Tuning the interplay of spin-orbit coupling and trigonal crystal-field effect in the Ising-like spin system ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$

In the ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$ (CCO) system, the large contribution of the orbital moment to the magnetization and the strong magnetocrystalline anisotropy (MCA) are considered to give rise to the Ising magnetism. In this study, the dominant role of both spin-orbit-coupling (SOC) and crystal-field (CF) effects behind this Ising character of magnetism is qualitatively elucidated from the temperature and field dependence of magnetization in the presence of hydrostatic pressures up to 1.04 GPa in a CCO single crystal (SC). The local trigonal prismatic environment is compressed with the application of high pressure, resulting in higher trigonal CF as compared to ambient conditions. It reduces the effect of SOC due to the initiation of orbital quenching that finally decreases orbital moment contributions to both the magnetization and the MCA, respectively. This interplay of triagonal CF and SOC effects is further shown from the detailed quantitative analysis of the field-dependent magnetization in different orientations of CCO and ${\mathrm{Ca}}_3{\mathrm{Co}}_{1.8}{\mathrm{Fe}}_{0.2}{\mathrm{O}}_6$ (CCFO) SCs at ambient pressure by employing a simple classical model and second-order perturbative analysis of SOC. The complete quenching of the orbital moment of ${\mathrm{Fe}}^{3+}$ ($S=5/2$) in CCFO weakens the MCA and also helps in deducing the SOC effect. Furthermore, the estimated anisotropic constants using density functional theory very well capture the Ising magnetism in CCO and deviation from it in CCFO compared to that of classical results.

Figure 1

(a) Variation of field-cooled magnetization as a function of temperature at 5 kOe, whereas the inset shows the zoomed-in view of $M$ (T) around long-range ordering. (b) Zoomed-in view of field-dependent magnetization in the low-field range at 15 K in two different hydrostatic pressures.

Figure 2

Plot of magnetization as a function of field at different hydrostatic pressures at (a) 15 K and (b) 10 K, respectively. The inset of panel (b) illustrates the pressure dependence of the saturation magnetization at 10 K.

Figure 3

Splitting of $d$ orbitals of cobalt ions at the trigonal prism site due to (a) the crystal-field effect and (b) both crystal-field and SOC effects.

Figure 4

[(a) and (b)] The hydrostatic-pressure dependence of long-range ordering temperature and critical field of field-driven 3D ferrimagnetic to 3D ferromagnetic transition. The inset of panel (a) shows the plot of the temperature derivative of magnetization vs temperature in different single-crystal orientations at 5 kOe.

Figure 5

Panels (a) and (b) illustrate the variation of magnetization as function of magnetic field for parallel ($M_{\parallel }$) and perpendicular ($M_{\perp }$) orientation of CCO and CCFO single crystals, respectively. For parallel and perpendicular alignments, the angle between the $c$ axis of the crystals and the applied field are $0^{\circ }$ and ${90}^{\circ }$, respectively, whereas panel (c) and its inset show the plot of the magnetic field vs the ratio of $M_{\perp }/M_{\parallel }$ in the high field range and the schematic diagram of the relative orientation of the magnetization ($M_{\parallel }$), the applied field ($H$), and the $c$ axis of the crystal.

Figure 6

[(a) and (b)] The spin and orbital resolved density of states calculated within $\mathrm{GGA}+U$ for Co and Fe atoms at the trigonal prismatic site in CCO and CCFO, respectively. The inset of panel (b) depicts the DOS for dopant Fe. Panel (c) shows the variation in the MCA as a function of the polar angle $\theta$ for CCO (blue solid circles) and CCFO (red solid circles) and fitted with a function $K_1{sin}^2\theta +K_2{sin}^4\theta$ (the black solid line). The calculations were performed by rotating the spin quantization axis within $\mathrm{GGA}+U+\mathrm{SOC}$ for $U_{\mathrm{eff}}=5$ eV.