Internal and External Field Effects upon Crystal Field Excitations in REFeO$_3$ (RE = Nd$^{3+}$, Er$^{3+}$, Yb$^{3+}$, Pr$^{3+}$, and Ho$^{3+}$)

We systematically investigated crystal-field (CF) excitations of Kramers (Nd$^{3+}$, Er$^{3+}$, Yb$^{3+}$) and non-Kramers (Pr$^{3+}$, Ho$^{3+}$) rare-earth ions in $REFeO_3$ using optimized CF simulations. Internal magnetic fields from the Fe$^{3+}$ and RE$^{3+}$ sublattices split the ground-state doublets of all Kramers ions (e.g. $NdFeO_3$, $ErFeO_3$, and $YbFeO_3$), generating low-energy excitations around 1 meV. In non-Kramers systems, low-energy excitations arise only when the ground state forms an accidental pseudo-doublet, as observed for Ho$^{3+}$ in $HoFeO_3$; such pseudo-doublets exhibit field-induced splitting analogous to Kramers ions. In contrast, true singlet ground states, exemplified by Pr$^{3+}$, show no zero-field splitting. Strong anisotropies are found in both internal- and external-field responses of the CF excitations in these $REFeO_3$. These results provide a unified explanation for the anomalous Zeeman splitting of CF ground states in $REFeO_3$.

💡 Research Summary

The paper presents a comprehensive study of crystal‑field (CF) excitations in the orthoferrite series REFeO₃ (RE = Nd³⁺, Er³⁺, Yb³⁺, Pr³⁺, Ho³⁺). Using the Stevens operator‑equivalent formalism, the authors construct detailed CF Hamiltonians for each rare‑earth ion, taking into account the low‑symmetry Cₛ point group of the 4c site, which requires fifteen independent Bₖᵠ parameters. These parameters are initially estimated from a point‑charge model and then refined by fitting to previously reported inelastic neutron scattering and Raman data (energies and relative intensities). A custom Python package, interfaced with the Mantid CF API and cross‑checked with PyCrystalField, is employed for the fitting and subsequent simulations.

For the three Kramers ions (Nd³⁺, Er³⁺, Yb³⁺) the time‑reversal symmetry guarantees that every CF level is at least doubly degenerate (Γ₃/Γ₄). The authors show that internal exchange fields arising from the Fe³⁺ sublattice and the ordered RE³⁺ moments act as an effective magnetic field on the 4f electrons. By applying internal fields along the crystallographic x, y, and z axes, they calculate the splitting of the ground‑state doublet. The simulations predict a low‑energy excitation in the 0.5–1.5 meV range, in excellent agreement with experimental observations of sub‑meV peaks in NdFeO₃, ErFeO₃, and YbFeO₃. The y‑axis internal field produces the largest shift (≈1.5 meV) and the steepest field dependence (≈0.2 meV T⁻¹), reflecting the anisotropic nature of the Fe‑RE exchange coupling. External magnetic fields are then added to the model. When a uniform external field of about 7.8 T is applied along any principal axis, it compensates the internal exchange‑field‑induced shifts, restoring the CF peaks to their zero‑field positions. This compensation does not directly measure the exchange field magnitude; rather, it demonstrates the delicate balance between internal and external magnetic contributions.

The non‑Kramers ions (Pr³⁺, Ho³⁺) possess singlet ground states (Γ₁ or Γ₂) under Cₛ symmetry, and therefore do not split under a magnetic field in the usual Kramers sense. However, the authors identify a special case for Ho³⁺ where two low‑lying singlets are accidentally close in energy, forming a pseudo‑doublet. In this situation, both internal and external magnetic fields lift the accidental degeneracy, producing a low‑energy excitation analogous to that observed for Kramers ions. Pr³⁺, by contrast, remains a true singlet with no zero‑field splitting, confirming the theoretical expectation.

A key outcome of the work is the demonstration of strong anisotropy in the magnetic‑field response of CF excitations. The direction of the applied field (x, y, or z) leads to markedly different energy shifts and splitting patterns, underscoring the role of the low‑symmetry crystal environment and the direction‑dependent exchange pathways. The authors’ unified CF model successfully reproduces the low‑energy spectra across the entire REFeO₃ series, providing a coherent explanation for previously puzzling observations of sub‑meV CF excitations in both Kramers and certain non‑Kramers orthoferrites.

In summary, the study establishes that (i) internal exchange fields universally split the ground‑state doublets of Kramers RE ions in REFeO₃, generating characteristic ~1 meV excitations; (ii) non‑Kramers ions exhibit similar low‑energy features only when an accidental pseudo‑doublet exists, as in HoFeO₃; (iii) true singlet ground states (e.g., Pr³⁺) show no such splitting; and (iv) the magnetic‑field response is highly anisotropic, reflecting the underlying Cₛ symmetry and exchange anisotropy. These insights not only resolve longstanding questions about low‑energy CF excitations in orthoferrites but also offer a predictive framework for designing rare‑earth based multifunctional materials with tailored magnetic and electronic properties.