Universal scaling behavior of the upper critical field in strained FeSe_0.7Te_0.3 thin films

The temperature dependence of the resistance is shown in figure 1 for FeSe_0.7Te_0.3 films grown on different substrates (i.e. bare LSAT, CaF₂-buffered LSAT and bare CaF₂) measured in magnetic fields up to 9 T for H ∣∣ c. The dashed lines indicate the fit of the normal state just above the superconducting transition temperature T_c by R(T) = R₀ + AT, where R₀ is the resistance extrapolated to T = 0 K (i.e. residual resistance) and A is a constant. T_c is defined as 90% of the resistance in the normal state. As shown in figure 1(a), the lowest T_c of 6.2 K without magnetic field is measured for the films on bare LSAT substrate. T_c is nearly double by employing a 25 nm CaF₂ buffer layer (figure 1(b)). Furthermore, the FeSe_0.7Te_0.3 thin films on bare CaF₂ substrate have the highest T_c ∼ 18.1 K, shown in figure 1(c) (the data for additional films can be found in figure A3 in the appendix). The applied magnetic field suppresses T_c resulting in a monotonous broadening of the transition attributed to different temperature dependencies of H_c2 and irreversibility fields (H_irr). The temperature dependence of the normalized resistance for different films is shown in figure A2. The value of the residual resistivity ratio defined by R(300 K)/R(20 K) is consistent with the results reported by other groups [24, 25], and is nearly substrate independent.

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As mentioned above, the particularities of the crystal structure have a strong effect on T_c in FBS [26, 27]. Recently we found that T_c of the FeSe_0.7Te_0.3 films is very sensitive to in-plane lattice parameter a [21]. T_c as a function of the a-axis is shown in figure 2 for a number of FeSe_0.7Te_0.3 films grown on the mentioned substrates. For comparison, values of optimally doped Ba(Fe_0.92Co_0.08)₂As₂ thin films on different substrates are also plotted [28]. The films have different a-axis lengths due to different in-plane compressive strain resulting mainly from the large thermal misfit between the substrates and the FeSe_1−xTe_x layer [21, 22, 28–32]. It is apparent that the superconducting transition temperature T_c decreases linearly with increasing a-axis lattice parameter for the FeSe_0.7Te_0.3 films, which is consistent with reports of other groups [33–35]. A linear dependence of T_c on the crystallographic a-axis was also found for Ba(Fe,Co)₂As₂ thin films [28]. However, FeSe_0.7Te_0.3 has a much steeper slope. The high sensitivity to strain in the FeSe system can be attributed to the presence of shallow Fermi pockets (with small Fermi energy ε_F) [34]. The strain shifts slightly the position of the bands with respect to the chemical potential resulting in a considerable change of the small ε_F value or even the appearance of a Lifshitz transition [34]. These changes of the electronic structure can affect T_c as was demonstrated before for the 122 system [30]. This allows us to vary T_c of the FeSe_0.7Te_0.3 films significantly solely by in-plane strain.

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Figure 3 shows H_c2 for the films on different substrates as a function of T_c for fields parallel to the c axis (the data for additional films can be found in figure A3(e) in the appendix) with 90% criteria for T_c. To describe the H_c2 curves for the FeSe_0.7Te_0.3 compound, the spin paramagnetic and the orbital pair-breaking effects should be taken into account [36–38]. A commonly used approach is the WHH theory generalized for multiband superconductors [39]. However, for the reliable determination of the paramagnetic and multiband effects, this analysis requires H_c2 values in a broad temperature range. Due to very high H_c2 values of this compound, only the data near T_c are accessible in our experiments. Therefore, we used alternatively an analysis based on the Ginzburg–Landau theory, which provides a simple analytical expression for the temperature dependence of H_c2(T) near T_c [40–42]. H_c2 and its slope near T_c can be estimated by the following equation:

$$\begin{eqnarray}&&{H}_{{\rm{c}}2}={H}_{{\rm{c}}2}(0)\left[\displaystyle \frac{1-{t}^{2}}{1+{t}^{2}}\right],\end{eqnarray} \tag{ 1 }$$

where t = T/T_c is the reduced temperature and H_c2(0) is the upper critical field values extrapolated to T = 0. We fitted our H_c2 data using equation (1), as shown by the dashed line in figure 3. The slope −dH_c2/dT at T_c is defined using equation (1) at t = 1 (figure 4). The analysis including paramagnetic effects results in quantitative changes of the slope (see figure A4 in the appendix) [43]. However, the functional dependence of the slope on T_c is qualitatively unchanged (figure 4 below and figure A4) in spite of relatively strong paramagnetic effects with the Maki parameter α_M = $\surd 2{{H}^{* }}_{{\rm{c}}2}/{H}_{{\rm{p}}}$ given in inset of figure A4 in the appendix, where ${{H}^{* }}_{{\rm{c}}2}$ is the orbital limited upper critical field and H_p is the paramagnetic critical field. We found also that α_M ∝ T_c in accord with the scaling behavior of H_c2 discussed in the next section. The large values of the α_M are consistent with previous studies of the FeSe_1−xTe_x system [44].

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The derivative of the upper critical field with temperature −dH_c2/dT versus T_c is shown in figure 4. As can be seen, the slope −dH_c2/dT of FeSe_0.7Te_0.3 thin films depends almost linearly on T_c in accordance with the behavior −dH_c2/dT ∝ T_c found in some other pnictides [12, 13]. We note that the choice of criteria does not change the observed linear dependence qualitatively. The −dH_c2/dT data using 50% of the resistance in the normal state as the criterion of T_c is shown in the inset of figure 4. In contrast to the 90% criterion, the slope does not reach zero by an extrapolation to T_c = 0, which indicates that −dH_c2/dT is affected by additional T_c independent contributions (such as sample inhomogeneity) not related to H_c2. Therefore, for further analysis we focused on the data obtained with the 90% criterion.

A linear behavior of −dH_c2/dT ∝ T_c/ν_F² is expected for a single-band superconductor in the clean limit, if the Fermi velocity (ν_F) is the same for different samples [39]. For our samples, the linear relation between T_c and lattice parameters (figure 2) indicates that the mechanism for a T_c suppression is related to the modification of the electronic properties, which should result in a variation of ν_F with strain. Therefore, for a clean limit we expect a deviation from the linear behavior −dH_c2/dT ∝ T_c. In a dirty limit, the slope −dH_c2/dT ∝ 1/D of an s-wave superconductor is independent on T_c, where D is an effective diffusivity constant [45]. It is known that T_c of a single-band s-wave superconductor can be suppressed by magnetic impurities. However, smaller D values (stronger impurity scattering) result in a lower T_c and higher −dH_c2/dT in contrast to the experimental observations. Therefore, for our films effectively a single-band s-wave superconductivity cannot reconcile the whole set of the experimental data.

In the case of superconductivity driven by a single leading interband interaction such as s_±, the minimal model that describes the system close to T_c is a two-band model [46]. The most of available experimental data are consistent with this picture. So far we have found that only hole overdoped Ba_1−xK_xFe₂As₂ system that breaks time reversal symmetry in the superconducting state cannot be described by a two-band model [47]. In this case, a three-band model is needed to describe the superconducting properties [48]. However, general scaling behaviors do not hold in this case (see introduction in [47]). Therefore, the observed university in scaling behavior of H_c2 relates our FeSe_0.7Te_0.3 films to the majority FBS with superconductivity driven by a single leading interband interaction.

It was shown that the pair-breaking parameters do not alter the slope for dirty superconductors with a sign changed order parameter such as s_± superconductors and for two symmetrical bands the slope is −dH_c2/dT  ∝ T_c/$\langle {W}^{2}{{\nu }_{{\rm{F}}}}^{2}\rangle ,$ where Ω is the variation of the gap along the Fermi surface [12, 16]. The symmetrical case can be excluded due to the expected variation of the Fermi velocity with strain, which results in a deviation from the linear behavior. The symmetrical case is also inconsistent with the available angle-resolved photoemission spectroscopy (ARPES) data [49]. The effect of impurities on a realistic s_± superconductor (non-symmetrical bands and non-zero intraband coupling) is rather complex and results for strong enough impurities in a transition to the s₊₊ superconducting state [50]. Therefore, an interpretation of the observed linear behavior based on a strong pair breaking effect is also doubtful for the 11 system.

In a clean two-band s-wave superconductor, the slope of H_c2 is defined by a combination of the Fermi velocities and coupling constants for different bands:

$$\begin{eqnarray}&&{\left.-\displaystyle \frac{{\rm{d}}{H}_{{\rm{c}}2}}{{\rm{d}}T}\right|}_{T{\rm{c}}}\propto \displaystyle \frac{{T}_{{\rm{c}}}}{({a}_{1}{v}_{1}^{2}+{a}_{2}{v}_{2}^{2})},\end{eqnarray} \tag{ 2 }$$

where ${a}_{1}=\tfrac{\sqrt{{({\lambda }_{11}-{\lambda }_{22})}^{2}+4{\lambda }_{12}{\lambda }_{21}}+{\lambda }_{11}-{\lambda }_{22}}{2({\lambda }_{11}{\lambda }_{22}-{\lambda }_{12}{\lambda }_{21})}$ and ${a}_{{\rm{2}}}=\tfrac{\sqrt{{({\lambda }_{11}-{\lambda }_{22})}^{2}+4{\lambda }_{12}{\lambda }_{21}}-{\lambda }_{11}+{\lambda }_{22}}{2({\lambda }_{11}{\lambda }_{22}-{\lambda }_{12}{\lambda }_{21})}$ are constants, which depend on the intraband λ₁₁, λ₂₂ and interband λ₁₂, λ₂₁ coupling constants [39]. a₁ ∼ a₂ holds in the case of an extreme s_± superconductivity with the dominant interband coupling, which presumably is the case for FeSe_1−xTe_x [51, 52], and ${a}_{1}\gg {a}_{2}$ for an extreme s₊₊ case with the dominant intraband coupling. The latter can be excluded based on the arguments for a single-band case since the leading band dominates the superconducting properties close to T_c. In the case of strong interband coupling, the universal behavior −dH_c2/dT ∝ T_c would indicate that the combination ${a}_{1}{v}_{1}^{2}+{a}_{2}{v}_{2}^{2}$ is nearly strain independent. It is known that the value of the Fermi velocities considerably varies between different bands in the 11 system [49]. Therefore, according to equation (2), −dH_c2/dT is dominated by the fastest Fermi velocity assuming sizeable interband coupling, which is expected in the case of the strongly anisotropic sign change superconducting gap [50, 51] or special s₊₊ case [53]. The linear scaling indicates that the fastest Fermi velocity is weakly sensitive to strain assuming a weak variation of the coupling constants with strain. In this case, T_c is mainly defined by the band/bands with low Fermi velocities forming small Fermi surface pockets. This is consistent with empirical conclusions based on the ARPES measurements of various pnictides [54]. The universality of the observed scaling −dH_c2/dT ∝ T_c for different FBS imposes constrain on the possible pairing mechanism and indicates a key role of the interband interactions.

The structural properties and composition of the films were analyzed using TEM. The data for the two representative films are shown in figure A1 indicating a homogeneous stoichiometry over the film thickness.

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In this section, we provide additional electrical resistivity data (not shown in the main text) measured in zero and applied magnetic field.

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To take into account paramagnetic pair-breaking effects we used an analysis based on the Ginzburg–Landau theory, which provides a simple analytical expression for the temperature dependence of H_c2(T) near T_c including paramagnetic effects [43]. As shown by Mineev et al H_c2 can be calculated for clean single-band superconductors using the following equation:

$$\begin{eqnarray}&&{H}_{{\rm{c}}2}=\displaystyle \frac{e\gamma {T}_{{\rm{c}}}^{2}}{a{\mu }^{2}}\left[-1+\sqrt{1+\displaystyle \frac{{\alpha }_{0}a{\mu }^{2}}{{(e\gamma {T}_{{\rm{c}}})}^{2}}\displaystyle \frac{{T}_{{\rm{c}}}-T}{{T}_{{\rm{c}}}}}\right],\end{eqnarray} \tag{ A.1 }$$

where T_c is the critical temperature at zero field and ${\alpha }_{0}={N}_{0},$ $a=7\zeta (3){N}_{0}/4{\pi }^{2},$ $\gamma =7\zeta (3){N}_{0}{{\rm{\nu }}}_{{\rm{F}}}^{2}/32{\pi }^{2}{T}_{{\rm{c}}}^{2}.$ Here, ${N}_{0}$ is the density of states at the Fermi level, v_F is the Fermi velocity and μ is the magnetic moment.

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The slope −dH_c2/dT at T_c is given by:

$$\begin{eqnarray}&&-\displaystyle \frac{{\rm{d}}{H}_{{\rm{c}}2}}{{\rm{d}}T}{| }_{{T}_{{\rm{c}}}}=\displaystyle \frac{{{\rm{\alpha }}}_{0}}{2e\gamma {T}_{{\rm{c}}}}\end{eqnarray} \tag{ A.2 }$$

with the Maki parameter defined as:

$$\begin{eqnarray}&&{\alpha }_{{\rm{M}}}^{2}=\displaystyle \frac{{\alpha }_{0}a{\mu }^{2}}{{(e\gamma {T}_{{\rm{c}}})}^{2}}.\end{eqnarray} \tag{ A.3 }$$

Substituting equations (A.2) and (A.3) to equation (A.1) we obtain

$$\begin{eqnarray}&&{H}_{{\rm{c}}2}=\displaystyle \frac{2}{{\alpha }_{{\rm{M}}}^{2}}\left(-\displaystyle \frac{{\rm{d}}{H}_{{\rm{c}}2}}{{\rm{d}}T}{| }_{{T}_{{\rm{c}}}}\right){T}_{{\rm{c}}}\left[-1+\sqrt{1+{\alpha }_{{\rm{M}}}^{2}\displaystyle \frac{{T}_{{\rm{c}}}-T}{{T}_{{\rm{c}}}}}\right].\end{eqnarray} \tag{ A.4 }$$

We fitted our upper critical field data using equation (A.4), as shown by the dash line in figure A4. The obtained slope of the upper critical field −dH_c2/dT is shown in figure A4 and the Maki parameter in the inset. Both quantities are proportional to T_c as expected (see also main text) [39].

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