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Figure 32. The binding energy of an on-center hydrogenic impurity as a function of the quantum dot radius for different magnetic fields. 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 32. The binding energy of an on-center hydrogenic impurity as a function of the quantum dot radius for different magnetic fields. 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.

curves deviate from each other reaching steady values as the dot radius increases. For weak spatial confinement (r0 ^ a*B), the binding energy converges asymptotically to the corresponding bulk values. In the limit of large dot radius, the binding energy for B = 0 approaches its bulk value sb = 1R*, as expected. An increase of the field strength decreases the magnetic length as compared with the dot radius; thus in this case the binding energy increases due to magnetic confinement that compels the electron to move "closer" to the on-center impurity.

Figure 33 shows the behavior of binding energy as a function of field strength for different dot radii of an on-center impurity. The variation of binding energy as the field increases is less sensitive as dot radius decreases, thus enforcing the previous comment regarding Figure 32.

The results for the asymmetric case (off-center impurity) are plotted in Figures 34 and 35 with zero magnetic field. Figure 34 shows the ground state energy for different sizes of quantum dot versus the relative position of the off-center impurity (a/r0). The ground state energy increases from its on-center impurity value (a/r0 = 0) as the impurity is shifted off the center; this is a general behavior. For larger radii the ground state energy becomes lower, and then, for closer distances to the wall, evolves from the 1s "free" hydrogen ground state (-1/2 Hartree) to the 2pz "free" hydrogen first excited state (-1/8 Hartree). These results are in good agreement with those of Brownstein [86], Gorecki and Byers Brown [85], and Marín et al. [65]. It is clear now that the trial wavefunction as chosen is flexible enough to describe the case when the size of the quantum dot becomes infinite and the nucleus is close to the surface. The latter corresponds to the case of a hydrogenic impurity close to a plane. This situation will be discussed later when the effect of a magnetic field on the ground state of a hydrogenic impurity near a semiconductor surface will be considered.

In Figure 35, the binding energy for different sizes of quantum dot versus the relative position of the off-center impurity (a/r0) was plotted. The binding energy decreases as the impurity approaches the boundary of a quantum dot. This behavior is due to the fact that wavefunctions vanish at the boundaries and thus their contributions to the binding energy of a quantum dot with an off-center impurity are smaller than a quantum dot with an on-center impurity. The results are similar to the case of a hydrogenic impurity placed in rectangular [81] or circular [77] cross-section quantum well wires, and in a spherical quantum dot [101].

The problem of a hydrogenic impurity near a crystal surface as a limit case of a hydrogenic impurity in a spherical quantum dot of very large radius will be considered next.

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