$D^{-}$ centers in uniaxially stressed Si and in $\text{Si}/{\text{SiO}}_2$ quantum wells

The spin-singlet ground states of a ${\text{D}}^{-}$ ion both in uniaxially stressed Si and in $\text{Si}/{\text{SiO}}_2$ quantum wells have been investigated for two types of donors, substitutional P and interstitial Li atoms, using a diffusion quantum Monte Carlo method. The valley-orbit interaction due to the singular donor ion potential is taken into account by diagonalizing the multiplet Hamiltonian matrix in the basis set of the ground-state wave functions assigned by the valley indexes. In the uniaxial compressive stress along the [100] direction, the binding energy of a negative P donor with ${\text{A}}_1$ symmetry decreases at first and then increases gradually with the stress-induced splitting in the valley energy. This nonmonotonic dependence is attributed to the disparity between the population probability in the stress-deepened valleys for the neutral donor and that for the negative donor. As for Li donors with E symmetry, the binding energy of a negative donor decreases linearly with the stress-induced splitting and then takes a constant value, caused by the transition of the ground state of the negative donor from the intervalley configuration to the intravalley configuration. These calculated behaviors agree with the experiment. In the quantum well in the [100] direction, the quantum confinement effect deepens the binding energy of a negative donor without the valley-orbit interaction. The same confinement effect is predicted for a Li negative donor ground state with E symmetry, and the binding energy increases monotonically with decrease in the well width. As for a P negative donor with ${\text{A}}_1$ symmetry, the enhanced binding energy is recovered at the well width of 5 nm although the valley-orbit interaction suppresses the enhancement of the binding energy in wider wells.

Figure 1

The schematic figure of a $\text{Si}/{\text{SiO}}_2$ quantum well and the equal energy surfaces in $k$ space in the vicinity of the conduction-band minima of the bulk Si.

Figure 2

(a) The ground-state energies $E_{\text{G}}\left({\text{D}}^0\right)$ of a neutral P donor, (b) the ground-state energies $E_{\text{G}}\left({\text{D}}^{-}\right)$ of a negative P donor, (c) the ground-state energies $E_{\text{G}}\left({\text{D}}^0\right)$ of a neutral Li donor, and (d) the ground-state energies $E_{\text{G}}\left({\text{D}}^{-}\right)$ of a negative Li donor with the VO interaction, as a function of stress-induced splitting $S$. The values calculated in the effective-mass approximation without the VO interaction are indicated by arrow heads (the solid arrow head for the intravalley configuration and the open arrow head for the intervalley configuration of a negative donor).

Figure 3

The binding energies $E_{\text{B}}\left({\text{D}}^{-}\right)$ (a thick solid line) of a negative donor for (a) P and (b) Li, as a function of stress-induced splitting $S$, with their experimental values (Ref. 23) (solid circles). Here, calculated result by Larsen (Ref. 25) is shown by a thin solid line and those by Oliveira (Ref. 26) are shown by a thin broken line (VO interaction for the inner orbit) and a dashed-dotted line (VO interaction for both the inner and the outer orbits), respectively. The arrow heads on the left and the right vertical axes indicate the calculated binding energy at zero stress and in the high stress limit, respectively.

Figure 4

The valley population probability of a neutral donor (broken line) and a negative donor (solid line) for Si:P, as a function of the stress-induced splitting $S$. Here, thick lines present the valley population probability for the stress-deepened valleys (1) and (2) and thin lines do those in other valleys, (3)–(6).

Figure 5

The ground-state energies $E_{\text{G}}\left({\text{D}}^{-}\right)$ of a Li negative donor in a magnified scale of Fig. 2d in the low stress region, as a function of stress-induced splitting $S$. The values calculated in the effective-mass approximation without the VO interaction are indicated by arrow heads (the solid arrow head for the intravalley configuration and the open arrow head for the intervalley configuration of a negative donor).

Figure 6

(a) The ground-state energies $E_{\text{G}}\left({\text{D}}^0\right)$ of a neutral donor without the VO interaction, in addition to the free-electron ground-state energies in the valley (1) (a solid line) and in the valley (3) (a broken line), (b) the ground-state energies $E_{\text{G}}\left({\text{D}}^{-}\right)$ of a negative donor with their bulk values (arrow heads) without the VO interaction, (c) the ground-state energies of a neutral P donor, and (d) the ground-state energies of a negative P donor with the bulk value of the lowest-energy state (solid lines), in the $\text{Si}/{\text{SiO}}_2$ quantum well. In (c) and (d), the lowest energy level is represented by the solid circle.

Figure 7

The binding energies $E_{\text{B}}\left({\text{D}}^{-}\right)$ of a negative donor of P and Li impurities in the $\text{Si}/{\text{SiO}}_2$ quantum well as a function of the well width, with the bulk value for P (solid line). The binding energy in the effective-mass approximation without the VO interaction is also plotted. The binding energies for the intervalley and intravalley configurations without the VO interaction in the bulk are also indicated by open and solid arrow heads, respectively. The $\times$ on the left axis indicates the calculated binding energy without the VO interaction for the two-dimensional electrons.

Figure 8

The valley population probability for a P neutral donor and a P negative donor in a $\text{Si}/{\text{SiO}}_2$ quantum well, as a function of the well width. The arrow head indicates the bulk value of 1/3.

Figure 9

(a) The single-electron probability amplitude $P\left(0\right)$ at the origin calculated in the effective-mass approximation without the VO interaction and (b) $V_{0}P\left(0\right)$, for a P neutral donor and a P negative donor in the $\text{Si}/{\text{SiO}}_2$ quantum well, as a function of the well width. In (a), the dotted, broken, and solid lines indicate the bulk values for a neutral donor, a negative donor with the intervalley configuration, and that with the intravalley configuration, respectively.