Ionization energy of donor and acceptor impurities in semiconductor nanowires: Importance of dielectric confinement

Calculations of the electronic states of donor and acceptor impurities in nanowires show that the ionization energy of the impurities is strongly enhanced with respect to the bulk, above all when the wires are embedded in a material with a low dielectric constant. In free-standing nanowires with diameter below $10\mathrm{nm}$, the ionization of the impurities at $300\mathrm{K}$ is strongly reduced and heavy doping is necessary to obtain conductive systems. These results imply that the critical density for metal-nonmetal transitions is not the same as in the bulk. Experiments are proposed to test the predictions.

Figure 1

Ionization energy $E_{\mathrm{I}}$ versus the wire radius $R$ for donor impurities ($\mathrm{P}=+$, $\mathrm{As}=\times$, $\mathrm{Sb}=●$) located along the axis of a free-standing SiNW $({ϵ}_{\text{out}}=1)$. The bulk values $E_{\mathrm{I}}^0$ are equal to $45\mathrm{meV}$ for P, $54\mathrm{meV}$ for As, and $39\mathrm{meV}$ for Sb. The solid curve represents Eq. (3) for a P impurity. Dashed line: shift of the conduction band edge due to quantum confinement. Doping efficiency (◆): probability of ionization towards the conduction band for a P impurity $\left(300\mathrm{K}\right)$.

Figure 2

Same as Fig. 1 but for ${ϵ}_{\text{out}}={ϵ}_{\text{in}}$ ($\mathrm{P}=+$, $\mathrm{As}=\times$, $\mathrm{Sb}=•$). Curves: fit of the TB values with $E_{\mathrm{I}}^0+a{(r_{\mathrm{B}}∕R)}^b$ (P, $a=57.33\mathrm{meV}$, $b=1.47$; As, $a=86.32\mathrm{meV}$, $b=1.55$; Sb, $a=40.60\mathrm{meV}$, $b=1.37$). Dashed line: shift of the conduction band edge due to quantum confinement.

Figure 3

Average extensions $\sqrt{⟨z^2⟩}$ (☉) and $\sqrt{⟨x^2⟩}$ (•) of the bound state of a P impurity located along the axis $z$ of a free-standing SiNW $({ϵ}_{\text{out}}=1)$ as a function of its radius $R$. The horizontal line represents the bulk limit $(r_{\mathrm{B}}∕\sqrt{3})$ equal to $1.55\mathrm{nm}$. Weight ${\mid \Psi \left(0\right)\mid }^2$ of the wave function on the P atom, the bulk value being equal to 0.014 (∎).

Figure 4

Ionization energy versus the radius $R$ for a P impurity located at different distances $r=0$ (+), $r=0.25R$ (×), $r=0.5R$ (☉), $r=0.75R$ (Δ) from the axis of a free-standing SiNW $({ϵ}_{\text{out}}=1)$. The symbols ▿ correspond to an impurity located at a fixed distance $\left(0.68\mathrm{\AA }\right)$ from the surface.

Figure 5

Ionization energy versus the wire radius $R$ for a B impurity (acceptor) located along the axis of a SiNW for ${ϵ}_{\text{out}}=1$ (☉, left axis) and for ${ϵ}_{\text{out}}={ϵ}_{\text{in}}$ (▿, right axis). The solid curve corresponds to [Eq. (3)] for $E_{\mathrm{I}}^0=45\mathrm{meV}$ and ${ϵ}_{\text{out}}=1$.