Introduction

A bulk magnetic transition in a quantum material is thermally-driven by spin interactions dictated by nano- and mesoscale structures, such as lattice composition, crystal structure, disorder and defects, lattice strain, chemical impurities and dopants. The parents of the iron-based superconductors with AFe2As2 (A = Ba, Sr, Ca, Eu) formula, known as ‘122’, are semi-metallic with all five Fe 3d-bands crossing the Fermi level1. These materials have a tetragonal structure at room temperature (a = b ≠ c), and at a Néel antiferromagnetic transition temperature (TN) there is a small tetragonal-to-orthorhombic structural distortion (Ts) where the unit cell rotates by ~45° within the ab-plane2,3. Below TN there is a sinusoidal modulation of Fe moments in the form of a spin-density wave (SDW)3 described by a wave vector q = (½ ½ 1) in the tetragonal structure, matching the nesting vector between the electron and hole pockets at the Fermi surface4. The Fe spin lattice is a ‘stripe’ arrangement (spins are antiparallel along a- and c-axes, and parallel along b-axis)5,6,7,8.

It has been found that strain can induce a nematic phase transition in 122 s9 that is seen as an electronic in-plane anisotropy10, driven by orbital (unequal occupation of dxz and dyz) or spin directional order (not long range) that causes breaking of the in-plane C4 rotational symmetry and splits the TN and Ts transitions. Moreover, higher TN in BaFe2As2 is linked to a more homogenous electronic structure11,12,13,14, and there are local structure variations3. Thermally annealing of BaFe2As2 is found to shift and sharpen the heat capacity anomaly15. With disorder via electron irradiation in BaFe2(As1−xPx)2 the magneto-structural transition is suppressed and the superconducting dome tracks the shift of the antiferromagnetic phase16, while in Ba1-xKxFe2As2 the antiferromagnetic and superconducting transition temperatures decrease17. For SrFe2As2, in addition to a large variability of TN values (195 to 220 K)6,13,18, superconducting signature (filamentary Tc = 21 K) can be found in strained crystals19. Furthermore, CaFe2As2 can be synthesized as entirely non-magnetic, or an antiferromagnet with a large TN, achieved by staggered alleviation of local Fe−As bonds with thermal annealing13,20,21. For this study, we hypothesize that the local lattice details, including disorder, of EuFe2As2 may affect its TN value. EuFe2As2 is unique among the 122 s for having both SDW order of Fe, and Eu local moment order below ~20 K with q = (0 0 L)22,23,24,25. The Eu local moments are weakly coupled to the Fe sublattice and are strong enough for a large and indirect spin-lattice coupling that can lead to a structural detwinning by a small magnetic field26 and can cause re-entrant non-bulk superconductivity27,28.

We have synthesized two stoichiometric EuFe2As2 crystals with onset of Fe ordering at TN = 195 K or TN = 175 K, and use X-ray and neutron diffraction, microscopy and spectroscopy techniques, and theory, to understand the reasons that give these magnetic ordering temperatures. Surprisingly, the higher TN crystal has a broader (usually associated with strain and disorder) specific heat peak, while the lower TN crystal has a sharp peak. How does TN relate to lattice and topological features, and does TN value correlate with averaged lattice parameters? Would higher TN mean more homogeneous electronic structure? What are the pressure results for diminishing antiferromagnetism and potentially deriving superconductivity in each of these crystals? Our results show that although TN(Fe) and Tc values are greatly sensitive to the lattice details and local arrangements of defect structures, Eu ordering is unaffected at TN(Eu) ≈ 20 K. This study demonstrates the apparent contradictory result that certain types of disorder can significantly increase the magnetic order, even in stoichiometric quantum material.

Experimental Procedure and Results

Single crystals of EuFe2As2 were grown out of a mixture of Eu and FeAs excess used as liquid flux12,29,30. Each of these mixtures was warmed to 1180 °C, then cooled (2 °C/h) followed by a decanting of the FeAs excess flux at 1090 °C. Two different reaction loading ratios were used to obtain EuFe2As2 single crystals with different TN. A loading ratio of Eu:FeAs = 1:5 gives an onset transition temperature of TN = 195 K (referred to as ‘crystal a’), while Eu:FeAs = 1:4 gives crystals with TN = 175 K (‘crystal b’). The chemical composition of these crystals was measured with a Hitachi S3400 scanning electron microscope operating at 20 kV. Three spots (each ∼80 μm area) were checked and averaged on each crystal; energy-dispersive X-ray spectroscopy (EDS) and site-refinement of single-crystal X-ray diffraction analyses indicate that both crystals are stoichiometric and are EuFe2As2. The phase purity of the crystals was checked by collecting data on an X’Pert PRO MPD X-ray powder diffractometer; structures were solely identified as tetragonal ThCr2Si2 structure type (I4/mmm, Z = 2).

Specific heat data were collected on EuFe2As2 single crystals, using a Quantum Design Physical Property Measurement System (PPMS); the C(T) results are shown in Fig. 1a. Each EuFe2As2 crystal exhibits two transitions: an ordering of the Fe lattice at higher temperature, followed by lower temperature ordering (≈20 K) due to Eu moments. The EuFe2As2 ‘crystal a’ exhibits the higher onset ordering temperature of TN = 195 K with a broader peak, compared to ‘crystal b’ giving a sharp lambda transition below TN = 175 K. The specific heat result of ‘crystal a’ looks similar to that reported in a pressed polycrystalline sample (with respect to peak width, height, and transition temperature)22.

Figure 1
figure 1

Data on EuFe2As2 ‘crystal a’ and ‘crystal b’. (a) Specific heat, C(T), results with transitions associated with Eu (TN = 21 K) and Fe ordering (TN = 195 K or 175 K). (b) Precession images showing the [HK0] and [H0L] reciprocal lattice planes reconstructed from single crystal X-ray diffraction patterns measured at room temperature. Smearing in diffraction spots are more prevalent in ‘crystal a,’ noted in [H0L] plane.

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To analyze crystal structures, single crystal X-ray diffraction data were collected at room temperature on a Rigaku XtaLAB PRO diffractometer (Mo Source, Kα = 0.71073 Å) equipped with a DECTRIS Pilatus 200 K area detector. Data collection and reduction used the CrysAlisPro program31. Crystal structure refinements were performed using the SHELX-2014 program32. Figure 1b (insets) shows the pictures of crystals for the data collection, stuck on top of needles. Crystals ‘a’ and ‘b’ appeared similar visually, with sheet morphologies; [001] direction is perpendicular to the plane of the plate. Figure 1b gives the precession images of [HK0] and [H0L] reciprocal lattice planes, showing lattice strain effects associated with smeared diffraction spots in the L direction for ‘crystal a’; this crystal has a relatively broad C(T) peak (TN = 195 K) from the Fe ordering. A broader peak of specific heat is usually associated with lattice strain and disorder. Table 1 shows the lattice parameters and refinement results of the X-ray structure measured at room temperature. The EuFe2As2 sample with higher TN (‘crystal a’) gives a lower overall a-lattice parameter and a shorter planar Fe−Fe distance of 2.7692(2) Å, indicating a contraction of 0.0053(4) Å for the in-plane Fe−Fe bond when compared to that of 2.7745(3) Å for the lower TN (‘crystal b’).

Table 1 Single-crystal X-ray diffraction refinement on the two crystals of EuFe2As2: ‘crystal a’ (TN = 195 K), ‘crystal b’ (TN = 175 K).
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Neutron diffraction measurements were carried out using the TOPAZ single crystal diffractometer at the ORNL Spallation Neutron Source (SNS), which uses the wavelength-resolved Laue technique with an extensive array of neutron time-of-flight area detectors for 3-dimentional Q space mapping of Bragg and diffuse scattering patterns originating from magnetic and nuclear phase transitions33. Pieces of ‘crystal a’ and ‘crystal b’ had dimensions of 3.25 × 2.50 × 0.20 mm3 and 2.75 × 2.63 × 0.30 mm3, respectively, and were attached on a MiTegen loop using super glue for data collection at room temperature and 95 K. Each of the samples was oriented with high precision for volumetric sampling of Bragg peaks in specific directions with the CrystalPlan program34,35. At room temperature, the (0 2 0)Tetragonal peak profiles for the two EuFe2As2 crystals are shown in Fig. 2, demonstrating the peak false-color maps and their corresponding peak intensity profiles along the crystallographic c-direction. The peak from ‘crystal a’ is broad and shows extensive diffuse lines along the L direction, which is consistent with that observed in X-ray diffraction, and gives broader and higher TN in C(T).

Figure 2
figure 2

Tetragonal (0 2 0) peak profiles for the two EuFe2As2 crystals at room temperature for (a) ‘crystal a’ and (b) ‘crystal b’, showing the peak false color maps, and their corresponding peak intensity profiles along the crystallographic c direction. Note the difference in scales on both axes. The peak from ‘crystal a’ is substantially broadened showing extended diffuse lines.

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As shown in Fig. 1b, the streaks for the (2 0 0) peak along L direction for ‘crystal a’ is much more pronounced (Fig. 2a) than that of ‘crystal b’ (Fig. 2b), which could be induced by strain, stacking fault or disorder in the bulk single crystal sample. Since the refined average c lattice parameter remains essentially unchanged for both crystals (Table 1), the much more pronounced streaks for ‘crystal a’ are likely from microstrain caused by randomly distributed defects in ‘crystal a’, which is evident from scanning tunneling microscopy/spectroscopy (STM/S) measurement (see Fig. 3 below). The split of the peaks along c also indicates twinning and multidomain structures of ‘crystal a’. In particular, we note from Fig. 2 (bottom panels) that the full-width at half maximum (FWHM) for the (2 0 0) peak along L direction is some 0.13–0.27 reciprocal lattice units larger than that of ‘crystal b.’ If one assumes that in the ordered ‘crystal b’ the finite (i.e. non-zero) FWHM is largely due to measurement resolution limits, the FWHM for ‘crystal a’ would correspond to a ~2–5% local variation in the lattice constant c.

Figure 3
figure 3

(a) The structure model of EuFe2As2 and the surface reconstruction models of arsenic 2 × 1 and europium √2 × √2 terminations. The black boxes outline the unit cells. (b,c) Topographic images of 2 × 1 surface reconstruction from ‘crystals a’ (−500 mV, 100 pA) and ‘crystal b’ (−1 V, 100 pA), respectively. (d,e) Topographic images of √2 × √2 surface reconstruction from ‘crystals a’ (−60 mV, 400 pA) and ‘crystal b’ (−20 mV, 800 pA), respectively. Insets show the atomic resolved images in 3 × 3 nm2 size and black boxes are the unit cells of √2 × √2 surface reconstruction. (f,g) Comparison of average LDOS from ‘crystal a’ (red) and ‘crystal b’ (blue), for surface reconstructions at 4.2 K.

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For these crystals, neutron diffraction results were additionally carried out at 95 K, to confirm the Fe magnetic peak below TN in accordance with the propagation vector q = (1 0 1)Orthorhombic = (½ ½ 1)Tetragonal and the stripe arrangement of the Fe spin lattice5,6,7,8 (see Fig. S1a in Supplementary). Additionally, variable temperature scans of the Eu magnetic peak at (0 3 0) for both crystals were performed below 30 K using the elastic diffuse scattering spectrometer CORELLI at SNS32 and showed the same temperature dependence of Eu magnetic peak as (0 3 0) with TN = 21 K (see Fig. S1b). In contrast, the Néel transition temperature for Fe is clearly lattice and disorder sensitive. The high-temperature phase transition is due to the ordering of the transition metal Fe moments, whereas the low-temperature phase transition is due to the ordering of the localized Eu moments22. The divergent behavior of the Fe and Eu sublattices from ‘crystal a’ to ‘b’ – the former sublattice sensitive to lattice disorder, the latter insensitive – is of interest here. One plausible hypothesis is that the Eu 4 f moments are much more localized, near the Eu nucleus, than the Fe moments that are more itinerant. This would suggest that the Eu 4 f moments are less subject to nano-strain than the Fe 3d moments, and the relatively isolated location of Eu between the layers (as opposed to the tightly bound FeAs layers) would enhance this possibility. Supporting this assertion is the relative insensitivity of Eu ordering temperature to pressure (see Fig. S4), again unlike the Fe sublattice ordering. Indeed the Eu magnetic order survives to higher pressure than the Fe order, despite occurring at much lower temperatures.

To further investigate the local origin of the bulk TN differences between the two EuFe2As2 crystals, surface topography and electronic structures were investigated using STM/S on in-situ low-temperature cleaved surfaces. The two sets of EuFe2As2 ‘crystal a’ and ‘crystal b’ were cleaved in an ultra-high vacuum at ~ 78 K and then immediately transferred to the STM/S head precooled to 4.2 K without breaking vacuum. The STM/S experiments were carried out using a scanning tunneling microscope with base pressure better than 2 × 10−10 Torr, with mechanically cut Pt-Ir tip. All Pt-Ir tips were conditioned on clean Au (1 1 1) and checked using the topography, surface state, and work function before each measurement. The STM/S were controlled by the SPECS Nanonis control system. Topographic images were acquired in constant current mode with bias voltage applied to samples, and tip grounded. All the spectroscopies were obtained using the lock-in technique with a modulation of 1 mV at 973 Hz on bias voltage, dI/dV. Current-imaging-tunneling-spectroscopy were collected over a grid of pixels at bias ranges around Fermi level using the same lock-in amplifier parameters. The survey on multiple large areas of both samples shows the coexistence surface reconstructions on ‘crystal a’ and ‘crystal b’, as shown in Fig. 3. Using a similar method as STM report on Co-doped BaFe2As236, the 2 × 1 and √2 × √2 reconstructed surfaces can be assigned to arsenic termination and europium termination, respectively, as shown in Fig. 3a. While the 2 × 1 surfaces of both crystals (Fig. 3b,c) are very similar, the √2 × √2 surfaces of the two crystals (Fig. 3d,e) are very different. The atomic resolved images in the insets of Fig. 3d,e from the well-ordered √2 × √2 reconstructed areas of the two crystals are similar, but the arrangements of the defects on the surfaces are rather different. In ‘crystal a’, surface defects (one of the defects is marked with a black arrow in Fig. 3d) are essentially randomly distributed on the surface, but in ‘crystal b’, large amount of defects prefer to form into chains (one of the chains is marked with a white arrow in Fig. 3e). By analyzing atomically resolved images around these areas, we found the chains are aligned on the antiphase boundaries. Because the surface reconstruction is 2 × 1 or √2 × √2, those domains can shift by 1 to form antiphase boundaries. Although antiphase boundaries also exist on the ‘crystal a’ surface (one of the antiphase boundaries is marked with a green arrow in Fig. 3d), defects in ‘crystal a’ do not segregate along the boundaries. The electronic properties revealed by the STS from the surfaces are consistent with the morphological observation (see also Fig. S2). In Fig. 3f, the average electronic local density of states (LDOS) of 2 × 1 surfaces (As termination) over large areas are almost identical for both crystals, but the LDOS from the two √2 × √2 surfaces (Eu termination) in Fig. 3g have significant differences. This shows that the different arrangements of the surface defects in the two crystals change the electronic structure dramatically. Given the preparation of these surfaces by cleaving the crystals in ultra-high vacuum at low temperatures, this difference is in electronic structure that is generally reflective of the effects of the defects in the bulk crystals.

For both EuFe2As2 crystals, high-pressure electrical-resistance measurements were performed using a diamond anvil cell (see Figs. S3 and S4), to explore the differences of pressure effects on the two crystals that have different TN, under the same experimental setup. Although the feature due to Eu ordering is not changed for either crystals up to ~4 GPa, Fe ordering is greatly sensitive to pressure and the rate of TN suppression for both crystals is similar. For ‘crystal b’ with smaller TN = 175 K, the drop in resistivity is noticed at lower pressure of 2.5 GPa, compared to ‘crystal a,’ with a drop in resistivity at 3.2 GPa. EuFe2As2 with sharper, but lower TN, is prone to a higher superconducting dome (S4); highest Tc value for ‘crystal a’ is 36 K, and for ‘crystal b’ is 41 K.

First Principles Calculations

In the effort to understand the sample-to-sample variation in Néel temperatures, we have conducted first principles theory calculations of the effect of nano-scale strain on the magnetic order, specifically on two questions. The first question is straightforward: is the observed variation in TN related to the slight difference in planar lattice constants in crystals ‘a’ and ‘b’? We assess based on calculations of the ‘ordering energy,’ specifically the difference in energy between the stripe ground-state and the checkerboard (all neighboring moments are anti-aligned) excited state. The second question, with more complex interpretation, is whether c-axis strain is relevant to the observed TN variation. The motivation for studying this question is the evident ‘shoulder’ in the neutron diffraction results presented in Fig. 2a, which could result from c-axis nano-scale strain, and thereby an effective distribution of local c-axis lattice parameters. Note that the bulk lattice constants for crystals ‘a’ and ‘b’ are identical (to within 0.001 Å), so that we are studying, in effect, the local nanoscale energetics rather than constructing an involved model of the bulk Néel point variation for a sample with a distribution of local lattice constants.

First principles calculations were performed using the linearized augmented plane wave (LAPW) density functional theory code WIEN2K37. As in our previous work14 we have used the local density approximation (LDA), with arsenic height and lattice parameters taken directly from our experimental XRD refinements (Table 1), except as stated below. An RKmax value of 8.0 – the product of the smallest sphere radius and largest planewave expansion wavevector – was used. To avoid the generally confounding complexities of 4 f physics (here we are focused on the Fe magnetic behavior) we have followed our previous theoretical work38 and performed the all-electron calculations with the isoelectronic substitution of Sr for the divalent Eu, but retained our lattice parameters and internal coordinates for EuFe2As2. The effect of these choices is to allow a direct evaluation of the Fe sublattice magnetic properties without the need to apply a correlated (i.e. LDA + U) approach to deal with the Eu 4 f electrons, which play little or no role in the Fe magnetism. Sphere radii of 2.5 Bohr were used for Sr, and between 2.26 and 2.30 for Fe (depending on volume), and between 2.15 and 2.22 for As. Identical sphere radii were used for the calculations at given volume, relevant for assessing the interlayer coupling.

With regards to the first question of whether changes in the magnetic ordering energy, or energetic difference between the stripe ground state and checkerboard excited state, result from the small difference in planar lattice parameters, we find only a small change (0.4%) in this energy, from 51.6 meV/Fe for ‘crystal a’, to 51.8 meV/Fe for ‘crystal b’. This means that the observed ~ 10% Néel temperature variation does not result from changes in this ordering energy. At the same time, one should note that the observed 0.2% change in lattice parameter from ‘crystal a’ to ‘b’ corresponds to an effective chemical pressure of several tenths of a GPA, and pressure is a well-known means of TN suppression. While it remains possible that this effective planar strain is relevant here, arguing against this is the fact that the smaller lattice parameter ‘crystal a’ has the larger Néel point, whereas the application of pressure, thereby yielding smaller lattice constants, generally lowers the TN.

Concerning the second question, the neutron scan of ‘crystal a’ along the [00 L] direction shows clear evidence of disorder, with a rather broadened peak around the (0 0 0) point and a substantial ‘shoulder’ feature extending nearly to the [001] point (Fig. 2a). While there are many possible sources of such disorder, ranging from vacancies and anti-site defects to small grain sizes, here we explore the idea that c-axis strain in ‘crystal a’ is ultimately responsible for the rather broad neutron diffraction peaks. The hypothesis here is that nanoscale c-axis strain could significantly change the interlayer magnetic coupling, since this coupling is known from previous work39 to depend rather substantially on pressure, or equivalently compression or expansion of the cell from the equilibrium value. The interlayer coupling is of course necessary to attain a finite ordering temperature, given the Mermin-Wagner finding that truly low-dimensional systems do not order at finite temperature. Since, as in many materials, the interlayer coupling here is relatively weak, it is plausible that nanoscale c-axis strain could change the coupling, which generally falls off quickly with increased distance.

We have therefore conducted calculations of this interlayer magnetic coupling. Here we report the energy difference, per unit cell, between the ground-state stripe magnetic order (which is antiferromagnetically coupled along the c-axis) and a state with ferromagnetic c-axis coupling. We have done this at the experimental lattice parameter of 12.104 Å, and in addition for c values 5% larger and smaller than this value. For the experimental c-lattice parameter, we find an interlayer coupling energy difference of 7.1 mRyd/u.c., which increases slightly to 7.7 mRyd for the +5% c value, but decreases sharply to some 4.5 mRyd for the −5% c value. These values are in qualitative agreement with the previous work on BaFe2As239, where it was found that the application of pressure decreases the interlayer coupling. We mention also that the calculated magnetic moments decrease substantially for the smaller cell, and correspondingly increase for the larger cell. In the antiferromagnetically coupled layer state, for the smallest cell these moments are 1.25 μB per Fe, for the experimental cell the value is 1.63 μB, and for the largest cell is 1.92 μB. Neutron diffraction results give an Fe moment of ~1 μB8, consistent with the general overstatement of magnetic order by first principles approaches in these materials.

These data suggest that interlayer strain can affect the magnetism in a substantial manner. In this scenario, the broadened specific heat peak in ‘crystal a’ would result from a distribution of local lattice parameters, and thereby a distribution of local interlayer exchange couplings, resulting in a spread of ordering points. However, one difficulty with this argument is that such a distribution would have a larger effect on the temperature width of the transition than the mean transition temperature, since the bulk c-axis lattice parameters in crystals ‘a’ and ‘b’ are identical (leaving aside such questions as the temperature dependence of the magnetic correlation length in such a disordered scenario). Furthermore the fact that the strained calculations show larger decreases in magnetic order (measured as the interlayer coupling) with compressive strain than increases with tensile strain suggests that the effect of a distribution of c-axis lattice parameters around the bulk value would be a decrease in Néel point, not an increase. If the reverse were true – that the increases in magnetic order with one form of strain were larger than the decreases with the opposite strain – one would in fact have a possible explanation for the TN enhancement. However, the calculations do not suggest this ladder possibility, and thus we must consider, as unsupported the hypothesis, that strain-related interlayer-magnetic exchange differences create the observed TN increase in ‘crystal a’.

Since the straightforward explanations of the TN increase, based on disorder-induced structure modification, do not explain this rather unusual behavior, it is necessary to consider other possibilities. For example, in materials near a magnetic instability, it has recently been posited40 that charge doping can induce a magnetic transition based on Stoner physics, and one could plausibly imagine a similar enhancement in magnetic character of an already magnetic material. However, the generally stoichiometric character of both samples studied here argues against such a possibility. An additional possibility is provided by the theoretical work of Gastiasoro et al.41, which finds (for superconductivity, and extended to magnetic order) density-of-states increase from either resonant states or Anderson localization leading to enhancements in ordering points. We here suggest a different possibility: spin fluctuations are known to be exceptionally strong in the iron arsenides, and in fact are a leading candidate for the interaction causing superconductivity. It is commonly believed42 that these fluctuations also play an important role in reducing the Néel point of these materials from a much higher ‘bare’ value. It is also known from several recent theory works43,44 that disorder can play a substantial role in weakening spin fluctuations and inducing magnetism, and this role is especially prominent45 in the case of stripe magnetic order, as is observed in the iron-arsenides. Putting such arguments together, we suggest that the disorder evident in our specific heat and neutron data is weakening the spin fluctuations and thereby enhancing the magnetic order.

One means of testing this theory would be to employ the quasi-particle interference techniques recently applied to LiFeAs46. While in that work it was the superconductivity (and its coupling to a bosonic excitation) that was studied, we anticipate that a measure suitably adapted to the case of the antiferromagnetic order here could also yield insight. If reduced spin fluctuations are in fact at work here, the momentum and energy dependent differential conductance g(k, ω) should show signatures of reduced scattering in the higher TN. In particular, generally sharper (as a function of momentum and energy) peaks are expected in the conductance in this material, when measured at the same temperature as the lower Néel point crystal. Future experimental studies would be highly beneficial are to assess this possibility.

Conclusion

We find that disorder-related lattice variability drives significantly different Fe Néel ordering temperatures in the stoichiometric EuFe2As2 crystals, with the highly unusual result that the more locally disordered crystal exhibits the higher Néel point. The diffuse scattering that is seen along the [00 L] direction in X-ray and neutron diffraction measurements for the higher TN crystal may well be the origin of the broadened specific heat peak and the suppressed superconducting dome. The surface topography and electronic structure studies show that there is a clear difference in electronic and defect structures between the two TN crystals, with defect states dominating and elevating the LDOS for the higher TN ‘crystal a’. Although the two crystals have a similar number of defects, their segregation around the antiphase boundaries in ‘crystal b’ largely decreases the number of random individual defects, which may cause a higher-Tc superconductor with pressure. This study thereby demonstrates the modification of TN in an iron arsenide by controlled disorder, thus explaining the observation of substantial ordering temperature variations for these stoichiometric quantum materials in the literature.