crystal-field excitations and quadrupolar fluctuations of 4f-electron systems studied by polarized light scattering

Because of the heaviness of $f$ elements and the compactness of $f$ orbits, the energy scale of crystal-field (CF) effect is much weaker than that of spin-orbit coupling (SOC) for $f$-electron systems. As a result, CF interaction plays an important role in the low-temperature physics of these materials. In particular, if the CF ground state processes orbital degeneracy, it supports multipole moments, with quadrupole moments being the most common one.

For investigation of the CF and quadrupolar physics of 4$f$-electron systems, polarization-resolved Raman spectroscopy is a well suited experimental technique. Besides supreme spectral resolution, this method offers the ability to disentangle the long-wavelength excitation spectra into individual symmetry channels. By controlling the polarization of the incident and scattered light, we can identify the symmetry of different CF levels and separate quadrupolar excitations from other signals. In particular, light scattering provides unique access to transitions involving two quanta change of angular momentum compared to scattering techniques using neutrons or electrons.

In this talk, we use three 4$f$-electron systems, CeB$_6$, TbInO$_3$, and YbRu$_2$Ge$_2$, to present the essential aspects of CF excitations and quadrupolar fluctuations. For CF excitations, we discuss the establishment of CF level scheme from measured Raman spectra. We illustrate how the 4$f$ orbit is split by SOC and CF potential, and show the relevant spectra of inter- and intra-multiplet excitations over a large energy range. Moreover, we explain how correlation effects manifest themselves through the spectral linewidth via self-energy effect. For quadrupolar fluctuations, we discuss both ferroquadrupolar (FQ) and antiferroquadrupolar (AFQ) cases. We demonstrate that inelastic light scattering can directly probe FQ fluctuations, and capture the FM correlations induced by the tendency towards AFQ ordering. We further show that the strength of exchange interactions between local quadrupoles can be evaluated by analyzing the Raman susceptibility.

We start with CeB$_6$ which has simple cubic lattice structure and 4$f^1$ Ce$^{3+}$ electronic configuration [Phys. Rev. Materials 3, 065003 (2019)]. Beside analyzing its CF level scheme, we show that the spectral linewidth of the CF excitations increases on cooling below the temperature at which the electric resistivity shows local minimum. This coincidence points to the relationship between the broadening of CF linewidth and Kondo physics. Next we discuss the two-dimensional spin-liquid candidate TbInO$_3$ [arXiv:2015.12656 (2021)]. It remains unclear whether its magnetic lattice has honeycomb or triangular symmetry at low temperature. Despite of its complex 4$f^8$ Tb$^{3+}$ electronic configuration, we demonstrate that the Tb ions have a non-Kramers doublet ground state, which supports the scenario of triangular magnetic lattice. We further relate the enhanced linewidth of the CF modes to the magnetic fluctuations near the spin-liquid ground state.

Then, we study the FQ ordering of YbRu$_2$Ge$_2$ [Phys. Rev. B 99, 235104 (2019)]. This FQ phase is a realization of electronic nematic states since the electronic properties spontaneously break the four-fold rotational symmetry of the tetragonal crystal. The electronic Raman susceptibility in quadrupolar symmetry channels exhibit nearly Curie-law behavior, indicating weak exchange interactions between local quadrupoles. It is the relatively strong coupling between the quadrupole moments and lattice strain fields, analogous to cooperative Jahn-Teller effect, that is essential for the FQ transition at finite temperature. CeB$_6$, instead, has an antiferroquadrupolar (AFQ) phase at low temperature. Because the quadrupoles order at a finite wave vector, their fluctuations cannot be directly probed by inelastic light scattering. However, above the transition temperature the tendency towards AFQ ordering induces ferromagnetic (FM) correlations which exhibit as long-wavelength magnetic fluctuations in Raman spectra.

We acknowledge grant support from NSF DMR-1709161 and DMR-2105001.