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barrier. In Figure 12, it is shown that this decrease is rather slow for zt > L/2; for instance, a donor placed 150 A away from a GaAs-Ga0 7AsAl03As quantum well (L = 94.8 A) still binds a state by «0.5RJ which is «2.5 meV [70].

Figure 11. Dependence of the on-edge hydrogenic donor binding energy in GaAs-Ga1-xAlxAs quantum wells versus the GaAs slab thickness L. m* = 0.067m0. k = 13.1. V0 = 212 meV (curve 1); 318 meV (curve 2); 424 meV (curve 3); and infinite (curve 4). 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.

Bulk GaAs Donor ground state

Bulk GaAs Donor ground state

Number of GaAs monolayers

Figure 10. Dependence of on-center hydrogenic donor binding energy in GaAs-Ga1-xAlxAs quantum wells versus the GaAs slab thickness L. Vb(x) = 0.85 [sg(Ga1-xAlxAs) - sg(GaAs)]. One monolayer is 2.83 A thick. 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.

### Number of GaAs monolayers

Figure 10. Dependence of on-center hydrogenic donor binding energy in GaAs-Ga1-xAlxAs quantum wells versus the GaAs slab thickness L. Vb(x) = 0.85 [sg(Ga1-xAlxAs) - sg(GaAs)]. One monolayer is 2.83 A thick. 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 12. Dependence of the hydrogenic donor binding energy in a quantum well versus the impurity position zi (a) in the case of a finite barrier well (V0 = 318 meV) and (b) in the case of an infinite barrier well [69]. There is an interface at -zi = L/2. L = a*B = 94.8 A. 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 12. Dependence of the hydrogenic donor binding energy in a quantum well versus the impurity position zi (a) in the case of a finite barrier well (V0 = 318 meV) and (b) in the case of an infinite barrier well [69]. There is an interface at -zi = L/2. L = a*B = 94.8 A. 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.

3.1.3. Excited Subbands: Continuum

The procedure followed for the bound state attached to the e1 subband can be generalized for excited subbands e2,e3,... as well as for the quantum well continuum.

However, it becomes rapidly cumbersome since a correct variational procedure for excited states requires the trial wavefunctions to be orthogonal to all the states of lower energies. For separable wavefunctions [Eq. (105)] this requirement is automatically fulfilled and one may safely minimize eVo (zi, A) to obtain a lower bound of ebvo (z). The new feature associated with the bound states attached to excited subbands is their finite lifetime. This is due to their degeneracy with the two-dimensional continua of the lower lying subbands. This effect, however, is not very large [67, 68] since it arises from the intersubband coupling, which is induced by the Coulombic potential: if the decoupling procedure is valid the lifetime of the quasi-bound state calculated with Eq. (105) should be long.

For impurities located in the barriers (an important practical topic with regard to the modulation-doping technique), we have seen that they weakly bind a state below the e1 edge. There exists (at least) a second quasi-discrete level attached to the barrier edge (i.e., with energy ~ V0 - R*). This state is reminiscent of the hydrogenic ground state of the bulk barrier. Due to the presence of the quantum well slab it becomes a resonant state since it interferes with the two-dimensional continua of the quantum well subbands v(ev < V0). Its lifetime t can be calculated using the expression n

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