The in-plane electronic anisotropy or electronic nematicity in iron-based superconductors (FeSCs) has triggered tremendous research interest because of their physical properties and possible connection to the microscopic mechanism of high-T_c superconductors.^[1–10] It largely surpasses the orthorhombic lattice distortion and has unexpected effect on the electronic properties of FeSCs, similarly in copper-oxide high-T_c superconductors.^[11] The tetragonal-to-orthorhombic structural transition associated with the nematic order transition would be expected to broke the C₄ rotation symmetry and result in the in-plane anisotropy of resistivity.^[12] However, due to the formation of twin domains below the structural transition, the intrinsic electronic anisotropy is usually obscured for transport measurements in as-grown samples. To explore the nematicity correctly, we must first detwin the sample by applying uniaxial pressure or strong magnetic field to achieve a real orthorhombic state.^[13–15]

EuFe₂As₂ is a peculiar parent compound of iron pnictides. In addition to the colinear antiferromagnetic order in the FeAs layers with T_N ≈ 188 K, it exhibits an A-type antiferromagnetic structure of the Eu²⁺ local moments when further cooling down to low temperatures below T_Eu ≈ 19 K (Fig. 4).^[16–18] Bulk superconductivity up to about 30 K can be induced by hydrostatic pressure,^[19] isovalent substitutions,^[20,21] and charge carriers dopings.^[22–24] Previous study revealed that magnetic fields within 1 T can fully detwin EuFe₂As₂ single crystal, much smaller than that for BaFe₂As₂.^[15,25] This could be attributed to the biquadratic coupling between Fe²⁺ and Eu²⁺ spins.^[26] Below T_Eu, a small magnetic field either along a or b direction can detwin the EuFe₂As₂ sample, while above T_Eu, only a field parallel to a direction can detwin this compound.^[15] In addition, the magnetic field could induce a metamagnetic transition with the Eu²⁺ local moments evolved from antiferromagnetic (AFM) to ferromagnetic (FM) order.^[15,25,27] Therefore, Eu based iron pnictides offer a unique platform to investigate the interplay of unconventional electronic properties, lattice, and local magnetism.

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. Magnetization results of twinned EuFe2As2. Field dependence of magnetization with the H parallel to ab plane at various temperatures (a) and the corresponding d M/H (b). The expanded M(H) plot at 4 K is shown in the inset of (a). The red and blue arrows represent the measuring setup with increasing field and decreasing field, respectively. Inset of (b) shows the magnetic phase diagram for twinned EuFe2As2, the data about TCurie gets from the neutron scattering result.[25] (c) Sketch of the magnetic field dependent twin distribution and spin configuration of EuFe2As2 below TEu.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Resistivity anisotropy of detwinned EuFe2As2. (a)–(c) Temperature dependence of in-plane resistivity (ρa, ρb) with no field, the field parallel to b, and the field parallel to c direction, respectively. The resistivity is measured by using Montgomery method. The uniaxial strain cell is shown in the inset of {(a)}. (d) The temperature dependent resistivity anisotropy.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. The in-plane magnetoresistance results of detwinned EuFe2As2. (a)–(c) The MR results with current I ∥ a while the magnetic field is parallel to a,b and c directions, respectively. (d)–(f) The corresponding MR results with I ∥ b.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. EuFe2As2 crystal and magnetic structure (Eu atoms (yellow) and spins (blue), Fe atoms (green) and spins (red), and As atoms (celadon)), and the spin configuration of Eu2+ dependent on an external magnetic field along a (a), b (b), and c direction (c), respectively. The black arrows are the applying external magnetic field and the length of the black arrows means the magnitude of the magnetic field. The gray arrows are the original spins of Eu2+.

In fact, the unique AFM order of the EuFe₂As₂ system favors a large in-plane electronic anisotropy, especially below T_Eu.^[6,13,28] Such property is very sensitive to magnetic field and thus can be easily tuned by the field strength and direction, which would be helpful to understand the relationship between magnetism and superconductivity in iron pnictides. Magnetoresistance (MR) is a very useful probe to investigate this issue by directly response to the coupling between itinerant electrons and local moments in magnetic materials.^[27,29,30]

Here, we report the in-plane magnetotransport results of EuFe₂As₂ single crystal which is fully detwinned by using uniaxial pressure. The MR behaviors simply follow the realignment of the Eu²⁺ spins when increasing the magnetic field along different crystallographic directions. Specifically, MR shows step-like behaviors as the spin configuration of Eu²⁺ is suddenly changed from AFM order to FM order by increasing the external magnetic field along a direction. Whereas, MR changes gradually as the spins of Eu²⁺ evolve gradually from AFM order to FM order with increasing external magnetic field along b or c direction. More interestingly, the sign of MR seems to be locked by the current direction below T_Eu under in-plane magnetic field, it is always negative when the current is parallel to a direction (I ∥ a) but almost positive when I ∥ b, which closely resembles to the anisotropic MR in La_0.7Sr_0.3MnO₃ films.^[31] Our results give clear evidence of the coupling between itinerant electrons and the local moments of Eu²⁺ in EuFe₂As₂, and thus provide a new way to tune the in-plane anisotropy as well as the electronic nematicity in FeSCs.

The single crystals of EuFe₂As₂ were grown by the self-flux method.^[21,27] The orientations of the crystals were determined by the x-ray Laue method and the samples were cut into 3.4 mm × 3.0 mm × 0.6 mm thin plates along ab direction in orthorhombic configuration. Resistivity measurements were performed on a Quantum Design physical property measurement system (PPMS@DynaCool-14T) at Beijing Normal University by using Montgomery method^[32] with high-quality EuFe₂As₂ single crystals. Besides, a 4-probe method was also executed on the same samples in order to verify the results of Montgomery method.

The magnetization results on the twinned samples were firstly presented. In Fig. 1(a), we applied a magnetic field within ab-plane and measured M(H) loops at different temperatures. Field-induced metamagnetic transition associated with the reorientation of local magnetic moments of Eu²⁺ is clearly observed in the magnetically ordered compounds. For example, the system is in an AFM ground state at 4 K, The magnetization displays a linear dependence on the in-plane magnetic field at low field [inset of Fig. 1(a)]. As the field increases, a partially polarized magnetic (PPM) state of Eu²⁺ arises at 0.6 T. This transition could be clearly observed in χ(H) = d M/d H as a sharp peak which is defined as H₁ [Fig. 1(b)]. The magnetization around H₁ exhibits a small hysteresis with the magnetic field [inset of Fig. 1(a)], implying a first-order transition. In the PPM intermediate state, the susceptibility displays a plateau between H₁ and H_crit. Beyond H_crit, the system evolves into a fully polarized magnetic state (FPM) or FM state. As the temperature increases above T_Eu, both H₁ and H_crit disappear which indicates a breakdown of the long-range magnetic order of Eu²⁺. The T–H phase diagram is shown in the inset of Fig. 1(b). T_Curie is the transition temperature of Eu²⁺ spins from paramagnetic order (PM) to ferromagnetic order. Because of the small magnetic hysteresis, the rapid increase in M around H₁ is unlikely due to a spin-flop transition, it is most likely a metamagnetic transition. The small hysteresis in our results is similar to previous magnetization measurement in Ref. [27], but it is unlike that in Ref. [15] in which there is almost no hysteresis in M(H). Irrespective of the small hysteresis, the magnetic field dependent M shows an abrupt jump in Ref. [15] which is associate with the spin-flip. As the magnetic field increases, the local moments flip from one easy axis to another equivalent easy axis without subsequent reorientation. Such spin flipping effect has been identified by infrared spectroscopy^[15] and neutron scattering experiment^[25] in EuFe₂As₂. The spin-flip indicates that anisotropic energy is larger than Zeeman energy in this system, even though both of them are responsible for the observed spin–lattice coupling.^[25]

As aforementioned the persistent detwinning effect could be induced by external magnetic field in EuFe₂As₂. Combining with the previous study,^[15] to a certain extent, H₁ and H_crit can reflect the spin-flip process and the detwinning process when increasing the magnetic field, as shown in Fig. 1(c). For H = 0, the crystal is twinned and the domains are equally distributed, the Eu²⁺ spins are ordered in A-type antiferromagnetism below T_Eu with the moments directions along a axis, as shown in the first panel of Fig. 1(c). After applying an external field, twin B with b ∥ H gets energetically favored and then grows. The spins of Eu²⁺ in twin B begin to deviate from their original direction to the direction of the external field, while the spins in twin A are robust and their directions are still maintained. With a further increasing magnetic field, the Eu²⁺ spins in twin A flip along the external field direction and the spins in twin B further deviate from their original direction. When the field exceeds H_crit, all Eu²⁺ spins are along the external field, and twin A and twin B merge together to a single FM domain. We note that Eu²⁺ is an S-state rare-earth ion possessing 4f⁷ electrons with M_sat = gS = 7.0 μ_B/f.u. for g = 2 and the total electron spin S = 7/2. M finally saturates to 6.9 μ_B/f.u. above 1.0 T at 4 K in our experiment, which is very close to the theoretical prediction.

Figure 2 shows the in-plane resistivity [Figs. 2(a)–2(c)] and the resistivity anisotropy [Fig. 2(d)] in detwinned EuFe₂As₂. The resistivity anisotropy is defined as δ = (ρ_b – ρ_a)/(ρ_a + ρ_b). Figure 2(a) shows the zero field measurements of in-plane resistivity. Linear temperature dependent ρ_a and ρ_b are found above the structure transition T_s and the AFM transition T_N (T_s ≈ T_N ≈ 188 K). Resistivity anisotropy is clearly visible already at temperatures above T_s [Fig. 2(d)]. This is very similar to the nematic fluctuations observed in other iron pnictides.^[1,2,13] As the temperature decreases, an obvious rise of ρ_b around T_s and a rapid decline of ρ_a are observed. As the temperature is lowered further, small kinks appear both in ρ_a and ρ_b at 19 K which are corresponding to the formation of AFM order of the Eu²⁺ spins. Figures 2(b) and 2(c) are the temperature dependent ρ_a and ρ_b with external field applying along b and c directions, respectively, where such kinks vanish due to the full suppression of AFM order of the Eu²⁺ spins when H = 3 T. We would like to emphasize that neither pressure nor magnetic field has much impact on T_s and T_N since they remain at 188 K for all measurements. We compare the resistivity anisotropy under different field geometries in Fig. 2(d). The maximum resistivity anisotropy is about 17 % around T_s. In the AFM ordered state of iron pnictides, the resistivity anisotropy is only approximately associated with uniaxial strain and detwinning ratio in the partially twinned state,^[32] the resistivity anisotropy of 17 % indicates that the sample is well detwinned. All the resistivity anisotropy increases dramatically at T_s and then decreases with temperature decreasing. The resistivity anisotropy in field is only a little smaller than the zero field case below T_s. However, they cross around T_Eu, this is likely due to the detwinning effect of the external field in the EuFe₂As₂ system.

The in-plane MR in the detwinned EuFe₂As₂ is illustrated in Fig. 3 which is defined as MR = (ρ(T,H) – ρ(T,H = 0))/ρ(T,H = 0). Figures 3(a)–3(c) are the MR results with current I ∥ a while the field is parallel to a, b, and c directions, respectively. Figures 3(d)–3(f) are the corresponding MR with current I ∥ b. In Fig. 3(a), the MR exhibits different behaviors below and above T_Eu. At 4 K, MR increases slightly as the magnetic field increases and reaches a maximum and then it has a step-like decrease with increasing the magnetic field further. The magnitude of MR drops to –4 % at H_crit ≈ 0.7 T and then gradually increases with increasing external field. Both the small positive MR at low field and the sudden drops of MR around the critical fields are only observed below T_Eu, which indicate that the itinerant electrons are intimately related to the change of ordering of the Eu²⁺ spins in this system as discussed below. The initial positive MR at the low magnetic field could be explained by the theory proposed in Ref. [33], in which a positive MR in the AFM state could be ascribed to that the conduction electrons are scattered by local magnetic moments. The abrupt decrease of MR is associated with the abrupt spin reorientation transition of the Eu²⁺ moments from AFM to FM configuration and could be understood through the superzone boundary effect.^[34] This phenomenon has been already found in some other rare-earth transition metal compounds.^[35,36]

In the I ∥ a with temperature below T_Eu case, the MR with external field along b and c directions [Figs. 3(b) and 3(c)] has two major differences in comparison to the H ∥ a case [Fig. 3(a)]. Firstly, there is not any initial small positive MR at the low magnetic field. Secondly, around the critical field, the MR does not decrease suddenly but almost decreases linearly as the magnetic field increases. Such differences could be attributed to different Eu²⁺ spins’ rearranged models. For the external field parallel to a direction case, the Eu²⁺ spins do not change below the critical field, they only flip suddenly at the critical field and then the spin moments are parallel to the direction of the external field [Figs. 1(c) and 4(a)]. For the external field parallel to b or c direction case, the Eu²⁺ spins start to rotate at the initial external field, and then the Eu²⁺ spins enter a PPM intermediate state. After the field exceeds the threshold value, all Eu²⁺ spins are completely realigned with FM configuration along b or c direction [Figs. 1(c),4(b), and 4(c)]. Figure 4 shows how the Eu²⁺ spins change with increasing magnetic field of different directions. Once applying the external field along b or c direction, the Eu²⁺ spins are in the PPM state rather than the real AFM order. That is the reason why we did not observe the positive MR at the initial field. With further increasing magnetic field, the MR decreases linearly, which is consistent with that the spins of Eu²⁺ reorientate gradually along the direction of the external field. We note that, above the critical field, the low temperature MR in the FM state starts to increase with increasing magnetic field. This could be explained as follows.^[34] The increased magnetic field will suppress the spin fluctuation in a FM system, and thus lead to the decreasing MR. However, if the cyclotron motion of the conduction electron plays a role, it will enhance the MR as the magnetic field increases.^[37] The charge carriers scattering time and cyclotron frequency are two critical parameters in determination of the contribution of cyclotron motion to the MR. The carriers scattering time becomes longer at low temperature and the cyclotron frequency increases within increasing external field. Therefore, under low temperature and high magnetic field, the cyclotron motion of charge carriers is the dominating contribution to MR. As a result, the MR increases linearly after the spins of Eu²⁺ enter the FM order. We can get the slope by fitting the linear increase of MR. In Fig. 3(b), the slope decreases as the temperature increases from 4 K (red dashed line) to 10 K (blue dashed line), it is a direct result of cyclotron motion on MR due to the increase of temperature. We also note that the slopes in Figs. 3(a) and 3(c) are smaller than that in Fig. 3(b) even though at the same temperature, that means the cyclotron motion has stronger influence on MR when the magnetic field is along b direction. The slopes vanish at 10 K in Figs. 3(a) and 3(c) within our measurement range, which also supports this argument.

Next, consider the I ∥ b case [Figs. 3(d)–3(f)]. MR becomes positive below T_Eu when applying external field within ab plane. This is quite different with that when I ∥ a. However, the changing trends of MR show similar behaviors around the critical field no matter what the current direction is. The typical features of MR could be the result of different orientate models of Eu²⁺ spins regardless of the current direction and the applying external field direction. A positive MR is clearly observed once the temperature is higher than T_Curie in Fig. 3(d). With applying the external field perpendicular to ab plane, the situation becomes much complicated below T_Eu [Fig. 3(f)]. If we neglect that, we can also find the common characteristic of all MR. When the temperature is above T_Eu and below T_Curie, all MR is negative and increases with increasing temperature at 3 T. This is consistent with that the magnitude of the Eu local moments keeps decreasing with increasing temperature in the FM state. The MR effect is not detectable at high temperature due to the oscillation of MR in our experiment.

To verify the availability of data by using the Montgomery method, we also measured MR by using the 4-probe method on the same detwinned samples [Fig. 5(a)]. The result is almost the same as that in Fig. 3(e). Thus, both methods give out the same MR result. We also measured the MR on the twinned EuFe₂As₂ [Fig. 5(b)]. The positive MR at the initial magnetic field is larger than that in Fig. 3(a). This could be attributed to the MR in twin B [Fig. 1(a)] in which the magnetic field and current are both parallel to b direction, this is most likely the situation in Figs. 3(e) and 5(a). With further increasing magnetic field, twin A is dominant [Fig. 1(c)] and the situation is similar to that in Fig. 3(a). Therefore, the MR in the twinned EuFe₂As₂ is almost the average result of the MR in twin A and twin B.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. The MR results by using 4-probe method. (a) The MR results on detwinned EuFe2As2 with both current and external field parallel to b direction. (b) The MR results with I ∥ H in twinned EuFe2As2.

Figure 6 summarizes the magnetic phase diagrams for the detwinned EuFe₂As₂. Figures 6(a) and 6(b) correspond to the reorientation models of Eu²⁺ spins for the magnetic field parallel to a direction and the magnetic field parallel to b and c direction, respectively. As shown in Fig. 6, the real AFM state only exists at a narrow range of magnetic field when the field is parallel to b or c direction, whereas it could insist up to the critical fields when the external field is parallel to a direction. Based on the critical field at the same temperature, we can find that a is the easy axis for magnetization while c is the hard axis for magnetization. Finally, we also note that H_crit in magnetization experiment [inset of Fig. 1(b)] locates between the critical field with magnetic field parallel to a direction and that with magnetic field parallel to b direction in magnetoresistance experiment (Fig. 6), so that it is an average of the two critical fields.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. Magnetic phase diagrams in detwinned EuFe2As2 with the field parallel to a direction (a) and the field parallel to b or c direction (b). A real AFM state only exists at a narrow range of magnetic field when the magnetic field is parallel to b or c direction (the purple area in (b)). PPM refers to the partially polarized magnetic state.

In summary, we study the in-plane resistivity and MR in the detwinned EuFe₂As₂. Previous study on the twinning EuFe₂As₂ proposed that the changes in MR can be attributed to the magnetic detwinning and electron–spin scattering.^[34] In our present MR study without the disturbance of twinning effect, we can conclude that the change of MR follows the configuration of the Eu²⁺ spins, which provides a direct evidence of the coupling between itinerant electrons and spins of Eu²⁺. Furthermore, within our experiment resolution, we did not observe strong MR when the temperature approaches T_N, this is due to the small ordered moments of Fe²⁺ compared with Eu²⁺. While the magnitude of MR is dependent on the field geometry, the sign of MR seems to only follow the current directions. Therefore, the charge carriers in EuFe₂As₂ appear to have a remarkably strong coupling with the spin order of Eu²⁺, which plays an important role in determining their transport properties.