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for

- L < z < L

The rectangular quantum well wire behavior is, according to the x, y directions, as two isolated single quantum wells under the action of an infinite barrier potential, where the boundary conditions are simply Xf'B(a) = 0 and YA'B(b) = 0. With this requirement the eigenfunctions are

The energy levels for the nth band are

w2h2 ,2 w2h2 2

2mAa2 2mBb2

For circular symmetry the differential equation for the envelope function is

2mB h2

The upper signs in Eq. (66) correspond to material A (we are considering that the potential VA is zero) and the lower signs correspond to material B, where the interface potential is VBB = V0. Equation (66) is the Bessel differential equation (+' —) and modified Bessel differential equation (—' +) respectively.

Integrating Eq. (66) for each material we obtain the Bessel function of the first kind, J(aAr), and the modified Bessel function, K(aBr), respectively; these functions and their derivatives must satisfy the continuity requirement for Rt in the r = r0 interfa section radius. That is,

R, in the r = r0 interface, where r0 is confinement circular aJ,(aAr0) = bKl(aBr0) and

where a, b are constants which are determined using the normalization condition for the wavefunction ft(0). Then we can calculate the energy levels for each n band solving the following transcendental equation;

Jl(aAr0) m*BKl(aBr0)

2.5.3. Quasi-Zero-Dimensional System

In this case the electron is totally or partially confined in the three spatial directions and since there is no direction for the free movement, the total energy does not depend on any of reciprocal space k wavevector components. Nor can we define, in a mathematical sense, the effective mass as a function of kl . However, since the electron still is under the action of the crystalline field, we can use the concept of effective mass as the change of its properties, as well as the effective mass scheme.

Therefore, if we continue with the final result obtained from the envelope function method or the effective mass approximation, the wavefunction can be written as

where we are assuming that the quasi-zero-dimensional system is a spherical quantum dot formed by two materials A and B, where the potential V0 is given by

0 0 < r < r0 (A material) V0 r > r0 (B material)

Taking into account the problem symmetry, it is reasonable to consider the spherical coordinates (r, 6, p). In these coordinates the Hamiltonian can be written as

where the L2 operator is

0 0

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