6.3.1  Hoffman’s Model

Hoffman was the first to acknowledge the energy balance of the coalescence stress formation [104]. He realized that adjacent islands snap together, in order to minimize the free surface energy at the cost of an increased elastic energy. He claimed that the zipping process (islands merging) takes place, when the distance between the islands reach a critical distance (Δ  ). Nix reviewed Hoffman’s work and added a simplified geometry model as depicted in Fig. 6.5.


Figure 6.5.: Hoffman model.

Nix restated Hoffman’s theory by looking into the energy balance of two islands before and after impingement [105]. Before impingement, the system free energy is given by

E1 = E0 − 2γsv ,

where E0   is the free surface energy per unit area of the top surface of the film and the substrate/film interface, and γ
 sv  is the free surface energy per unit area of the islands’ lateral surfaces. After islands merge the system free energy is given by

                   (   )2
E2 = E0 + γgb + Mf   Δ--   .

Before island impingement there are two independent surfaces, one for each island, with an amount of energy per surface area of γsv  . After impingement part of the surfaces energy (2γsv  ) is exerted for the formation of the interface between them (grain boundary). This energy amount per unit area is represented by γgb  . The remainder of the surface free energy is converted by the islands’ stretching with elastic energy, which is represented by the second term of (6.2). Actually, this is just the conservation of the free energy (E2 − E1 = 0  ), which can be used to calculate Δ  . The resulting Δ  is

    [                   ]1∕2
Δ =  4R (2γsv − γgb)  E       .

The coalescence stress can then be computed from Hooke’s law using Δ  as in

        (   )   [             ]1∕2
          Δ          2γsv − γgb
σ = Mf   2R-  =   Mf ---2R-----    .

Hoffman’s model overestimates the coalescence stress in the film, mainly because of the simple geometry assumed for the islands. However, it is a reasonable approach for low adatom materials. Moreover, it is a simple model when compared to the alternatives and it can be useful as a quick estimate of the upper bound of the coalescence stress.