Nuclear magnetic relaxation and superfluid density in Fe-pnictide superconductors: An anisotropic \pm s-wave scenario

We discuss the nuclear magnetic relaxation rate and the superfluid density with the use of the effective five-band model by Kuroki et al. [Phys. Rev. Lett. 101, 087004 (2008)] in Fe-based superconductors. We show that a fully-gapped anisotropic \pm s-wave superconductivity consistently explains experimental observations. In our phenomenological model, the gaps are assumed to be anisotropic on the electron-like \beta Fermi surfaces around the M point, where the maximum of the anisotropic gap is about four times larger than the minimum.

One of the confused points in the experiments for Fe-based superconductors is that the results of the nuclear magnetic relaxation rate seem inconsistent with the superfluid density observations. The nuclear magnetic relaxation rate has the lack of the coherence peak below T c and exhibits the low temperature powerlaw behavior (1/T 1 ∝ T 3 ). 31,32,33,34,35 This is seemingly the evidence of unconventional superconductivity with line-node gaps. However, some experiments report that the superfluid density (i.e., penetration depth) does not depend on the temperature at low temperatures, which means that the pairing symmetry is fully-gapped s-wave symmetry. 22,23,24,25,26,27 The ±s-wave pairing symmetry is theoretically proposed as one of the candidates for the pairing symmetry in Fe-pnictide superconductors. 40,41,42,43,44,45,46,47,48,49 The ±s-wave symmetry means that the symmetry of pair functions on each Fermi surface is s-wave and the relative phase between them is π. Very recently, several theoretical groups suggested that the ±s-wave symmetry explains the lack of the coherence peak and the low temperature power-law behavior in the nuclear magnetic relaxation rate, with introducing impurity scatterings. 37,38,39 Part of their scenarios is based on the fact that, in a ±s-wave phase, substantial low-energy states appear in the density of states in the case of a unitary-limit scattering, while only higher-energy density of states near gap edges is modified when approaching to the Born limit. 39,50 To theoretically investigate the superconductivity, it is necessary to consider a model for the electronic structure. There are many theoretical studies, especially by band calculations, to understand the unique electronic and magnetic properties of those Fe-pnictide superconductors. 42,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69 In addition, an effective five-band model was elaborated by Kuroki et al., 42 where the five bands originate predominantly from 3d orbitals at the Fe atomic site. A simpler two-band Hamiltonian was also proposed as a tractable minimal model, which reproduces the structure of Fermi surfaces obtained by band calculations. 70,71,72,73,74 However, Arita et al. 45 claimed that the five bands are necessary for describing correct band dispersions around the Fermi level. They also suggested that an anisotropic ±s-wave superconductivity is realized in a Fe-pnictide superconductor. 45 In this paper, we investigate the nuclear spin-lattice relaxation rate and the superfluid density on the basis of the realistic effective five-band model. We will show that an anisotropic ±s-wave pair function explains consistently the experimental results even in assuming a rather clean system. This paper is organized as follows. The effective fiveband model and the pair functions are introduced in Sec. II. We then discuss the nuclear spin-lattice relaxation rate (Sec. III), the superfluid density (Sec. IV), and the density of states (Sec. V). Finally, the conclusion is given in Sec. VI. In the appendix, we describe the derivation of the nuclear spin-lattice relaxation rate on the basis of the quasiclassical theory of superconductivity.

III. NUCLEAR SPIN-LATTICE RELAXATION RATE
The nuclear spin-lattice relaxation rate 1/T 1 T is given as 75,76,77,78 Here, and The brackets · · · FS mean the Fermi-surface average, where dS F,i is the area elements on each Fermi surface. ω n = πT (2n + 1) is the Matsubara frequency. We use units in whichh = k B = 1. We assume the smearing factor η = 0.1T c . 79 We set 2∆ 0 /T c = 4, which is a representative value near the BCS value 3.53. The coherence factor is represented as 1+W F F /W GG . The contribution of W F F is related to the coherence effect, which becomes zero in the case of unconventional pair functions such as d-wave one. First, we consider conventional s-wave pair functions (Φ a < 0). In Fig. 3, we show the results for the isotropic s-wave (Φ a = −1, Φ βmin = 1) and the anisotropic swave (Φ a = −1, Φ βmin = 0.2). The coherence peaks appear below T c for both pair functions because of non-zero W F F , meaning that these pair functions cannot explain the experiments.
Second, we consider ±s-wave pair functions (Φ a > 0). Figure 4(a) shows the result in the case of the isotropic ±s-wave pair function (Φ a = 1, Φ βmin = 1) and Fig. 4(b) shows the result in the case of the anisotropic ±s-wave function (Φ a = 0.2, Φ βmin = 0.2) whose k-dependence is similar to the result of the RPA calculation by Arita et al. 45 In both cases, the coherence peak below T c is  suppressed, since W F F is almost zero. In the five band model, the difference of the density of states between the Fermi surfaces α 1,2 and β 1,2 is small. Therefore, the cancellation of the ±s-wave pair functions between α and β is almost perfect, resulting in W F F ≈ 0. On the other hand, the temperature dependence at low T is inconsistent with the experiments in both cases. The exponential behavior appears in the isotropic ±s-wave case. 37 The temperature dependence is concave down in the anisotropic ±swave case with Φ a = 0.2 and Φ βmin = 0.2 as seen in Fig.  4(b). We next consider another parameter set for the anisotropic ±s-wave pair function below.
We search for the most suitable pair function with Φ a and Φ βmin . We check the two points as follows: (i) the lack of the coherence peak below T c and (ii) the low temperature power-law behavior 1/T 1 T ∝ T 2 . We show the temperature dependence of 1/T 1 T in the cases of the various pair functions in Fig. 5. First, we fix the β gap anisotropy Φ βmin = 0.2 and examine the Φ a (the α gap amplitude) dependence as shown in Fig. 5(a). With increasing Φ a from the value Φ a = 0.2, the exponent (i.e., the slope in Fig. 5) approaches to the experimental result ∼ 2. The best coincidence is attained at Φ a = 1. Second, we fix Φ a = 1 and examine the Φ βmin dependence as shown in Fig. 5(b). With decreasing the anisotropy (i.e., increasing Φ βmin ), the deviation becomes larger for Φ βmin > 0.25. Hence, the experimental results are best reproduced when Φ a = 1 and Φ βmin = 0.25. That is, the maximum pair amplitudes on the Fermi surfaces α 1,2 and β 1,2 are of the same order (Φ a = 1), and the ratio of the minimum to the maximum of the pair amplitude on β 1,2 is 0.25 (Φ βmin = 0.25). We show the comparison of our calculation with the experimental result of 75 As-NQR for LaFeAsO 0.6 (Ref. 34) in Fig. 6. Indeed, this anisotropic ±s-wave pair function explains the observed low-temperature power-law behavior 1/T 1 ∝ T 3 .

IV. SUPERFLUID DENSITY
Let us confirm whether the above anisotropic ±s-wave pair function can also explain the observed temperature dependence of the superfluid density. The superfluid density ρ xx is given by 81,88 Here, ρ 0 denotes the superfluid density at the zero temperature and v Fx is the Fermi velocity component in the (π, 0) direction. As shown in Fig. 7, the superfluid density ρ xx (T ) for the anisotropic ±s-wave pair function (Φ a = 1, Φ βmin = 0.25) does not depend on the temperature in the low temperature region. When we increase 2∆ 0 /T c , the result approaches to that of the isotropic s-wave case. Indeed, the anisotropic ±s-wave pair function can explain the fully-gapped behavior observed in the experiments. 22,23,24,25,26,27 In contrast to it, pair functions with line nodes such as d-wave one lead to a strong temperature dependence near the zero temperature in general.

V. DENSITY OF STATES
Finally, we show the density of states N s (E) for the anisotropic ±s-wave pair function (Φ a = 1, Φ βmin = 0.25) with 2∆ 0 /T c = 4 in Fig. 8. It is calculated by N s (E) = N n Re g ↑↑ (iω n → E+iη) FS , 79 where N n is the normal-state density of states at the Fermi level and g ↑↑ is defined in Eq. (3.8). The density of states is gapped in is the minimum gap on the Fermi surfaces β 1,2 ). This is the reason why the superfluid density does not depend on the temperature in the low temperature region. In the region Φ βmin ∆ 0 (= 0.5T c ) < ∼ |E| < ∼ ∆ 0 (= 2T c ), the density of states has a linear energy dependence. Therefore, the nuclear magnetic relaxation rate exhibits the line-nodes-like power-law behavior. The density of states also has the single peak structure near the gap edge at |E| = 2T c = ∆ 0 , since the gap maxima on the Fermi surfaces α 1,2 and β 1,2 now coincide with each other owing to Φ a = 1. Note here that the maximum gap amplitudes on α 1,2 and β 1,2 are ∆ 0 |Φ a | and ∆ 0 , respectively.
In addition, the density of states for the anisotropic ±s-wave pair function is a monotonically-increasing function of the energy (|E| < ∆ 0 ) as seen in Fig. 8, while the unitary-scattering-induced density of states and the multi-gapped density of states are nonmonotonic in some cases. 39,50 This difference would be observed by spectroscopy experiments.

VI. CONCLUSION
With the use of the five band model, we calculated the nuclear magnetic relaxation rate 1/T 1 and the superfluid density ρ xx and showed that the anisotropic ±s-wave pair function can explain the seemingly contradictory experimental results on Fe-pnictide superconductors. That is, the anisotropic ±s-wave pair function reproduces consistently 1/T 1 ∼ T 3 and the T -independence of ρ xx at low T .
Our scenario is similar to the theories by Parker et al., 37 Chubukov et al., 38 and Bang and Choi 39,40 in the sense that ±s-wave pair functions are considered in all theories. However, impurity effects are essential for those previous theories. 37,38,39 The impurity scattering rate is relatively large in Refs. 37 and 38. A unitary-limit impurity scattering or an impurity scattering intermediate between Born and unitary limits 50 is essential in Refs. 37 and 39. In contrast, we have assumed a rather clean system and not considered a unitary-limit or an intermediate phase-shift scattering. On the other hand, it was pointed out that a fitting resulted in quite big value 2∆ 0 /T c ≈ 7.5 within a model in Ref. 40. In our model, rather strong gap anisotropy on the β Fermi surfaces 45 has been introduced, which enables us to explain 1/T 1 ∼ T 3 even in a clean system and with relatively reasonable value 2∆ 0 /T c ∼ 4. This is a distinguished feature of our scenario.
It should be noted that while some of experimental groups have reported the fully-gapped behavior of the superfluid density, part of measurements showed somewhat strong temperature dependence indicating gap nodes. 22,23,24,25,26,27,28,29,30 Those results seem to depend on kinds of materials and doping level, but it is still unclear what is the essential origin of such scattered observations between materials. The difference might mean that the pairing symmetry changes between materials or that the degree of gap anisotropy on the β Fermi surfaces changes, albeit there are no microscopic theories suggesting them at present. In any case, it is an interesting issue left for feature studies.

APPENDIX
In this Appendix, we describe the procedure for deriving the nuclear spin-lattice relaxation rate T −1 1 (r, T ) on the basis of the quasiclassical Green function theory. 82,83,84,85,86,87,88,89 The derived formula has been utilized in Sec. III and in Refs. 75,76,77,78. Quasiclassical theory -We start with the Green functions defined as 86 Here, the brackets · · · denote the thermal average. We use units in whichh = k B = 1. We writě Throughout this Appendix, "hat" (Â) denotes the 2 × 2 matrix in the spin space, and "check" (Ǎ) denotes the 4 × 4 matrix composed of the 2 × 2 particle-hole space and the 2 × 2 spin one. The quasiclassical Green functionǧ is defined aš where the integration is performed with respect to the energy variable ξ k in the k space, Here, ε(k) is the quasiparticle dispersion relation and µ is the chemical potential. We have defineď According to a conventional procedure, the k-space integration is approximated as Here, an isotropic spherical Fermi surface is assumed for clarity. The extension to general cases can be done straightforward by replacing the solid-angle integration dΩ/4π with the Fermi surface average · · · FS . N F is the total density of states at the Fermi level.
In the case of spin-singlet superconductivity, the Eilenberger equation is solved in a spatially uniform system and the solution for the quasiclassical Green function is 90 Here, the Pauli matrices areσ = (σ x ,σ y ,σ z ) in the spin space.
Relaxation Rate -The nuclear spin-lattice relaxation rate T −1 1 (r, T ) is obtained from the spin-spin correlation function χ −+ (x, x ′ ). 91 We define x ≡ (r, τ ), and set τ ′ = 0. We apply a static external magnetic field along a certain axis and take the spin quantization axis parallel to this. χ −+ (x, x ′ ) is given as Let us consider a Fourier transformation with respect to τ . In what follows, A and B stand for the Green functions. The Fermi-and Bose-Matsubara frequencies are ω n = πT (2n + 1) and Ω n = πT (2n), respectively. The Fourier transformation is Setting r ′ = r (i.e.,r = 0), we have Here, we have referred to Eq. (A.3) and have replaced dΩ/4π with · · · FS . Next, let us consider the spectral representation of the quasiclassical Green functions: Utilizing the formula (f (ω) is the Fermi distribution function) Setting iΩ m → Ω + iδ (δ → 0 + ), where we have used 1 In the normal state, the spectral function of the quasiclassical Green function is a 11 = a 22 = 1 for diagonal components in the particle-hole space (i.e., the density of states is unity in units of N F ) and is a 12 = a 21 = 0 for off-diagonal components (because the order parameter is zero). We then obtain at T = T c , where i, j = {1, 2}. Letting iω n → E ± iη (η > 0), To calculate the relaxation rate in Eq. (A.36), we need to consider the spin-space matrix elements presented in Sec. III.