Importance of dynamic lattice effects for crystal field excitations in the quantum spin ice candidate ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$

We explore dynamic interactions between the crystal lattice and magnetic degrees of freedom in a frustrated magnetic system using the example of a pyrochlore quantum spin-ice candidate ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. Using Raman scattering spectroscopy we demonstrate that crystal electric field excitations of Pr3+, which define the magnetic properties of ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$, cannot be understood within a model of a static lattice. We identify vibronic interactions with a phonon which lead to a splitting of a doublet crystal field excitation at around 55 meV. We also observe an unconventional behavior of a splitting of the non-Kramers ground state doublet of ${\mathrm{Pr}}^{3+}$, revealed by observing excitations to the first excited singlet state $E_g^0\to A_{1g}$ at around 10 meV. The splitting has a strong temperature dependence, where the doublet structure is most prominent between 50 and 100 K, and the weight of one of the components strongly decreases on cooling contrary to simple thermal population tendency. We suggest a static or dynamic deviation of ${\mathrm{Pr}}^{3+}$ from the position in the ideal crystal structure can be the origin of the effect, with the deviation strongly decreasing at low temperatures.

Figure 1

Temperature dependence of the Raman scattering spectra of ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ in the temperature range from 6 to 300 K in a parallel polarization configuration $(x,x)$ in the spectral region of the CEF. The spectra are shifted along the $y$ axis for clarity. CEF excitations are marked by green triangles. The full measured spectral range up to 125 meV can be found in SM [20].

Figure 2

(a) Raman spectra of CEF excitation of ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ at 14 K. (b) A scheme of the CEF levels of ${\mathrm{Pr}}^{3+}$ in ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. (c) Diagram of the vibronic coupling between the phonons and CEF states. (d) ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ phonon dispersion obtained by DFT calculations. The energy range close to the vibronic features is shown. Dashed blue lines mark the experimentally observed CEF vibronic excitations. The vibronic coupling is expected to occur at the intersection point of the nonsplit CEF doublet (solid blue line) and the $T_{2g}$ phonon branch (red curve). (e) Atomic displacements of the $T_{2g}$ phonon mode. Upper panel: View of the unit cell. Lower panel: ${\mathrm{PrO}}_8$ octahedron site viewed from the [111] direction.

Figure 3

(a) Temperature dependence of the Raman scattering spectra in the energy range of the lowest-lying CEF excitation $E_g^0\to A_{1g}$. (b) Upper panel: Positions of the two components of the transition (${\omega }_a$ and ${\omega }_b$). Middle panel: Linewidths of the two components of the excitations (${\omega }_a$ and ${\omega }_b$). Lower panel: The ratio of the spectral weight of ${\omega }_b$ to ${\omega }_a$. (c) The scheme of the excitation $E_g^0\to A_{1g}$ illustrates that the shape of the electronic density distribution $\rho$ of the $E_g^0$ level, such as splitting, can define the shape of the observed spectral excitation line. Histograms demonstrate the density of the ground state splitting $\rho \left(\omega \right)$ obtained by the deconvolution of the Raman scattring spectra at 14, 40, and 80 K.