Exploring the Structural Complexity of Intermetallic Compounds by an Adaptive Genetic Algorithm

Solving the crystal structures of novel phases with nanoscale dimensions resulting from rapid quenching is difficult due to disorder and competing polymorphic phases. Advances in computer speed and algorithm sophistication have now made it feasible to predict the crystal structure of an unknown phase without any assumptions on the Bravais lattice type, atom basis, or unit cell dimensions, providing a novel approach to aid experiments in exploring complex materials with nanoscale grains. This approach is demonstrated by solving a long-standing puzzle in the complex crystal structures of the orthorhombic, rhombohedral, and hexagonal polymorphs close to the ${\mathrm{Zr}}_2{\mathrm{Co}}_{11}$ intermetallic compound. From our calculations, we identified the hard magnetic phase and the origin of high coercivity in this compound, thus guiding further development of these materials for use as high performance permanent magnets without rare-earth elements.

Figure 1

Lowest energy structures of ${\mathrm{ZrCo}}_{5+x}$ from AGA searches. The solid (black) boxes indicate the unit cell for each structure. The largest unit cell used in our GA search contains 117 atoms at the composition of ${\mathrm{ZrCo}}_{5.5}$. The dotted (red) boxes indicate the common structure motif in the obtained structures. The dashed (brown) box in the ${\mathrm{ZrCo}}_{5.25}$ structure shows the supercell of the monoclinic structure corresponding to the experimentally observed “orthorhombic” structure with 150 atoms per unit cell. Top views of different layers in the common structure motif, labeled as no. 1, no. 2, and no. 3, respectively, are plotted on the right.

Figure 2

(a) Atomic structure with monoclinic symmetry. (b) Atomic structure with orthorhombic symmetry. (c) Comparison of the simulated XRD spectra from the predicted orthorhombic and monoclinic structure models (red and purple) with experiments (black and green, $10\mathrm{m}/\mathrm{s}$ indicates the wheel speed). The blue line shows the XRD spectrum from fcc-Co structure. Cu $\mathrm{K}\alpha$ line and a broadening factor $B(2\theta )=3.1\times {10}^{-3}/\cos \theta $ were used in the simulation [22].

Figure 3

(a) Atomic structure proposed for the rhombohedral phase. (b) Comparison of simulated XRD spectrum from the rhombohedral structure model (red) with experiments (black and blue, wheel speeds are given). Cu $\mathrm{K}\alpha$ line and a broadening factor $B(2\theta )=3.1\times {10}^{-3}/\cos \theta$ were used in the simulation [22]. (c, d) Experimental and simulated [23] SAED patterns along [010] direction. (e) HREM image taken along the [010] zone axis. The red arrow indicates the repeat distance along the $c$ axis. Inset within the red box is the simulated HREM image, and the structure model (large atoms are Zr and small atoms are Co) is laid on top.

Figure 4

(a) Convex hull of formation energies in Zr-Co system with Zr atomic percent $<34\%$. The formation energy is defined relative to the hcp Co and ${\mathrm{ZrCo}}_2$ phases: $E_F(Z{r}_{m}C{o}_n)=[E(Z{r}_{m}C{o}_n)-xE(Co)-yE(ZrC{o}_2)]/(m+n)$, where $x=n-2m$, $y=m$. (b) Free energy of three ${\mathrm{ZrCo}}_5$ structures: the rhombohedral, hexagonal model, and ${\mathrm{Ni}}_5\mathrm{Zr}$-type structure. The atomic structure of the hexagonal phase is shown as the inset of (b).