Anomalous vibrational dynamics in the Mg${}_2$Zn${}_{11}$ phase

We present a combined experimental and theoretical study of the structure and the lattice dynamics in the complex metallic alloy Mg${}_2$Zn${}_{11}$, by means of neutron and x-ray scattering, as well as ab initio and empirical potential calculations. Mg${}_2$Zn${}_{11}$ can be seen as an intermediate step in structural complexity between the simple Laves-phase MgZn${}_2$ on one side, and the complex 1/1 approximants and quasicrystals ZnMgAl and Zn(Mg)Sc on the other. The structure can be described as a cubic packing of a triacontahedron whose center is partially occupied by a Zn atom. This partially occupied site turned out to play a major role in understanding the lattice dynamics. Data from inelastic neutron scattering evidence a Van Hove singularity in the vibrational spectrum of Mg${}_2$Zn${}_{11}$ for an energy as low as 4.5 meV, which is a unique feature for a nearly-close-packed metallic alloy. This corresponds to a gap opening at the Brillouin zone boundary and an interaction between a low-lying optical branch and an acoustic one, as could be deduced from the dispersion relation measured by inelastic x-ray scattering. Second, the measured phonon density of states exhibits many maxima, indicating strong mode interactions across the whole energy range. The origin of the low-energy modes in Mg${}_2$Zn${}_{11}$ and other features of the vibrational spectra are studied, using both ab initio and empirical potential calculations. A detailed analysis of vibrational eigenmodes is presented, linking features in the vibrational spectrum to atomic motions within structural building blocks.

Figure 1

Structure of Mg${}_2$Zn${}_{11}$ showing the constituent shells of the Pauling triacontrahedron, with Zn5 icosahedron (left), followed by a distorted dodecahedron of Mg and Zn4 atoms (middle), and a Zn3 icosahedron (right), connected by Zn2 octahedra. The view is along the $c$ axis, with the $a$ and $b$ axes in plane. Zn atoms are represented in red (light gray), Mg atoms in blue (dark gray).

Figure 2

Structure of Mg${}_2$Zn${}_{11}$ showing the constituent shells of the Tsai-type cluster, with Zn2 octahedron (left), strongly distorted Zn3/Zn4 dodecahedron (middle), and Mg icosahedron (right) connected by Zn5 icosahedra. The view is along the $c$ axis, with the $a$ and $b$ axes in plane. Zn atoms are represented in red (light gray), Mg atoms in blue (dark gray).

Figure 3

Structure of Mg${}_2$Zn${}_{11}$ (a) represented as cubic packing of Pauling triacontahedral clusters linked by Zn octahedra and (b) represented as filled double icosahedral shells connected by Zn icosahedra, with Mg atoms in blue (dark gray) and Zn atoms in red (light gray). The shapes of the atoms represent the atomic displacement parameters.

Figure 4

Local environment polyhedra. The polyhedra in the second row are rotated by ${90}^{°}$ around the vertical axis with respect to the first row, except in the leftmost figures. These show the two icosahedral environments, Zn1 (top) and Zn5 (bottom). The other columns depict the Zn2, Zn3, Zn4, and Mg environments, from left to right. Zn atoms are represented in red (light gray) and Mg atoms in blue (dark gray).

Figure 5

Room-temperature GVDOS from inelastic neutron scattering for Mg${}_2$Zn${}_{11}$ [blue (dark gray line) with peak labels $A$ to $G$ as in Table ] and MgZn${}_2$ [red (light gray) line].

Figure 6

$S(Q,\omega )$ from inelastic neutron scattering for Mg${}_2$Zn${}_{11}$ (top) and MgZn${}_2$ (bottom) at 300 K.

Figure 7

GVDOS from inelastic neutron scattering for Mg${}_2$Zn${}_{11}$ at temperatures of $T=100$ K (black line) and 300 K [red (light gray line)], IN4 data. Room temperature IN6 data with peak labels (Table ) are shown for reference [dashed blue (gray) line].

Figure 8

Intensity distribution of the $Q$-integrated $S(Q,\omega )$ data as a function of energy for Mg${}_2$Zn${}_{11}$ (integrated $Q$ range 2.75 to 3.25 Å${}^{-1}$), at different temperatures for neutron energy loss. The solid curves are fits to the experimental data.

Figure 9

Dispersion relation measured by inelastic x-ray scattering in Mg${}_2$Zn${}_{11}$ along the direction $(\xi ,4,0)$. Triangles and diamonds stand for acoustic and optical excitations. The dispersion is overlaid on a color coded intensity distribution of the simulated $S(Q,\omega )$ response function. The calculated curves are corrected for a frequency shift $\gamma =0.2$ following Eq. (10).

Figure 10

Intensity distribution of the $S(Q,\omega )$ response function measured by inelastic x-ray scattering along $(\xi ,8,0)$. The red (solid) curves stand for the fits resulting from the modes depicted as blue (dashed) lines. The gap opening at the zone boundary is nicely visible and marked by an arrow (top panel). The optical excitation at 11 meV is clearly visible in the bottom panel.

Figure 11

Dispersion relation measured by inelastic x-ray scattering in Mg${}_2$Zn${}_{11}$ along the direction $(4+\xi ,\xi ,-\xi )$ Triangles and diamonds stand for acoustic and optical excitations. The dispersion is overlaid on a color-coded intensity distribution of the simulated $S(Q,\omega )$ response function. The calculated curves are corrected for a frequency shift $\gamma =0.2$ following Eq. (10).

Figure 12

Intensity distribution of the $S(Q,\omega )$ response function measured by inelastic x-ray scattering along $(4+\xi ,\xi ,-\xi )$. Black dots stand for experimental data, while the red (solid) curves stand for the calculation, corrected for a frequency shift $\gamma =0.2$ following Eq. (10). The calculated intensities are scaled to fit the experimental data.

Figure 13

Specific heat in Mg${}_2$Zn${}_{11}$: upper (red) curve, calculation; lower (black) curve, experiment.

Figure 14

Electronic density of states of Mg${}_2$Zn${}_{11}$ with fully occupied Zn1 site (a), empty Zn1 site (b), and partially occupied (1/3) Zn1 site (c). The solid vertical line shows the Fermi level energy. Models (a) and (b) contain 39 and 38 atoms in the unit cell, respectively. The most stable model (c) has 115 atoms in a triple unit cell, with one out of three Zn1 sites occupied by Zn.

Figure 15

Comparison of experimental [blue (dark gray)] and calculated GVDOS [red (light gray)] for MgZn${}_2$ at 300 K. The calculated GVDOS is corrected by frequency shift $\gamma =0.1$ [see Eq. (10)].

Figure 16

Comparison of experimental and calculated GVDOSs for Mg${}_2$Zn${}_{11}$, for double-supercell models with full or half occupancy of the Zn1 site. A uniform frequency shift of $\gamma =0.07$ [see Eq. (10)] is applied. Peaks on the IN6 GVDOS are labeled following Table .

Figure 17

Simulated dispersion curve along the direction ($4+\xi ,\xi ,-\xi$) using EAM potentials (see text). Top: Fully occupied unit cell. Bottom: 2 $\times$ 2 $\times$ 2 supercell with $50\%$ vacancies.

Figure 18

Comparison of the experimental GVDOS [red (light gray)] and the GVDOS calculated with MD for the 2 $\times$ 2 $\times$ 2 supercell with $50\%$ vacancy at the cluster center (black) at 100 and at 300 K. $\gamma =0.1$ [see Eq. (10)]; peak labels as in Table .

Figure 19

Partial densities of states for the Mg${}_2$Zn${}_{11}$ Wyckoff sites (2 $\times$ 1 $\times$ 1 supercell with one vacancy at Zn1) (black curves). GVDOS obtained from the IN6 instrument [red (light gray) curve]. Peak labels are as in Table . The partial DOSs are weighted by site multiplicities.

Figure 20

Participation ratio and density of states of Mg${}_2$Zn${}_{11}$, both after uniform frequency shift correction $\gamma =0.07$ [see Eq. (10)]. Eigenmodes with frequency and participation from the $A$–$G$ windows, shown as red rectangles in the upper panel, are selected for graphical representation in Figs. 21 and 22.

Figure 21

Time-averaged particle density plots for slabs projected along the twofold axis of Mg${}_2$Zn${}_{11}$. Spatial particle motions are due to time evolution of vibrational eigenstates obtained from the DFT dynamical matrix (for a 2 $\times$ 1 $\times$ 1 supercell with two Zn1 sites containing one Zn and one vacancy). The selected range of eigenmodes, represented in windows $A$–$D$, corresponds to the marked rectangular areas in the energy-participation plot (Fig. 20). The Mg atom density is shown in blue–scale (gray and marked by a circle) for densities up to $5\%$ of maximal density, otherwise white; for Zn the same is valid in red-scale (gray, without circle). The left column panels contain the “cubic” flat layer at $z=0$, together with the puckered layer $0.163⩽z⩽0.343$. The right column panels contain the same puckered layer, and the “icosahedral” flat layer at $z=0.5$ (so the puckered layer is shown in both columns).

Figure 22

Time-averaged density as in Fig. 21, but for windows $E$–$G$. The bottom panel is a plot of the atomic structure along the same axis, and using the same layers. Size of the circles is proportional to the height $z$ of an atom. Wyckoff sites are shown in different colors (different gray scale plus marks).