¹¹⁹Sn NMR study in the normal state of the superconductor Ca₃Ir₄Sn₁₃

Recently, the superconductivity in the (Ca, Sr)₃(Ir, Rh)₄Sn₁₃ system has attracted attention due to unusual physical properties in the normal state. The superconductivity is driven by pressure-induced structural phase transition accompanied by charge-density-wave (CDW) ordering [1–8]. Ca₃Ir₄Sn₁₃ is one of the ternary stannide compound with superconducting transition temperature $T_c = 7\ \text{K}$ [9,10]. Yang et al. report a systematic magnetic, thermodynamic and transport measurements of this compound [11]. They found a hump in the temperature dependence of the magnetic susceptibility of Ca₃Ir₄Sn₁₃, as observed in the itinerant magnetic systems, such as YCo₂ [12], TiBe₂ [13] and Fe₃Mo₃N [14], where the hump is ascribed to the spin fluctuation effect [11]. On the one hand, the specific heat and thermal conductivity measurements show that the superconductivity gap of Ca₃Ir₄Sn₁₃ is nodeless [11,15], and ¹¹⁹Sn-NMR measurements in the superconducting state also indicate it is a BCS superconductor with multi isotropic superconducting gaps [16]. On the other hand, some groups had pointed out that there is no clear evidence supporting the existence of spin fluctuations and the localized magnetic moments on the μSR time scale, which means the hump at $T^{*} = 45\ \text{K}$ does not have a magnetic origin [7,17]. Furthermore, Klintberg et al. conducted X-ray diffraction analysis for a (Sr, Ca)₃Ir₄Sn₁₃ single crystal, and found a superlattice distortion at $T^{*}$ [1]. Their results indicate that such a lattice distortion is associated with a CDW transition of the conduction electron at $T^{*}$. Their result is consistent with the Seebeck coefficient measurement in Ca₃Ir₄Sn₁₃, which suggests the Fermi surface reconstruction and the opening of CDW gap at the superlattice transition temperature [18]. The sign change in the Hall coefficient observed in this system provides direct evidence for the Fermi surface reconstruction during the superlattice phase transition [4]. The CDW scenario has also been observed in the isostructural La₃Co₄Sn₁₃ [19], Ce₃Co₄Sn₁₃ [20] compounds by ⁵⁹Co NMR, Sr₃Ir₄Sn₁₃ [8] and Sr₃Rh₄Sn₁₃ [5] compounds by ¹¹⁹Sn NMR measurements, respectively. However, there is little microscopic evidence for the structural transition in the Ca₃Ir₄Sn₁₃ system.

Nuclear magnetic resonance (NMR) is a local probe of electronic states through hyperfine interactions and is highly sensitive to the low-energy spin fluctuations. It is useful to understand the microscopic characteristics of electronic systems. In this paper we report the NMR measurements of ¹¹⁹Sn nuclei in the powdered single-crystal sample Ca₃Ir₄Sn₁₃ in a wide temperature range from 2 to 280 K. The temperature dependence of the Knight shift K and the temperature-divided nuclear spin lattice rate $1/T_1T$ exhibit a peak corresponding to the anomaly in the temperature dependence of susceptibility and resistivity at $T^*$. Interestingly, we find that the Knight shift K and $1/T_1T$ do not show linear relation with the magnetic susceptibility in the whole temperature as those observed in the YCo₂ [12] and Sr_1−xCa_xCo₂P₂ [21] system, which is the signature for the existence of ferromagnetic spin fluctuations in the framework of self-consistent renormalization (SCR) theory. On the other hand, we find that the slopes of the $K\text{-}\chi$ plot and $1/T_1T\text{-}\chi$ plot show an abrupt change at $T^{*}$, which suggests that the change of the electronic structure of the conduction electron is not the magnetic but the CDW scenario at $T^{*}$ in Ca₃Ir₄Sn₁₃. At the same time, our results also hint the existence of three-dimensional ferromagnetic spin fluctuations in the normal state of Ca₃Ir₄Sn₁₃ depending on the SCR theory.

The powdered single-crystal samples used in the present experiments are the same as used for resistivity and susceptibility measurements in ref. [11]. For the observation of ¹¹⁹Sn NMR signals, a phase-coherent pulsed NMR spectrometer was utilized. The measurements were carried out by monitoring the spin-echo intensity as a function of frequency at a fixed external magnetic field. We have also measured the Knight shift, K, and the nuclear spin-lattice relaxation rate, $1/T_{1}$, of the ¹¹⁹Sn (nuclear spin $I = 1/2$, the nuclear gyromagnetic ratio $^{119}\gamma/2\pi = 15.867\ \text{MHz/T}$) nuclei in the sample Ca₃Ir₄Sn₁₃. The ¹¹⁹Sn nuclear spin-lattice relaxation rate $1/T_{1}$ was determined by fitting the time dependence of spin-echo intensity after a saturation of nuclear magnetization with a single component of T₁ in the whole temperature range.

Figure 1 shows the temperature dependence of ¹¹⁹Sn NMR spectra measured at external magnetic field $H = 8.0010\ \text{T}$. A significant feature of the spectra is that there are one shoulder at the low-frequency side and two peaks at the high-frequency side as shown in fig. 1 by arrows. The shoulder at the low-frequency side shows temperature-independent behaviour, the other two peaks shift to the high frequency at first, and then begin to shift to the low frequency at about 45 K.

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The crystal structure for Ca₃Ir₄Sn₁₃ above $T^{*}$ is of Yb₃Rh₄Sn₁₃ type with the space group $Pm\overline{3}n$. Ca and Ir atoms occupy the 6d and 8e sites, respectively. Two non-equivalent crystallographic Sn atoms, termed Sn(1) and Sn(2), locate at 2a and 24k sites, respectively. The bonding length of Ir-Sn(2) is the shortest among the nearest-neighbor atomic distance in Ca₃Ir₄Sn₁₃. It is thus natural to expect that a non-negligible $p\text{-}d$ hybridization between Sn(2) 5p and the Ir 4d states exist in this compound. Thus, we arrange the shoulder originating from Sn(1) sites, and the other two peaks from Sn(2) site.

The inset of fig. 2(a) shows a frequency swept spectrum at the magnetic field $H = 8.0010\ \text{T}$ for the ¹¹⁹Sn NMR at 30 K. This is a typical powder pattern spectrum for $I = 1/2$. The isotropic $K_\mathrm{iso}$ and anisotropic $K_\mathrm{ani}$ Knight shift calculated by $K_\mathrm{iso} = ( K_{\parallel} + 2K_{\perp} )/3$ and $K_\mathrm{ani} = ( K_{\parallel} - K_{\perp} )/3$ are shown in fig. 2(a). The temperature dependence of $K_\mathrm{iso}$ is quite similar to the temperature dependence of bulk susceptibility χ, which displays a broad peak at about 45 K as shown in the inset of fig. 2(b). This confirms that the broad peak appearing around $T^*$ in the temperature dependence of magnetic susceptibility is intrinsic in this system. Since $K_\mathrm{iso}\ \gg\ K_\mathrm{ani}$, the overall shift is nearly isotropic in this compound.

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Generally, the measured Knight shift can be written as

$$\begin{equation} K(T) = K_0 + \frac{A_\mathrm{hf}}{N_A\mu_\mathrm{B}} \chi(T). \end{equation} \tag{ 1 }$$

Here, K₀ is the temperature-independent part arising from orbital susceptibility and diamagnetic susceptibility of the core electrons, and, $\frac{A_\mathrm{hf}}{N_A\mu_\mathrm{B}} \chi(T)$ is the spin part Knight shift, arising from the temperature-dependent susceptibility due to the Ir-4d spins contribution. $A_\mathrm{hf}$ is the hyperfine coupling constant due to the electron nuclear hyperfine and dipolar interaction, N_A is the Avogadro number and $\mu_\mathrm{B}$ is the Bohr magneton.

Taking into account that the $K_\mathrm{ani}$ are almost temperature independent, only the $K_\mathrm{iso}$ are plotted vs. bulk susceptibility χ (the so-called $K\text{-}\chi$ plot) in fig. 2(b). It is not like the $K\text{-}\chi$ plot observed in nearly ferromagnetic materials, such as the YCo₂ [12], and Sr_1−xCa_xCo₂P₂ [21] system, where the linear relation can be found in the whole temperature range with the same slope, suggesting the magnetic origin of the broad peak, but show different slope in the $K\text{-}\chi$ plot below and above $T^*$. This kind of phenomenon was also observed in Ce₃Co₄Sn₁₃ by ⁵⁹Co NMR, where the CDW is suggested to be the likely mechanism [20].

Usually, we can deduce the hyperfine coupling constant from the linear slope of the $K\text{-}\chi$ plot. Thus, the linear slope changes around $T^*$, demonstrating a significant modification of the hyperfine coupling constant accompanied by the transition. The hyperfine coupling constants estimated are $23.15\ \text{kOe}/\mu_\mathrm{B}$ Ir for the higher-temperature range and $128.20\ \text{kOe}/\mu_\mathrm{B}$ Ir for the lower-temperature range, respectively. The relatively large magnitude of the hyperfine coupling constant cannot be understood simply by classical dipolar-dipolar interactions, hence, the transferred hyperfine fields due to hybridization between Ir 5d and Sn 5s, p orbitals are considered to be the underlying mechanism. The abrupt change of the hyperfine coupling constant suggests the variation of the electronic state at about 45 K. Hence, our result shows that the anomaly at $T^*$ do not have a magnetic origin, and supports the CDW scenario.

In order to confirm whether ferromagnetic spin fluctuations exist or not in the normal state for Ca₃Ir₄Sn₁₃, the measurement of spin dynamics by means of the spin-lattice relaxation rate $1/T_1$ is useful. In a general form, $1/T_{1}T$ can be written as the summation of the wave vector q of the imaginary part of the dynamical electronic susceptibility $\chi^{\prime\prime}(q,\ \omega_{n})$,

$$\begin{equation} \frac{1}{T_{1}T} =\frac{\gamma_{n}^{2}k_{B}}{\mu_{B}^{2}\hbar}\sum_{q}|A_\mathrm{hf}(q)|^{2}\frac{\chi^{\prime\prime}(q, \omega_{n})}{\omega_{n}}, \end{equation} \tag{ 2 }$$

where $A_\mathrm{hf}(q)$ and $\omega_n$ represent the wave vector q-dependent hyperfine coupling constant and the NMR frequency, respectively. Apparently, $1/T_1T$ measures the average of various q-modes of the low-frequency spin fluctuations. The physical properties of itinerant magnetism have been discussed on the basis of the SCR theory of spin fluctuations and the theoretical results suggest that $1/T_1T$ can be expressed as $1/T_1T\propto\chi$ for a system where the three-dimensional ferromagnetic spin fluctuations are dominant. This had been confirmed in some itinerant magnetic compounds, such as Y(Co_1−xAl_x)₂ [12,22], Sr_1−xCa_xRuO₃ [23] and MnSi [24].

We measured the T₁ of ¹¹⁹Sn at the $K_{\perp}$ position of the ¹¹⁹Sn NMR spectrum as shown in the inset of fig. 2(a) and determined the T₁ by fitting the time dependence of spin-echo intensity to a single exponential curve after the inversion pulse. Typical recovery curves are shown in the inset of fig. 3. Good fits are obtained throughout all the temperature range in the normal state.

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We present the temperature dependence of $1/T_1T$ in fig. 3. A clear anomaly can be found at $T^*$ as that observed in the temperature dependence of the Knight shift. For $T>T^{*}$, $1/T_1T$ increases continuously with decreasing temperature and shows Curie-Weiss behaviour. On the other hand, $1/T_1T$ is almost constant below $T^{*}$. This behaviour is quite different as observed in Sr₃Rh₄Sn₁₃ system, where $1/T_1T$ is constant below and above $T^*$ [5]. According to eq. (2), $1/T_1T$ is proportional to $\chi^{\prime\prime}(q, \omega_{n})$, we infer that the dynamical spin susceptibility in this system is different with Sr₃Rh₄Sn₁₃ system.

Furthermore, our result reminds us that the results of the the temperature dependence of the electrical resistivity measured at different fields in the temperature range $T_c < T < 45\ \text{K}$, where the resistivity shows linear temperature dependence, that is, non–Fermi-liquid behaviour at zero magnetic field H [11], and then, the system develops into a Fermi-liquid state with increasing $H\ (=14\ \text{T})$ gradually. At temperatures $T_c < T < 45\ \text{K}$, $1/T_1$ is linear with temperature in high magnetic fields, that is $1/T_1T$ is constant as shown in fig. 3. This is satisfied with the so-called Korringa relation, which is expected of a Fermi-liquid state. Of course, in the low magnetic field, the $1/T_1T$ should show the temperature dependence because of the deviation of the electronic state from the Fermi-liquid state. This had also been confirmed by the low field NMR measurements as shown in fig. 3. Both the macro- and the microscopic results suggest that the non–Fermi-liquid behavior may play an important role in Ca₃Ir₄Sn₁₃. Usually, the external magnetic field would suppress the spin fluctuations. The reason for the enhancement of $1/T_1T$ with the increase of magnetic field in the temperature $T_c < T < 45\ \text{K}$ is still unknown up to the present. However, we can conclude that this unusual behaviour does not have a magnetic origin.

In order to confirm the existence of magnetic spin fluctuations in the normal state of Ca₃Ir₄Sn₁₃, we plot the $1/T_{1}T$ vs. bulk susceptibility χ with temperature as an implicit parameter. As shown in fig. 4, good linear relations were found between $1/T_{1}T$ and χ in the high-temperatures range. This indicates that the ferromagnetic spin fluctuations are dominant and of three-dimensional characteristics in the normal state of Ca₃Ir₄Sn₁₃ above $T^*$. On the one hand, NMR measurements show that the ferromagnetic spin fluctuations are dominant above $T^*$, on the other hand, the specific heat and thermal conductivity measurements indicate that Ca₃Ir₄Sn₁₃ may be a BCS-type superconductor. Because of the change of electronic state at $T^*$, the ferromagnetic spin fluctuations above $T^*$ may be irrelevant to the superconductivity in this material.

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In summary, we report the ¹¹⁹Sn NMR investigation in the Ca₃Ir₄Sn₁₃ system. The temperature dependence of the Knight shift K and $1/T_1T$ exhibit a peak corresponding to the anomaly in susceptibility and resistivity at $T^*$. Detailed analysis indicates that the hyperfine coupling constant shows the drastic change at $T^*$, which suggests the reconstruction of Fermi surface, that is, the changes of electronic state at $T^*$. The existence of the three-dimensional ferromagnetic spin fluctuations above $T^*$ in the NMR time scale in the normal state of Ca₃Ir₄Sn₁₃ is supported by the linear relation in $1/T_1T$ with χ in the framework of the SCR theory.

This work is supported by the NSF of Zhejiang Province (No. LY14A040007), the start-up fund of Hangzhou Normal University (No. 2011QDL43) the Grants-in-Aid for Scientic Research (Nos. 19350030, 22350029) from the Japan Society for Pro-motion of Science. BC acknowledges helpful discussions with F. L. Ning.