Surface Phonon - Quantum Description

Quantum Description

Surface phonons are represented by a wave vector along the surface, q, and an energy corresponding to a particular vibrational mode frequency, ω. The surface Brillouin zone (SBZ) for phonons consists of two dimensions, rather than three for bulk. For example, the face centered cubic (100) surface is described by the directions ΓX and ΓM, referring to the direction and direction, respectively.

The description of the atomic displacements by the harmonic approximation assumes that the force on an atom is a function of its displacement with respect to neighboring atoms, i.e. Hooke's law holds. Higher order anharmonicity terms can be accounted by using perturbative methods.

The positions are then given by the relation

ü_i,α =

where i is the place where the atom would sit if it were in equilibrium, m_i is the mass of the atom that should sit at i, α is the direction of its displacement, u_i,α is the amount of displacement of the atom from i, and are the force constants which come from the crystal potential.

The solution to this gives the atomic displacement due to the phonon, which is given by

where the atomic position i is described by l, m, and κ, which represent the specific atomic layer, l, the particular unit cell it is in, m, and the position of the atom with respect to its own unit cell, κ. The term x(l,m) is the position of the unit cell with respect to some chosen origin.