Two-Gap Superconductivity in CaFe_{0.88}Co_{0.12}AsF Revealed by Temperature Dependence of the Lower Critical Field H_{c1}^c(T)

Gap symmetry and structure are crucial issues in understanding the superconducting mechanism of unconventional superconductors. Here we report an in-depth investigation on the out-of-plane lower critical field $H_{c1}^{c}$ of fluorine-based 1111 system superconductor CaFe$_{0.88}$Co$_{0.12}$AsF with $T_c$ = 21 K. A pronounced two-gap feature is revealed by the kink in the temperature dependent $H_{c1}^c(T)$ curve. The magnitudes of the two gaps are determined to be $\Delta_1$ = 0.86 meV and $\Delta_2$ = 4.48 meV, which account for 74% and 26% of the total superfluid density respectively. Our results suggest that the local antiferromagnetic exchange pairing picture is favored compared to the Fermi surface nesting scenario.


Introduction
Superconducting (SC) mechanism is the central issue in the study of unconventional superconductors. Since the discovery of Fe-based superconductors (FeSCs) 1 , many efforts have been made on this problem 2 . At the early stage, itinerant mechanism based on the weak correlation was accepted widely and the Fermi surface (FS) nesting (abbreviate as nesting scenario) was believed to be very crucial for the superconductivity 3,4 . Later on, this scenario was challenged by other studies [5][6][7][8] , especially by the discovery of K x Fe 2−y Se 2 system without hole type Fermi surface near the Γ point 9-12 . Consequently, the local antiferromagnetic exchange pairing scenario (abbreviate as local scenario), considering a stronger electron correlation, attracts more and more attentions [13][14][15][16] .
Despite the distinct mechanisms mentioned above, the prospective physical manifestations may be rather subtle. For example, both of them predicted a sign-changed s-wave (S±) gap symmetry.
However, the Fermi surfaces with a better nesting condition tend to have a stronger pairing amplitude and larger SC gap in the itinerant mechanism [17][18][19] , while according to the local scenario, a larger SC gap should open on the smaller Fermi surface 13 . Typically approximations were made in the theoretical models and a precise comparison to the experimental results is difficult. In the case of 122 system Ba 0.6 K 0.4 Fe 2 As 2 , the larger SC gap was found to open on the Fermi surfaces with a smaller size and a better nesting condition [19][20][21] , which couldn't discriminate these two theoretical proposals. Therefore, currently more delicate experiments are required.
Recently clear progresses were made on the single-crystal growth of the fluorine-based 1111 system of FeSCs, CaFeAsF 22 and the Co doped counterparts 23 , and systematic investigations 2 have been carried out on this system [24][25][26][27][28][29][30][31] . Especially, it was found that the smaller FS around the Γ point (see the α FS in Fig. 4) is much smaller than other FSs around M point and consequently shows a worse nesting condition 27,32 , as compared with the other larger FSs, which should benefit the identification of the abovementioned itinerant and local mechanisms.
In this paper, we present a detailed investigation on the temperature dependence of the outof-plane lower critical field H c c1 (T ) of the high-quality CaFe 0.88 Co 0.12 AsF single crystals. The lower critical field reflects the information of penetration depth and superfluid density, which has been used to investigate the intrinsic SC properties of FeSCs [33][34][35]

Results
The dc magnetic susceptibility χ for the CaFe 0.88 Co 0.12 AsF sample was measured under a magnetic field of 10 Oe in zero-field-cooling and field-cooling modes, which is presented in Fig. 1(a).
The χ − T curve shows a sharp SC transition, which reflects the homogeneity and high quality of our sample. The onset transition temperature T c is about 21 K. The absolute value of magnetic susceptibility χ is over 95% after the demagnetization was considered, indicating a high supercon- i.e., H c c1 . Field dependence of such a deviation ∆M is displayed in Fig. 2(b). Two criteria, ∆M = 5 × 10 −5 emu and 2.5 × 10 −5 emu equivalent to 2 Oe and 1 Oe respectively, are adopted for the determination of H c c1 . As revealed by the two dashed lines in Fig. 2(b), obviously the variation of criterion will affect the obtained H c c1 values. Nevertheless, as shown in Fig. 3, the evolution behavior with temperature is not affected by the criterion. In addition, we found that the temperature dependent tendency from our measurements is also consistent with that obtained by the magnetic torque experiments 26 , as displayed by the green asterisks. So we will focus on the analysis of the normalized values H c c1 (T )/H c c1 (0), which are more solid and reliable.
It is known that typically the FeSCs are in the local limit 33 Here λ ab is the penetration depth within the ab plane. Moreover, the Fermi surfaces in the present system are nearly ideal cylinders 27,32 and the in-plane Fermi velocity is rather isotropic within the k x − k y plane. In this case, ρ ab s of the ith Fermi surface can be given by 36 where f (E) is the Fermi function and ∆ i is the value of the energy gap in the ith Fermi surface.
The temperature dependence of ∆ i was calculated based on the simple weak-coupling BCS model.
Evidently, the kink feature around T c /2 in Fig. 3 could not be described by an isotropic single gap model. In order to simplify the discussion, here we adopt a two-gap model and the total normalized superfluid density can be expressed as Here dS F,i indicates an integral over the ith Fermi surface and v ab F is the component of Fermi velocity within the ab plane 36 . By tuning the values of ∆ i and w i , a simulating curve well describing the experimental was obtained, as shown by the blue solid curve in Fig. 3. This consistency between our data and the fitting curve suggests that the two-gap model has grasped key features of this system. The two dashed lines reveal the contributions from the two components with ∆ 1 = 0.86 meV, w 1 = 0.74, and ∆ 2 = 4.48 meV, w 2 = 0.26.

Discussion
Investigating the weighting factor w i allows us to seek out the locations of the different superfluid components on the FSs. For the roughly isotropic FSs and isotropic v ab F , w i is only determined by the v ab F value and the size of the ith FS. Although the detailed correlation is diverse, both the nesting and local scenarios imply that the gap value is determined by the shape and size of the FSs 13, 17-19 , which can be derived from the calculated electronic structures. As shown in Fig. 4, roughly the five FSs can be divided into two groups from the viewpoint of FS shape and size: the small α FS and the large ones (β/γ/δ/η) with similar sizes. Thus we only need to simply discuss and compare the two groups. By checking the energy dispersion of the calculated band structure, we estimated that the in-plane v ab F on the α FS is 1.25∼2 times of that on the large ones (β/γ/δ/η).
As for the FS size, however, α FS is only 1/4 of the latter. Moreover, the number of the larger FSs is four, while only one small FS is present. Considering the above factors, the weighting factor of the α FS should be clearly smaller than that of the others. Consequently we ascribe the w 2 and ∆ 2 to superfluid on the α FS.
The α FS, which has a rather bad nesting condition with other FSs, carried the superfluid with a larger gap. Evidently, this is inconsistent with the nesting scenario for the pairing mechanism.
Based on the local antiferromagnetic exchange pairing, which considered the local antiferromagnetic exchange of nearest neighboring and next nearest neighbor irons, a simple gap function was proposed 13 : 6 The distribution of the gap value |∆(k)| in the brillouin zone was displayed in Fig. 4(a)  To summarize, we conduct magnetization measurements on CaFe 0.88 Co 0.12 AsF single crystals, and the out-of-plane lower critical field H c c1 is extracted. It is found that the temperature dependent H c c1 exhibits a pronounced kink around T c /2, which can be described by a two-gap model. Importantly, the lower superfluid density with a rather large gap is attributed to the small α FS, from which the local antiferromagnetic exchange pairing mechanism is identified to be a better candidate for understanding the unconventional superconductivity of FeSCs. Moreover, our data follow the Uemura plot quite well, indicating a low-superfluid-density feature resembling the hole-doped high-T c cuprates.

Methods
Sample preparation. High quality CaFe 0.88 Co 0.12 AsF single crystals were grown using CaAs as the self-flux 22,23 . The detailed growth conditions and the characterizations of the samples can be seen in our previous reports 23 .
Magnetization measurements. The magnetization measurements were carried out on the magnetic property measurement system (Quantum Design, MPMS 3). The magnetic fields were applied along the c axis of the single crystal in all the measurements.
Band structure calculations. The first-principles calculations presented in this work were performed using the all-electron full potential linear augmented plane wave plus local orbitals method 42 as implemented in the WIEN2K code. 43 The exchange-correlation potential was calculated using the generalized gradient approximation as proposed by Pedrew, Burke, and Ernzerhof. 44 The calculations for the parent compound were performed using the experimental crystal structure 22     and Sr 1−x La x CuO 2 (SrLaCuO)) and the oxygen-based 1111 system (F-doped LaFeAsO and SmFeAsO ((La,Sm)FeAsO-F)) are taken from Refs. [38,39]. The data of K-doped BaFe 2 As 2 (Ba122-K), MgB 2 , and NbSe 2 are taken from Refs. [33,40,41], respectively.
The result of CaFe 0.88 Co 0.12 AsF (CaFeAsF-Co) is from the present work.