Crystal-field Paschen-Back effect on ruby in ultrahigh magnetic fields

Zeeman spectra of the R lines of ruby (${\mathrm{Cr}}^{3+}$: $\alpha \text{−}{\mathrm{Al}}_2{\mathrm{O}}_3$) were studied in ultrahigh magnetic fields up to 230 T by magnetophotoluminescence measurements. The observed Zeeman patterns exhibit nonlinear behaviors above 100 T, evidencing the breakdown of the previously reported Paschen-Back effect for $\mathbf{B}\perp c$ geometry. We adopted the crystal-field multiplet theory including the cubic crystal field (${\mathcal{H}}_{\mathrm{cubic}}$), the trigonal crystal field (${\mathcal{H}}_{\mathrm{trig}}$), the spin-orbit interaction (${\mathcal{H}}_{\mathrm{SO}}$), and the Zeeman interaction (${\mathcal{H}}_{\mathrm{Z}}$). It is found that the nonlinear splitting of the R lines is owing to the hybridization between the ${}^{2}E$ and ${}^{2}T_1$ states, which leads to the quantization of these Zeeman levels with the orbital angular momentum. Our results suggest that the exquisite energy balance among ${\mathcal{H}}_{\mathrm{cubic}}$, ${\mathcal{H}}_{\mathrm{trig}}$, ${\mathcal{H}}_{\mathrm{SO}}$, and ${\mathcal{H}}_{\mathrm{Z}}$ realized in ruby offers a unique opportunity to observe the onset of the crystal-field Paschen-Back effect toward the high-field extreme.

Figure 1

(a) Absorption spectrum of ruby in visible region and the local structure of $\mathrm{Cr}{\mathrm{O}}_6^{9-}$ with the $C_3$ symmetry. The corresponding excited state from the ground state ${}^{4}A_2$ is denoted for each absorption peak. (b), (c) Energy diagrams of the Zeeman splitting of the R lines for (b) $\mathbf{B}\parallel c$ and (c) $\mathbf{B}\perp c$ geometries. A small initial splitting of 0.05 meV in the ${}^{4}A_2$ state is neglected. Optically allowed transitions for $\mathbf{E}\perp c$ are denoted by A, B,..., P. In the PB region for $\mathbf{B}\perp c$, the excited levels asymptotically approach the dashed straight lines.

Figure 2

(a) Block diagram of the experimental setup for the magneto-PL measurements using the STC system. The dashed lines show the trigger signals. (b) Schematic view of the ${}^4\mathrm{He}$-flow-type cryostat and the photo of the ruby single crystal used in this work.

Figure 3

(a) PL spectra of the R lines up to 230 T for $\mathbf{B}\parallel c$ ($\mathbf{B}\perp c$) in the upper (lower) panel. The spectra were recorded around 200 K except for the 230 T data. (b), (c) Zeeman patterns of the R lines for (b) $\mathbf{B}\parallel c$ and (c) $\mathbf{B}\perp c$ geometries. Filled circles show the experimental peak positions. Red and blue lines show the theoretical curves obtained from the crystal-field multiplet theory (see text for details). Inset in (b) presents the average of the peak shifts with the magnetic field, reflecting the change in the mean energy of the four Zeeman levels splitting from the ${}^{2}E$ states. Error bars in (c) are taken from the full width at half maximum (FWHM) of the fitted Lorentzian functions.

Figure 4

[(a) and (b)] Calculated energy diagrams of the four Zeeman levels splitting from the ${}^{2}E$ states with the magnetic field up to 250 T for (a) $\mathbf{B}\parallel c$ and (b) $\mathbf{B}\perp c$ geometries. [(c) and (d)] The corresponding field-derivatives of the energy for (c) $\mathbf{B}\parallel c$ and (d) $\mathbf{B}\perp c$ geometries.

Figure 5

Square of the coefficient $|c_k{|}^2$ for each of the $\left|\left(t_2^3{}^{2}E\right)\pm \frac{1}{2}{u}_{\pm }\right.〉$ and $\left|\left(t_2^3{}^{2}T_1\right)\pm \frac{1}{2}{a}_{\pm ,0}\right.〉$ components in the wave function of four Zeeman levels related to the R lines with the magnetic field up to 250 T. Data for $\mathbf{B}\parallel c$ and $\mathbf{B}\perp c$ are shown in (a)–(d) and (e)–(l), respectively. The figures are arranged in the same raw for each of the four levels in order of $|{\phi }_{2+}\rangle$, $|{\phi }_{2-}\rangle$, $|{\phi }_{1+}\rangle$, and $|{\phi }_{1-}\rangle$ from the top. Arrows $\uparrow$ ($\downarrow$) represent the spin $M_S=+\frac{1}{2}$ ($-\frac{1}{2}$) of the bases. For clarity, some plots show the summation of two components that mainly contribute to the same Kramers doublet [40].

Figure 6

[(a)–(d)] Magnetic-field dependences of the relative peak intensities obtained from the multi-Lorentzian fits for the experimental results [(a) and (b)] and the calculation based on Eq. (A1) [(c) and (d)]. In the calculation, the temperature is fixed to $T=200$ K to derive the Boltzmann distribution within the ${}^{2}E$ state. (e) Plots of the calculated intensity ratios for all of the individual peaks for $\mathbf{B}\perp c$ geometry, which are decomposed from the plots in (d).

Figure 7

[(a) and (b)] Calculated energy diagrams with the magnetic field up to 5000 T for (a) $\mathbf{B}\parallel c$ and (b) $\mathbf{B}\perp c$ geometries. [(c) and (d)] Enlarged view of the Zeeman patterns focusing on the ${}^{2}E$ and ${}^{2}T_1$ states. They correspond to the region surrounded by yellow squares in (a) and (b). Expected asymptotic lines, $E={\mu }_{\mathrm{B}}(k{M}_L+2M_S)B$, are displayed in red.