Introduction of Surface Stress Anisotropy

    Surface stress anisotropy can come from structural variants or misfit of low-symmetry lattices.  A typical example for the first case is the Si (100) surface, where the pairing row reconstruction with the formation of Si dimers introduces an anisotropy in the surface stress, as shown in (a).  The stress along the dimer bonds is tensile, while the stress perpendicular to the dimer bonds is less tensile or compressive.  The difference can be 1~3 N/m.  Figure (b) illustrates the origin of anisotropy for the latter case.  Suppose we deposit a square lattice of atoms on the (110) surface of a substrate with cubic structure, such as Cu(110) surface.  To form a coherent lattice, the atoms in the epilayer need to be stretched more in the  direction than in the  direction.  Hence the surface stress in the  direction is larger.  It should be noted that even when the substrate has high symmetry structure and is almost elastic isotropic, surface stress anisotropy may still be very important if the atoms are deposited on a low-symmetry surface.  Figure (b) also shows that surface stress in the two directions can be both compressive or have different signs - tensile in one direction and compressive in the other.  Surface stress can be obtained from experimental measurements or first principal calculations.  The nature of surface stress is a kind of elastic effect and it will influence both the shape of patterns and the special ordering considerably.

    When the surface stress is anisotropic, f should be a second order tensor.  It has two principal directions within the plan of the epilayer.  We can always rotate the coordinates so that shear components disappear.  Hence we can set  without loosing any generosity.  The two principal surface stresses play equivalent roles.  Furthermore, when both the principal stresses change signs, the evolution equation does not change.  Consequently, we need only consider surface stresses represented on the  plane within the sector bounded by  and , as shown in the following figure.  The five representative states a, b, c, d, and e correspond to , respectively.  A dimensionless parameter  is defined to measure the anisotropy of surface stress.  All the simulations start from the same random initial condition, i.e. average concentration plus the same fluctuation within 0.001 from the average.   and Q are taken to be 2.2 and 1, Possion’s ration n   is taken to be 0.3.  The size of each figure is 256b´256b. 

Introduction of Surface Stress Anisotropy

    Surface stress anisotropy can come from structural variants or misfit of low-symmetry lattices.  A typical example for the first case is the Si (100) surface, where the pairing row reconstruction with the formation of Si dimers introduces an anisotropy in the surface stress, as shown in (a).  The stress along the dimer bonds is tensile, while the stress perpendicular to the dimer bonds is less tensile or compressive.  The difference can be 1~3 N/m.  Figure (b) illustrates the origin of anisotropy for the latter case.  Suppose we deposit a square lattice of atoms on the (110) surface of a substrate with cubic structure, such as Cu(110) surface.  To form a coherent lattice, the atoms in the epilayer need to be stretched more in the  direction than in the  direction.  Hence the surface stress in the  direction is larger.  It should be noted that even when the substrate has high symmetry structure and is almost elastic isotropic, surface stress anisotropy may still be very important if the atoms are deposited on a low-symmetry surface.  Figure (b) also shows that surface stress in the two directions can be both compressive or have different signs - tensile in one direction and compressive in the other.  Surface stress can be obtained from experimental measurements or first principal calculations.  The nature of surface stress is a kind of elastic effect and it will influence both the shape of patterns and the special ordering considerably.

    When the surface stress is anisotropic, f should be a second order tensor.  It has two principal directions within the plan of the epilayer.  We can always rotate the coordinates so that shear components disappear.  Hence we can set  without loosing any generosity.  The two principal surface stresses play equivalent roles.  Furthermore, when both the principal stresses change signs, the evolution equation does not change.  Consequently, we need only consider surface stresses represented on the  plane within the sector bounded by  and , as shown in the following figure.  The five representative states a, b, c, d, and e correspond to , respectively.  A dimensionless parameter  is defined to measure the anisotropy of surface stress.  All the simulations start from the same random initial condition, i.e. average concentration plus the same fluctuation within 0.001 from the average.   and Q are taken to be 2.2 and 1, Possion’s ration n   is taken to be 0.3.  The size of each figure is 256b´256b.