Selective mass enhancement close to the quantum critical point in BaFe2(As1−xPx)2

A quantum critical point (QCP) is currently being conjectured for the BaFe2(As1−xPx)2 system at the critical value x c ≈ 0.3. In the proximity of a QCP, all thermodynamic and transport properties are expected to scale with a single characteristic energy, given by the quantum fluctuations. Such a universal behavior has not, however, been found in the superconducting upper critical field H c2. Here we report H c2 data for epitaxial thin films extracted from the electrical resistance measured in very high magnetic fields up to 67 Tesla. Using a multi-band analysis we find that H c2 is sensitive to the QCP, implying a significant charge carrier effective mass enhancement at the doping-induced QCP that is essentially band-dependent. Our results point to two qualitatively different groups of electrons in BaFe2(As1−xPx)2. The first one (possibly associated to hot spots or whole Fermi sheets) has a strong mass enhancement at the QCP, and the second one is insensitive to the QCP. The observed duality could also be present in many other quantum critical systems.

In the present Supplementary material we show the doping dependence of the c-axis lattice parameter for BaFe 2 (As 1−x P x ) 2 films grown on MgO and LaAlO 3 (LAO) substrates, the temperature dependences of the electrical resistance in magnetic fields, the criteria used for the determination of the SDW transition temperature T N and the superconducting critical temperature T c in static magnetic fields as well as the upper critical field H c2 in pulsed magnetic fields. We estimated the fluctuation effect on the superconducting transition width and that on the evaluated values of the slope of the upper critical field. Finally, we provide tables with a list of the fitting parameters described in the main text.
Composition of the films. The P doping level of the thin films given in the main text were calculated using the relation between the c-axis lattice parameter and the P doping obtained in previous studies (Fig. S1). [S1] 12.4 12.5 12.6 12.7 12.8 12. FIGURE S1. (a) Relation between the c-axis lattice parameter and the P doping level x of BaFe 2 (As 1−x P x ) 2 films grown on MgO and LaAlO 3 (LAO) substrates. The data are taken from previous studies. [S1] Evaluation of T N and T c . The spin density wave transition temperatures T N of the BaFe 2 (As 1−x P x ) 2 films were defined by a standard procedure developed for single crystals (Fig. S2). [S2] 0 . An example of the temperature dependence of the resistance of BaFe 2 (As 1−x P x ) 2 films with SDW transition at low temperatures (left) and its derivative (right). The peak position of the derivative is assigned as T N . The criterion is based on the comparison between the neutron scattering and transport data for the BaFe 2 As 2 system. [S2] The temperature dependencies of the upper critical fields H c2 given in the main text were obtained from resistivity measurements in static and pulsed magnetic fields (Figs. S3, S4, and S5). S6b, and S7b). To exclude possible effects of the irreversibility field H irr on the transition width, in the main text we used T c,90 to plot H c2 (for discussion see below). Two-band model for H c2 . The doping evolution of the temperature dependencies of H c2 was described by the two-band model for a clean superconductor as proposed by Gurevich [S9, S10]. Its expression for B ∥ c is given by  (for sufficiently small measurement currents) and not by H c2 which is related to the condition when the normal vortex cores overlap. Therefore, R = 0 corresponds to the irreversibility line H irr rather than to H c2 . In general, H irr can differ considerably from H c2 . [S3, S4] Therefore, we avoided to employ this R = 0 criterion. We note that the high-field data presented in Ref. [S5] reflect the measurements of H irr , which actually differs from H c2 . TABLE S1. The parameters obtained from the fit of H c2 temperature dependences shown in Fig. 2 main text. Ω sf = 100 K. The crystallographic c-axis length is given in nm, the Fermi velocities are given in 10 6 cm s −1 and the transition temperature is in K.
Taking T c = 30 K and the largest E F ∼ 100 meV in our system according to ARPES data. [S7] Alternatively, the Gi parameter can be estimated directly from the superconducting parameters: ∆T c /T c = Gi = (Γk B T c /H cm (0) 2 ξ 3 0 ) 2 /2. [S3, S8] Taking the experimental values for T c = 30 K, H c2 = 470 kG, H c1 = 600 G (field along crystallographic c-axis), [S5] λ ≈ 3· 10 −5 cm, ξ c = [Φ 0 /2πH c2 ] 0.5 ≈ 3· 10 −7 cm and the anisotropy Γ ∼ 2 -4, one arrives at the same estimates. However, close to T c the c-axis coherence length ξ c (T ) diverges as (1 − T /T c ) −0.5 . Therefore, in close vicinity to T c , where ξ c (T ) > D film , one can consider the films as 2D superconductors, where D film ≈ 100 nm is the films thickness. For optimally doped films with high H c2 values one can estimate that ξ c (T ) ∼ D film holds only for a very narrow temperature range of ∆T /T c ≈ 0.001.
On the other hand, this range is about 0.5 K for overdoped films with low T c ∼ 10 K and small H c2 ∼ 10 kG. However, due to the low T c , the fluctuation effect is rather weak ∆T c /T c = Gi ≈ (T c /E F ) ∼ 10 −3 even in 2D case.
Finally we list parameters obtained from the fits of H c2 temperature dependences shown in the main text (Tabs. S1, and S2), for the case of a zero intraband coupling λ 11 = λ 22 = 0.