Application of the modified analytic embedded atomic method in W(100) surface phonon spectrum

Abstract

Based on the theory of surface lattice dynamics, the surface phonon spectrums along three symmetrical directions of <inline-formula><tex-math id="M4">\begin{document}$\bar \varGamma \bar L$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M4.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$\bar L\bar M$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M5.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$\bar \varGamma \bar M$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M6.png"/></alternatives></inline-formula> are simulated for the W(100) surface by using the modified analytic embedded atom method. The polarization vectors at different symmetrical points are also calculated. According to the criterion and marking method of surface mode, the surface modes along different symmetrical directions are drawn, the distribution range and mode coupling of surface modes are discussed as well. The vibration frequencies of surface modes calculated by us have been compared to available experimental datum and some theoretical values correspondingly. The results display that the present results are general agreement with the referenced experimental or theoretical results. Based on the calculated polarization vector, the surface vibration states are constructed for the atomic layers in the neighboring surface. And the polarization and local features of the surface modes along different symmetrical directions are analyzed. The results show that there are some coupling phenomena between surface mode dispersion, such as avoid crossing and independence crossing. The avoid crossing is found between the surface-mode branch S₁ and the surface-mode branch S₂ near <inline-formula><tex-math id="M7">\begin{document}${\bar \zeta _y} = 0.32$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M7.png"/></alternatives></inline-formula> along <inline-formula><tex-math id="M8">\begin{document}$\bar L\bar M$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M8.png"/></alternatives></inline-formula> direction. In the region, going from <inline-formula><tex-math id="M9">\begin{document}$\bar L$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M9.png"/></alternatives></inline-formula> to <inline-formula><tex-math id="M10">\begin{document}$\bar M$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M10.png"/></alternatives></inline-formula>, S₁ changes from <i>y</i> polarization to <i>z</i> polarization, and S₂ changes from <i>z</i> polarization to <i>y</i> polarization. The independence crossings exist between surface-mode branch S₁ and surface-mode branch S₂ at <inline-formula><tex-math id="M11">\begin{document}${\bar \zeta _x} = 0.5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M11.png"/></alternatives></inline-formula> along <inline-formula><tex-math id="M12">\begin{document}$\bar \varGamma \bar L$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M12.png"/></alternatives></inline-formula> direction, and surface-mode branch S₂ and surface-mode branch S₃ at <inline-formula><tex-math id="M13">\begin{document}${\bar \zeta _x} = 0.5$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M13.png"/></alternatives></inline-formula> along <inline-formula><tex-math id="M14">\begin{document}$\bar L\bar M$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191910_M14.png"/></alternatives></inline-formula> direction, respectively. Before and after the crossings, the polarization and local features of the surface modes have not changed. Inspection of the polarization vectors, the coupling phenomena are iconically demonstrated.