Temperature-dependent renormalization of magnetic interactions by thermal, magnetic, and lattice disorder from first principles

We put forth an ab initio framework to calculate local moment magnetic interaction parameters, renormalized to treat both the lattice and magnetic systems as a function of temperature $T$. For bcc Fe, magnetic and lattice thermal disorders act in opposition, the former strengthening the Heisenberg-like interactions, while the latter decreasing them. Below $T_C$, $J$ stays nearly independent of $T$, while around and above $T_C$, it exhibits a sharp decrease. This remarkable behavior reflects an intricate spin-lattice coupling and its evolution with $T$, in which magnetic interactions and interatomic bonds are each renormalized by the other. This finding is consistent with magnetization data and with the observed softening of magnon and phonon modes at high temperatures. Magnetization as well as magnon and phonon mode softening are discussed.

Figure 1

Magnetic exchange parameter $J_1$ for Fe as a function of temperature for various degrees of inclusion of lattice and magnetic thermal disorder. Points are calculated values while lines are guides to the eye. The solid red curve includes realistic, self-consistent coupled thermal disorder in both the lattice and the magnetic systems. The black dashed curve includes only magnetic thermal disorder and no lattice thermal disorder. The purple dotted curve includes lattice thermal disorder but no magnetic thermal disorder. Curves are lightened between 1174 and 1743 K to signify that Fe is in a fcc phase between 1185 and 1667 K [44].

Figure 2

Normalized magnetization per atom curves, $M\left(T\right)/M\left(0\right)$, for Fe calculated via Monte Carlo simulations. The red curve uses the full $J_1\left(T\right)$ (red curve from Fig. 1). The black curve uses the $J_1\left(T\right)$ which completely excludes lattice thermal disorder (black curve in Fig. 1). $M\left(0\right)$ is the magnetization per atom at $T=0$ K. Curves are for 6750-atom simulations. Points are calculated values, and lines are guides to the eye.

Figure 3

Magnetic exchange parameter $J_1$ as a function of temperature with direct effects of magnetic thermal disorder artificially excluded (blue dotted) but indirect effects still present. Also the complementary case of direct effects included but indirect effects excluded (green dashed). The red curve is the full calculation (red curve from Fig. 1).