Self-Consistent Two-Gap Approach in Studying Multi-Band Superconductivity of NdFeAsO0.65F0.35

High quality single crystals of NdFeAsO0.65F0.35 (the superconducting transition temperature Tc ≃ 30.6 K) were studied in zero-field (ZF) and transverse-field (TF) muon-spin rotation/relaxation (μSR) experiments. An upturn in muon-spin depolarization rate at T ≲ 3 K was observed in ZF-μSR measurements and it was associated with the onset of ordering of Nd electronic moments. Measurements of the magnetic field penetration depth (λ) were performed in the TF geometry. By applying the external magnetic field Bex parallel to the crystallographic c-axis (Bex||c) and parallel to the ab-plane (Bex||ab), the temperature dependencies of the in-plane component (λab-2) and the combination of the in-plane and the out of plane components (λab,c-2) of the superfluid density were determined, respectively. The out-of-plane superfluid density component (λc-2) was further obtained by combining the results of Bex||c and Bex||ab set of experiments. The temperature dependencies of λab-2, λab,c-2, and λc-2 were analyzed within the framework of a self-consistent two-gap model despite of using the traditional α-model. Interband coupling was taken into account, instead of assuming it to be zero as it stated in the α-model. A relatively small value of the interband coupling constant Λ12 ≃ 0.01 was obtained, thus indicating that the energy bands in NdFeAsO0.65F0.35 are only weakly coupled. In spite of their small magnitude, the coupling between the bands leads to the single value of the superconducting transition temperature Tc. The penetration depth anisotropy γλ = λc/λab was found to increase upon cooling, consistent with most of Fe-based superconductors, and their behavior is attributed to the multi-band nature of superconductivity in NdFeAsO0.65F0.35.


I. INTRODUCTION
Iron-based superconductors (IBS's) remain a subject of intensive research due to a comparable large value of the transition temperature T c . It reaches up to 55 K for the RFeAsO 1−x F x IBS family (R corresponds to the lanthanides La, Sm, Ce, Nd, Pr, and Gd), [1][2][3][4][5] and approaches T c ≃ 100 K in a single layer of FeSe on the SrTiO 3 substrate. 6 Emergence of superconductivity at such high temperatures raises a puzzling question about the gap symmetry, which can further determine the pairing mechanism for the superconducting state.
The superconductivity in IBS's appears in close proximity to the magnetism offered by d-orbitals of Fe, hence one can expect the unconventional nature of the superconducting state. The electronic band structure calculations manifest that superconductivity in IBS's originates from multiple disconnected Fermi surface sheets derived from Fe d-orbitals, thus reflecting the possibility of a complex nature of the superconducting gap structure. 7,8 There are already different scenarios proposed for the gap structure in IBS's including two-gap, s-wave, d-wave, isotropic, anisotropic, and surprisingly, p-type wave symmetry of the superconducting order parameter. [9][10][11][12][13][14][15][16][17][18] Even after several years of discovery of IBS's, an unified picture of the gap structure is not reached, contradicting the case of cuprate high-temperature superconductors, where almost all superconducting families represent a nodal pair-ing state (see e.g. Ref. 19 and references therein). In the context of conflicting results, there is still a need of comprehensive tools to understand the gap symmetry of IBS's. The magnetic penetration depth and its anisotropy carry important information about the low lying quasiparticles and hence can shed light on the gap structure of IBS's. This paper presents a detailed muon-spin rotation/relaxation (µSR) investigations of high quality single crystals of NdFeAsO 0.65 F 0.35 grown with high pressure and high temperature cubic anvil technique. Very few investigations were carried out in the direction of exploring the symmetry of order parameter for NdFeAsO 1−x F x (Nd-1111). As an example, a single gap without nodes at the Γ hole pocket was revealed through angle resolved photoemission spectroscopy (ARPES), 20 a nodal type gap structure was concluded through the linear behavior of the lower critical field B c1 at low temperatures. 21 A multi-band nature of superconductivity seems to be a more generic feature for Nd-1111 as most measurements point towards a two superconducting gaps without nodes as, e.g., the magnetic penetration depth measured through Tunnel Diode Resonator (TDR) technique, 22 ARPES, 23 the point contact Andreev reflection spectroscopy, 24,25 conductance, 26 and B c1 measurements. 27 In most cases, however, the analysis of the multiple gap behavior was performed within the framework of a phenomenological α-model, [28][29][30][31][32][33][34][35][36][37] which assumes zero coupling between the energy bands. In fact, the zero-coupling requires that the temperature dependencies of the energy gaps, as well as the values of the superconducting transition temperatures, can not be identical and should vary from one to another energy band. Speaking in a broader way, there is a clear need of different set of data and an analysis taking into account the coupling between the bands. In the present paper we approached to a so-called self-consistent model, [38][39][40][41] and used it in order to analyze the magnetic penetration depth data obtained in the TF-µSR experiment on high-quality NdFeAsO 0.65 F 0.35 single crystalline samples. Within our analysis, the energy bands with two different superconducting order parameters were assumed to be coupled and the gap equations were solved selfconsistently by considering the presence of the 'interband' and 'intraband' coupling strengths.
The paper is organized as follows: In Sec. II the sample preparation procedure, the results of magnetization measurements and the details of µSR experiments are briefly discussed. The experimental results obtained in zero-field (ZF) and transverse-field (TF) µSR experiments are described in Sec. III: the subsection III A comprises studies of the magnetic response of NdFeAsO 0.65 F 0.35 , and the subsection III B describes the results of the field-shift experiments, as well as the measurements of the temperature dependencies of the magnetic field penetration depth. The self-consistent two-gap model and the temperature evolution of the penetration depth anisotropy are presented in Sec. IV. The conclusions follow in Sec. V.

A. Sample Preparation
Bulk single crystals of NdFeAsO 1−x F x with nominal fluorine content x = 0.35 were grown at ≃ 3 GPa and ≃ 1450 o C from NaAs/KAs flux by using the cubic anvil high-pressure and high-temperature technique. The detailed description of the sample preparation procedure is given in Ref. 42. The individual crystals obtained after the sample grow had a typical size of approximately 0.5x0.5x0.03 mm 3 .

B. Magnetization measurements
The magnetization measurements were carried out on a Quantum Design MPMS-5 system. Figure 1 shows the temperature variation of the normalized magnetic moment [M (T )/M (T = 5 K)] measured simultaneously on about thirty NdFeAsO 0.65 F 0.35 single crystals. These crystals were further used in µSR experiments. The external field B ex = 0.5 mT was applied parallel to the ab-plane of the crystals. Measurements were performed in the zero-field cooled (ZFC) mode. A sharp diamagnetic signal is seen across the superconducting transition, which confirms the bulk nature of the superconductivity. The superconducting transition temperature T c ≃ 30.6 K was determined from the cross point of the two lines extrapolated from the high temperature normal state and the low temperature superconducting state, respectively (see Fig. 1).

C. Muon-spin rotation/relaxation experiments
Muon-spin rotation/relaxation (µSR) measurements were carried out in a temperature range of 1.5 to 50 K at the GPS (General Purpose Surface) (πM3 beam line) and DOLLY (πE1 beam line) spectrometers at the Paul Scherrer Institut (PSI), Villigen, Switzerland. In this technique, 100% spin-polarized muons are implanted uniformly through the sample volume, where they decay with the lifetime of 2.2 µs and the relevant decay positrons are detected successively. Muons act as sensitive magnetic probes. The spin of the muon precesses in the local magnetic field B µ with a frequency ω µ = γ µ B µ (γ µ is muon gyromagnetic ratio, γ µ /2π = 135.53 MHz/T). The detailed description of µSR technique and its applications for studying the superconducting and magnetic samples can be found in Refs. 43-50. A specific sample holder was designed in order to perform µSR experiments on thin single crystals of NdFeAsO 0.65 F 0. 35 . A mosaic of about 200 single crystals was sandwiched between two sheets made of several 0.125 nm thick Kapton layers. 51 The first few Kapton layers decelerate the muons from incoming beam and served the role of a degrader. The outgoing muons from the degrader were slow enough to stop inside the sample. The last few layers were used to stop the muons which still manage to pass through the sample. A schematic picture of the sample holder can be found in the Ref. 52. The data were analyzed using the free software package MUSRFIT, Ref. 53.

III. EXPERIMENTAL RESULTS
A. The magnetic response of NdFeAsO0.65F0.35: ZF-µSR experiments The µSR experiments in zero-field (ZF-µSR) were performed in order to study the magnetic response of the NdFeAsO 0.65 F 0.35 sample. In two sets of experiments the initial muon-spin polarization P (0) was applied parallel to the c−axis and the ab−plane, respectively. Few representative muon-time spectra for P (0) c and P (0) ab orientations are shown in Figs. 2 (a) and (c). The experimental data were analyzed by separating the µSR response on the sample (s) and the background (bg) contributions: Here A 0 is the initial asymmetry of the muon-spin ensemble. A s (A bg ) and P s (t) [P bg (t)] are the asymmetry and the time evolution of the muon-spin polarization of the sample (background), respectively. The background contribution accounts for muons missing the sample and/or stopped in Kapton layers. 52 In ZF-µSR experiments the sample contribution was described by assuming the presence of the nuclear and the electronic magnetic moments: Here the term within the square brackets is the Gaussian Kubo-Toyabe function with the relaxation rate σ GKT , which is generally used to describe the nuclear magnetic moment contribution in ZF-µSR experiments (see, e.g., Refs. 43-49, and references therein). The exponential term with the relaxation parameter Λ represents the contribution of randomly distributed magnetic impurities and/or disordered magnetic moments. 35,54 The temperature evolution of the exponential relaxation rate Λ for P (0) c and P (0) ab set of experiments are presented in panels (b) and (d) of Fig. 2. During fits the Gaussian Kubo-Toyabe relaxation σ GKT entering Eq. 2 was assumed to be dependent on the orientation, but independent on temperature, respectively. From the data presented in Figs. 2 (b) and (d) two important points emerges: (i) No detectable change in the relaxation rates Λ is observed across the superconducting transition temperature, which rules out the possibility of any spontaneous magnetic field below T c . This means that the timereversal symmetry breaking is not an immanent feature of NdFeAsO 0.65 F 0.35 studied here.
(ii) An increase in Λ is seen below 3 K for both orientations, which is probably associated with the onset of ordering of Nd magnetic moments. A similar upturn was seen in measured frequency shift [δf (T )] obtained by means of TDR technique and was explained with the ordering of the local magnetic moments of Nd below 4 K. 22 Further evidence comes form the powder Neutron diffraction experiment, where below ≃ 1.96 K, a long range antiferromagnetic order was apparent and it was associated to the combined magnetic ordering of Fe and Nd magnetic moments in the parent compound NdFeAsO. The data presented in Fig. 3 (b) and (d) reveal that for both field orientations the main part of the signal, accounting for approximately 70% of the total signal amplitude, remains unchanged within the experimental accuracy. Only the symmetric sharp peak follows exactly the applied field. It is attributed, therefore, to the residual background signal from muons missing the sample (see also Ref. 54 where the µSR field-shift experiments were initially introduced). The field-shift experiments clearly demonstrates that for both B ex c and B ex ab field orientations, the flux-line lattice in NdFeAsO 0.65 F 0.35 sample is strongly pinned.
The field distribution caused entirely by the flux-line lattice was further obtained by subtracting the symmetric background peak. The corresponding P (B) ′ s are represented in Fig. 4 by blue curves. It is worth noting that, for both field orientation P (B) distributions possess the basic features expected for an arranged flux-line lattice. The cutoff at low fields, the pronounced peak at the intermediate field and the long tail in the high field directions are clearly visible.

Analysis of Bex c and Bex ab set of TF-µSR data
The distribution of the internal magnetic fields P (B) in the superconductor in the FLL state is uniquely determined by two characteristic lengths: the magnetic field penetration depth λ and the coherence length ξ. For an isotropic extreme type-II superconductor (λ ≫ ξ) and for fields much smaller than the upper critical field B c2 (B ex ≪ B c2 ) the P (B) is almost independent on ξ and it could be calculated from the spatial variation of the internal magnetic field B(r) (r is the spatial coordinate). 56,57 In the present work the magnetic field distribution P (B), measured by means of TF-µSR, was analyzed assuming B(r) is being described within the framework of Ginzburg-Landau approach. [56][57][58][59] The spatial distribution of magnetic fields in the mixed state of a type-II superconductor is calculated via the Here B is the average magnetic field inside the superconductor, G is the reciprocal vector, r represents the vector coordinate in a plane perpendicular to the applied magnetic field and B G is the Fourier component. Within the Ginzburg-Landau model B G is obtained via: 58 φ 0 is the magnetic flux quantum, S = φ 0 / B represents the area of the FLL unit cell, b = B /B c2 , K 1 (u) is the modified Bessel function, with . For the hexagonal FLL, the reciprocal lattice is . m and n are the integer numbers.
The internal field distribution within the 'ideal' fluxline lattice was obtained as: Here dA(B ′ ) is the elementary area of the FLL with a field B ′ inside, and the integration is performed over a quarter of the flux-line lattice unit cell. 60 The FLL disorder, the broadening of the TF-µSR line due to the nuclear depolarization and the contribution of the electronic moments were considering by convoluting P id (B) with Gaussian and Lorentzian functions. 34,35,57,61 Finally, the following depolarization function was fitted to the measured TF-µSR data: Here φ is phase of the muon-spin ensemble, Λ represents the relaxation rate associated with the electronic moments, and σ g is associated with the FLL disorder and the nuclear moments contributions, respectively. In our calculations Λ was fixed to the values obtained in ZF-µSR experiments (see Sec. III A and Fig. 2). The results of the fit of Eq. 1 with the sample part described by Eq. 6 to the B ex c and B ab set of data are presented in  Fig. 2), the data points below 5 K were excluded from consideration.
In order to elucidate the pairing states in NdFeAsO 0.65 F 0.35 , the experimental data were analyzed by means of a two-gap model, with both gaps having an s-wave symmetry. Despite of considering a similar BCS type temperature dependence for both the gaps, as in phenomenological α-model, [28][29][30][31][32][33][35][36][37] the temperature dependencies of the two gaps (∆ 1 and ∆ 2 ) were obtained through a self-consistent coupled gap equations: 38,40,41 Here, N 1 (0) and N 2 (0) are the partial density of states for each band at the Fermi level. V 11 (V 22 ) and V 12 (V 21 ) are the intraband and the interband interaction potentials, respectively. A simplification of the above expressions is further done by using the notation for the coupling constant, The advantage of using the above introduced simplifications is that: (i) Within the notation of Kogan et al. 39 (ii) The number of the free parameters, which were initially 8 in Eq. 7 [namely: ω D1 , ω D2 , N (0) 1 , N (0) 2 , V 11 , V 12 , V 21 , and V 22 ], reduces to 4 in Eq. 8 [namely: ω D , Λ 11 , Λ 12 , and Λ 22 ]. 40,41 With the known temperature variation of ∆ 1 (T ) and ∆ 2 (T ), a rigorous analysis of λ −2 is carried out by separating it into two components: 39,41 ω is the weight factor for the larger gap ∆ 1 and λ −2 i (T )/λ −2 i (0) is the superfluid density component of the i−th band. The superfluid density component is related to the superconducting energy gap via the expression: 62  Fig. 6.
From the analysis of the magnetic penetration depths based on the self-consistent two-gap model three following important points emerge: (i) The interband coupling constant Λ 12 ≃ 0.01 is relatively small, indicating the fact that the two bands are nearly decoupled. However, the value of Λ 12 is significant enough to assign a single T c for each gap along both the According to the London model, the inverse squared magnetic field penetration depth for the isotropic superconductor is proportional to the superfluid density in terms of λ −2 ∝ ρ s = n s /m * (ρ s is the superfluid density, n s is the charge carrier concentration and m * is the effective mass of the charge carriers). For an anisotropic superconductor, as NdFeAsO 0.65 F 0.35 , the magnetic penetration depth is also anisotropic and is determined by an effective mass tensor: 64 Here, m * i is the effective mass of charge carrier flowing along i-th principal axis. For a magnetic field applied along i-th principal axis of the effective mass tensor, the effective penetration depth is given as: 64 By using Eq. 12 the out of plane component of the magnetic penetration depth, λ −2 c , was further obtained from λ −2 ab (T ) and λ −2 ab,c (T ) data shown in Fig. 5 as: The resulting dependence of λ −2 c on temperature is shown in Fig. 7. The theoretical temperature variation of λ −2 c (T ) was also obtained from the theory curves for λ −2 ab (T ) and λ −2 ab,c (T ), as they described in Figs. 5 (a) and (b), and it is represented by solid black line. It is evident that the curve obtained by means of two-gap model replicates the experimental data very well, which indicates that the magnetic penetration depth along c-axis is well analyzed with two-gap s + s-wave model. For the zero-temperature value of the out-of plane component the value λ −2 c (0) ≃ 0.48 µm −2 is obtained.
C. Magnetic penetration depth anisotropy, γ λ Figure 8 (a) shows the temperature evolution of the magnetic penetration depth anisotropy obtained with the experimental data presented in Fig. 5 and Eqs. 12, 13: γ λ (T ) increases with decreasing temperature from γ λ ≃ 1.8 at T = T c to γ λ ≃ 6.3 close to T = 0 K. The theoretical curve obtained with two-band model is represented by the solid black line. The temperature variation of anisotropy is reproduced well with this theoretical curve, which further confirms the multi-band nature of superconductivity in the studied oxypnictide material. It is worth to mention, that λ −2 ab (T ) and λ −2 ab,c (T ), obtained within the present study, were measured on a mosaic of about 200 NdFeAsO 0.65 F 0.35 single crystalline samples. For such a big number of simultaneously measured crystals a certain misalignment will definitively take place. Consequently, our results put a lower limit on the determination of γ λ . Figure 8 (b) compares γ λ obtained in the present study with that measured by means of torque magnetometry by Weyeneth et al.
in Ref. 65. In both cases λ λ increases with decreasing T .
A similar qualitative behavior of γ λ (T ) was observed in Sm-and Nd-1111 system by means of torque magnetometry; 65,66 in Ba(Fe 1−x Co x ) 2 As 2 by means of TDR; 67 in Ba 1−x K x Fe 2 As 2 , 34 SrFe 1.75 Co 0.25 As 2 , 35 FeSe 0.5 Te 0.5 , 68 CaKFe 4 As 4 , 37 by means of µSR etc. In all these works the pronounced temperature dependence of γ λ was attributed to the multiple gap nature of superconductivity.
As a further step, γ λ is compared with the anisotropy of the upper critical field γ Bc2 for NdFeAsO 1−x F x , as obtained form resistivity 69,70 and specific heat measurements. 71 According to the phenomenological Ginzburg-Landau theory, these two anisotropies should be equal for a single gap superconductor: 62,72  follows: (i) No changes in the relaxation rate were observed in ZF-µSR spectra across the superconducting transition, thus ruling out the possibility of any spontaneous magnetic field below T c .
(ii) An upturn in exponential muon-spin depolarization rate at T 3 K, is detected in ZF-µSR measurements. It is most probably associated with the onset of ordering of Nd electronic moments. (iii) Measurements of the magnetic field penetration depth (λ) were performed in the TF geometry. By applying the external magnetic field B ex parallel to the crystallographic c-axis and parallel to the ab-plane, the temperature dependencies of the in-plane component λ −2 ab and the combination of the in-plane and the out of plane components λ −2 ab,c of the superfluid density were determined, respectively. The out-of-plane component λ −2 were analyzed within the framework of a self-consistent two-gap model despite of using the traditional α-model. Interband coupling is taken into account instead of assuming it to be zero as is assumed in the α-model. The values of intraband and interband coupling constats were determined to be: Λ 11 ≃ 0.368, Λ 22 ≃ 0.315, and Λ 12 ≃ 0.01. A relatively small value of the interband coupling constant Λ 12 indicates that the energy bands in NdFeAsO 0.65 F 0.35 are nearly decoupled.

VI. ACKNOWLEDGMENTS
This work was performed at Swiss Muon Source (SµS), Paul Scherrer Institute (PSI, Switzerland). RK, AM, and NDZ thank Bertram Batlogg for fruitful discussions on the early stage of this study. The work of RG was supported by the Swiss National Science Foundation (SNF-Grant No. 200021-175935).