On the origin of the two thermally driven relaxations in diluted spin ice Dy_1.6Y_0.4Ti₂O₇

The geometrically frustrated spin systems have garnered intensive attention because of their peculiar magnetic ground states and spin dynamics originating from the frustrated interactions [1]. For the same reason, spin dynamics in these systems show rich behaviors. As a typical geometrically frustrated spin system, Dy₂Ti₂O₇ (DTO) has a pyrochlore structure with Dy³⁺ spins residing on the corner-sharing tetrahedral lattice. Dy³⁺ spins are confined along the local (111) directions due to strong crystal field [2, 3], rendering a well separated spin ground state doublet at about 200 K below the first excited state. DTO shows absence of magnetic order down to 0.3 K, as evidenced by specific heat measurements [4, 5]. This is quite surprising considering the value for the nearest-neighbor dipolar interaction [6] D_nn ∼ 2.35 K and nearest-neighbor exchange interaction [7] J_nn ∼− 1.24 K, as one would expect a magnetic ordering at around 1 K, the same magnitude as the Curie–Weiss temperature θ_W. The dipolar interaction is found to be the origin of the spin ice physics [7], so DTO is also termed dipolar spin ice.

Efforts of both ac susceptibility [3, 8, 9] and muon spin relaxation [10] have been devoted to the understanding of spin relaxation mechanisms in the cooperative paramagnetic state [11]. At low temperatures (below 4 K), experiments [3, 9] and theoretical simulation [12] identified a spin freezing process, with a much stronger frequency dependence than those of spin glass and a different field dependence [13]. Recent advances in the investigation of the ground state excitation of spin ice materials revealed a collective excitation equivalent to the magnetic monopole [14–17], with its zero field relaxation behavior consistent with the ac susceptibility measurement [15]. At temperatures higher than 13 K, ac susceptibility revealed a spin freezing peak at both zero and finite magnetic fields [3, 18]. Spin relaxation in this temperature range was found to be dominated by single ion relaxation [10, 19, 20], with the relaxation barrier set by the crystal electric field. At the intermediate temperature range (4–13 K), an apparently temperature-independent relaxation mode was found to dominate and was attributed to quantum tunneling [10, 20]. In line with this finding, field-induced behaviors in this material are also unconventional [20, 21]. In particular, in the diluted spin ice compound Dy_1.6Y_0.4Ti₂O₇ (DYTO), two spin relaxation modes were found [20]: one being thermally activated, and the other attributed to quantum tunneling. The coexistence of both thermal and quantum spin relaxations at the same temperature range is intriguing, since spins are expected to choose the faster relaxation channel, instead of having more than one relaxation mode.

This unresolved issue arouses our interest in a more thorough study of this phenomenon. Here we report a detailed analysis of the ac susceptibility of diluted spin ice DYTO, focusing on the temperature range 9–13 K in which two relaxation modes coexist. The frequency spectra were fitted by a superposition of two Cole–Cole relaxation functions, from which the characteristics of the spin relaxation are extracted. We found that the relaxation times for both processes increase as temperature decreases, indicating that both are thermally activated, instead of one being thermal and the other of quantum origin respectively, as reported in earlier studies [20]. Moreover, the intrinsic frequencies for both processes were found to be the same, with a magnitude that is two orders of magnitude smaller than that of a single spin process; while the activation barriers are different. We propose that the two processes correspond to two different relaxation modes of correlated spin clusters.

Polycrystalline DYTO samples were prepared by the standard solid-state reaction method [18, 22]. X-ray diffraction showed that the samples are of pure pyrochlore structure. Ac susceptibility was measured in a Quantum Design physical property measurement system (PPMS) with an excitation field H_ac = 10 Oe and frequency f (10 ≤ f ≤ 10⁴ Hz).

As shown in figure 1(a), at zero field, the frequency spectrum shows a well defined single peak, consistent with previous studies. The relaxation mechanism has been identified to be a single ion process [19, 20]; i.e., spins relax between the two degenerate ground states through the excitation to the first excited state. In contrast, the behavior at nonzero field is very different. The frequency spectra of DYTO in figure 1(b) show the coexistence of two relaxation peaks at H = 5 kOe: one at the lower frequency side shifts to higher frequencies with increasing temperatures, while the other at the higher frequency side shows no apparent shift at different temperatures. These two modes have been attributed to thermal spin relaxation and quantum spin relaxation, respectively [20]. The origin of the presumed quantum relaxation remains unclear, but it is believed that the quantum tunneling between the degenerate ground state spin doublet is facilitated by the transverse component of the effective field [19, 20, 23]. At a higher field of 1 T, the frequency spectra return to a single peak profile, as shown in figure 1.

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The relaxation time for both processes at 0.5 T has been extracted by Snyder et al previously [20], by taking the local maxima f_p in the frequency spectra. Relaxation time is then obtained by τ = 1/(2πf_p). However, it is important to note that when two peaks coexist in the frequency spectra this method is not accurate, due to the interference of the peak positions between the two [24]. Here we apply a more reliable extraction of the relaxation time for both processes by fitting the spectra using a superposition of two Cole–Cole functions. The formula we use is shown in equation (1).

$$\begin{eqnarray} {\chi }_{\mathrm{ac}}&=&{\chi }_{s,1}+\frac{({\chi }_{0,1}-{\chi }_{s,1})}{(1+(\mathrm{i}2 \pi f{\tau }_{1})^{1-{\alpha }_{1}})^{{\beta }_{1}}}+{\chi }_{s,2}\nonumber\\ &&+~\frac{({\chi }_{0,2}-{\chi }_{s,2})}{(1+(\mathrm{i}2 \pi f{\tau }_{2})^{1-{\alpha }_{2}})^{{\beta }_{2}}}. \end{eqnarray} \tag{ 1 }$$

It consists of two generalized Cole–Cole functions, i.e. Havriliak–Negami formulas [25]. χ_s is the adiabatic susceptibility (χ_ac at the high frequency limit) and χ₀ is the isothermal susceptibility (χ_ac at the zero frequency limit). The subscripts 1 and 2 denote the parameters for the two sets of peaks respectively. Compared to the original Cole–Cole formula, the Havriliak–Negami formula includes two parameters, α and β, which describe the distribution of the relaxation time around the medium relaxation time, and the asymmetry of the frequency spectrum. When α = 0 and β = 1, the Havriliak–Negami formula retains the original Debye form that describes the relaxation with a single relaxation time; For α ≠ 0 and β = 1, it becomes the Cole–Cole form [26]. The plot of χ'' versus χ' is a circular arc with the length of (1 − α)π, which intersects with the χ' axis at χ_s and χ₀; when α = 0 and β ≠ 1, it is the Davidson–Cole formula, and the χ'' − χ' plot shows an upturned tail at the high frequency side [27].

As is known from previous studies, the relaxation time of this system shows sizable distribution, so α ≠ 0. Also, the χ'' − χ' plots in figures 1 and 2 show no upturn at the high frequency side. Therefore, in the following fittings, we use α ≠ 0 and β = 1. Figure 2 shows the fitting to the data; discrete symbols are the measured data, and red solid lines are the best fitting with equation (1) using the least squares method. It can be seen that the double peak features are well reproduced by the double Cole–Cole formula. Contributions from the two processes are also shown in figure 2. Their peak positions indeed show considerable differences compared with the local maxima positions taken directly on the original data, which is the advantage of our analysis.

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The optimized fitting parameters are summarized in figures 3 and 4. Both α₁ and α₂ increase with decreasing temperature, signaling the broadening of the distribution of relaxation time at lower temperatures. Most importantly, both τ₁ and τ₂ increase with decreasing temperature, suggesting that both modes are thermally driven. This is in sharp contrast to the previous understanding that one of the modes is thermally driven and the other is quantum [20]. The relaxation time τ₂ is of the order of 10⁻³ s; it increases from 0.5 × 10⁻³ s at 13 K to 9 × 10⁻³ s at 9 K. The relaxation time τ₁ is much shorter, with a weaker but monotonic temperature dependence, increasing from 1.1 × 10⁻⁴ s at 13 K to 2.0 × 10⁻⁴ s at 9 K.

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For high temperature (T > 13 K) relaxation at zero field, the single relaxation mode corresponding to the single ion relaxation dominates this diluted and also the parent compound Dy₂Ti₂O₇. At nonzero fields, however, two relaxation modes emerge. With both relaxation times τ₁ and τ₂ growing with decreasing temperature, it is reasonable to check whether both modes can still be assigned to the single ion process. The thermally activated process is fitted using τ = τ₀e^E_A/k_BT, where τ₀ is the relaxation time at the high temperature limit and E_A is the relaxation energy barrier. The fittings to the experimental data are shown in figure 5. The energy barrier E_A for τ₁ and τ₂ are 15.2 K and 56.5 K, respectively. These values can only serve as an order of magnitude estimate due to the limited temperature range available for the fitting. It is very interesting to note that both fittings result in an intercept at around lnτ₀ =− 10.1, corresponding to an intrinsic relaxation time of 4 × 10⁻⁵ s, which is much longer than the typical relaxation time of the order of 10⁻⁹ s for the single ion process. The very close intrinsic relaxation time τ₀ for both processes provides clues for their mechanism.

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The intrinsic frequencies that are several orders of magnitude lower than that of the single ion process suggest that the relaxation modes are associated with correlated spin ensembles. The nearly identical frequencies for the two modes further suggest that the two relaxation processes may originate from the same correlated spin ensemble, i.e. spin clusters. As temperature is lowered to T < 13 K, there is a crossover from single ion relaxation to the relaxation of correlated spin clusters. The development of spin correlation and formation of spin clusters is feasible in DTO due to the large dipole–dipole interactions, facilitated by the presence of a DC magnetic field. The two modes could be the result of the relaxations of clusters with magnetization mainly parallel to the field, and those with magnetization mainly perpendicular to the field, respectively. Thus the two modes are orthogonal and independent, which can be observed simultaneously. Otherwise it would be difficult to understand the coexistence of two modes, since as mentioned earlier the system should choose the faster relaxation mode. An alternative explanation for the weaker temperature dependence of τ₁ could be thermally assisted quantum tunneling [23]. However, this is inconsistent with the similar intrinsic frequency of τ₁ and τ₂.

In fact, characteristics of the two relaxation modes have readily emerged in figure 1 in the temperature dependence of the peak magnitude. At zero field, the peak magnitude increases as T decreases, while at 1 T the dependence is reversed. Interestingly, at 0.5 T, the set of peaks at the higher frequency side shows the same temperature dependence as the zero field case, whereas the set of peaks at the lower frequency side has the same temperature dependence as the 1 T case. We also determined the relaxation time at zero and 1 T and plotted their temperature dependence in figure 5 for comparison, by taking the peak position in the spectra since both cases show a well defined single peak. It appears clearly that the τ at 0 T behaves similarly to τ₁, and the behavior of τ at 1 T resembles that of τ₂. This shows that the two relaxation modes at 0.5 T actually inherit from the zero field and high field ones. In our proposed picture, τ₁ represents the relaxation of the spin clusters with magnetization perpendicular to the field, while τ₂ is for the relaxation of the spin clusters with magnetization parallel to the field. With increasing field, the spin clusters tend to be polarized along the field direction; therefore, the relaxation mode associated with τ₂ will dominate as seen in the 1 T case. Our physical picture can also explain the difference in relaxation energy barrier for the two modes: the τ₂ process requires the flip of the magnetization of the clusters, whereas the τ₁ process only partially reorients the clusters. Therefore, E_A should be smaller for the τ₁ process, which is indeed what we have found.

The existence of spin clusters has been proposed in previous studies of spin frustrated materials, in both zero field [23] and nonzero field [21], although the detailed mechanism remains unclear. Our findings provide further evidence of the existence of spin clusters and their cooperative behavior in these systems. We note that a previous study on the single-crystalline DTO samples shows a stronger frequency and magnetic field dependence [28] than that of polycrystalline samples, which is in line with our results, as in the cluster picture the formation of spin clusters in single-crystalline samples is facilitated due to the fewer grain boundaries compared with the polycrystalline case. Meantime, our cluster picture seems to provide a plausible explanation for the long-standing issue where the spin relaxation time τ observed by both ac susceptibility [20] and μSR measurement [10] shows the same temperature dependence (τ increases with decreasing temperature for T > 13 K and T < 4 K, and has a plateau in τ(T) for 4 < T < 13 K) but with time scales differing by three orders of magnitude. With the existence of spin clusters, the magnitude of the relaxation time depends on the relaxation mechanism and the size of the clusters. It is therefore possible that the spin clusters have a wide distribution of size, and ac susceptibility and μSR selectively probe different spin ensembles with their respective frequency ranges.

To summarize, we have investigated the ac susceptibility of the spin diluted Dy_1.6Y_0.4Ti₂O₇ compound. The coexistence of two relaxation modes in the temperature range between 9 and 13 K was investigated quantitatively, by fitting the two-peak frequency spectra using a double Cole–Cole formula. The resultant relaxation times indicate that both are thermally activated, in contrast to previous studies. Both relaxation modes show the same intrinsic frequency, but different activation barriers. We propose that the existence of spin clusters accounts for the relaxation in this regime.

This work is supported by the US NSF-DMR0547036, DMR1104994, the National Basic Research Program of China (grant No 2012CB927404), and the NSFC (grant No 11174247).