Complex conductivity of FeSe1–x Te x (x = 0 – 0.5) films

We measured the complex conductivity, σ, of the FeSe1−x Te x (x = 0 – 0.5) films below T c which show a drastic increase of the superconducting transition temperature, T c, when the nematic order disappears. Since the magnetic penetration depth, λ (> 400 nm) of Fe(Se, Te) is longer than the typical thickness of the film (∼100 nm), we combined the coplanar-waveguide-resonator- and cavity-perturbation techniques to evaluate both the real and imaginary parts of σ. Films with the nematic order showed a qualitatively different behavior of the quasiparticle scattering time compared with those without the nematic order, suggesting that the nematic order influences the superconducting gap structure. On the other hand, the proportionality between the superfluid density, n s/m* (∝ λ−2), and T c was observed irrespective of the presence or absence of the nematic order. This result indicates that the amount of the superfluid has a stronger impact on T c of Fe(Se, Te) than the presence or absence of the nematic order itself.


Introduction
An iron chalcogenide superconductor, FeSe, has been intensively studied [1,2,3] because of its various intriguing properties: the potential ability for the high-transition-temperature superconductivity, the absence of the magnetic order under ambient pressure, and the exotic electronic states due to the extremely small Fermi surface. The superconducting transition temperature, T c , can be largely enhanced above 40 K from 9 K [4] by the intercalation [5], the carrier doping using the electron-double-layer-transistor [6,7], and the synthesis of a monolayer film [8,9]. The nematic phase without the magnetic order in FeSe is ideal for studying the origin of the nematicity [10] and the relationship between the nematicity and the superconductivity [11]. Also, its small Fermi surface (ϵ F < 10 meV) can easily be tuned by the hydrostatic pressure [12], the chemical pressure by the isovalent substitution [13,14,15], and the in-plane lattice strain [16]. Since the changes in the Fermi surface influence all of the superconducting phase, the nematic phase, and the magnetic phase, various different techniques have been applied to investigate the electronic phase diagram and the exotic superconductivity of FeSe.
Among the above-mentioned techniques to control the electronic state, the chemical isovalent substitution is advantageous since experiments can be performed under the ambient pressure. The S-substitution shrinks the lattice of FeSe, corresponding to the positive chemical pressure.  With increasing S content, the nematic transition temperature, T N , decreased, and T c once slightly increased and then decreased [17]. Although there were no significant changes in T c when the nematic order disappeared, some abrupt changes in the superconducting gap have been observed in the measurement of thermal properties [18] and the scanning tunneling microscopy/spectroscopy [19]. Thus, the nematic order or its fluctuation still may have some influences on the superconducting state. On the other hand, systematic investigations of Tesubstituted FeSe, which is subjected to the negative chemical pressure, were fewer than Fe(Se,S) since the systematic synthesis of bulk Fe(Se,Te) had been hindered by the phase separation region until recently [20]. Although the superconducting gap structure of FeSe [21,22] is distinctly different from Fe(Se,Te) [23], it is still the subject of importance how the superconducting gap evolves with increasing Te content. In advance of the systematic synthesis of bulk Fe(Se,Te), we have succeeded in growing the single-crystalline thin films of Fe(Se,Te) in the whole composition using a pulsed laser deposition technique [15,24]. While T N of the Fe(Se,Te) films decreased by the Te substitution, T c was largely enhanced after the nematic order disappeared ( Fig. 1 [a]) [15]. This enhancement of T c is in contrast to the Fe(Se,S) films [25] and bulk Fe(Se,Te) [20], indicating that the effect of nematicity on T c is complicated in these materials. Rather, the correlation between the carrier density and T c was observed by the magneto-transport measurements in the normal state [26]. However, the superconducting property of these films is yet to be fully understood. To clarify the effect of the Te substitution and the nematic order on the superconducting state, we investigated both the dynamics of the quasiparticle and the response of the superfluid in Fe(Se,Te) films.
In this paper, we report a systematic measurement of the complex conductivity, σ, of the FeSe 1−x Te x (x = 0 − 0.5) films below T c . Since the magnetic penetration depth, λ, is several times as long as the typical thickness of the film (∼ 100 nm), the measurement technique used for bulk crystals cannot be applied. Thus, to evaluate both the real and imaginary parts of σ, we combined the coplanar-waveguide-resonator-and cavity-perturbation techniques. The quasiparticle scattering time, τ , calculated from the real part of σ increased at low temperatures as was observed in bulk FeSe [27] and FeSe 0.4 Te 0.6 [28]. Besides, 1/τ of the film with the nematic order showed a quantitatively different behavior from those without the nematic order, suggesting the change of the superconducting gap structure due to the nematic order. On the other hand, the proportionality between the superfluid density, n s (∝ λ −2 ), and T c was observed irrespective of the presence or absence of the nematic order. This result indicates that while the nematic order affects the superconducting gap structure, the amount of the superfluid has a stronger influence on T c of Fe(Se,Te) than the nematic order.

Sample
All the films were grown on CaF 2 substrates (∼ 5 × 5 × 0.5 mm 3 ) by a pulsed laser deposition method using a KrF laser. Details of the film growth were described elsewhere [29,30]. The thicknesses of the grown films were measured by a stylus profiler. The electrical resistivity was measured with a standard four-probe method using a physical property measurement system (Quantum Design, PPMS).

Experiments
To measure λ of the FeSe 1−x Te x films, we fabricated the ∼ 5 × 5 mm 2 film into the coplanar waveguide resonator ( Fig. 2 [a]) by Ar ion milling and focused ion beam (FIB). The Ar ion milling was used to fabricate the whole structure, and the 50 µm gap between the resonator and the microwave input/output port was etched using FIB. The width of the resonator, w, the gap between the resonator and the ground, s, and the length of the resonator, l, were designed to be 120 µm, 30 µm, and 6.2-9.9 mm, respectively. Figure 2 (b) shows the fabricated resonator on the FeSe 0.8 Te 0.2 film. The resonator was mounted onto the printed circuit board, which was connected to the resonator by Al wirebonding. They are cooled down to 2 K using PPMS. The transmitted power was measured by a network analyzer (Keysight, N5222A).
Then, λ was calculated from the resonance frequency as follows. For the half-wavelength coplanar resonator, where L is the inductance per unit length and C is the capacitance per unit length. Here, using an electromagnetic simulation software (WIPL-D), we confirmed that the coupling between the resonator and the input port had negligible effects on f c . For a superconductor, where L m is the magnetic inductance and L k is the kinetic inductance which corresponds to the response of the superfluid [31]. L k is a quadratic function of λ as where µ 0 is the vacuum permeability, g(s, w, d) is a geometrical factor, d is the thickness of the film [31,32]. From eq. (1), eq. (2), and eq. (3), λ is expressed as All parameters in eq. (4) can be determined from the shape of the resonator (d, w, s) and the measurements of f c and C. The length was measured by an optical microscope (Keyence,   Fig. 2 (b). The inset is the resonance spectrum at 2 K. (b) Te content versus T c, zero and λ 0 . [28,34,35] and FeSe [21,27] are also shown.
VHS-6000), and C was measured using an impedance analyzer (Hewlett-Packard, 4192A) in the frequency range, 10-1000 kHz. The measured C was in good agreement with the calculated C assuming that the relative permittivity of CaF 2 is 6.5 [33]. The obtained λ(T ) was extrapolated to 0 K assuming that λ(T ) = λ 0 + AT n , where λ 0 is the penetration depth at 0 K, A and n are constants.
The dynamics of the quasiparticle in the Fe(Se,Te) films was measured using the cavity perturbation technique. For a thin film (d < λ), the cavity perturbation formula for the analysis of a bulk crystal cannot be applied. In such a case, the measured quantity is the effective impedance, Z eff (Z s , d), where Z s is the surface impedance. The formulas of Z eff corresponding to the various situations have been derived [36,37,38], which depend on the configuration of the electromagnetic field and the sample.
A flake of the FeSe 1−x Te x films was cut from the coplanar resonator after the measurement of λ. The flake (∼ 0.5 × 0.5 mm 2 ) was mounted onto the sapphire rod at the center of the cavity resonator (Fig. 2 [c]). The TE 011 mode (44 GHz) of the resonator was used, and the magnetic field of the TE 011 mode was perpendicular to the film. In this configuration, at low temperatures, Z eff can be expressed as where ω is the angular frequency [38]. Experimentally, the effective surface resistance, R eff , is determined by where G is the geometric factor, Q sample is the quality factor of the cavity with the sample, Q blank is the quality factor of the cavity without the sample. Here, we have confirmed that the effect of the CaF 2 substrate was negligible by the measurement of the substrate alone. Also, the effective surface reactance, X eff is where f c,sample is the resonance frequency with the sample and f c,blank is the resonance frequency without the sample.  To obtain Z s by solving eq. (5), eq. (6) and eq. (7), we determined G and X eff (T 0 ) as follows. At low temperatures where σ 1 << σ 2 , from eq. (5) and X s ≈ µ 0 ωλ. Thus, X eff (T ) can be calculated by substituting λ(T ) measued by the coplanar resonator into eq. (8). Here, X eff (T 0 ) was obtained using eq. (8) and λ(T 0 ) measured by the coplanar resonator. On the other hand, G was determined by the curve fitting which satisfies X coplanar eff (T ) ≃ X cavity eff (T ) in the temperature range, 0.2 − 0.5T c . After we determined G and X eff (T 0 ), we numerically solved eq. (5) and obtained Z s . Figure 1 (b) shows the temperature dependence of the resistivity of the FeSe 1−x Te x (x = 0.1 − 0.5) films. T c increased almost twice from x = 0 to x = 0.2, which is consistent with the previous report [15]. Then, T c gradually decreased with increasing Te content. Also, the resistivity at T c,onset changed from ∼ 100 µΩ cm at x = 0 to ∼ 400 µΩ cm at x = 0.5, which was the typical value for these films, [24]. Fig 3 (a) shows the resonance spectrum and the temperature dependence of the resonance frequency, f c , of the FeSe 0.8 Te 0.2 film, respectively. We calculated λ using eq. (4) from f c in each film. Figure 3 (a) shows Te content versus T c,zero and λ 0 . The negative correlation between T c and λ 0 seems to exist irrespective of the presence or absence of the nematic order. Then, we plotted T c as a function of λ −2 0 (Fig. 3 [b]). T c has a positive correlation with λ −2 0 , which corresponds to the superfluid density, n s . The observed correlation between T c and n s is consistent with the relationship between T c and the carrier density of the Fe (Se,Te) and Fe(Se,S) films in the normal state [26]. These results indicate that n s (carrier density) plays a more crucial role in determining T c of the Fe(Se,Te) films than the presence or absence of the nematic order or its fluctuation. Of note, the trend between T c and λ −2 0 were similar to that of the bulk Fe(Se,Te) [28,34], while the discrepancy existed in FeSe [21,27], the origin of which is unclear at present.

Results and Discussions
Then, we show the result of the measurement of the dynamics of the quasiparticle using the cavity perturbation technique. In Figs. 4 (b)(c), the temperature dependence of Q −1 and f c of the FeSe 0.6 Te 0.4 film was shown as a representative. The peak in Q −1 around T c is considered to originate from a drastic change of the elctromagnetic field around the film [38]. Figure 5  FeSe 1−x Te x (x = 0 − 0.5) films as a function of reduced temperature. The real part of the complex conductivity, σ 1 , was calculated from σ 1 = 2ωµ 0 R s X s /(R 2 s + X 2 s ) 2 . When calculating σ 1 , we subtracted residual resistance from R s which was estimated from the linear extrapolation of R s to 0 K. Assuming the two-fulid model and Drude-like single-carrier normal fluid, the quasiparticle scattering time, τ , can be expressed as whereσ =σ 1 + iσ 2 = µ 0 ωλ 2 0 (σ 1 + iσ 2 ) is the dimensionless conductivity [28]. Here, we should be careful this single-carrier treatment because Fe(Se,Te) is actually a multi-band superconductor. Since FeSe has highly anisotropic gaps in both hole and electron pockets [22] while FeSe 1−x Te x (x > 0.5) has nodeless gaps in both pockets [23], τ of the electron pocket is expected to show similar temperature dependence to that of the hole pocket in Fe(Se,Te). Hence, in eq. (9), we assumed that the temperature dependence of τ in both pockets could be represented using a single τ as a first approximation.
The decrease in 1/τ at low temperatures was observed in all films ( Fig. 6 [a]), which indicates the rapid suppression of the inelastic scattering of the electron. It was consistent with the result of bulk FeSe [27] and FeSe 0.4 Te 0.6 [28]. Besides, the slope of 1/τ seems to be different among these films as shown in Fig. 6 (a). To obtain more insights, we performed the curve fitting with 1/τ = aT n + b, where a, b, n are some positive constants. Figure 6 (b) shows n of each film as a function of the maximum temperature used for the curve fitting, T fit max . n showed a different behavior among these films when we decreased T fit max . While n was nearly 1 in x = 0, 0.1 film below T fit max = 0.5T c , n tended to increase with lowering T fit max in the other films. Also, in the bulk FeSe [27,39], which is in the nematic phase as the same as x = 0, 0.1 films, n was nearly equal to 1. In the bulk FeSe 0.4 Te 0.6 [28], which do not show the nematic order, n increased over 2 with decreasing T fit max . As was pointed out by Li et al. [39], the T -linear behavior (n = 1) in 1/τ may be the consequence of the gap structure with the line nodes or deep gap minima [39,40,41]. On the other hand, the n > 2 behavior could be the sign of the nodeless superconducting gap since the exponential decrease of 1/τ is expected in the nodeless superconductor [42,43]. Hence, considering the change in n in the samples with different Te content, we consider that the superconducting gap structure changes from the line nodes or deep films as a function of the reduced temperature. 1/τ of FeSe (grown by the vapor transport) [27] and FeSe 0.4 Te 0.6 (grown by the flux method) [28] are also shown. (b) The exponent, n, in the equation, 1/τ = aT n + b, determined from curve fitting. The maximum temperature for the fitting was varied from 0.25T /T c to 0.5T /T c . The orange circles and blue squares are 1/τ of the bulk FeSe [27,39]. The pink triangles are 1/τ of the bulk FeSe 0.4 Te 0.6 [28].
minima in the nematic phase to the nodeless outside the nematic phase. The result is consistent with those of other measurement techniques claiming that bulk FeSe has the line nodes or the deep minima [22,44] while bulk FeSe 1−x Te x (x > 0.5) shows nodeless superconducting gap [23].

Conclusion
In conclusion, we measured the complex conductivity of the FeSe 1−x Te x (x = 0 − 0.5) films below T c combining the coplanar-waveguide-resonator-and cavity-perturbation techniques. In the presence of the nematic order, the dynamics of the quasiparticle was qualitatively different from the samples without the nematic order. The difference indicates that the nematic order strongly affects the formation of nodes or gap minima in its superconducting gap structure. On the other hand, the proportionality between T c and λ −2 0 was found irrespective of the presence or absence of the nematic order, suggesting that the amount of the superfluid has a more direct influence on T c of Fe(Se,Te) than the nematic order itself.