Terrace ledge kink

The Terrace Ledge Kink (TLK) model, which is also referred to as the Terrace Step Kink (TSK) model, describes the thermodynamics of crystal surface formation and transformation, as well as the energetics of surface defect formation. It is based upon the idea that the energy of an atom’s position on a crystal surface is determined by its bonding to neighboring atoms and that transitions simply involve the counting of broken and formed bonds. The TLK model can be applied to surface science topics such as crystal growth, surface diffusion, roughening, and vaporization.

History

The TLK model is credited as having originated out of two German papers by Kossel [Kossel, W., Extenoling the Law of Bravais [M] . Nach Ges Wiss Gottingen 1927, 143.] and Stranski [Stranski, I. N., Zur Theorie des Kristallwachstums. Z. Phys. Chem 1928, 136, 259-278.] wherein the thermodynamic stability of step edges were discussed.

Definitions

Depending on the position of an atom on a surface, it can be referred to by one of several names. Figure 1 illustrates the names for the atomic positions and point defects on a surface for a simple cubic lattice.

Figure 2 shows a scanning tunneling microscopy topographic image of a step edge that shows many of the features in Figure 1.

Thermodynamics

The energy required to remove an atom from the surface depends on the number of bonds to other surface atoms which must be broken. For a simple cubic lattice in this model, each atom is treated as a cube and bonding occurs at each face, giving a coordination number of 6 nearest neighbors. Second-nearest neighbors in this cubic model are those that share an edge and third-nearest neighbors are those that share corners. The number of neighbors, second-nearest neighbors, and third-nearest neighbors for each of the different atom positions are given in Table 1.Oura, K.; Lifshits, V. G.; Saranin, A. A.; Zotov, A. V.; Katayama, M.; Yates, J. T., Surface Science: An Introduction. Springer-Verlag: Berlin, 2003.]

This can be understood as the breaking of all of the kink atom’s bonds to remove the atom from the surface and then reforming the adatom interactions. This is equivalent to a kink atom diffusing away from the rest of the step to become a step adatom and then diffusing away from the adjacent step onto the terrace to become an adatom. In the case where all interactions are ignored except for those with nearest neighbors, the formation energy for an adatom would be the following, where $phi$ is the bond energy in the crystal is given by Equation 2.

This can be extended to find the equilibrium concentration of other types of surface point defects as well. To do so, the energy of the defect in question is simply substituted into the above equation in the place of the energy of adatom formation.

References