Linkage between scattering rates and superconductivity in doped ferropnictides

We report an angle-resolved photoemission study of a series of doped ferropnictides. We focus on the energy dependent scattering rate Gamma(E) of the inner hole pocket, which is the hot spot in these compounds, as a function of the 3d count. We obtain a linear-in-energy non-Fermi-liquid like scattering rate, independent of the dopant concentration. The main result is that slope Gamma(E)/E, which can be related to a coupling constant, scales with the superconducting transition temperature. This supports the spin fluctuation model for superconductivity for these materials. In the optimally hole-doped compound, the slope of the scattering rate of the inner hole pocket is about three times bigger than the Planckian limit Gamma(E)/E~1. This signals very incoherent charge carriers in the normal state which transform at low temperatures to a coherent superconducting state.

Angle-resolved photoemission spectroscopy (ARPES) measures the energy (E), momentum (k), and temperature (T ) dependent spectral function A(E, k, T ) [1] multiplied by a transition matrix element and the Fermi function and convoluted with the energy and momentum resolution [2]. The coherent part of the spectral function is given by Γ is the scattering rate of the photo hole in the valence band, and * k is the renormalized dispersion. * k = k − Σ(E, k, T ), and Γ(E, k, T ) = −2Z(E, k, T ) Σ(E, k, T ). Here k is the bare particle energy, Σ is the complex self-energy and Z is the renormalization function [1]. For small scattering rates, i.e., Γ E, together with a linear dispersion * k , the spectral function is a Lorentzian for energy distribution curves at constant momentum (EDC) as well as for momentum distribution curves at constant energy (MDC). In this case the spectral function describes the Fermi liquid behavior of the quasi-particles and Γ is related to their inverse life time. The spectral function at constant momentum has a maximum at * k and a full width at half maximum of Γ. Very often the description of the spectral function is extended to higher scattering rates, i.e., Γ is comparable to the binding energy E [3,4].
In the standard procedure for the evaluation of the scattering rates from the ARPES data, MDCs are fitted by a Lorentzian and the FWHM momentum width is multiplied by the velocity d * /dk yielding Γ(E, T ) [2,4]. Very often non-Lorentzian MDCs or EDCs are realized in the ARPES data. For example a large Γ(E, T ) leads to long tails in the energy distribution curves. Non-linear dispersions lead to asymmetric shapes in the MDCs. Furthermore, large scattering rates lead to contributions from the unoccupied part of the band causing an apparent back-dispersion for k < k F in hole pockets [5] similar to dispersions in the superconducting state (see Fig. 1 (a) of the main paper). All this may lead to incorrect scattering rates when using the standard procedure.
To derive more exact data for the scattering rates, we developed a new elaborate evaluation method. We fit the two-dimensional E − k intensity distribution at once, using as parameters Γ(E, T ) at each energy point, as well as parameters describing the dispersion of * k . Usually a parabolic dispersion gives reasonably fits. Higher orders in k did not improve the fits presented in the main paper. Other fit parameters describe a weakly momentum and energy dependent background which is added to the spectral function. The sum is multiplied with the Fermi function and the product is then convoluted with the energy and momentum resolution. We mention that very close to the Fermi level, the derived Γ values are very sensitive to the exact position of the Fermi level.
From the analysis of the ARPES data described above, we obtain the energy dependent total scattering rate Γ(E, T ) which is the sum of the elastic scattering rate Γ el (E) and the inelastic scattering rate Γ in (E, T ). To obtain the inelastic scattering rates we have to subtract the elastic scattering rates. Γ el (E) is usually assumed to be constant [4] which is reasonably for a small energy range. However, this holds only for a linear dispersion when where w 0 is the momentum width at the Fermi level related to the constant inverse mean free path between scattering sites inducing elastic scattering. To obtain the inelastic lifetime broadening for a parabolic dispersion, one has to subtract the energy dependent Γ el (E) from Γ(E, T ). w 0 can be calculated from the data near E F because for small temperatures there Γ in (0, 0) is zero and thus Γ el (0) = Γ(0, 0) = v(0)w 0 . Thus we derive for the energy dependence of the elastic scattering Γ el (E) = v(E)w 0 = Γ(0, 0)(v(E)/v(0)). The velocities are taken from the measured dispersion. We report an angle-resolved photoemission study of a series of doped ferropnictides. We focus on the energy dependent scattering rate Γ(E) of the inner hole pocket, which is the hot spot in these compounds, as a function of the 3d count. We obtain a linear-in-energy non-Fermi-liquid like scattering rate, independent of the dopant concentration. The main result is that slope Γ(E)/E, which can be related to a coupling constant, scales with the superconducting transition temperature. This supports the spin fluctuation model for superconductivity for these materials. In the optimally hole-doped compound, the slope of the scattering rate of the inner hole pocket is about three times bigger than the Planckian limit Γ(E)/E ≈ 1. This signals very incoherent charge carriers in the normal state which transform at low temperatures to a coherent superconducting state.
Introduction. There is an ongoing debate about the mechanism for unconventional superconductivity in ironbased superconductors (FeSCs) [1]. The most popular model is connected with spin fluctuation scattering processes between hole pockets in the center and electron pockets at the border of the Brillouin zone (BZ), together with a sign change of the superconducting order parameter: the s ± superconductivity [2,3]. While the cuprates have just one band near the Fermi surface, the situation is, however, rather complicated for FeSCs. In the latter there are three hole and two electron pockets, and several of them have sections with different orbital character. Therefore, it is rather difficult to receive a microscopic picture for the mechanism of high-T c superconductivity in these materials.
Our previous ARPES studies, mainly on electrondoped compounds [4][5][6] and on LiFeAs [7], indicate that the strongest scattering rate Γ(E, T ≈ 0), which is equal to the inverse lifetime broadening, occur between sections of the inner hole pocket and the inner electron pocket, both having the same orbital character (Fe 3d xz,yz ). At these hot spots a linear-in-energy scattering rate was detected which, for the cuprate superconductors, was described by the conjecture of a marginal Fermi liquid [8]. However, Brouet et al. [9] received conflicting experimental results on scattering rates in FeSCs. They obtained a quadratic-in-energy dependence of the scattering rates Γ(E, T ≈ 0) corresponding to a Fermi liquid behavior. Moreover, combined density functional plus dynamical mean field theory (DFT+DMFT) calculations derive by construction a quadratic energy dependence of Γ(E, T ) [5,7,10,11].
In our previous studies we have just conjectured that strong scattering processes mediate superconductivity.
There was no direct comparison of the scattering rates with superconducting properties such as the superconducting transition temperature T c . In the present contribution we compare the doping dependent strength of the scattering rates of the inner hole pocket with T c . We obtain for both a rather similar doping dependence and we therefore conclude that the scattering rates of the inner hole pockets are related to superconductivity. From this comparison we also derive that superconductivity is determined by a combination of correlation effects and nesting conditions. Furthermore, using our new elaborated evaluation technique for the analysis of Γ(E) from the ARPES data of optimally hole-doped FeSCs close to the Fermi level, we conclude that the charge carriers at the hot spots have not Fermi liquid character, but can be well described by the marginal Fermi liquid model. Finally, from our experimental scattering rates we conclude that the hot spot charge carriers are completely incoherent, i.e., Γ(E πk B T, T )/E ≈ 3 well above the Planckian limit, where Γ(E πk B T, T )/E = 1 [12].
Experimental. Single crystals were grown using the self-flux technique [13][14][15]. ARPES measurements were conducted at the 1 2 and 1 3 -ARPES end stations attached to the beamline UE112 PGM at BESSY. All data presented in this contribution were taken in the normal state at temperatures between 5 and 50 K. The achieved energy and angle resolutions were between 4 and 15 meV and 0.2 • , respectively. Polarized photons with energies hν = 20−130 eV were employed to reach different k z values in the BZ and spectral weight with a specific orbital character [16,17]. Inner potentials between of 12 and 15 eV were used to calculate the k z values from the photon energy. Results. We focus mainly on data of hole-doped compounds. In Fig. 1 we exemplary show ARPES data of optimally hole-doped K 0.4 Ba 0.6 Fe 2 As 2 (T c = 38 K), recorded at a temperature T = 50 K along the Γ-M direction (direction between centers of the hole and the electron pocket) using vertically polarized photons with an energy of 62 eV. Under these experimental conditions we record the spectral weight of the inner hole pocket at the Γ point having predominantly Fe 3d yz character [16]. In Fig. 1(a) we show an energy momentum distribution map.
Usually Γ(E) is analyzed using the standard evaluation method, i.e., multiplying the momentum distribution (MDC) width with the velocity [18,19]. In the present contribution we analyze Γ(E, T ) using an elaborate fitting procedure. The two-dimensional intensity distribution is described by the spectral function multiplied by the Fermi function [20]. The product is then convoluted with the finite energy and momentum resolution. The resulting function is fitted to the data using an all-at-once approach. Only in this way reliable results at low energies can be derived. Our fitting method is described in more detail in the supplementary material [21].
The fit result and the corresponding residual for the spectral weight are presented in Fig. 1(b) and (c), respectively. A waterfall plot of the data together with the fit result in the energy range 0.0 to 0.12 eV is shown in Fig. 1(d). The derived energy dependence of the scattering rate Γ(E, T = 50 K) is depicted in Fig. 1(e). The data can be well fitted by a marginal Fermi liquid behavior for the inelastic part Γ in (E, T ) = λ[E 2 + (πk B T ) 2 ] 1/2 [8] plus a slightly energy dependent elastic scattering rate Γ el (E) = Γ el (0)v(E)/v(0) [21] due to impurities or defects at the surface [20] (see the dashed green line in Fig. 1(e)). Here v(E) = d * k /dk is the velocity. From the fit we derive a dimensionless coupling constant λ = 3.0, Γ(0, T = 50 K) = 0.084 eV and Γ el (0) = 0.044 eV. The difference Γ(0, T = 50 K) − Γ el (0) = 0.040 eV agrees well with the contribution Γ in (0, T = 50 K) corresponding to a finite temperature T = 50 K equal to λπk B T = 0.042 eV. Γ el (0) = 0.044 eV together with the Fermi velocity v F = 0.9 eVÅ corresponds to a mean free path of ≈ 60 Å, a value which is quite common in ARPES experiments, also for not strongly correlated materials [18,23]. This means that the finite width at E F is not necessarily connected with many-body properties. We also fitted the data with a Fermi liquid behavior Γ in (E, T ) = λ[E 2 + (πk B T ) 2 ] (see blue line in Fig. 1(e)). The comparison with the experimental data clearly shows that the charge carriers for this hot spot do not exhibit Fermi liquid behavior.
• BaFe2As2. The filled markers correspond to antiferromagnetic compounds. The ARPES data are compared with the range of the superconducting phase (SC, blue), the spin density wave phase (SDW, magenta), and the range where both phases overlap (yellow) [14].
taining both contributions from a finite energy and a finite temperature (T = 50 K), with the same coupling constant λ. This gives us some confidence that the models for the fit we used, i.e., the marginal Fermi liquid model and the assumption for the elastic scattering, is reasonable.
In Fig. 2 we present the results for the inelastic scattering rates by plotting the slopes Γ in (E πk B T, T )/E ≈ λ for various hole doping concentrations. We complement these results with similar ARPES data of electron doped systems and related compounds, partially presented in previous publications [4-7, 16, 24-26]. A weak minimum occurs for the undoped BaFe 2 As 2 sample. A large difference of the scattering rate between optimally hole and electron doped compounds was already described in a previous publication [5]. We see a clear maximum near optimal hole doping. There is a remarkable decrease of the scattering rates in the hole over-doped compounds, although an enhancement of correlation effects is expected because there the 3d count is close to the halffilled 3d shell. The ARPES results are compared with the range of the superconducting phase (blue), the antiferromagnetic phase (magenta), and the range were both phases overlap (yellow) [14].
Discussion. Regions with a linear increase of the scattering rates as a function of energy have been already detected in other FeSCs and related compounds [4][5][6][7]22] They were discussed in terms of momentum and not orbital dependent strong correlation effects. We emphasize that for the hot spots in FeSCs, a marginal Fermi liquid behavior is observed independent from the doping con-centration in the range of the 3d count from 4 to 6. This is different from the nodal point in cuprates where for the scattering rate a continuous superposition of a linear and a quadratic energy dependence [27] or a T n dependence with n changing from one to two was discussed [28], when moving from optimal to overdoped compounds.
It is interesting to note that the largest slope of the scattering rate occurs near the optimally hole-doped compound. We conclude that for this dopant concentration, the charge carriers are completely incoherent because the scattering rate is about three times bigger than the energy and therefore well above the Planckian limit where Γ in (E, T ≈ 0) = E [12]. Moreover, it is also remarkable that the slope, which to our knowledge is the largest slope ever detected by ARPES, is about three times bigger than that in optimally doped cuprate superconductors along the diagonal direction [28][29][30][31]. It is certainly a challenge to understand how the completely incoherent hot spots in the normal state transform into a coherent superconducting state.
Using Fermi's Golden rule and a local approximation, the scattering rate is related to the on-site interaction U eff , which is determined by on-site Coulomb interaction U and the Hund's exchange coupling J H , and the susceptibility which determines the relaxation of the photoelectron to lower energies by an electron-hole excitations [32,33]. According to Avigo et al. [6], the top of the inner hole pocket and the bottom of the inner electron pocket are separated in these compounds by ≈ 0.1 eV. In a rigid band approximation, the Fermi level moves ≈ 0.5 eV per dopant electron/hole. Assuming that intraband transitions would be only possible when hole and electron pockets cross the Fermi level, they could only occur in a range of a 3d count of ± 0.2 around the undoped sample. This is in line with the maximum of T c for the hole doped system but too large for the electron doped systems.
Superconductivity at higher dopant concentration could be explained by the fact that the equation for the superconducting transition temperature yields also solutions for excitations away from the Fermi level in a range of the coupling energies. If these excitations are spin fluctuations, which have an energy range between 0.01 to 0.2 eV [34,35], we can understand that for hole-doped compounds there is also a finite T c in the over-doped compounds. On the other hand, the decrease of T c can be understood by a decrease of the spin susceptibility, also detected by inelastic neutron scattering [35], because in the overdoped compounds, the electron or hole pocket has moved far above or below the Fermi level, respectively. This reduction of the scattering rate is clearly detected for large hole doping. On the other hand, thermal properties have derived a strong enhancement of the effective mass when going to higher hole doping [14]. This indicates that the related flat bands close to the Fermi level, detected in KFe 2 As 2 by ARPES [36] and by quantum-oscillation experiments [37], are causing the largest band mass enhancement, but not superconductivity.
In the same way the reduced scattering rate at high electron doping together with a reduced T c can be understood by a larger distance of the top of the hole pocket relative to the Fermi level [6,38]. The stronger reduction of T c compared to the hole doped compounds was explained for Ba(Fe 1−x Co x ) 2 As 2 by a large pair-breaking by Co scatterers [14].
Also for a 3d count of four, i.e., BaCr 2 As 2 [22,39], reduced scattering rates (see Fig. 2) or correlation effects are detected. A reduction of correlation effects at higher electron doping was also derived from ARPES results on the mass enhancement in BaCo 2 As 2 [40] and BaNi 2 As 2 [41]. As explained in Ref. [5] the larger scattering rate for the hole doped systems, when compared to the electron systems, can be explained by an enhancement of U eff due to the larger Hund exchange coupling in the proximity to the half-filled 3d shell [42,43]. The observation of a finite scattering rate at high dopant concentrations can be explained by a scattering of the inner hole pocket with other bands close to the Fermi level.
Recently, there were several combined density functional/dynamical mean-field theory (DFT + DMFT) calculations on the scattering rates in FeSCs [5,7] and calculation also taking non-local effects into account [10,11]. Inherent to DFT+DMFT calculations, the calculated scattering rates always showed a Fermi liquid behavior, except one calculation which presented a linear in energy dependence down to the lowest energies [10]. For LiFeAs at 0.1 eV, scattering rates of about Γ = 2Z Σ = 0.015 were calculated. Σ is the imaginary part of the selfenergy and Z is the renormalization factor. This is much smaller than the experimental value Γ in (E = 0.1 eV, T = 50 K) = 0.4 eV for K 0.4 Ba 0.6 Fe 2 As 2 [see Fig. 1(e)]. Even when we take into account that in the latter compound correlation effects are three times bigger, there remains a difference between experiment and theory of a factor of about seven.
In summary, our results on the 3d count dependence of the scattering rates clearly show that correlation effects alone are not the most important tuning parameter in these systems. Rather a combination of 3d count dependent correlation effects and susceptibility can explain the scaling of the scattering rate and the superconducting transition temperature presented in Fig. 2.