Dipolar spin ice states with fast monopole hopping rate in CdEr$_2$X$_4$ (X = Se, S)

TL;DR

This study demonstrates that CdEr₂X₄ (X=Se, S) exhibits dipolar spin ice states with significantly faster monopole hopping rates than traditional spin ices, enabling new insights into spin ice dynamics and ground states.

Contribution

The paper reveals that CdEr₂X₄ supports dipolar spin ice states with ultra-fast monopole hopping, distinct from known spin ices, and provides a detailed analysis of the underlying mechanisms.

Findings

CdEr₂X₄ exhibits dipolar spin ice states with Pauling entropy.

Monopole hopping rates in CdEr₂X₄ are three orders of magnitude faster than in Dy₂Ti₂O₇.

Fast dynamics are primarily due to rapid monopole hopping, not increased monopole population.

Abstract

Excitations in a spin ice behave as magnetic monopoles, and their population and mobility control the dynamics of a spin ice at low temperature. CdEr$_2$Se$_4$ is reported to have the Pauling entropy characteristic of a spin ice, but its dynamics are three-orders of magnitude faster than the canonical spin ice Dy$_2$Ti$_2$O$_7$. In this letter we use diffuse neutron scattering to show that both CdEr$_2$Se$_4$ and CdEr$_2$S$_4$ support a dipolar spin ice state -- the host phase for a Coulomb gas of emergent magnetic monopoles. These Coulomb gases have similar parameters to that in Dy$_2$Ti$_2$O$_7$, i.e. dilute and uncorrelated, so cannot provide three-orders faster dynamics through a larger monopole population alone. We investigate the monopole dynamics using ac susceptometry and neutron spin echo spectroscopy, and verify the crystal electric field Hamiltonian of the Er$^{3+}$ ions using inelastic neutron scattering. A quantitative calculation of the monopole hopping rate using our Coulomb gas and crystal electric field parameters shows that the fast dynamics in CdEr$_2$X$_4$ (X = Se, S) are primarily due to much faster monopole hopping. Our work suggests that CdEr$_2$X$_4$ offer the possibility to study alternative spin ice ground states and dynamics, with equilibration possible at much lower temperatures than the rare earth pyrochlore examples.

Tables

Table 1. Table 1: The fitted Wybourne CEF parameters (meV) and ground state doublets for CdEr 2 Se 4 and CdEr 2 S 4 .

	$B_2^0$	$B_4^0$	$B_4^3$	$B_6^0$	$B_6^3$	$B_6^6$
CdEr2Se4	$-25.70$	$-107.73$	$-97.74$	$25.31$	$-19.06$	$9.51$
CdEr2S4	$-29.18$	$-122.72$	$-113.66$	$25.97$	$-21.89$	$14.41$

Table 2. Table S1: Refinement results of the X-ray diffraction data for CdEr 2 Se 4 and CdEr 2 S 4 . The listed parameters are the refined size of the unit cell a 𝑎 a , fractional position x 𝑥 x of the X = 𝑋 absent X= Se or S ions, and the goodness-of-fit R p subscript 𝑅 p R_{\textrm{p}} , R wp subscript 𝑅 wp R_{\textrm{wp}} , and χ 2 superscript 𝜒 2 \chi^{2} .

	$a$ (Å)	$X$ ($x$)	$R_{\text{p}}$	$R_{\text{wp}}$	${\chi }^2$
CdEr2Se4	11.6097(1)	0.2566(1)	13.0	14.6	2.16
CdEr2S4	11.1527(1)	0.2589(2)	19.7	19.1	1.89

Table 3. Table S2: CEF levels for CdEr 2 Se 4 (column 2) and CdEr 2 S 4 (column 3) fitted from inelastic neutron scattering experiments. CEF levels for CdEr 2 Se 4 fitted from magnetization measurements from Ref. Lago  et al. ( 2010 ) are listed in column 1. Energy unit is meV.

	CdEr2Se4 Lago et al. (2010)	CdEr2Se4	CdEr2S4
${\Gamma }_5^{+}\oplus {\Gamma }_6^{+}$	0	0	0
${\Gamma }_4^{+}$	3.64	3.95	5.08
${\Gamma }_4^{+}$	7.22	5.65	6.91
${\Gamma }_5^{+}\oplus {\Gamma }_6^{+}$	7.65	8.93	10.47
${\Gamma }_4^{+}$	8.70	9.76	11.35
${\Gamma }_4^{+}$	21.70	26.28	30.46
${\Gamma }_4^{+}$	23.31	29.11	33.78
${\Gamma }_5^{+}\oplus {\Gamma }_6^{+}$	23.81	29.21	34.00

Table 4. Table S3: The Wybourne CEF parameters for the CdRE 2 Se 4 compounds scaled from the refined CdEr 2 Se 4 values. The sizes of the unit cell used in the scaling calculations are also listed.

	CdDy2Se4	CdHo2Se4	CdTm2Se4	CdYb2Se4
unit cell (Å)	11.467	11.638	11.560	11.528
Wybourne CEF parameters (meV)
$L_2^0$	$-27.88$	$-26.59$	$-24.82$	$-23.96$
$L_4^0$	$-124.91$	$-114.88$	$-100.99$	$-95.04$
$L_4^3$	$-113.33$	$-104.23$	$-91.63$	$-86.23$
$L_6^0$	$30.68$	$27.70$	$23.28$	$21.53$
$L_6^3$	$-23.10$	$-20.85$	$-17.53$	$-16.21$
$L_6^6$	$11.53$	$10.40$	$8.75$	$8.09$

Equations

$$\mathcal{H}$$

$$\mathcal{H}=\sum _{lm}{B}_l^m{\hat{O}}_l^m+H\cos (\varphi ){\hat{J}}_x+H\sin (\varphi ){\hat{J}}_y\text{,}$$

$$A_l^m(R^{\prime })=\frac{a^{l+1}(R)}{a^{l+1}(R^{\prime })}{A}_l^m(R)\text{,}$$

$$B_l^m=A_l^m⟨r^l⟩{\theta }_l\text{.}$$

$$\displaystyle\text{Dy}^{3+}\ |\pm\rangle=\$$

$$+$$

$$\displaystyle\text{Ho}^{3+}\ |\pm\rangle=\$$

$$+$$

$$\displaystyle\text{Tm}^{3+}\ |\phi\rangle=\$$

$$\displaystyle\text{Yb}^{3+}\ |\pm\rangle=\$$

$$+0.885|3.5,\mp 2.5⟩\text{.}$$

$$\mathcal{H}=$$

$$\displaystyle+Dr_{0}^{3}\sum_{\langle n,\nu;n^{\prime},\nu^{\prime}\rangle}\Bigl{[}\frac{S_{n,\nu}\cdot S_{n^{\prime},\nu^{\prime}}}{|r_{n,\nu;n^{\prime},\nu^{\prime}}|^{3}}-\frac{3\left(S_{n,\nu}\cdot r_{n,\nu;n^{\prime},\nu^{\prime}}\right)\left(S_{n^{\prime},\nu^{\prime}}\cdot r_{n,\nu;n^{\prime},\nu^{\prime}}\right)}{|r_{n,\nu;n^{\prime},\nu^{\prime}}|^{5}}\Bigr{]}$$

$$=$$

$$\sum _{{\nu }^{\prime },\beta }{J}_{k;\nu ,\alpha ;{\nu }^{\prime },\beta }{u}_{k;{\nu }^{\prime },\beta }^{(\rho )}={\lambda }_k^{(\rho )}{u}_{k;\nu ,\alpha }^{(\rho )},$$

$$k_{\mathrm{B}}{T}_{\mathrm{c}}$$

$${\chi }_{k;\nu ,\alpha ;{\nu }^{\prime },\beta }=\frac{N{\mu }^2}{V}\sum _{\rho }\frac{u_{k;\nu ,\alpha }^{(\rho )}{u}_{k;{\nu }^{\prime },\beta }^{(\rho )\ast }}{3k_{\mathrm{B}}{T}_{\mathrm{MF}}-2{\lambda }_k^{(\rho )}},$$

$$\frac{d\sigma }{d\Omega }$$

$$\times {\chi }_{k;\nu ,\alpha ;{\nu }^{\prime },\beta }\cos \left[\tau \cdot \left(r_{\nu }-r_{{\nu }^{\prime }}\right)\right]-Kf(Q{)}^2,$$

Full text

The spin relaxation rate of canonical spin ices was a particular problem in the spin representation. From high to low temperature it changes from thermally activated, to a temperature independent plateau, to a re-entrant thermally activated regime Snyder et al. (2004); Jaubert and Holdsworth (2009); L. D. C. Jaubert and P. C. W. Holdsworth (2011); Ruminy et al. (2017). At high temperature, above the monopole regime, Orbach processes describe the thermally activated relaxation rate Ruminy et al. (2017). The plateau and re-entrant thermally activated regimes are not readily explained in the spin representation, but can now be understood as the hopping of monopoles by quantum tunneling in screened and unscreened regimes of the Coulomb gas respectively Jaubert and Holdsworth (2009); L. D. C. Jaubert and P. C. W. Holdsworth (2011). In the unscreened regime, the relaxation rate depends on the monopole density $\rho$ with the hopping rate $u$ as the coefficient: $f\propto u\rho$ when the system is near equilibrium Ryzhkin (2005); Castelnovo et al. (2011); Paulsen et al. (2014).

Although the monopole charge $Q_{m}$ and chemical potential $v_{0}$ can be calculated exactly from the spin model, the value of the monopole hopping rate $u$ is not well-understood and is usually treated as a fitting parameter Jaubert and Holdsworth (2009); L. D. C. Jaubert and P. C. W. Holdsworth (2011); Takatsu et al. (2013). For Dy2Ti2O7, $u$ is fitted to be $\sim 10^{3}$ Hz at $T<12$ K, which has been experimentally confirmed through the Wien effect Giblin et al. (2011). Recently, Tomasello et al found that this hopping rate can be estimated by the splitting of the crystal-electric-field (CEF) ground state doublet under an internal transverse magnetic field of 0.1–1 T Tomasello et al. (2015). To verify the universality of this approach, it is beneficial to compare the monopole dynamics in other dipolar spin ice compounds.

The newly proposed spin ice state in the spinel CdEr2Se4 provides such an opportunity Lau et al. (2005); Lago et al. (2010); Reig-i Plessis et al. (2017). In this compound, Er3+ ions constitute the pyrochlore lattice, and bulk measurements have revealed the Pauling entropy and local Ising character for the Er3+ spins Lau et al. (2005); Lago et al. (2010); both are strong indicators of the existence of the spin ice state although microscopic evidence is required to confirm the dipolar character necessary for deconfined, interacting monopoles. Of special importance is the low-temperature dynamics in CdEr2Se4, which was revealed to be three-orders faster than that of the pyrochlore titanate Dy2Ti2O7 Lago et al. (2010). The origin of this increase and its compatibility with monopole dynamics in CdEr2Se4 remains unclear.

In this letter, we explore spin ice states and monopole dynamics in CdEr${}_{2}X_{4}$ ($X$ = Se, S). Using inelastic neutron scattering to study the CEF transitions and neutron diffuse scattering to study the spin correlations, we confirm the existence of dipolar spin ice states in CdEr${}_{2}X_{4}$. Through ac susceptibility measurements, we reveal fast monopole dynamics in the whole quantum tunneling regime. Comparison with a calculation of the splitting of the Er3+ CEF ground state doublet under perturbative transverse fields reveals the increase of the monopole hopping rate as the main contribution to the fast dynamics. Thus our work explains the fast monopole dynamics in CdEr${}_{2}X_{4}$ and provides general support to this monopole hopping mechanism in dipolar spin ices.

Our powder samples of CdEr2Se4 and CdEr2S4 were synthesized by the solid state reaction method sup . To reduce neutron absorption, the 114Cd isotope was used. X-ray diffraction measurements confirmed the good quality of our samples, with the Er${}_{x}X_{y}$ impurities less than 1 %. Inelastic neutron scattering experiments were performed on IN4 with 1.21 and 2.41 Å incident neutron wavelengths at Institut Laue-Langevin (ILL). Polarized neutron diffuse scattering experiments were performed on CdEr2Se4 using D7 with a 4.8 Å setup at ILL. Non-polarized neutron diffuse scattering experiments were performed on CdEr2S4 using DMC with a 2.46 Å setup at SINQ of Paul Scherrer Insitut (PSI). Neutron spin echo experiments were performed on IN11 at ILL. AC susceptibilities $\chi$ in the frequency range of $1$–$1\times 10^{3}$ Hz were measured with the Quantum Design MPMS SQUID at Laboratory for Scientific Developments and Novel Materials of PSI. AC susceptibilities in the frequency range of $2.5\times 10^{4}$–$5.5\times 10^{6}$ Hz were measured using a bespoke induction ac susceptometer.

Fig. 1 presents the inelastic neutron scattering results of the CEF transitions in CdEr2Se4 and CdEr2S4. Altogether 6 peaks are observed at the base temperature for both compounds, which is consistent with the Stokes transitions within the Er3+ ${}^{4}I_{15/2}$ manifold under $D_{3d}$ symmetry. Using the McPhase program Rotter (2004), we fitted the measured spectra with the CEF Hamiltonian $\mathcal{H}=\sum_{lm}B_{l}^{m}\hat{O}_{l}^{m}$, where $\hat{O}_{l}^{m}$ are the Stevens operators and $B_{l}^{m}$ are the corresponding coefficients. The fitting results are shown in Fig. 1 as the solid lines and Table S2 lists the fitted CEF parameters and ground state wavefunctions. The energies of the CEF levels are presented in the inset of Fig. 1, and also in the supplemental materials sup . For both compounds, the ground states transform as the $\Gamma_{5}^{+}\oplus\Gamma_{6}^{+}$ dipole-octupole doublet Huang et al. (2014); Li et al. (2016). Specifically, the wavefunctions for both of the ground state doublets are dominated by the $|15/2,\pm 15/2\rangle$ components and have almost the same anisotropic $g$-factors of $g_{\perp}=0$ and $g_{\parallel}=16.4$, which is consistent with the previous report for CdEr2Se4 Lago et al. (2010). Thus our inelastic neutron scattering results confirm the Ising character of the Er3+ spins in CdEr2Se4 and CdEr2S4. Scaling our parameters sup suggests other members of the series may be: Heisenberg-like (Dy, Yb); non-magnetic (Tm); or Ising-like with low-lying excited states (Ho), a situation of interest for forming a quantum spin ice Molavian et al. (2007)

Although the Pauling entropy is a strong signature of the spin ice state in CdEr2Se4 Lago et al. (2010), it only characterizes the spin configurations at the length scale of a single tetrahedron. To realize a magnetic Coulomb gas with interacting monopoles, it is essential to have a dipolar spin ice state with power-law spin correlations, which can be verified through measurements of the spin correlations Fennell et al. (2009). Fig. 2 presents the quasi-static spin-spin correlations in CdEr2Se4 obtained from polarized neutron diffuse scattering Ehlers et al. (2013). Broad peaks are observed at 0.6 and 1.4 Å*-1*, and the overall pattern is very similar to that of the known dipolar spin ices Kadowaki et al. (2002); Mirebeau and Goncharenko (2004); Hallas et al. (2012). Sharp peaks with very weak intensities are also discernible near 1.1 Å*-1* and can be attributed to the magnetic Bragg peaks of ErxSey impurities sup .

To fit the observed spin-spin correlations in CdEr2Se4, we performed single-spin-flip Monte Carlo simulations for the dipolar spin ice model with exchange couplings up to the second neighbors Yavors’kii et al. (2008):

[TABLE]

Here, $\vec{n}_{i}$ is the unit vector along the local $\langle 111\rangle$ axes with the positive direction pointing from one diamond sublattice of the tetrahedra center to the other, $\sigma_{i}=\pm 1$ is the corresponding Ising variable, $J_{1}$ and $J_{2}$ are the exchange interactions for nearest neighbors (NN) $\langle ij\rangle$ and second-nearest neighbors $\langle\langle ij\rangle\rangle$, respectively, $r_{0}$ is the NN distance, and $D=\mu_{0}(\langle\hat{J}_{z}\rangle g\mu_{B})^{2}/(4\pi r_{0}^{3})$ is the dipolar interaction, 0.62 and 0.69 K for CdEr2Se4 and CdEr2S4, respectively. With the ALPS package Bauer et al. (2011), we implemented the Hamiltonian (1) on a $6\times 6\times 6$ supercell with periodic boundary conditions. The dipolar interaction was truncated beyond the distance of 3 unit cells. The spin-spin correlations were evaluated every 100 sweeps during the $4\times 10^{5}$ sweeps of measurement. Assuming the effective NN coupling $J_{\textrm{eff}}=J_{1}+5D/3$ to be equal to 1 K at which temperature the CdEr2Se4 specific heat maximum was observed Lago et al. (2010); den Hertog and Gingras (2000); Zhou et al. (2012), we fixed $J_{1}$ to $-0.03(1)$ K and only varied $J_{2}$ in the fitting process. As is shown in Fig. 2, the model with $J_{2}=0.04(1)$ K fits the measured spin correlations very well. We found no need to include $J_{3}$, which appears in other dipolar spin ices Yavors’kii et al. (2008). Although the exact value of $J_{2}$ might be susceptible to both the supercell size and the dipolar cutoff, our simulations do confirm the dominance of the dipolar interactions in CdEr2Se4. Non-polarized neutron diffuse scattering results for CdEr2S4 are shown in the Supplemental Material sup , which have similar $Q$-dependence as that of CdEr2Se4 and can be fitted by the dipolar spin ice model as well. In this way, we establish the existence of the dipolar spin ice state in CdEr2Se4 and CdEr2S4.

With the fitted CEF ground states and coupling strengths, we can determine the monopole parameters. The monopole charge $Q_{m}=2\langle\hat{J}_{z}\rangle g\mu_{B}/\sqrt{3/2}r_{0}$ can be calculated to be 3.28 and 3.42 $\mu_{B}/$Å for CdEr2Se4 and CdEr2S4, respectively Castelnovo et al. (2008). The chemical potential $v_{0}=2J_{1}+(8/3)(1+\sqrt{2/3})D$, which is half of the energy cost to create and unbind a monopole-antimonopole pair Zhou et al. (2011), is 2.93 K for CdEr2Se4 and 3.84 K for CdEr2S4. Although the chemical potentials in CdEr${}_{2}X_{4}$ are lower than that in Dy2Ti2O7 (4.35 K), they are still more than two times higher than the energy cost $E_{\textrm{unbind}}=(8/3)\sqrt{2/3}D$ to unbind a monopole-antimonopole pair, locating both compounds in the same weakly correlated magnetolyte regime as Dy2Ti2O7 Zhou et al. (2011).

Monopole dynamics in the low and high frequency regimes can be probed with ac-susceptibility Snyder et al. (2004); Quilliam et al. (2011); Yaraskavitch et al. (2012); K Matsuhira and Y Hinatsu and T Sakakibara (2001) and neutron spin echo spectroscopy Ehlers et al. (2003, 2004), respectively, and the representative results for CdEr2Se4 are shown in Fig. 3a and b. Fig. 3c summarizes the temperature dependence of the characteristic relaxation time $\tau=1/2\pi f$ in CdEr${}_{2}X_{4}$, where the results for $\tau>1\times 10^{-3}$ s are extracted from the peak positions of the imaginary part of the ac-susceptibility $\chi^{\prime\prime}(T)$, the results for $10^{-5}>\tau>10^{-7}$ s are obtained by fitting $\chi(\omega)$ to the Cole-Cole model Bovo et al. (2013), and the results for $\tau<10^{-8}$ s are obtained by fitting the neutron spin echo intermediate scattering function with $S(Q,t)/S(Q,0)=A\operatorname{exp}[-t/\tau(T)]$ Ehlers et al. (2003, 2004). The relaxation time in Dy2Ti2O7 Yaraskavitch et al. (2012); K Matsuhira and Y Hinatsu and T Sakakibara (2001) is also shown in Fig. 3c for comparison.

Firstly, we observe that at $T>10$ K, the relaxation time in CdEr${}_{2}X_{4}$ obeys the Orbach law of $\tau=\tau_{0}[\operatorname{exp}(\Delta/k_{B}T)-1]$ Ruminy et al. (2017), with the parameters $\tau_{0}=3.93(9)\times 10^{-11}$ s and $\Delta=77.1$ K for CdEr2Se4, and $\tau_{0}=2.73(5)\times 10^{-11}$ s and $\Delta=96.3$ K for CdEr2S4. The fitted excitation energies $\Delta$ in CdEr${}_{2}X_{4}$ are much smaller than that of Dy2Ti2O7 ($\Delta>230$ K), which is due to their lower CEF excited states Ruminy et al. (2017).

The Orbach behavior of the relaxation rate does not extend to the lowest temperature. Instead, at $T$ in-between 2 and 5 K, a plateau region with $\tau\sim 4.9\times 10^{-7}$ s, which was inaccessible in the previous susceptibility measurements Lago et al. (2010), is observed for both CdEr2Se4 and CdEr2S4, reminiscent of the $\tau\sim 2.6\times 10^{-4}$ s quantum tunneling plateau in Dy2Ti2O7 Snyder et al. (2004); Jaubert and Holdsworth (2009); K Matsuhira and Y Hinatsu and T Sakakibara (2001). Such a similarity extends to even lower temperatures where the relaxation time starts rising again. As can be seen in Fig. 3c, at $T<1$ K, the relaxation time in CdEr${}_{2}X_{4}$ can be described by the Arrhenius law of $\tau_{0}\operatorname{exp}(\Delta/k_{B}T)$, with parameters $\tau_{0}=1.01(1)\times 10^{-10}$ s and $\Delta=10.07$ K for CdEr2Se4, and $\tau_{0}=2.9(1)\times 10^{-10}$ s and $\Delta=10.2(6)$ K for CdEr2S4. Due to the limited data points for CdEr2Se4, the fitted $\Delta$ value from Ref. Lago et al. (2010) has been used. The activation energies in CdEr${}_{2}X_{4}$ are very close to that of Dy2Ti2O7, where the Arrhenius law with $\tau_{0}=3.07\times 10^{-7}$ s and $\Delta=9.93$ K has been observed in a similar temperature regime Yaraskavitch et al. (2012).

Despite the similar temperature evolution, the absolute values of the monopole relaxation rates in CdEr${}_{2}X_{4}$ are about $10^{3}$ times higher than that in Dy2Ti2O7 for the whole measured quantum tunneling region, which cannot be simply accounted for by the difference of the monopole densities $\rho$. Assuming $\rho(T)\propto\operatorname{exp}(-v_{0}/k_{B}T)$, the monopole densities in CdEr${}_{2}X_{4}$ are no more than $10$ times higher than that of Dy2Ti2O7 in the investigated quantum tunneling region. According to the $f\propto u\rho$ relation of the Debye-Hückel theory, there must be a two-orders increase of the monopole hopping rates $u$ in CdEr${}_{2}X_{4}$.

Following Tomasello et al. Tomasello et al. (2015), we analyze the perturbation effect of an internal transverse magnetic field on the CEF ground state doublet in CdEr2Se4 and CdEr2S4. Due to the similar NN couplings Zhou et al. (2011), we expect similar internal field strengths in CdEr${}_{2}X_{4}$ and Dy2Ti2O7 Sala et al. (2012). The perturbed Hamiltonian can be written as:

[TABLE]

where the $y$ direction is along the $C_{2}$ axis and $\phi$ is the angle between the transverse field $H$ and the $x$ direction (see the inset of Fig. 4). Similar to the Dy3+ ions in Dy2Ti2O7 Tomasello et al. (2015), the Kramers degeneracy of the Er3+ ions causes a third-order dependence of the ground state splitting on the field strength in the perturbative regime: $\Delta E=\alpha\left[1+A\operatorname{cos}(6\phi)\right]H^{3}$. Using the McPhase program Rotter (2004), we directly diagonalize the Hamiltonian (2) and fit the coefficients to be $\alpha=2.80\times 10^{-4}$ ($1.95\times 10^{-4}$) [meV/T3] and $A=0.136$ (0.098) for CdEr2Se4 (CdEr2S4). For Dy2Ti2O7, using the CEF parameters of Ref. Ruminy et al. (2016), the coefficients are calculated to be $\alpha=2.14\times 10^{-6}$ [meV/T3] and $A=0.183$. As is compared in Fig. 4 for magnetic field along the $x$ direction, the CEF ground state splittings in CdEr${}_{2}X_{4}$ are indeed $\sim 10^{2}$ larger than that in Dy2Ti2O7 under the same transverse magnetic field. This higher susceptibility to transverse magnetic field is a property of the full CEF Hamiltonian of CdEr${}_{2}X_{2}$ as compared to that of Dy2Ti2O7.

Our results suggest that to explain the much faster dynamics of CdEr${}_{2}X_{4}$ vs Dy2Ti2O7 only similar monopole populations combined with a much faster monopole hopping rate in the former are required, and also support the single-ion quantum tunnelling process proposed in Ref. Tomasello et al. (2015) as a general monopole hopping mechanism in dipolar spin ices. Meanwhile, it should be noted that other factors may also contribute to the high monopole hopping rates in CdEr${}_{2}X_{4}$. For example, the non-vanishing components of $|J,J_{z}\rangle$ with $|J_{z}|\leq 7/2$ in CdEr${}_{2}X_{4}$ ground state doublet might induce multipolar interactions that can further increase the monopole hopping rates Rau and Gingras (2015).

In summary, neutron scattering investigations of the spin correlations in CdEr${}_{2}X_{4}$ ($X$ = Se, S) confirm they are the first spinels that realize dipolar spin ice states. High temperature Orbach behavior gives way to fast (compared to Dy2Ti2O7) monopole hopping dynamics at low temperature. Comparison of monopole populations calculated using Coulomb gas parameters estimated from the diffuse scattering experiments and bulk properties, and monopole hopping rates calculated using the CEF Hamiltonian derived from our inelastic neutron scattering data, show that the main contribution to the fast monopole dynamics of CdEr${}_{2}X_{4}$ is due to the much larger hopping rate. The reproduction of the very different relaxation rates in CdEr${}_{2}X_{4}$ and Dy2Ti2O7 using realistic parameters supports the general application of this method to the description of monopole hopping processes in dipolar spin ices.

Cd$R_{2}X_{4}$ (and Mg$R_{2}X_{4}$ Reig-i Plessis et al. (2017)) afford new possibilities in the study of frustrated magnetism on the pyrochlore lattice, with single ion ground states sup , interactions, and dynamics that contrast with the well known pyrochlore oxides Gardner et al. (2010). One immediate benefit of the fast monopole hopping rate in CdEr${}_{2}X_{4}$ is that it enables the study of the magnetic Coulomb phase in a broader frequency regime. In particular, nonequilibrium phenomena such as the Wien effect Paulsen et al. (2014, 2016), which appear in Dy2Ti2O7 at temperatures well below those measured here (by susceptibility), may be modified. On the other hand, if the timescale of dynamics is taken as a measure of the quantum contribution to the dynamics of a spin ice, going from slow and classical (Dy2Ti2O7) to fast and quantum (e.g. Tb- or Pr-based quantum spin ice candidates Gingras and McClarty (2014)), CdEr${}_{2}X_{4}$ offer an intermediate case that may help in the extrapolation of our understanding of the former to that of the latter. Finally, CdEr${}_{2}X_{4}$ offer the possibility to look for a new ground state of dipolar spin ice McClarty et al. (2015a). As is discussed in the supplemental materials sup , for Dy2Ti2O7, an antiferromagnetic ordering transition at $\sim 0.1$ K is expected Melko et al. (2001); McClarty et al. (2015a); Ruff et al. (2005); Henelius et al. (2016), but is experimentally inaccessible due to the relatively high freezing temperature of $\sim 0.65$ K Snyder et al. (2004); Pomaranski et al. (2013). For CdEr2Se4, our parameters predict a ferromagnetic ordering transition at $\sim 0.37$ K and a comparable freezing temperature of $\sim 0.29$ K sup . This means that both the ferromagnetic ground state and new monopole interactions caused by the bandwidth of the spin ice states may be experimentally accessible sup . Further dynamical and thermodynamic measurements at low temperatures would be required to conclude whether the spin ice state that we observed at 0.07 K is an equilibrium state and to explore the possible ordering transition in CdEr${}_{2}X_{4}$.

.1 Sample preparation

The polycrystalline samples of CdEr${}_{2}X_{4}$ ($X=$ Se, S) were prepared by solid state synthesis from binary Er and Cd selenides and sulfides. The binary Cd$X$ were synthesized from the elemental Cd-114, while the Er${}_{2}X_{3}$ were prepared from the high purity Er chips (99.9 %, Chempur) and elemental S (99.999 %, Strem Chemicals) or Se (99.999 %, Alfa Aesar). Selenium was additionally purified by zone melting. To reduce the oxide impurity Er2O${}_{2}X$ which easily forms in the open air, all preparation procedures (quartz ampoule filling, reacted mixture regrinding, and pellets pressing) were performed in an argon box with an O2 and H2O content of $\sim 1$ ppm. To reach full homogeneity, at least three sintering cycles of synthesis of the binary Er and Cd chalcogenides were performed. The phase purity of the binary compounds was checked by x-ray powder diffraction. Finally, the ternary 114CdEr${}_{2}X_{4}$ were prepared by two consecutive synthesis at 800 ∘C for one week each.

The single crystals of CdEr${}_{2}X_{4}$ were grown by the chemical transport reactions method. As starting materials the preliminary synthesized polycrystalline powders were used. For the growth, several transport agents were probed, including chlorine, bromine and iodine. We found that only the iodine is suitable for the growth of the ternary phase, while in the case of chlorine or bromine the final product contained mainly binary Cd and Er chalcogenides. The growth process was performed in a two-zone furnace with the hot part temperature of 950 ∘C and a temperature gradient of about 40 ∘C. The time for one crystal growth experiment was between 1 and 1.5 months. As a result, the octahedron-like single crystals with dimension up to 1.5 mm of the edge were obtained.

.2 Sample characterizations and impurities in CdEr2Se4

The purity content and crystal structure of the samples were checked by conventional X-ray powder diffraction on polycrystalline samples and crashed single crystals. Fig. S1 shows the refinement results for the 114CdEr2Se4 and 114CdEr2S4 polycrystalline samples using cubic $Fd\overline{3}m$ symmetry expected for the normal spinel structure. Tab S1 lists the refined size of the unit cell, fractional position for the Se or S ions, and the goodness-of-fit parameters. No inversion between Cd and Er can be observed. No peaks from impurities are detectable, implying their tiny amount. However, at low temperatures, weak magnetic Bragg peaks of the ErxSey impurities S. Calder and T. Fennell and W. Kockelmann and G. C. Lau and R. J. Cava and S. T. Bramwell (2010) are discernible in the neutron diffuse scattering experiment shown in Fig. 2 of the main text.

The extrinsic origin of the weak Bragg peaks is evident in their different temperature dependence compared with the broad diffuse scattering. Fig. S2 shows the non-polarized neutron diffraction results measured on D20 at ILL with the 20 K measurement subtracted as the background. The setup with 2.41 Å incoming neutron wavelength was employed. As can be seen in the inset of Fig. S2a and Fig. S2b, intensities of the sharp peaks saturate at temperatures below 0.8 K while the broad peaks from the diffuse scattering continue their growth, evidencing their different origins.

.3 CEF levels for CdEr2Se4 and CdEr2S4

Table S2 lists the crystal-electric-field (CEF) levels for CdEr2Se4 and CdEr2S4 fitted from inelastic neutron scattering experiments. The CEF levels for CdEr2Se4 fitted from magnetization measurements Lago et al. (2010) are also listed, showing strong deviations at high energies.

.4 Scaled CEF levels for CdRE2Se4

With the classical spin ice state established in CdEr${}_{2}X_{4}$ ($X=$ S, Se), it is tempting to look for the quantum spin ice state in the CdRE${}_{2}X_{4}$ series. Until now, four other compounds with RE = Dy, Ho, Tm, and Yb have been successfully synthesized Lau et al. (2005); Yaouanc et al. (2015); Yoshizawa et al. (2015); Higo et al. (2017). Their CEF parameters can be approximated using the scaled values of CdEr2Se4 Bertin A. and Chapuis Y. and Dalmas de Réotier P. and Yaouanc A. (2012):

[TABLE]

where $R$ and $R^{\prime}$ represent different rare-earth ions, $a$ is the size of the unit cell and the values for the CdRE2Se4 compounds are listed in Tab. S3, and $A_{l}^{m}$ are the Hutchings CEF parameters that can be transformed from the Stevens CEF parameters $B_{l}^{m}$ following the relation:

[TABLE]

In this expression, $\langle r^{l}\rangle$ is the expectation value of the $l$-th power of the $f$-electron radius and can be found in Ref. Freeman and Desclaux (1979), $\theta_{l}$ is the Stevens factor that can be found in Ref. Jensen and Mackintosh (1991).

Tab. S3 lists the scaled Wybourne CEF parameters for the CdRE2Se4 compounds. The calculated CEF levels are shown in Fig. S3, and the ground states are:

[TABLE]

Firstly, it is clear that the ground state of Tm3+ is a singlet due to its non-Kramers character, while for spin ice, a doublet ground state is required. For the remaining compounds where a doublet CEF ground state is realized, spins in CdDy2Se4 and CdYb2Se4 exhibit Heisenberg-like character, with $g$-factors of $g_{\perp}=5.69$, $g_{\parallel}=6.38$ for Dy3+ and $g_{\perp}=2.16$, $g_{\parallel}=3.97$ for Yb3+. For the Ho3+ spin, an Ising character with $g_{\perp}=0$, $g_{\parallel}=4.43$ is observed, which satisfies the local Ising condition to realize the spin ice state. Therefore, from the CEF point of view, CdHo2Se4 might be the most promising compound to realize the quantum spin liquid state. However, it should be noted that the lowest CEF excited states in CdDy2Se4, CdTm2Se4, and CdHo2Se4 are lying at energies below $\sim 2$ meV. Such low-lying excited levels might renormalize the spin couplings and make our single-ion analysis inappropriate Molavian et al. (2007); Yaouanc et al. (2015).

.5 Dipolar spin ice state in CdEr2S4

Ice-correlations similar to that of CdEr2Se4 are also observed in CdEr2S4. Fig. S4 presents the non-polarized neutron diffuse scattering results for CdEr2S4 measured on DMC at PSI with the setup of 2.46 Å incoming neutron wavelength. The 50 K measurement has been subtracted as the background. Similar to the CdEr2Se4 results shown in Fig. 2 of the main text, broad peaks at $\sim 0.6$, 1.4, and 2.5 Å*-1* are observed at low temperatures, which suggest similar ice-correlations in CdEr2S4.

Mean-field calculations were performed to confirm the ice-correlations in CdEr2S4 Kadowaki et al. (2002). Denoting the $\alpha$ component ($\alpha=x,y,z$) of the $\nu$-th unit-length spin ($\nu=1,2,3,4$) in the $n$-th primitive unit cell as $S_{n,\nu,\alpha}$, the Hamiltonian on the pyrochlore lattice can be explicitly expressed as:

[TABLE]

where the $E_{a}$ term represents the easy-axis anisotropy and $\hat{n}_{\nu}$ is the unit vector along the easy axis of the $\nu$-th spin. Fourier transform of the real-space coupling $J_{n,\nu,\alpha;n^{\prime},\nu^{\prime},\beta}$ leads to the $12\times 12$ coupling matrix $J_{k;\nu,\alpha;\nu^{\prime},\beta}$ in reciprocal space, which is then diagonalized:

[TABLE]

where $\lambda_{k}^{(\rho)}$ with $\rho=1,2,...,12$ denotes the eigenvalues and $u_{k}^{(\rho)}$ denotes the corresponding eigenvectors. The global maximum of $\lambda_{k}^{(\rho)}$ determines the long-range order transition under the mean-field approximation, with the transition temperature $T_{c}$ as:

[TABLE]

The paramagnetic susceptibility at $T_{\textrm{MF}}>T_{c}$ can be approximated by the eigenvalues and eigenfunctions:

[TABLE]

where $N$ is the total number of the unit cell, $V$ is the volume of the system, and $\mu$ is the size of the magnetic moment. The cross section of the magnetic scattering can be expressed as:

[TABLE]

where $P$ is a constant, $f(Q)$ is the magnetic form factor, $\tau$ is the reciprocal lattice vector, and $r_{\nu}$ denotes the position of the $\nu$-th atom in the first primitive cell. The additional term of $Kf(Q)^{2}$ accounts for the subtracted spin correlations at 50 K.

To account for the Ising character of the Er3+ spin, a high anisotropy of $E_{a}=100$ K was used. The dipolar interactions with $D=0.69$ K was truncated beyond the length of 5 unit cells. Since $T_{c}$ is an effective temperature that can be different from the real ordering temperature, the mean-field temperature $T_{\textrm{MF}}$ was used as a fitting parameter Kadowaki et al. (2002). Fig. S4 presents the fitted results with $J_{1}=0$. In this case, $T_{c}$ is calculated to be 1.8 K, and the fitted $T_{\textrm{MF}}=1.9$, 2.3, 3.8, and 15.8 K for the data measured at 1.6, 4, 8, and 20 K, respectively. Thus the ice-correlation is proved to exist in CdEr2S4.

In our mean-field calculation, the variance of $J_{1}$ does not affect the goodness-of-fit as long as $J_{1}/3+5D/3>0$. To obtain the monopole chemical potential, we measured the specific heat $C_{p}$ for CdEr2S4 single crystals. As is shown in Fig. S5, the $C_{p}(T)$ maximum of CdEr2S4 is at $\sim 1.4$ K, which enable us to fix the monopole chemical potential to 3.84 K in CdEr2S4 (see main text).

.6 Freezing and ordering temperatures in CdEr2Se4

The true ground state of a dipolar spin ice has again become a topical question Pomaranski et al. (2013). An ordering transition is expected due to the bandwidth of the dipolar spin ice microstates Melko et al. (2001), the transition temperature and eventual ground state being controlled by $D/J_{2}$ Ruff et al. (2005); McClarty et al. (2015a); Henelius et al. (2016). In Dy2Ti2O7 a transition to antiferromagnetic order is expected at $\sim 0.1$ K Yavors’kii et al. (2008); McClarty et al. (2015a); Henelius et al. (2016), far below temperatures at which equilibration is easily possible, i.e. 0.65 K Snyder et al. (2004).

The freezing temperature of 0.29 K for CdEr2Se4 is estimated from the 0.65 K freezing temperature of Dy2Ti2O7. With the known monopole parameters, the temperature dependence of the monopole density $\rho(T)$ can be obtained using the Debye-Hückel theory Zhou et al. (2011). Assuming the monopole hopping rate $u$ to be temperature independent and $u\rho$ to be the same at the freezing temperature for CdEr2Se4 and Dy2Ti2O7, we first calculate the monopole density in Dy2Ti2O7 at 0.65 K, then divide by 100 to account for the increased value of $u$ in CdEr2Se4, and finally locate the freezing temperature in CdEr2Se4 by following its $\rho(T)$ relation using Debye-Hückel theory.

The ordering temperature $T_{c}=0.37$ K for CdEr2Se4 is estimated from the $J_{2}/D$-$T_{c}/D$ phase diagram in Ref. McClarty et al. (2015b). Considering the $-3$ times difference in the definition of $J_{2}$, the ratio $J_{2}/D$ is found to be $-0.2$, which leads to $T_{c}=0.37~{}K$ under a linear extrapolation of the boundary between the classical spin ice and ferromagnetically long-range ordered phases. This means that in CdEr2Se4 there is a chance not only to investigate a spin ice with an alternative ground state; but also to study the effect on monopole dynamics of reaching a temperature comparable to the bandwidth of the spin ice states where new monopole-monopole interaction terms may appear.

.7 Note for the ground state doublet splitting in Dy2Ti2O7

Private communications with the authors of Ref. Tomasello et al. (2015) confirm a typo for the fitted parameter $\alpha$ in Eq. (9) of their publication. The correct value should be $\alpha=3.29\times 10^{-6}$ [meV/T3], which we reproduce using the same CEF parameters. The similar but different values of $\alpha=2.14\times 10^{-6}$ [meV/T3] and $A=0.183$ reported in our main text are calculated with refined CEF parameters Ruminy et al. (2016), which were not available to the authors of Ref. Tomasello et al. (2015).