First-principles theory of multipolar order in neptunium dioxide

We provide a first-principles, materials-specific theory of multipolar order and superexchange in ${\text{NpO}}_2$ by means of a noncollinear local-density approximation $+U$ $\left(\text{LDA}+U\right)$ method. Our calculations offer a precise microscopic description of the triple-$\mathit{q}$ antiferro ordered phase in the absence of any dipolar moment. We find that, while the most common nondipolar degrees of freedom (e.g., electric quadrupoles and magnetic octupoles) are active in the ordered phase, both the usually neglected higher-order multipoles (electric hexadecapoles and magnetic triakontadipoles) have at least an equally significant effect.

Figure 1

(Color) Charge density cross sections of ${\text{NpO}}_2$ (computed with $U=4\text{eV}$ and $J=0\text{eV}$) for (a) and (b): nonmultipolar-ordered state with cubic fcc symmetry in the [100], respectively, [110] plane, and (c) and (d): magnetic multipolar state revealing the (111) $3\mathit{q}$-antiferroelectric long-range order in the [100], respectively, [110] plane.

Figure 2

(Color) (a) Orbital-projected DOS of a multipolar state computed with $U=4\text{eV}$, $J=0\text{eV}$. Plotted is the $\text{Np}5f$, $j=5/2$, DOS projected on components of the irreducible $D_{3d}$ basis, and the $\text{Np}j=7/2$ and $\text{O}p$ character DOS. (b) Orbital-projected DOS of the multipolar phase obtained with $U=4\text{eV}$ and $J=0.2\text{eV}$. (c) Calculated $\text{LDA}+U$ one-electron orbital occupation numbers and off-diagonal $\left({\Gamma }_6^{\left(1\right)},{\Gamma }_6^{\left(2\right)}\right)$ matrix elements as a function of exchange $J$ at $U=4\text{eV}$. (d) Calculated expectation values of the multipolar (quadrupolar, octupolar, hexadecapolar, and triakontadipolar) order parameters in ${\text{NpO}}_2$ as a function of $J$.

Figure 3

(Color) Computed charge and spin distributions of the $\text{Np}5f$ electrons in ${\text{NpO}}_2$. (a) Nonmultipolar-ordered state, computed with $U=4$ and $J=0\text{eV}$. (b) Magnetic multipolar-ordered state, for $U=4$ and $J=0\text{eV}$. (c) Magnetic multipole state, for $U=4$ and $J=0.04\text{eV}$. (b) Magnetic multipole state, for $U=4$ and $J=0.2\text{eV}$. The charge distribution is depicted by the three-dimensional hypersurface and the direction of the current flow is shown by the thin lines. The colors on the three-dimensional charge distribution depict the local spin magnetization projected on the $z$-quantization axis. Red and blue colors label, respectively, a net up or down spin. Green color labels a vanishing net spin magnetization.