Impact of hyperfine contributions on the ground state of spin-ice compounds

We examined the magnetic ground state of the pyrochlore spin-ice compounds ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ by means of specific-heat, magnetization, and ac-susceptibility measurements in the mK regime. At these low temperatures, we observe an unexpected large specific heat and corresponding entropy, which diminish in applied magnetic fields. This evidences the presence of additional states beyond the electronic spin and orbital degrees of freedom. We can qualitatively explain the large specific heat by the coupling of the nuclear spins of ${}^{141}\mathrm{Pr}$ and ${}^{165}\mathrm{Ho}$ with their electronic counterparts, which leads to a complex hyperfine-coupled term scheme. With increasing fields, the nuclear and electronic spins decouple leaving only the electronic excitations in the measured temperature window. At intermediate fields, a rather evolved term scheme emerges that may explain the unusual hysteretic magnetization and a remarkable state with a negative magnetization found for ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Our findings bring deep insights to the complex ground state of pyrochlore spin-ice compounds and their low-energy excitations.

Figure 1

(a) Low-temperature specific heat of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ as function of temperature, compared to literature data for ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ [3], all scaled in mol per rare-earth atom. Dashed lines of corresponding color show the subtracted phonon background estimated from data up to 40 K. Real (b) and imaginary part (c) of the zero-field dynamic magnetic susceptibility of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ as function of temperature.

Figure 2

Temperature-dependent magnetic part of the specific heat of (a) ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (b) ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ measured in several external magnetic fields. Entropy change as function of temperature for (c) ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (d) ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. $\mathrm{\Delta }{S}_{1/2}=Rln\left(2\right)$, expected for a spin-1/2 system, and $\mathrm{\Delta }{S}_{\mathrm{ice}}=R/2ln(3/2)\approx 0.2R$ are indicated by black and greens bars, respectively.

Figure 3

(a) Scheme of the effective localized magnetic moments for the case of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ (for ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ analogous, with a hyperfine split of $59.5a$). In weak magnetic fields (left), the localized electronic magnetic moment, $\vec{J}$, and the nuclear spin, $\vec{I}$, couple to the total angular momentum, $\vec{F}=\vec{J}+\vec{I}$. At high fields (right), $\vec{J}$ and $\vec{I}$ decouple. (b) Simplified level scheme of the ${\mathrm{Pr}}^{3+}$ ion. In low fields (left), the multiplet is described by $(F,m_F)$. At high fields (right), $\vec{J}$ and $\vec{I}$ decouple to a split electronic ground-state doublet with maximum $m_J$, whose two states are further split in $2m_I+1$ equidistant hyperfine states [37]. Levels with $m_J\ne \pm J$ are shifted to higher energies by the CEF. (See text for further details.)

Figure 4

Comparison of the measured (red/blue circles) specific heat of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ at zero field with the calculated (red/blue line) results using the single-ion $F$ multiplet model shown in Fig. 3 and Eq. (3). The hyperfine constants are set to $a^{\text{Pr}}=10$ mK for the Pr and to $a^{\text{Ho}}=42$ mK for the Ho compound.

Figure 5

Magnetization of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ as function of (a) magnetic field and (b) temperature. At small fields and low temperatures, the magnetic moment is negative. Panels (c) and (d) show the magnetization of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ as function of field and temperature, respectively. The saturation value at 4 T, about $1.0{\mu }_B$, is slightly smaller than the expected magnetic moment per Pr ion for a two-in-two-out spin configuration of the pyrochlore lattice and we observe no sign for a transition into a kagome-ice phase as indicated around 0.5 T in the ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ data at $T=0.05$ K shown in (a).