{'fraction_non_alphanumeric': 0.04405423658371863, 'fraction_numerical': 0.03600187022702478, 'mean_word_length': 4.016940317956737, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Superconductivity and normal state properties of single-crystalline Tl0.47Rb0.34Fe1.63Se2 as seen via 77Se and 87Rb NMR

We report both 77Se and 87Rb NMR studies on Tl0.47Rb0.34Fe1.63Se2 single-crystalline superconductors (T_c about 32 K). Singlet superconductivity is decisively determined by a sharp drop of the Knight shift K(T) below T_c, after subtracting the superconducting diamagnetic effect. However, the Hebel-Slichter coherence peak below T_c is not observed in the spin-lattice relaxation rate 1/T_1, even with a low in-plane NMR field of 2.6 Tesla. Just above T_c, no evidence of low-energy spin fluctuation is found in the spin-lattice relaxation rate on both the 77Se and the 87Rb sites. Upon warming, however, the Knight shifts and the spin-lattice relaxation rates of both nuclei increase substantially with temperature. In particular, the Knight shift is nearly isotropic and follow a function fit of K=a+bT^2 from T_c up to 300 K. These normal state properties may be an indication of thermally activated spin fluctuations. Our observations should put a strong constraint to the theory of magnetism and superconductivity in the newly discovered iron-based superconductors.


I. INTRODUCTION
As the second type of high-temperature superconductors after cuprates, the iron-based superconductors 1-4 have attracted a lot of research interests in recent years. The iron pnictide superconductors are remarkable in that they originate from antiferromagnetic (AFM) semimetals 4 , and have multiple electron and hole bands on the Fermi surface which are gapped in the superconducting state 5,6 . Lately, a new family of iron selenide superconductors, with nominal chemical formulas A y Fe 2−x Se 2 or (Tl, A) y Fe 2−x Se 2 (A=K, Rb, Cs) 7-11 , have been discovered with a T c as high as 33 K. Comparing with iron pnictides, this new family show qualitatively different properties in the lattice structure, the band structure and the magnetic properties as described below, and thus enrich and challenge our understanding of the iron-based superconductivity.
First, the composition of this type of superconductors is identified to be close to A 0.8 Fe 1. 6 Se 2 by the x-ray and the neutron diffraction structure refinement studies 12,13 . This chemical stoichiometry leads to an almost perfect Fe-vacancy order in superconductors with a √ 5× √ 5×1 supercell at low temperatures [12][13][14][15][16] . Second, the Angleresolved Photoemission Spectroscopy (ARPES) studies found that the hole bands centered at the Γ point sink below the Fermi level 17,18 , which is different from previous iron pnictide and chalcolgenide superconductors [19][20][21] . Third, nodeless superconducting gaps are observed on the electron pockets on the Fermi surface 17,[19][20][21] . NMR studies on nominal K 0.8 Fe 2−x Se 2 show a singlet pairing symmetry from the Knight shift, whereas the Hebel-Slichter coherence peak is not observed in the spin-lattice relaxation rate 22 . The s ± gap symmetry, proposed in iron pnictides with interband transitions, may not be applicable here because of the absence of the hole band on the Fermi surface. Finally, an antiferromagnetic order with a large magnetic moments (2.3-3.3 µ B /Fe) has been determined by neutron diffraction 13,23 for all of the new iron selenide superconductors, and possibly coexists with bulk superconductivity as indicated by the µSR and other bulk measurements 13,24,25 . From NMR, however, the Curie-Weiss upturn in 1/T 1 T , as an indication of low-energy spin fluctuations observed in most iron pnictides [26][27][28][29][30][31][32][33] , is not seen in K 0.8 Fe 2−x Se 2 22 .
In the previous NMR studies, the superconducting diamagnetic effect is not counted, which may preclude a decisive evidence for a singlet pairing 22,34,35 . It should also be noted that the large NMR magnetic field used in those studies may suppress the Hebel-Slichter coherence peak. Furthermore, a large increase of the Knight shifts and the spin-lattice relaxation rates with temperature are also observed in the normal state with an unknown origin. So far NMR data are only available in K 0.8 Fe 2−x Se 2 . NMR studies in the same structure family are demanding to verify and understand these observations.
In this paper, we report both 77 Se and 87 Rb NMR studies on a single-crystalline Tl 0.47 Rb 0.34 Fe 1.63 Se 2 superconductor. The paper is organized as following. First, we compare the Knight shifts of 77 Se and 87 Rb in the superconducting state. The superconducting diamagnetic effect are evaluated, thanks to the differences in the hyperfine coupling strength of the two nuclei. Second, we investigate the spin-lattice relaxation rate under different NMR fields below T c . The coherence peak is not observed even with a 2.6 T in-plane field. Finally, we study the Knight shift and the spin-lattice relaxation rate in the normal state, and discuss the possible spin fluctuations. Detailed comparisons with K 0.8 Fe 2−x Se 2 are also given in the paper. Se NMR spectra at different temperatures. For data below 28 K, the circles represent the integrated spectral intensity, and the solid lines are Gaussian fit to the data.

II. EXPERIMENTAL TECHNIQUES AND RESULTS
The Tl 0.47 Rb 0.34 Fe 1.63 Se 2 single crystals were synthesized by the Bridgeman method 11 . The chemical composition, determined by the inductively coupled plasma (ICP) analysis, is consistent with the Fe-vacancy order in a general stoichiometric structure of A 0.8 Fe 1. 6 Se 2 (or A 2 Fe 4 Se 5 ) 14 . In this paper, we primarily report our results on one single crystal with dimensions of ∼5×3×1mm 3 (T c ≈32K at zero field). The superconducting transition was monitored in situ by the ac susceptibility measurements with the NMR coil. In Fig. 1(a), T c is shown at about 31 K with an 11.62 T field oriented along the crystalline c axis.
The 77 Se (S=1/2) NMR is performed under 11.62 T NMR field, and 87 Rb (S=3/2) NMR is performed under 11.62 T and 2.6 T field. We also studied the field anisotropy effect with field along both the crystalline c axis and the ab plane. Typical 77 Se spectra with H//c are shown in Fig. 1(b). A linewidth about 20 kHz is observed at T =35 K, and narrows down as temperature increases. Below T c , the spectra broaden and shift to lower frequencies. The 87 Rb has one center transition and two satellites (ν q ≈1.4 MHz, data not shown).
The Knight shifts 77 K(T ) and 87 K(T ) are obtained from K(T ) = (f − γ n B)/γ n B, where f is the measured resonance frequency at magnetic field B, and 77 γ n = 8.131 MHz/T and 87 γ n = 13.931 MHz/T are the gyromagnetic ratios of two respective nuclei. The Knight shift data of 77 Se and 87 Rb are shown in Fig. 2  The spin-lattice relaxation rate 1/T 1 is measured by the inversion-recovery method, and is obtained from the fitting I(t)/I(0) = 1 − ae −t/T 1 for 77 Se and I(t)/I(0) = 1−a(0.1e −t/T1 −0.9e −6t/T 1 ) for 87 Rb (a ≥ 1). The fitting works nicely at all measured temperatures, indicating a single phase. The 1/ 77 T 1 and 1/ 87 T 1 data are shown in Fig. 4 and Fig. 5, respectively.

III. THE NMR KNIGHT SHIFT AND THE SINGLET SUPERCONDUCTIVITY
The 77 Se Knight shift data of Tl 0.47 Rb 0.34 Fe 1.63 Se 2 are shown in Figure 2(a). The measurements were performed with field along either the ab-plane (hollow diamonds) or the c-axis (hollow circles). Above T c , the 77 K for both field orientations increases monotonically with temperature. In fact, both sets of data can be nicely fit by a simple function K s (T ) = a + bT 2 with temperature from T c to 300 K, as shown in Fig. 2(b). The fitting parameters are given as a ≈ 0.1% and b ≈ 6 × 10 −6 %/K 2 for H//ab, and a ≈ 0.1% and b ≈ 5.5 × 10 −6 %/K 2 for H//c. The low-temperature part of 77 K(T ) is highlighted in Figure 2(c) to show a fast drop of 77 K from T c to T c /2 (about 15 K). Below 15 K, the spectral intensity is very small due to RF screening, and we believe our Knight shift is not well determined because that vortex core also contributes to the spectrum.
The 87 Rb Knight shift data of Tl 0.47 Rb 0.34 Fe 1.63 Se 2 are shown in Figure 3(a). The measurements were performed with field along the crystalline c-axis. Above T c , the 87 K also increases with temperature, following the same a + bT 2 behavior (see the solid line fitting).
We first discuss the Knight shift data in the superconducting state and its implication to the pairing symmetry. Below T c , the total shift K includes three parts K =K d +K s +K c , where K d , K s and K c represent the superconducting diamagnetic shielding, the spin, and the chemical (or orbital) contribution to the frequency shift. In principle, K c does not change with temperature, while K d decrease from zero to negative below T c , and K s measures local electron susceptibility. Therefore, below T c , the change of the shift follows ∆K(T )=∆K s (T )+∆K d (T ) with ∆K defined as ∆K=K(T )−K(T c ). Usually, the K s and the K d are difficulty to separate. In the current compound, due to a weaker hyperfine coupling on the interlayer 87 Rb site, as shown by the smaller Knight shift, K s and K d can be estimated by comparing the 77 K and the 87 K under the same field condition (see Fig. 3(b)), as described below.
Let's first look at the Knight shifts at a specific temperature T =16 K. The 87 K decreases by about 0.028 ± 0.004% from T c to 16 K, which gives the diamagnetic shift of −0.028%<∆K d <0. It is reasonable to assume that K d is the same for both nuclei from the same diamagnetic shielding, therefore the value of 77 K s + 77 K c for 77 Se should be moved up by the same amount 0.028 ± 0.004% (or less) from the 77 K(T ) (0.015 ± 0.004%). This gives 77 K s + 77 K c ≤0.043 ± 0.008% at 16 K, which is far below the guide line of the normal state 77 K.
The K d and K s can also be calculated with a higher precision as following. We first estimate 87 K s / 77 K s ≈ 0.4 from the change of the Knight shift in the temperature range between T c and 300 K, because the 77 K and the 87 K increases with the same T 2 behavior. Then the value of K d and K s are separated from the 77 K and the 87 K data at each temperature below T c , using 87 K s / 77 K s =0.4. We plot the resulting value of 77 K s + 77 K c below T c in Fig. 3(b) (the solid diamonds). The data show that the diamagnetic shift is less than 0.02% even at the lowest temperature.
A sharp drop in 77 K just below T c , after the subtraction of the diamagnetic effect, is clearly seen, indicating decisively a singlet superconductivity. This confirms the previous results in K 0.8 Fe 2−x Se 2 , although the diamagnetic effect was not considered there 22 . The singlet pairing puts a strong constraint to the superconducting mechanism, and narrows down the gap to s-wave, d-wave, or other even orbital symmetry.

IV. THE SPIN-LATTICE RELAXATION RATE AND THE COHERENCE PEAK
We further study the spin-lattice relaxation rate below T c to understand the pairing symmetry. Usually, the 1/T 1 far below T c follows an activation behavior for a conventional s-wave superconductor. Unfortunately, the strong RF screening of our single crystal limits our studies of T 1 to a narrow temperature range below T c .
The 1/ 77 T 1 data with field oriented along the c axis is shown in Fig. 4. Above T c , the 1/ 77 T 1 increases with temperatures. Below T c , a sharp drop of 1/ 77 T 1 is clearly shown by the low-temperature highlight in Fig. 4(b). In Fig. 5, we show the 1/ 87 T 1 data of 87 Rb with a 2.6 T field oriented along the ab plane. We are able to measure 1/ 87 T 1 below T c due to the higher natural abundance of 87 Rb. A sharp drop of 1/ 87 T 1 is also prominent as shown in Fig. 5(b).
The above data show that the Hebel-Slichter coherence peak is absent in the 1/T 1 just below T c for both nuclei. Since the H ab c2 (0) is estimated to be over 100 Tesla for the (Tl,Rb) y Fe 2−x Se 2 superconductors 36 , the coherence peak is unlikely to be suppressed with the 2.6 T in-plane field. The absence of the coherence peak in the current compound is consistent with the result reported in K 0.8 Fe 2−x Se 2 22,34,35 . Usually, the absence of coherence peak is interpreted as evidence for d-wave or other non s-wave pairing symmetry. In most iron pnictides, the coherence peak is not observed by far [37][38][39][40][41][42][43] , which is proposed as a consequence of the s ± pairing symmetry with interband impurity scattering [44][45][46][47][48] . In the new iron selenide superconductors, however, the s ± scenario with interband transitions may not be applicable, because the hole band is absent on the Fermi surface close to the Γ point. The absence of the coherence peak suggest that the system is different from a conventional s-wave superconductor and needs to be further addressed. It may be worthwhile to note that an s ± symmetry is proposed in K 0.8 Fe 2−x Se 2 considering the coupling between the bonding and the antibonding states on the Fermi surface 49 .

V. THE LOW-TEMPERATURE SPIN DYNAMICS
Next we discuss the magnetic properties from the Knight shift and the spin-lattice relaxation rate data in the normal state. In Fig. 4(a), the 1/ 77 T 1 T is plotted versus temperature to compare with the Fermi liquid behavior (1/T 1 T = const.). Our data show that 1/ 77 T 1 T reaches a constant value when the temperature is close to T c . 1/ 87 T 1 T also shows a similar temperature dependence as seen in 5(a). The Knight shift data, 77 K and the 87 K, also level off toward T c (see Fig. 2(a) and Fig. 3(a)). The level off of both the Knight shift and the 1/T 1 T suggest that the system approaches a normal Fermi liquid behavior at low temperatures, with a constant density of states at the Fermi level.
The absence of the Curie-Weiss upturn in 1/ 77 T 1 T in the current material and also in K 0.8 Fe 2−x Se 2 22 suggests that the spin fluctuations are weak in both compounds, although their T c are also high. These distinctive behaviors of the spin-lattice relaxation rate in the new iron selenides does not seem to support that the superconductivity has a magnetic origin.
However, we should be aware that spin fluctuations may exist in other forms. For example, some spin fluctuations at the wave vector (±π, ±π) cannot be detected in the current spin-lattice relaxation rate, due to the cancellation effect on the Se and the Rb sites which are located at the center of the Fe square. Furthermore, as we show below, the spin fluctuations may exist in the high-energy modes according to our high-temperature data.

VI. THE HIGH-TEMPERATURE SPIN DYNAMICS
To further compare with a Fermi liquid behavior, we plot (T 1 T ) −0.5 against K(T ) in Fig. 6(a) and (b) for both nuclei. Usually, a Fermi liquid follows the Korringa relation (T 1 T ) −0.5 ∝ K s . As shown in Fig. 6(a), a linear relation between (T 1 T ) −0.5 and K(T ), as demanded by the Korringa relation, is approximately satisfied for 77 Se in the low-temperature limit. However, the Korringa relation is not obeyed in the high temperature range, indicating a deviation from the Fermi liquid behavior. Here we estimated K c ≈ 0 based on the value of K at T ≪ T c , where K s (T ) = 0 is expected (see Fig. 3(b)). For 87 Rb, the Korringa relation does not hold either at high temperatures.
Now we look closely on the high-temperature behavior of the Knight shift and the spin-lattice relaxation rate. The 1/T 1 T of two nuclei increases substantially with temperature. The Knight shift data 77 K and the 87 K increase by over a factor of three, when the temperature increases from T c to the room temperature, as shown in Fig. 2 and Fig. 3. The Knight shift 77 K of K 0.8 Fe 2−x Se 2 with H//ab is also plotted in Fig. 2 (b), and it falls on the same line of the current compound. This suggests a universal electronic structure in the new iron selenide superconductors. In contrast, in most iron pnictides, the Knight shift shows a much smaller increase with a linear-T behavior 27 . The anisotropy of the Knight shift is small with K ab s /K c s ≈ 1.1 in the studied temperature range, as indicated by the slightly different slope in the T 2 plot of 77 K (see Fig. 2). This anisotropy may come from an anisotropic susceptibility or an anisotropic Hyperfine coupling. As a local probe, the Knight shift data should indicate an intrinsic behavior of the magnetic susceptibility. In contrast, the magnetic susceptibility reported on Tl 0.58 Rb 0.42 Fe 1.72 Se 2 shows a T 2 behavior with H c, but not with H ab 36 , which may be affected by some unknown contributions. Furthermore, the ratio 87 K/ 77 K ≈ 0.4 indicates that the hyperfine field on the interlayer Rb site is comparable to that on the Se site, implying a three-dimensional like system.
Such a large increase of the Knight shift is unlikely caused by charge excitations. In this compound, the charge excitation scenario will require an increase of the electron density over a factor of three up to the room temperature, which has not been indicated by any ARPES or transport measurements. Spin fluctuations is another possible scenario. The temperature behavior of the Knight shift suggest that a finite spin gap (or pseudogap) may exist, so that the spin fluctuations are gapped out at the low temperatures and thermally activated at high temperatures. However, the experimental evidence for the physical origin of such a spin gap is lacking. Our evidence of nearly isotropic, three-dimensional-like Knight shift should be important constraints to possible theories.
We note that a magnetic transition at about 500 K has been reported by recent neutron scattering 23 and susceptibility measurements 25 on (Tl, A) y Fe 2−x Se 2 superconductors. Unfortunately, the long-range antiferromagnetism has not been resolved by NMR so far. Further NMR measurements are demanding to address these issues.

VII. SUMMARY AND ACKNOWLEDGMENT
To summarize, our 77 Se and 87 Rb Knight shifts of Tl 0.47 Rb 0.34 Fe 1.63 Se 2 show a decisive bulk evidence for singlet superconductivity. However, the Hebel-Slichter coherence peak is not seen in the spin-lattice relaxation rate at very low in-plane field, which suggests that the system is probably not a conventional s-wave superconductor. In the normal state, evidence for low-energy spin fluctuations is not observed close to T c , which questions the correlation between spin fluctuations and the superconductivity. With increasing temperature, however, the Knight shift increases substantially with a T 2 behavior up to room temperature, which is in contrast to most iron pnictides. The spin-lattice relaxation rate also shows a large increase with temperature. The Knight shift and the spin-lattice relaxation may suggest spin excitations with a finite gap (or pseudogap), which is thermally excited at high temperatures.
Our resolved chemical composition of Tl 0.47 Rb 0.34 Fe 1.63 Se 2 suggests that the A 2 Fe 4 Se 5 stoichiometric structure is formed in our superconducting compound. Furthermore, the similar NMR data of Tl 0.47 Rb 0.34 Fe 1.63 Se 2 and K 0.8 Fe 2−x Se 2 indicates the same electronic structure and pairing symmetry in both compounds, which points to a common origin of superconductivity in this new structure family of the iron-based superconductors.