Sample preparation and electrical measurements

Epitaxial NdFeAs(O,F) thin films of around 90 nm thickness were fabricated on MgO(001) single crystals by molecular beam epitaxy (MBE). For fluorine doping, the NdOF over-layer technique has been employed in which the mother compound NdFeAsO is deposited, followed by a NdOF over-layer. Both NdFeAsO and NdOF were deposited at 800 °C. More details about sample preparation can be found in Ref. Reference Kawaguchi, Uemura, Ohno, Tabuchi, Ujihara, Takeda and Ikuta9.

For the electrical transport measurements, a microbridge of 30 µm width and 1 mm length was structured by laser cutting. The V(I) characteristics as well as the temperature dependence of the resistance, R(T), were measured in four-point geometry in a Quantum Design 14 T PPMS with Keithley current source 2460 and Nanovoltmeter 2482A. The current and voltage leads were attached at the contact pads via Pogo pins. The film investigated had an onset T _c^on of 44.2 K, which is determined as the extrapolation of largest slope in R(T) toward the temperature-dependent normal state resistance R _n(T), corresponding to around 90% of R _n [Fig. 1(a)]. The upper critical field B _c2(T) was determined with the same 90% R _n criterion. The V(I) characteristics were measured up to a voltage level of 0.1 mV for constant magnetic fields (B||c) between 1 and 13 T (step size 2 T) at temperature steps of 0.4 K between 4 and 35 K. The lnV(lnI) curves were fitted with first- and second-order polynomials with different fit procedures to extract glass–liquid temperature T _g (disappearance of quadratic term) and the n value (n = dlnV/dlnI) near the critical current density J _c, which itself was determined using an electric field criterion of 1 µV/cm.

Figure 1.

Temperature dependence of the resistance, R(T), for self-field as well as between 1 and 13 T (step size 2 T) in (a) normal and (b) Arrhenius representation. The green lines are fits according to Eq. (1) and the red lines according to R = R _n exp(−U _eff/kT)^s. (c) Scaling behavior of the resistance curves at fields between 1 and 13 T (step size 2 T) according to Ref. Reference Liu, Wu, Xu, Wu and Shao24. The data of the critical region show power law behavior with exponent s = ν(z−1) changing from 8.8 to 10.5 due to the T-dependence of z [Fig. 4(c)]. Consequently, true scaling of all curves is not observed.

Results and discussion

The resistive transition to the superconducting state of the film [Fig. 1(a)] shows the typical broadening with increasing magnetic fields (B||c) due to thermally activated flux motion. The activation energy U ₀ for flux motion is usually determined by linear fits to the relevant region in the respective Arrhenius plots [Fig. 1(b)]. This analysis is, however, only valid for activation energies which are either constant or linearly dependent on temperature. In those cases, the derivative U’ = –dlnR/dT ⁻¹ is temperature-independent. As shown in Fig. 2(a), this is not the case for the investigated sample. To evaluate this further, we follow here the analysis by Zhang et al.^[Reference Zhang, Ren and Zhao^6] based on the early work of Palstra et al.^[Reference Palstra, Batlogg, Dover, Schneemeyer and Waszczak^10]: for activated flux motion, the resistance is approximated by

(1) $$R(T,B) = ({\rm 2}R_0U(B,T)/kT){\rm exp}(-U(B,T)/kT),$$

where the prefactor 2R ₀U(B,T)/kT (k…Boltzmann constant) is generally T-dependent, in contrast to the usual analysis. If U(B,T) is separated as U = U ₀(B) (1–t)^q, t = T/T _c, the temperature-dependent derivative U’ for each magnetic field reads as follows:

(2) $$\eqalign{ U{\vskip-5.8pt \hskip-1pt,}(T) & = -d{\rm ln}R/dT^{-{\rm 1}} \cr & = [U_0(B)({\rm 1}-t)^q-{\rm} kT][{\rm 1} + qt/({\rm 1}-t)].} $$

Whereas the exponent q is field-dependent in general as proposed by Zhang et al.,^[Reference Zhang, Wen and Wang^11] it is set constant here in order to reduce arbitrariness. The green line in Fig. 2(a) is a fit to the U’ data at 13 T with Eq. (2), and the green lines in Figs. 1(a) and 1(b) show the corresponding fits with Eq. (1) to all R(T) data. Best fits were achieved with a T _c value of 44.0 K (corresponding well to the onset T _c value) and q = 1.55. The zero-temperature activation energy U ₀ is shown in Fig. 2(c) in comparison to values from the usual linear fits to the Arrhenius plots. Both show B ⁻¹ dependence in magnetic fields above 4 T, indicative of collective pinning. For lower magnetic fields, the logarithmic slope of U ₀(B) is gradually decreasing, which is expected for the transition to the single vortex pinning regime. The Arrhenius plot values (corresponding to q = 1) are underestimations but may be regarded as U ₀ values near the GLT. U ₀ is very comparable to literature data (q = 1) on a NdFeAs(O,F) single crystal,^[Reference Jaroszynski, Hunte, Balicas, Jo, Raicevic, Gurevich, Larbalestier, Balakirev, Fang, Cheng, Jia and Wen^12] whereas the SmFeAsO_0.85 single crystal^[Reference Lee, Bartkowiak, Kim and Lee^5] and the SmFeAs(O,F) thin film^[Reference Iida, Hänisch, Tarantini, Kurth, Jaroszynski, Ueda, Naito, Ichinose, Tsukada, Reich, Grinenko, Schultz and Holzapfel^13] show around three times higher values and the LaFeAs(O,F) thin film^[Reference Kidszun, Haindl, Reich, Hänisch, Iida, Schultz and Holzapfel^14] one order of magnitude lower values. This is closely correlated to T _c [being around 25 K (La), 45 K (Nd), and 55 K (Sm)] but individual differences in defect density and pinning strength (e.g., F-doped versus O-deficient samples^[Reference Liu, Chai, Kim, Stewart and Kim^7] and F-content dependence^[Reference Shahbazi, Wang, Shekhar, Srivastava, Lin, Zhu and Dou^15]) certainly play a role. Polycrystalline samples of Fig. 2(c)’s Ln1111 compounds usually exhibit slightly larger values of U ₀ in the literature (due to the averaging effect between different crystallite orientations and increased density of strongly pinning defects). At medium fields, U ₀ is inversely proportional to B for all samples in Fig. 2(c), at high fields U ₀(B) is well approximated by B ^−a(1−B/B ₀)^b[Reference Thompson, Sorge, Cantoni, Kerchner, Christen and Paranthaman^16] with a = b = 1 and reasonable values for B ₀, which is close to the irreversibility field at 0 K.

Figure 2.

Derivatives of R(T) at B = 13 T. (a) Apparent activation energy U ’(T), (b) derivative D = (dlnR/dlnT)⁻¹. The green lines are fits according to Eq. (2), and the red lines according to Eq. (8). Both derivatives are related as U ’ = T ²/D. (c) Magnetic field dependence of the thermal activation energy for flux motion U ₀ for B||c of the NdFeAs(O,F) film (red) in comparison to NdFeAs(O,F)^[Reference Jaroszynski, Hunte, Balicas, Jo, Raicevic, Gurevich, Larbalestier, Balakirev, Fang, Cheng, Jia and Wen^12] and SmFeAsO_0.85^[Reference Lee, Bartkowiak, Kim and Lee^5] single crystals as well as SmFeAs(O,F)^[Reference Iida, Hänisch, Tarantini, Kurth, Jaroszynski, Ueda, Naito, Ichinose, Tsukada, Reich, Grinenko, Schultz and Holzapfel^13] and LaFeAs(O,F)^[Reference Kidszun, Haindl, Reich, Hänisch, Iida, Schultz and Holzapfel^14] epitaxial thin film. Open symbols: films, closed symbols: single crystals. q = 1: from linear fits of the Arrhenius plots (with T _c = 44.2 K), q = 1.55: determined via fits of U ’ with Eq. (2) (with T _c = 44.0 K). The former values can be regarded as apparent activation energies near the glass–liquid transition. At medium fields, U ₀ is inversely proportional to B, at high fields U ₀(B) is well approximated by B ^−a(1−B/B ₀)^b with a = b = 1 and reasonable values for B ₀. The errors in U ₀ are roughly the symbol size.

The theory of vortex GLT^[Reference Koch, Foglietti, Gallagher, Koren, Gupta and Fisher^1] predicts that resistance and current (or more generally resistivity ρ and current density J) can be scaled according to

(3) $$V/I(T-T_{\rm g})^{-\nu (z + {\rm 2}-d)},$$

(4) $$I/T^\alpha (T-T_{\rm g})^{\nu ({\rm 1}-d)}, $$

where ν and z are the static and the dynamic critical exponent, respectively, and d the dimensionality of the system. Equation (4) was modified by the factor 1/T ^α with α = [0,1] in order to check for these both scaling methods found in the literature. Yamasaki et al.^[Reference Yamasaki, Endo, Kosaka, Umeda, Yoshida and Kajimura^17] pointed out that if the scaling parameter J ₀ (crossover current density) is proportional to kT as predicted by Koch et al., α should be 1, as is found for several compounds such as YBa₂Cu₃O₇,^[Reference Safar, Foltyn, Jia and Maley^18] Co-doped Ba122,^[Reference Hao, Zhang, Teng, Yin, Jiao, Cao and Li^4] SmFeAsO_0.85,^[Reference Lee, Bartkowiak, Kim and Lee^5] and MgB₂.^[Reference Kim, Kang, Choi, Kim, Kim and Lee^19] Many other studies (including on similar samples as listed above and also interestingly by Koch et al. themselves^[Reference Koch, Foglietti and Fisher^20]) find good scaling instead for α = 0 (i.e., without the extra factor 1/T). This can be explained by J ₀ being proportional to a temperature-independent thermal-fluctuation activation energy U _tf instead of kT.^[Reference Yamasaki, Endo, Kosaka, Umeda, Yoshida and Kajimura^17] For the sample in this study, scaling was also only possible for α = 0.

The V(I) curve exactly at T _g is following a perfect power law with exponent

(5) $$n_{{\rm Tg}} = (z + {\rm 1})/(d-{\rm 1}). $$

We determined this temperature by the disappearance of the quadratic term of second-order polynomial fits to the lnV(lnI) curves, adopted from Ref. Reference Heinig, Redwing, Nordman and Larbalestier21. Figure 3 shows the set of V(I) curves (a), their logarithmic derivatives near T _g (b), and the scaling according to Eqs. (3) and (4) for d = 3 (3D case) (c) for B = 13 T exemplarily. The behavior of the other data sets is accordingly and of the same quality. The V(I) curve closest to T _g not only shows the smallest variance in n value for different fit procedures but also the largest range of constant logarithmic derivative [Fig. 3(b)].

Figure 3.

(a) V(I) curves, (b) their logarithmic derivatives, and (c) their scaling near T _g of the investigated NdFeAs(O,F) film exemplarily for B = 13 T.

From scaling behavior alone, it is not possible to discern between the different cases: 3D vortex glass (VG), (quasi-)2D vortex glass, and Bose glass (BG, for the presence of strong columnar pinning centers), since their respective scaling parameters are related to each other via:

(6) $${\rm BG}-{\rm VG}\, (3{\rm D}):\nu _{{\rm BG}} = {\rm 2}/{\rm 3} \nu _{{\rm VG}},z_{{\rm BG}} = ({\rm 3}z_{{\rm VG}} + {\rm 1})/{\rm 2,}$$

(7) $$\hskip-2.6pc{\rm 2D}-{\rm 3D}\, ({\rm VG}):\nu _{{\rm 2D}} = {\rm 2}\nu _{{\rm 3D}},z_{{\rm 2D}} = (z_{{\rm 3D}}-{\rm 1})/{\rm 2}{\rm.} $$

Certain types of glass phases have to be ruled out instead by their parameters lying outside the expected range, i.e., ν _VG = 1…2 and z _VG > 4 and usually <6 for vortex glass, and ν _BG ~ 0.8…1.8 and z _BG ~ 6…9 for Bose glass.^[Reference Trastoy, Ruoco, Ulysse, Bernard, Palau, Puig, Faini, Lesueur, Briatico and Villegas^22] For the NdFeAs(O,F) film of this study, a Bose glass could be ruled out in that way (z > 9) and also because strong correlated c-axis-aligned defects are unlikely in such MBE-grown films. Typical defects are uncorrelated regions of differing stoichiometry in the range of 10…20 nm as measured via atom probe tomography (not shown here). Discerning between 2D and 3D via the scaling parameters alone is not possible for the investigated sample (2D: ν > 2 and z in the expected range, 3D: ν in the range but z > 6). On the other hand, the slope of B _c2||c near T _c, dB _c2/dT|_Tc = −2.1 T/K, and a T _c of 44.2 K result in an orbital upper critical field B _c2^orb(0 K) = 64 T and therefore in a coherence length ξ _ab = 2.25 nm. With a typical anisotropy parameter γ for this type of NdFeAs(O,F) films with thickness >50 nm of around 4.5,^[Reference Tarantini, Iida, Sumiya, Chihara, Hatano, Ikuta, Singh, Newman and Larbalestier^23] the coherence length in c-direction can be estimated to ξ _c ~ 0.5 nm. The crossover temperature between 2D and 3D behavior in layered superconductors, T _cr = (1–2(ξ _c/d)²)T _c, where d here is the inter-layer distance, i.e., the sample's c-axis parameter of 0.8543 nm, is therefore around 14 K. Even though close to the transition to 2D behavior, the measured glass–liquid temperatures of this study are within the 3D region above T _cr, which leaves 3D vortex glass scaling as best option, and the large (and T-dependent) z values have a different reason as discussed below.

As visible in Fig. 3(b), T _g at 13 T is 15.2 ± 0.4 K (the maximum error in T _g was estimated by the step size in T), and n _Tg = 4.45. With Eq. (5), z is 7.9 ± 0.1 at 13 T. The resistance will go to zero for T→ T _g as

(8) $$R \sim (T-T_{\rm g})^{s},{\rm with\ exponent}\,s = \nu (z + {\rm 2}-d), $$

within the critical region (Vogel–Fulcher relationship). That means, T _g and z being known, ν can be evaluated from the behavior of R near T _g. If Eq. (8) holds, the derivative D = (dlnR/dT)⁻¹ goes linearly to zero for T→T _g with slope 1/s. This is shown in Fig. 2(b) for 13 T. The value of s in that case is 8.8 ± 0.1, and the according value of ν = 1.27 ± 0.05. The two derivatives U’ and D are related as U’ = T ²/D. That means they can easily be converted in the other form and cross-checked for consistency [dashed lines in Figs. 2(a) and 2(b)] to reduce the errors in U ₀ and s (and hence ν). The temperature T* (23.7 ± 0.2 K at 13 T) at which both derivatives start to fail to fit the data determines the upper bound of the critical region. It is worth mentioning that solely from R(T) curves T _g can only be estimated with relatively large error bars in many cases, and T _g and T* are usually overestimated as well as s and hence ν underestimated.

As Liu et al. showed, an effective activation energy U _eff = kT (T _c−T)/(T−T _g) for thermal activation of flux motion in the critical region leads to the observed power law dependence of R near T _g and is consistent with the vortex-glass theory.^[Reference Liu, Wu, Xu, Wu and Shao^24] That means, R can be scaled in the critical region with a temperature scale (T−T _g)/(T _c−T _g). Figure 1(c) compares these dependencies for the measured fields between 1 and 13 T. Due to an increasing z value for increasing T (or decreasing B), s (expected values 2.7…8.5 for a 3D vortex glass) is not constant and perfect scaling is not observed. However, the critical region between T _g and T* can be identified clearly. A similar scaling approach by Andersson et al.^[Reference Andersson, Rydh and Rapp^25] with temperature scale [T(T _c−T _g)/T _g(T _c−T)]−1 also does not lead to scaling for this sample and, furthermore, does not resolve the critical region correctly. The latter scaling approach has nonetheless been successfully applied to Ba122 single crystals^[Reference Ghorbani, Wang, Shahbazi, Dou, Choi and Lin^26] and (Li_1−xFe_xOH)FeSe.^[Reference Yi, Wang, Tang, Peng, Qiu, Xu, Sun, Luo and Yu^27] The red lines in Fig. 1(b) are fits according to R = R _n exp(−U _eff/kT)^s[Reference Liu, Wu, Xu, Wu and Shao^24] with U _eff given above.

The results of this study are summarized in Fig. 4: Three clearly distinct regions in the magnetic field–temperature phase diagram [Fig. 4(a)] of this Nd1111 film are visible: The GLT [T _g or B _g(T)] separates vortex glass and vortex liquid. The vortex liquid shows a critical region near T _g as well as a thermal-activation region, which are separated by T* [i.e., B*(T)]. Nonetheless, the resistance within the critical region can also be described by thermal activation with an effective pinning potential U _eff as described above. The dashed line separating thermal activation and transition region (determined by inhomogeneity broadening and fluctuations) denotes the temperatures at which Eqs. 1 and 2 fail to fit R(T) [Fig. 1(a)]. Whereas the static exponent ν [Fig. 4(b)] is constant with temperature (or field) at around 1.35 within the error bars (regarding scaling behavior), the dynamic critical exponent z [Fig. 4(c)] is increasing with temperature (or as often visualized in the literature decreasing with increasing field) as mentioned above. Forcing z to be constant leads to non-scaling of V(I) and offsets in T _g. Similarly high values of z (while ν is more or less constant) have been observed for YBa₂Cu₃O₇ films by Voss-de Haan et al.^[Reference Voss-de Haan, Jakob and Adrian^28] The authors pointed out that reducing the range in electrical field of the V(I) curves would lead to lower z values. The dashed red line is a fit according to B _g ~ (1−T/T _g0)^m with m = 1.5 ± 0.1, where T _g0 is the GLT temperature in zero applied field. An exponent m around 1.5 is often found for sufficiently high magnetic fields.

Figure 4.

(a) B−T phase diagram for B||c of a 90 nm-thick NdFeAs(O,F) film on MgO(001) together with the temperature dependencies of the critical exponents (b) ν and (c) z.

The theory of GLT has not been unquestioned. Coppersmith et al.^[Reference Coppersmith, Inui and Littlewood^29] showed very early that the (scaling) V(I) curves could in principle be explained by pure thermal activation of flux line motion, where the counterargument, however, is the universality of the scaling parameters.^[Reference Koch, Foglietti and Fisher^20] Similar results were achieved, e.g., by Hu et al.^[Reference Hu, He, Ning, Chen, Yin, Lu, Xu, Guo, Wang, Li and Yin^30] for theoretical V(I) curves on YBa₂Cu₃O₇ films. Strachan et al.^[Reference Strachan, Sullivan, Fournier, Pai, Venkatesan and Lobb^31] showed that often a unique glass–liquid temperature cannot be deduced from logarithmic derivatives of the V(I) curves nor from successful scaling, which might explain the large range of the critical parameters in the literature. They found a peak in dlnV/dlnI instead of a constant value at T _g. Such an effect, however, was not seen for our sample [Fig. 3(b)]. Sullivan et al.^[Reference Sullivan, Frederiksen, Repaci, Strachan, Ott and Lobb^32] finally pointed out that the V(I) curves near the transition can be heavily altered by noise effects. We therefore ensured to measure R(T) as well as V(I) with minimum noise and to avoid possible spurious effects as well as to disregard any data which may be affected by run-away temperature. All of the above critics come to the conclusion that scaling alone is not sufficient evidence for the presence of a vortex phase transition. Other authors on the contrary conclude that a GLT and thermal activation of flux motion do simply not necessarily have to exclude each other,^[Reference Liu, Wu, Xu, Wu and Shao^24, Reference Andersson, Rydh and Rapp^25] see above. Matsushita et al.^[Reference Matsushita, Tohdoh and Ihara^33] developed a model that describes the influence of disorder and J _c variance on the scaling behavior of the V(I) curves, relating the critical exponents ν and z with new parameters m = (z−1)/2 (J _c variance) and δ = 2ν (T-dependence of J _c), which leads to, in general, T-dependent parameters ν and z. This model can explain the large and T-dependent z values for our sample.

In the end, however, it is a matter of interpretation whether one regards the transport behavior as GLT altered by thermal activation and disorder or as thermal activation mimicking a phase transition, since it is not decidable from the data alone according to the literature. Whether possibly cleaner Nd1111 (and other Ln1111) film samples with maximized T _c would show glass–liquid behavior with constant critical parameters in the expected range is an interesting task for future investigations.