping induces ρ [ µ Ω c m ] ∆ E F [eV] for Ba(Fe Ru ) As ρ a AFM ρ b AFM
FIG. 4. Change in ρ a ( b ) for Ru-122 with x = 0 . E F of the Fermi energy. The cyan,black, green and purple cuts correspond to the cuts in theband structures in Fig. 3. this energy shift and explains the possible sign changein ∆ ρ . Increasing the energy to ∆ E F = 0 .
10 eV (greenline) significantly increases ∆ ρ that gets comparable tothe Co-122 case for x = 0 .
075 in Fig. 3 (A - E). Fur-ther increasing the shift to ∆ E F = 0 .
20 eV (purple line)moves again away from this peculiar band situation, de-creasing ∆ ρ as it is the case for overdoped Co-122 with x > . ρ in the low tem-perature phase of the iron pnictides can be explained interms of an in-plane AMR. Additional contributions ofextended impurity states obviously cannot be ruled out.Furthermore, we do not disagree with work consideringthe impact of higher temperatures in the nematic phase[17–19, 21], rather we provide new insights from first-principles calculations on real materials. These disprovedin particular recent suggestions that the anisotropic bandstructure has no impact on ∆ ρ [15] and highlighted theimportance of the dispersion with k z .In conclusion, this work is the first to present first prin-ciples transport calculations of the resistivity anisotropyin the low temperature antiferromagnetic phase of theiron pnictide superconductors. We show for three exem-plary systems resistivity values in good agreement withexperiment. It turned out that it is sufficient to dis-cuss ∆ ρ in the antiferromagnetic phase in terms of ananisotropic magnetoresistance. This AMR can be tunedby either disorder scattering from impurities or by anenergy shift due to respective electron or hole doping.Most important, it was mandatory to account for the k z dispersion in order to understand the results based onthe anisotropic band structure, showing the crucial im-pact of the three-dimensional character of the electronicstructure for the iron pnictides which future studies haveto account for. Furthermore, resistivity calculations forvarying energy reveal that the sign of ∆ ρ is no funda-mental property of the pnictides but rather a peculiarityof the anisotropic band structure.We thank Ilja Turek for valuable discussions. We ac-knowledge financial support within the research groupFOR 1346, within priority program SPP 1538 and fromCENTEM PLUS (L01402). ∗ [email protected][1] J.-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon,Z. 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1, 2
Sebastian Wimmer, and Hubert Ebert Department Chemie, Physikalische Chemie, Universit¨at M¨unchen,Butenandtstr. 5-13, 81377 M¨unchen, Germany NewTechnologies-Research Center, University of West Bohemia, Pilsen, Czech Republic (Dated: October 10, 2018) a r X i v : . [ c ond - m a t . s up r- c on ] A ug . COMPUTATIONAL DETAILS All calculations have been performed self-consistently and fully relativistically within the fourcomponent Dirac formalism using the coherent potential approximation (CPA) alloy theory asimplemented in the Munich SPR-KKR program package [1, 2]. The treatment of disorder via theCPA was already shown to be a powerful and accurate tool, fully comparable to recent supercellcalculations that have to take the average over many configurations [3, 4]. Doing so, would howevergive no direct access to the in-plane resistivity anisotropy as it is possible within the presentedapproach. Furthermore, the CPA allows for an investigation of isovalent doped Ru-122 which is notaccessible by e.g. a virtual crystal approximation (VCA). Furthermore, the Dirac formalism impliesby construction a full inclusion of effects induced by spin-orbit coupling, which was recently shownto be important for the iron pnictides [5]. The crystal structure is based on the orthorhombic 4-Feunit cell in its experimental magnetic configuration [6], implying antiferromagnetic chains alongthe a and c axes and ferromagnetic chains along the b axis. The magnetic moments are oriented in-plane along the a axis. The lattice parameters where chosen according to experimental X-ray dataand using a linear interpolation to account for the influence of substitution based on available data[7–10]. More details on the procedure can be found in a previous publication [3]. For all electronicstructure calculations the local density approximation (LDA) exchange-correlation potential wasapplied, using the parameterization of Vosko, Wilk and Nusair [11]. The longitudinal resistivitywas calculated using the Kubo-Greenwood equation within KKR-CPA [12, 13], for the symmetricpart of the conductivity tensor (1): σ µµ = ~ πV Tr h ˆ j µ Im G + ˆ j µ Im G + i c , (1)with G + being the retarded Green’s functions, V the cell volume and ˆ j µ the µ component of theelectric current operator. In the relativistic formulation the latter reads ˆ j µ = −| e | cα µ , with theelementary charge e , the speed of light c and α µ being one of the 4 × α matrices. Thebrackets h i c indicate a configurational average with respect to an alloy in terms of the CPA.
2. VERTEX CORRECTIONS
The CPA vertex corrections [12, 13] are included in all resistivity calculations, ensuring properaveraging of the product of two Green’s functions in equation (1). Still, the overall impact ofthe vertex corrections is comparably small (in the order of at most 7 %) which indicates thatincoherent scattering processes are less important. This is in agreement with having mainly d character for the electrons at the Fermi level. Furthermore, this means that it is a reasonable2pproximation to look at the effective single particle Bloch spectral function in order to investigatethis resistivity anisotropy [14]. However, it is interesting to note that the vertex corrections itselfare also highly anisotropic as can be seen in Fig. S1. Here, ρ a and ρ b of the AFM state are shown forall investigated compounds, including vertex corrections (VC) and also without vertex corrections(nVC). It is obvious that the impact of vertex corrections on ρ a is almost zero in all cases whilethe influence on ρ b is especially for Co-122 and Ru-122 much more pronounced. This finding isconsistent with a stronger impact of disorder on the AFM coupling along b . (A) ρ [ µ Ω c m ] x of Ba(Fe − x Co x ) As ρ a VC ρ b VC ρ a nVC ρ b nVC (B) ρ [ µ Ω c m ] x of (Ba − x K x )Fe As ρ a VC ρ b VC ρ a nVC ρ b nVC (C) ρ [ µ Ω c m ] x of Ba(Fe − x Ru x ) As ρ a VC ρ b VC ρ a nVC ρ b nVC FIG. S1. In-plane resistivity of the AFM state for (A) Co-122, for (B) K-122 and for (C) Ru-122 dependingon the substitution level x . The results are split into ρ a and ρ b either including vertex corrections (VC)with solid lines or without any vertex corrections (nVC) shown in dashed lines.
3. LIFSHITZ TRANSITION OF THE PARENT COMPOUND
Lifshitz transitions are characterized as a topological change of the Fermi surface.[15] Thereare several kinds of Lifshitz transitions discussed for the iron pnictides and these are of significantimportance for the underlying physics and superconductivity.[16–19] The topological change ofthe Fermi surface for the BaFe As mother compound when going from the nonmagnetic to theantiferromagnetic state can be also understood in terms of a Lifshitz transition, introducing stronganisotropy into the AFM electronic structure.[3, 4] The corresponding electronic structure forBaFe As is shown in Fig. S2 for the (A - E) AFM state and the (F - J) nonmagnetic case. Forcomparison, all calculations are presented with respect to the same orthorhombic 4-Fe unit cellwhich implies a back-folding of bands into a smaller Brillouin zone, compared to the 2-Fe cell. Seealso Ref. [20] for further information on the magnetic Brillouin zone in BaFe As . The upper rowsshow the band structure for different values of k z , while the lower rows depict the respective Fermisurface cuts. As in the main paper, the paths ΓY and ΓX correspond to paths in k -space whichare parallel the b and a axes, respectively. 3 aFe As - antiferromagnetic AFM(A) k z = 0 . E n e r g y [ e V ] Y Γ X (B) k z = 0 . k || ΓY k || ΓX (C) k z = 0 . k || ΓY k || ΓX (D) k z = 0 . k || ΓY k || ΓX (E) k z = 1 . (cid:48) Z X (cid:48) maxmin
YΓ X Y (cid:48)
Z X (cid:48)
BaFe As - nonmagnetic NM(F) k z = 0 . E n e r g y [ e V ] Y Γ X (G) k z = 0 . k || ΓY k || ΓX (H) k z = 0 . k || ΓY k || ΓX (I) k z = 0 . k || ΓY k || ΓX (J) k z = 1 . (cid:48) Z X (cid:48) maxmin
YΓ X Y (cid:48)
Z X (cid:48)
FIG. S2. Bloch spectral functions of orthorhombic BaFe As for different values of k z in πc , with ΓY andΓX corresponding to paths parallel to the b and a axes, respectively. Shown for (A - E) the AFM state andfor (F - J) the nonmagnetic case. The upper panel presents the band structure and the lower panel showsthe corresponding cut of the Fermi surface. In order to allow direct comparison, all images are shown forthe 4-Fe unit cell with a back-folding of bands onto Γ. Going from a 2-Fe unit cell to the 4-Fe cell implies that the hole pockets and the electron pocketsof the 2-Fe cell are back folded onto each other in one Γ point as can be seen for the nonmagneticband structure of a 4-Fe unit cell in Fig. S2 (F - J). An additionally emerging antiferromagneticorder leads to a hybridization of these hole- and electron pockets as shown in Fig. S2 (A - E), which4as discussed in detail by e.g. Liu et al. [16] in terms of a Lifshitz transition. It is possible to discussthe emergence of a non-zero ∆ ρ already at this point, although earlier attempts were less fruitful.[21,22] The reason for this apparent incompatibility is that the iron pnictides are clearly 3D materialswhose bands show typically a strong k z dispersion, which is often not sufficiently accounted for.[23]This becomes obvious in Fig. S2 where especially in the AFM case a strong electronic anisotropyemerges which does significantly depend on the value of k z . For the nonmagnetic case in Fig. S2(F - J) the corresponding electronic structure is, when considering the k z dispersion, more or lessisotropic. The situation is completely different for the AFM state in Fig. S2 (A - E). With asimilar argumentation as used in the main paper one can qualitatively expect a ∆ ρ with a higher ρ b along the path ΓY for e.g. k z = 0 . k z = 0 .
0. For k z = 0 .
75 a minimum of anelectron pocket at E F leads to a minimum of the quasiparticle velocity, increasing the resistivity.Using an even higher resolution in the k z dispersion (45 different values were calculated) showsclearly that these two distinct features emerge only along ΓY and never along ΓX. This can alreadyqualitatively explain the occurrence of ∆ ρ in the undoped BaFe As . This effect increases withsubstitution, as disorder-induced anisotropic band broadening leads to a simultaneous reduction ofthe quasiparticle lifetime as discussed in the main paper. Electron or hole doping do further movethe Fermi level and can either increase or decrease the anisotropy, as was shown in Fig. 4. ∗ [email protected][1] H. Ebert, D. K¨odderitzsch, and J. Min´ar, Rep. Prog. Phys. , 096501 (2011).[2] H. Ebert et al., The Munich SPR-KKR package , version 6.3, http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR, 2012.[3] G. Derondeau, S. Polesya, S. Mankovsky, H. Ebert, and J. Min´ar, Phys. Rev. B , 184509 (2014).[4] G. Derondeau, J. Braun, H. Ebert, and J. Min´ar, Phys. Rev. B , 144513 (2016).[5] S. V. Borisenko, D. V. Evtushinsky, Z.-H. Liu, I. Morozov, R. Kappenberger, S. Wurmehl, B. B¨uchner,A. N. Yaresko, T. K. Kim, M. Hoesch, T. Wolf, and N. D. Zhigadlo, Nature Physics , 311 (2016).[6] Q. Huang, Y. Qiu, W. Bao, M. A. Green, J. W. Lynn, Y. C. Gasparovic, T. Wu, G. Wu, and X. H.Chen, Phys. Rev. Lett. , 257003 (2008).[7] M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, and R. P¨ottgen, Phys. Rev. B ,020503 (2008).[8] M. Rotter, M. Pangerl, M. Tegel, and D. Johrendt, Angew. Chem. Int. Ed. , 7949 (2008).[9] A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D. J. Singh, and D. Mandrus, Phys. Rev. Lett. ,117004 (2008).[10] A. Thaler, N. Ni, A. Kracher, J. Q. Yan, S. L. Bud’ko, and P. C. Canfield, Phys. Rev. B , 014534(2010).
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