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Sub-kelvin Andreev reflection spectroscopy of superconducting gaps in FeSe

Point contact Andreev reflection studies have been conducted on FeSe single crystals by lowering the temperatures down to 0.5 K. The point contact Andreev reflection spectra were analyzed in the framework of the two-band model. As a result, the presence of two anisotropic superconducting gaps in FeSe were certainly established and their BCS-like temperature dependencies were obtained. The weights of each gap have been determined and the anisotropy parameter has been calculated. It is shown, that sub-kelvin temperatures are necessary to ascertain details of the superconducting gap structure, especially for multiband materials when Andreev reflection spectroscopy is applied.

Introduction may be equally well described by theoretic curves using an anisotropic single-gap model and a model with two superconducting gaps. It is known that the temperature is responsible for the resolution in PCAR spectroscopy 1 . Therefore, measurements at lower temperatures are desirable to get more correct information about the absolute value and anisotropy of the SC gaps. Thus, in the present study PCAR measurements were carried out at lower temperatures down to 0.5K.

Method
The "soft" PCs were made by placing a tiny drop of a silver paste between the FeSe sample and a 0.1 mm copper wire. The same technique was used in [11]. Four of these contacts made to one crystal are shown in Fig.1inset. A standard 20-lead chip was used as the sample substrate, enabling simultaneous measurements on several "soft" PCs during one cooling down period. Fig. 1 demonstrates four wires each soldered with two leads on a chip in order to realize "pseudo" four probe measuring configuration since the sample was also electrically coupled with two separate leads. The chip itself was clamped to a cold plate of a 3 He refrigerator at ambient conditions. After cooling down the temperature of the chip was measured and controlled with a multi-channel controller, which regulates the temperature in the 0.5-20 K range.
The FeSe samples where grown by flux technique method. The description of the crystal growth can be found in [12]. In our experiments we measured PCAR spectra (differential resistance dV/dI(V)) through a "soft" PC by a standard modulation method. The schematic picture of measuring setup is shown in Figure 4.3 of Ref. [13]. FeSe sample clamped within the copper clip by means of the brass screw and the nut. To create good thermal contact the screw itself was soldered to the base plate of the 20-lead chip and coupled electrically with two such leads. Inset (left top corner): Enlarged image of the FeSe sample with four "soft" contacts attached. Figure 2 shows temperature a), c) magnetic field b), d) variation of the dV/dI spectra for two representative "soft" PCs. PC spectra measured at the lowest temperatures show a doubleminimum structure characteristic for the Andreev reflection. This structure is suppressed as the temperature increases, as it is seen in Figure 2a) and c). At first, it transforms into a single minimum which vanishes approaching the mark of 12 K similarly to our previous study [11]. This Tc is a few degrees higher than the bulk critical temperature Tc=9.4K [12]. Higher values of the local Tc's were observed in our previous paper [10] where we explained different surface and bulk properties on account of interfacial effects. Figures 2 b) and d) show the magnetic field dependencies of the PC spectra from a) and c) measured at the lowest possible temperature T = 0.5 K. Both sets reveal similar field dependencies. In magnetic fields up to 8 T the filling of the gap structure is evident and surprisingly, the suppression of the double-minimum intensity is not followed with the moving of the gap minima to lower energies, as it is usually observed in classical BCS superconductors. Unfortunately, the largest magnetic field B = 8 T is much lower than the upper critical magnetic field of the studied samples, which is above 20T at T=0.5K [14]. Thus, we cannot follow the full suppression of the superconducting gap(s). Our PCAR spectra, shown in Figure 2, reveal additional spectral features in the form of symmetric side maxima above the gap structure at around 3 mV. Fig. 2a), b) clearly depicts these maxima, while in Fig.  2c), d) they look just like shoulders. The presence of side maxima above the gap structure of the dV/dI spectra is a typical signature of the suppression of superconductivity with high current density and/or temperature increase due to the transition to the thermal regime with a bias rise of a non ballistic, likely diffusive PC [15,16]. Temperature and magnetic field variation of the differential resistance dV/dI for two "soft" PCs measured in the range of the lowest temperature of 0.5K to 13K and in the magnetic field up to 8T at 0.5K.

Results and discussion
The measured dV/dI spectra have been fitted to the Blonder-Tinkham-Klapwijk (BTK) equation [11,17], which describes the Andreev reflection of quasiparticles at the ballistic PC between a normal metal and a superconductor taking into account the superconducting gap Δ, the PC barrier strength Z and the spectral smearing parameter Γ. We focused on the spectra which revealed the lowest presence of the disturbing side maxima (from Fig. 2c) and d)). We also fitted our data with three modifications of the BTK model, taking into account an anisotropic s-wave gap, and the weighted sums of two isotropic or anisotropic gaps. While fitting data, we considered the so-called scaling parameter S which is included to fit the intensity of the calculated and experimental curves, that is S=(dV/dI)exp/(dV/dI)theor. For instance, S = 1 means that the calculated curve also fits the measured dV/dI in absolute values 2 , in other words, it fits its intensity.
In the case of fitting to the anisotropic one gap model, the energy gap and the spectral smearing parameter were defined with the anisotropy functions Δ=Δ 0 (1 +α cos 4Θ) [6], Γ= Γ 0 (1 +α cos 4Θ), where α is the anisotropy parameter. The fitting parameters are shown in the legends for Figure 3b. The fitting curve is shown as a solid red curve. The quality of fit is sufficient, however, the scaling parameter S >>1.
When we applied a weighted sum of two anisotropic gap contributions, the number of fitting parameters redoubled due to the contribution of the second gap. We reduced the number of fitting parameters, supposing that Z was the same for both gaps because this parameter reflects primarily the physical character of the interface, rather than details of the electronic structure as discussed in [16]. Also, the same anisotropy α has been predicted for both gaps. While fitting data we used the weight parameter w, which characterizes the weight of the small gap and (1-w) the large gap contribution. The fitting result is shown in Figure 3a. The scaling parameter S reveals the value close to 1.
In the case of the isotropic two-gap model we simply put α =0, as described in our previous paper [11]. The fitting curve plotted in Figure 3 c) as a red curve describes our experimental data similarly to the anisotropic one gap fit (see Figure 3c), but then again with the scaling parameter S>>1.  Fig. 3 offers the possibility to compare fitting results of the used one-and two-gap models at T = 0.5 K. All models may help to describe sufficiently accurate experimental curves. The dominant gap value in all curve fittings is close to the position of the dV/dI spectral minimum, which is positioned just above 1 mV in the measured spectra. However, only the anisotropic two gap fit gives the scaling parameter S near to 1. It is important to notice that in this fit we obtained the smallest value for the spectral smearing parameters Γ 0 S and Γ 0 L. Therefore, the anisotropic two-gap model may in our opinion fit our spectra more accurate. Discrepancies may be visible only between 3 and 6 mV, where the non-Andreev-reflection like side maxima disturb the spectra. These maxima are strongly sensitive to the measuring temperature and the applied magnetic field. This is visible in Figure 2 where the maxima are shifting to lower energies (into the gap) at increased temperatures and magnetic fields. The presence of this disturbing effect may strongly affect the study of the gap features of the dI/dV spectra at higher temperatures. Therefore, we were able to perform an adequate fit and to get temperature dependence of the fitting parameters only up to 6 K. Figure 4 shows all of our fitting results. Figure 4a) plots the temperature dependencies of the superconducting gaps with discrete symbols. The gray part of it shows areas of the possible extrapolations of the data with standard BCS curves. Possible values of the critical temperature Tc are between the bulk value 9.4K and 12K when the SC minimum disappears. Note, that a non-Andreev V-type shape of zero-bias minimum above 9K (see Fig. 2c) most likely testifies to transition to the gapless superconducting fluctuation region. Therefore, BCS extrapolation of the gaps to the bulk critical temperature is more accurate. It should also be mentioned that the anisotropy parameter α has a value 0.31 at low temperature, which is close to 0.34 reported by [6].
The fitting parameters of the spectral weight w, spectral smearing Γ 0 S and Γ 0 L and gap anisotropy α reveal evident temperature dependencies above 3 K (see Fig.4). These dependencies are probably connected with the strong temperature dependence of the side maxima (positioned at 3 mV at T 0.5K), which are moving into the gap at increased temperatures disabling correct fitting of the spectral features. The weak temperature dependence of the scaling parameter S (between 1.6 and 1), shown in Figure 4b) as red symbols and an almost temperature independent value of the barrier strength (black symbols in Figure 4b)) is a confirmation of the validity of our fitting model with the weighted sum of two Andreev reflection contributions with anisotropic energy gaps. The magnetic field dependence of the PC spectra, shown in Figure 2 b) and d) cannot be observed in a single band system, thus, it represents another proof of the multiband superconductivity in FeSe samples. With regard to the measurements in a magnetic field, the available field strength was too weak to suppress superconductivity and the field effect was mainly observed in the broadening of the curves (see Fig. 2b, d). As it was shown in [11], at least a two times larger field is needed to practically suppress the double-minimum structure. Anyway, some noteworthy results are as follows: similar to what was found in Ref. [11], the SC gap value is robust 3 with respect to the magnetic field strength up to 8T although contribution w of the small gap vanishes, which is consistent with the observations made in our previous paper [11]. Besides, anisotropy parameter α turned out to be field independent.

Conclusion
We have performed PCAR measurement in FeSe single crystals in the sub-kelvin temperature range. It allowed us to perform high resolution Andreev-reflection measurements in this compound. Analyzing our PCAR dV/dI data we have shown that two anisotropic superconducting gaps are responsible for the superconducting properties of FeSe. The temperature dependencies of the superconducting gaps determined by dV/dI curve fitting strongly support the anisotropic two-band scenario of superconductivity in this compound .