For 1D model, we still choose the quantum well of GaAs-Al0:3Ga0:7As, and the number of total grid points which we discretize is 800, and the number of grid points in the well is 60. We delete the grid points which are outside the well, and show the result in Table 11.1.
Table 11.1
The number of total grid points is 800, and the number of grid points in the well is 60.
Number of deleted points the smallest eigenvalue
0 0.067648381
2 0.067648381
100 0.067648381
200 0.067648381
300 0.067648381
400 0.067648381
500 0.067648381
600 0.067649395
700 0.070597346
If we delete the grid points which are in the well, the result is shown in Table 11.2 as follows:
Table 11.2
The number of total grid points is 800, and the number of grid points in the well is 60.
Number of deleted points the smallest eigenvalue
0 0.0676483810
1 0.0691423424
2 0.0706856067
3 0.0722803122
4 0.0739287095
11.2 2D Model
With the experience of 1D model, we deal with 2D problem similarly. For 2D model, we still consider the GaAs-Al0:3Ga0:7As quantum wire with two forms, quadrangular wire and triangular wire.
11.2.1 Quadrangular Wire
The GaAs is embedded in the center of Al0:3Ga0:7As, and the domain length is still 80nm, the wire length is still 6nm, as shown in Figure 11.1.
Figure 11.1 Structure schema of deleted point for the quadrangular wire.
For each side, the number of total grid points which we discretize is 800, and the number of grid points in the wire is 60. We delete the grid points which are outside the wire symmetrically, and show the result in Table 11.3.
Table 11.3
For each side, the number of total grid points is 800, and the number of grid points in the wire is 60.
Number of deleted points the smallest eigenvalue
0 0.133454599
100 0.133454599
200 0.133454599
300 0.133454599
400 0.133454599
500 0.133454603
600 0.133459353
700 0.139817185
11.2.2 Triangular Wire
We use the same skill to triangular quantum wire which the domain length is 80nm and the high length of triangular wire is 3nm, as shown in Figure 11.2.
Figure 11.2 Structure schema of deleted point for the triangular wire.
Again, the number of total grid points which we discretize for each side is still 800, and the number of grid points for the high of triangular wire is 30. We show the numerical result in Table 11.4.
Table 11.4
For each side, the number of total grid points is 800, and the number of grid points for the high of triangular wire is 30.
Number of deleted points the smallest eigenvalue
0 0.220000492
100 0.220000492
200 0.220000492
300 0.220000492
400 0.220000492
500 0.220000689
600 0.220002040
700 0.230964566
11.3 3D Model
For 3D model, we consider the GaAs-Al0:3Ga0:7As quantum dot with two forms, quadrangular dot and truncated octagonal-based pyramid dot.
11.3.1 Quadrangular Dot
The GaAs is embedded in the center of Al0:3Ga0:7As, and the domain length is 80nm; the dot length is 6nm. For each side, the number of total grid points which we discretize is 80, and the number of grid points in the dot is 6. We delete the grid points which are outside the dot symmetrically, like 2D model, and show the result in Table 11.5.
Table 11.5
For each side, the number of total grid points is 80, and the number of grid points in the dot is 6.
Number of deleted points the smallest eigenvalue
0 0.222975522
10 0.222975522
20 0.222975522
30 0.222975523
40 0.222975524
50 0.222975662
60 0.223003735
70 0.228506895
11.3.2 Truncated Octagonal-Based Pyramid Dot
The GaAs embedded in the center of Al0:3Ga0:7As is a truncated octagonal-based pyramid., and the domain length is 80nm; the dot length is 6nm. For each side, the number of total grid points which we discretize is still 80, and the number of grid points in the dot is 6. We still delete the grid points symmetrically, and get the result in Table 11.6.
Table 11.6
For each side, the number of total grid points is 80, and the number of grid points in the dot is 6.
Number of deleted points the smallest eigenvalue
0 0.226957440
10 0.226957440
20 0.226957440
30 0.226957439
40 0.226957439
50 0.226957574
60 0.226984064
70 0.232934269
12 Conclusion
In the …rst part, we know that it is the best choise to let the discretization of the grid point at the interface to be quadratic. With this discretization, we can get the accuracy of O(h2) to the smallest eigenvalue. In the second part, to avoid the calculations which are nonnecessary, we delete some grid points, and we …nd that the deleted grid points which are outside the quantum well, wire, and dot are much better than those grid points which are inside the well, wire and dot.
References
[1] R. L. Burden and J. D. Faires. Numerical Analysis. pp 673-683, Brooks/Cole Publishing Company, 1997.
[2] J. H. Davies. The Physics of Low-Dimensional Semiconductors. pp. 80-117, Cambridge University Press, 1998.
[3] D. R. Fokkema, G. L.G. Sleijpen, and H. A. van der Vorst. Jocobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J.
Sci. Comput., 20(1):94-125, 1998.
[4] S. Gilbert and F. George J.. An Analysis of The Finite Element Method, Englewood Cli¤s, NJ/Prentice-Hall, 1973.
[5] L. Harris, D. J. Mowbray, M. S. Skolnick, M. Hopkinson, and G. Hill. Emis-sion spectra and mode structure of InAs/GaAs self-organized quantum dot lasers. Appl. Phys. Lett., 73:969-971, 1998.
[6] P. Harrison. Quantum Wells, Wires and Dots. pp. 17-70, John Wiley and Sons, Ltd., 2000.
[7] S. Maimon, E. Finkman, G. Bahir, S. E. Schacham, J. M.Carcia, and P.
M. Petro¤. Intersublevel transitions in InAs/GaAs quantum dots infrared photodetectors. Appl. Phys. Lett., 73:2003-2005, 1998.
[8] G. L. Sleijpen and H. A. van der Vorst. A Jacobin-Davidson iteration method for linear eigenvalue problems. SIAM J. Sci. Matrix Anal. Appl., 17(2):401-425, April, 1996.
[9] J. C. Strikwerda. Finite Di¤erence Schemes and Partial Di¤erential Equa-tions. pp. 13-23 and 53-65, Brooks/Cole Publishing Company, 1989.
[10] J. C. Strikwerda. Finite Di¤erence Schemes and Partial Di¤erential Equa-tions. pp. 351-363, Brooks/Cole Publishing Company, 1989.
[11] K. Zhang, J. Falta, Th. Schmidt, Ch. Heyn, G. Materlik, and W. Hansen.
Pure Appl. Chem., Vol. 72, Nos. 1-2, pp. 199-207, 2000.