Spin and orbital structure of uranium compounds on the basis of a $j\text{‐}j$ coupling scheme

Key role of orbital degree of freedom to understand the magnetic structure of uranium compounds is discussed from a microscopic viewpoint by focusing on typical examples such as $\mathrm{U}\mathrm{M}{\mathrm{Ga}}_5$ ($\mathrm{M}=\mathrm{Ni}$ and Pt) and the mother compound $\mathrm{U}{\mathrm{Ga}}_3$. By analyzing an orbital degenerate Hubbard model constructed from the $j\text{‐}j$ coupling scheme in the cubic system, we obtain the phase diagram including several kinds of magnetic states. Especially, in the parameter region corresponding to actual uranium compounds, the phase diagram suggests successive transitions among paramagnetic, magnetic metallic, and insulating Néel states, consistent with the experimental results for $\mathrm{Au}{\mathrm{Cu}}_3$-type uranium compounds. Furthermore, taking account of tetragonal effects such as level splitting and reduction of hopping amplitude along the $z$ axis, an orbital-based scenario is proposed to understand the change in the magnetic structure from G- to A-type antiferromagnetic phases, experimentally observed in $\mathrm{U}\mathrm{Ni}{\mathrm{Ga}}_5$ and $\mathrm{U}\mathrm{Pt}{\mathrm{Ga}}_5$.

Figure 1

Configurations of $f$-electrons in the $j\text{‐}j$ coupling scheme for (a) $\mathrm{Ce}{\mathrm{In}}_3$ and (b) $\mathrm{U}{\mathrm{Ga}}_3$. Note that up and down arrows denote pseudospin states in order to distinguish the two states of the Kramers doublet.

Figure 2

Fermi-surface sheets of the ${\Gamma }_8$ tight-binding model for $⟨n⟩=1$. Note that we obtain two Fermi-surface sheets, (a) and (b), due to the multiorbital nature of the model. Note also that both sheets are depicted in the electron picture. The bounding box indicates the first Brillouin zone for a simple cubic lattice. The $\Gamma$ point is located at the center of the box, while the apices denote $R$ points.

Figure 3

(a) Spin correlation $S\left(\mathbf{q}\right)$ as a function of $J$ for $U^{\prime }=6$. (b) Spin structures in the FM, A-type AF, C-type AF, and G-type AF phases.

Figure 4

(a) Orbital correlation as a function of $J$ for $U^{\prime }=6$. Meanings of the symbols are the same as those in Fig. 3a. Orbital arrangements (b) for $J\lesssim 0.7$ and (c) for $J\gtrsim 0.7$.

Figure 5

(a) Spin correlation as a function of $U^{\prime }$ for $J=0$. (b) Orbital correlation as a function of $U^{\prime }$ for $J=0$. (c) Phase diagram for $\mathrm{U}{\mathrm{Ga}}_3$ obtained by the exact diagonalization. The region of $J>U^{\prime }$ is ignored, since it is unphysical. See Fig. 3b for the definitions of abbreviations. Here “PM-G” indicates the PM phase with enhanced $(\pi ,\pi ,\pi )$ spin correlation.

Figure 6

Spin (a) and orbital (b) correlation functions vs $\Delta$ for $t_z=0.8$, with $J=0$ and $U^{\prime }=3.5$. (c) Phase diagram of the magnetic structure in the $(\Delta ,t_z)$ plane for $J=0$ and $U^{\prime }=3.5$. (d) Ferro-orbital pattern in the A-type AF phase.

Figure 7

Electron configurations in the $j\text{‐}j$ coupling scheme for $\mathrm{Au}{\mathrm{Cu}}_3$-type compounds, (a) $\mathrm{Ce}{\mathrm{In}}_3$, (b) $\mathrm{Pr}{\mathrm{In}}_3$, and (c) $\mathrm{Nd}{\mathrm{In}}_3$.

Figure 8

Electron configurations in the $j\text{‐}j$ coupling scheme for rare-earth hexaborides, (a) $\mathrm{Ce}{\mathrm{B}}_6$, (b) $\mathrm{Pr}{\mathrm{B}}_6$, and (c) $\mathrm{Nd}{\mathrm{B}}_6$.