## Abstract

A quantum critical point (QCP) is currently being conjectured for the BaFe_{2}(As_{1−x
}P_{
x
})_{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 BaFe_{2}(As_{1−x
}P_{
x
})_{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.

## Introduction

In most of unconventional superconductors, a quantum critical point (QCP) of charge or spin density wave (CDW/SDW) states lies beneath the superconducting dome^{1,2,3,4}. Low-energy quantum fluctuations in the vicinity of a QCP lead to non-Femi liquid (nFL) behavior in the normal state and a strong enhancement of the effective electron mass (*m**). A good example is given by heavy fermion superconductors. In some of these systems the maximum superconducting transition temperature (*T*_{c}) coincides with the position of the expected QCP of the magnetic phase^{4}. The presence of a QCP beneath the superconducting dome is evidenced by a strong enhancement of the superconducting specific heat jump Δ*C*/*T*_{c} at *T*_{c} and the slope of the upper critical field $\left|{H}_{\mathrm{c}2}^{\prime}\right|=\left|\mathrm{d}{H}_{\mathrm{c}2}/\mathrm{d}T\right|$ normalized by the critical temperature in the vicinity of *T*_{c}^{5}.

In multi-band iron-based superconductors (FBS), the maximum of *T*_{c} is usually linked to the expected position of a QCP of the SDW phase^{6}. Evidence for a zero-temperature second order magnetic transition with pronounced quantum fluctuations was found for optimally doped BaFe_{2}(As_{1−x
}P_{
x
})_{2} by various measurements in the normal state^{7,8,9,10,11,12}. Therefore, it is considered to be a classical example of unconventional superconductivity emerging in the vicinity of a magnetic state^{13, 14}. However, no doping dependence of the scattering rates expected for a QCP scenario was observed in recent angle-resolved photoemission spectroscopy (ARPES) studies^{15}. In the superconducting state, a divergent quasiparticle effective mass (*m**) above the QCP of the SDW phase was suggested based on specific heat^{16} and penetration depth measurements^{17, 18} as well as predicted by theoretical studies^{19, 20}. However, *H*_{c2} at low *T* and its slope near *T*_{c} are insensitive to the QCP^{21}. This behavior is seemingly in contradiction to many other experimental observations. To resolve this puzzle we investigated in detail the temperature dependence of *H*_{c2} for BaFe_{2}(As_{1−x
}P_{
x
})_{2} single-crystalline thin films in a wide range of P-doping. The obtained data can be described in an effective two-band model with qualitatively different doping dependences of the Fermi velocities (*v*_{F}). Namely, *v*_{F1} is indeed nearly featureless across the QCP implying a doping independent ${m}_{1}^{\u204e}$. On the other hand, *v*_{F2} is strongly doping-dependent, in accord with the almost divergent logarithmic enhancement of ${m}_{2}^{\u204e}$ observed in many other experiments.

## Results

### Electronic phase diagram of BaFe_{2}(As_{1−x
}P_{
x
})_{2}

BaFe_{2}(As_{1−x
}P_{
x
})_{2} epitaxial thin films were grown by molecular beam epitaxy (MBE)^{22, 23}. The investigated MBE thin films have high crystalline quality with *T*_{c} values above 30 K at optimal doping level. Some of the films were prepared by pulsed laser deposition (PLD). The PLD films have slightly reduced *T*_{c} at similar doping levels compared to the films prepared by MBE as shown in inset of Fig. 1a. This result is consistent with previous studies^{24}. To construct the phase diagram of our thin films, we analyzed the temperature dependence of the resistivity for various doping levels shown in Fig. 1a. The phase diagrams of the BaFe_{2}(As_{1−x
}P_{
x
})_{2} thin films and single crystals^{14, 25} are shown in Fig. 1b. The whole phase diagram for the thin films prepared on MgO substrates is shifted to lower doping levels compared to that of the single crystals. The shift of the phase diagram, as it was shown in previous studies, is substrate-dependent due to different in-plane strain^{22, 23, 26,27,28}. In particular, the in-plane tensile strain for the films grown on MgO modifies slightly the position of the bands resulting in the observed difference between the phase diagrams of thin films and single crystals^{28}. On the other hand, the amount of strain for the films grown on LaAlO_{3} (LAO) is negligibly small resulting in the same phase diagram as for single crystals^{23}.

We assumed that the temperature dependences of the resistivity (Fig. 1a) can be described by *ρ* = *ρ*_{0} + *AT*^{n} in the normal state above the superconducting and magnetic transition temperatures. This general expression has been frequently employed in the quantum critical region, where *n* = 1 at the QCP and *n* = 2 in a Fermi liquid (FL) state^{8, 14}. The contour plot in Fig. 1b illustrates the temperature and doping dependences of the exponent $n=\frac{T{d}^{2}\rho \phantom{\rule{-.15em}{0ex}}/\phantom{\rule{-.15em}{0ex}}d{T}^{2}}{d\rho \phantom{\rule{-.15em}{0ex}}/\phantom{\rule{-.15em}{0ex}}dT}+1$, as calculated using experimental temperature dependences of the resistivity. In this analysis we exclude the data close to the SDW transition, where $d\rho /dT\underset{\u02dc}{<}0$ (white region in Fig. 1b). The region in the phase diagram with nFL behavior is similar to the single crystals: the exponent *n* shows a V-shape; however, it shifts to lower doping level. This allows us to estimate the critical doping level for thin films on MgO substrates as ${x}_{\mathrm{c}}\approx 0.25\pm 0.03$, which is slightly lower than ${x}_{\mathrm{c}}\approx 0.3$ reported for single crystals^{14}. For the films prepared on LAO substrate we assumed that the position of the QCP coincides with the one for the single crystals due to the close similarity between their phase diagrams as discussed above.

### Upper critical field

The temperature dependences of *H*_{c2} for BaFe_{2}(As_{1−x
}P_{
x
})_{2} thin films with various doping levels for fields parallel to the *c*-axis are shown in Fig. 2. The temperature dependence of *H*_{c2} is strongly affected by the amount of doping. To compare the data of samples with different doping levels, we plot the reduced field ${h}_{\mathrm{c2}}=\frac{{H}_{\mathrm{c2}}}{-{H}_{\mathrm{c2}}^{\prime}{T}_{\mathrm{c}}}$ versus the reduced temperature *t* = *T*/*T*_{c} in Fig. 2b, where ${H}_{\mathrm{c2}}^{\prime}$ is the extrapolated slope of *H*_{c2} at *T*_{c}. For the strongly overdoped, and slightly underdoped samples, 0.15 < *x* < 0.21, the experimental *h*_{c2} data are close to the prediction of the single-band Werthamer-Helfand-Hohenberg (WHH) model which includes only the orbital pair-breaking effect^{29}. For other doping levels, the experimental *h*_{c2} data deviate from the single band fit. The doping dependence of *h*_{c2}(0) extrapolated to zero temperature is shown in the inset of Fig. 2b. The *h*_{c2}(0) values exhibit a broad maximum around optimal doping *x*_{c}. Additionally, *h*_{c2}(0) is strongly enhanced in the coexistence state between SC and magnetism, where the SDW transition temperature *T*_{N} > *T*_{c}.

The doping evolution of the temperature dependences of *H*_{c2} can be described by the two-band model for a clean superconductor as proposed by Gurevich^{30, 31} assuming dominant interband coupling, ${\lambda}_{12}{\lambda}_{21}\gg {\lambda}_{11}{\lambda}_{22}$, as expected for *s*_{±} superconductors. The expression for $B\parallel c$ is given in the Supplementary material Eq. S1. A small value of the intraband coupling constants *λ*_{11} = *λ*_{22} ~ 0.1 affects the resulting Fermi velocities within 10%, only around optimal doping (see Fig. S7) and has a negligible effect for overdoped samples. Therefore, to reduce the number of fitting parameters, we adopted zero intraband pairing constants *λ*_{11} = *λ*_{22} = 0. In this case, the superconducting transition temperature is related to the coupling constants by ${T}_{\mathrm{c}}=1.14\phantom{\rule{.1em}{0ex}}{\mathrm{\Omega}}_{\mathrm{sf}}{\mathrm{e}}^{(-1/{\lambda}_{12}{\lambda}_{21})}$. We considered two different values of the characteristic spin fluctuation energy Ω_{sf}: 100 K and 62 K, in order to take into account the observed softening of the spin fluctuations spectrum at the QCP^{32}. We assumed also that the paramagnetic pair breaking is negligibly weak, ${\alpha}_{\mathrm{M}}\ll 1$, as suggested by the small electronic susceptibility of BaFe_{2}(As_{1−x
}P_{
x
})_{2}, where the Maki parameter ${\alpha}_{\mathrm{M}}={2}^{\mathrm{1/2}}{H}_{\mathrm{c2}}^{\mathrm{orb}}/{H}_{\mathrm{p}}$, defined by the ratio between the orbital critical field ${H}_{\mathrm{c2}}^{\mathrm{orb}}$ and the Pauli limiting field *H*_{p}, quantifies the strength of the paramagnetic pair breaking (see also the Supplementary material). This assumption is consistent with a relatively small Knight shift of BaFe_{2}(As_{1−x
}P_{
x
})_{2}^{12}. The result of the fit is shown in Fig. 2, and the obtained fitting parameters are given in the Supplementary Tables (Tables S1 and S2).

## Discussion

The doping dependencies of ${\left|{H}_{\mathrm{c2}}^{\prime}/{T}_{\mathrm{c}}\right|}^{0.5}$ extrapolated to *T*_{c}, and the ${H}_{\mathrm{c}2}^{0.5}/{T}_{\mathrm{c}}$ extrapolated to *T* = 0 K are shown in Fig. 3a. According to the BCS theory for clean superconductors, these values are proportional to the quasiparticle effective mass (*m**). As it was pointed out in ref. 21, ${\left|{H}_{\mathrm{c}2}^{\prime}/{T}_{\mathrm{c}}\right|}^{0.5}$ should have a peak-like maximum at the QCP of the SDW phase since *m** is strongly enhanced near optimal doping on the whole Fermi surface according to various experimental data^{7, 16, 17}. However, this is not the case: ${\left|{H}_{\mathrm{c}2}^{\prime}/{T}_{\mathrm{c}}\right|}^{0.5}$ and ${H}_{\mathrm{c2}}{\mathrm{(0)}}^{0.5\phantom{\rule{-.25em}{0ex}}\phantom{\rule{-.25em}{0ex}}}/{T}_{\mathrm{c}}$ are nearly featureless at optimal doping (*x*_{c} ~ 0.25) in accord with ref. 21. Both the single crystals and our MBE films have high *T*_{c} values of about 30 K at optimal doping indicating similar low impurity scattering rates. The slightly higher ${\left|{H}_{\mathrm{c}2}^{\prime}/{T}_{\mathrm{c}}\right|}^{0.5}$ values of the single crystals compared to those of the MBE films are probably related to the different experimental methods used for the evaluation of *H*_{c2}. Also, *H*_{c2} of the PLD films follows the same trend in spite of a lower *T*_{c} and residual resistivity ratio (inset of Fig. 1a) as compared to the MBE films. Therefore, we believe that the observed doping dependence of *H*_{c2} is not affected essentially by impurity scattering rates and related instead mainly to the changes of *v*_{F} and the coupling constants.

*H*_{c2} of multi-band unconventional *s*-wave superconductors with dominant interband coupling is limited by the largest *v*_{F} in the usually considered pronounced *s*_{±}-regime^{30, 31}. Therefore, in the case of a strong *v*_{F} asymmetry between different bands, the larger Fermi velocity (*v*_{F1} in our notation) dominates *H*_{c2} around optimal doping. In this case one can write ${({H}_{\mathrm{c}2}^{\prime}/{T}_{\mathrm{c}})}^{0.5}\propto {H}_{\mathrm{c}2}{\mathrm{(0)}}^{0.5\phantom{\rule{-.25em}{0ex}}\phantom{\rule{-.25em}{0ex}}}/{T}_{\mathrm{c}}\propto {v}_{\mathrm{F}1}^{-1}\propto {m}_{1}^{\u204e}$. This explains the observed weak doping dependence of these quantities (Fig. 3). The obtained doping dependences of the (normalized reciprocal) *v*_{F1} and *v*_{F2} are shown in Fig. 3b. The 1/*v*_{F1} values are indeed smaller than 1/*v*_{F2} and show a weaker doping dependence. In contrast, 1/*v*_{F2} is strongly enhanced around optimal doping. The Ω_{sf} value affects the Fermi velocities quantitatively but their qualitative doping dependence is conserved. The corresponding normalized effective mass *m**/*m*_{b} obtained from the de Haas-van Alphen (dHvA) experiments^{7, 16} (m_{b} is the quasiparticle mass taken from the band structure calculations) follows the same trend if the small shift of the QCP along the doping axis (Δ*x* = −0.05) due to the strain is taken into account (see Fig. 1b). The logarithmic divergence at *x*_{c} = 0.25 is an indication for the reduction of *v*_{F2} caused by the quantum fluctuations associated with a QCP of the SDW phase^{8, 16}. A strong reduction of *v*_{F2} is observed also at *x* < 0.15 which roughly corresponds to the doping level where *T*_{N} > *T*_{c} (Fig. 1b, see also Tables S1 and S2 in the Supplementary material). This behavior may be associated with the reconstruction of the Fermi surface due to the presence of the coexisting SDW phase^{15, 33, 34}.

Some of the multi-band heavy fermion superconductors show a similar behavior around the magnetic QCP as the BaFe_{2}(As_{1−x
}P_{
x
})_{2} system. The measured enhancement of the effective mass depends also essentially on the experimental method^{35}. Also, a seemingly conflicting behavior between the dHvA, ARPES and transport data was discussed for cuprate superconductors around optimal hole doping^{36}. It was proposed that for the suggested nodal electron pocket induced by bidirectional charge order in high fields, the mass enhancement is very anisotropic around the small Fermi surface. It was argued that the corners of that pocket exhibit a large enhancement without any enhancement along the diagonal nodal direction. Such an angle-dependent mass enhancement is interpreted as a destruction of the Landau quasiparticles at ‘hot spots’ on the large Fermi surface at a proximate QCP. Moreover, another recent theoretical work questioned the paradigm of the universal nFL behavior at a QCP^{37}. It was shown that at the nematic QCP the thermodynamics may remain of FL type, while, depending on the Fermi surface geometry, either the entire Fermi surface stays cold, or at most there are ‘hot spots’. Therefore, one may speculate that the complex behavior observed in FBS and in particular for BaFe_{2}(As_{1−x
}P_{
x
})_{2} can be related to the superposition of *two distinct* QCPs associated with the SDW phase and the nematic order^{38}. The evidence for two distinct QCPs was indeed reported for the Ba(Fe_{1−x
}Ni_{
x
})_{2}As_{2} system^{39}. Recently, a band-dependent mass enhancement toward the QCP was suggested from the high-field specific heat measurements of overdoped BaFe_{2}(As_{1−x
}P_{
x
})_{2} single crystals^{40}. Thus so far, the available experimental data emphasize the relevance of multi-band effects for a proper and complete understanding of the quantum criticality of BaFe_{2}(As_{1−x
}P_{
x
})_{2} and related systems. Further experimental and theoretical investigations including possible strong coupling interactions since the suggested bosonic frequencies (spin fluctuations) exceed the superconducting critical temperature by a factor of three,only, retardation effects might be important, would be helpful to develop a microscopic scenario of the QCP for the title compound and other multi-band systems.

## Methods

### Samples

BaFe_{2}(As_{1−x
}P_{
x
})_{2} single crystalline thin films with various P doping levels *x* were grown by MBE with a background pressure of the order of 10^{−7} Pa. All elements were supplied from solid sources charged in Knudsen cells. Pure elements were used as sources for Ba, Fe, and As. The P_{2} flux was supplied from a GaP decomposition source where Ga was removed by two trapping caps placed on the crucible. The details of the sample preparation are given in refs. 22 and 23. Some of the films on MgO (100) substrate were prepared by PLD with a KrF excimer laser (248 nm). In this case, we used polycrystalline BaFe_{2}(As_{1−x
}P_{
x
})_{2} as the PLD target material. The preparation process took place in an ultra-high vacuum chamber with a similar base pressure of 10^{−7} Pa. Before the deposition, the substrate was heated to 850 °C. Then the BaFe_{2}(As_{1−x
}P_{
x
})_{2} layer was grown with a laser repetition rate of 3 Hz. The layer thickness was adjusted via the pulse number at constant laser energy. To improve the sample’s homogeneity and thickness gradient, the substrate was rotated during the whole deposition process. Phase purity and crystalline quality of the films were examined by X-ray diffraction (XRD). The *c*-axis lattice parameters were calculated from the XRD data using the Nelson Riley function. It depends linearly on the P-doping (determined by electron probe micro-analysis (EPMA)) for the films grown on the same substrate^{23}. In this work, we mainly investigated films prepared on MgO (100) substrate. At high doping levels, also several films on LaAlO_{3} (100) substrate have been used. The P-doping levels given in the paper have been determined using the *c*-axis lattice parameter values according to the data in ref. 23 as shown in the Supplementary material Fig. S1.

### Resistivity measurements

The temperature dependence of the electrical resistivity was measured by a four-contact method in a Quantum Design physical property measurement system (PPMS) in magnetic fields up to 14 T. Examples of the temperature dependence of the resistivity in zero and applied magnetic fields are shown in Supplementary material (Figs. S2–S7). The high-field measurements were performed in DC magnetic fields up to 35 T at the National High Magnetic Field Laboratory, Tallahassee, FL, USA. The high-field transport measurements in pulsed magnetic fields up to 67 T were performed at the Dresden High Magnetic Field Laboratory at HZDR and at the National High Magnetic Field Laboratory, Los Alamos, NM, USA. The superconducting transition temperature *T*_{
c
}, as given in the paper, was determined using *T*_{c,90} as shown in the Supplementary material (Figs. S6 and S7). Other criteria, such as 50% of the normal state resistance, yield qualitatively the same temperature dependence of *H*_{c2}. The SDW transition temperature *T*_{N} was defined as the peak position of the temperature derivative of the resistivity curves in analogy to the procedure applied for bulk single crystals^{41}, see Supplementary material Fig. S2.

The measurements were performed in magnetic fields applied along the crystallographic *c*-axis of the films, which coincides with the normal direction of the films surface. Therefore, the *H*_{c2} data presented in the paper depend on the in-plane coherence length ${\xi}_{\mathrm{ab}}$ only, which is unaffected by the film thickness *D*_{film} ~ 100 nm. Additionally, ${\xi}_{c}>d/2$ is satisfied for all doping levels, where *d* is the spacing between the neighboring FeAs layers. The estimates given in the Supplementary material indicate that the fluctuation effects close to *T*_{c} can be neglected in our case. We assume that the transition width is related to small inhomogeneities in the P distribution and to a difference between *H*_{c2}(*T*) and *H*_{irr}(*T*), where *H*_{irr} is the irreversibility field. In particular, *H*_{irr}(*T*) is noticeably affected by flux pinning at low temperatures and high magnetic fields. Thus, our consideration of BaFe_{2}(As_{1−x
}P_{
x
})_{2} thin films as 3D superconductors and the neglect of 2D corrections and fluctuation effects are indeed justified.

## Additional information

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## Acknowledgements

This work was supported by DFG (GR 4667/1-1). S.-L.D., D.E., I.C. and I.M. thank the VW foundation for financial support. D.E. also thanks RSCF-DFG Grant. The work at NHMFL was supported by the National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida. K.I. acknowledges the Open Partnership Joint Projects of JSPS Bilateral Joint Research Projects. We acknowledge the support of the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL). I.C. and I.M. thank the support RSF, grant No. 16-42-01100 and RFBR grant No. 15-03-99628a. We acknowledge fruitful discussions with D. Daghero, C. H. Lee, T. Terashima and J. Wosnitza. The publication of this article was funded by the Open Access Fund of the Leibniz Association.

## Author information

### Affiliations

#### Institute for Solid State Physics, TU Dresden, 01069, Dresden, Germany

- V. Grinenko
- & F. Kurth

#### IFW Dresden, Helmholtzstrasse 20, 01069, Dresden, Germany

- V. Grinenko
- , K. Iida
- , F. Kurth
- , D. V. Efremov
- , S.-L. Drechsler
- , I. Morozov
- , J. Hänisch
- & R. Hühne

#### Department of Crystalline Materials Science, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

- V. Grinenko
- , K. Iida
- , I. Nakamura
- , R. Fujimoto
- , T. Hatano
- & H. Ikuta

#### Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation

- I. Cherniavskii
- & I. Morozov

#### Karlsruhe Institute of Technology, Institute for Technical Physics, Hermann-von-Helmholtz-Platz 1, 76344, Eggenstein-Leopoldshafen, Germany

- J. Hänisch

#### Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01314, Dresden, Germany

- T. Förster

#### NHMFL, Florida State University, Tallahassee, FL, 32310, USA

- C. Tarantini
- & J. Jaroszynski

#### MPA-CMMS, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

- B. Maiorov
- & M. Jaime

#### Department of Applied Physics, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo, 184-8588, Japan

- A. Yamamoto

### Authors

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### Contributions

V.G., K.I. and F.K. designed the study. V.G. analyzed *H*_{c2} data, and wrote the manuscript. D.V.E. and S.-L.D. provided theoretical support in data analysis. I.C., I.M. and A.Y. prepared the PLD targets. Thin PLD films were prepared by V.G., K.I. and F.K. High-field measurements were performed by J.H., T.F., C.T., J.J., B.M., M.J., F.K., K.I. and V.G. V.G. performed transport measurements at magnetic fields up to 14 T. I.N., R.F., T.H. and H.I. prepared and characterized MBE thin films. K.I., J.H., H.I. and R.H. supervised the project. All authors discussed the results and implications and commented on the manuscript.

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding author

Correspondence to V. Grinenko.

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