Uncovering field-induced magnetic phase transition by direct observation of the crystal electric-field splitting in a rare-earth magnetic insulator

TL;DR

This work directly maps the crystal electric field (CEF) level structure of CsErSe$_2$ under magnetic fields using far-infrared and Raman magneto-optical spectroscopy to extract the Stevens coefficients and exchange scales that define the single-ion Hamiltonian. By combining these parameters with a minimal XXZ exchange in a Weiss mean-field, the authors predict a field-induced ground-state level crossing at $B_C\approx5$ T for $\mathbf B\parallel c$, producing a metamagnetic-like jump in $M_c$ confirmed by AC susceptibility and magnetization data, and they identify a secondary crossing in the first excited state at $B_C^*\approx1.4$ T. Neutron scattering reveals low-temperature stripe antiferromagnetic order with a moment around $3.27\mu_B$, indicating coexistence of long-range order with predominantly single-ion–driven physics. Overall, the results show that accurate CEF parameterization is essential to understand and predict the rich magnetic behavior of rare-earth insulators under applied fields. The study highlights how narrowly spaced CEF levels can drive nontrivial, field-tuned phenomena with potential implications for designing and interpreting spin models in $4f$ systems.

Abstract

An indispensable step toward understanding magnetic interactions in rare-earth magnets is to determine the spatially anisotropic single-ion properties set by crystal electric field (CEF) physics. The CEF Hamiltonian yields a discrete energy spectrum governed by a set of parameters reflecting the local site symmetry of the magnetic ion. However, experimentally determining these parameters, especially for ones at low-symmetry sites remains highly challenging. In this work, we directly measure the CEF level splittings of CsErSe2 under magnetic fields using optical spectroscopy. This enables us to determine the CEF parameters and to predict the metamagnetic-like transition arising from a level-crossing in the ground state. We also identify a level-crossing in the first excited state that leads to a non-monotonic Zeeman splitting, which strongly influences the temperature and field dependence of the magnetization. Our results highlight the capacity of single-ion physics to drive rich and unanticipated phenomena in rare-earth magnetic insulators under applied magnetic field.

Figures (9)

• Figure 1: The field dependent CEF energy levels of single Er$^{3+}$ ion in CsErSe$_{2}$ under applied magnetic field (a) $\mathbf B\parallel b$ and (b) $\mathbf B\parallel c$, calculated from Eq. (\ref{['eq:ham']}) using the obtained parameters from this study. Eight doublets at zero field ($B=0$) (shaded area on the left side of each plot) are clustered such that five lower levels lie below around 5 meV and the rest above 23 meV. Note that the ground state wave function changes due to the level crossing at $B_C \approx 5$ T, and another crossing between the upper Zeeman-split level of the ground state and the lower of the first excited occurs at $B_C^* \approx 1.4$ T in $\mathbf B\parallel c$ shown in (b) (c) Crystal structure of CsErSe$_{2}$ adopts $P6_3/mmc$ space group Xing2020acs (d) Er$^{3+}$ (red) triangular layers formed by edge-sharing ErSe$_6$ octahedra, exhibiting $D_{3d}$ site symmetry (e) The calculated magnetization $M_b$ ($\mathbf B\parallel b$) and $M_c$ ($\mathbf B\parallel c$) with mean-field approximation (see the text) at $T=$ 25 mK are shown a function of applied field, which will be discussed in Sec. \ref{['mo']} and Sec. \ref{['chiac']}. The inset displays the same quantities at lower field range with $M_b$ and $M_c$ calculated at $T=2$ K added. Magnetization curves calculated with the MF approximation at different temperatures are Fig. \ref{['fig:mag']} in the Appendix.
• Figure 2: (a,b) Field dependence of normalized FIR absorption spextra at $T=5.5$ K below 120 cm$^{-1}$ and magnetic field applied in $\mathbf B\parallel b$ (a) and $\mathbf B\parallel c$ (b). (c) Raman shift, denoted as $E$, as a function of field in $\mathbf B\parallel c$ measured at $T \approx 5$ K are shown. The field-independent Raman active phonon mode at 49.3 cm$^{-1}$ corresponds to the $E_{2g}^1$ mode is clearly visible (See Table \ref{['tb:phonon']}). The peak positions in (a), (b) and (c) are marked with circles in (d), triangles (e) and crosses in (f). Solid lines in (a-f) indicate transitions from the ground state to field-split excited CEF levels. \ref{['fig:schem']}(a-b). Gray-shaded areas mark the regions $B_{\rm ext} > 0$ and $E > 0$ for clarity.
• Figure 3: (a) Schematics of two lowest energy levels under magnetic field in $\mathbf B\parallel c$, where two field scales marked as $B_C$ and $B_C^*$ refer the level crossings at the ground (blue) and the 1st excited states (red line) respectively. (b) Measured AC magnetic susceptibility, $\chi^{\rm ac}_c$ (solid lines) are plotted as a function of field in $\mathbf B\parallel c$ at various $T$'s as indicated. Dotted lines display $\frac{dM_c}{dB}$ using the calculated $M_c (B)$ as shown in Fig. \ref{['fig:schem']}(e) Both $\chi^{\rm ac}_c$ data and calculated $\frac{dM_c}{dB}$ are plotted with a constant offset for clarity. Green and pink dashed lines are guides to the eyes for the evolution of the features near $B_C^*$ and $B_C$, respectively (See text). (c) $\chi^{\rm ac}_c (T)$ measured at 0.05 T as a function of $T$ is shown in the red solid line, where the spontaneous ordering is identified as a peak at $110$ mK. The dashed line shows the calculated $T$ dependence of $\frac{dM_c}{dB}$ in ${B\rightarrow0}$, which clearly does not capture the long-range ordering. (d) Calculated $M_c$ (dotted lines) and the numerical integration of the $\chi^{\rm ac}_c(B)$ data (solid line) at $T = 25$ mK and 830 mK are plotted . They are all normalized by the values at $B=12$ T for comparison. Inset: Experimentally measured magnetization $M_c$'s (symbols) are plotted together with the calculated one at 2 (blue), 6 (green), and 10 K (red) in solid lines, which agree well.
• Figure 4: (a) Neutron diffraction on CsErSe$_{2}$ , showing magnetic Bragg peaks with propagation vector $\mathbf k = (\frac{1}{2},0,0)$ appearing at $T = 50$ mK which is about 1/2 of T$_N$ [Fig. \ref{['fig:mag']}(c)]. (b) Magnetic refinements to the temperature-subtracted data reveal Er stripe antiferromagnetism with a weak staggered canting along $c$ with a refined moment 3.27(15)$\mu_B$.
• Figure 5: (a) Simulated spectrum including selection rules [Eq. (\ref{['eq:fermi']})] and excitations from thermally populated states (1st and 2nd excited states under field). (b) Transitions with a probability larger than $\rho_c = 1.7\times10^{-3}$ (within the spectroscopic resolution) are shown in lines. Different line types show where the excitations are from. The highest-lying three doublets (6 states) with $E>160$ cm$^{-1}$ have transition probabilities less than 10$^{-4}$, making it impossible to detect.