Thermally activated transport and vortex-glass phase transition of K_0.78Fe_1.70Se₂ single crystals

Since the discovery of SmOFeAs [1], the intense interest in iron-based superconductor research has lasted for a long time. On the one hand, the transition temperature(T_c) of Gd${}_{1-x}$ Th_xFeAsO [2] has reached as high as 56 K, far surpassing the MacMillan limit, which makes iron-based superconductors the second system of high temperature superconductors (HTS). On the other hand, although the structure of the iron-based superconductors highly resembles that of cuprates, the transport property behavior differs. Therefore, iron-based superconductors are regarded as interesting subjects to explore, in the hope of increasing the understanding of the mechanism of superconductivity of cuprates. Amongst them, the FeSe compound, which is solely built by alternatively intercalating the FeSe layers, has the simplest structure. Because of the simple layered structure, it is considered as an ideal subject for doping and intercalation. In previous reports, we mentioned the potential application of A_xFe${}_{2-y}$Se₂ (A = K, Rb, Cs, and Tl) superconductors. The series display great robustness against a magnetic field, with its ${{\rm{H}}}_{c2}$ reaching over 180 T [3], which is rather rare in HTS. This gives rise to both the great potential of practical applicability in extreme conditions, and important references for designing a new type of high ${{\rm{H}}}_{c2}$ superconductors. It is even more striking when one realizes that all compounds of this family share the feature of phase separation [4]. Actually, 80% of the volume is anti-ferromagnetic, displaying nonsuperconductivity, according to the Mossbauer experiment [5]. It seems both entrancing and confusing when one tries to comprehend the mechanism behind the surprisingly high ${{\rm{H}}}_{c2}$ with the superconducting phase occupying only 20% of the whole volume. All of the above renders it especially essential to investigate its flux behavior to understand the mechanism.

The vortex-glass (VG) model [6, 7] is usually applied to describe the flux behavior of strong pinning systems, and is obeyed by all cuprates found so far [8–10]. The theory proposes a critical temperature(T_M). The superconductor undergoes a vortex-solid to vortex-liquid phase transition when temperature rises above T_M, i.e. there exists a VG melting line, which draws the outline of the transition boundary. The line could be determined by the temperature-dependent irreversible line obtained from magnetization measurement. A scaling rule is often utilized to judge the suitability of the VG model for the material. It is worth mentioning that although the applicability of the VG theory is confirmed by cuprates, its suitability for superconductors other than cuprates, especially iron-based superconductors, has been doubted [11]. All of the above renders the clarification of the applicability of the VG model for the A_xFe${}_{2-y}$Se₂ system is rather important and necessary to understand the mechanism of the flux dynamics of iron-based superconductors.

In this article, we present a comprehensive investigation of the transport property ($\rho -T$) and current-voltage characteristics (I − V) of high quality single crystals ${{\rm{K}}}_{0.78}$ Fe${}_{1.70}$ Se₂. Measurements are performed with the magnetic field perpendicular and parallel to the ab-plane of the sample, respectively, within the field range of 0–14 T. Typical tail effects are observed, in accordance with the characteristic of strong pinning, which effectively hinders the flux flow. A phase diagram is drawn according to the analysis of the flux flow behavior and scaling of the I − V isotherms.

Single crystals of ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ were grown following the procedure mentioned previously [12] via the self-flux method. High-purity powder of iron (purity 3N) and selenium (purity 3N), and high-purity potassium slices (purity 3N), used as starting materials, were mixed stoichiometrically (K : Fe : Se = 1.0 : 1.8 : 2.0) and thoroughly ground. The mixture was then pelletized and sealed into an evacuated quartz tube. The tube was sealed into a larger quartz tube followed by evacuation, in case it cracked during heating. The furnace was heated to 600 °C, and maintained for 10 hours, then the temperature was raised to 1050 °C, and maintained for 24 hours. With a rate of 2 °C/h, the mixture was slowly cooled down to 600 °C before furnace cooling. The obtained crystals were shiny on the surface. They were characterized by both powder x-ray diffraction (XRD) and the single crystal XRD with Cu ${K}_{\alpha }$ radiation at room temperature. With the aid of energy dispersive x-ray spectrometry (EDX), the actual composition of the sample is determined to be ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$As₂. Four different areas were chosen in each sample when conducting the EDX measurement, and the average value was calculated to determine the actual composition. Via a standard four-probe method, the DC resistivity was measured with a constant current of 1 mA. In the resistivity measurement, the samples were first cut into rectangle shapes with typical dimensions of about 5 × 3 × 0.3 mm³. In the I-V measurement, one unavoidable issue is the Joule heat, which causes obvious lift of the temperature, and therefore, damages the reliability of the isotherm. In order to solve this problem, we tailored the single crystal into a proper size to restrict the resistance to a small value. All isotherms employed in the scaling remain in a narrow temperature range of less than 0.1 K.

Figures 1(a) and (b) present the temperature dependence of the resistivity of the ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ under a varied magnetic field ranging from 0–14 T, with the field oriented parallel and perpendicular to the c-axis. The detailed orientation procedure has been reported elsewhere [3]. In the zero field, with a criterion of 10%–90% of the normal state resistivity (${\rho }_{n}$), the sample shows a superconducting transition at around 30.46 K, with a narrow transition width of less than 1 K. When ${{\rm{H}}}_{//c}$, the curve near the onset of the transition is evidently broadened as the field increases. A typical tail effect is also easily observed near the completion of the transition, and becomes more evident under larger fields. Interestingly, none of the above phenomena exists when ${{\rm{H}}}_{\perp c}$. The case highly resembles the condition of the iso-structure (Ba, K)Fe₂As₂ [13]. The tail effect is commonly seen in cuprates [14, 15], which could be ascribed to many reasons, e.g. the anisotropy of the superconductivity, the connectivity, and most importantly in this case, the vortex motion. As shown in the upper left insets of figures 1(a) and (b), the configuration of the vortex differs with different field directions [3]. The vortex appear as 'pancake' vortices when ${{\rm{H}}}_{//c}$, and as vortex strings when ${{\rm{H}}}_{\perp c}$. This profoundly affects the vortex dynamics and thermal fluctuations. Both the broadening behavior and the tail effect are tuned by thermal fluctuations, the magnitude of which is characterized by the Ginzburg parameter [16]. The intersecting point of the extrapolating lines of the normal state resistivity and the steep transition regime, T_c under different field strengths and orientations can be determined from figure 1, and therefore, the ${{\rm{H}}}_{c2}$ and anisotropic ratio (γ) are calculated. The temperature dependence of γ is displayed in the lower left inset of figure 1(b). γ monotonously decreases with descending temperature, which is in accordance with previous reports of both FeSe [17, 18] and FeAs-211 [13] samples. The ${{\rm{H}}}_{c2}$ and γ behavior is argued to be attributed to the Pauli paramagnetic effect, which is aroused and tuned by the Fe vacancies [19].

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Analysis of the ρ-T curves is based on the thermally assisted flux flow (TAFF). In this regime, the resistivity can be described by

$$\begin{eqnarray}&&\rho =(2{\rho }_{c}U/T){\rm{\exp }}(-U/T)={\rho }_{0f}{\rm{\exp }}(-U/T)\end{eqnarray} \tag{ 1 }$$

where U is the thermally activation energy. Assuming the prefactor ${\rho }_{0f}=2{\rho }_{c}U/T$ to be temperature independent, we could simplify the equation to an Arrhenius relation $\mathrm{ln}\rho (T,H)\,=$ ${\mathrm{ln}\rho }_{0}(H)-{U}_{0}(H)/T$, where ${\rm{ln}}{\rho }_{0}(H)={\rm{ln}}{\rho }_{0f}+{U}_{0}(H)/{T}_{c}$ is a temperature independent constant and ${U}_{0}(H)$ is the apparent activation energy, which plays the role of effective pinning barrier. Moreover, it is noted that $\partial {\rm{ln}}\rho /\partial {T}^{-1}={U}_{0}(H)$, which renders the linear relation of ${\rm{ln}}\rho$ versus ${T}^{-1}$ in the TAFF region with the slope equaling ${U}_{0}(H)$ and the intercept equaling ${\rm{ln}}{\rho }_{0}(H)$. It is worth noting that ${U}_{0}(H)$ is often as large as 10⁴ K for iron-based superconductors [20, 21], as in our case. By presuming a temperature-dependent prefactor [22], ${U}_{0}(H)$ is even reported to reach 10⁵ K. However, it must be mentioned that ${U}_{0}(H)$ extracted from TAFF model is always one order greater than that determined by the irreversible region from flux creep or a modified VG model [23–25]. It it noted that ${U}_{0}(H)$ is closely dependent on current density according to the results of the magnetization relaxation [23, 26], thus the current density of the measurement might exert an influence on the obtained effective ${U}_{0}(H)$ values. Only when the superconducting current J = 0 in the flux flow region, should one expect ${U}_{0}(H)$ deduced from TAFF model to be of the same order as ${U}_{0}(H)$ ($J\to 0$) obtained from flux creep.

The Arrhenius plots of the $\mathrm{ln}\rho (T,H)$-$1/T$ relation under different magnetic fields, namely from 0.3–14 T, are displayed in figures 2(a) and (d), corresponding to ${{\rm{H}}}_{//c}$ and ${{\rm{H}}}_{\perp c}$, respectively. It is clearly observed that the Arrhenius relation fits the experiment data very well. Every two extrapolations of the $\mathrm{ln}\,\rho (T,H)$ lines cross at the same point, approximately equal to T_c, which is also seen in other iron-chalcogen superconductors [27]. In the upper right insets of figures 2(a) and (d), we display the linear fits of ${\mathrm{ln}\rho }_{0}(H)$ versus ${U}_{0}(H)$, from which we deduce the prefactor ${\rho }_{0f}$ and T_c under two magnetic field orientations. Field dependence of the effective pinning barrier is investigated, the ${U}_{0}(H)$-T relation is shown in the lower left insets of figures 2(a) and (d).

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When ${{\rm{H}}}_{//c}$, it is clear that the relation follows a power law of ${U}_{0}(H)\sim {H}^{-\alpha }$, with various α values as the magnetic field increases. Below 5 T, the dependence relation is described by ${U}_{0}(H)\sim {H}^{-0.43}$, while ${U}_{0}(H)\sim {H}^{-0.70}$ when ${{\rm{H}}}_{//c}$ is above 5 T. The case is similar when ${{\rm{H}}}_{\perp c}$. It should be noted that such behavior highly resembles that which happens to anisotropic cuprates [28, 29] and NdFeAsO${}_{0.7}$F${}_{0.3}$ single crystals [30]. For cuprates such as BSCCO, the behavior was well explained by the dimensional-crossover of the vortices, where the crossover field (H${}_{3{\rm{D}}-2{\rm{D}}}$) required for a 3D-2D cross was rather small, approximately 2 T. However, for the ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ sample, the ${{\rm{H}}}_{3{\rm{D}}-2{\rm{D}}}$ is much larger. As ${{\rm{H}}}_{3{\rm{D}}-2{\rm{D}}}\propto 1/{(\gamma \delta )}^{2}$, where γ is around 4–5 according to figure 1, and the superconducting interlayer distance δ is around 0.8 nm, ${{\rm{H}}}_{3{\rm{D}}-2{\rm{D}}}$ for the ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$As₂ sample should be around 500 T. Therefore, no dimensional-crossover happens during the $\rho -T$ measurement. On the other hand, ${U}_{0}(H)$ undergoes an increase instead of a reduction as the magnetic field ascends, thus the ${U}_{0}(H)$-T relation cannot be explained under the presumption of elastic creep theory [31, 32]. Instead, a plastic creep model [33] offers a reasonable explanation. As a larger field is applied to the sample, the correlation length undergoes a reduction as a result of plastic deformation and vortex entanglement. Therefore, vortex coupling is weaken and ${U}_{0}(H)$ needed becomes smaller. From the lower inset of figures 2(a) and (c), we can see it clearly. We would take figure 2(a) for instance. The vortices entangle in weakly pinned vortex-liquid when the field is weaker. As in the case of ${U}_{0}(H)\sim {H}^{-0.43}$ with smaller α, ${U}_{0}(H)$ drops more slowly with an increasing field, which is shown as red circles. While it comes to strongly pinned vortex-liquid when a stronger field is applied and ${U}_{0}(H)$ drops faster with larger α, as in the case of ${U}_{0}(H)\sim {H}^{-0.70}$, which is displayed as blue squares. It has to be mentioned that in the cross of α value, the intrinsic pinning centers mainly composed of point defects play the crucial role, the entanglement of vortex lines around which introduce lateral vortex wandering in the strongly pinned vortex-liquid phase.

In isotropic FeSe and relatively slightly anisotropic doped Fe-chalcogenides, a tiny value of α is often seen in the low magnetic field regime (usually lower than 1 T) [34, 35], before a crossover appears at higher fields. It is often utilized as a rule to judge the mechanism of flux creep that if $\alpha \approx 0$ in the low field region, and U₀ remains almost irrelevant to H, then single flux creep is dominant in the regime. As the magnetic field ascends, U₀ becomes highly dependent on H with a much larger α. The vortex spacing is significantly smaller than the penetration depth, and collective vortex creep becomes the main mechanism. According to our data, the crossover of α resembles that of SmFeAsO${}_{0.85}$ [36], as U₀ is obviously related to H until 0.3 T.

In figures 2(b) and (e), the relations of activation energy versus temperature are displayed for ${{\rm{H}}}_{//c}$ and ${{\rm{H}}}_{\perp c}$, respectively. U is extracted from the slope of the Arrhenius plot as $U=-d({\rm{ln}}\rho )/d(1/T)$. For any field value, one can observe that from a certain critical temperature ${T}^{* }$, the corresponding U plot starts to climb fast with decreasing temperature, in accordance with previous reports for cuparates [37]. The upturn of U denotes the upper temperature limit of VG critical state region [37].

To further verify the ${T}^{* }$ value, we plot the ${dT}/{d}{\rm{ln}}\rho$ versus T relation in figures 2(c)and (f), corresponding to ${{\rm{H}}}_{//c}$ and ${{\rm{H}}}_{\perp c}$, respectively. ${T}^{* }$ under varied fields is marked by downward arrows, below which a linear behavior is observed for each field. The slope of the linearity renders the value of exponent $s=\nu (z-1)$ for each field, where z is the dynamic critical exponent, and ν is the static exponent, while the critical temperature T_M, as denoted by upward arrows, are determined by the 0 line of ${dT}/{d}{\rm{ln}}\rho$, which we will expand on a bit later.

The existence of a VG state in the ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ sample is revealed by both the $\rho -T$ behavior and the upturn of the U-T curve. But to further confirm it, we investigate E-J isotherms under a varied magnetic field ranging from 2–14 T, in different field orientations. The data are displayed in double logarithmic coordinates in the insets of figures 3(a) and (b), for ${{\rm{H}}}_{//c}$ and ${{\rm{H}}}_{\perp c}$, respectively. As in the case of cuprates [8–10], a straight line could be found amongst the isotherms, which corresponds to the critical temperature T_M. T_M is around 25 K when ${{\rm{H}}}_{//c}$, and 26.2 K when ${{\rm{H}}}_{\perp c}$, as marked by arrows in the insets of figures 3(a) and (b). According to the VG model, the vortex-glass state melts into vortex-liquid once the temperature rises above T_M. To clarify the existence of the phase transition, we apply a universal scaling to the isotherm data. Upon estimation of the dynamic critical exponent z, static exponent ν and dimension d (2 or 3), the E- J isotherms under transformation collapse into two distinct branches, denoted by ${E}_{+}(T\gt {T}_{M})$ and ${E}_{-}(T\lt {T}_{M})$. It must be mentioned that the quality of the scaling, i.e. whether the isotherms with the same curvature sign are neatly scaled onto a single branch, highly depends on the choice of scaling parameters. The estimation of parameter ν is based on the behavior of the linear resistivity ${\rho }_{{\rm{ln}}}$. ${\rho }_{{\rm{ln}}}\approx {(T-{T}_{M})}^{\nu (z+2-d)}$ at the edge of entering into the VG phase, while z is deduced from $E(J,T)\,\approx {J}^{(z+1)(d-1)}$ near T_M. The coordinates of the scaling are written as

$$\begin{eqnarray}&&{\rho }_{{\rm{scale}}}=E/J\displaystyle \frac{1}{| T-{T}_{M}{| }^{\nu z}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{J}_{{\rm{scale}}}=\displaystyle \frac{J}{| T-{T}_{M}{| }^{\nu }}.\end{eqnarray} \tag{ 3 }$$

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Dozens of different parameter sets were carefully tried out for each magnetic field value and orientation to obtain the best scaling. In figures 3(a) and (b), we only display the most appropriate scaling results for the isotherms measured under the 10 T field from ${{\rm{H}}}_{//c}$ and ${{\rm{H}}}_{\perp c}$, respectively. As is easily observed, isotherms of temperature higher (lower) than T_M all collapse onto ${E}_{+}$ (${E}_{-}$) smoothly. Under a varied magnetic field ranging from 2–10 T, the critical exponents of the best scaling are all confined in the narrow regime of $1.0\lt \nu \lt 1.8$ and $3.2\lt z\lt 4.6$. It should be clarified that only isotherms under ${T}^{* }$ were used for the scaling, as marked in figures 2(b) and (d). The reason was explained above. Also, a threshold E_g was taken for the isotherms before scaling, as the VG model only describes the behavior in the regime neighboring the vortex-glass to vortex-liquid transition. Thus, the portion related to the high dissipation region, i.e. the portion of curves with corresponding $E\gt {E}_{g}$ should be excluded from the scaling [13, 38]. Actually, data with a higher E value belong to the region of the 'unpinned vortex-liquid state', where the influence mechanism of the pinning center towards vortex behavior is different.

It is necessary to point out that different from other iron-based superconductors, the E-J curves measured for ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ seems to be based on filamentary superconductivity, as we could observe low J_c values as shown in figure 3. In the preparation of ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ single crystals, it was found that the relation between the superconducting and the insulating phase is very delicate. The fabrication of superconducting ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ samples demands strictly controlled procedures, either an overdose of iron or improper annealing will bring a larger percentage of insulating phase. Interestingly, superconductivity could be recovered by either iced quenching or vacuum annealing, reflecting the tuning effects of iron vacancies. According to previous reports, the weak coupling between the superconducting islands could be weakened severely under magnetic penetration in the high temperature region [4]. However, great robustness against the magnetic field is clearly detected in our sample, and VG behavior is revealed based on the superconducting filaments. All of the above makes the interpretation of the exact view of phase separation and its influence on the flux dynamics of the A_xFe${}_{2-y}$Se₂ family both essential and urgent.

In figures 4(a) and (b), we display a static vortex phase diagram of ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ extracted from the evolution of H_c and T_M (H_M) with the varying magnetic field. The diagram is similar to that of the cuprates, where effects of thermal fluctuation and pinning were both included. ${H}_{c2}$ denoted by green triangles is determined by 90% of ${\rho }_{n}$ as mentioned, above which the sample enters into the normal state. Well below ${H}_{c2}$, a vortex-solid to vortex-liquid phase transition is observed. Marked by red squares, the characteristic magnetic field H_M, which is deduced from the scaling, divides the VG critical state into vortex-glass and vortex-liquid regimes. The upper field limit of the VG model applicable regime is given by ${H}^{* }$, as shown in yellow dots. ${H}^{* }$ is inversely obtained from ${T}^{* }$, which is decided by $-d({\rm{ln}}\rho )/d(1/T)$ versus the T curves, as displayed in figures 2(b) and (d). The ${H}^{* }$ line cut the vortex-liquid phase into the pinned liquid phase and unpinned liquid phase. The latter outranges the description of the VG theory, and therefore is excluded from the scaling. Finally, investigation of the ${U}_{0}(H)$-T relation reveals the existence of a crossover field, which is around 5 T when ${{\rm{H}}}_{//c}$ and around 7 T when ${{\rm{H}}}_{\perp c}$. According to the plastic-flux-creep model, it separates the pinned vortex-liquid phase into a strongly and weekly pinned vortex-liquid phase by different dynamical characteristics.

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To summarize, based on TAFF and the VG model, we have studied the vortex motion of ${{\rm{K}}}_{0.78}$Fe${}_{1.70}$Se₂ single crystals under ${{\rm{H}}}_{//c}$ and ${{\rm{H}}}_{\perp c}$ up to 14 T. Due to different vortex configurations, a drastic contrast exists between the transport property of the two magnetic field orientations. The investigation of E-J isotherms and the $\rho -T$ relation reveal the existence of a VG state. As in the case of cuprates, a vortex-glass to vortex-liquid state transition is observed well below ${H}_{c2}$ in the phase diagram and confirmed by E- J scaling under both field orientations. The $\rho -T$ curves clearly display Arrhenius behavior with a crossover field extracted from the ${U}_{0}(H)$-T relation, which divides the pinned vortex-liquid phase into weekly and strongly pinned regions according to a plastic-flux creep model. The former region is associated with weak vortex entanglements under lower fields, while the magnetic field increases, the point defects compose strong intrinsic pinning center, causing strong vortex entanglements and lateral vortex wandering. The effective pinning barrier extracted from the TAFF region is around 10⁴ K, consistent with the strong pinning nature of the system. Variance of magnetic strength and orientations brings about a complex change in pinning, and therefore, abundant vortex behavior in A_xFe${}_{2-y}$Se₂, which makes it a convenient system to study diverse vortex phases in layered superconducting systems.

This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11404002, 11404003 and 11074001) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars from the State Education Ministry, and the '211 Project' of Anhui University (J01001319-J10113190007).