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Figure 14. Energy of the ground and 2pz states of the hydrogenic impurity confined in a penetrable quantum well wire with the nucleus on the axis as a function of wire radius and potential barrier heights. Reprinted with permission from [205], J. L. Marín et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

with r the position vector of the electron relative to the origin on the wire axis. Then the suitable Hamiltonian is

where Vb(p) is the same as that given by Eq. (115).

The trial wavefunction for the ground state energy can then be written as t1s =

ti = A(Po - aP) exp(-ar') 0 S P S Po to = B exp(-ar') exp(-ßP) Po S P < o

where, as before, A and B are normalization constants and a, 3 are the variational parameters involved in the calculation. Again, a, 3 and A, B are, respectively, related by Eqs. (121) and (122).

When the nucleus of hydrogenic impurity is off the symmetry axis, it is not possible to find the wavefunctions, for the 2px and 2py states, that satisfy the orthonormality condition as is required by the method. Therefore we will restrict to the 2pz state only. The suitable Hamiltonian is the same that given by Eq. (116), and the variational wavefunction is given by i fa = C(P0-yp)exp(-yr')z 0<p<P0 = 1 (132)

where, as before, C, D are normalization constants and y, £ are the variational parameters involved in the calculation. Again, y, £ and C, D are, respectively, related by Eqs. (126) and (127).

In Figure 16, we show the energy of the ground and 2pz states as a function of (b/p0) for p0 = 1.0 Bohr and several heights of potential barrier.

In Figure 17a, we display the variation of the hydrogenic impurity binding energy sb as a function of (b/p0) for p0 = 1.0 Bohr and several heights of potential barrier.

3.2.3. Optical Properties

To predict the absorption peak due to 1s-2p transitions as a function of the wire radius and confining degree, we have calculated the transition energy between these states as well as their oscillator strengths.

The f0 oscillator strength is defined as f0 = 3\(1s\l-0 |2p)|2 Ae

where er is the incident light polarization vector and As = s2p - s1s is the transition energy between 2p and 1s states.

The 1s-2pz oscillator strength f0 (for er || z) is shown in Figure 18 as a function of wire radius for several heights of potential barrier, when the nucleus is fixed on the center of the wire. For a given value of the finite barrier potential the oscillator strength increases from 0.139, the characteristic value of the "free" hydrogen atom, as the wire radius is reduced.

Finally, in Figure 19, we show the oscillator strength f0 (for er || z) as a function of relative nucleus position (b/p0)

0 0

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