Lattice dynamics of neodymium: Influence of $4f$ electron correlations

Incorporation of strong electron correlations into the density functional theory (DFT) for the electronic structure calculations of light lanthanides leads to a modification of interatomic forces and consequently the lattice dynamics. Using first-principles theory we demonstrate the substantial influence of the $4f$ electron correlations on the phonon dispersion relations of Nd. The calculations are verified by an inelastic x-ray scattering experiment performed on a single-crystalline Nd(0001) film. We show that very good agreement between the calculated and measured data is achieved when electron-electron interactions are treated by the $\mathrm{DFT}+U$ approach.

Figure 1

Ab initio calculated (a) dispersion relations and (b) PDOS of Nd within GGA and GGA+$U$ using the optimized lattice constants. The high-symmetry points in units of $2\pi /a$ are $\mathrm{\Gamma }=(0,0,0), K=(\frac{1}{3},\frac{1}{3},0), M=(\frac{1}{2},0,0), A=(0,0,\frac{1}{2}), H=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$, and $\mathrm{L}=(\frac{1}{2},0,\frac{1}{2})$. Shaded areas correspond to the Gauss convolution with $\text{FWHM}=1$ meV.

Figure 2

(a) Schematic presentation of the investigated sample. (b) LEED image (obtained at 100  eV) of the Nd(0001) surface. The hexagonal $ab$ plane is indicated by broken white lines. (c) XRD scan along $00L$ in dhcp $4H$ reciprocal lattice units.

Figure 3

Comparison between the GGA+$U$ calculated (colored lines) and measured by IXS (black circles; error bars are comparable to the symbol size) phonon branches close to (a), (b) $\mathrm{\Gamma }\to K\to M$, (c) along $\mathrm{\Gamma }\to A$, and (d), (e) close to $\mathrm{\Gamma }\to M\to \mathrm{\Gamma }$ directions. The intensity scale is shown in (f). The directions, polarizations, and exact coordinates of the measured branches are listed in Table 2. The phonon polarization vector $\overrightarrow{e_p}$ (black arrows), direction of the momentum transfer (red dashed arrows), and dhcp Brillouin zone (black hexagon/structure) are shown as insets. Details about the calculation of the intensity of the phonon branches is given in [39].