Vibrational free energy and phase stability of paramagnetic and antiferromagnetic CrN from ab initio molecular dynamics

We present a theoretical first-principles method to calculate the free energy of a magnetic system in its high-temperature paramagnetic phase, including vibrational, electronic, and magnetic contributions. The method for calculating free energies is based on ab initio molecular dynamics and combines a treatment of disordered magnetism using disordered local moments molecular dynamics with the temperature-dependent effective potential method to obtain the vibrational contribution to the free energy. We illustrate the applicability of the method by obtaining the anharmonic free energy for the paramagnetic cubic and the antiferromagnetic orthorhombic phases of chromium nitride. The influence of lattice dynamics on the transition between the two phases is demonstrated by constructing the temperature-pressure phase diagram.

Figure 1

Phonon-dispersion relations for the paramagnetic cubic phase of CrN calculated using a supercell containing 108 Cr atoms and 108 nitrogen atoms at 1000 K.

Figure 2

Phonon-dispersion relations for the antiferromagnetic phase of CrN calculated using a supercell containing 144 Cr atoms and 144 nitrogen atoms at 1000 K.

Figure 3

Vibrational free energy as a function of temperature for cubic and orthorhombic phases.

Figure 4

Difference in Gibbs free energy between the cubic and orthorhombic phases of CrN as a function of pressure calculated at different temperatures. The dashed lines represent the Gibbs free energy differences, including only a magnetic mean-field entropy together with the potential energies from the DLM-MD calculations. The bold lines describe results considering the effect of magnetic disorder and the full consideration of vibrations including the entropy term.

Figure 5

Pressure-temperature phase diagram of CrN. The phase diagram calculated by means of the DLM-MD-TDEP method with Helmholtz free energy $F$ defined in Eq. (5) is shown with a red solid line. Experimental phase diagrams are from Refs. [1] (open circles) and [11] (circles). The green dashed line corresponds to the phase diagram obtained by static calculations in Ref. [4] using Eq. (10). The red dotted line shows the phase diagram calculated from Eq. (6), but with the magnetic entropy term taken from magnetic moments of static calculations [4]. The blue dashed-dotted line shows the diagram calculated with $F$ defined in Eq. (11). The orange double dashed-dotted line gives the diagram calculated with $F$ defined in Eq. (12). See text for the discussion.

Figure 6

Average magnitudes of local magnetic moments on Cr sites as a function of lattice parameter calculated by means of DLM-MD simulations at different temperatures $T$: 0 K (black dashed-dotted line), 300 K (blue triangles), 1000 K (green open circles), and 2000 K (red circles).

Figure 7

Phonon densities of states for cubic and orthorhombic phases at constant pressure.

Figure 8

Convergence of one the components of the force constant for the paramagnetic phase with simulation time at different temperatures.

Figure 9

Free energy of lattice vibrations $F_{\mathrm{vib}}$ for the paramagnetic phase as a function of MD simulation time calculated at different temperatures at equilibrium volume.