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Figure 9.7. Photoluminescence spectrum of an array of 60-nm-diameter quantum dots formed by lithography, compared with the spectrum of the initial as-grown multiple quantum well. The intense peak at 0.7654eV is attributed to localized excitons (LE) in the superlattice (SL). The spectra were taken at temperature 4 K. [From T. P. Sidiki and C. M. S. Torres, in Naiwa (2000), Vol. 3, Chapter 5, p. 251.]

3-nm-thick layers of Si and Sio^Geoj patterned into quantum-dot arrays consisting of 300-nm-high columns with 60 nm diameters and 200 nm separations.

9.3. SIZE AND DIMENSIONALITY EFFECTS 9.3.1. Size Effects

Now that we have seen how to make nanostructures, it is appropriate to say something about their sizes relative to various parameters of the system. If we select the type III-V semiconductor GaAs as a typical material, the lattice constant from Table B. 1 (of Appendix B) is a = 0.565 nm, and the volume of the unit cell is (0.565)3 = 0.180nm3. The unit cell contains four Ga and four As atoms. Each of these atoms lies on a face-centered cubic (FCC) lattice, shown sketched in Fig. 2.3, and the two lattices are displaced with respect to each other by the amount j £ | along the unit cell body diagonal, as shown in Fig. 2.8. This puts each Ga atom in the center of a tetrahedron of As atoms corresponding to the grouping GaAs4, and each arsenic atom has a corresponding configuration AsGa4. There are about 22 of each atom type per cubic nanometer, and a cube-shaped quantum dot lOnm on a side contains 5.56 x 103 unit cells.

The question arises as to how many of the atoms are on the surface, and it will be helpful to have a mathematical expression for this in terms of the size of a particle with the zinc blende structure of GaAs, which has the shape of a cube. If the initial cube is taken in the form of Fig. 2.6 and nanostructures containing n3 of these unit cells are built up, then it can be shown that the number of atoms Ns on the surface, the total number of atoms Nr, and the size or dimension d of the cube are given by here a = 0.565 nm is the lattice constant of GaAs, and the lattice constants of other zinc blende semiconductors are given in Table B.l. These equations, (9.1)-(9.3), represent a cubic GaAs nanoparticle with its faces in the x-y, y-z, and z-x planes, respectively. Table 9.1 tabulates Ns, Nr, d, and the fraction of atoms on the surface Ns/Nr, for various values of n. The large percentage of atoms on the surface for small n is one of the principal factors that differentiates properties of nanostructures from those of the bulk materia!. An analogous table could easily be constructed for cylindrical quantum structures of the types illustrated in Figs. 9.2 and 9.6.

Comparing Table 9.1, which pertains to a diamond structure nanoparticle in the shape of a cube, with Table 2.1, which concerns a face-centered cubic structure nanoparticle with an approximately spherical shape, it is clear that die results are qualitatively the same. We see from the comparison that the FCC nanoparticle has a greater percentage of its atoms on the surface for the same total number of atoms in

Table 9.1. Number of atoms on the surface Nfe, number in the volume Ny, and percentage of atoms Ns/Nv on the surface of a nanoparticie*
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