Ferromagnetic (Ga,Mn)As Layers and ... - OPUS Würzburg

Chapter 4
Finite Element Simulations of Strain
Relaxation
To study the complex interaction between crystalline and magnetic properties of
(Ga,Mn)As, a fundamental understanding of the mechanisms governing the strain
relaxation behavior in this material is essential. In the following, we present finite element
calculations which constitute a powerful tool in the investigation of the structures
involved in this work.
Due to the considerable processing time of samples containing large stripe arrays,
it is necessary to develop a method which allows reliable predictions about the relaxation
in the patterned structures. With such a method, it is possible to optimize
critical parameters before growth and patterning of the actual sample. The principal
focus lies on predicting the extend and shape of strain relaxation, achieving homogeneity
of strain throughout the structure, and high reproducibility of samples due to
limited dependence of experimental results on small fluctuations in sample growth or
processing.
The simulations are based on the stress/strain equations of elastic continuum mechanics,
which will be discussed in the following section. For the actual calculations,
we use the finite element simulation software FlexPDE (version 5.0.7). In this program
a three-dimensional grid is defined, with individual material parameters for selected
regions. For the numerical calculation, the grid is filled with a tetrahedral finite element
mesh of points at which the equations are solved. The result is finally presented
as diagrams. Simulated values on arbitrary cut planes through the 3D volume can be
exported in tabular form.
4.1 Derivation of the Equation System
4.1.1 The Strain Coefficients
Consider the elastic properties of a crystal as a homogeneous continuous medium rather
than as a periodic array of atoms [Kitt 05]. Further, we consider only small strains
such that Hooke’s Law stating that strain is directly proportional to the stress is valid.
Let three orthogonal vectors ˆx, ŷ, ẑ of unit length be the basis of our coordinate
system. After a small deformation of the solid, the axes are distorted in orientation
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