Fermi-level oscillation in $n$-type $\delta$-doped $\mathrm{Si}$: A self-consistent tight-binding approach

We have used the empirical tight-binding method within the antibonding orbital model to compute the self-consistent potential profile and Fermi level position in $n$-type $\delta$-doped $\mathrm{Si}$. This model describes the six valleys in the $\mathrm{Si}$ conduction band adequately. We include exchange-correlation effects under the local density approximation. The comparison of our results to empirical pseudopotential calculations shows very good agreement, while effective mass approximation calculations agree only in the low doping regime. At ultra high densities, an oscillatory behavior of the Fermi-level position as a function of the doping concentration is predicted.

Figure 1

Minimum of the self-consistent potential for a $n$-type $\delta$ layer. The full width half maximum (FWHM) of the donor distribution is $16.0\mathrm{\AA }$. The inset illustrates several symbols used in the text (all quantities are negative except for $E_F$) for a potential profile corresponding to a carrier concentration of $9.82\times {10}^{-3}{\mathrm{\AA }}^{-2}$.

Figure 2

Position of $\mu$ as a function of the doping for a FWHM of the dopant distribution of $16.0\mathrm{\AA }$ with (solid) and without (dashed) exchange-correlation terms, at $0\mathrm{K}$.

Figure 3

Position of $\mu$ as a function of the doping for structures with different separation between the $\delta$ layers.

Figure 4

Position of $\mu$ as a function of the doping for structures with different FWHM of the dopant distribution.

Figure 5

Position of $\mu$ as a function of the doping for different temperatures and a FWHM of the dopant distribution of $19.2\mathrm{\AA }$.

Figure 6

Position of $\mu$ as a function of the doping for ABOM and two implementations of EMA (see text for details).

Figure 7

Comparison of ABOM with empirical pseudopotential method (EPM) results (from Ref. 23). The open circle labels results including short range Coulomb interaction effects, while full squares do not include this correction.