Parametrization of $\mathrm{LSDA}+U$ for noncollinear magnetic configurations: Multipolar magnetism in ${\mathrm{UO}}_2$

To explore the formation of noncollinear magnetic configurations in materials with strongly correlated electrons, we derive a noncollinear $\mathrm{LSDA}+U$ model involving only one parameter $U$, as opposed to the difference between the Hubbard and Stoner parameters $U-J$. Computing $U$ in the constrained random phase approximation, we investigate noncollinear magnetism of uranium dioxide ${\mathrm{UO}}_2$ and find that the spin-orbit coupling (SOC) stabilizes the $3\mathbf{k}$ ordered magnetic ground state. The estimated SOC strength in ${\mathrm{UO}}_2$ is as large as 0.73 eV per uranium atom, making spin and orbital degrees of freedom virtually inseparable. Using a multipolar pseudospin Hamiltonian, we show how octupolar and dipole-dipole exchange coupling help establish the $3\mathbf{k}$ magnetic ground state with canted ordering of uranium $f$ orbitals. The cooperative Jahn-Teller effect does not appear to play a significant part in stabilizing the noncollinear $3\mathbf{k}$ state, which has the lowest energy even in an undistorted lattice. The choice of parameter $U$ in the $\mathrm{LSDA}+U$ model has a notable quantitative effect on the predicted properties of ${\mathrm{UO}}_2$, in particular on the magnetic exchange interaction and, perhaps trivially, on the band gap: The value of $U=3.46\mathrm{eV}$ computed fully ab initio delivers the band gap of 2.11 eV in good agreement with experiment, and a balanced account of other pertinent energy scales.

Figure 1

Schematic plots of the $\langle 001\rangle$-AFM, $3\mathbf{k}$ (longitudinal) and $\langle 110\rangle$-AFM noncollinear spin configurations considered in this study.

Figure 2

Schematic view of (a) spin and (b) orbital moments in a $3\mathbf{k}$ (longitudinal) magnetic unit cell of ${\mathrm{UO}}_2$. Panel (c) shows the total energy as a function of the canting angle $\theta$ along the $\langle 001\rangle$-AFM–$3\mathbf{k}$–$\langle 110\rangle$-AFM transformation pathway. Blue and green arrows show the spin and orbital moments, respectively. The inset explains the definition of the spin canting angle $\theta$ adopted in our analysis.

Figure 3

Spin quantization axes $x$, $y$, and $z$ and the geometry of the exchange-coupled pair for neighboring ${\mathrm{U}}^{4+}$ exchange ions. There are six inequivalent exchange-coupled pairs AB per a magnetic unit cell.

Figure 4

Variation of the spin-orbit coupling constant $\lambda$ as a function of nuclear charge $Z$ compared with literature data for oxygen [96], iron [97], rhodium [98], iridium [95, 99], and uranium (this work).

Figure 5

Total energy as a function of the canting angle $\theta$ computed for several different values of parameter $U$.

Figure 6

Band structures of ${\mathrm{UO}}_2$ computed for the $\langle 001\rangle$-AFM, $3\mathbf{k}$ and $\langle 110\rangle$-AFM ordered configurations assuming $U=3.46\mathrm{eV}$.

Figure 7

Charge densities of ${\mathrm{UO}}_2$ in a (110) plane computed for the $\langle 001\rangle$-AFM, $3\mathbf{k}$, and $\langle 110\rangle$-AFM spin-ordered configurations. All the three plots were computed assuming that $U=3.46\mathrm{eV}$.

Figure 8

Comparison between the calculated ($\mathrm{DFT}+U^{\mathrm{cRPA}}+\mathrm{SOC}$) and fitted (outfit using Mironov's parameters) magnetic canting energies $\mathrm{\Delta }E\left(\theta \right)$. In our fit we employed an extension of the Mironov model to all the U-U interactions and optimized the fit with respect to the dominant quadrupolar terms $q_1$ and $q_2$, by keeping all the other terms fixed to the corresponding values obtained by Mironov [23] (see Table 5).

Figure 9

Decomposition of the magnetic energy $\mathrm{\Delta }E\left(\theta \right)$ into four components $H_1$ [DD interaction, $D$ & $E$, Eq. (17)], $H_2$ [bilinear exchange, $J$'s, Eq. (17)], $H_3$ [four-spin exchange, $j$'s, Eq. (19)], and $H_4$ [quadrupolar, $q$'s, Eq. ( (20)]. The right panel shows the difference between the quadrupolar term derived directly from Mironov's data and our fitted curve.