The Vortex Glass-Liquid Transition in Fe 1.02 Se Crystal

The vortex-glass (VG) to vortex-liquid (VL) transition is studied in flux-grown Fe 1.02 Se crystal with nanosized hexagonal phase inclusions. These non-superconducting impurities effectively pin the vortices and shift lightly the irreversibility line to higher fields and temperatures in comparison with single crystal. It is shown that the interplay between vortex pinning and thermal fluctuations enable the observation of VG-VL transition. The existence of this transition was proved by the scaling presentation of current-voltage characteristics at two different magnetic fields. The obtained scaling parameters are practically field independent. The values of the dynamic z exponent are in the range predicted by the VG model, while the values of static ν exponent are a little smaller. This is not considered as a lack of the universality of the model, but rather as a consequence of the type of pinning and special domain morphology of the crystal resembling the granularity in polycrystalline samples. Copyright © VBRI Press.


Introduction
Superconductors are a special group of materials with unique electric and magnetic properties, specific functions and various applications. Up to now 32 classes of organic and inorganic superconducting compounds (including more than 1000 alloys, intermetallic compounds, ceramics, magnesium diboride, heavy fermions, iron-based superconductors etc.) are known [1]. The superconducting mechanism in conventional low-Tc superconductors is described by the Bardeen-Cooper-Schrieffer (BCS) theory based on electron -phonon interactions. The H-T phase diagram of these superconductors is very close to that predicted by the mean -field theory. The vortex-lattice melting coincides with the upper critical field Hc2(T) line and practically there is no observable temperature range below Hc2(T) with vortex-liquid (VL) phase (except for temperatures very close to Tc). Superconductors with properties not explained by the conventional theory because of their high critical temperature or other reasons suggesting different mechanism, form the group of the unconventional superconductors. Among them the most investigated are electron-and holedoped cuprates and iron-based superconductors: pnictides and chalcogenides, discovered ten years ago. In high-Tc superconductors, due to their specific characteristics (high Tc, small coherence length -ξ, large penetration depth -λ, large anisotropy) thermal (dynamic) and quenched (static) disorder significantly influence the vortex phase diagram and phase transitions. In the vortex matter the most obvious type of disorder is introduced by thermal fluctuations. Thermally activated jumps of the vortices described by the Anderson creep model, predict a resistivity state and reestablish the dissipation in the entire mixed state. Quenched disorder, when present in the samples, destroys the long-range order in the vortex lattice changing it in an arrangement determined by the randomly distributed pinning centers [2]. This vortexglass (VG) state, similar to the magnetic order of spin glasses, is a real superconducting state with long range phase coherence and strictly zero resistivity in contrast to the flux creep model [3]. Different types of static disorder results in different types of vortex phase transitions [4]. Parallel columnar defects cause a Boseglass to superfluid transition in the vortex matter. The intrinsic planar structure in high-Tc superconductors in magnetic field directed parallel to the CuO2 layers leads to a "smectic-crystal"vortex-liquid transition. In the presence of random point defects, a vortex-glass to vortex-liquid phase transition was predicted to occur in the vortex matter [2, 3, 5-7] etc. In this framework, transport measurements bring up significant information in the process of studying the magnetic properties and phase transitions in the vortex matter [7].
In this article it is shown that the necessary conditions (pinning centers and thermal fluctuations) for the VG -VL phase transition observation are satisfied in the investigated Fe1.02 Se crystal. Nano-sized inclusions from the non-superconducting hexagonal phase obtained during synthesis are effective pinning centers, improve the irreversibility line and application potential. In spite of the low Tc of this superconductor thermal fluctuations play a considerable role. The VG -VL transition is confirmed by scaling of current-voltage, V-I, characteristics. The demonstrated scaling behavior extends the universality of the predicted VG theory to the binary chalcogenide system and shows the similarity of vortex matter behavior of Fe1.02Se with that in other iron-based superconductors and cuprates.

Experimental
The Fe1.02Se crystals were grown using a NaCl/KCl flux technique from pre-sintered powders mixed in the indicated nominal composition. All procedures (weighing, grinding, pressing) were carried out in an argon filled glove box. The synthesis was performed in evacuated (10 -3 Torr) and sealed quartz ampoules. More details of the crystals synthesis can be found elsewhere [8].
The crystal structure of the obtained samples was determined by XRD analysis using a Bruker D8 Advance diffractometer with Cu Kα radiation (λ = 1.54056 Å) and a LynxEye detector. The phase identification was performed with the Diffrac-plus EVA using an ICDD-PDF2 Database. The samples' morphology is characterized by a dual-beam scanning electron focused ion beam system (SEM/FIB LYRA I XMU, TESCAN), equipped with an EDX detector (Quantax 200, Bruker).
The transport measurements V(I) characteristics at different temperatures and R(T) dependencies at different magnetic fields) were performed using the Quantum Design 14T PPMS. The standard four-point geometry was used (Fig. 1a). In order to minimize the Joule heating effect, the DC current was applied for a very short period of 0.002 sec. The voltage signal was detected within an error margin of several nano-volts. The magnetic field was applied perpendicular to the current direction in the (101) plane of the sample (Fig.  1b). The sample's cross section is approximately S=0.229 x 0.02 cm 2 = 0.00458 cm 2 and the distance between the voltage contact pads is L=0.098 cm. The electric field (E), current density (J) and resistivity (ρ) have been determined from the basic relations: E=V/L; J=I/S and ρ=E/J.

Results and discussion
The primary reflexes in the X-ray diffractogram of the Fe1.02Se crystal indicate the presence of two phases. To identify them, additional investigations on a small piece of powdered crystal were performed. It was found that the main phase is a tetragonal one, P4/nmm, and its content is ~ 82%. The detected impurity phase is a hexagonal one, its quantity is about 18% with an average grain size of crystallites of ~46 nm. The coexistence of two crystallographycally different phases presumes a complex crystal morphology described as an aggregate of domains. However, it is well known that in cuprate superconductors nano-sized particles from various phases (Y-211, La-211, CeO2) could serve as effective pinning centers and enhance the critical current. In our case, the nano-sized nonsuperconducting inclusions naturally obtained during the preparation process of the investigated crystal are expected to pin efficiently the vortices in the mixed state. In Fig. 2a, the ρ(T) dependencies are presented for several magnetic fields in the range of 0-14T. From these curves the critical temperature Tc(ρ=0) and the superconducting transition width (10% -90%ρn) are determined, where ρn is the normal state resistivity (Fig. 2a). For increasing magnetic field, the ρ(T) curves shift to lower temperatures, and a broadening of the superconducting transition (from Δ T=1.77 K to ΔT = 2.58 K, respectively) is observed. The enhanced Tc ~ 10.6 K of the crystal, compared to Tc ~ 8.1K for a single phase FeSe crystal, is explained by the internal stress due to the presence of the impurity hexagonal phase. The increased transition width is usually accepted as an indication of thermally activated flux dynamic. A weak magnetoresistive (MR) effect is observed, when the magnetic field is increased from zero to 14 T. According to the relation: MR = [R(H,T) -R0 (H = 0,T)] / R0 (H = 0,T), the MR effect is estimated to be ~11% at T=15 K. A MR effect was reported in compensated semimetals [9] and in highquality FeSe and FeSe1-xSx single crystals obtained by the vapor transport method [10]. This MR effect is closely related to the multi-band structure of the FeSe compound. FeSe is a compensated semimetal with two types of carriers: electrons and holes [11,12]. In such materials, the MR effect is expected. The well-known structural transition determined in high-quality single crystals at Ts~86 K [10] and a nematic state established at T<Ts change dramatically the band structure and the properties of FeSe [13,14]. In this context, Ts is an important characteristic and its determination is essential. It has been found that in FeSe1-xSx single crystals Ts decreases to 49 K due to the chemical pressure caused by partial isovalent substitution of Se by S [10]. The R(T) dependence at H = 0 and its derivative dρ/dT are presented in the Fig. 2b. The anticipated structural transition in the investigated crystal occurs at Ts~60 K. By analogy this low Ts value could be related to internal pressure resulting from the presence of the hexagonal impurity phase. The magnetic field dependence of the pinning activation energy U(H) was determined as well. In the thermally activated flux flow regime (TAFF) and in the low current density limit the resistance is given by the relation: R(H,T) = R0 exp (-U (H,T)/kBT), where kB is the Boltzmann constant [15,16]. The experimental data for R(T) at different magnetic fields are presented in the Arrhenius plot, i.e. lnR plotted as a function of 1/T. The pinning activation energy is determined from the slope of the linear part of the curves for the given magnetic field. The obtained result for the U(H) dependence is presented in the Fig. 2c. The magnetic field dependence of the pinning activation energy is described by the relation: U(H) ∞ H -α with a crossover field Hcr=2.5 T from single-vortex pinning regime to the collectivepinning regime. Pinning investigations of another crystal from the same batch revealed the presence of bulk pinning also [17]. The U(H) dependence showing a similar trend, yields a crossover field Hcr=2.8 T. The strength of the pinning force density is defined by the ratio Jc/J0, where Jc is the depinning critical current and J0 is the depairing critical current [18]. Using the Bean critical state model, we established that Jc is in the order of 10 4 A/cm 2 (10 8 A/m 2 ) at temperatures below 6 K [19] for these crystals. Based on experimental results for FeSe [20] the J0 value is accepted to be ~10 11 A/m 2 . Thus, for the dimensionless ratio Jc/J0 we obtain the value ~10 -3 which is similar to the results obtained for YBCO superconductors (10 -3 -10 -2 ) [18]. This supports the assumption for the presence of weak pinning in these crystals, which is an important condition for the formation of disorder in the vortex matter.
Using the 10% and 90% ρn criterion, the irreversibility field Hirr(T/Tc) and upper critical field Hc2(T/Tc) lines are obtained, respectively, and presented in Fig. 2d. The Hirr(T) line determines the region of application, limiting the applied magnetic field at which the material can sustain its supercurrent, while Hc2(0) gives information on the pair-breaking mechanisms in magnetic field, the coherence length and anisotropic parameter. A very small anisotropy is characteristic for our samples even at temperatures close to Tc [17]. Large values of the Ginzburg-Landau parameter κ~72 are obtained for FeSe, [21] which further increase to κ ~104 in Ag doped FeSe. [20]. For comparison, the Hirr(T/Tc) and Hc2(T/Tc) line for a single-phase FeSe single crystal [21] are presented in Fig. 2d as well. The Hirr(T/Tc) line for the investigated sample is lightly shifted to higher temperatures (especialy for high fields) compared to the Hirr(T/Tc) line of the single crystal. This is another indication for the presence of effective pinning in the investigated FeSe crystal. In such strong type-II superconductors with effective pinning centers, a rich vortex phase diagram and phase transitions are expected to exist.    . For iron chalcogenides (Fe-11) the enhancement of Gi is a result of its low carrier density and its large effective mass m ~ (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) m0, where m0 is the free-electron mass [23]. Thus, thermal fluctuations are essential in cuprates (Gi ~10 -1 for YBCO [24]) and iron based superconductors (in particular for chalcogenides -Gi~10 -3 for FeSe [24]) although in the former they are mainly a result of the high Tc, and in the latter due to their low carrier density and large effective mass.
Comparing the Gi values for FeSe and the obtained pinning force density Gid = (Jc/J0) 3 for the investigated crystal we found that Gid << Gi which ensures the observation of the vortexglass to vortex-liquid transition [18].
Transport measurements were performed to confirm the existence of the mentioned transition in the mixed state. The current-voltage characteristics were measured in a wide temperature range (5K -12.5K) with an interval of 0.5 K at fixed magnetic field. In Fig. 3a and Fig. 4a the ρ-J curves for different temperatures are presented at H = 3T and 6T. In both cases, a crossover from a negative curvature (at low temperatures) to a positive curvature (at higher temperatures) is observed. The negative curvature of ρ-J isotherms is a signature of a vortex glass state [7]. At low temperatures the ρ-J isotherms almost overlap. In this case pinning barriers are high enough and almost prevent thermal fluctuations, which results in very small changes in the current. With increasing temperature, the distance between the ρ-J curves grows and the curvature of isotherms falls down until it disappears completely. Each ρ-J curve (in double logarithmic scale) was fitted by a quadratic polynomial function starting from low temperatures. The absolute value of the coefficient in the quadratic term decreases with increasing temperature until it reaches zero. The lowest temperature at which the quadratic term disappears and the ρ-J curve is linearly fitted is denoted as Tg. This procedure is used to determine Tg values at H=3T and H=6T. Going back to Fig. 2d it is important to mention that the two blue star points (Tg=9.0K, H=3T) and (Tg=8.0K, H=6T) determined from this fitting procedure are part of the line below which a vortex-glass state with zero resistance is realized. The Hirr(T/Tc) line obtained from ρ(T) curves at different magnetic fields using the 0.1ρn criterion is lightly shifted to higher fields and temperatures in comparison to (Tg, H) blue points. This indicates that the Tg value is properly determined in the frame of the given error (maximum error for Tg is estimated by the step size in T). At temperatures above Tg a positive curvature characterizing ρ-J isotherms, reflects the vortex liquid state and especially the TAFF regime [7]. A further increase of the temperature establishes the flux flow regime with ρ-J isotherms directed parallel to the x axis.  According to the VG theory [5] the existence of a continuous phase transition in the vortex matter can be proved by the collapse of E-J characteristics in two master curves from both side of Tg after scaling. At this phase transition a power low divergence for the correlation length ξg ~ |T-Tg| -ν and relaxation time τ ~ ξg z ~ |T-Tg| -νz is expected, where z and ν are dynamic and static scaling exponents, respectively. The vortex correlation length ξg represents the topological difference between fluctuating vortices in the VG state and their equilibrium configuration, while τ is the time needed for vortex relaxation [4]. According to [7] ξg can be estimated by using the relation: where Jc is the critical current density at T<Tg and D is the vortex dimensionality. At H=3 T and T=2 K the critical current density was determined to be Jc= 2.4.10 3 A/cm 2 [19] and the estimated value for the vortex correlation length is ξ g = 23.6 nm. This value of ξg is large enough compared with the lattice parameter c of the unit cell of the superconducting tetragonal phase (c=5.514 Å), confirming the 3D vortex dimensionality. At these conditions (H=3 T and T=2 K) the intervortex spacing a ~ (Ф0/B) 1/2 is approximately 26.2 nm and ξg<a as it should be in the VG state. For increasing temperature, the critical current decreases and ξg increases. At temperatures T>Tg as a result of a larger correlation length the opposite relation (ξg>a) is valid in the VL state.
For the scaling procedure, the universal scaling function is used: Y=(E/J)|1-T/Tg| ν(D-z-2) versus X= (J/T)|1-T/Tg| ν (1-D) . A precise determination of the scaling exponents is needed for a successful scaling. The predicted power law dependence for ρ(J) at T=Tg : [5] ρ (J, T=Tg) ~ J (z-1) / (D-1) was used for the determination of z. The dependence (3) presented in double logarithmic scale is a straight line and its slope is equal to (z-1)/(D-1). From log(ρ) vs log(J) curves at T=Tg=9.0 K (for H=3T) and Tg=8.0K (for H=6T) the slope of the lines was found (Fig. 3a and Fig. 4a). Assuming 3D dimensionality the following values were determined for the dynamic exponents: z=5.84±0.02 (H=3T) and z=5.40± 0.01(H=6T). The static exponent ν was determined from the resistivity vs temperature dependence (at H=3T and H=6T) at low current. In the vortex liquid state, T>Tg the linear resistivity at low current is given by: This dependence in double logarithmic scale is presented in Fig. 3b and Fig. 4b for H = 3T and H = 6T, respectively. From the slope of the linear part in these figures the following values for the static exponents are determined: ν = 0.26 ± 0.02 for H = 3T and ν = 0.30 ± 0.02 for H = 6T.  According to the vortex glass model [2] the scaling exponents should be field independent and their values should be z ≈ 4-7 and ν ≈ 1-2, respectively. Similar values are obtained for the investigated Fe1.02Se crystal in the field range between 3 T and 6 T. For the z exponent these values belong to the predicted interval, while for ν exponent a little bit smaller values are obtained than those predicted by the model. This is a result from the type of disorder and normal state resistivity [25] in the crystal. The investigated Fe1.02Se sample has a complex morphology described as an aggregate of domains. It's Jc(T) dependence is quadratic and characteristic for a SNS (superconductornormal metal-superconductor) path of the current through the sample [19]. In this context, the crystal might possess a similar granularity as polycrystalline samples reflecting the resistivity and static exponent respectively.

Conclusion
The investigated Fe1.02Se crystal contains nano-sized non-superconducting inclusions of a hexagonal impurity phase. These inclusions act as effective pinning centers and shift the Hirr(T) line to higher fields and temperatures in comparison to the Hirr(T) line of a single-phase single FeSe crystal. In spite of the low Tc of the investigated compound, thermal fluctuations are significant because of its large effective mass and its low carrier density. The interplay of pinning and thermal fluctuations results in a vortex-glass phase evidenced by the negative curvature of ρ-J isotherms at T<Tg and a vortex-liquid phase with a positive curvature of ρ-J isotherms at T>Tg. Using the universal scaling function of the VG model, the E-J dependencies are scaled in the two master curves for both states, thus confirming the VG-VL phase transition. The determined critical exponents are practically field independent with the z values in the model predicted interval and ν values a little bit smaller than the predicted by the model.