Abstract
Surface phonon-dispersion relations for ${\mathrm{ZrB}}_{2}(0001)$ and ${\mathrm{NbB}}_{2}(0001)$ are measured by high-resolution electron-energy-loss spectroscopy. In addition to the acoustic surface modes, phonon modes of the graphitelike boron sheet are observed. The boron phonon modes are more complex and softer on ${\mathrm{NbB}}_{2}(0001)$ than on ${\mathrm{ZrB}}_{2}(0001).$ This is consistent with the metal- and boron-layer termination of ${\mathrm{ZrB}}_{2}(0001)$ and ${\mathrm{NbB}}_{2}(0001)$ surfaces, respectively. The phonon dispersion is analyzed by a force-constant model. By a simple force-constant model, the boron phonon modes are reproduced fairly well. The obtained force constants show that the B-B bond strength is comparable to the $\mathrm{B}\ensuremath{-}M$ bond. The B-B bond in ${\mathrm{NbB}}_{2}$ is stronger than in ${\mathrm{ZrB}}_{2}.$
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