The electronic phase diagrams of the Eu(Fe0.81Co0.19)2As2 superconductor

The magnetic and superconducting properties of an Eu(Fe0.81Co0.19)2As2 single crystal are investigated by means of ac magnetic susceptibility, dc magnetization, specific heat, transverse resistivity and Hall effect measurements in magnetic fields up to 9 T, applied parallel and perpendicular to the c-axis. The compound exhibits the coexistence of magnetism and superconductivity (SC), characterized by structural distortion (SD) and/or spin-density-wave (SDW) ordering at TSD/SDW = 78 ± 4 K, canted-antiferromagnetic (C-AF) ordering at the Néel temperature TN = 16.5 ± 0.5 K and SC at the critical temperature Tc = 5.3 ± 0.2 K at zero field. Upon applying fields both the C-AF and SC states evolve in an unconventional manner. Magnetic field distinctly affects the spin canting, resulting in separation of the C-AF into two new phases: the C-AF and ferromagnetic (F) ones. The unusual behavior of the SC state produces field-induced SC in the H⊥c configuration as an outcome of the weakening orbital pair-breaking effect. From the experimental data we derive the field-temperature phase diagrams for Eu(Fe0.81Co0.19)2As2. A comparison of experimental results is made with theory developed for type II superconductors and then some important thermodynamic parameters characteristic of the superconducting state of Eu(Fe0.81Co0.19)2As2 are deduced such as the specific heat jump at Tc, ΔCp(Tc)/γnTc, the electron–phonon coupling constant λe–ph, the upper critical field Hc2, coherence length ξ, the Fermi wave-vector kF, effective mass m*, Hall mobility μH, magnetic penetration depth λ and the Ginzburg–Landau parameter κ.

3 et al [2] or a canting of the Eu 2+ moments from the c-axis was suggested by Guguchia et al [6] and Błachowski et al [7]. It is worth mentioning that Jiang et al [2], Niclas et al [4] and Guguchia et al [6] found a spin reorientation from an AF to a ferromagnetic (F) arrangement upon application of external magnetic fields or hydrostatic pressure. The only two reports of field-temperature H-T magnetic phase diagrams that we are aware of are those obtained for superconducting Eu(Fe 0.89 Co 0.11 ) 2 As 2 [2] and nonsuperconducting Eu(Fe 0.9 Co 0.1 ) 2 As 2 [6] crystals. The diagram of Eu(Fe 0.89 Co 0.11 ) 2 As 2 is complex, not only due to as many as five different types of phase regimes: paramagnetism, SC, F ordering and coexistence of SC and AF or F states, but also due to magnetocrystalline anisotropy. For magnetic fields parallel to the ab-plane, SC and F states coexist in a wide temperature range. On the other hand, for magnetic fields parallel to the c-axis, SC and AF states coexist in a wide range of magnetic fields. In this paper, we investigate H-T magnetic phase diagrams of other superconducting Eu-based superconductors, i.e. Eu(Fe 0.81 Co 0.19 ) 2 As 2 . The compound exhibits multiple phase transitions, from paramagnetic (P) through structural distortion (SD) or SDW ordering at T SD/SDW = 80 K to a canted-antiferromagnetic (C-AF) state at T N = 16.5 ± 0.5 K and finally to SC at T c = 5.15 ± 0.05 K at zero field. As in the published data for Eu(Fe 0.89 Co 0.11 ) 2 As 2 [2] and for Eu(Fe 0.8 Co 0.1 ) 2 As 2 [6], we found a separation of the C-AF phase into two new phases: the AF and F phases by the external magnetic field. However, the main point of our work is to show the field effect on superconducting properties of Eu(Fe 0.81 Co 0.19 ) 2 As 2 , namely an applied field in the H ⊥c configuration can induce SC. Moreover, based on the measurements of the temperature and magnetic field dependences of ac susceptibility, dc magnetization, specific heat, ac electrical resistivity, magnetoresistance and Hall effect, we determine the most important thermodynamic parameters characteristic of the superconducting state of Eu(Fe 0.81 Co 0.19 ) 2 As 2 .

Sample preparation and characterization
Single crystals of Eu(Fe 0.81 Co 0.19 ) 2 As 2 were grown using tin as a flux. A mixture of high-purity elements (Eu : 3N, Fe, Co, Sn : 4N and As : 5N) with the atomic ratio Eu : Fe : Co : As : Sn = 1 : 1.65 : 0.5 : 2 : 30 was placed in an alumina crucible and sealed at a pressure of 0.3 bar under a purified Ar atmosphere in a quartz ampule. The ampule was heated to 1050 • C, kept at this temperature for 12 h to ensure the complete dissolving of all components in molten Sn and then, slowly cooled (2-3 • C h −1 ) down to 600 • C. Finally, the liquid Sn flux was decanted and the remaining Sn was etched away with hydrochloric acid. The obtained single crystals are in the form of irregular platelets with the crystallographic c-axis perpendicular to their surface.
In order to examine the quality of the grown crystals, transmission and scanning electron microscopy (TEM and SEM) were performed with a Philips CM-20 Super Twin microscope operating at 200 kV and a FEI Nova NanoSEM 230, respectively. Figure 1 shows the SEM picture of the largest grown crystal, which has dimensions of 0.7 × 2.5 × 3.5 mm 3 . The average chemical composition of grown crystals was determined by collecting EDX spectra at selected points of the surface, corresponding to the chemical formula Eu(Fe 0.81±0.02 Co 0.19±0.02 ) 2 As 2 . No impurity elements, e.g. Sn, are observed in the EDX spectra (see figure 2). X-ray powder diffraction of powdered small crystals was performed at room temperature using an X Pert PRO x-ray diffractometer with monochromatized CuK α radiation. All the observed Bragg lines on the x-ray diffraction (XRD) pattern could be indexed on the basis of the tetragonal ThCr 2 Si 2 -type structure (space group I4/mmm) with the lattice parameters a = 0.391 15(4) nm  The EDX spectrum of Eu(Fe 0.81 Co 0.19 ) 2 As 2 is shown as a plot of x-ray counts versus energy. Energy peaks corresponding to various elements in the sample are indicated. and c = 1.208 05(2) nm (see figure 3). The absence of un-indexed peaks indicates an upper limit of the impurity phases of less than 3 wt%. The unit cell volume of our sample is slightly larger (about 0.2%) than that reported for Eu(Fe 0.8 Co 0.2 ) 2 As 2 [5].

Experimental techniques
Measurements of ac magnetic susceptibility χ ac were made using an Oxford Instruments susceptometer at external fields up to 9 T. An ac field with an amplitude of 10 Oe and a frequency of 1 kHz was applied. Dc magnetization M measurements were carried out with a Quantum Design magnetic property measurement system MPMS-5 SQUID magnetometer in the temperature range of 1.8-400 K and at magnetic fields up to 5.5 T. The strength of the low fields was controlled by an Applied Physics Systems fluxgate model 150-6325 magnetometer. The ac-susceptibility and magnetization data were collected in the zero-field-cooled sample mode with field applied parallel and perpendicular to the c-axis. Specific heat data were obtained on a Eu(Fe 0.81 Co 0.19 ) 2 As 2 crystal of 2.5 mg by the 2-τ -thermal relaxation method in a Quantum Design physical property measurement system 9 T-PPMS platform. The data were collected twice at each temperature in the temperature range of 1.8-300 K and at fields up to 9 T. The specific heat of an addendum with a small amount of Apiezon N grease was determined previously. Measurements of the transverse resistivity and Hall effect were carried out simultaneously using a standard six-wire ac technique with a frequency of 47 Hz and a current j = 5 mA parallel to the ab-plane. The electron transport data were collected on two samples by employing a Quantum Design horizontal rotator in a Quantum Design PPMS platform. Sample 1 (0.7 × 1.5 × 2.5 mm 3 ) and sample 2 (0.4 × 0.6 × 1 mm 3 ) were cut from two different crystals originating from the same batch. The gold-wire contacts were mounted using DuPont silver conductive paint and the contact resistances were less than 0.8 . The transverse resistivity (H ⊥ j) and the Hall coefficient R H were measured rotating the sample by 90 • and 180 • , respectively. The 0 • and 90 • resistivity data correspond to the H c and H ⊥ c data, respectively. The 0 • and 180 • R H data were used to determine the Hall coefficient R H = 1 Figure 4. Temperature dependence of the inverse dc-magnetic susceptibility of Eu(Fe 0.81 Co 0.19 ) 2 As 2 single crystal at a magnetic field of 5 T applied perpendicular to the c-axis. The Curie-Weiss behavior in the temperature range of 80-400 K is shown by the solid line. The inset shows the low-temperature data collected at a field of 1 mT parallel and perpendicular to the c-axis. The anomaly at 16.8 K indicated by the solid arrow is due to a C-AF ordering of Eu 2+ moments, while the drop at low temperatures is ascribed to SC.

Magnetic susceptibility and magnetization
The inset of figure 4 shows the temperature dependence of dc-magnetic susceptibility χ(T ) ≡ M/H of the Eu(Fe 0.81 Co 0.19 ) 2 As 2 single crystal at a low magnetic field of 1 mT, applied parallel and perpendicular to the C-AF ordering of the Eu 2+ moments. It is seen that χ H c (T ) flattens off below T N , whereas χ H ⊥c (T ) is more temperature dependent. Such anisotropic behavior of the dc-χ (T ) curves suggests that the magnetic easy axis is close to the ab-plane. Similar anisotropic behavior of the dc-magnetic susceptibility was reported for Eu(Fe 0.89 Co 0.11 ) 2 As 2 [2] and Eu(Fe 0.9 Co 0.1 ) 2 As 2 [6]. However, in contrast to Eu(Fe 0.89 Co 0.11 ) 2 As 2 and Eu(Fe 0.9 Co 0.1 ) 2 As 2 , in which no further anomaly was detected in the magnetic state, the investigated Eu(Fe 0.81 Co 0.19 ) 2 As 2 sample additionally exhibits a clear drop in χ(T ) around 5 K. Taking into account the ac magnetic susceptibility and electrical resistivity data (see below), we ascribe the drop in χ(T ) curves to the onset of SC. For T > 25 K the χ H ⊥c (T ) and χ H c (T ) data are of approximately the same magnitude. We show in figure 4 the inverse susceptibility χ −1 H ⊥c (T ) measured up to 400 K at a field of 5 T. The same behaviour of susceptibility at temperatures above 80 K was obtained at a field of 0.5 T. Our data can be fitted well with the Curie-Weiss law with an effective moment µ eff = 8.05(3)µ B fu −1 and a paramagnetic Curie temperature p = 25.3(4) K. We also obtain µ eff = 7.90(3)µ B fu −1 and p = 25.5(6) K for the χ −1 H c (T ) data. A similar difference between the effective moments for H c and H ⊥ c was previously observed for EuFe 2 As 2 and Eu(Fe 0.8 Co 0.1 ) 2 As 2 [6].   curve below T N we can determine a crossover field H cr at which the second field derivative of the magnetization d 2 M/dH 2 reaches a minimum. As an example, the derivative d 2 M/dH 2 versus H at 2 K is shown in the insets of figures 5 and 6. It is found that the value of H cr decreases as the temperature increases. The temperature dependence of H cr (T ) for H c and H ⊥ c is gathered in the H-T phase diagrams (see below). To describe the physical meaning of H cr we consider the field dependence of M(H ) below and above H cr . For H < H cr the magnetization increases linearly with increasing field, whereas for H > H cr the magnetization saturates. The linearity of M(H ) in the low-field regime indicates that the ground state of Eu(Fe 0.81 Co 0.19 ) 2 As 2 must be AF. On the other hand, the saturation at high fields points to a field-induced ferromagnetic (FI-F) state. Therefore, H cr may denote the field at which the Eu 2+ moments start flipping. It is worth noting the large values of the slopes dM H c /dH and dM H ⊥c /H and the distinct difference between them. This feature may signify a canting of the Eu 2+ moments. For Eu(Fe 0.81 Co 0.19 ) 2 As 2 a faster increase of M versus H with field is observed for H ⊥ c, corresponding to a larger susceptibility χ H ⊥c than χ H c . In fact, at 2 K, χ H ⊥c amounts to 10.7 cm 3 mol −1 compared to χ H c = 7.5 cm 3 mol −1 . The tilt angle from the c-axis can be estimated using the relation θ = arctan( χ H ⊥c χ H c ). The experimental value is θ = 54.8 • for Eu(Fe 0.81 Co 0.19 ) 2 As 2 , being consistent with those deduced from the Mössbauer experiment of θ = 37 ± 5 • for Eu(Fe 0.815 Co 0.185 ) 2 As 2 and θ = 60 ± 8 • for Eu(Fe 0.805 Co 0.195 ) 2 As 2 [7]. The magnetic sublattice of the Eu 2+ moments in Eu(Fe 0.81 Co 0.19 ) 2 As 2 is shown schematically in figure 7, where spins at the Eu position are shown to be canted with respect to the c-axis.
Let us now discuss the origin of the spin-canting and spin-flip transitions upon applied fields. We first evaluate interatomic distances in the Eu(Fe 0.81 Co 0.19 ) 2 As 2 unit cell, assuming atomic positions in this compound to be the same as those in the parent EuFe 2 As 2 compound [9]. In the Eu 2+ layers, each magnetic ion is surrounded by four nearest neighbors at 0.391 nm and two next-nearest neighbors at 0.551 nm. The magnetic interactions between them can be described using two respective exchange integrals J ab1 and J ab2 , shown by dash-dotted lines in figure 7. Along the c-axis, the Eu 2+ ions are separated by a very long distance of 0.638 nm; thus the magnetic coupling J c between the Eu 2+ ions located on different layers will no longer be direct. There are three possibilities of magnetic exchange over long distances between magnetic ions: via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [10][11][12], via a superexchange path Eu-As-Eu or Eu-Fe/Co-Eu and finally via the Fert-Levy interaction [13]. These interactions may be expressed as the respective exchange constants J RKKY , J superex and J FL . According to the neutron diffraction experiment on EuFe 2 As 2 [10], both J ab1 and J ab2 are ferromagnetic, whereas the coupling J c is AF, and thus we can expect that all the coupling integrals J RKKY , J superex and J FL in Eu(Fe 0.81 Co 0.19 ) 2 As 2 are AF.
In order to explain long-range AF ordering in EuFe 2 As 2 and its Co-doped Eu(Fe 1−x Co x ) 2 As 2 alloys, one takes into account the conduction-electron-mediated RKKY interaction. However, the RKKY interaction alone seems to be an insufficient factor for the spin-canting in Eu(Fe 0.81 Co 0.19 ) 2 As 2 . Generally, spin-canting can arise from two different sources. The first one is due to the Dzyaloshinskii-Moriya (DM) interaction [14,15]. The second one is due to the magnetocrystalline anisotropy because of different preferential directions for the magnetic moments located on different sublattices. The crystal structure of Eu(Fe 0.81 Co 0.19 ) 2 As 2 is characterized by only one atomic position for the Eu 2+ ions and therefore only the DM interaction will be considered. Classically speaking, the DM interaction describes an antisymmetric, anisotropic exchange coupling between two magnetic moments on a lattice bond i j. Due to a relativistic spin-orbit coupling the DM Hamiltonian has the form: Figure 7. Schematic structure of Eu(Fe 0.81 Co 0.19 ) 2 As 2 . The magnetic sublattice of the Eu 2+ moments is shown. Magnetic moments are ferromagnetically coupled within the ab-plane but antiferromagnetically coupled between adjacent planes. Direct ferromagnetic couplings between the Eu 2+ spins in the ab-plane are represented by the exchange constants J ab1 and J ab2 . The indirect exchange paths between the two Eu 2+ spins, Eu· · · As· · · Eu and Eu· · · (Fe/Co)· · · Eu, are shown by dotted and dashed lines, respectively.
Clearly, the Hamiltonian depends on the direction of spins i and j, which in fact leans on the exchange path S i · · · S j . An inspection of the unit cell of Eu(Fe 0.81 Co 0.19 ) 2 As 2 shows that angles of the bonds Eu-As-Eu or Eu-(Fe/Co)-Eu are always 180 • . Moreover, the position of As or (Fe/Co) located inside the exchange path excludes the inversion center of the S i · · · S j bonds. The paths of indirect exchange between two Eu 2+ spins, Eu· · · As· · · Eu and Eu· · · (Fe/Co)· · · Eu, furnishing the exchange constant J c are pictured in figure 7 by dotted and dashed lines, respectively. This observation suggests that the crystal structure of both EuFe 2 As 2 and Eu(Fe 0.81 Co 0.19 ) 2 As 2 is favorable for spin-canting. After all, within the DM interaction picture it is difficult to comprehend why there is absence of spin-canting in the Co-undoped EuFe 2 As 2 . Thus, in order to interpret the spin canting in Eu(Fe 0.81 Co 0.19 ) 2 As 2 and other Co-doped Eu(Fe 1−x Co x ) 2 As 2 alloys, we need to take into account the role of nonmagnetic Co atoms and hence invoke the model developed by Fert and Levy [14]. The authors have investigated the indirect interaction between two localized spins S i and S j mediated by spin s of a conduction electron. According to Fert and Levy, the enhancement of the anisotropy field may arise from an additional term in the RKKY interaction, which is of the DM type and is due to spin-orbit scattering of the conduction electrons by the nonmagnetic impurities. The role of nonmagnetic Co ions here is to cause such interactions. The doped Co atoms introduce such a spin-orbit scattering and basically break the inversion symmetry with respect to the midpoint between the two spin sites, allowing the realization of a DM-type interaction. In our opinion, the substitution of magnetic Fe ions by nonmagnetic Co ions permits the realization of the Fert-Levy interaction and this might explain why the spin-canting occurs at low temperatures in the Co-doped Eu(Fe 1−x Co x ) 2 As 2 alloys and not in EuFe 2 As 2 .
In for H ⊥ c. The jump in the M 2 (H/M) curves is due to the spin-flip of the Eu 2+ moments. The high-field data at 2 K furnish an estimate of ordered moments M ord in the FI-F state. We obtained M ord,H c = 7.14 µ B fu −1 and M ord,H ⊥c = 7.36 µ B fu −1 at 5 T, which are larger than the theoretical value of gS = 7 µ B per Eu atom. This finding suggests that the Fe moments possibly give some contribution to the observed M ord values. Remarkably, we found that M ord,H ⊥c > M ord,H c ; thus we suspect that the Fe moments are aligned within the ab-plane. A similar difference between M ord,H ⊥c and M ord,H c was reported for the parent EuFe 2 As 2 compound [9]. The results of the ac magnetic susceptibility measurements at several dc magnetic fields are presented in figures 9 and 10. At zero dc field the ac susceptibility displays a pronounced maximum in its real part χ (T ) at the Néel temperature T N = 16.5 K, followed by a change in the slope dχ (T )/dT at the superconducting phase transition T c ∼ 5 K. The occurrence of SC and AF ordering of a canted type is supported by maxima presented at the respective temperatures in the imaginary component of the ac magnetic susceptibility χ (T ). Besides, the χ (T ) curves show an additional maximum inside the superconducting state. Usually, the χ (T ) maximum at T c corresponds to intragrain dissipation in the material and the maximum at a lower temperature coincides with the temperature where weak links between grains or Josephson-type junctions start carrying super-currents. The sheet-type structure of the investigated crystals (see figure 1) seems to be the main source of the weak link behavior. Applied magnetic fields parallel or perpendicular to the c-axis significantly affect the magnetic Eu 2+ sublattice. As can be seen in figures 9 and 10, the χ (T )-maximum at T N splits into two anomalies upon application of magnetic fields; one shifts down to lower temperatures and the other shifts upwards to higher temperatures. We attribute this behavior to a change in the C-AF structure. The field dependence of the low and high temperature maxima corresponds to a change of AF and F components, respectively. Previously, such a change in the magnetic ground state from AFM to ferromagnetism (FM) under applied magnetic fields was found in Eu(Fe 0.89 Co 0.11 ) 2 As 2 [2] and in Eu(Fe 0.9 Co 0.1 ) 2 As 2 [6]. A conspicuous influence of external magnetic fields on the observed physical properties emerges in the vicinity of T c . The applied field reduces the value of χ (T ) and finally the real component of the ac susceptibility starts showing a diamagnetic signal at fields above 0.6 T. We may recall that naked diamagnetism was not observed previously in either Eu(Fe 0.89 Co 0.11 ) 2 As 2 [2] and Eu(Fe 0.8 Co 0.2 ) 2 As 2 [8]. This may indicate that SC in our samples is more robust. A comparison of the field dependence of T c (H ) for H c and H ⊥ c reveals a distinct difference between them. When applying the external magnetic field parallel to the c-axis (figure 9), the superconducting transition temperature is found to decrease with increasing field, as can be expected for a type II superconductor. At a field above 2 T, SC vanishes. On the other hand, for the magnetic field applied perpendicular to the c-axis, after an initial decrease, T c shifts upwards to higher temperatures, as high as 7.2 K at 0.6 T. With further increasing field T c decreases, i.e. the T c versus H dependence recovers the behavior of an ordinary type-II superconductor. The anisotropic properties of the superconducting state are evidenced by the difference in magnitude of H c2 (T ) for H c and H ⊥ c. As is visible in figure 10, the diamagnetism still persists strongly at a field of 2 T. This fact means that the upper critical field for fields applied perpendicular to the c-axis H ⊥c c2 is much larger than that for fields applied parallel H H c c2 . Therefore, our observation of the diamagnetic signal provides evidence that magnetism and SC have a close relationship.

Specific heat
The temperature dependences of specific heat divided by the temperature of Eu(Fe 0.81 Co 0.19 ) 2 As 2 measured at 0 and 9 T are shown in figure 11. These data overlap each another above 100 K, indicating a dominating lattice contribution to the specific heat. Assuming that the specific heat is additive we analyze the high-temperature C p (T ) as the sum of the lattice C ph and the conduction electron C el contributions. The latter was taken as linearly temperature dependent C el = γ HT T . At room temperature (RT), the specific heat amounts to about 134 J mol K −1 . The difference between the RT-C p and the Dulong-Petit value of 124.7 J mol K −1 implies that the coefficient of the high-temperature electronic specific heat γ HT is approximately 0.03 J mol K −2 . Assuming all phonon modes are acoustic, C ph of Eu(Fe 0.81 Co 0.19 ) 2 As 2 can be modeled by the Debye function [16]: where R is the molar gas constant, n D is the number of Debye vibrators and D is the Debye temperature. For T > 100 K, fixing γ HT = 30 mJ mol K −2 we are able to fit the C p (T ) data (dashed line) with D = 330 ± 10 K. This value is quite large compared to those found in AFe 2 As 2 (A = Ca, Sr and Ba) compounds ( D = 292 K [17] where µ * is the Coulomb pseudopotential, so one can estimate the λ e-ph value for Eu(Fe 0.81 Co 0.19 ) 2 As 2 . Putting T c = 5.15 K, D = 330 K and typical values of µ * = 0.1-0.15 [24,25] into equation (2) we obtained λ e-ph = 0.6-0.7, being appreciably enhanced above the usual Bardeen-Cooper-Schrieffer (BCS) weak-coupling value of 0.4. Lowtemperature C p /T data at zero field (open circles) are dominated by a pronounced λ-type anomaly at 16.8 K, corroborating the Néel temperature of Eu(Fe 0.81 Co 0.19 ) 2 As 2 . As we have noted above, the Néel temperature of Eu(Fe 0.81 Co 0.19 ) 2 As 2 is comparable in magnitude to that of the parent compound EuFe 2 As 2 (T N = 19 K), but the nature of the AF ordering in those compounds is essentially different. Besides the difference in spin arrangements, we may point out also a disparity in the entropy S at T N . For Eu(Fe 0.81 Co 0.19 ) 2 As 2 we found the S at T N to amount to only ∼14.7 J (mol K) −1 , corresponding to 85% of R ln 8 expected for J = 7/2, which was observed in EuFe 2 As 2 [26,27].
The evidence for a superconducting phase transition in the zero-field specific heat of Eu(Fe 0.81 Co 0.19 ) 2 As 2 is not clear-cut. As can be seen from figure 11 the C p /T curve does not show any discontinuity but only a knee around T c . In a BCS-type conventional superconductor, a sharp discontinuity at T c is generally observed in the C p (T ) curve. This difference in C p (T ) behavior between the Eu(Fe 0.81 Co 0.19 ) 2 As 2 and BCS-type superconductors may be for several possible reasons: (i) a dominating magnetic contribution, (ii) the possibility of gapless SC or (iii) unconventional C p (T ) dependence. On the other hand, the absence of naked specific jump C p (T )/T c is not surprising if we remember the similar behavior of specific heat found in other FeAs-based superconductors. We recall the data reported by Budko et al [28]. According to the authors the magnitude of C p (T )/T c at T c is strongly dependent on the T c value as If we assume that the specific heat of Eu(Fe 0.81 Co 0.19 ) 2 As 2 also follows the scaling behavior (see figure 3 of [28]), we may expect the ratio C p (T )/T c to amount to 1.6 J mol K −2 . Such a small change in the specific heat, especially in magnetic materials such as Eu(Fe 0.81 Co 0.19 ) 2 As 2 , is not easy to observe in the raw data. Since we have no reasonable way of determining C p (T )/T c from the zero-field C p data, we will try to evaluate C p (T )/T c by combining the 0 T and 9 T data (open diamonds). Because an applied field of 9 T suppresses the AF order down to temperatures below 2 K and pushes the F phase transition to temperatures higher than 35 K, the magnon contribution to the 9T C p at around T c should be considerably quenched. Therefore, we may use the 9T C p data as the reference in the first approximation. In figure 12(a), we show the temperature dependence of the difference C p (T, 0 T) − C p (T, 9 T ) divided by temperature. Apparently, the C p (T )/T c at T c ratio shows an anomaly around T c . Clearly, this anomaly must be associated with the superconducting phase transition and may prove the bulk effect of SC. By extrapolating the two branches of the C p /T versus T curve below and above T c (see figure 12(a)) we evaluate the C p (T )/T c at T c ratio to be about 0.18 J mol K −2 . The Sommerfeld ratio γ n in the normal state of Eu(Fe 0.81 Co 0.19 ) 2 As 2 amounts to 0.35 J mol K −2 , corresponding to the ratio C p (T )/(γ n T c ) = 0.5, considerably smaller than the BCS weak coupling limit of 1.43. Usually, the deviation of the specific heat jump from the BCS theory might mean a nonconventional SC. However, one should take care in the case of Eu(Fe 0.81 Co 0.19 ) 2 As 2 , since the phonon and magnetic contributions to the total specific heat are not yet well defined. In figure 12(b), the Sommerfeld ratio C p (T )/T at 2 K in magnetic fields up to 9 T is shown. C p (T )/T at 2 K amounts to 0.477 J mol K −2 at zero field. With increasing magnetic field, C p (T )/T decreases to lower values, and at 9 T the 2 K C p (T )/T arrives at a value of 44 J mol K −2 . A large decrease in C p (T )/T at 2 K under magnetic fields implies that the magnon contribution to the specific heat is considerably suppressed by applied fields. Figure 13 displays the temperature dependence of the ac-electrical resistivity during cooling for the single-crystalline Eu(Fe 0.81 Co 0.19 ) 2 As 2 (sample 1) at zero field. The investigated sample exhibits three successive phase transitions, more clearly evidenced by respective anomalies in the temperature derivative of the ρ(T ) curve shown in the lower inset of figure 13. The high-temperature anomaly at T SD/SDW is attributed to an SD and/or SDW transition, while two anomalies at lower temperatures T N = 16.5 K and T c = 5.15 K correspond to AF and SC transitions, respectively. We note that the SD/SDW phase transition temperature T SD/SDW depends on the thermal history of the sample. During the cooling cycle T SD/SDW is equal to 82 K, while in the heating cycle it amounts to 87 K (see the upper inset). The SD/SDW transition and its thermal history are confirmed once by the resistivity measurements on sample 2, for which we obtain T SD/SDW = 74 and 85 K during cooling and heating cycles, respectively. The onset of SC in our samples sets in at 8 K, being consistent with the published critical temperature T c for the samples Eu(Fe 1−x Co x ) 2 As 2 with a similar stoichiometry, i.e. of 9.5 and 7.5 K for x = 0.185 and 0.195, respectively [7]. Moreover, a comparison of the overall temperature dependence of ρ(T ) reveals a systematic change in T SD/SDW with a change of the Co concentration. If Eu(Fe 0.815 Co 0.185 ) 2 As 2 shows the SD/SDW transition at 100 K and Eu(Fe 0.8 Co 2 ) 2 As 2 shows no SDW transition, our samples with x = 0.81 exhibit an SD/SDW transition at T SD/SDW with an average value 78 ± 4 K from the two samples.

Electron transport properties
In order to investigate the nature of the superconducting transition, the resistivity around the superconducting transition was measured in more detail. The obtained data are presented in figures 14(a) and (b) for magnetic fields applied parallel and perpendicular to the c-axis, respectively. The critical transition T c is defined as the midpoint of the resistivity jump of the resistivity curve. At zero field, sample 1 exhibits the superconducting phase transition at T c = 5.15 K, while sample 2 has a higher T c value of 5.51 K, yielding an average value T c = 5.3 ± 0.2 K. We must note here that the superconducting phase transition T c is affected differently by applied fields H c or H ⊥ c. For H c, the value of T c of both Eu(Fe 0.81 Co 0.19 ) 2 As 2 samples decreases as the applied field strength increases. The field dependence of T c for Eu(Fe 0.81 Co 0.19 ) 2 As 2 measured upon applying fields perpendicular to the c-axis is complex. After an initial decrease of T c with increasing fields up to 0.15 T, T c undoubtedly increases with further increasing fields and at 0.4 T T c attains a value of 6.59 K in sample 1 or 6.64 K in sample 2. Thus, the magnetic field in the H ⊥ c configuration acts very strongly on SC, resulting in increasing T c up to at least 27%. This unusual FI-SC is an intrinsic property of Eu(Fe 0.81 Co 0.19 ) 2 As 2 since it is observed in both ac magnetic susceptibility and electrical resistivity measurements.
It is worth noting that there exist a close relationship between the field dependences of T c (H ) and superconducting transition width T c (H ). The width T c (H ) is defined as T c = T 90% − T 10% , where T 90% and T 10% are the temperatures corresponding to 90 and 10% of the resistivity jump. At zero field, T c (H ) amounts to 2.96 K. For H c, T c (H ) decreases smoothly and at µ 0 H = 1 T the relative decrease of T c (H ) is 42%. On the other hand, for H ⊥ c, T c (H ) rapidly decreases below H cr and levels off at a value of 1.07 K at higher fields. The relative decrease of T c (H ) for this field configuration is equal to 64%, and is substantially larger than that for H c. Very recently, we pointed out that the drop in T c (H ) of Eu(Fe 0.81 Co 0.19 ) 2 As 2 is associated with a weakness of the orbital pair-breaking effect [30]. We suggested that this phenomenon would account for FI-SC in Eu(Fe 0.81 Co 0.19 ) 2 As 2 . We may add that such FI-SC behavior was not observed in the closely stoichiometric Eu(Fe 0.8 Co 0.2 ) 2 As 2 compound [8]. Instead, an external field applied parallel to the c-axis causes re-entrance resistance in the superconducting state. The authors have interpreted the unusual behavior to be a weakening of the vortex pining induced by rotation of the Eu magnetic moments.
In figure 15(a), we show the temperature dependence of the Hall coefficient R H of Eu(Fe 0.81 Co 0.19 ) 2 As 2 at 5 and 9 T. R H (T ) amounts to 0.76 × 10 −9 m 3 C −1 at room temperature and is comparable to that of Eu(Fe 0.8 Co 0.2 ) 2 As 2 measured at 13 T [6]. However, there exists a difference in R H value at low temperatures between two samples, x = 0.19 and 0.2. At 1.8 K, the R H of our sample (x = 0.19) at 5 T reaches a value of −2.3 × 10 −9 m 3 C −1 . The application of a larger field of 9 T results in an increase in R H (T ), and at 1.8 K the 9 T R H (T ) attains a value of −1.92 × 10 −9 m 3 C −1 . This value is still smaller than that of Eu(Fe 0.8 Co 0.2 ) 2 As 2 measured at 13 T [5] and that of the parent EuFe 2 As 2 compound at 8 T [26], both the latter compounds have the same value of R H of ∼−1 × 10 −9 m 3 C −1 at 2 K. Except for the visible field effect below 30 K, R H of Eu(Fe 0.81 Co 0.19 ) 2 As 2 shows an anomaly at T SD/SDW = 86 K. This behavior may be attributed to a change in the Hall mobility µ H (T ). Neglecting the temperature dependence of the magnetic contribution to the total Hall coefficient, the Hall mobility can be evaluated by calculating the ratio µ H (T ) = |R H (T )|/ρ(T ). Our data reveal that µ H (T ) remarkably increases with decreasing temperature. In the three temperature intervals 300 K-T SD/SDW , T SD/SDW -T N , and T N -1.8 K, µ H (T ) changes with the respective rates µ H (T ) T = 9, 24 and 26%. Based on the proportionality µ H = τ/m * and due to the close relationship between the effective mass of quasi-particles m * and the electron correlation strength, one would expect an increase in the scattering time of quasi-particles τ with decreasing temperature.
The temperature and field dependences of R H (T ) shown in figure 15 imply that in addition to the ordinary contribution to the Hall effect R 0 , which corresponds to the effect of the Lorentz force on the motion of the free carriers, there exists also an anomalous contribution to the Hall effect R s arising from the spin-orbit coupling between localized moments and itinerant electrons. The Hall coefficient of such a magnetic material can be written as [29] R H (T ) where χ(T ) = M(T ) µ 0 H is the volume susceptibility. Taking the volume susceptibility χ(T ) at 5 T, we have fitted the Hall data. The agreement between theoretical calculations and experimental data, presented as solid lines in figure 15(a), indicates that equation (3) holds for Eu(Fe 0.81 Co 0.19 ) 2 As 2 in the temperature range 200-300 K. The values of R 0 and R s were obtained as 3.08 × 10 −10 and −3.55 × 10 −8 m 3 C −1 , respectively. A large ratio R s /R 0 of the order of 100 implies that the high-temperature R H (T ) is dominated by spin-orbit coupling. The charge-carrier concentration n estimated in a one-band model from R 0 is 2.13 × 10 28 m −3 or 1.97 electrons per Eu(Fe 0.81 Co 0.19 ) 2 As 2 formula unit. The field dependence of the Hall coefficient R H of Eu(Fe 0.81 Co 0.19 ) 2 As 2 at 1.8 K is shown in figure 15(b). It is apparent that R H weakly increases with increasing field. In figure 15(b), we also show the fit of equation (3) to the Hall data. The best fit for data below 5 T was obtained with R 0 = −(1.80 ± 0.05) × 10 −9 m 3 C −1 and R s = −(2.56 ± 0.06) × 10 −9 m 3 C −1 . The R 0 value corresponds to the number of carriers in simple one-band approximation, n = 3.47 × 10 27 m −3 or 0.16 electron per Eu(Fe 0.81 Co 0.19 ) 2 As 2 formula unit. We may access quantitatively the Hall mobility using the formula: µ H = R 0 /ρ. For the 5 T data at 1.8 and 300 K, we find that µ H = 38.8 and 2.7 cm 2 V s −1 , respectively. These values are of the same order of magnitude as those found in heavy-fermion systems CeCu 2 and CePd 3 [31]. This means that the charge carriers in Eu(Fe 0.81 Co 0.19 ) 2 As 2 may have enhanced the effective masses m * .

Magnetic phase diagrams and estimation of thermodynamic parameters
The field dependences of T c , T N and T C are summarized in the temperature versus magnetic field phase diagrams shown in figures 16(a) and (b) for H c and H ⊥ c, respectively.
At zero field, Eu(Fe 0.81 Co 0.19 ) 2 As 2 undergoes three successive phase transitions: from a paramagnetic phase (phase I), to a C-AF phase (phase II) and finally to an unconventional superconducting phase. In the latter phase, in fact, the SC coexists with the C-AF, i.e. SC+C-AF, (phase III). Under applied fields both the C-AF and SC phases evolve unusually. In addition to the C-AF phase, there appears a new FI-F phase (phase IV). This phase originated from the Eu 2+ spin-canting and exists above the spin-flop field H cr . The boundary lines between C-AF and FI-F phases are determined by the ac-susceptibility (empty diamonds) and dc-magnetization (empty squares) data. As T → 0 the critical field µ 0 H c 0.55 T and µ 0 H ⊥ c 0.45 T. Because SC remains in the new FI-F state, we may designate this phase as SC+FI-F (phase V). We recognize that phase VI, i.e. FI-SC, exists for H ⊥ c only. An interesting point of comparison to the H-T phase diagram of Eu(Fe 0.81 Co 0.19 ) 2 As 2 is given by the Eu(Fe 0.89 Co 0.11 ) 2 As 2 superconductor [2]. Our Eu(Fe 0.81 Co 0.19 ) 2 As 2 sample exhibits SC coexisting only with magnetism, in contrast to Eu(Fe 0.89 Co 0.11 ) 2 As 2 , where for some range of H-T SC exists alone, i.e. without magnetism. Another difference between the two superconductors is that FI-SC occurs in Eu(Fe 0.81 Co 0.19 ) 2 As 2 upon H ⊥ c. In the following, we shall concentrate on estimation of some thermodynamic parameters of Eu(Fe 0.81 Co 0.19 ) 2 As 2 . An examination of the H c2 (T ) in the vicinity of T c reveals a linearity between H and T c ,

Summary
We have synthesized single-crystalline Eu(Fe 0.81 Co 0.19 ) 2 As 2 adopting the tetragonal ThCr 2 Si 2type structure at room temperature. The measurements of the ac magnetic susceptibility, dcmagnetization, specific heat, magnetoresistivity and Hall coefficient reveal that the compound undergoes multiple phase transitions, from paramagnetism via SD and/or SDW ordering at T SD/SDW = 78 ± 4 K, to C-AF at T N = 16.5 ± 0.5 K and finally to SC at T c = 5.3 ± 0.2 K at zero field. From the field dependence of T N , T C and T c we propose the magnetic phase diagrams and compare to the literature data for other compositions of the Eu(Fe 1−x Co x ) 2 As 2 solid solutions. We found an FI separation of the C-AF phase into two new phases: C-AF and F. We have discussed the spin-canting in Eu(Fe 0.81 Co 0.19 ) 2 As 2 and in other Co-doped in the framework of the Fert-Levy model, which includes RKKY interaction and spin-orbit coupling between conduction electrons and nonmagnetic ions. We discovered that the external field applied perpendicular to the c-axis may increase T c significantly, at least up to 27%. Since the SC is intimately associated with the observed magnetic phase transitions of the Eu 2+ moments, we may interpret the phenomenon by a weakening of the orbital-breaking effect as a result of the alignment of magnetic Eu 2+ moments within the ab-plane. Because of the fact that the SC is basically related to Fe ions, further investigation of possible effects of small magnetic fields on the magnetic Fe moments and/or their fluctuations affecting the FI-SC is highly desirable. Finally, some physical parameters characteristic of the superconducting state of Eu(Fe 0.81 Co 0.19 ) 2 As 2 , such as the specific heat jump at T c , C p (T c )/γ n T c , the electron-phonon coupling constant λ e-ph , the upper critical field H c2 , effective mass m * , Hall mobility µ H , the penetration depth λ, coherence length ξ and the Ginzburg-Landau parameter κ were determined. The evaluated thermodynamic parameters of Eu(Fe 0.81 Co 0.19 ) 2 As 2 are listed in table 1. The notable information in the table is that Eu(Fe 0.81 Co 0.19 ) 2 As 2 exhibits small values of ξ and µ H but large values of λ and m * . This finding is indicative of a metal with heavily renormalized electrons and may call for further studies in order to explore the mechanism leading to the coexistence of magnetism and heavy-fermion SC in Eu(Fe 1−x Co x ) 2 As 2 superconductors.