Single-particle states in spherical $\mathrm{Si}∕\mathrm{Si}{\mathrm{O}}_2$ quantum dots

We calculate the ground and excited electron and hole levels in spherical Si quantum dots inside $\mathrm{Si}{\mathrm{O}}_2$ in a multiband effective mass approximation. The Luttinger Hamiltonian is used for holes, and the strong anisotropy of the conduction electron effective mass in Si is taken into account. As the boundary conditions for the electron and hole wave functions, we use the continuity of the wave functions and the flux at the boundaries of the quantum dots.

Figure 1

(Color online) Dependence of positions of the electron energy levels above the bottom of the conduction band of bulk Si on the quantum dot diameter. The numbers near the lines indicate the total degeneracy (including the spin degeneracy) of the corresponding levels.

Figure 2

(Color online) Electron wave function dependence on the cylindrical coordinates (a) $z$ and (b) $\rho$ for the lowest five electron levels of a quantum dot with diameter of $2\mathrm{nm}$. The wave function ${\xi }_4^e$ of the fourth level with magnetic number $m=\pm 1$ has also an angular dependence $e^{im\varphi }$.

Figure 3

(Color online) Dependence of the positions of the hole energy levels below the top of the valence band of bulk Si on the quantum dot diameter. The numbers near the lines indicate the total degeneracy (including the spin degeneracy) of the corresponding levels. The types of the levels and values of the total angular momentum $F$ are also shown.

Figure 4

(Color online) Dependence of the ground-state electron-hole recombination energy as a function of diameter of nanocrystal (solid line). The dashed line shows the same energy without taking into account the exciton shift. For comparison, the experimental data obtained from photoluminescence spectra (Refs. 37, 38, 39, 40), measured for Si nanocrystals inside $\mathrm{Si}{\mathrm{O}}_2$, are presented.

Figure 5

(Color online) Probabilities $P_{r,gr}$ of radiative transitions between the ground electron and hole states assisted by emission of a TO phonon (dashed line), an LO phonon (dash-dot line), and their sum (thick solid line) as well as the probability of direct (zero phonon) transition (dot line), as functions of nanocrystal diameter. Experimental points (Ref. 40) are shown as well.