NMR relaxation and rattling phonons in type-I Ba₈Ga₁₆Sn₃₀ clathrate

Clathrates are materials with oversized polyhedral cages and guest atoms loosely bound inside. Type-I clathrates are based on the $A_{8}X_{46}$ structure, where $X$ stands for the cage atom and $A$ is the guest atom (Ba, Eu, Sr etc.). Group IV clathrates with cages formed by Ga, Ge, Sn and Si atoms are particularly well studied examples. Recent investigations of such intermetallic clathrates showed excellent thermoelectric propertiesSales et al. (2001); Nolas et al. (1998); Deng et al. (2010); Snyder and Toberer (2008); Christensen et al. (2010). This is especially interesting for the group IV elements given their great potential in the modern semiconductor industry Fujita et al. (2006); Avila et al. (2006). Research on the framework and caged atomic motions has also shown anomalous vibrational properties and their connection to the electronic behavior Biswas and Myles (2007); Nowaczyk et al. (2009). Understanding the guest atom vibration modes, often called ”rattling”, has been considered to be one of the most important ways to reveal the essence of these phenomena. Many methods including Raman scattering Takasu et al. (2006, 2010), inelastic neutron scattering Christensen et al. (2008), optical conductivity Mori et al. (2011, 2009), NMR relaxation Gou et al. (2005) and theoretical calculations Dong et al. (2000); Madsen and Santi (2005) have been reported to analyze the rattling atoms in clathrates using different models.

During the last few years, ultra-low lattice thermal conductivity ($\kappa_{L}$) and glasslike thermal behavior have been discovered in particular in type-I Ba₈Ga₁₆Sn₃₀ Deng et al. (2010); Avila et al. (2008); Suekuni et al. (2010). The rattling of guest atoms in the larger of its two structural cages has been confirmed to have close connection to those properties Avila et al. (2008); Suekuni et al. (2010). Anharmonic oscillators have also been used as trial models for the rattling phonons to analyze the thermoelectric properties and NMR relaxation behavior of other similar materials Mahan and Sofo (1993); Yoshida et al. (2007); Dahm and Ueda (2007). In this paper, we discuss the NMR relaxation behavior and guest atomic motion of Ba₈Ga₁₆Sn₃₀ clathrates. A Raman process involving local vibrational modes, which is responsible for the NMR quadrupolar relaxation, will be discussed. Simple one and two dimensional anharmonic potentials will be introduced to model the anharmonic local oscillators. The shape and energy levels of these potential wells will be obtained by matching the simulations to the NMR experimental results.

The clathrate samples were prepared using the self-flux method, following a technique reported previously Suekuni et al. (2008). Because of the existence of a type-l/type-VIII dimorphism in Ba₈Ga₁₆Sn₃₀, a carefully controlled annealing process is needed during the sample making procedure. For type-I Ba₈Ga₁₆Sn₃₀, the pure elements were mixed together based on the nominal composition followed by an initial arc melting in argon environment. Annealing in an evacuated quartz tube at 900 ^oC for 50 hours was then applied, followed by a controlled slow cooling to 500 ^oC in 80 hours Suekuni et al. (2008); Y. Rodriguez et al. (2010). X-ray diffraction (XRD) measurements were performed on a Bruker D8 X-ray Powder Diffractometer, and wavelength dispersion spectroscopy (WDS) measurements were done in a Cameca SX50 spectrometer. Rietveld refinements of the XRD results were performed using EXPGUI Toby (2001), and the result confirmed the composition and structure of type-I Ba₈Ga₁₆Sn₃₀ with no type-VIII reflections detected and with 1% (per mol) Ba(Ga,Sn)₄ minority phase obtained in the fit Y. Rodriguez et al. (2010).

NMR experiments were carried out under external magnetic fields of 8.8 T and 7 T in a temperature range from 4.2 K to 295 K using a pulse spectrometer and a homemade multi-temperature detecting probe. The nuclei measured are ⁷¹Ga and ⁶⁹Ga with different gyromagnetic ratios $\gamma$ and quadrupole moments Q, where ${}^{71}\gamma=8.1355$ rad/s G^-1, ${}^{69}\gamma=6.4208$ rad/s G^-1 (ref. Carter et al., 1977), ${}^{71}Q=10.7$ fm², and ${}^{69}Q=17.1$ fm² (ref. Harris and Becker, 2002). The inset to Fig.1 shows the central portion of the ⁷¹Ga NMR lineshapes at three temperatures under 8.8T. No significant change in average shift vs temperature has been observed in the lineshape mapping. The small changes at the base of the lineshape vs. temperature are due to unreacted Ga metal. The ⁷¹Ga lineshape is a superposition of two close peaks, which are due to different sites of the framework atoms Y. Rodriguez et al. (2010). Here we consider only the behavior at the center of the resonance, which is due to a superposition of different local configurations. The weighted center shift of this resonance is about 0.033% at 295 K. For comparison, the ⁶⁹Ga lineshape under the same conditions, has a weighted shift of 0.023%. The shift includes magnetic and quadrupole terms, which can be expressed as ${\delta}f=K+BQ^{2}$, where $K$ stands for the magnetic shift and $Q$ is the quadrupole moment of the nucleus. From the observed field-dependence we extracted $K=0.039\%$ as the center-of-mass magnetic shift and a negative quadrupole shift. In this case $K$ is mainly a Knight shift due to conduction electrons, with some contribution due to chemical shifts. Note that we did not observe any significant change in $K$, such as those observed in Na-Si type II clathrates Gryko et al. (1998), indicating excitations involving sharp electronic features in the system. As reported previously we have also performed a structural analysis using $ab$-$initio$ calculations of the low-temperature NMR shifts for this sample, modeling in particular the first-order quadrupole broadening at the base of the lineshape according to the distribution of Ga framework occupation. This agreed with the experimental lineshape quite well Y. Rodriguez et al. (2010).

NMR spin-lattice relaxation measurements were performed at the central transition frequency at the center of the lineshape. The relaxation time, $T_{1}$, is a fitted value based on a magnetic relaxation mechanism using a standard multi-exponential function for recovery of the central transition Gou et al. (2005). The quadrupole relaxation process entails a different relaxation function, which however leads only to an overall scaling of the $T_{1}$, and does not affect any of the dynamical fitting parameters described below. The signal is also a superposition of different framework sites, however we fitted to a single average $T_{1}$ as parameter. Fig. 1 shows the resulting rates for both ⁷¹Ga and ⁶⁹Ga under a field of 8.8 T. A clear peak at a temperature around 10 K can be observed for both nuclei.

The isotopic ratio, ${}^{69}T_{1}/^{71}T_{1}$, under 8.8 T is shown in the inset of Fig. 2. According to hyperfine relaxation theory Abragam (1982), if the relaxation mechanism contains only a magnetic part, $T_{1}$ should be inversely proportional to ${\gamma}^{2}$ which gives ${}^{69}T_{1}/^{71}T_{1}{\cong}1.67$, while if the quadrupolar relaxation is in control, $T_{1}$ should be inversely proportional to $Q^{2}$ which gives ${}^{69}T_{1}/^{71}T_{1}{\cong}0.4$. The experimental ratio for our sample is consistent over a wide temperature range and is close to the quadrupole moment ratio. Thus, the relaxation is mainly controlled by the quadrupole mechanism, indicating that lattice vibrations are the most important contribution. As our experimental data are a mixture of magnetic and quadrupole parts, it is necessary to separate them for further investigations. The corresponding relationships are Abragam (1982),

where $T_{1}$ is the overall experimental relaxation time while $T_{1M}$ and $T_{1Q}$ represent the magnetic and quadrupole parts. According to Eq. (1) and (2), the relaxation rates were separated into two contributions as shown in Fig. 2. Again, the result confirms the dominant role of the quadrupole relaxation rate.

At higher temperatures, as values of $K^{2}T_{1}T$ do not change much, therefore the sample appears to follow a Korringa-like relation Abragam (1982); Shastry and Abrahams (1994), which would normally indicate the influence of metallic electrons if $1/T_{1}$ were magnetic. However a recent model Dahm and Ueda (2007) for relaxation dominated by anharmonic localized vibrations indicates such behavior as a high-temperature limit, along with a low-temperature peak much as observed here. Our later simulation based on this anharmonic model will be compared with the quadrupolar relaxation rates we have separated.

From refinements of x-ray diffraction spectra for Ba₈Ga₁₆Sn₃₀, the guest Ba(2) atom has location probability concentrated near four equivalent off-center positions with off-center dynamic displacements around 0.4 ÅAvila et al. (2008). Our first principles calculations Y. Rodriguez et al. (2010) also gave similar values for the static displacement of the Ba atom due to cage asymmetry. A 1-D double well potential was also introduced by Dahm and Ueda to analyze this kind of problem, and has shown good agreement for the pyrochlore case Dahm and Ueda (2007). To model our data, therefore we use the Hamiltonian,

where $M$, $p$, and $x$ are the mass, momentum and spatial coordinate of the guest atom Ba Mahan and Sofo (1993); Dahm and Ueda (2007). An effective localized phonon frequency, $\omega_{0}$, and thermal average of $x^{2}$ were introduced in a self-consistent quasiharmonic approximation giving $M{\omega}_{0}^{2}=a+b{\langle}x^{2}{\rangle}_{{\omega}_{0},T}$, where

and the relationship between $\omega_{0}$ and $T$ is given by,

where ${\omega}_{00}={\omega}_{0}(T=0)$, and $\beta=b{\hbar}/M^{2}{\omega}_{00}^{3}$ is a dimensionless anharmonicity factor.

As the relaxation is dominated by the quadrupole term, a two-phonon Raman process can be used to describe the NMR relaxation. This can be expressed Dahm and Ueda (2007) as

with

where $V_{2}$ is the second spatial derivative of the electric field gradient, ${\omega}_{L}$ is the nuclear Lamor frequency, $A(\omega)$ is the phonon spectral function, $n(\omega)$ is the Bose function, $D(\omega)$ is the retarded phonon propagator, $\Gamma_{0}$ is a phonon damping rate and ${\omega}_{r}^{2}$ is the renormalized phonon frequency determined by the phonon self-energy, ${\omega}_{r}^{2}={\omega}_{0}^{2}+2{\omega}_{0}$Re$\Pi(\omega)$. Here, we assume the real part of the phonon self energy, Re$\Pi(\omega)$, to be temperature independent as assumed in [17]. By carefully choosing parameters, we obtained a good fit to our data as shown in Fig. 3. The matching results clearly show that the spin-lattice relaxation mechanism can be explained by the rattling phonon model.

The corresponding values for the parameters are ${\omega}_{00}=20$ K, $\beta=50$, ${\Gamma}_{0}=12$ K and ${\omega}_{r}(T=0)=19.5$ K. The potential well is given by the calculated expression,

where $V(x)$ is in J with $x$ given in meters. Also from equations (4) and (5), when $T=296$ K, ${\omega}_{0}$ ${\cong}$ 11 THz and ${\langle}x^{2}{\rangle}_{{\omega}_{0},T}^{1/2}$ ${\cong}$ 0.12 Å. Since in 2D ${\langle}r^{2}{\rangle}_{{\omega}_{0},T}=2{\langle}x^{2}{\rangle}_{{\omega}_{0},T}$, this corresponds to a rms guest atom displacement of 0.17 Å, which is not far from the values reported previously Avila et al. (2008); Y. Rodriguez et al. (2010). Solving the Schrödinger equation numerically, the energy levels of this double well potential can be calculated as shown in Fig. 4. The energy difference between the lowest two states, ${\Delta}E_{12}$ ${\cong}$ 30 K, is much smaller than those for higher energy levels, which agrees with recent reported results from other methods Mori et al. (2011), but with a larger ${\Delta}E_{12}$.

This model can also be extended to a simplified 2-D potential by using ${\langle}r^{2}{\rangle}_{{\omega}_{0},T}={\langle}x^{2}{\rangle}_{{\omega}_{0},T}+{\langle}y^{2}{\rangle}_{{\omega}_{0},T}=2{\langle}x^{2}{\rangle}_{{\omega}_{0},T}$ in equation (4). We correspondingly modified the relationship in equation (5). Then, following the same procedure, we obtain a fitting similar to the 1-D model. The result is shown in the inset of Fig. 3, with the fitted values ${\omega}_{00}=20$ K, $\beta=25$, ${\Gamma}_{0}=12$ K and ${\omega}_{r}(T=0)=19.5$ K. The corresponding potential is $V(r)=-8.98r^{2}+5.52\times 10^{22}r^{4}$, where $V(r)$ is in J with $r$ given in meters. The average displacement is still 0.17 Å, which indicates a consistency of the model compared with the 1-D case.

Compared to previous Ga NMR results for Sr₈Ga₁₆Ge₃₀, also identified to behave as an anharmonic rattler system Nolas et al. (1998); Dong et al. (2000), it seems initially surprising that the $(T_{1}T)^{-1}$ in Sr₈Ga₁₆Ge₃₀ does not show a similar phonon-dominated behavior but instead follows a Korringa law quite closely for several decades of temperature Gou et al. (2005). However, a previous report for Sr-Ge clathrates used density functional theory to extract potential well parameters for Sr in the large cage Dong et al. (2000), giving a 2D anharmonic potential much like the model used here. The resulting potential has a very small quadratic term, but also a much smaller anharmonicity parameter $(\beta)$ than found here, and inserting the calculated parameters into the relaxation theory described above yields a smaller quadrupole contribution to $(T_{1}T)^{-1}$, rising slowly with temperature without exhibiting a peak as in Fig. 3. Also in the analysis reported for the elastic response of Eu- and Sr-filled Ge clathrates Zerec et al. (2004) a four-well potential was used to model the vibrational response for Eu, but for Sr a harmonic Einstein oscillator model provided satisfactory agreement. Thus it is consistent that the $(T_{1}T)^{-1}$ in Sr₈Ga₁₆Ge₃₀ is dominated by interactions with the charge carriers, while the quadrupole-dominated peak observed in Ba₈Ga₁₆Sn₃₀ is indicative of the much larger anharmonicity for rattler atoms.

In our measurements, we prepared a second sample of type-I Ba₈Ga₁₆Sn₃₀ in the same way, whose ${}^{71}T_{1}^{-1}$ exhibits a low-temperature maximum that is nearly identical to that of Fig. 1. In fitting to the model oscillator potential, the position of the ${}^{71}T_{1}^{-1}$ peak is particularly sensitive to ${\omega}_{00}$, which is close to the spacing of the two lowest levels in Fig. 4. The ability to model this behavior with a single set of parameters attests to the lack of irregularity of the cage potential, despite the presence of quasi-random framework substitution. This is apparently due to the Sn-based cage size, providing space for the relatively unconstrained motion of the Ba(2) atoms without allowing for a permanent distortion Avila et al. (2008). Thus, type-I Ba₈Ga₁₆Sn₃₀ can be viewed as possessing a more or less uniform array of strongly anharmonic local oscillators. The NMR relaxation times are particularly sensitive to the low-frequency anharmonic motion of these atoms, and thus provide an excellent probe for this behavior.

In conclusion, type-I Ba₈Ga₁₆Sn₃₀ NMR lineshapes and spin-lattice relaxation rates indicated the presence of a strong quadrupole relaxation mechanism. Analysis showed this behavior to be due to a strongly anharmonic rattler-type motion of the caged Ba atoms. Fitting using a 1-D double well potential with strong anharmonicity showed good agreement with the experimental data, which offers a good explanation for the rattling behavior and the relaxation mechanism.