Modeling of crystal growth in heteroepitaxial systems

This paper investigates the elastic deformation of the structure containing InAs nanoclusters in a pyramid, grown on the substrate GaAs. So far the data have not been grown quantum dots (QDs), one of the reasons is significant difference of periods, which reaches 7%. Ideally atomic plane on the border of QDs and substrate must continuously sews. Due to the difference in lattice periods crosslinking occurring deformations and mechanical stresses, the magnitude of which is proportional to the number of atomic planes, the size of base of the pyramid. Therefore, when you reach a certain size islands (quantum dots), they may experience mechanical stresses pcr sufficient for the appearance of structural defects - dislocations, fractures.


Introduction.
One of the methods of forming quantum dots is based on semiconductor nanostructures self organization during its epitaxial growth [1]. Mechanical stresses in epitaxial film of future quantum dot (QD) material and in its islands on the surface of the substrate are critical in the transition from the film's growth to growth of islets (Stranski-Krastanov mechanism [2,3]).These stresses are important in the further growth of QD in size, changing of its shape and its distribution on the substrate. Notable among material for QD occupies the binary semiconductors of the III V AB family such as InAs, InSb [4]. The problem for the cultivation of such QD is the absence of complementary substrates with a close lattice period. Thus, the lattice mismatch periods in QD and in the coming per period substrate for InAs/GaAs system equals to about 7%, and for InSb/GaAs system -14.5% [4,5]. As a result, in the QD-substrate interface there are occurred the mechanical stresses and deformations, the magnitude of which increases with the number of cross-linked atomic planes of that fundamental size of QD, which is often the pyramid . Therefore, when one reaches a certain size of QD, then the mechanical stresses cr p appear, the value of which is sufficient for the appearance of structural defects namelyfractures, dislocations, etc. [5,6].
In this regard there is still the urgent task to find the mechanical stresses and deformations in InAs/GaAs systems and to estimate the size of QD, the achievement of which in the process of growth process there is running the conditions for the formation of structural defects. It is divided into two tasks: i) to find the stresses and deformations occurred in the InAs/GaAs heteroepitaxial system; ii) to obtain an estimate of the size of the QD the achievement of which there are appeared the mechanical stresses sufficient for the formation of defects.
shorter than the previous one from both edges (as is when atoms in planes are strictly one above the other), in this case it should be realized the condition l na  where npairwise, a -InAs lattice period and then we receive 2 hl  where hthe height of the pyramid lthe length of the base of the pyramid. The sides of the base are oriented along the crystallographic axes [010] and in 2 xl  and 2 yl  [100]. The sides of the pyramid has the orientation (010). The substrate is represented as the parallelepiped of sizes ,,  Considering the elastic properties of the system, we will not take its full account of atomic structure, despite the fact that QD as well as substrate contains thousands of atoms. QD for instance shown at figure 1, at 20 n  contains 1 771 atoms [8]. This assumption can consider the system as a complete environment. We also believe that as a zero approximation for the deformation description in the system can be applied the linear theory of elasticity.
Within these assumptions the stresses are satisfied with the Lame equation: where ij p -(mechanical stress tensor for anisotropic linear environment is given by Hooke's law: where ijkl C -elastic constants tensor of environment, strain tensor is defined as 2 and their value and periods of crystalline structure for GaAs and InAs are shown in Table 1 taken from [9].
where  and  -Lame parameters while Hooke's law is : The structure of the matrix (3) and (4) is the same that is why for assessing of elastic constants tensor deviations for GaAs and InAs from tensor for isotropic environment we will choose the following value: which is zero for isotropic environment. Due to Table 1 it is followed that for InAs 0.02   , and for GaAs 0.05   . These values are somewhat smaller than the non-complementarity lattices and significantly less than the relative deviation of the elastic modules of its crystals.
So we make the third step: the deformations in QD-substrate we will describe with the equations of elasticity theory for two isotropic environments. Then the equation (1, 2) takes the form of: condition is met if the atomic planes of InAs and GaAs crystal lattice are oriented in the same crystallographic direction and deformed without altering the symmetry. Figure 2 а) it is shown the fragment of QD-substrate geometric contact where the atomic planes [010] in QD and in the substrate (see the figure 1) are shown as the vertical lines. Here due to our choice are stitched only average atomic planes. On the figure 2 b) is shown the same fragment where formally and without gaps all atomic planes are stitched. The average planes of both environments haven't been displaced and the displacement of other planes is increasing together with its distance from the axis of the system. Under such terms the mechanical stresses will increase around the interface. At some distance from the axis of the system they can reach such values when it is energetically favorable to pass the part of atomic plane near the interface. It can lead to structural defect - figure 2c). ,, z u x y z on the verge of a square QD and substrate obtain the following homogeneous boundary conditions: , ,0 , ,0 where the coordinates  

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,, x y l l  are within the QD foundations and multiplier 21 ll   in the arguments of deformations in substrate appears due to the fact that for the same number of atomic planes N in QD and substrate their lateral dimensions 1,2 1,2 l Na  will be different due to different lattice periods 12 aa  ,due to the Table 1. The second and third conditions set continuity of nuclear axes of which are formed the atomic planes in QD and in the substrate while the normal passing through the interface along each of these axes. So we get the following correlation : We are considering the model of QD-substrate in which the lateral dimensions of substrate are much larger than the QD's dimensions which has the triangular cross section. In the longitudinal direction the system is infinite so the two-dimensional case is considered - figure 3. Equation (6) for the deformation together with the boundary conditions on the interface and the outside of the system (7-9) are solved numerically using FlexPDE.
In figure 4 it is shown the z -component of mechanical stresses. We emphasize that within the tops of the foundations the mechanical tensions reach its maximum value. In turn, this will be the most likely QD area for the appearance of dislocations.

4.Conclusion
1. It has been established at the ideal stitching of atomic QD planes and substrates the only source of mechanical stresses and deformations is a mismatch of lattice period. 2. Analytical estimates and numerical calculations show that the maximum stresses in the system occur at the edges of QD-substrate interface, so there will be created conditions for the emergence of defects. 3.The maximum tension of elastic modulus QD order scans occur when its transverse size up to several tens of periods in the event that non-period lattice gratings is about 7%