Hall-plot of the phase diagram for Ba(Fe1-xCox)2As2.

The Hall effect is a powerful tool for investigating carrier type and density. For single-band materials, the Hall coefficient is traditionally expressed simply by , where e is the charge of the carrier, and n is the concentration. However, it is well known that in the critical region near a quantum phase transition, as it was demonstrated for cuprates and heavy fermions, the Hall coefficient exhibits strong temperature and doping dependencies, which can not be described by such a simple expression, and the interpretation of the Hall coefficient for Fe-based superconductors is also problematic. Here, we investigate thin films of Ba(Fe1-xCox)2As2 with compressive and tensile in-plane strain in a wide range of Co doping. Such in-plane strain changes the band structure of the compounds, resulting in various shifts of the whole phase diagram as a function of Co doping. We show that the resultant phase diagrams for different strain states can be mapped onto a single phase diagram with the Hall number. This universal plot is attributed to the critical fluctuations in multiband systems near the antiferromagnetic transition, which may suggest a direct link between magnetic and superconducting properties in the BaFe2As2 system.

It is widely believed that for most unconventional superconductors, Cooper pairs are mediated by spin or orbital fluctuations. A very good example is given by Co-doped BaFe2As2, one of the most studied Fe-based superconductors (FBS), in which the neutron resonance peak was observed1. Note that this resonance peak is hardly elucidated by electron-phonon interaction. Typically, the parent compound of an Fe-based superconductor (FBS) shows a spin-density wave (SDW) phase at low temperatures. This magnetic instability is linked to the Fermi surface (FS) nesting between hole-like pockets centered at the Γ-point and electron-like pockets at M-points in the Brillouin zone2. Upon carrier doping, the nesting condition is deteriorated and the superconductivity appears at a given doping level. The emergence of superconductivity in the vicinity of SDW immediately pushed the idea of strong spin fluctuations providing the main glue for Cooper pairing in FBS.

External pressure3, chemical pressure4, and strain in thin films5 may change the nesting conditions similarly to carrier doping. For the latter case, tensile or compressive in-plane strain with biaxial and uniaxial components is induced by the lattice and/or thermal expansion mismatch between film and substrate. Therefore, tensile or compressive in-plane strain may act as control parameter for the phase diagram by selecting a specific substrate.

Here, we report a systematic study of Ba(Fe1−Co)2As2 epitaxial thin films grown on MgO(001) and CaF2(001) single crystalline substrates by pulsed laser deposition (PLD). The former substrate induces tensile strain, whereas the latter yields compressive one. Using transport data, we construct the phase diagram for Ba(Fe1−Co)2As2 thin films under different strain states (i.e., tensile and compressive in-plane strain). The resultant phase diagrams show that the Néel temperature (TN) and the superconducting transition temperature (Tc) at a given Co doping level depend strongly on the direction of in-plane strain. Both TN at zero doping and Tc at optimal doping level are enhanced by in-plane compressive strain in comparison to single crystals. Moreover, the whole phase diagram is shifted in the direction of higher Co doping. For tensile strain, TN at zero doping is reduced and Tc at optimal doping level is almost unchanged, and the whole phase diagram is shifted to lower doping level. Finally, we demonstrate that the phase diagrams for films on both substrates and single crystals (i.e., the relaxed samples) including magnetic and superconducting regions can be mapped onto a single phase diagram with the Hall number as new variable. Our findings may suggest a direct link between magnetism and superconductivity in FBS.

Results

Structural properties

All Ba(Fe1−Co)2As2 films (0 ≤ x ≤ 0.15) were epitaxially grown on MgO(001) and CaF2(001) substrates with high phase purity. The epitaxial relation is (001)[100]film||(001)[100]MgO and . More information on the structural analyses by x-ray diffraction can be found in the Supplementary Information. As shown in Fig. 1a, the lattice constant a of the Ba(Fe1−Co)2As2 films on CaF2 substrates (Ba-122/CaF2) is shorter than that of the bulk samples6, whereas the opposite relation holds for the Ba(Fe1−Co)2As2 films on MgO substrates (Ba-122/MgO). As expected, an elongation of the c-axis for Ba-122/CaF2 and a shrinkage of the c-axis for Ba-122/MgO were observed due to the Poisson effect, as shown in Fig. 1b. These structural changes are due to the biaxial strain with average uniaxial components over sample volume, ϵ = ϵ. Here, the average lattice deformations in the tetragonal phase are defined as ϵ = (afilm − aPLD target)/aPL Dtarget and ϵ = (cfilm − cPLD target)/cPLD target along the a- and c-axis. In this way, the biaxial in-plane tensile strain acts similarly to uniaxial pressure along the c-axis, while the in-plane compressive strain acts as negative pressure along the c-axis. The average lattice deformations at room temperature (RT) for Ba-122/MgO along the a- and c-axis are ϵ = 5.9 × 10−3 and ϵ = −4.4 × 10−3, respectively. The corresponding values for Ba-122/CaF2 films are ϵ = −5.8 × 10−3 and ϵ = 5.3 × 10−3. It is noted that the lattice deformation for both films are almost constant irrespective of Co contents (Supplementary Fig. S3). The origin of the biaxial strain is discussed in the Supplementary Information.

Figure 1

Co doping dependence of lattice parameters.

(a) In-plane lattice constant a of Ba(Fe1−Co)2As2 thin films on MgO and CaF2 substrates, Ba(Fe1−Co)2As2 single crystals (Ba-122 sc)6, and PLD target as a function of Co doping. The lines are a guide to the eye. (b) The corresponding out-of-plane lattice constants c for the same samples. The lines are a guide to the eye. (c) The As position (z) for the strained films and PLD target materials as a function of the Co doping. The data of bulk single crystals are taken from refs 7, 8, 9, 10. The solid green line shows the average As position for unstrained samples.

Figure 1c summarizes the As position (z) in the unit cell for the strained films and PLD targets. The literature data for single crystals are also shown in the same figure78910. It is apparent that the As coordinate is nearly independent of strain and Co doping.

Resistivity and phase diagram

The evolution of the in-plane longitudinal resistivity (ρ) curves in zero magnetic field as a function of temperature for the Ba(Fe1−Co)2As2 films on MgO and CaF2 substrates are displayed in Fig. 2a,b, respectively. The low-temperature state in the films on both substrates changes upon doping similar to the bulk material: from antiferromagnetic to superconducting, followed by metallic state6. However, the doping levels at which the phase transitions occur depend on the strain state. For Ba-122/MgO, Co doping of x = 0.02 induces superconductivity with a Tc of 7.5 K. Additionally a sudden drop of the Hall coefficient around 100 K due to the SDW transition was observed (see Fig. 2c). Hence, for Ba-122/MgO with x = 0.02 superconductivity coexists with antiferromagnetism. On the other hand, Ba-122/CaF2 with the corresponding composition did not show superconductivity down to the lowest temperature available in our experiments (i.e., ∼2 K).

Figure 2

Transport properties of Ba-122/MgO and Ba-122/CaF2 thin films.

Resistivity data for Ba(Fe1−Co)2As2 thin films on (a) MgO and (b) CaF2 substrates. Broken lines are the fitting curves using ρ = ρ0 + AT in the paramagnetic (PM) state. Hall coefficient of Ba(Fe1−Co)2As2 films on (c) MgO and (d) CaF2 substrates as a function of temperature.

Based on the resistivity data, the phase diagrams of both Ba-122/MgO and Ba-122/CaF2 are constructed, Fig. 3a,b. For comparison, the single crystal data are plotted in the same figures. Here, Tc was determined by the superconducting onset temperature (Supplementary Fig. S7), whereas TN was defined as a peak position of the temperature derivative of the resistivity curves in analogy to bulk single crystals (Supplementary Information in the section of criterion for TN)111213. It is noted that the peak position of the temperature derivative of the resistivity is related to the magnetic transition according to x-rays and neutron diffraction measurements11. Zero resistivity temperature and middle point of superconducting transition may be influenced by flux pinning effect. Therefore, we chose the onset temperature of resistivity as a criterion of the Tc. It is clear from Fig. 3a that tensile strain (Ba-122/MgO) slightly reduces TN and shifts the superconducting dome to lower doping levels compared to the single crystals. A similar shift of the superconducting dome by in-plane tensile strain was observed in P-doped Ba-122 on MgO substrates14. To the contrary, biaxial in-plane compressive strain (Ba-122/CaF2) effectively pushes the phase diagram to a higher doping level in comparison with single crystals (Fig. 3b). Qualitatively, the shift of the phase diagram can be understood by examining the electronic band structure. At zero doping level, the ab-initio calculations show that compressive biaxial in-plane strain makes the band structure more two-dimensional with good nesting (see the section Discussion), resulting in a higher AFM transition temperature. Tensile strain shows the opposite effect which makes the band structure more three-dimensional, and consequently TN decreases. For single crystals, a similar development of the FS takes place. Related angle-resolved photoemission spectroscopy (ARPES) measurements showed that upon Co doping the electronic states in the vicinity of the Fermi level become more three-dimensional15. Therefore, the two effects (charge doping and in-plane strain) determine the shift of the phase diagram along the doping axis.

Figure 3

Electronic phase diagram of Ba(Fe1−Co)2As2.

The electronic phase diagram of thin films grown on (a) MgO and (b) CaF2 substrates. For comparison, the single crystal data626 are also shown in the figures as dotted lines. TN and Tc denote the antiferromagnetic and the superconducting transition temperatures, respectively. SDW, PM, and SC are the spin density wave, paramagnetic, and superconducting phases, respectively. Value for the exponent n taken from the resistivity data ρ = ρ0 + AT in the paramagnetic state: (c) Ba-122/MgO and (d) Ba-122/CaF2.

The temperature dependence of the resistivity for the films in the paramagnetic (PM) state was fitted using ρ = ρ0 + AT and the resultant fitting curves are shown in Fig. 2a,b. This expression has been widely used for analyzing the resistivity in the quantum critical region, e.g. refs 16 and 17. The dependence of the power-law exponent n on Co doping (i.e., x) is presented in Fig. 3c,d. For Ba-122/MgO, the exponent n has a minimum value close to unity at x ∼ 0.05. This may be assigned to the AFM quantum critical point (QCP), where the AFM transition temperature goes to zero. For Ba-122/CaF2, the QCP is observed at x ∼ 0.075 (Fig. 3d). The presence of the AFM QCP for Co-doped Ba-122 has been proposed recently by specific heat, thermal expansion, and nuclear magnetic resonance measurements181920. The observed simultaneous shift of the QCP and the maximum T for the strained thin films suggests the relationship between critical magnetic fluctuations and superconductivity in FBS.

In the case of thin films, the substrate may essentially weaken the orthorhombic distortion, as the Ba-122 grains are rigidly fixed at the interface by the substrate. This mechanism is responsible for the strain in the thin films (Supplementary Information). Another evidence for the tetragonal crystal structure being maintained in thin films grown on the substrates comes from the angular dependence of in-plane magnetoresistance (MR) measurements. An example of the in-plane MR for Ba-122/MgO and Ba-122/CaF2 with the same Co doping level of x = 0.04 in an applied field of 14 T at various temperatures is shown in Fig. 4a,b. As stated above, Ba(Fe1−Co)2As2 thin films are grown on MgO(001) substrates with cube-on-cube configuration, whereas the basal plane of Ba(Fe1−Co)2As2 is rotated by 45° on CaF2(001). We applied the current along the tetragonal [110] direction for Ba-122/CaF2 and along the tetragonal [100] direction for Ba-122/MgO, respectively. According to refs 21 and 22, a magnetic field of B = 14 T parallel to the ab-plane can partially detwin Ba(Fe1−Co)2As2 single crystals, leading to a twofold symmetry of the in-plane MR curves below the temperature at which the nematicity sets in. When bias current and magnetic field are parallel to the orthorhombic [100] or [010] axis, the in-plane MR curves show the maximum values. On the other hand, the position of the peak is shifted by 45°, if the bias current (I) flows along orthorhombic [110] axis (zero angle corresponds to B || I. In this case, the peak values are much smaller than for the former geometry (i.e., current and magnetic field || orthorhombic [100] or [010]). However, our results contradict the one obtained from single crystals. Below 100 K, the measured in-plane MR curves for both films clearly follow an almost perfect sinusoidal angle dependence without phase shift. If the oscillation were defined by the nematic domains oriented by the applied field as in the case of single crystals2122, the MR signal for Ba-122/MgO should be shifted by 45° with respect to the one for Ba-122/CaF2. Additionally, the amplitude of MR signals for both films are quite small compared to those of single crystals. This indicates that the substrate completely blocks the rotation of the nematic/magnetic domains. However, the appearance of oscillation in the MR at a certain temperature T+ indicates some changes of the FS topology or scattering rates. The T+ is rather high compared to TN and preserved at doping levels above the QCP of the SDW phase. By analogy with refs 23 and 24, T+ may be related to the nematic phase or fluctuating magnetic domains. This temperature is presumably increased by uniaxial strain if compared to relaxed Ba(Fe1−Co)2As2 single crystals.

Figure 4

Angular dependence of in-plane magnetoresistance data.

The angular dependence of in-plane magnetoresistance (MR) data (Δρ/ρ) in the presence of a magnetic field (14 T) for (a) Ba-122/MgO and (b) Ba-122/CaF2. The sketch gives the orientation of the crystallographic axes for (c) Ba-122/MgO and (d) Ba-122/CaFv in orthorhombic notations.

Effective carrier density plot of the phase diagram

The temperature dependencies of the Hall coefficients (RH) measured at 9 T for Ba-122/MgO and Ba-122/CaF2 films are shown in Fig. 2c,d. For both parent compound films (i.e., x = 0), RH is weakly decreasing with decreasing temperature until the SDW transition occurs, similarly to the observation in single crystals2526. In contrast to single crystals, however, RH changes sign from negative to positive. This behavior can be understood qualitatively by the effect of strain on carrier mobilities. The non-doped Ba-122 is a compensated metal with equal electron and hole carrier densities. Therefore, a small change of the mobilities can strongly affect the experimental value of the effective nH (especially in AFM state with reconstructed Fermi surfaces). Only a small amount of Co addition to the system leads to a drastic change in RH at low temperature. For both films with x = 0.02, RH is decreased sharply with decreasing temperature below TN due to a large change in the carrier concentration and mobility. This behavior is similar to that observed in single crystals2526.

In order to quantify the effect of strain on the electronic properties, we consider the effective carrier number per Fe, nH(Tc) and nH(TN), as , where V is the unit cell volume estimated from Fig. 1a,b. Now, we re-plot Tc and TN as a function of nH(Tc) and nH(TN), as shown in Fig. 5a,b. For comparison, the single crystal data from refs 6 and 26 are also shown in the graph. As can be seen, Tc for both strained thin films and single crystals can be mapped onto a master curve by nH(Tc) as a new variable. Note that nH scales both the position of the superconducting dome and the absolute value of Tc, whereas the carriers numbers27 and structural parameters (i.e., bonding angle and anion height)28 scale only either the position of the superconducting dome or the absolute value of Tc.

Figure 5

Effective carrier density plot of the superconducting (Tc) and Néel (TN) temperatures of Ba(Fe1−Co)2As2.

(a) Superconducting (Tc) and (b) Néel (TN) temperatures as a function of nH(Tc) and nH(TN). For comparison, Ba-122 single crystal data taken from ref. 26. are also plotted. SDW and PM are the spin density wave and paramagnetic phases, respectively. The labels show the characteristic range of nH in SDW and PM phases.

On the other hand, TN is vaguely independent of nH(TN) for non-zero doping, i.e., a magnetic transition occurs, when nH(T) approaches about 0.05 carriers/Fe.

Discussion

The shift of the superconducting dome and the AFM transition temperature with uniform in-plane biaxial strain can be understood qualitatively by considering the effect of the strain on the FS shape, its orbital weight, and composition. First of all, our local (spin) density approximation (L(S)DA) calculation shows that the strain and the Co doping affect mainly the hole FS pockets located at the zone center, whereas the electron pockets at the zone corners are nearly unchanged. These results are consistent with the ARPES measurements for doped and undoped Ba-122 single crystals reported recently29. Additionally, it was observed that an L(S)DA and generalized gradient approximation (GGA) calculation gives a reasonable prediction of the effect of the Co doping on the band structure29. This observation strengthens our theoretical approach.

Analyzing the changes of the hole FS pockets, we found that TN correlates well with the value of the k dispersion of the Fe 3d xz/yz orbitals on the hole FS pockets. As can be seen in Fig. 6, the dispersion along k of the undoped Ba-122 film on MgO increases compared to that of the undoped bulk sample (i.e., relaxed) due to tensile strain. TN of the former is lower than that of the latter, as shown in Fig. 3a. Simultaneously, the shape of the corresponding FS sheets is getting more three-dimensional. The same trend is observed for different Co doping but with fixed strain state. In contrast, compressive strain alone (i.e., Ba-122/CaF2 with fixed Co doping) reduces the k dispersion of the Fe 3d xz/yz orbitals, which leads to the enhancement of TN. The observed three-dimensional effects of the FS are responsible for the suppression of the FS nesting conditions found by the ARPES study29. One can also see from Fig. 3a,b that the degree of shift in the superconducting dome along the doping axis correlates well with TN; lower/higher TN pushes the SC dome towards the underdoped/overdoped region. Such a tendency may be understood if Cooper pairing and magnetic ordering is controlled by a common parameter.

Figure 6

Fermi surface of Ba(Fe1−Co)2As2.

Evolution of the Fermi surface (FS) of Ba(Fe1−Co)2As2 at the Γ point as a function of Co doping and strain. The color code corresponds to a relative orbital weight per Fe-atom. The detailed theoretical approach can be found in Methods section.

The result of the band structure calculation is insufficient for the interpretation of the observed behavior shown in Fig. 5b. The scaling indicates that the value of the effective carrier number nH(TN) at the phase transition is not only related to the electronic structure but also strongly affected by critical fluctuations which are not included in the band structure calculations. As was shown by Kontani et al., nH scales with the antiferromagnetic (AF) correlation length (ξ−2) in the case of strong AF spin fluctuations30. Therefore, the value of nH should approach zero at the phase transition. However, nH(TN) tends to a finite value of ∼0.05 carriers/Fe (excluding films with zero doping level) at the magnetic transition as can be seen in Fig. 5b. This seeming contradiction may be explained by considering the multiband nature of the FS. The Hall coefficient is defined as RH = σ/H(σσ), where σ and σ are the full conductivities summed up over all bands. Therefore, the Hall number is given by , where σ is the conductivity of a hole (h) or an electron (e) band, i or j is the summation index running over different hole or electron bands, respectively, and n is the corresponding carrier density31. Hence, as in the case of resistivity32, the bands with the largest conductivities contribute most to the Hall number. At the phase transition, the conductivities of the interacting bands or part of the bands tend to zero due to strong scattering of their quasi-particles on the critical fluctuations. Therefore, nH is simply determined by those parts of the FS which are less sensitive to critical fluctuations. This explains why the magnetic transition occurs at a finite nH value. Good scaling for Tc is observed for each side of the superconducting dome, since those regions are far from the magnetic transition lines. In this case, the value of nH(Tc) is sensitive to the distance to the SDW line. Therefore, the scaling for Tc can be interpreted by the indication of a strong interplay between Tc and the magnetic fluctuations slightly above Tc, which are sensitive to the carrier doping and strain as well. A possible disorder effect on both Tc and TN cannot be separated from the spin fluctuation effect, since impurity scattering is also included in nH.

In conclusion, we have shown that strain essentially affects the phase diagram of the generic system Ba(Fe1−Co)2As2. The biaxial in-plane strain is responsible for a nearly rigid shift of the whole phase diagram including the magnetic and superconducting regions along the electron doping. This behavior is explained by band structure calculations in which biaxial in-plane strain affects the FS similar to Co doping. Moreover, the superconducting dome is rigidly connected to the position of the SDW line. The direct relationship between the paramagnetic normal state and TN, as well as the relationship between Tc and the preceding state above Tc are given by the unusual plot of TN and Tc with the Hall number at those temperatures. This emphasizes a crucial role of the critical fluctuations for superconductivity and magnetism in FBS. It is important to check whether a similar plot exists for other families of the FBS, too. Also, a microscopic explanation of the observed unusual behavior is still lacking. We believe that our experimental results will stimulate future theoretical investigations.

Methods

Ba(Fe1−Co )2As2 films on MgO(001) and CaF2(001) substrates

Ba(Fe1−Co)2As2 films of around 100 nm thickness have been grown on MgO(001) and CaF2(001) substrates by pulsed laser deposition (PLD). PLD targets made by a solid state reaction with various Co levels ranging from 0 ≤ x ≤ 0.225 were ablated by a KrF excimer laser with a laser repetition rate of 7 Hz. Prior to the deposition, the substrates were heated to 850 °C. A base pressure of around 10−8 mbar at 850 °C was achieved, which increased to 10−7 mbar during the deposition. The thicknesses of all films (∼100 nm) were measured by scanning electron microscope images of cross-sectional focused ion beam (FIB) cuts. In a previous investigation on Ba(Fe0.92Co0.08)2As2 films (nominal composition x = 0.08) by energy dispersive X-ray spectroscopy, we determined the Co content to 0.74 ± 0.017, indicating good agreement with the nominal value33.

Structural analyses by x-ray diffraction

The c-axis texture and phase purity were investigated by x-ray diffraction in Bragg-Brentano geometry with Co-Kα radiation. In-plane orientation of Ba-122/MgO and Ba-122/CaF2 was investigated by using the 103 pole in a texture goniometer operating with Cu-Kα radiation. In order to precisely evaluate the lattice parameter a of Ba-122/MgO and Ba-122/CaF2, high resolution reciprocal space maps (RSM) around the 109 reflection were performed with Cu-Kα radiation.

Temperature evolution of the lattice constants c for Ba-122/MgO and Ba-122/CaF2 was investigated by x-ray diffraction in Bragg-Brentano geometry with Cu-Kα radiation in flowing He gas atmosphere. Diffraction patterns were acquired at elevated temperatures from 298 K to 773 K (Supplementary Fig. S4).

Determination of As position (z)

The As position of the PLD target materials was refined by Rietveld analysis using powder x-ray data. For the thin films, z was calculated by using the experimental lattice constants a and c, shown in Fig. 1a,b, and the optimized As position in the paramagnetic state, which is the same method as described in ref. 32.

Transmission electron microscopy (TEM)

The samples for TEM analysis were prepared using a FIB (SMI3050MS2) by cutting and milling the identical films used for transport measurements. The microstructure near the interface of Ba-122/MgO with x = 0.06 was analyzed using JEOL JEM-2100F (Supplementary Fig. S5b).

In-plane transport measurements

Prior to the micro bridge fabrication, the temperature dependence of the resistance for all films was measured by a 4-probe method, in which small pins are aligned co-linear on the film surfaces. After the measurements, the films were photolithographically patterned and ion-beam-etched to fabricate a small bridge of 100 μm width and 0.41 mm length for transport measurements. No changes in transport properties after the micro bridge fabrication have been found. Longitudinal and transverse resistance were measured with four-probe configuration by a Quantum Design physical property measurement system (PPMS) up to 14 T.

Theoretical analysis

To understand the impact of the strain on the electron band structure, we performed density functional theory (DFT) calculations of the Fermi surface (FS) for both Ba(Fe1−Co)2As2 thin films and bulk single crystals. Our calculations were carried out within the local (spin) density approximation (L(S)DA) using the Full Potential Local Orbital band structure package (FPLO, http://www.fplo.de)34. The Co doping was taken into account within the virtual crystal approximation. As can be seen in Fig. 1c, z is almost constant irrespective of both strain and Co doping. Therefore, the absolute As position is just proportional to the lattice parameters. A k-mesh of 12 × 12 × 6 k-points in the whole Brillouin zone was employed. The calculations were performed using the FPLO Ab-Initio Simulation Package within the Perdew, Burke and Ernzerhof (PBE) functional for the exchange-correlation potential. The calculated hole FS for the Co doping level x = 0 and 0.1 are summarized in Fig. 6.

Additional Information

How to cite this article: Iida, K. et al. Hall-plot of the phase diagram for Ba(Fe1–Cox)2As2. Sci. Rep. 6, 28390; doi: 10.1038/srep28390 (2016).