## E

where p is the 1s bulk hydrogenic wavefunction of the barrier. If the distance d separating the impurity from the quantum well edge is much larger than a*B, Eq. (112) simplifies to

where Pv is the total integrated probability of finding the carrier in the vth state (energy ev) in any of the two barriers of the quantum well. One sees from Eq. (113) that the lifetime t is strongly dominated by the escape processes to the excited subband ev whose energy is nearest to V0 - R*. As an example let us take a GaAs-GaAlAs quantum well with thickness L = 50 A, V0 = 0.2 eV and assume that d = 3a*B (i.e., d « 300 A). We then get t « 3 x 10-6 s. The quasi-bound state can thus be considered, to a reasonable approximation, as stationary (H/t ~ 4 x 10-8R*).

3.1.4. Excited Impurity Levels Attached to the Subband

The Schrodinger equation [Eq. (102)] has several bound states below e1; their binding energies have been calculated by several groups [72, 73]. The calculated energy difference between the on-center donor ground state (quasi 1s) and the excited states (quasi 2px, 2py) agrees with the far-infrared absorption and magnetoabsorption data [74]. On-edge donor levels have also been investigated.

### 3.1.5. Acceptor Levels in a Quantum Well

The problem of acceptor levels in semiconductor quantum wells is much more intricate than the equivalent donor problem. This is due to the degenerate nature of the valence bands in cubic semiconductors. In quantum wells this degeneracy is lifted (light and heavy holes have different confinement energies).

However, the energy separation between the heavy and light hole subbands is seldom comparable to the bulk acceptor binding energies. Thus many subbands are coupled by the combined actions of the Coulombic potential and the quantum well confining barrier potential. No simple decoupling procedures appear manageable.

Masselink et al. [75] used variational calculations to estimate the binding energy of the acceptor level due to carbon (a well-known residual impurity in MBE grown GaAs layers). Their results, shown in Figure 13, agree remarkably with the experiments of Miller et al. [76]. One notices in Figure 13 the same trends versus quantum well thickness as displayed by Coulombic donors (Figs. 10 and 11). The binding energy first increases when L decreases (increasing

Figure 13. Dependence of the on-center carbon binding energy versus well width in GaAs-Ga1-xAlxAs quantum wells. Vb(x) = 0.15 [ss(Ga1-xAlxAs) - ss(GaAs)] is the assumed hole confining barrier height. The open circles are the experimental values obtained by Miller et al. [76]. 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.

Figure 13. Dependence of the on-center carbon binding energy versus well width in GaAs-Ga1-xAlxAs quantum wells. Vb(x) = 0.15 [ss(Ga1-xAlxAs) - ss(GaAs)] is the assumed hole confining barrier height. The open circles are the experimental values obtained by Miller et al. [76]. 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.

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