Finite temperature magnetic interactions from first principles

Abstract

Density functional theory has demonstrated remarkable predictive power in calculating magnetic properties at zero temperature. At finite temperatures, thermally excited phonons may affect magnetism. Efficient ab-initio methods to calculate the temperature dependence of magnetic exchange interactions are still lacking despite the importance of room temperature magnetism for applications. Exchange is controlled by an interplay between metal-ligand hybridization, Hubbard repulsion, band gap, interatomic distances and bond angles, all of which change with temperature. Here we present a method to calculate the exchange interactions at finite temperatures from first principles using only two supercell calculations and quantify these mechanisms. Changes in bond angles and the band gap are identified as a primary factors. In NiO with 180-degree bonds exchange decreases with temperature, while in Cr$_2$O$_3$ with the bond angles away from 180 degrees the exchange increases by 10% at room temperature.

Figures (3)

• Figure 1: Principal mechanisms driving temperature dependence of magnetic exchange constants: (a) thermal linear expansion elongates the bonds, resulting in the reduction of M-O-M hopping integrals $t_{MO}=t_0-\beta T$ and the exchange constants $J\propto(t_{MO}^2/\Delta)^2/U$; (b) reduction of octahedral tilts with temperature results in enhanced $t_{MO}$ and stronger antiferromagnetic $J$; (c) thermal liberations of magnetic O$_2$ molecules in CsO$_2$ enhance $t$ and $J$ with increasing temperature. (d) Thermal phonons lead to stronger hybridization of Ni with ligands and expansion of the Wannier function (in green) drives the decrease of Hubbard $U$ and increase of $J$ with temperature.
• Figure 2: (a) Crystal structure of Cr$_2$O$_3$ with Cr and oxygen atoms in blue and red, respectively. $J_1$ and $J_2$ are the two nearest-neighbor Heisenberg exchange constants. (b) Crystal structure of NiO. Ni ions are in yellow, while oxygens are in red. Heisenberg exchange $J$ is on a 180$^\circ$ Ni-O-Ni bond. (c) Brillouin zone and a path along which the phonon dispersion is plotted. (d) Calculated phonon dispersion in NiO along the reciprocal space path shown in panel (c). (e) Phonon density of states in NiO.
• Figure 3: (a) Increase of total energy with increasing temperature. Red dots and blue triangles show the total energy of +ZG and -ZG configurations. The solid black line is the function $\frac{3}{2}Nk_{\mathrm{B}}T$. (b) Temperature dependent change of the band gap $\Delta$ with temperature. (c) Temperature dependence of the deviation of the average Ni-O-Ni bond angle in NiO from 180$^\circ$. The inset shows the Feynman diagram corresponding to the Katsnelson-Lichtenstein exchange constant $J$ with thermal phonon correction. (d) Temperature dependent change of magnetic exchange for NiO and Cr$_2$O$_3$. At $T=0$, $J_1=12$, meV, $J_2=10$ meV, $J=20$ meV.