First-principles prediction of phase transition of ${\mathrm{YCo}}_5$ from self-consistent phonon calculations

A recent theoretical study has shown that hexagonal ${\mathrm{YCo}}_5$ is dynamically unstable and distorts into a stable orthorhombic structure. In this paper, we show theoretically that the orthorhombic phase is energetically more stable than the hexagonal phase in the low-temperature region, while the phonon entropy stabilizes the hexagonal phase thermodynamically in the high-temperature region. The orthorhombic-to-hexagonal phase transition temperature is $\sim 165$ K, which is determined using self-consistent phonon calculations. We investigate the magnetocrystalline anisotropy energy (MAE) using self-consistent and non-self-consistent (force theorem) calculations with spin-orbit interaction (SOI) along with the Hubbard $U$ correction. Then, we find that the orthorhombic phase has MAE, orbital moment, and orbital moment anisotropy values that are similar to those of the hexagonal phase when the self-consistent calculation with SOI is performed. Since the orthorhombic phase still gives magnetic properties comparable to those found in experiments, the orthorhombic distortion is potentially realized in the low-temperature region, which awaits experimental exploration.

Figure 1

Crystal structure of the hexagonal (a) and orthorhombic (b) phases of ${\mathrm{YCo}}_5$. The Y and Co atoms are represented by the large and small spheres, respectively. The atomic Wyckoff position for different atoms is marked with the corresponding colors. (c) Top view of the crystal structures of hexagonal (top panel) and orthorhombic (lower panel) ${\mathrm{YCo}}_5$. (d) Schematic diagram of the double-well potential obtained by displacing atoms along the direction of the $L$-point soft phonon mode of the hexagonal phase. $Q_L$ is the normal coordinate amplitude, and $\mathrm{\Delta }E$ represents the static energy difference between the two phases.

Figure 2

Temperature-dependent anharmonic phonon dispersion of hexagonal (a) and orthorhombic (b) phases of ${\mathrm{YCo}}_5$. The color map shows the self-consistent phonon solutions in the temperature range 100–300 K, and the dashed curves are harmonic lattice dynamics results. Imaginary frequency is shown as negative. (c) Calculated difference of the free energies, $\mathrm{\Delta }F\left(T\right)$ (see text). The horizontal dashed line is the difference of the static energies $-\mathrm{\Delta }E=23.24$ meV/f.u. Here, the phonon calculations are performed with $\sigma =0.075$ eV.

Figure 3

(a) Crystal orbital Hamilton population (COHP) calculation of the ${\mathrm{Co}}_{2c}-{\mathrm{Co}}_{2c}$ bond in the hexagonal phase and ${\mathrm{Co}}_{8h}-{\mathrm{Co}}_{8h}$ bond in the orthorhombic phase. (b) Projected density of states (PDOS) for $3d$ states at the ${\mathrm{Co}}_{2c}$ and ${\mathrm{Co}}_{8h}$ sites. The solid and dashed curves represent the minority and majority spin states, respectively. The data for hexagonal ${\mathrm{YCo}}_5$ are adapted from Ref. [17].

Figure 4

Dependence of magnetocrystalline anisotropy energy $K_{\mathrm{u}}^{\mathrm{DFT}}$ (a), orbital moment anisotropy $\mathrm{\Delta }{m}_{\mathrm{o}}$ (b), and orbital moment $m_{\mathrm{o}}$ (c) for the hexagonal and orthorhombic phases of ${\mathrm{YCo}}_5$ under the $\mathrm{GGA}+U$ scheme. The solid and translucent symbols represent the quantities mentioned above obtained using the self-consistent and force theorem approaches, respectively. The unfilled diamonds show the theoretical results from Nguyen et al. [13] based on the force theorem, and the horizontal dash-dotted lines, labeled as Expt. 1, and Expt. 2, show the experimental results from Ref. [8] and Ref. [15], respectively. Note that the $\mathrm{\Delta }{m}_{\mathrm{o}}$ and $m_{\mathrm{o}}$ of the Y atom are not included in (b) and (c) for comparison with the experimental data.