Anomalous transverse resistance in 122-type iron-based superconductors

The study of transverse resistance of superconductors is essential to understand the transition to superconductivity. Here, we investigated the in-plane transverse resistance of Ba0.5K0.5Fe2As2 superconductors, based on ultra-thin micro-bridges fabricated from optimally doped single crystals. An anomalous transverse resistance was found at temperatures around the superconducting transition, although magnetic order or structure distortion are absent in the optimal doping case. With the substitution of magnetic and nonmagnetic impurities into the superconducting layer, the anomalous transverse resistance phenomenon is dramatically enhanced. We find that anisotropic scattering or the superconducting electronic nematic state related with the superconducting transition may contribute to this phenomenon.

ScieNtific REPORTS | (2019) 9:664 | DOI: 10.1038/s41598-018-37152-y temperature T s , and almost simultaneously, a symmetry breaking from C 4 to C 2 happens within the electronic structure [17][18][19][20] . The electronic nematic state is generally considered to be related to the structure, although the relation between the electronic ordering and structure transition is still a "Chicken and Egg Problem" 17,18 . Up to now, abundant researches have been carried out on the magnetic order and electronic nematic states in the iron-based superconductors, while the corresponding studies on the ATR still lack. Therefore, it is greatly important to explore the dynamics of electron pairing and transport, which yields to the origin of magnetic order and electronic nematicity.
In this work, we investigated the in-plane transverse resistance of Ba 0.5 K 0.5 Fe 2 As 2 . This hole-doped 122-type compound was selected because high-quality single crystals are available. The transport measurements were performed on ultrathin single-crystalline micro-bridges, which are shown in Fig. 1. A pronounced ATR was found at temperatures around the superconducting transition, although the magnetic order is absent in the case of optimal doping. With the substitution of magnetic or nonmagnetic impurities into the superconducting layer, the ATR is significantly enhanced. The anomalous Hall effect, vortex motion, or electronic nematic state can hardly be regarded as the origin of the observed ATR, and a possible origin will be discussed.

Results
Anomalous Transverse Resistance. Figure 2 shows the temperature dependence of R xx and R xy for the BK micro-bridge. The midpoint of the resistive transition, as determined from R xx is at 39.1 K, and the transition width is about 1.4 K. The values of ρ xx (~12 µΩ cm at 40 K) are about one order of magnitude smaller than bulk crystals 21,22 , indicating the high quality of the crystal and the improved measurement setup as described in previous work [23][24][25] . We emphasize that in the traditional four-probe measurement technique, a high current bias is often necessary to enhance the measurement signal, due to the extremely low resistivity and the large size of cross-section area. Consequently, heating effects and measurement errors may obscure the intrinsic transport properties of the crystals. For the micro-bridges used in this work, the resistance can be up to tens of ohm, being considerably higher than the interfacial contact resistance between the sample and the thin film gold electrodes (~0.1 Ω at room temperature). We note that the interfacial contact resistance has been well improved because of the in-situ fabrication and the annealing processes as introduced in a recent work 26 , by which the Schottky contact from the interface can be well eliminated.
R xy vs. T exhibits a very different profile from R xx . In principle, R xy should be completely zero, once the electrodes for the voltage measurements are perfectly symmetric. For the micro-device, however, an asymmetric structure often happens, resulting in a finite resistance R xy . For the present case, a weak resistance of about 0.008 Ω is observed at 40 K, which is two orders of magnitude less than R xx (~1.39 Ω at 40 K). ATR peaks occur at temperatures below the T c -onset, which is the central finding of this work. The negative peak of R xy at 39.1 K is about 3.33 times the value of the normal state resistance at 40 K (see Fig. 2(a)). Overall, R xy vs. T exhibits two sign reversals. At the same time, for the temperature region up to room temperature the R xy vs. T curve shows a similar profile as R xx vs. T (see Fig. 2(b)), indicating that in the normal state R xy is due to the asymmetric structure of the voltage electrodes. However, for the temperatures below the T c -onset, the dramatically different profile of the R xy vs. T curve from those of R xx vs. T suggests that the anomalous peaks should be related to the superconducting transition. We have measured more than ten samples, all of them demonstrate anomalous peak in the R xy (T) curves, while the value of the peaks is considerably different.

Effect of Applied Current.
To understand the link between the anomalous peaks and the superconducting transition, we applied various currents to suppress the superconductivity. Figure 3(a) demonstrates the temperature dependent R xy under applied currents ranging from 0.01 to 5.0 mA. The anomalous peak is strong under weak current (0.01 mA) and is gradually suppressed by the applied current. Interestingly, the positive peak can be completely suppressed by a current of 2.5 mA, while the negative peak still exists for all currents. For comparison, the temperature dependence of R xx under applied currents ranging from 0.01 to 5.0 mA is also given in Fig. 3(b). The current gradually suppresses the superconductivity as those of R xx -T curves, while the anomalous peak is absent. It is worth noting that the normal resistance of R xy is enhanced by the increasing current density, and such phenomenon was also found on the Ba(Fe 1-x Co x ) 2 As 2 bulk crystals once the applied currents are up to the critical points 27,28 . Here, the current density is considerably high as about 1 MA/cm 2 for I = 5 mA. The resistance enhancement is not due to the heating effect, because the R xy -T curves can be completely repeated by increasing or decreasing the temperatures. Instead, a spatial variation of superconductivity may induce such effect 27,28 .
Effect of Magnetic Fields. Figure 4(a,b) give the temperature dependence of R xy under out-of-plane magnetic fields between 0 and 9 T, applied at different angles θ relative to the direction of bias current. The negative peak in R xy is suppressed dramatically, but this suppressing effect works weakly on the positive peaks. Since in magnetic fields the Hall effect will contribute to R xy , the magneto-resistance under the field along the c-axis (θ = ± 90°) is calculated as = + to eliminate the Hall contribution. Here, + R xy B and − R xy B are the magneto-resistance under positive (θ = 90°) and negative (θ = −90°) magnetic fields, respectively. In fact, + R xy has two contributions, which are the normal magneto-resistance ΔR(B) due to Lorentz force and the change in the   Figure 4(c) shows + R xy vs. T curves calculated from Fig. 4(a,b). Here, the positive peak is sensitive to the magnetic fields along the c-axis, while the negative peak is more resistive. For the , there is no contribution from magneto-resistance or transverse resistance terms, thus basically the − R xy can be considered as Hall resistance. The − R xy vs. T curves demonstrate no anomalous peaks as shown in Fig. 4(d), suggesting that the anomalous peak is independent of Hall effect. Sample Geometry. To further investigate the effect of asymmetry on the voltage leads V+ and V−, we also fabricated a nano-scaled bridge by focused ion beam (FIB) milling technique as shown in Fig. 5(a,b). The length of the bridge is about 700 nm. Figure 5(c) gives the corresponding temperature dependent R xx and R xy . Note that the anomalous peak for the R xy vs. T curve also appears just below the T c -onset, while no such anomalous peak is visible in the R xx vs. T curve, which is consistent with the micro-bridge samples as Fig. 2. Therefore, we can conclude that the anomalous peak is unlikely to arise from the geometry of samples. Some intrinsic effects related to the superconducting transition seem to be responsible for the anomalies. Impurity Doping Effects. Although the present Ba 0.5 K 0.5 (Fe,M) 2 As 2 single crystals are of high quality in both crystalline structure and superconductivity 22 , the distribution of supercurrent can hardly be completely homogenous within the crystals, especially at the superconducting transition region. Thus, the effect of impurities or disorder should be seriously considered in the present micro-and even nano-scaled samples. To understand the impurity effects within the superconducting Fe 2 As 2 layers, we substituted the Fe-sites by both magnetic Co and nonmagnetic Zn ions with a weak doping level of 2.5%, and measured the R xy for the same micro-device geometry as the sample shown in Fig. 1. The temperature dependent R xy for Ba 0.5 K 0.5 (Fe,M) 2 As 2 doped with Co and Zn are shown in Fig. 6(a,b), respectively. The R xy vs. T curves of both impurity-doped samples are quite similar: First, dramatically large anomalous peaks are observed under a low applied current of 0.01 mA, for which the peak value ( ) R R / xy peak xy normal is about 33.24 for the Co-doped and 27.49 for the Zn-doped samples, being substantially larger than those of impurity-free samples (see Fig. 2). In fact, we have measured tens of samples, and the peak values for these samples vary from 10 to 100. Therefore, we can conclude that the substitution of impurity ions on the superconducting layer can enhance the anomalous transverse resistance. The magnetic field-suppression effect on the anomalous peaks is in accordance with the impurity-free samples. Here, we take the Co-doped sample as an example. Figure 7(a,b) show the transverse resistance under magnetic fields along θ = 90° and −90°, respectively. The magneto-resistance = + is shown in Fig. 7(c). The Hall resistance has been fully eliminated, which can be confirmed by the magnetic field independent normal resistance, while the anomalous peaks are suppressed by the fields. However, for the Hall resistance , the anomalous peak is absent (see Fig. 7(d)). Such profiles of the magnetic field-dependent magneto-resistance and Hall resistance are in accordance with the impurity-free samples, indicating the anomalous peak is a common property.

Discussion
The anomalous peaks in the R xy vs. T curves below the T c -onset may have various origins, which will be discussed in this session, including vortex motion, inhomogeneous distribution of the superconducting phase, and the intrinsic origins.
Generally Possibilities. Since the anomalous peaks always occur within the superconducting transition region, vortex motion is one of the most likely possibilities for the origin, via the so-called vortex Hall effect. Hagen et al. found a sign reversal for the Hall voltage just below the superconducting transition of both high-T c YBa 2 Cu 3 O 7 and low-T c Nb thin films 29 , which was attributed to vortex motion. However, the magnetic field is needed here to drive the vortex motion with a velocity v L . For the present experiments, the transverse voltage was  In the case of zero field, the present transverse voltage may be due to the anomalous Hall effect (AHE) 30,31 . The magnetization of the sample itself can contribute to R xy once the material transforms to the ferromagnetic state, resulting in AHE. Nevertheless, the present optimal-doped Ba 0.5 K 0.5 (Fe,M) 2 As 2 is always in a paramagnetic state, which has been well confirmed by various measurements as neutron scattering 32 and μSR 33 . Therefore, one can hardly obtain an AHE effect in these samples.
On the other hand, the anomalous transverse voltage may be attributed to spin-orbit coupling, which can be of either an extrinsic origin due to disorder-related spin-dependent scattering of the charge carriers, or of an intrinsic origin due to a spin-dependent band structure of the conducting electrons 13 . The spin for AHE effects basically originates from electron-orbit coupling, which is a relativistic quantum mechanical effect 1,11 . However, the behavior of strongly current dependence and existence only in superconducting transition region can hardly be explained by the spin-orbit coupling model.

Inhomogeneous Distribution.
In the thin film systems, a non-uniform transport current can be considered as a possible explanation due to inhomogeneity of material or superconductivity [4][5][6][7] . Such ATR was widely found in the high-T c and conventional superconductors, and the corresponding model have been proposed to explain at least part of the anomalous behavior. Since one can hardly avoid the problem of superconductivity spatial variation in thin films due to the island-like growth mechanism in most fabrication techniques, even for the molecular beam epitaxy method, it is rather challenge to grow a single-crystalline thin film with thickness up to few hundred nanometers. In the high quality Ba 0.5 K 0.5 Fe 2 As 2 single crystals, although the external inhomogeneous can be considerably reduced and the superconducting Fe 2 As 2 layer is a pure structure, the distribution of K ions in the Ba layer will induce inhomogeneous as well 34 . In a real compound, a finite number of micro-defects or impurities often exist in the lattice, which can be identified from the finite residual resistivity at 0 K (see more details in ref. 24 ).
With substitution of atomic impurities into the superconducting Fe 2 As 2 layers, the impurities are found to enhance the ATR phenomenon as introduced above. Substituting magnetic Co and nonmagnetic Zn onto the Fe-site in the Fe 2 As 2 layers, the impurity ions can suppress the superconductivity within a region of a coherence length 22,24,25,35 , and then disorganize the superconducting distribution. Particularly, the impurity scattering centers are aligned with a certain direction, for instance the antiferromagnetic a-axis, resulting in highly anisotropic impurity states 36 . Such anisotropic scattering can induce the transport nematicity, i e. the ATR as shown in Fig. 6. Nevertheless, the impurity fails to explain the sign-reversal of the ATR as shown in Figs 2-5. Actually, the sign-reversal ATR happens in many samples. For instance, the Co-doped sample also demonstrates both positive and negative ATR below the T c -onset as shown in the R xy vs. T curves in Fig. 8.

Electronic Nematicity. A nematicity of the electronic state has been observed in both cuprates and
iron-based superconductors [18][19][20][37][38][39] . Although the nematicity will shift in angle with the different doping levels, it breaks the C 4 symmetry in all samples. Similar to the situation of the cuprate superconductors, the anomalous transverse voltage can be ascribed to this anisotropic electronic state 5 . The resistivity can be defined in E = ρJ, where E is the electric field, J is the applied current density, and the resistivity along the principal axes is in matrix form. One can rotate the resistivity matrix as, , the off-diagonal term is non-zero. Therefore, as long as the applied current is not aligned with one of the principal axes, there will be a transverse voltage without magnetic field. For the present results, since the samples are in the optimally doped state, the structure distortion and the corresponding electronic nematic ordering along the ρ a and ρ b can be basically ignored. Thus, we can hardly apply the previous mechanism for cuprate superconductors onto the present results. Particularly, the present ATR phenomenon appears at temperatures just below the T c -onset, namely, with the occurrence of superconductivity. Therefore, another nematic ordering related to the superconducting transition should be taken into account for the ATR.
Nematic Superconducting State. In our recent work, nematic ordering was found in the optimal-doped Ba 0.5 K 0.5 Fe 2 As 2 superconductors at temperatures below the T c -onset, resulting in a superconductivity related nematicity 26 . Such nematicity demonstrates an electronic state ordering, and of course, will lead to anisotropic transverse resistance similar to the case of conventional nematic order in the normal state. However, the sample for the present measurements is rather small (few tens of micrometers), thereby, it is extremely hard to measure the angular-dependent ATR similar to the measurements on the (La,Sr)CuO 4 thin films 40 . Despite the fact that serials of micro-bridges can be fabricated with different angles corresponding to current and lattice direction, it is almost impossible to obtain a systematic angular-dependent anomalous transverse resistance because one can hardly calibrate the longitudinal resistance contribution, which is due to asymmetries in the electrode configurations, as discussed in Section A. Anomalous Transverse Resistance in the Result part.
The support for such a scenario of anomalous transverse resistance is that it is manifested in the domain of nucleation of superconductivity, where the nematic superconducting state also appears. Moreover, as discussed in reference cited above, nematic superconductivity arises in sufficiently thin samples of Ba 0.5 K 0.5 (Fe,M) 2 As 2 and is not expected to survive in bulk materials. This is fully in line with the phenomenology of the anomalous transverse resistance presented here, which gradually disappears with the increase of the thickness of the samples. What concerns the mechanism, we foresee two roots for the appearance of the anomalous transverse resistance in the nematic superconducting state, (i) via the anisotropic scattering from the nucleated superconducting domains in the normal/superconductor transition region and (ii) via the time-reversal symmetry breaking accompanying with the mixing of three components of the superconducting order parameter in the nematic superconducting state. To establish the relevant mechanism further experimental and theoretical studies will be needed.

Methods
The synthesis method for Ba 0.5 K 0.5 Fe 2 As 2 and Ba 0.5 K 0.5 (Fe,M) 2 As 2 (M = Zn and Co) single crystals is described elsewhere 22 . Because of the high-pressure synthesis technique, the impurity ions (Zn or Co) could be homogenously distributed into the superconducting Fe 2 As 2 layers to avoid disorders or other external defects. Here, the impurity-free optimally doped (Ba 0.5 K 0.5 Fe 2 As 2 ) crystal was selected for which the antiferromagnetic order and the structure distortion are absent 26 . Ba 0.5 K 0.5 Fe 2 As 2 , nonmagnetic impurity Zn-doped (Ba 0.5 K 0.5 Fe 1.95 Zn 0.05 As 2 ), and magnetic impurity Co-doped (Ba 0.5 K 0.5 Fe 1.95 Co 0.05 As 2 ) superconductors are abbreviated as BK, BKZn and BKCo, respectively. The fabrication of micro-bridge is described in refs [23][24][25] . The samples were etched by ion beam milling for a few seconds to remove surface layers, and subsequently a gold film was deposited for electrodes via magnetron sputtering. Because fabrication steps were performed in different chambers, the samples were transported in a high-vacuum tube (<10 −10 torr), resulting in a quasi in-situ fabrication process (AdNaNo-Tek Ltd.). The micro-bridges have a width (W) of 5 μm, a length (L) of 20 μm, and the thickness is confirmed from the longitudinal resistivity (ρ xx ) at room temperature 26 . The R xx and transverse resistance (R xy ) were measured as a function of temperature in the Physical Properties Measurement System − 9 T, Quantum Design. Figure 1 shows a scanning electron microscope (SEM) image of a typical micro-bridge. Here the current was applied along the Fe-As bond direction.