Phonon-mediated spin-flipping mechanism in the spin ices ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

To understand emergent magnetic monopole dynamics in the spin ices ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, it is necessary to investigate the mechanisms by which spins flip in these materials. Presently there are thought to be two processes: quantum tunneling at low and intermediate temperatures and thermally activated at high temperatures. We identify possible couplings between crystal field and optical phonon excitations and construct a strictly constrained model of phonon-mediated spin flipping that quantitatively describes the high-temperature processes in both compounds, as measured by quasielastic neutron scattering. We support the model with direct experimental evidence of the coupling between crystal field states and optical phonons in ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$.

Figure 1

General features of QENS in spin ices, as exemplified by ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. (a) We see a $|\vec{Q}|$-independent $S\left(\right|\vec{Q}|,\omega )$ response, with QENS around the elastic line and transitions among excited crystal field states. (b) An example of a resolution-convoluted fit of the quasielastic Lorentzian (QENS) and two CFEs (T1 and T2). (c) The general evolution of the QENS and excited-state transitions, along with the resolution regimes used in the measurements (${\lambda }_{1,2,3}=11$, 6, and 4.3 Å in this case).

Figure 2

(a) The QENS width $\mathrm{\Gamma }$ as a function of temperature and (b) relaxation time as a function of inverse temperature, measured with different neutron wavelengths shown by the symbols. Solid lines are from the model described in the text; dotted lines in (b) indicate the resolution limit of the different settings of the spectrometer. The same quantities for ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are shown in (c) and (d), incorporating QENS (FOCUS, this study) and neutron spin echo (NSE, [43]), compared to the full model (FM), the first term of the model ($\mathrm{\Delta }=26.3$ meV), and an Arrhenius law (AL) [43].

Figure 3

Schematic overview of the interaction of CFEs and phonons. The line color of the CFEs represents the combined values of the quadrupolar transition-matrix elements between the members of the crystal field ground state and excited states. Quadrupolar operators $Q_{\mu }$ and zone center phonon modes ($u$) of (a, b) ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and (c, d) ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are sorted by their symmetries (a, c) $A$ and (b, d) $E$. The relevant phonon modes (represented by solid lines) are quasidegenerate with a CFE supporting a strong quadrupolar matrix element of the correct symmetry. Phonons that are not involved are shown by dotted lines.

Figure 4

CFE and phonon interactions in a single crystal of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Two CFEs ($E=22$ and 26.3 meV) and a phonon ($E=31$ meV) can be seen. The CFEs can be measured at the zone center ($l=8$) and boundary ($l=7$) and shift upward between 5 and 200 K. (a) The CFE at $E=22$ meV disperses upward by 0.5 meV at the zone center, where it intersects with an optical phonon [51]. (b, c) The intensity of the two CFEs follows the magnetic form factor along $(0,0,l)$, except for the CFE at $E=22$ meV which is boosted at zone centers with strong phonon structure factors ($l=4n$) (scan positions of (a) are indicated by colored points).