Simulation of thermodynamic properties of magnetic transition metals from an efficient tight-binding model: The case of cobalt and beyond

Atomic scale simulations at finite temperature are an ideal approach to study the thermodynamic properties of magnetic transition metals. However, the development of interatomic potentials explicitly taking into account magnetic variables is a delicate task. In this context, we present a tight-binding model for magnetic transition metals in the Stoner approximation. This potential is integrated into a Monte Carlo structural relaxations code where trials of atomic displacements as well as fluctuations of local magnetic moments are performed to determine the thermodynamic equilibrium state of the considered systems. As an example, the Curie temperature of cobalt is investigated while showing the important role of atomic relaxations. Furthermore, our model is generalized to other transition metals highlighting a local magnetic moment distribution that varies with the gradual filling of the $d$ states. Consequently, the successful validation of the potential for different magnetic configurations indicates its great transferability and makes it a good choice for atomistic simulations sampling a large configuration space.

Figure 1

Total energy (in red) and magnetic moment (in black) as a function of the number of MC steps at 800 K starting from a fcc random spin configuration.

Figure 2

Total energy (a) NM and (b) FM calculations and (c) total magnetic moment as a function of atomic volume for hcp (green), fcc (in black), and bcc (in red) at 0 K. Dashed lines represent DFT calculations and full lines TB calculations.

Figure 3

Total magnetic moment average of fcc (in black) and bcc (in red) systems as a function of temperature on Ising-type lattice.

Figure 4

Total magnetic moment average of (a) fcc and (b) bcc system as a function of temperature with different approximations: only longitudinal spin fluctuations (dashed line with cross), longitudinal spin fluctuations $+$ lattice vibrations (dotted line with square), and longitudinal spin fluctuations $+$ lattice vibrations $+$ lattice expansion (full line with circle).

Figure 5

Average linear thermal expansion coefficient for fcc Co as a function of temperature (FM and NM states). The blue vertical line indicates the $T_C$.

Figure 6

Local magnetic moment distribution of 500 configurations of the fcc system below $T_C$ at 1300 K (top) and above $T_C$ at 1400 K (bottom) calculated by TB-FMA. One magnetic configuration of each temperature is represented with a color code scale from $-3{\mu }_B$ to $3{\mu }_B$.

Figure 7

Total energy as a function of the total magnetic moment for the fcc phase calculated by TB-FMA. The lattice parameter is kept constant and equal to its value at 0 K.

Figure 8

Local magnetic moment distribution of 500 configurations of bcc system at 100 K (top), 500 K (middle), and 2000 K (bottom) from TB-FMA model (left column) and from EIM model (middle column) with the corresponding magnetic configuration (right column). In this EIM model, the magnetic moment is decomposed in three directions: $x$ in green, $y$ in red, and $z$ in black.

Figure 9

Discontinuity of the thermal expansion coefficient at the Curie temperature as a function of the number of $d$ electrons. $\mathrm{\Delta }a$ is defined as $a_{\mathrm{PM}}-a_{\mathrm{FM}}$, respectively, above and before the Curie point.

Figure 10

Local magnetic moment distribution of 500 configurations of the fcc system below and above $T_C$ for different $N_{\mathrm{d}}$.

Figure 11

Energy with respect to the ground state as a function of normalized magnetic moment for different $N_{\mathrm{d}}$ at 0 K.