Crystalline electric field of Ce in trigonal symmetry: ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ as a model case

The crystalline electric field (CEF) of ${\mathrm{Ce}}^{3+}$ in trigonal symmetry has recently become of some relevance, for instance, in the search of frustrated magnetic systems. Fortunately, it is one of the CEF cases in which a manageable analytic solution can be obtained. Here, we present this solution for the general case, and use this result to determine the CEF scheme of the new compound ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ with the help of $T$-dependent susceptibility and isothermal magnetization measurements. The resulting CEF parameters $B_2^0=34.4\mathrm{K}, B_4^0=0.82\mathrm{K}$, and $B_4^3=67.3\mathrm{K}$ correspond to an exceptionally large CEF splitting of the first and second excited levels, 374 K and 1398 K, and a large mixing between the $\left|\pm \frac{5}{2}\right.〉$ and the $\left|\mp \frac{1}{2}\right.〉$ states. This indicates a very strong easy plane anisotropy with an unusually small $c$-axis moment. Using the same general expressions, we show that the properties of the recently reported system ${\mathrm{CeCd}}_3{\mathrm{As}}_3$ can also be described by a similar CEF scheme, providing a much simpler explanation for its magnetic properties than the initial proposal. Moreover, a similar strong easy plane anisotropy has also been reported for the two compounds CeAuSn and ${\mathrm{CePdAl}}_4{\mathrm{Ge}}_2$, indicating that the CEF scheme elaborated here for ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ corresponds to an exemplary case for ${\mathrm{Ce}}^{3+}$ in trigonal symmetry.

Figure 1

Left: Optical microscope image of a ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ single crystal grown using the self-flux method [23]. Right: The Ce-atom (gray) coordination in ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$, which emphasizes its trigonal point symmetry. There are six Ir neighbors (distance 3.21 Å) and twelve Ge neighbors ($2\times 6$, distances 3.25 Å and 3.26 Å).

Figure 2

Left: Inverse magnetic susceptibility of ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ measured at $B=1$ T applied along the [100] and [001] crystallographic axes. The gray lines are linear fits to the Curie-Weiss law at temperatures between 400 and 500 K. Right: Field dependence of the magnetization measured at 1.8 K along the [100] and [001] axes.

Figure 3

Proposed CEF scheme for ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ with ${\mathrm{\Delta }}_1=374\mathrm{K}$ and ${\mathrm{\Delta }}_2=1398\mathrm{K}$.

Figure 4

Zoom of the inverse magnetic susceptibility of ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ at low (left) and high (right) temperatures. The gray lines are linear fits to the Curie-Weiss law which evidence a kink at about 10 K (left) possibly due to a small additional paramagnetic contribution and an upturn above 500 K due to a diamagnetic contribution from the sample holder.

Figure 5

Left: Inverse magnetic susceptibility of ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$ measured at $B=1$ T after subtraction of the diamagnetic ${\chi }_0$ contribution. The gray lines are linear fits to the Curie-Weiss law between 400 and 600 K which yield an effective moment of 2.54 ${\mu }_B$, as expected for ${\mathrm{Ce}}^{3+}$. The CW temperatures are ${\theta }_W^a=5\mathrm{K}$ and ${\theta }_W^c=-362\mathrm{K}$ which give [from Eq. (1)] $B_2^0=38.1\mathrm{K}$. Right: Field dependence of the magnetization measured at 1.8 K. In both plots, the green lines are the results of the CEF calculations after having fitted ${(\chi -{\chi }_0)}^{-1}$ in the ranges 1.8–6 K and 200–600 K for $B\parallel \left[001\right]$ and in the whole measured range for $B\parallel \left[100\right]$.

Figure 6

Left: Inverse magnetic susceptibility of ${\mathrm{CeCd}}_3{\mathrm{As}}_3$. Right: Field dependence of the magnetization measured at 1.9 K. The data were taken from Ref. [20]. The green lines are our CEF calculations with parameters given in the text and in Table 1.

Figure 7

Calculated total and partial density of states (DOS) of ${\mathrm{CeIr}}_3{\mathrm{Ge}}_7$. Both Ir (red) and Ge(1) (blue) contribute almost equally to the DOS at the Fermi level. The inset shows the comparison to the fictitious isostructural Rh system (orange).