: dx + ev and the binding energy is ebva(zi) = ev„ _ Min e (zi, Ä)

One should be aware that the decoupling procedure [Eqs. (105)—(110)] runs into difficulties in the limits L ^ x and L ^ 0. In the former case the energy difference £V0+i — eV0 becomes very small and many subbands become admixed by the Coulombic potential. Clearly, for infinite L one cannot describe the bulk is hydrogenic bound state by a separable wavefunction. In the latter case (L ^ 0) one finds similar problems due to the energy proximity between the ground quantum well subband e1 and the top of the well V0. Consequently Xj_(z) leaks more and more heavily in the barrier. At L = 0 the only sensible result is to find a 1s bulk hydrogenic wavefunction corresponding to the barrier-acting material (in this model the latter has a binding energy equal to Ry). Once again this state cannot have a wavefunction like that of Eq. (105). If V0 is infinite the result at L = 0 is qualitatively different. The quantum well only has bound levels whose energy separation increases like L—2 when L decreases. The smaller the well thickness, the better the separable wavefunction becomes. At L = 0, one obtains a true two-dimensional hydrogenic problem whose binding energy is 4R* whereas A = a*B/2.

To circumvent the previous difficulties and obtain the exact limits at L = 0 and L = x for any V0 one may use [69-71]

where N is a normalization constant, A is the variational parameter, and attention is focused on the ground bound state attached to the ground quantum well subband 61. Calculations are less simple than with the separable wavefunc-tion [Eq. (105)]. Comparing the binding energy deduced from Eqs. (105)-(111) one finds, for infinite V0, that the separable wavefunction gives almost the same results as the nonseparable wavefunction if L/a*B < 3. This is the range where for most materials the quantum size effects are important.

Other variational calculations have been proposed [72,73]. For example, instead of using a nonlinear variational parameter one uses a finite basis set of fixed wavefunctions (often Gaussian ones) in which H is numerically diagonal-ized. The numerical results obtained by using a single nonlinear variational parameter compare favorably with these very accurate treatments.

3.1.2. Results for the Ground Impurity State Attached to the Ground Subband

Figures 9 to 11 give a sample of some calculated results for the ground impurity state attached to the ground subband. Two parameters control the binding energy:

(i) Thickness dependence of the impurity binding energy: the dimensionless ratio L/aB indicates the amount of two-dimensionality of the impurity state. If L/a* > 3 or L/a*B < 0.2 and V0 ~ 3 eV in GaAs-Ga1—xAlxAs the problem is almost three-dimensional. This is either because the subbands are too close (L/a*B > 3) or because the quantum well continuum is too close (L/a*B < 0.2). The on-center donor binding energy increases from R*y(L ^ x) to reach a maximum (L/aB < 1) whose exact L location and amplitude depend on Vb. Finally it decreases to the value Ry at L = 0 [72, 73]. If V0 is infinite, the maximum is only reached at L = 0 and has a value of 4R* [69].

(ii) Position dependence of the impurity binding energy: the impurity binding energy monotonically decreases when the impurity location zt moves from the center to the edge of the well and finally deep into the

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