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Fermi surface and effective masses in photoemission response of the (Ba1−xKx)Fe2As2 superconductor

The angle-resolved photoemission spectra of the superconductor (Ba1−xKx)Fe2As2 have been investigated accounting coherently for spin-orbit coupling, disorder and electron correlation effects in the valence bands combined with final state, matrix element and surface effects. Our results explain the previously obscured origins of all salient features of the ARPES response of this paradigm pnictide compound and reveal the origin of the Lifshitz transition. Comparison of calculated ARPES spectra with the underlying DMFT band structure shows an important impact of final state effects, which result for three-dimensional states in a deviation of the ARPES spectra from the true spectral function. In particular, the apparent effective mass enhancement seen in the ARPES response is not an entirely intrinsic property of the quasiparticle valence bands but may have a significant extrinsic contribution from the photoemission process and thus differ from its true value. Because this effect is more pronounced for low photoexcitation energies, soft-X-ray ARPES delivers more accurate values of the mass enhancement due to a sharp definition of the 3D electron momentum. To demonstrate this effect in addition to the theoretical study, we show here new state of the art soft-X-ray and polarisation dependent ARPES measurments.


LIFSHITZ TRANSITION
A Lifshitz transition is characterized as a topological change of the Fermi surface. [1] For the iron pnictide superconductors this type of transition is of crucial importance as it is believed to mark the onset of superconductivity. [2][3][4][5] The K-doped (Ba 1−x K x )Fe 2 As 2 is a famous example for such a Lifshitz transition around the X point, leading to the discussed propeller topologies seen in ARPES experiments. [5][6][7][8] As known from experimental data [6][7][8] and the experiments performed within this work these topological features are already clearly visible for the optimally doped (Ba 0.6 K 0.4 )Fe 2 As 2 . This is expected, following the argumentation that the Lifshitz transition suppresses the magnetic order due to a reduced nesting and thus induces superconductivity. [2,9] However, the observed Lifshitz transition on the basis of the LDA was so far discussed only for high doping concentrations x ≈ 0.9. [5,10] It is also known that it is difficult to prepare homogeneous samples of over-doped (Ba 1−x K x )Fe 2 As 2 [5] which might explain discrepancies between various experiments about the onset of the Lifshitz transition. [6,8,10,11] One remarkable and important paper from Khan and Johnson used the CPA to investigate the Lifshitz transition for (Ba 1−x K x )Fe 2 As 2 and they found a similar emergence of these propeller-like topologies for the heavily over-doped (Ba 0.1 K 0.9 )Fe 2 As 2 . [5] However, with our new findings this result is now fully understandable. Using only a LDA based approach the relevant bands around X are still around 0.1 eV below the Fermi level (E F ). One can use over-doping with K in order to decrease E F . The disadvantage of this approach is however that not only the relevant bands around X but the whole band structure is moved. Using a LDA+DMFT based approach one can see that correlation effects alter the electronic structure around X already for optimally doped (Ba 0.6 K 0.4 )Fe 2 As 2 so that the Lifshitz transition can emerge for lower doping concentration, in agreement with experiments. [6][7][8] In order to show how this topology is affected by the correlation strength we show the corresponding BSF and FS in interaction J = 0.9 eV for Fe. Best agreement with experiment can be found for U = 3.0 eV as used and discussed in the main paper but it is also obvious that bands responsible for the Lifshitz transition are directly controlled by the Coulomb interaction U . Thus, we show that the origin of the important Lifshitz transition in (Ba 1−x K x )Fe 2 As 2 can be fully explained by correlation effects.
Still, it is also interesting to note that recent work on the ARPES spectra of the electron doped Ba(Fe 1−x Co x ) 2 As 2 was successful using only a LDA approach. [12] The strength of correlation effects in the iron pnictides seem to vary with electron or hole doping.

EXTENDED FERMI SURFACE CUT
In correspondence to Fig. 3 of the main manuscript we present additionally extended Fermi surface cuts of (Ba 0.6 K 0.4 )Fe 2 As 2 for experimental data and theoretical calculations, respectively.

DERIVED EFFECTIVE MASS ENHANCEMENT
In order to derive the mass enhancements shown in Tab. I the corresponding bands shown in character whereas the band at X considered here is almost a 2D band. Furthermore, one should note that in the ARPES spectra the two outer bands around Γ are hardly distinguishable due to intensity loss connected to the ARPES response. As a technical detail, the calculations used an imaginary energy of 0.025 eV for the initial states and of 5.0 eV for the final states.
The resulting mass renormalization is shown in Fig. S4 (A -C) with the LDA bands from ARPES is affected by these ARPES response effects and thus it is not an intrinsic property of the quasiparticle valence band structure or spectral function. Why the ARPES response is affected by the k z broadening ∆k z is depicted schematically in Fig. S5. As discussed in the main paper, ∆k z can be reduced by choosing a higher photon energy, although the resulting effects can be never fully avoided for 3D materials like the iron pnictides. The ARPES signal is formed by averaging of the valence band dispersion within the ∆k z broadening interval. Near the extremes of the k z dispersion, this averaging results in asymmetry of the ARPES weight and shifting of the resulting spectral peak away from the extreme towards the band interior (for a more detailed picture including lifetime broadening of the valence states see Ref. [13]). Concerning its k || dependence of the ARPES spectra this shift causes an apparent reduction of the bandwidth and an increase of m * .