Presented by Gusakov Yuri
(
home page; e-mail)
Technion El. Engineering Department
September 2000

Abstract

Strain distribution in and around pyramidal Quantum Dot (QD) is modeled by continuum elasticity theory and is approximated by three-dimensional finite element method (3D FEM). The numerical calculation is performed using MSC/NASTRAN finite element software. In order to check reliability of the results obtained for pyramidal QD, an additional example which has analytical solution was first solved numerically using the MSC/NASTRAN software.


Introduction
Formulation of the problem
Numerical technique
MSC/NASTRAN software
An example with analytical solution
Results for Quantum Dot (QD)
References


Introduction

The present work is a project for the Computational Physics course. Taking into account the requirements of the course (work should be concentrated on the computational part of a problem) I describe here the mathematical model only of the real problem. Recently people have learned to grow structures, Quantum Dots (QD), in which a charged particle is confined in all three dimensions.1-4 It was discovered experimentally that such QDs have a pyramidal form. Naturally, by growth conditions, QDs are highly strained. Moreover the strain field is also distributed into material surrounding the QD. Therefore, if we are going to study theoretically the electronic properties of such structures, as a first step the strain distribution in and around the QD have to be calculated. Calculation of strain in the pyramidal QD is the main task of this work.

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Formulation of the problem

In order to build a model for the real QD structure a number of assumptions is needed.
1)
First, assume that there is no coupling between adjacent dots, i.e. surrounding material is distributed to infinity.
2)
Secondly, assume that the Dot has a pyramidal form which is symmetrical under rotation by 90o about z-axis (Fig. 1).
3)
Finally, I use the continuum limit elastic theory, which means that the strain is assumed to be pseudomorphic (no plastic deformations).
For numerical calculations I take one quarter of the structure with the following dimensions: for the pyramid x,y,z change from 0 to 0.4; for the surrounding cube (it can also be sphere or another form which is large enough compared to the pyramid) x,y change from 0 to 1 and z changes from -0.5 to 1. Measurement units can be considered as dimensionless. We will see further that such cube can be considered as a large enough one (relative to dimensions of the pyramidal QD).



Fig. 1 A quarter of the pyramidal QD immersed into barrier material

QD's crystalline material has lattice constant different from that of the surrounding barrier material. Because of this, the strain field is present inside the whole structure. The system of equations describing the problem is well known 5 and is written below in terms of displacements (1.1), with appropriate number of boundary conditions (1.2).

In the case of pyramidal QD it is not clear what boundary conditions in terms of stress, strain or displacements should be applied. But we can use an equivalent approach. Let's the whole structure consists of only one kind of crystalline material and is first unstrained. The initial temperature can be taken as reference. We can imagine that we heat (or cool) the pyramidal QD region without heat transfer. Assume that the same thermal expansion coefficient is associated with the whole structure. Final temperature of the QD should be chosen so that its lattice constant will reach that one of the real QD if it was not immersed into barrier material. Using this equivalent approach with additional boundary conditions (for example, if we want that the QD will keep its pyramidal form) we will obtain the same strain field. Concluding this section we see that by solving the system of differential equations (1.1) for displacements ui(x,y,z) all relative quantities can be obtained. Particularly, six independent strain tensor components, each being function of three variables: eij(x,y,z), are obtained by the following relations:


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Numerical method

In order to solve the system of differential equations (1.1) described above I choose the Finite Element Method (FEM). This is a numerical method widely used for variety of problems and is most suitable for problems with irregular boundaries (as is in my case). There are o lot of good books on FEM.6-8 I briefly describe here the main idea of the FEM. Finite Element analysis seeks to approximate the behavior of an arbitrary shaped structure under general loading and constraint conditions with an assembly of discrete finite elements. Finite elements have regular (or nearly regular) geometric shapes and known solutions. The behavior of the structure is obtained by analyzing the collective behavior of the elements.
The FEM solution flowchart is:
1)
Find the quadratic functional that corresponds to the system of differential equations.
2)
Subdivide the region into subregions that span the region of the problem.
3)
Write relations that interpolate values of the function (the solution) at the nodes to give values of the function at points within the element. The interpolation relations are chosen to be zero outside the element so there is a purely local effect. The sum of these relations weighted by the nodal values is used as an approximation to the solution of the problem.
4)
Substitute this weighted sum into the quadratic functional and minimize with respect to each unknown weighting factor by setting derivatives to zero. The quadratic functional breaks into a sum of integrals over each element. This leads to a set of linear equations. The solution to the original system of differential equations is obtained by solving the set of linear equations.
For example,
Fig. 2 shows the mesh I use for the QD structure; Fig. 3 and Fig. 4 show meshes (rough and fine) of a hollow sphere.

Fig. 2 Meshing the QD structure
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MSC/NASTRAN software

MSC/NASTRAN is a general purpose finite element analysis computer program. MSC/NASTRAN addresses a wide range of engineering problem-solving requirements (e.g., statics, dynamics, nonlinear behavior, thermal analysis, or optimization). NASTRAN software is available on tx and aluf servers. In order to use NASTRAN, an input file in ASCII format which reflects geometry and boundary conditions of a problem must be constructed. For example using fine mesh it will perhaps be necessary to divide a structure by 10000 finite elements (really, it is not fine but medium mesh). In this case the input file will contain at least 10000 lines defining the corners of each element. This work can be done easily by PATRAN, which is pre- and post-processor of NASTRAN. Using PATRAN it is also easy to interpret results. In brief, the procedure is as follows:
1)
Using PATRAN preprocessor we create the geometry of a problem and apply loads (pressure, temperature, forces and so on) and constraints conditions.
2)
Once the input file, e.g. filename.bdf, is constructed (and edited if necessary), we run NASTRAN:

nastran scr=yes filename.bdf

3)
The file containing results is filename.f06 (it is in ASCII format). Using PATRAN postprocessor we can easily work with results (for example, plot).
Documentation for NASTRAN is available in the library of the Taub Computer Center. Documentation for PATRAN is on-line and available on tx. In order to initialize PATRAN write in x-terminal window on tx: patran&
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An example with analytical solution (hollow sphere)

Before proceeding with QD structure it is useful to check how fine the mesh should be to give a desired precision. For this goal any problem which has analytical solution can be used. I take a hollow sphere subjected to internal pressure. Taking into account spherical symmetry of this problem we can significantly reduce computer effort. Consider 1/8 part of the hollow sphere (
Fig. 3).

Fig. 3 Rough mesh of the hollow sphere

With boundary conditions that points at plane edges are constrained there (can not move in perpendicular direction) we have an equivalent formulation of the original problem (another way to construct boundary conditions is allowing only radial displacements for each point). Taking for instance Rin=0.5 m, Rout=1.5 m, Young modulus E=30.0E6 N/m2, Poisson ratio 0.3, Pin=10.0 N/m2 we obtain the analytical result for displacements of internal points:

Numerical solution using rough mesh (
Fig. 3) is 0.8828E-7 m and is 20% smaller, while using finer mesh (Fig. 4) the result is 1.1373E-7 m and is only 0.3% smaller, then analytical solution. The input file corresponding to Fig. 3 you can see here. You can also see the output file (results) here. Fig. 5 shows displacements plotted using PATRAN.

Fig. 4 Fine mesh of the hollow sphere

Fig. 5 Displacements (plotted using PATRAN)
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Results for Quantum Dot

Using mesh that was constructed for QD (see
Fig. 2) I apply the following boundary conditions:
1)
By the symmetry of the problem, points on XZ (YZ) edge can not move in Y (X) direction.
2)
In order to keep the pyramidal form of the QD all points belonging to the external surfaces of the pyramid are allowed to move but don't change the pyramidal form of the QD.
The input file for calculating strain distribution in and around QD using NASTRAN software is available here. I use the following quantities:
1)
Young modulus E=1.0E+11 N/m2;
2)
Poisson ratio 0.3;
3)
Reference temperature 10.0 oC;
4)
Temperature inside the QD 20.0 oC;
5)
Thermal expansion coefficient 0.0227 m/oC;
Using this thermal expansion coefficient and heating the QD by 20.0 oC the lattice constant of Si (5.431 angstrom) is increased to the lattice constant of Ge (5.658 angstrom). But all these parameters can be changed by changing one line in the input file (the line is MAT1,1,1.E11,,0.3,,2.27E-2,10.0).
The output (.f06) file is very large and is not available here by reason of saving disk space. The following figures show strain distribution and displacements in the QD structure.









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References

1 J.M.Moison, et al, Appl.Phys. Lett. 64,196 (1994)
2 J.Y.Marzin, et al, Phys. Rev. Lett. 73, 716 (1994)
3 G.Medeiros-Ribeiro, et al, Appl. Phys. Lett. 66, 1767 (1995)
4 M.Grundmann, et al, Phys. Rev. Lett. 74, 4043 (1995)
5 Adel S. Saada, Elasticity: Theory and Applications, 1974
6 R.H. Gallagher, Finite Element Analysis Fundamentals, Prentice-Hall, Inc., New Jersey, 1975
7 Zienkiewicz, Olgierd C., The Finite Element Method, 3th edition, 1991
8Huebner, Kenneth H., The Finite Element Method for Engineers, 1982