Finite-temperature properties of strongly anharmonic and mechanically unstable crystal phases from first principles

We introduce a methodology for constructing anharmonic vibrational Hamiltonians that are parametrized from first-principles electronic-structure calculations and can be used to study high-temperature properties of crystalline materials. Our method provides an accurate description of the Born-Oppenheimer potential energy surface of a crystal that can be systematically refined and is invariant to space-group symmetries of the ideal reference crystal and finite rigid-body rotations and translations. These features make it ideally suited for Monte Carlo or molecular dynamics simulations to predict finite-temperature thermodynamic properties, structural phase transitions, and thermal conductivity. We use this method to construct an anharmonic Hamiltonian for ZrH${}_2$, which exhibits a high-temperature cubic phase that undergoes a symmetry-breaking second-order transition to one of three equivalent tetragonal phases upon cooling. Although density functional theory predicts a zero-Kelvin dynamical instability of cubic ZrH${}_2$, we find via Monte Carlo simulation that the cubic phase can be anharmonically stabilized at high temperature and predict a cubic-to-tetragonal transition temperature that is in good agreement with extrapolation from experiments. We also calculate finite-temperature free energies for the cubic and tetragonal phases, finding that they are consistent with the phenomenological Landau theory of second-order phase transitions.

Figure 1

Total electronic energy of homogeneous deformations of the ideal ZrH2 primitive cell as a function of $c/a$, measured with respect to $c/a=1$. The curve was calculated from density functional theory.

Figure 2

(a) Conventional cubic unit cell of fcc ZrH${}_2$. (b) Eight conventional cells of fcc ZrH${}_2$, with emphasis on tetrahedra formed by Zr nearest neighbors.

Figure 3

Illustration of the symmetrized collective cluster deformations (CCDs) of the NN tetrahedral cluster of Zr atoms in fcc ZrH${}_2$.

Figure 4

(a) The correspondence of high-symmetry directions in the $Q_2$-$Q_3$ plane to different tetragonal variants. (b) The energy surface corresponding to homogeneous deviatoric deformations of the ideal crystal at $Q_1=0$, interpolated from DFT energy calculations.

Figure 5

Average lattice parameters calculated from Monte Carlo simulation, compared to experimental results for ZrH${}_x$ ($x<2$).[17, 18, 19] Experimental data are unavailable for $x\approx 2$ at high temperature, due to the very high partial pressure of H required. Monte Carlo data are shown for a $24\times 24\times 24$ simulation cell, as well as a $40\times 40\times 40$ simulation cell. Figure adapted from Ref. [20].

Figure 6

Ensemble averages calculated for CDAs of the Zr NN tetrahedron, along a stress-free temperature path passing through the cubic-tetragonal transition. (a) Ensemble average of $Q_2$ and $Q_3$ and (b) square root of the ensemble averages of $Q_2^2$ and $Q_3^2$.

Figure 7

Fixed-strain free energy as a function of $c/a$ for a range of temperatures. The free energy becomes fully convex around 1400 K.

Figure 8

Electronic free energies, as a function of $c/a$, calculated for homogeneous deformations of the ZrH${}_2$ primitive. Calculations were performed using electronic Fermi smearing, as implemented in vasp, for several different temperatures over a range that demonstrates the stabilization of cubic ZrH${}_2$ due to electronic entropy.