Hall-plot of the phase diagram for Ba(Fe1-xCox)2As2
Kazumasa Iida, Vadim Grinenko, Fritz Kurth, Ataru Ichinose, Ichiro Tsukada, Eike Ahrens, Aurimas Pukenas, Paul Chekhonin, Werner Skrotzki, Angelika Teresiak, Ruben Huehne, Saicharan Aswartham, Sabine Wurmehl, Ingolf Moench, Manuela Erbe, Jens Haenisch, Bernhard Holzapfel, Stefan-Ludwig Drechsler, Dmitri V. Efremov
HHall-plot of the phase diagram for Ba(Fe − x Co x ) As Kazumasa Iida,
1, 2, ∗ Vadim Grinenko,
1, 2, † Fritz Kurth,
2, 3
Ataru Ichinose, Ichiro Tsukada, Eike Ahrens,
2, 5
Aurimas Pukenas, Paul Chekhonin, Werner Skrotzki, Angelika Teresiak, Ruben H¨uhne, Saicharan Aswartham, Sabine Wurmehl,
2, 5
Ingolf M¨onch, Manuela Erbe,
2, 7
Jens H¨anisch,
2, 7
Bernhard Holzapfel, Stefan-Ludwig Drechsler, and Dmitri V. Efremov Department of Crystalline Materials Science Graduate School of Engineering,Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan IFW Dresden, P.O. Box 270116, 01171 Dresden, Germany Dresden University of Technology, Faculty for Natural Science and Mathematics, 01062 Dresden, Germany Central Research Institute of Electric Power Industry,2-6-1 Nagasaka, Yokosuka, Kanagawa 240-0196, Japan Dresden University of Technology, 01062 Dresden, Germany IFW Dresden, 01171 Dresden, Germany Karlsruhe Institute of Technology, Institute for Technical Physics,Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany (Dated: October 13, 2018) a r X i v : . [ c ond - m a t . s up r- c on ] J un BSTRACT
The Hall effect is a powerful tool for investigating carrier type and density. For single-band mate-rials, the Hall coefficient is traditionally expressed simply by R − = − en , where e is the charge ofthe carrier, and n is the concentration. However, it is well known that in the critical region near aquantum phase transition, as it was demonstrated for cuprates and heavy fermions, the Hall coef-ficient exhibits strong temperature and doping dependencies, which can not be described by sucha simple expression, and the interpretation of the Hall coefficient for Fe-based superconductors isalso problematic. Here, we investigate thin films of Ba(Fe − x Co x ) As with compressive and tensilein-plane strain in a wide range of Co doping. Such in-plane strain changes the band structure ofthe 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 singlephase diagram with the Hall number. This universal plot is attributed to the critical fluctuations inmultiband systems near the antiferromagnetic transition, which may suggest a direct link betweenmagnetic and superconducting properties in the BaFe As system. INTRODUCTION
It is widely believed that for most unconventional superconductors, Cooper pairs are mediated by spinor orbital fluctuations. A very good example is given by Co-doped BaFe As , one of the most stud-ied Fe-based superconductors (FBS), in which the neutron resonance peak was observed [1]. Note thatthis resonance peak is hardly elucidated by electron-phonon interaction. Typically, the parent compoundof Fe-based superconductor (FBS) shows a spin-density wave (SDW) phase at low temperatures. Thismagnetic 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 zone [2]. Upon carrier doping, the nestingcondition is deteriorated and the superconductivity appears at a given doping level. The emergence of su-perconductivity in the vicinity of SDW immediately pushed the idea of strong spin fluctuations providingthe main glue for Cooper pairing in FBS.External pressure [3], chemical pressure [4], and strain in thin films [6] may change the nesting condi-tions similarly to carrier doping. For the latter case, tensile or compressive in-plane strain with biaxialand uniaxial components is induced by the lattice and/or thermal expansion mismatch between film andsubstrate. Therefore, tensile or compressive in-plane strain may act as control parameter for the phasediagram by selecting a specific substrate.Here, we report a systematic study of Ba(Fe − x Co x ) As epitaxial thin films grown on MgO(001) andCaF (001) single crystalline substrates by pulsed laser deposition (PLD). The former substrate inducestensile strain, whereas the latter yields compressive one. Using transport data, we construct the phasediagram for Ba(Fe − x Co x ) As thin films under different strain states (i.e., tensile and compressive in-plane strain). The resultant phase diagrams show that the N´eel temperature ( T N ) and the superconductingtransition temperature ( T c ) at a given Co doping level depend strongly on the direction of in-plane strain.Both T N at zero doping and T c at optimal doping level are enhanced by in-plane compressive strain incomparison to single crystals. Moreover, the whole phase diagram is shifted in the direction of higherCo doping. For tensile strain, T N at zero doping is reduced and T c at optimal doping level is almostunchanged, and the whole phase diagram is shifted to lower doping level. Finally, we demonstrate thatthe phase diagrams of all considered strained films and single crystals (i.e., the relaxed samples) includingmagnetic and superconducting regions can be mapped onto a single phase diagram with the Hall numberas new variable. Our findings may suggest a direct link between magnetism and superconductivity inFBS. 2 ESULTSStructural properties
All Ba(Fe − x Co x ) As films ( ≤ x ≤ . ) were epitaxially grown on MgO(001) andCaF (001) substrates with high phase purity. The epitaxial relation is (001)[100] film (cid:107) (001)[100] MgO and (001)[110] film (cid:107) (001)[100]
CaF . More information for the structural analyses by x-ray diffractioncan be found in the Supplementary Information. As shown in Fig. 1a, the lattice constant a of theBa(Fe − x Co x ) As films on CaF substrates (Ba-122/CaF ) is shorter than that of the bulk samples [6],whereas the opposite relation holds for the Ba(Fe − x Co x ) As films on MgO substrates (Ba-122/MgO).As expected, an elongation of the c -axis for Ba-122/CaF and a shrinkage of the c -axis for Ba-122/MgOwere observed due to the Poisson effect, as shown in Fig. 1b (i.e., unit cell volume is roughly con-served). These structural changes are due to the biaxial strain with average uniaxial components oversample volume, (cid:15) xx = (cid:15) yy . Here, the average lattice deformations in the tetragonal phase are de-fined as (cid:15) xx = ( a film − a PLD target ) /a PLD target and (cid:15) zz = ( c film − c PLD target ) /c PLD target alongthe a - and c -axis. In this way, the biaxial in-plane tensile strain acts similarly to uniaxial pressurealong the c -axis, while the in-plane compressive strain acts as negative pressure along the c -axis. Theaverage lattice deformations at room temperature (RT) for Ba-122/MgO along the a - and c -axis are (cid:15) xx = 5 . × − and (cid:15) zz = − . × − , respectively. The corresponding values for Ba-122/CaF films are (cid:15) xx = − . × − and (cid:15) zz = 5 . × − . It is noted that the lattice deformation for bothfilms are almost constant irrespective of Co contents (Supplementary Fig. S3). The origin of the biaxialstrain is discussed in the Supplementary Information.Fig. 1c summarizes the As position ( z ) in the unit cell for the strained films and PLD targets. Theliterature data for single crystals are also shown in the same figure [7–10]. It is apparent that the Ascoordinate is nearly independent on strain and Co doping. c ( n m ) x in Ba(Fe x Co x ) As Ba-122 scBa-122/CaF Ba-122/MgO Ba-122PLD target a ( n m ) x in Ba(Fe x Co x ) As Ba-122/MgOBa-122/CaF Ba-122 sc Ba-122PLD target a b z x in Ba(Fe x Co x ) As Ba-122/CaF Ba-122/MgOBa-122 PLD targetBa-122 scz~0.354 c FIG. 1. Co doping dependence of lattice parameters: (a) In-plane lattice constant a of Ba(Fe − x Co x ) As thin filmson MgO and CaF substrates, Ba(Fe − x Co x ) As single crystals (Ba-122 sc) [6], and PLD target as a function of Codoping. 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 functionof the Co doping. The data of bulk single crystals are taken from Refs. [7–10]. The solid green line shows the averageAs position for unstrained samples. esistivity and phase diagram The evolution of the in-plane longitudinal resistivity ( ρ xx ) curves in zero magnetic field as a function oftemperature for the Ba(Fe − x Co x ) As films on MgO and CaF substrates are displayed in Figs. 2a and2b, respectively. The low-temperature state in the films on both substrates changes upon doping similarto the bulk material: from antiferromagnetic to superconducting, followed by metallic state [6]. However,the doping levels at which the phase transitions occur depend on the strain state. For Ba-122/MgO, Codoping of x = 0 . induces superconductivity with a T c of 7.5 K. Additionally a sudden drop of the Hallcoefficient around 100 K due to the SDW transition was observed (see Fig. 2c). Hence, for Ba-122/MgOwith x = 0 . superconductivity coexists with antiferromagnetism. On the other hand, Ba-122/CaF withthe corresponding composition did not show superconductivity down to the lowest temperature availablein our experiments (i.e., ∼ are con-structed, Figs. 3a and 3b. For comparison, the single crystal data are plotted in the same figures. Here, T c was determined by the superconducting onset temperature (Supplementary Fig. S7), whereas T N wasdefined as a peak position of the temperature derivative of the resistivity curves in analogy to bulk sin-gle crystals (Supplementary Information in the section of criterion for T N ) [12, 13]. It is noted that thepeak position of the temperature derivative of the resistivity is related to the magnetic transition accord-ing to x-rays and neutron diffraction measurements [12]. Zero resistivity temperature and middle pointof superconducting transition may be influenced by flux pinning effect. Therefore, we chose the onsettemperature of resistivity as a criterion of the T c . It is clear from Fig. 3a that tensile strain (Ba-122/MgO)slightly reduces T N and shifts the superconducting dome to lower doping levels compared to the singlecrystals. A similar shift of the superconducting dome by in-plane tensile strain was observed in P-doped ac db -2-1012 R H ( x - m / C ) T (K) x =0 x =0.75 x =0.02 x =0.1 x =0.04 x =0.15 x =0.06Ba-122/MgO -2-1012 R H ( x - m / C ) T (K) x =0 x =0.1 x =0.02 x =0.15 x =0.04 x =0.175 x =0.06 x =0.2 x =0.075 x =0.225Ba-122/CaF ρ xx ( m Ω c m ) T (K) x =0 x =0.1 x =0.02 x =0.15 x =0.04 x =0.06 x =0.075 Ba-122/MgO 0.70.60.50.40.30.20.10.0 ρ xx ( m Ω c m ) T (K) x =0 x =0.15 x =0.02 x =0.04 x =0.2 x =0.06 x =0.225 x =0.075 Ba-122/CaF FIG. 2. Transport properties of Ba-122/MgO and Ba-122/CaF thin films: Resistivity data for Ba(Fe − x Co x ) As thin films on (a) MgO and (b) CaF substrates. Broken lines are the fitting curves using ρ = ρ + AT n in theparamagnetic (PM) state. Hall coefficient of Ba(Fe − x Co x ) As films on (c) MgO and (d) CaF substrates as afunction of temperature. ) ef-fectively 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 struc-ture. At zero doping level, the ab-initio calculations show that compressive biaxial in-plane strain makesthe band structure more two-dimensional with good nesting (see the section Discussion), resulting ina higher AFM transition temperature. Tensile strain shows the opposite effect which makes the bandstructure more three-dimensional, and consequently T N decreases. For single crystals, a similar develop-ment of the FS takes place. Related angle-resolved photoemission spectroscopy (ARPES) measurementsshowed that upon Co doping the electronic states in the vicinity of the Fermi level become more three-dimensional [15]. Therefore, the two effects (charge doping and in-plane strain) determine the shift of thephase diagram along the doping axis.The temperature dependence of the resistivity for the films in the paramagnetic (PM) state was fittedusing ρ = ρ + AT n and the resultant fitting curves are shown in Figs. 2a and 2b. This expression has beenwidely used for analyzing the resistivity in the quantum critical region, e.g., [16, 17]. The dependence ofthe power-law exponent n on Co doping (i.e., x ) is presented in Figs. 3c and 3d. For Ba-122/MgO, theexponent n has a minimum value close to unity at x ∼ . . This may be assigned to the AFM quantumcritical point (QCP), where the AFM transition temperature goes to zero. For Ba-122/CaF , the QCP isobserved at x ∼ . (Fig. 3d). The presence of the AFM QCP for Co-doped Ba-122 has been proposedrecently by specific heat, thermal expansion, and nuclear magnetic resonance measurements [18–20]. Theobserved simultaneous shift of the QCP and the maximum T c for the strained thin films may suggest therelationship 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 strainin the thin films (Supplementary Information). Another evidence for the tetragonal crystal structure being FIG. 3. Electronic phase diagram of Ba(Fe − x Co x ) As : The electronic phase diagram of thin films grown on(a) MgO and (b) CaF substrates. For comparison, the single crystal data [6, 26] are also shown in the figures asdotted lines. T N and T c denote the antiferromagnetic and the superconducting transition temperatures, respectively.SDW, PM, and SC are the spin density wave, paramagnetic, and superconducting phases, respectively. Value forthe exponent n taken from the resistivity data ρ = ρ + AT n in the paramagnetic state: (c) Ba-122/MgO and (d)Ba-122/CaF . with the same Co doping level of x = 0 . in an applied field of 14 T at various temperatures is shownin Figs. 4a and 4b. As stated above, Ba(Fe − x Co x ) As thin films are grown on MgO(001) substrateswith cube-on-cube configuration, whereas the basal plane of Ba(Fe − x Co x ) As is rotated by 45 ◦ onCaF (001). We applied the current along the tetragonal [110] direction for Ba-122/CaF and along thetetragonal [100] direction for Ba-122/MgO, respectively. According to Refs. [13, 22], a magnetic fieldof B = 14 T parallel to the ab -plane can partially detwin Ba(Fe − x Co x ) As single crystals, leading toa 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-planeMR curves show the maximum values. On the other hand, the position of the peak is shifted by 45 ◦ , ifthe bias current ( I ) flows along orthorhombic [110] axis (zero angle corresponds to B (cid:107) I ). In this case,the peak values are much smaller than for the former geometry (i.e., current and magnetic field (cid:107) or-thorhombic [100] or [010]). However, our results contradict the one obtained from single crystals. Below100 K, the measured in-plane MR curves for both films clearly follow an almost perfect sinusoidal angledependence without phase shift. If the oscillation were defined by the nematic domains oriented by theapplied field as in the case of single crystals [13, 22], the MR signal for Ba-122/MgO should be shiftedby 45 ◦ with respect to the one for Ba-122/CaF . Additionally, the amplitude of MR signals for both filmsare quite small compared to those of single crystals. This indicates that the substrate completely blocksthe rotation of the nematic/magnetic domains. However, the appearance of oscillation in the MR at acertain temperature T + indicates some changes of the FS topology or scattering rates. The T + is ratherhigh compared to T N and preserved at doping levels above the QCP of the SDW phase. By analogy withRefs.[23, 24], T + may be related to the nematic phase or fluctuating magnetic domains. This temperatureis presumably increased by uniaxial strain if compared to relaxed Ba(Fe − x Co x ) As single crystals. -505 ∆ ρ / ρ x - ( % )
200 K150 K100 K70 K50 K
Ba-122/MgO( x =0.04) -202 ∆ ρ / ρ x - ( % ) φ (deg.)
300 K200 K170 K150 K 70 K
Ba-122/CaF ( x =0.04) a/ba/b φ H=14 TI a/ba/b φ H=14 TI cc a cb d FIG. 4. Angular dependence of in-plane magnetoresistance data: The angular dependence of in-plane magnetoresis-tance (MR) data ( ∆ ρ/ρ ) in the presence of a magnetic field (14 T) for (a) Ba-122/MgO and (b) Ba-122/CaF . Thesketch gives the orientation of the crystallographic axes for (c) Ba-122/MgO and (d) Ba-122/CaF in orthorhombicnotations. ffective carrier density plot of the phase diagram The temperature dependencies of the Hall coefficients ( R H ) measured at 9 T for Ba-122/MgO and Ba-122/CaF films are shown in Figs. 2c and 2d. For both parent compound films (i.e., x = 0 ), R H is weaklydecreasing with decreasing temperature until the SDW transition occurs, similarly to the observation insingle crystals [25, 26]. In contrast to single crystals, however, R H changes sign from negative to positive.This behavior can be understood qualitatively by the effect of strain on carrier mobilities. The non-dopedBa-122 is a compensated metal with equal electron and hole carrier densities. Therefore, a small changeof the mobilities can strongly affect the experimental value of the effective n H (especially in AFM statewith reconstructed Fermi surfaces). Only a small amount of Co addition to the system leads to a drasticchange in R H at low temperature. For all films with x = 0 . , R H is decreased sharply with decreasingtemperature below T N due to a large change in the carrier concentration and mobility. This behavior issimilar to that observed in single crystals [25, 26].In order to quantify the effect of strain on the electronic properties, we consider the effective carriernumber per Fe, n H ( T c ) and n H ( T N ) , as e | R H | × V , where V is the unit cell volume estimated fromFigs. 1a and 1b. Now, we re-plot T c and T N as a function of n H ( T c ) and n H ( T N ) , as shown in Figs. 5aand 5b. For comparison, the single crystal data from Refs. [6, 26] are also shown in the graph. As canbe seen, T c for both strained thin films and single crystals can be mapped onto a master curve by n H ( T c ) as a new variable. Note that n H scales both the position of the superconducting dome and the absolutevalue of T c , whereas the carriers numbers [27] and structural parameters (i.e., bonding angle and anionheight) [28] scale only either the position of the superconducting dome or the absolute value of T c .On the other hand, T N is vaguely independent of n H ( T N ) for non-zero doping, i.e., a magnetic transi-tion occurs, when n H ( T ) approaches about 0.05 carriers/Fe. DISCUSSION
The shift of the superconducting dome and the AFM transition temperature with uniform in-planebiaxial strain can be understood qualitatively by considering the effect of the strain on the FS shape, itsorbital weight, and composition. First of all, our local (spin) density approximation (L(S)DA) calculationshows 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
FIG. 5. Effective carrier density plot of the superconducting ( T c ) and N´eel ( T N ) temperatures of Ba(Fe − x Co x ) As :(a) Superconducting ( T c ) and (b) N´eel ( T N ) temperatures as a function of n H ( T c ) and n H ( T N ) . For comparison, Ba-122 single crystal data taken from Ref. [26] are also plotted. SDW and PM are the spin density wave and paramagneticphases, respectively. The labels show the characteristic range of n H in SDW and PM phases. T N correlates well with the value of the k z dispersion of the Fe d xz/yz orbitals on the hole FS pockets. As can be seen in Fig. 6, the dispersionalong k z of the undoped Ba-122 film on MgO increases compared to that of the undoped bulk sample(i.e., relaxed) due to tensile strain. T N 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 sametrend is observed for different Co doping but with fixed strain state. In contrast, compressive strain alone(i.e., Ba-122/CaF with fixed Co doping) reduces the k z dispersion of the Fe d xz/yz orbitals, whichleads to the enhancement of T N . The observed three-dimensional effects of the FS are responsible for thesuppression of the FS nesting conditions found by the ARPES study [29]. One can also see from Figs. 3aand 3b that the degree of shift in the superconducting dome along the doping axis correlates well with T N ; lower/higher T N pushes the SC dome towards the underdoped/overdoped region. Such a tendencymay be understood if Cooper pairing and magnetic ordering is controlled by a common parameter.The result of the band structure calculation is insufficient for the interpretation of the observed behaviorshown in Fig. 5b. The scaling indicates that the value of the effective carrier number n H ( T N ) at the phasetransition is not only related to the electronic structure but also strongly affected by critical fluctuationswhich are not included in the band structure calculations. As was shown by Kontani et al ., n H scales withthe antiferromagnetic (AF) correlation length ( ξ − ) in the case of strong AF spin fluctuations [30]. There-fore, the value of n H should approach zero at the phase transition. However, n H ( T N ) tends to a finite value ∼ . carriers/Fe (excluding films with zero doping level) at the magnetic transition as can be seen inFig. 5b. This seeming contradiction may be explained by considering the multiband nature of the FS. TheHall coefficient is defined as R H = σ xy /H ( σ xx σ yy ) , where σ xy and σ xx ( yy ) are the full conductivitiessummed up over all bands. Therefore, the Hall number is given by n H = | ( (cid:80) i σ h,i + (cid:80) j σ e,j ) (cid:80) i σ h,i /n h,i − (cid:80) j σ e,j /n e,j | ,where σ h ( e ) ,i ( j ) is the conductivity of a hole (h) or an electron (e) band, i or j is the summation indexrunning over different hole or electron bands, respectively, and n h ( e ) is the corresponding carrier den-sity [31]. Hence, as in the case of resistivity [32], the bands with the largest conductivities contributemost to the Hall number. At the phase transition, the conductivities of the interacting bands or part of thebands tend to zero due to strong scattering of their quasi-particles on the critical fluctuations. Therefore, n H is simply determined by those parts of the FS which are less sensitive to critical fluctuations. Thisexplains why the magnetic transition occurs at a finite n H value. Good scaling for T c is observed for eachside of the superconducting dome, since those regions are far from the magnetic transition lines. In thiscase, the value of n H ( T c ) is sensitive to the distance to the SDW line. Therefore, the scaling for T c canbe interpreted by the indication of a strong interplay between T c and the magnetic fluctuations slightlyabove T c , which are sensitive to the carrier doping and strain as well. A possible disorder effect on both T c and T N cannot be separated from the spin fluctuation effect, since impurity scattering is also includedin n H .In conclusion, we have shown that strain essentially affects the phase diagram of the generic systemBa(Fe − x Co x ) As . The biaxial in-plane strain is responsible for a nearly rigid shift of the whole phasediagram including the magnetic and superconducting regions along the electron doping. This behavioris explained by band structure calculations in which biaxial in-plane strain affects the FS similar to Codoping. Moreover, the superconducting dome is rigidly connected to the position of the SDW line. Thedirect relationship between the paramagnetic normal state and T N , as well as the relationship between T c and the preceding state above T c are given by the unusual plot of T N and T c with the Hall number atthose temperatures. This emphasizes a crucial role of the critical fluctuations for superconductivity andmagnetism in FBS. It is important to check whether a similar plot exists for other families of the FBS,8oo. Also, a microscopic explanation of the observed unusual behavior is still lacking. We believe thatour experimental results will stimulate future theoretical investigations. METHODSBa(Fe − x Co x ) As films on MgO(001) and CaF (001) substrates Ba(Fe − x Co x ) As films of around 100 nm thickness have been grown on MgO(001) and CaF (001)substrates by pulsed laser deposition (PLD). PLD targets made by a solid state reaction with various Colevels ranging from ≤ x ≤ . were ablated by a KrF excimer laser with a laser repetition rate of7 Hz. Prior to the deposition, the substrates are heated to 850 ◦ C. A base pressure of around 10 − mbarat 850 ◦ C is achieved, which increases to 10 − mbar during the deposition. 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(Fe . Co . ) As films (nominal composition x = 0 . )by energy dispersive X-ray spectroscopy, we determined the Co content to 0.74 ± Structural analyses by x-ray diffraction
The c -axis texture and phase purity were investigated by x-ray diffraction in Bragg-Brentano geometrywith Co-K α radiation. In-plane orientation of Ba-122/MgO and Ba-122/CaF was investigated by usingthe 103 pole in a texture goniometer operating with Cu-K α radiation. In order to precisely evaluatethe lattice parameter a of Ba-122/MgO and Ba-122/CaF , high resolution reciprocal space maps (RSM)around the 109 reflection were performed with Cu-K α radiation.Temperature evolution of the lattice constants c for Ba-122/MgO and Ba-122/CaF was investigatedby x-ray diffraction in Bragg-Brentano geometry with Cu-K α 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 Figs. 1aand 1b, and the optimized As position in the paramagnetic state, which is the same method as describedin Ref. [32]. Transmission electron microscopy (TEM)
The samples for TEM analysis were prepared using a FIB (SMI3050MS2) by cutting and milling theidentical films used for transport measurements. The microstructure near the interface of Ba-122/MgOwith x = 0 . was analyzed using JEOL JEM-2100F [Supplementary Fig. S5(b)].9 n-plane transport measurements Prior to the micro bridge fabrication, the temperature dependence of the resistance for all films wasmeasured by a 4-probe method, in which small pins are aligned co-linear on the film surfaces. Afterthe measurements, the films were photolithographically patterned and ion-beam-etched to fabricate asmall bridge of 100 µ m width and 0.41 mm length for transport measurements. No changes in transportproperties after the micro bridge fabrication have been found. Longitudinal and transverse resistance weremeasured 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 functionaltheory (DFT) calculations of the Fermi surface (FS) for both Ba(Fe − x Co x ) As z isalmost constant irrespective of both strain and Co doping. Therefore, the absolute As position is justproportional to the lattice parameters. A k -mesh of × × k -points in the whole Brillouin zonewas employed. The calculations were performed using the FPLO Ab-Initio Simulation Package withinthe Perdew, Burke and Ernzerhof (PBE) functional for the exchange-correlation potential. The calculatedhole FS for the Co doping level x = 0 and 0.1 are summarized in Fig. 6. FIG. 6. Fermi surface of Ba(Fe − x Co x ) As : Evolution of the Fermi surface (FS) of Ba(Fe − x Co x ) As at the Γ point as a function of Co doping and strain. The color code corresponds to a relative orbital weight per Fe-atom. Thedetailed theoretical approach can be found in Methods section. cknowledgement The authors thank Helge Rosner and Andrey Chubukov for fruitful discussions, Juliane Scheiter forpreparing FIB cuts, as well as Michael K¨uhnel and Ulrike Besold for their technical support. The researchhas received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) undergrant agreement numbers 283141 (IRON-SEA) and number 283204 (SUPER-IRON). K.I. acknowledgessupport by JSPS Grant-in-Aid for Challenging Exploratory Research Grant Number 15K13336. A partof the work was supported by the DFG under the projects SPP1458 and GRK1621. S. W. acknowledgessupport by the DFG under the Emmy-Noether program (Grant no. WU595/3-1).
Authors contribution
K.I. and V.G. designed the study and wrote the manuscript together with D.V.E. and S.-L.D. The PLDtargets were prepared by F.K., S.A., E.A. and S.W. Thin films were prepared by K.I. and F.K. K.I.,F.K., A.T. and R.H. conducted x-ray experiments. A.P., P.C., and W.S. analyzed local strain of thin filmsby high-resolution EBSD. A.I. and I.T. conducted TEM investigation. M.E. and I.M. have developedmicro bridge processing. K.I., F.K., V.G. and J.H. measured transport properties. D.V.E. and S.-L.D.developed a theoretical model and calculated the band structure. B.H., R.H., K.I., V.G., D.V.E., and S.-L.D. supervised the projects. All authors discussed the results and implications and commented on themanuscript at all stages.
ADDITIONAL INFORMATION
The authors declare no competing financial interests. Correspondence and requests for materials shouldbe addressed to K. I. and V. G. 11
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Phys. Rev. B , 1743-1757 (1990). upplemental InformationHall-plot of the phase diagram for Ba(Fe − x Co x ) As Structural characterization by x-ray diffraction
Our Ba(Fe − x Co x ) As (Ba-122) thin films on MgO(001) and CaF (001) substrates have been grownby pulsed laser deposition (PLD). The composition of the films is almost identical to that of theBa(Fe − x Co x ) As PLD targets, indicative of a successful stoichiometric transfer [S1]. X-ray diffractionpatterns for Ba(Fe − x Co x ) As thin films on MgO and CaF substrates as a function of the Co contentare summarized in Fig. S1. Almost all peaks are assigned as 00 l reflections of Ba(Fe − x Co x ) As andsubstrate, indicating a c -axis texture of Ba(Fe − x Co x ) As . For both cases, the 002 reflection of Fe isobserved. The 008 peak position shifts towards higher angles with Co doping (Figs. S1b and S1d), as aresult of a smaller lattice constant c with increasing Co content. φ scans of the 103 peak for Ba-122 films on both MgO and CaF are summarized in Figs. S2a andS2b, respectively. For both films, φ scans of the 220 substrate reflection were also measured. Sharp andstrong reflections from Ba-122 are observed at every 90 ◦ . These results highlight that Ba(Fe − x Co x ) As thin films are grown epitaxially. Here, the respective epitaxial relation for Ba-122 on MgO and CaF substrates are (001)[100] film (cid:107) (001)[100] MgO and (001)[110] film (cid:107) (001)[100]
CaF . The films on CaF (Ba-122/CaF ) with high doping regime ( x ≥ . ) contained a small amount of in-plane 45 ◦ rotateddomains. Log [I n t en s i t y ( a r b . un i t ) ] θ (deg.) x =0.150.10.060.040.020 008 reflection Log [I n t en s i t y ( a r b . un i t ) ] θ (deg.)
002 004 006 008
Fe002 M g O , M g O , x =0.150.10.060.040.020 a b Log [I n t en s i t y ( a r b . un i t ) ] θ (deg.) x =0.15 x =0.075 x =0.225 x =0
008 reflection
Log [I n t en s i t y ( a r b . un i t ) ] θ (deg.) x =0.15 x =0.075 x =0.225 x =0 C a F , F e , C a F ,
002 004 006 008 c d
FIG. S1. The θ − θ scan of the Ba(Fe − x Co x ) As thin films on (a) MgO(001) and (c) CaF (001) substrates. The θ − θ scans in the vicinity of the 008 reflection for (b) MgO(001) and (d) CaF (001) substrates, respectively. Aclear shift of the diffraction peak is observed with increasing Co content. shows smaller full-width at half-maximum values of both out-of-plane and in-plane re-flections compared to Ba-122/MgO, as shown in Figs. S2c and S2d. Here, the rocking curve of the 004reflection was measured for all Ba-122 films. The ∆ ω and average ∆ φ values are mainly constant re-gardless of Co content. ∆ φ ( deg . ) x in Ba(Fe x Co x ) As
103 reflectionBa-122/CaF Ba-122/MgO ∆ ω ( deg . ) x in Ba(Fe x Co x ) As
004 reflectionBa-122/CaF Ba-122/MgO I n t en s i t y ( a r b . un i t ) φ (deg.) x =0 x =0.15 x =0.175Ba-122/CaF CaF I n t en s i t y ( a r b . un i t ) φ (deg.) Ba-122/MgO x =0 x =0.06 x =0.15 MgO a bdc
FIG. S2. 103 φ -scans for Ba-122 thin films with different Co levels on (a) MgO and (b) CaF single crystallinesubstrates. The 220 φ -scans for MgO and CaF substrates are also shown in the same graphs. (c) Full width athalf maximum ( ∆ ω ) value of the 004 reflection of Ba-122 thin films as a function Co content prepared on varioussubstrates. (d) Average full width at half maximum ( ∆ φ ) value of the 103 reflection of Ba-122 thin films as a functionCo content prepared on various substrates. Lattice deformation in a tetragonal phase for both Ba-122/MgO and Ba-122/CaF is summarized inFigs. S3a and S3b. Clearly, the lattice deformation for both films are almost constant regardless of Cocontent. -1.0-0.50.00.51.0 ε xx ( x - ) x in Ba(Fe x Co x ) As -1.0-0.50.00.51.0 ε zz ( x - ) c -axis a -axisBa-122/CaF -1.0-0.50.00.51.0 ε xx ( x - ) x in Ba(Fe x Co x ) As -1.0-0.50.00.51.0 ε zz ( x - ) a -axis c -axisBa-122/MgO a b FIG. S3. The lattice deformation in a tetragonal phase along the crystallographic a - and c -axis for Ba-122 thin filmson (a) MgO and (b) CaF substrates. rigin of biaxial strain The origin of the biaxial strain in thin films may be understood by considering the evolution of thelattice constants c with temperature. The temperature dependence of x-ray diffraction patterns in thevicinity of the 004 reflection of Co-doped Ba-122 films with the same Co doping, x = 0 . , but grownon different substrates MgO and CaF , are presented in Figs. S4a and S4b. Our measurements have beenconducted in flowing He gas. For both Ba-122 films, the diffraction peaks are shifted toward lower angle,indicative of the elongation of the lattice constant c due to thermal expansion. For Ba-122/MgO, thediffraction intensity is observed to decrease around 673 K and almost disappeared at 773 K, indicatingthat the Ba-122 phase was decomposed. On the other hand, for Ba-122/CaF films the diffraction peakwas still observed even at 773 K, although the peak height is significantly reduced. From the temperaturedependent x-ray measurements, the lattice parameters a of MgO and CaF substrates have also beenevaluated, as shown in Fig. S4c. The evaluated values of the linear thermal expansion coefficients, α ,are given in Table S1. Based on those results, the temperature dependencies of the lattice constants c for the two films are presented in Fig. S5a. For comparison, the temperature dependence of the c -axislength of a Ba-122 single crystal is also shown. These data were calculated using the experimentallattice constant at room temperature and the thermal expansion coefficient for the single crystal with x = 0 . [S2]. The lattice constants c for the Ba-122/CaF and the single crystal are close to each otherat high temperature, indicating that the respective in-plane lattice parameters a are also close to each otherat that temperature. Upon cooling, the difference in the c -axis length between the Ba-122/CaF and thesingle crystal increases, which is attributed to a large difference in the thermal expansion coefficients ofthe CaF substrate and Ba-122 (Table S1). This effect is mainly responsible for the compressive strain of (cid:15) xx = − . × − in Ba-122/CaF thin films. Log [ i n t en s i t y ( a r b . un i t ) ] θ (deg.) MgO298 K423 K573 K673 K773 K
Log [ i n t en s i t y ( a r b . un i t ) ] θ (deg.)
298 K423 K573 K673 K773 KCaF a b L / L K T (K) L / L
298 K =0.993 +22.4X10 -6 T CaF L / L
298 K =0.996 +12.7X10 -6 T MgO a ( n m ) CaF MgO c FIG. S4. Temperature dependence of the x-ray diffraction patterns in the vicinity of the 004 reflection for (a) Ba-122/MgO and (b) Ba-122/CaF . (c) Temperature dependence of the lattice constants a for MgO and CaF substratesand the corresponding normalized values at 298 K. On the other hand, the two traces of the Ba-122/MgO film and the single crystal are almost parallel,presumably due to a small thermal expansion mismatch (Table S1). Unlike Ba-122/CaF films [S3, S4], aclean interface between the Co-doped Ba-122 film and the MgO substrate has been observed by transmis-sion electron microscopy, as shown in Fig. S2b and reported in Ref. [S5]. In this case, the lattice misfityields a tensile strain in the thin films. However, the measured magnitude of the strain, (cid:15) xx = 5 . × − ,is smaller than expected for the relatively large lattice misfit of around -6 % at room temperature. This3 ABLE S1. The linear thermal expansion coefficient, α , of the MgO and CaF substrates at 298 K. The value forBa(Fe . Co . ) As along the crystallographic a -axis at 300 K was taken from Ref. S2. MgO CaF Ba(Fe . Co . ) As α ( × − K − ) 12 . . . larger difference is caused by strain relaxation since our Ba-122 films have a thickness of about 100 nm,which is beyond the critical thickness for relaxation. Indeed, our previous investigations on the Ba-122/Febilayer system revealed a critical thickness of around 30 nm for a lattice misfit of -2.5% [S6], whereas thecorresponding value of Ba-122/MgO results in a few atomic layers [S5]. Therefore, the presence of asmall amount of biaxial strain, (cid:15) xx = (cid:15) yy = 5 . × − , indicates that residual strain exists beyond thecritical thickness, which was also observed in III-V semiconductors [S7] and P-doped Ba-122 films onMgO substrates [S8, S9]. Additionally, we have found nanoscale oscillation of uniaxial strain compo-nents using high resolution electron backscatter diffraction (HR-EBSD) [S10], which will be discussedlater. This strain state can be described by the modulation of (cid:15) xx and (cid:15) yy having opposite sign in the rangeof about ± . % with (cid:15) zz ∼ . We assume that this strain component originates from the formation of alow-angle grain boundary network during the coalescence of slightly rotated nanoscale islands nucleat-ing on the mismatched MgO surface during film growth. These strain inhomogeneities were detected byHR-EBSD using a comparable experimental and evaluation procedure provided in reference [S11]. c ( n m ) T (K)DepositiontemperatureBa-122/MgOBa-122/CaF Ba-122sc
MgO Ba-122[001] a b
FIG. S5. Temperature dependence of the out-of-plane lattice constants and microstructure: (a) Temperature depen-dence of the lattice constants c for a Ba-122/MgO and a Ba-122/CaF film with x = 0 . . The single crystal data(dotted lines) were estimated from Ref. [S2]. (b) TEM picture in the vicinity of the interface between the Co-dopedBa-122 film ( x = 0 . ) and the MgO substrate. The local strain by HR-EBSD
Local strain components in a Co-doped Ba-122 ( x = 0 . ) thin film on MgO were measured by highresolution electron backscatter diffraction (HR-EBSD). HR-EBSD was performed as line scans in a ZeissUltra 55 scanning electron microscope using 20 kV acceleration voltage and 10 nm step size. The EBSDpatterns have been recorded and analyzed subsequently with an in-house written software, based on thealgorithm developed by Wilkinson et al [S10]. Shown in Fig. S6 is the local strain distribution of the4rystallographic a , b , and c direction (normal strains) relative to a chosen reference position as a functionof position. Variations of strain for all directions are within ± . %. S t r a i n ( % ) µ m) Reference point a , b , c ∆ a / a ∆ b / b ∆ c / c FIG. S6. The local strain distribution of the crystallographic a , b , and c direction as a function of position. Determining the superconducting transition temperature, T c ρ ( m Ω • c m ) T (K) T c =27.1 K FIG. S7. The resistivity curve of a 10 % Co-doped Ba-122 thin film on a CaF substrate. T c was determined to27.1 K. The superconducting onset transition temperature was defined as the intersection between the linear fitof the normal state resistance and the steepest slope of the superconducting transition (see Fig. S7). Zeroresistivity temperature and middle point of superconducting transition are influenced by flux pinningeffect. Therefore, we chose the onset temperature of resistivity as a criterion of the T c . Our optimallyCo-doped Ba-122 superconducting films showed an exact match of the zero resistivity temperature T c , and the onset T c from magnetization measurements, proving high quality of our films [S1]. Determining the magnetic transition temperature, T N The peak position of the temperature derivative of the resistivity is related to the magnetic transitionaccording to x-rays and neutron diffraction measurements [S12]. Therefore, the magnetic transition tem-5erature, T N , was defined as the peak position, as shown in Fig. S8. In contrast to bulk single crystals (i.e.,unstrained material), the kink above T N related to the structural/nematic transition is absent. Recently,a similar behavior was observed under application of uniaxial strain to the ab -plane of Co-doped Ba-122 single crystals [S13]. Therefore, the uniaxial component in the films obscures the nematic/structuraltransition (Fig. S6). However, inhomogeneous strain does not affect T N and T c noticeably, since thetransitions are rather sharp in the strained films. d ρ xx / d T ( a r b . un i t ) T (K) Ba-122/CaF T N T N x =0 x =0.02 x =0.04 x =0.06 T N d ρ xx / d T ( a r b . un i t ) T (K) x =0 x =0.02 x =0.04 T N T N Ba-122/MgO a b
FIG. S8. The temperature derivative of the resistivity curves of (a) Ba-122/MgO and (b) Ba-122/CaF . The peakposition is assigned as T N . ∗ Electronical address: [email protected] † Electronical address: [email protected][S1] Daghero, D. et al.
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