Angular dependence of vortex instability in a layered superconductor: the case study of Fe(Se,Te) material

Anisotropy effects on flux pinning and flux flow are strongly effective in cuprate as well as iron-based superconductors due to their intrinsically layered crystallographic structure. However Fe(Se,Te) thin films grown on CaF2 substrate result less anisotropic with respect to all the other iron based superconductors. We present the first study on the angular dependence of the flux flow instability, which occurs in the flux flow regime as a current driven transition to the normal state at the instability point (I*, V*) in the current-voltage characteristics. The voltage jumps are systematically investigated as a function of the temperature, the external magnetic field, and the angle between the field and the Fe(Se,Te) film. The scaling procedure based on the anisotropic Ginzburg-Landau approach is successfully applied to the observed angular dependence of the critical voltage V*. Anyway, we find out that Fe(Se,Te) represents the case study of a layered material characterized by a weak anisotropy of its static superconducting properties, but with an increased anisotropy in its vortex dynamics due to the predominant perpendicular component of the external applied magnetic field. Indeed, I* shows less sensitivity to angle variations, thus being promising for high field applications.


Flux Flow Instability
The instability of the superconducting state has been studied so far as an interesting phenomenon of vortex dynamics as well as for its relevant impact on the lossless electric current transport in type-II superconductors. No matter is the mechanism triggering the instability, its fingerprint consists of voltage jumps that can be observed in current-driven current-voltage characteristics of the superconducting material. Such jumps can be ascribed to several possible mechanisms, each of which shows its own peculiar feature in the I −V curve branch above the critical current. Here we make a list of the conventional and more exotic ones in connection with their observable fingerprints. Thermal runway 4 is well known, since high currents induce a power dissipation in the film that is high enough to destroy the superconducting state, leading to an abrupt increase of sample temperature above T c . Hot spot effect 4 is related to a localized normal domain (hot-spot) maintained by Joule heating, usually such domain appears where there is a maximum current concentration; the I −V curve manifests an counterclockwise hysteresis. Electron overheating 1 is due to the finite heat removal rate of the power dissipated into the sample, depending on the film-substrate interface transparency to phonons, indeed non-equilibrium phonons leave the film without being reabsorbed; therefore the heat removal rate is determined by the strength of the electron-phonon coupling constant rather than by interface properties. Vortex system crystallization 5 may occur if the system has enough time to arrange itself into a coherently moving perfect crystal at large velocities; the ordering of the vortex lattice at large applied currents show a jumplike transition between pinned static state and homogeneously moving lattice. Self-organized criticality 9 is marked by voltage instabilities that could appear near the pinning-depinning transition by thermally activated jumps of vortices, which trigger a chain reaction of vortex movements leading to avalanches of diverging size. Phase-slip centers 11 (PSC) and/or lines 12 (PSL) occur at currents larger than a certain instability current I * , a system of transverse alternating normal and superconducting domains is formed, and a voltage-step structure in the I −V curve appears; these segments of constant dynamic resistance have a slope independent from the magnetic field strength. PSC appears when the uniform superconducting state is destroyed since the transport current reaches the GL pair-breaking current (I GL c < I < I c2 ); PSL appears when the steady viscous flux flow of Abrikosov vortices is disrupted at currents I m < I GL c . The normal state is reached at current higher than the upper critical current I > I c2 I m . Theoretical approaches within this scenario include the fundamental theory of Larkin and Ovchinnikov 7 and the hot electrons instability model by Kunchur 6 . The first predicts the instability from the flux flow regime at temperatures close to T c , caused by the shrinking of the vortex core due to the quasiparticles escaping when a sufficiently high vortex velocity is reached. The second main feature is the expanding of the vortex core as a consequence of the electronic temperature increase which adds quasiparticles within the vortex core at sufficient high electric field but at temperatures far from T c . Nevertheless, both approaches ignores material pinning influence on flux flow instability, since originally they are derived in totally absence of any pinning mechanism. Moreover, just recently, pinning effects have been taken into account in the hot electron flux flow instability by Shklovskij 10 .
We have recently demonstrated that some of the aforementioned instability mechanisms may compete depending on the material under investigation, so that in Fe(Se,Te) superconductor the flux flow instability can be considered halfway between

Sample characterization
Pinning properties Transport measurements were performed in order to characterize the pinning properties of this material and their anisotropy. Indeed, the anisotropy factor can be deduced from the critical currents vs field dependence measured in the two field orientation, that is the ratio of I c (0 • , H) to I c (90 • , H). In the Figure 1A the critical current versus H curves are displayed at different magnetic field orientations θ = 0 • , 45 • , 90 • , and at fixed temperature T = 10 K. The same plot includes the instability currents as well. It is clear that the I * values are slightly dependent on the intensity and the orientation of the magnetic field. By the way, the I c values reflects the expected behavior with the parallel in field values always greater than the perpendicular ones. Furthermore, the critical current density is J c = 2 · 10 5 A/cm 2 at 10 K. The inset shows the anisotropy factor γ J versus field dependence, i.e. I c (0 • )/I c (90 • ), which results between 1 and 2 up to 5 T.

Anisotropy properties
The high quality of Fe(Se,Te) thin films is also confirmed by the H − T phase diagrams measured in perpendicular and parallel orientation of the applied magnetic field, as reported in Figure 1B. In particular the critical temperature value T c estimated by the 50% of the normal state resistance in zero field is 18.5 K, with a transition width below 1 K. The transition width is defined as the difference between the temperature values corresponding to 90% and 10% of the normal state resistance R N . These are also the two criteria used to identify the upper critical field H c2 and the irreversibility lines, respectively, in the phase diagrams shown in Figure 1B. In Figure 2 the temperature dependence of the resistance is displayed at different angles from θ = 0 • to θ = 90 • with a field intensity of 2 T.

Vortex critical velocity
By rotating the external field from θ = 90 • to θ = 0 • , the vortex velocity results from an in-plane vector velocity towards an out-of-plane vector velocity. In both cases, the resulting longitudinal electric field is detected from the two yellow dots used as voltage taps, as displayed in Figure 1B of the main text. The average vortex critical velocity is given by the measured critical voltage V * = v * · µ 0 H · l. In Figure 4A, the v * (H) behavior is presented at different values of field orientation equals to 0 • , 45 • , 90 • , and at the fixed temperature of 10 K. Clearly, in Figure 4B and C, the typical dependence of v * ∝ H −1/2 is observed, which is the expected behavior of the intrinsic electronic nature of flux flow instability 3 . Moreover, it results that v * increases from parallel (i.e. θ = 0 • ) to perpendicular (i.e. θ = 90 • ) orientation of the applied magnetic field. Consequently, in this latter case vortex lattice can move faster. In the inset of Figure 4A the corresponding pinning force dependence is also shown in order to enlighten that the stronger pinning results when the field is applied in the parallel direction, in agreement with the magnetic