Energy of a strained vicinal surface

Macroscopically, the surface is assumed to be smooth (as indicated by the dashed line in the figure) and to have a local surface energy density Us at any point on the surface, determined by the orientation of the tangent plane and the level of elastic strain in the crystal at that point. A quantitative interpretation of the free energy of a strained crystal surface in terms of the 

Suppose that the reference surface 0 = 0 is the 100 surface of a cubic crystal, and that the background strain in this state is a spatially uniform equilibrium elastic field consistent with a traction free surface. As 0 increases from zero, the surface strain e is given by (8.126) where mi = COS0, m2 = sin0. The local state of strain would be altered as a result of long-range elastic effects if 0 would be spatially nonuniform in the present instance, such spatial variation is not taken into account and the surface strain is determined by 0 and the uniform strain field e . For the case in which 6 2 = substitution of the expression for e into (8.139), followed by 

Continuum theory provides no basis for estimating the sub-continuum parameters j3, /3i and (3 for any particular material, even though these parameters are well-defined characteristics of behavior. They derive strictly from the discreteness of the material, so values can be estimated only when the model of the material includes that discreteness. Such estimates require the adoption of a particular interatomic potential to simulate the material and, in the case of the surface, an atomic structure for that surface. 

Interpret the surface energy expression (8.140) for the case of a strained vicinal surface on a Si crystal near the high symmetry (001) orientation. 

The angle 6 implied by these parameter values at strain is about 8.5°. The actual crystal can assume only those angles characteristic of (lOn) orientations, where n = 5,9,13. for this reconstruction, and 8.5° is about midway between the (105) and (109) orientations. Long range elastic effects, which have not been taken into account in the discussion in this section, would tend to drive the angle toward the larger angle orientation. 

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