Finite-temperature elastic constants of paramagnetic materials within the disordered local moment picture from ab initio molecular dynamics calculations

We present a theoretical scheme to calculate the elastic constants of magnetic materials in the high-temperature paramagnetic state. Our approach is based on a combination of disordered local moments picture and ab initio molecular dynamics (DLM-MD). Moreover, we investigate a possibility to enhance the efficiency of the simulations of elastic properties using the recently introduced method: symmetry imposed force constant temperature-dependent effective potential (SIFC-TDEP). We have chosen cubic paramagnetic CrN as a model system. This is done due to its technological importance and its demonstrated strong coupling between magnetic and lattice degrees of freedom. We have studied the temperature-dependent single-crystal and polycrystalline elastic constants of paramagentic CrN up to 1200 K. The obtained results at T = 300 K agree well with the experimental values of polycrystalline elastic constants as well as the Poisson ratio at room temperature. We observe that the Young's modulus is strongly dependent on temperature, decreasing by $\sim 14\%$ from T = 300 K to 1200 K. In addition we have studied the elastic anisotropy of CrN as a function of temperature and we observe that CrN becomes substantially more isotropic as the temperature increases. We demonstrate that the use of Birch law may lead to substantial errors for calculations of temperature induced changes of elastic moduli. The proposed methodology can be used for accurate predictions of mechanical properties of magnetic materials at temperatures above their magnetic order-disorder phase transition.

Figure 1

Schematic representation of the DLM-MD process. Starting with a random arrangement of magnetic moments, an MD calculation is run for $N_{\mathrm{MD}}^{\mathrm{SF}}=t_{\mathrm{SF}}/t_{\mathrm{MD}}$ steps during which the magnetic configuration is allowed to evolve according to the solution of the electronic structure problem. At the $t_{\mathrm{SF}}$, the moments are flipped and rearranged in another random magnetic configuration, while positions and velocities of all the atoms are preserved. Then the MD simulation continues. The figure shows only one layer of Cr atoms from the CrN supercell and the N atoms are not shown here. The spin-flip time is chosen to be $t_{\mathrm{SF}}=5$ fs. The Born-Oppenheimer MD time step is chosen as $t_{\mathrm{MD}}=1$ fs. The phonon time scale is also shown as $t_{\mathrm{ph}}>100$ fs. The lengths of the time arrows demonstrate that the time scale of the collective atomic vibrational modes are much slower than the time scales of both the electronic and transverse local magnetic moments ($t_{\mathrm{MD}}\le t_{\mathrm{SF}}<t_{\mathrm{ph}}$).

Figure 2

Calculated temperature-dependent single-crystal elastic constants of PM CrN obtained by using different theoretical methods. The error bars correspond to the standard deviation of the molecular dynamics simulations with a 95% confidence interval.

Figure 3

Calculated Voigt-Reuss-Hill averages [Eqs. (13, 14, 15)] of polycrystalline elastic constants of PM CrN from top to bottom (a) bulk modulus, (b) Young's modulus, (c) shear modulus, and (d) Poisson ratio, as a function of temperature. The error bars correspond to the standard deviation of the molecular dynamics simulations with a 95% confidence interval.

Figure 4

Calculated Voigt-Reuss-Hill anisotropy $A_{\mathrm{VR}}$, and Zenner elastic shear $A_Z$ of PM CrN as a function of temperature. The error bars correspond to the standard deviation of the molecular dynamics simulations with a 95% confidence interval.