Anomalous critical fields and the absence of Meissner state in Eu(Fe_0.88Ir_0.12)₂As₂ crystals

The as-grown crystals were easily cleaved along the basal planes, displaying shiny metallic luster on the surface. Indeed, the XRD pattern of the crystal flake lying on the sample holder shows only (00l) reflections, as shown in figure 1. The full-width at half-maximum (FWHM) is less than 0.15°, indicating good crystallinity. The c-axis parameter, calculated by a least-squares fit, is c = 12.037(2) Å, obviously smaller than that of the undoped EuFe₂As₂ (12.136 Å [12]). Similar result was observed in Ru-doped EuFe₂As₂ [10]. The exact Ir content was determined by EDXS, which gives the chemical formula of Eu(Fe_0.88(4)Ir_0.12(1))₂As₂, consistent with the 1:2:2 stoichiometry expected.

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Figure 2 presents the temperature dependence of in-plane resistivity for the Eu(Fe_0.88Ir_0.12)₂As₂ crystal (sample A). No resistivity anomaly associated with the Fe-site SDW ordering can be seen. Furthermore, ρ_ab(T) shows a linear temperature dependence in a broad temperature range above T_sc. These data strongly suggest that the Ir-doping level is in the optimal regime, since the linear temperature dependence of normal-state resistivity is only found for samples with optimal doping in the related systems such as Sr(Fe_1−xIr_x)₂As₂ [31] and Ba(Fe_1−xCo_x)₂As₂ [35]. A sharp superconducting transition appears below T^onset_sc = 22.2 K, and zero resistance is achieved at 21 K. No re-entrant resistance was observed below T_sc. For the crystals with x = 0.10 and 0.13, re-entrant resistance was detected, such as the case of the Eu(Fe_1−xIr_x)₂As₂ polycrystalline samples [33]. This result further indicates that the x = 0.12 crystal is optimally doped with regard to SC. The superconducting transition width, conventionally defined by ΔT_sc = T(90%ρ_n) − T(10%ρ_n), where ρ_n is the normal-state resistivity just above T_sc, is as small as 0.7 K, suggesting good quality of the sample with neat homogeneity.

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Figure 3 shows the result of in-plane magneto-resistance measurement. As expected, the superconducting transition shifts downwards upon increasing magnetic fields. For H∥c, the decrease in T_sc is more pronounced under a magnetic field higher than 2 T, qualitatively consistent with the superconducting anisotropy in a layered system described by the Ginzburg–Landau theory [36]. However, the T_sc value of H∥c is anomalously higher than that of H∥ab for μ₀H < 1.5 T, as shown in figure 3(c).

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By using the criteria of 90% of ρ_n which determines the transition temperature under a magnetic field [T_c(H)], we obtained the temperature dependence of upper critical fields H_c2(T) shown in figure 3(d). The values of the initial slope, μ₀ dH^c_c2/dT|_T=Tsc = −0.79(7) T K⁻¹ and μ₀ dH^ab_c2/dT|_T=Tsc = −0.68(7) T K⁻¹, are much smaller than that of the analogous superconductor SrFe_1.5Ir_0.5As₂ (− 3.3 T K⁻¹) [31], Ba(Fe_0.9Co_0.1)₂As₂ (− 4.9 T K⁻¹) [37] and other iron-based superconductors [38]. The suppression of H_c2 in Eu(Fe_0.88Ir_0.12)₂As₂ can be ascribed to the indirect exchange interactions between Eu²⁺ spins (via Fe 3d electrons) which produces a strong internal field, although the Eu-spin ordering temperature is 2 K lower (it is also noted that the external field tends to enhance the FM of the Eu²⁺ spins [8, 19]). By a rough linear extrapolation, the upper limits of μ₀H^c_c2(0) and μ₀H^ab_c2(0) are estimated to be 15.7(3) and 18.9(5) T, respectively. Compared to μ₀H_c2(0) ∼ 50 T for Ba(Fe_0.9Co_0.1)₂As₂ with the same T_sc = 22 K [37], the upper critical field of Eu(Fe_0.88Ir_0.12)₂As₂ was about 30 T smaller at zero temperature. Similar suppression of H_c2(T) was also reported in the pressure-induced superconductor EuFe₂As₂ (T_sc = 30 K under 2.5 GPa) [39], which shows μ₀H^c_c2(0) = 22 T and μ₀H^ab_c2(0) = 25 T. In a zeroth-order approximation, which assumes that SC in the FeAs layers and FM in the Eu sublattice are decoupled, the decrease in H_c2 is simply ascribed to the internal exchange field. This point is supported by the ¹⁵¹Eu Mössbauer spectroscopy which indicates a hyperfine field of 25–30 T on the Eu nuclei in the EuFe₂As₂-related systems [14, 23]. The decoupling nature between SC and FM has been ascribed to the multiorbital feature of Fe 3d electrons, which simultaneously allows SC mainly from the d_yz and d_zx orbitals and FM due to an effective Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions among the Eu²⁺ spins mediated mainly through the d_x²−y² and d_z² electrons [22]. The extremely high intrinsic upper critical field in iron pnictides guarantees the survival of SC in the presence of the internal exchange field.

It is noted that the upper critical fields show abnormal temperature dependence featured by a convex curvature just below T_sc = 22 K, followed by a concave curvature. Thus H_c2(T) curves have an inflection point at ∼20 K, which is clearly seen in the derivative of H_c2(T) (figure 3(e)). This anomalous H_c2(T) behavior is related to the magnetic ordering of Eu²⁺ spins (to be discussed in the next section).

The anisotropy ratio of upper critical fields, defined by γ = H^ab_c2/H^c_c2, is about 1.3 at low temperatures, resembling other 122-type Fe-based superconductors [37, 40]. However, the temperature dependence γ(T) is qualitatively different. Normally, γ increases monotonically with temperature [37, 40], but here for Eu(Fe_0.88Ir_0.12)₂As₂, γ decreases significantly for T > 13 K and, even an inverse anisotropy (γ < 1) appears around 20 K. This anomalous γ(T) behavior is likely to be associated with the re-orientation of the internal field. Generally, the internal magnetic field (H_in) includes a dominant exchange field (H_ex) plus a spontaneously magnetized one, H_in = H_ex + 4πM [41]. Assuming that the internal field has two components H^c_in and H^ab_in, under a zeroth-order approximation, the intrinsic upper critical fields in the absence of internal field (denoted as H*_c2) should be H^c*_c2 = H^c_c2 + H^c_in and H^ab*_c2 = H^ab_c2 + H^ab_in. Thus the apparent anisotropy ratio is

$$\begin{equation} \gamma=\frac{H_{{\rm c}2}^{ab}}{H_{{\rm c}2}^c}=\frac{H_{{\rm c}2}^{ab*}-H_{\rm in}^{ab}}{H_{{\rm c}2}^{c*}-H_{\rm in}^c}=\frac{\gamma_*-H_{\rm in}^{ab}/H_{{\rm c}2}^{c*}}{1-H_{\rm in}^c/H_{{\rm c}2}^{c*}}, \end{equation} \tag{ 1 }$$

where γ_* = H^ab*_c2/H^c*_c2. Since the in-plane internal field is along the a-axis in the parent compound [12, 13, 34], H^c_in could be much less than H^ab_in or H^c*_c2 (thus H^c_in/H^c*_c2 ≪ 1) near T_m. In this situation, γ ≈ γ_* − H^ab_in/H^c*_c2, which explains the anomalously small γ-value around 20 K. At lower temperatures, the Eu-spin re-orientates toward the crystallographic c-axis [14, 18, 23], which means that H^c_in is comparable to, or even higher than H^ab_in. Therefore, γ recovers to a value larger than 1.0 at low temperatures.

Figure 4 shows the temperature dependence of magnetic susceptibility for Eu(Fe_0.88Ir_0.12)₂As₂ crystals under an external field of 10 Oe. For the ZFC susceptibility with the field along c-axis (χ^c_zfc), a diamagnetic transition occurs below T_sc = 22.1 K, consistent with the above resistivity measurement. However, χ^c_zfc increases sharply with temperature decreasing below 20 K, suggesting a strong magnetism opposing the superconducting diamagnetism. Below 17.4 K, χ^c_zfc decreases steadily. The superconducting magnetic shielding fraction is as large as 600% at 2 K. If the demagnetization factor (N_d ∼ 0.84) is taken into account, the corrected shielding fraction 4πχ(1 − N_d) is close to 100%, suggestive of bulk SC. For the FC susceptibility (χ^c_fc), however, no diamagnetic signal is present at all. Instead it increases with decreasing temperature until being 'saturated' at 17.4 K. The susceptibility with the field along the basal plane (χ^ab) shown in figure 4(b) indicates no signature of superconducting transition for both FC and ZFC modes, similar to that observed in Eu(Fe_0.75Ru_0.25)₂As₂ crystals [10]. This phenomenon suggests that the Eu²⁺ spins tend to lie flat and ordered within the basal plane around T_sc, which suppresses SC. In addition, no magnetic shielding (i.e. no diamagnetic signal in the ZFC mode) effect appears for H∥ab down to 2 K, which reflects absence of zero resistance for the current along the c-axis, consistent with the 'anisotropic SC' reported in [10].

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Here, let us discuss the possibility of the Eu-spin ordering. In an analogous EuFe₂P₂ ferromagnet, a ferromagnetic transition appears at 29 K, followed by another magnetic transition at 26 K featured by the canting of Eu moments toward the c-axis [42]. The FM as well as the spin canting at low temperatures was confirmed by neutron diffraction [17]. Similar magnetic ordering is anticipated in Eu(Fe_0.88Ir_0.12)₂As₂. Figure 4(c) shows the temperature dependence of anisotropy in χ_fc. The anisotropic ratio increases steeply below 20 K, suggesting that the Eu²⁺ spins start to lie flat in the basal plane. The Curie–Weiss fitting using equation χ = χ₀ + C/(T − θ) with the high-temperature susceptibility data (figure 4(d)) gives an effective moment of 8.0 μ_B f.u.⁻¹ (consistent with the Eu²⁺ spin state of S = 7/2), and a paramagnetic Curie temperature of θ = 22 K. The positive θ value, which is close to the magnetic ordering temperature, suggests a ferromagnetic exchange interaction between the Eu²⁺ spins. We thus propose that the Eu moments lie ferromagnetically in the ab plane below T^ab_m = 20 K. At 17.4 K, a clear kink is seen for χ^ab and χ^ab_fc/χ^c_fc. This kink can be understood as a consequence of Eu-spin tilting or canting [14, 15, 19] as observed in many EuFe₂As₂-related system. However, the canted AFM [18, 19, 33], which is expected to show pure AFM behavior for H∥ab, is unlikely because χ^ab increases below T^tilt_m (note that χ^ab of EuFe₂As₂ crystals decreases steeply below the magnetic ordering temperature [34]). Besides, the FM-like hysteresis in magnetization with the field along the basal plane (see figure S1 in the supplementary data (available from stacks.iop.org/NJP/15/113002/mmedia)) also suggests FM component in the basal plane, which is incompatible with the canted AFM. The 'saturation' in χ^c_fc below T^tilt_m = 17.4 K (figures 4(a) and (d)) is probably due to the formation of antiparallel ferromagnetic domains that are difficult to be 'magnetized'. The peak at a lower temperature ∼15 K seems to be related to the freezing of the magnetic (micro)domains, resembling the recently reported re-entrant spin-glass behavior in the EuFe₂(As_1−xP_x)₂ system [43]. Note that the heat capacity data (not shown here) show an abrupt jump at T^tilt_m rather than at T^ab_m, suggesting that the full magnetic ordering occurs below T^tilt_m, and the magnetic ordering at T^ab_m is basically two dimensional in nature. Indeed, the M(T) curves for different fields parallel and perpendicular to the basal plane (see figure S2 in the supplementary data) show that T^ab_m increases with magnetic field. Meanwhile, T^tilt_m decreases with the field for both H∥c and H∥ab, which can be explained by spin tilting/canting. It is pointed out that this scenario of spin re-orientation fully supports the above explanation for the anomalous anisotropy in upper critical field for T > T^tilt_m (figure 3(f)).

Since T^ab_m and T^tilt_m are respectively 2 and 4.6 K lower than T_sc, superconducting Meissner effect should be detected in the temperature range of T^tilt_m < T < T_sc for the applied field parallel to the c direction. Indeed, a significant Meissner signal (magnetic expulsion in the paramagnetic background) is present under an ultra-low magnetic field (H < 0.25 Oe), as shown in figure 5. For T < T^ab_m, however, χ_fc increases with decreasing temperature, indicating a spontaneous field³ that penetrates the superconductor. According to the discussions above, the internal field is much larger than the intrinsic lower critical field, i.e. H_in ≫ H*_c1. This means that the Meissner state is absent below T^tilt_m even under zero external field. Nevertheless, the Meissner effect, which describes a phenomenon where the applied magnetic field is expelled from a superconductor during its transition to superconducting state, is reflected from the evidence that the susceptibility at low temperatures is smaller for lower external field applied. This can be seen in figure 5(d), which shows relatively strong 'diamagnetic' signal by subtracting χ^0.25Oe_fc from χ^0.1Oe_fc. This effect was also reported in Eu(Fe_0.89Co_0.11)₂As₂ system [8].

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The isothermal magnetization curve [M(H)] below T_m (or T_sc) reflects its magnetic (or superconducting) characteristics for a ferromagnet (or a superconductor). For an FMSC, both characteristics could be present. Figure 6(a) shows the M(H) data at 4 K in a broad range of magnetic fields parallel to the c-axis for the Eu(Fe_0.88Ir_0.12)₂As₂ crystal. Indeed, the M–H loop is basically featured by a ferromagnetic-like magnetization superposed by a type-II superconducting loop. The equilibrium magnetization (M_eq), calculated by averaging the field-up magnetization (M_up) and field-down one (M_dn), is similar to the as-measured M(H) curve of the analogous ferromagnets EuFe_1.8Ni_0.2As₂ [16] and EuFe₂P₂ [42] where SC was absent. The saturated magnetization M_s = 6.9 μ_B f.u.⁻¹ is consistent with the ferromagnetic ordering of the Eu²⁺ spins (the theoretical ordered moment is gJ = gS = 7.0 μ_B f.u.⁻¹). Since the ferromagnets EuFe_1.8Ni_0.2As₂ and EuFe₂P₂ show only tiny magnetic hysteresis with a coercive field of 10–30 Oe [16, 42] that can be ignored in a rough approximation, we made a subtraction of M_eq from M_up (or M_dn), shown in figure 6(b). The resultant ΔM(H) indeed resembles a magnetization loop of a non-ideal type-II superconductor with strong flux pinning. Therefore, our low-T magnetization data unambiguously indicate the coexistence of SC and FM in Eu(Fe_0.88Ir_0.12)₂As₂. The steeper increase in ΔM_c below the saturated field H_sat can be explained below. Since the effective field along the c-axis can be expressed as H* = H^c_in + H = H^c_ex + 4πM_c + H, the slope of ΔM_c is

$$\begin{equation} \frac{\partial(\Delta M_{c})}{\partial H}=\frac{\partial(\Delta M_{c})}{\partial H^{*}}\frac{\partial H^{*}}{\partial H}=\frac{\partial(\Delta M_{c})}{\partial H^{*}}\left(1+4\pi \frac{\partial M_{c}}{\partial H}\right). \end{equation} \tag{ 2 }$$

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When H < H_sat, |∂(ΔM_c)/∂H| > |∂(ΔM_c)/∂H*| and for H > H_sat, |∂(ΔM_c)/∂H| = |∂(ΔM_c)/∂H*|.

To examine the detailed magnetization in the superconducting state, the M(H) curve in the field range of −1 < H < 1 kOe is shown in figure 7. The initial magnetization curve M_ini(H) shows the magnetic shielding effect. However, unlike common type-II superconductors, M_ini(H) has no linear region, and the lower critical field H_c1 cannot be defined⁴. Similar observation was also found in Eu(Fe_0.75Ru_0.25)₂As₂ crystals [10]. Furthermore, the equilibrium magnetization M_eq does not show a typical peak-like anomaly associated with the transition from the Meissner state (0 < H < H_c1) to the mixed state (H_c1 < H < H_c2). This simply points to the loss of Meissner state in the present system, consistent with the above low-field susceptibility measurement. The M_ini(H) slope at zero-field limit also agrees with the low-field magnetic susceptibility data above.

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In general, the Meissner state will be lost if the internal magnetic field H_in surpasses the intrinsic lower critical field H*_c1. Considering the strong internal field which suppresses H_c2 significantly, the absence of Meissner state in the Eu(Fe_0.88Ir_0.12)₂As₂ FMSC is not surprising. Here we should mention that the loss of Meissner state was previously reported in the superconducting ferromagnets UCoGe [44] and RuSr₂GdCu₂O₈ [45]. These superconducting ferromagnets with T_m > T_sc show much lower ferromagnetic internal field.