The effect of spin-orbit coupling on dynamical properties of Pb(111) and Tl(0001) surfaces

Abstract

The neighboring elements in the periodic table, Pb and Tl, are both sp-type metals and are both strong coupling superconductors. In spite of these similarities the two metals exhibit quite different behavior when SOC is taken into account. In particular, while the spin-orbit interaction has a profound impact on the lattice dynamics of bulk Pb1 the influence of SOC on phonon frequencies of bulk Tl2 is rather weak. To verify whether the effects of SOC on lattice dynamics are similar in the case of surfaces we present an ab initio study of dynamical properties of the Pb(111) and Tl(0001) surfaces. The calculations based on the density-functional theory were carried out using a linear response technique and a mixed-basis pseudopotential method. We show that the effect of spin-orbit coupling on the surface dynamics of Tl(0001) is only slightly larger compared to the bulk case. The SOC-induced modifications of the calculated phonon frequencies are, in general, smaller than 2.5% except for the Rayleigh mode which stiffens by 7%-10% at the surface Brollouin zone boundary. In the case of the Pb(111) surface we can compare in detail the calculated surface phonons with the high-resolution helium-atom scattering data3 for a 50-layer Pb(111) film, which was shown experimentally to be a good representative for the semi-infinite surface. We find that the inclusion of spin-orbit coupling considerably softens the phonon spectrum of the Pb(111) surface. We also show that the inclusion of SOC is crucial for a proper quantitative description of the surface modes and, thereby, significantly improving the agreement with experiment as was found previously in the bulk case.