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2.3.2 NonLinear Surface Reactions
In general, the surface velocity (2.25) is not a linear function. As an example, the sticking probability in (2.31) can depend on the local arriving particle flux, if higher order surface kinetics are assumed [21]

(2.32) 
Here
is the order of the reaction. Obviously, in the case of nonlinear reaction kinetics (
) the sticking coefficient depends on the flux.
Other examples, where nonlinear surface reactions need to be incorporated, are processes with ionenhanced etching. There physical (ions) and chemical (neutrals) components act in a synergistic manner, so that the etch rate is larger than that obtained by summing up their individual contributions.
The Langmuir adsorption model is able to capture this behavior [91]. The idea is to assume an absorbed state of byproducts. The fraction of surface sites covered by these byproducts is called the surface coverage
. The etch rate is then composed of three contributions

(2.33) 
The first term corresponds to chemical etching which is proportional to the coverage, the second term represents physical sputtering with the total sputter rate
, and the last term is due to ionenhanced etching, which is proportional to the coverage and the total ionenhanced etching rate
. The total yields are both calculated using a weight function similar to that used in (2.27). Two different total yields are introduced, since in the general case the physical sputter yield and the ionenhanced etching yield are not equal. The constants
,
, and
in (2.33) are model parameters.
For the coverage
a balanced equation can be set up as follows

(2.34) 
The first term describes the adsorption of chemical components, which is proportional to the total arriving flux
of neutrals and the fraction of empty surface sites
. The second and the third term are losses due to chemical and ionenhanced etching, respectively, which are both proportional to the coverage. The constants
,
, and
are again model parameters.
A common approach is to assume that the coverage is always in a steady state
. Therefore, the coverage can be explicitly expressed as a function of the rates
and
,

(2.35) 
and can be plugged into (2.33). The result is a nonlinear function of the rates for the surface velocity.
Next: 2.3.3 TransportIndependent Surface Reactions
Up: 2.3 Surface Kinetics
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Otmar Ertl: Numerical Methods for Topography Simulation