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Figure 28. The binding energy as a function of (a) side length of the square wire and (b) radius of the cylindrical wire. For both cases E = 0, 10, and 20 kV/cm (from top to bottom). Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

The behavior of the binding energy under a different electric field is shown in Figure 28a and b for wires of square and circular cross-sections, respectively. The electric field is taken to be applied along the positive axis direction with 6 = 0. Thus, the electron shifts toward the negative part of the axis. It is seen from these figures that the binding energy is a more sensitive function of the electric field for wires of cylindrical cross-section.

The impurity position dependence of the binding energy is shown in Figure 29a when the impurity position is changed along the diagonal of the square wire (L = ^a*B) for different electric fields. The impurity position dependence of the binding energy is shown in Figure 29b when the impurity position is changed radially for a cylindrical wire (r0 = a*B) for different electric fields. As expected the binding energy becomes smaller for impurities located at the boundary of the wires since for this position the quantum well walls prevent the electron from feeling the full power of the Coulomb attraction.

The behavior of the binding energy is also checked as the wire dimension is increased. Figure 30a shows the binding energy as a function of impurity position along the x-axis of the square wire (L = ^2a*B) for different electric fields. Now, the binding energy is generally smaller since the walls are not pushing the electron closer to the Coulomb center.

Figure 28. The binding energy as a function of (a) side length of the square wire and (b) radius of the cylindrical wire. For both cases E = 0, 10, and 20 kV/cm (from top to bottom). Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

The trial function in this case is taken to be

where ( is the variational parameter. The Hamiltonian becomes

The trial function for the bound electron is taken to be

where A is the variational parameter.

The calculations are carried out for the model system of GaAs/Alx Ga1-xAs for which the effective Bohr radius is a% = 98.7 Á and the effective Rydberg is R* = 5.83 meV for x = 0.3. The results are in perfect agreement with previous calculations without the electric field. For example, the binding energies are found to be almost identical for wires of circular and square cross-sections if wire dimensions are taken to be comparable. The binding energies are different for finite and infinite barrier potentials. For infinite barrier potentials, the binding energy sb tends to infinity as the diameter of the wire tends to zero. It tends to smaller finite values for finite barrier potentials.

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