E ndn(r0 -

LCn n=0

p n p can be solved numerically. Once the nth eigenenergy en(l) is known, the A, B, C, and D [hence ^(r)] are known with the use of the normalized condition of ^(r). This ^(r) depends on the value of l, the quantum well, Coulomb potential, and energy en(l). We should point that we have neglected the difference of the electron effective mass between GaAs and Ga1—xAlxAs in the Hamiltonian and the matching conditions. If the effective mass difference is considered, similar formulas can be obtained.

If there is no Coulomb potential in the Hamiltonian of Eq. (144) (i.e., w = 0), using the same formulas, we can obtain wavefunction w = 0) and quantum levels en(l, w = 0) of an electron in the quantum well. In fact, Eqs. (148) and (151) become the spherical Bessel function and Hankel function if w = 0. The equation of eigenenergies e(l, w = 0) is k0 + K0 tan(k0 r0) = 0 if l = 0 (159)

iklhl(iKl r0)jl-1(klr0) + Kihi_1(Kir0)jl(klr0) = 0 if l > 1 (160)

3.4.1. Quantum Levels and Binding Energies

A numerical calculation for GaAs-Ga1-xAlxAs spherical quantum dots of the r0 between 0.15aB and 7.0aB with different V0 has been performed. In Table 3, the quantum levels of an electron in a spherical quantum dot with different r0 and V0 have been shown. The levels en(l) are indicated by two symbols n and l as shown in Eq. (162). n is equal to the number of the root of Eqs. (158) or (159) and (160) in order of increasing magnitude (i.e., n = 1, 2, 3,... and hence n — 1 is the radial quantum number as usual). l is the usual notation (i.e., s, p,d,...). Thus we have 1p, 1d, 2s, 1/ levels (states) and so on if the n and l are used as the level notation, and we have 1s, 2p, 3d, 2s, 4/ levels, and so on, if the principal quantum number, which is equal to n + l, and l are used as the notation. It is interesting to point out that when V0 approaches infinity en(l) = (ßln/ro)2

where ¡iln is the nth root of the lth-order spherical Bessel function. In Table 3, it is shown that the different values of en(l) are obtained as r0 is equal to 1a*B and 2.5a*B, respectively. It is also shown that the values of quantum levels are different between infinite and finite barrier heights. The differences increase as the r0 and finite V0 decrease. There are an infinite number and a finite number of bound states for a spherical quantum dot with infinite and finite barrier heights, respectively. There is no bound state if r0 < Rc = 0.5w/(V0)1/2 [94]. However, the order of en(l) is the same for both infinite and finite barriers [i.e., the unique level sequence 1s, 1p(2p), 1s(3d), 2s, 1/(4/), and so on]. We should note that the level order is different between both cases of a spherical quantum dot and Coulomb field, in which the level order of an electron is 1s, 2s, 2p, 3s, 3p, 3d, and so on if the principal and orbital quantum numbers are used as the level notation. It is because of the lack of the deep attractive region in the vicinity of the center of a spherical quantum dot. For the motion of an electron in a Coulomb field, the quantum levels are only dependent on the principal quantum number np and degenerate with respect to both l (orbital quantum number) and m (magnetic quantum number). The total degree of degeneracy of a quantum level with np is equal to np (excluding spin degeneracy). For an electron in a spherical quantum dot, however,

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