Elemental Dilution Effect on the Elastic Response due to a Quadrupolar Kondo Effect of the Non-Kramers System Y_1−xPr_xIr₂Zn₂₀

Figures 2(a) and 2(b) show the temperature dependence of the elastic constants \(C_{v}=(C_{11}-C_{12})/2\) and \(C_{44}\), respectively, of the Pr-37% system. The data are normalized and presented as relative change using absolute values at 2.4 K: \(C_{v}=4.64\times 10^{10}\) J/m³ and \(C_{44}=4.45\times 10^{10}\) J/m³. \(C_{v}\) shows a remarkable frequency-dependent upturn at around 40 K in the frequency range between 18 and 110 MHz and Curie-like softening below 5 K. As shown in the lower part of Fig. 2(a), the background-subtracted ultrasonic attenuation coefficient exhibits a local maximum at which the elastic constant \(C_{v}\) shows the upturn. The maximum shifts to higher temperatures with increasing ultrasonic frequencies. Here, a frequency-dependent background \(\Delta\alpha_{0}(\omega, T)\) exhibiting monotonic increase with temperature is subtracted. On the other hand, \(C_{44}\) displayed in Fig. 2(b) increases monotonically on cooling at least down to 2.3 K with no obvious frequency dependence. These mode-selective frequency dependences are reminiscent of the ultrasonic dispersion (UD) found in cage compounds such as \(R_{3}\)Pd₂₀Ge₆ (R = rare earth)²⁶⁾ and the filled-skutterudites \(RT_{4}X_{12}\) (T = Os, Ru, and Fe; X = P, Sb, and As).^27–29) Some examples of the UD in the Pr-based compounds including the present system are summarized in Table I. The origin of these UDs is considered as a “rattling” motion due to the cage-like crystal structure.²⁷⁾ Here, the rattling means a thermally activated oscillation of the guest atom in an oversized atomic cage.^30,31) Pr atoms at the site 8a and Zn atoms at the site 16c in the present Y\(_{1-x}\)Pr_xIr₂Zn₂₀ system are surrounded by highly symmetric cages as shown in Figs. 1(b) and 1(c). For the analysis of the UD in \(C(\omega, T)\) and \(\alpha(\omega, T)\), we adopt a Debye-type dispersion²⁶⁾ as \begin{align} C(\omega, T) &= C(\infty, T) - \frac{C(\infty, T)-C(0, T)}{1+\omega^{2}\tau^{2}}, \end{align} (1) \begin{align} \alpha(\omega, T) &= \frac{C(\infty, T)-C(0, T)}{2\rho{v_{\infty}}^{3}}\frac{\omega^{2}\tau}{1+\omega^{2}\tau^{2}}. \end{align} (2)

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Figure 2. (Color online) Relative change of the elastic constants (a) \(C_{v} = (C_{11}-C_{12})/2\) and (b) \(C_{44}\) of the Pr-37% system as a function of temperature. The ultrasonic attenuation coefficients for the \(C_{v} = (C_{11}-C_{12})/2\) mode are shown in the same panel (a). Three different measurement frequencies are shown in different colors. Black-dashed lines in panel (a) indicate the phonon background of the high- and low-frequency limit, \(C_{\text{phonon}}(\infty, T)\) and \(C_{\text{phonon}}(0, T)\), respectively. Solid lines are the calculation results analyzed based on a Debye-type dispersion to reproduce UD1 (see text).

»View table Table I. Ultrasonic dispersions observed in the Pr-based cubic systems. Resonant temperature means the temperature which satisfies the resonant equation [\(\omega\tau(T) =1\)] at around \(\omega = 100\) MHz. The parameters for the UD of PrIr2Zn20 (Pr-100% system) are estimated by us from the data described in the original paper.13)

Here, ω is the angular frequency of the ultrasonic wave and τ is the relaxation time of the system. \(C(\infty, T)\) and \(C(0, T)\) denote the high- and low-frequency limits for the frequency-dependent elastic constant \(C_{v}(\omega, T)\), respectively. Since the contribution of the phonons is obviously large compared to that of the CEF excitation at around 40 K, we estimated that \(C_{\text{phonon}}(0, T)=4.600\times 10^{10}-9.28T^{2}+4.176T^{4}\) J/m³ from the experimental data and \(C_{\text{phonon}}(\infty, T)\) is vertically shifted by \(0.025\times 10^{10}\) J/m³ as shown with the broken lines in Fig. 2(a). The colored solid lines in Fig. 2(a) are the results of a calculation assuming an Arrhenius-type relaxation represented by \(\tau(T)=\tau_{0}\exp(E/k_{B}T)\), which roughly reproduces the experimental results. We obtained a characteristic relaxation time \(\tau_{0}=2.4\times 10^{-11}\) s and an activation energy \(E=250\) K from the Arrhenius plot as shown in Fig. 3(b).

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Figure 3. (Color online) (a) Relative change of the elastic constant \(C_{v}\) [\(= (C_{11}-C_{12})/2\)] and the ultrasonic attenuation coefficient of the Pr-37% system at 8 T (\(B\parallel [001]\)) as a function of temperature. The results at three different measurement frequencies are shown in different colors. Each arrow shows the resonant point (\(\omega\tau=1\)). (b) Arrhenius plot of the logarithm of the relaxation time \(\ln\tau\) vs the reciprocal temperature \(1/T\) for UD1 at \(B = 0\) (red) and 8 T (blue). The solid line shows the fitted straight line to the data in \(B = 0\).

In addition, we investigated the magnetic-field dependence of UD1. Figure 3(a) exhibits results for \(C_{v}\) obtained at selected frequencies (19, 65, and 109 MHz) and magnetic field of \(B = 8\) T. As shown in Fig. 3(b), the characteristic relaxation time and the activation energy are independent of the magnetic field, which is ascribed to the lattice vibrations.

As shown in the previous report by Ishii et al.,¹³⁾ a UD at around 2 K was observed in the Pr-100% system. The activation energy of this UD was estimated to be about 10 K. Assuming that the origin of this UD is same as for UD1 of the Pr-37% system, the energy scale of the UD decreases as the Pr concentration increases. Previous powder x-ray diffraction measurements pointed out that the lattice constant of Y\(_{1-x}\)Pr_xIr₂Zn₂₀ increases as the Pr concentration increases.²⁵⁾ Thus, this Pr concentration dependence can be interpreted as a size increase of the atomic cage together with an energy decrease of the vibration as the Pr concentration increases. Since no UD was observed for the Pr-3.4% system below 150 K,²¹⁾ it may be possible that such a UD exists at higher temperatures for the Pr-3.4% system. However, the UD in the Pr-100% system reported by Ishii et al. shows some differences from that of the Pr-37% system, such as that UD is observed for both \(C_{v}\) and \(C_{44}\) modes.

On the other hand, a mode-selective UD was observed previously for PrRh₂Zn₂₀, having the same crystal structure as PrIr₂Zn₂₀.³²⁾ This UD is named as “UDH” in the original paper.³²⁾ (PrRh₂Zn₂₀ has the other UD named as “UDL”, which has a similar energy scale to UD2 described in the Sect. 3.3 later.) The activation energy of UDH is \(E=440\) K, which is close to the present values of UD1 (\(E=250\) K). Recently, low-energy optical-phonon excitations at ∼7 meV were observed by inelastic x-ray scattering (IXS) in Pr\(T_{2}\)Zn₂₀ (T = Rh, Ir). It was concluded that such low-energy phonon modes originate from the oscillation of Zn atoms (site 16c) encapsulated in the Pr₂Zn₁₂ cage [shown in Fig. 1(c)].^34,35) First-principle calculations for LaRu₂Zn₂₀ (same structure as PrIr₂Zn₂₀) suggests that the vibrations of the \(\Gamma_{3}\) (\(E_{\text{g}}\)) and \(\Gamma_{5}\) (\(T_{\text{2g}}\)) modes in the (111) plane have very low energies. As their frequencies are so low and sensitive to the parameters of the calculation, thus, the symmetry of vibrations has not been revealed yet.³⁶⁾ The present ultrasonic results suggest that this vibration corresponds to the \(\Gamma_{3}\) mode. Further experiments are needed to clarify the origin of the ultrasonic dispersion.

The main panel of Fig. 4 shows the temperature dependences of \(C_{v}\) at various magnetic fields \(B\parallel [001]\) below 20 K. These temperature dependences of \(C_{v}\) can be analyzed by localized CEF models with the following Hamiltonian,^37,38) \(\mathcal{H}=\mathcal{H}_{\text{CEF}}+\mathcal{H}_{\text{Zeeman}}+\mathcal{H}_{\text{MS}}+\mathcal{H}_{\text{MM}}\) (see details in the Appendix).

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Figure 4. (Color online) Relative change of the elastic constant \(C_{v}\) of the Pr-37% system as a function of temperature at various magnetic fields \(B\parallel [001]\). Data above and below 0.5 K were measured with frequency of 110 and 201 MHz, respectively. Solid lines in (a) and (b) show the calculation results, respectively, without and with the contribution of the electric hexadecapole \(H_{v}\) (\(\lambda = 0.006\)). The black broken line indicates the background elastic constant (high-frequency limit). The inset in (b) shows the relative change of \(C_{v}\) as a function of magnetic field at \(T=0.020\), 0.920, and 10 K. The black solid curve in the inset is the CEF calculation for 20 mK. \(B^{*}\) is the characteristic magnetic field (see text). In addition, magneto-acoustic quantum oscillations were observed above 9 T at 20 and 920 mK (as shown in the enlarged view).

In principle, it is necessary to consider the hyperfine interaction between nuclear spins and localized quadrupoles, since the nuclear spin of the ¹⁴¹Pr nucleus (natural abundance is 100%) is non-zero (\(I = 5/2\)). In our previous study, we took into account the effect of hyperfine interactions, which could explain the appearance of a local minimum of the elastic constant at low magnetic fields.²¹⁾ The present results do not show such a local minimum. In fact, our calculations without hyperfine interaction already reproduce the experimental results well (as described in detail later), therefore, we conclude that the contribution of the hyperfine interaction is sufficiently small to be negligible at least in the present system.

We use the CEF parameters for PrIr₂Zn₂₀: \(B_{4}^{0}=-0.0109\) K and \(B_{6}^{0}=-0.4477\) mK in the present CEF Hamiltonian \(\mathcal{H}_{\text{CEF}}\) for the Pr-37% sample, since they as well can be used for the analysis for the ultrasonic data of the Pr-3.4% sample.^21,39)

The CEF level scheme based on these parameters is as follows; \(\Gamma_{3}\) (0 K)–\(\Gamma_{4}\) (27 K)–\(\Gamma_{1}\) (66 K)–\(\Gamma_{5}\) (67 K). The third term of the Hamiltonian describes the interaction between the elastic strain and an effective multipole, and the fourth term describes the intersite multipole–multipole interaction. Under the assumption with an effective multipole \(\widetilde{O}_{v}=O_{v}+\lambda H_{v}\) with \(\Gamma_{3}\) symmetry, we compare the results with and without the contribution of \(H_{v}\) in Figs. 4(a) and 4(b). Here, \(O_{v}=J_{x}^{2}-J_{y}^{2}\) is the \(\Gamma_{3}\)-type electric quadrupole and \(H_{v}=1/4[\{7J_{z}^{2}-J(J+1)-5\}(J_{+}^{2}+J_{-}^{2})+(J_{+}^{2}-J_{-}^{2})\{7J_{z}^{2}-J(J+1)-5\}]\) is the \(\Gamma_{3}\)-type electric hexadecapole. λ means the contribution ratio of the hexadecapole to the quadrupole. The calculation with \(\lambda=0.006\) [Fig. 4(b)] reproduces the experimental data much better than the calculation without including the electric hexadecapolar contribution [Fig. 4(a)], in particular, in the range of \(B > 8\) T. Thereby, it is necessary to consider not only the contribution of the electric quadrupole \(O_{v}\) but also that of the electric hexadecapole \(H_{v}\) to reproduce the data. Here, such non-negligible contribution of the electric hexadecapole (\(\lambda\neq 0\)) has already been demonstrated by ultrasonic measurements of Pr systems with a \(\Gamma_{3}\)–\(\Gamma_{4}\) state.⁴⁰⁾ The elastic constant \(C_{\Gamma_{3}}^{4f}(T, B)\) can be expressed by the following formula:⁴¹⁾ \begin{equation} C_{\Gamma_{3}}^{4f}(T, B) = C_{\Gamma_{3}}^{0}(T)-\frac{Ng_{\Gamma_{3}}^{2}\chi_{\Gamma_{3}}(T, B)}{1-{g^{\prime}}_{\Gamma_{3}}\chi_{\Gamma_{3}}(T, B)}. \end{equation} (3) Here, \(N=8x/a(x)^{3}=1.03\times 10^{27}\) m⁻³ is the number of Pr ions per unit volume in the Pr-37% sample, estimated from the lattice constant \(a(x)\) at room temperature, and \(C_{\Gamma_{3}}^{0}\) is the background elastic constant which originates from the phonon contribution [\(C_{\text{phonon}}(\infty, T)\), black broken line in Fig. 4]. \(g_{\Gamma_{3}}\) and \(g_{\Gamma_{3}}^{\prime}\) are the coupling constants of the multipole-strain interaction and the multipole–multipole interaction, respectively. Then, \(\chi_{\Gamma_{3}}\) is the susceptibility of the effective multipole \(\widetilde{O}_{v}\).

The coupling constants, \(g_{\Gamma_{3}}=40\) K and \(g^{\prime}_{\Gamma_{3}}=-0.05\) K are obtained from the fit to the data at various magnetic fields above 1 K, since a QKE and a change in the ground state due to a low-energy phonon excitations affect the data at lower temperatures (as described in detail below). The present parameter set for the calculation reproduces the magnetic-field dependence above 5 T, as shown in the inset of Fig. 4(b). Here, we define the characteristic magnetic field \(B^{*}\) below which the localized \(4f\)-electron model is no longer valid. \(B^{*}\) can be estimated as 5 T at the lowest temperature of 20 mK. It should be noted that magneto-acoustic quantum oscillations were observed above 9 T at 20 and 920 mK. This phenomenon proves the high quality of the present single crystal.

In addition, we reanalyzed previously reported data for the Pr-3.4%²¹⁾ system and the Pr-100%¹³⁾ system in zero field by considering hexadecapolar contributions using the same parameter \(\lambda=0.006\) (Fig. 5). In case of the Pr-3.4% system, \(g'_{\Gamma_{3}}\) is equal to zero, indicating that the contribution of the electric hexadecapole is negligible. On the other hand, the electric hexadecapolar contribution cannot be ignored in the Pr-100% system. The analysis as well \(\lambda = 0.006\) reproduces the data well for all Pr concentrations. Figure 6(a) shows \(g_{\Gamma_{3}}\) obtained with \(\lambda = 0.006\) (calc. 1) and \(\lambda = 0\) (calc. 2) for each Pr concentration. Figure 6(b) shows \(g^{\prime}_{\Gamma_{3}}\) correspondingly. The hexadecapolar contribution suppresses the Curie-like softening. Thus, the values of \(g_{\Gamma_{3}}\) are larger than those considering only the contribution of the electric quadrupole. The value of \(g_{\Gamma_{3}}\) in the present Pr-37% system is ∼1.5 times larger than for \(\lambda = 0\). In any case, the absolute values of both \(g_{\Gamma_{3}}\) and \(g^{\prime}_{\Gamma_{3}}\) increase linearly with Pr concentration.

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Figure 5. (Color online) Relative change of the elastic constants \(C_{v}\) with phonon background subtracted of the Pr-3.4% system¹³⁾ (left) and the Pr-100% system²¹⁾ (right) as a function of temperature. The red and blue lines show the calculations without and with the contribution of the electric hexadecapole \(H_{v}\) (\(\lambda=0.006\)).

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Figure 6. (Color online) The coupling constants of (a) the multipole-strain interaction \(g_{\Gamma_{3}}\) and (b) the multipole–multipole interaction \(g_{\Gamma_{3}}^{\prime}\) as a function of Pr concentration x for Y\(_{1-x}\)Pr_xIr₂Zn₂₀.^13,21) Blue closed markers (calc. 1) and red open diamonds (calc. 2) are parameters obtained by fits with (\(\lambda=0.006\)) and without (\(\lambda=0\)) electric hexadecapolar contribution, respectively. (c) T–x phase diagram of Y\(_{1-x}\)Pr_xIr₂Zn₂₀. \(T_{0}\) is the characteristic temperature of the QKE determined from specific-heat experiments.²⁰⁾ \(T^{*}\) is the temperature at which the behavior of the ultrasound mode \(C_{v}\) deviates from the CEF model.²¹⁾

Figure 7 shows the temperature dependence of the elastic constant \(C_{v}\) at the frequencies of 65, 110, and 201 MHz below 10 K.⁴²⁾ The purple dashed line at higher temperatures is obtained using the CEF calculation as described above. \(C_{v}\) at 110 MHz deviates from the calculation below \(T^{*}\sim 1\) K. Here, we defined \(T^{*}\) as the characteristic temperature below which the localized CEF model seems to be no longer applicable.

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Figure 7. (Color online) Relative change of the elastic constants \(C_{v}\) of the Pr-37% system as a function of temperature below 10 K. The purple broken lines show the CEF calculation using \(\lambda=0.006\). The black broken lines show the elastic constants estimated for the high-frequency limit, \(C_{v}(\infty)\) [the line in (a) is proportional to \(\ln T\), though that in (b) follows \(\sqrt{T}\)]. Solid lines are the calculations assuming a Debye-type dispersion to reproduce UD2.

In the present Pr-37% system, \(C_{v}\) shows frequency dependences at around 0.5 K, while \(C_{44}\) does not show a frequency dependence also at this low-temperature region (not shown). It is different from that \(C_{v}\) in the Pr-3.4% system shows no frequency dependence down to 0.04 K in the frequency range from 63 to 237 MHz.²¹⁾

Here, for UD2, we apply the same Arrhenius-type analysis as for UD1 at high temperatures. The solid lines in Figs. 7(a) and 7(b) show the results based again on a Debye-type dispersion [Eq. (1)]. Here, we assume that the high-frequency limit \(C(\infty, T)\) has (a) a logarithmic temperature dependence based on the single-site quadrupolar Kondo model and (b) a square-root temperature dependence based on the quadrupolar Kondo (virtual) lattice model.²²⁾ For both calculations, we assume that the low-frequency limit \(C(0, T)\) is calculated using the localized CEF model. The latter analysis reproduces the experimental data better, in particular, for the data at high frequency. This suggests that the quadrupolar Kondo (virtual) lattice model should be applied in the Pr-37% system, where the interaction between localized quadrupoles is not negligible, unlike the Pr-3.4% system.

We obtain the relatively slow characteristic relaxation time \(\tau_{0} = 3.1\times 10^{-9}\) s and the low activation energy \(E\approx 550\) mK. This energy scale is much smaller than that of UD1, indicating that UD1 and UD2 have different origins. The energy scale of UD2 is as well much smaller than that of UD observed in the Pr-100% system (\(E\sim 10\) K). On the other hand, a similar low-energy UD was observed in the non-Kramers compound PrMg₃ (\(\tau_{0}=0.85\times 10^{-9}\) s, \(E=500\) mK).³³⁾

PrMg₃ shows no long-range quadrupolar order at least down to 20 mK.¹⁰⁾ Previously, Araki et al. pointed out possible formation of a vibronic state, which is a quantum state caused by the coupling between the non-Kramers doublet and the phonons of the surrounding lattice near absolute zero.³³⁾ It has been theoretically predicted that a Kondo-like singlet can be formed by the coupling of the \(\Gamma_{3}\) doublet to dynamical Jahn–Teller phonons.^43,44) Based on these experimental and theoretical proposals and the similarities to the present case, it is possible that a vibronic state is formed in the NFL region of the Pr-37% system. From the perspective shown in Fig. 1(b), a Pr atom is surrounded by the high-symmetry cage of sixteen Zn atoms, which suggests the existence of an isolated low-energy phonon at the Pr site (interpreted as Einstein phonon). Due to this structural situation, it is relatively easy to form a “vibronic state” coupling between the low-energy phonons and the localized electric quadrupoles to release the entropy of the CEF ground-state doublet \(\Gamma_{3}\).

As we discussed, the elastic response of Y\(_{1-x}\)Pr_xIr₂Zn₂₀ at very low temperature changes drastically depending on the Pr-concentration x. Figure 6(c) summarizes the change in the quantum state as a function of x. Here, \(T_{0}\) is the characteristic temperature of NFL behavior, which is obtained from the specific-heat measurements.²⁰⁾ \(T^{*}\), the characteristic temperature of the QKE observed by our ultrasonic measurements, lies close to \(T_{0}\). In the Pr-37% system, it is possible to form a vibronic state, whereas a square-root temperature dependence of \(C_{v}\) expected for the quadrupolar Kondo lattice model was observed below 0.15 K. The present results suggest that the Pr–Pr interactions become non-negligible with increasing x, leading to a crossover from a quantum ground state of single-site QKE to a quadrupolar Kondo lattice, which may occur for Pr concentrations between 3.4 and 37%.

In the present paper, we analyze the temperature dependence of \(C_{v}\) at very low temperatures assuming that UD2 originating from electron–phonon coupling and the temperature dependence proportional to \(\sqrt{T}\), caused by the quadrupolar Kondo (virtual) lattice effect are independent. Such analysis qualitatively explains the present experimental results. However, whether these quantum states are independent or entangled is still an open question. To address the issue, it is necessary to construct a new theory including a localized quadrupolar moment and conduction electrons as well as phonons to describe the quantum ground state of Y\(_{1-x}\)Pr_xIr₂Zn₂₀. Therefore, it would be highly desirable to investigate the Pr concentration dependence in more detail through a series of experiments on Y\(_{1-x}\)Pr_xIr₂Zn₂₀.

Finally, we discuss the multiple effects of the elemental dilution. The Y-dilution changes the following three factors, i) the number density of the localized Pr moments, ii) the lattice constant, and iii) the atomic disorder. These three contributions can affect the magnitude of the Ruderman–Kittel–Kasuya–Yosida (RKKY)-type interaction among the localized quadrupoles. As the Pr concentration decreases, the lattice constant decreases.²⁵⁾ However, as the AFQ order observed in the Pr-100% system collapses in Y-containing systems, the effective RKKY-type interaction between the Pr³⁺ ions becomes weakened with the Y dilution. Indeed, the absolute value of the interaction between multipoles \(g^{\prime}_{\Gamma_{3}}\) [Fig. 6(b)] decreases linearly with decreasing Pr concentration. Then, the characteristic temperature of the QKE, \(T^{*}\) (or \(T_{0}\)) increases as the Pr concentration increases up to 37%. We assume that the change of \(T^{*}\) is ascribed to the atomic disorder which loweres the site symmetry of Pr³⁺ and locally splits the ground-state doublet \(\Gamma_{3}\). Since the atomic disorder is generally largest for \(x=0.5\) (\(\text{Y}:\text{Pr}= 1:1\)), this assumption is consistent with the Pr-concentration dependence of the characteristic temperatures. It is necessary to investigate the Pr-concentration dependence of the energy scale of the low-energy UD and compare it with that of the QKE characteristic temperature, which would serve for understanding the relationship between the QKE and the vibronic ground state.