Tight-Binding Calculations of the Optical Response of Optimally P-Doped Si Nanocrystals: A Model for Localized Surface Plasmon Resonance

We present tight-binding calculations in the random-phase approximation of the optical response of Silicon nanocrystals (Si NCs) ideally doped with large concentrations of phosphorus (P) atoms. A collective response of P-induced electrons is demonstrated, leading to localized surface plasmon resonance (LSPR) when a Si NC contains more than $\approx 10\mathrm{P}$ atoms. The LSPR energy varies not only with doping concentration but also with NC size due to size-dependent screening by valence electrons. The simple Drude-like behavior is recovered for NC size above 4 nm. Si NCs containing a large number of deep defects in place of hydrogenic impurities do not give rise to LSPR.

Figure 1

Energy of the absorption peak with the highest intensity versus doping concentration for a Si NC. The diameter ($d=2R$) of the NC is 1.8 (plus), 2.2 (cross), 3.2 (star), or 4.0 nm (square). The solid line represents the LSPR energy predicted by Eq. (1) with $m_e^{\mathrm{Si}}=0.3m_0$, ${\epsilon }_b^{\mathrm{Si}}=11.7$ (bulk Si values) and ${\epsilon }_m=1$.

Figure 2

Absorption spectra [$\mathrm{Im}\mathbf{(}\alpha (E)/R^3\mathbf{)}$] of (a) 1.8 nm Si NCs (${\mathrm{Si}}_{87}$) doped with 0, 2, 8, 16, and 26 P atoms. (b) Same for 2.0 nm Si NCs (${\mathrm{Si}}_{175}$) doped with 0, 4, 9, 18, and 52 P atoms. The corresponding doping levels are also indicated.

Figure 3

(a) Absorption spectra [$\mathrm{Im}\mathbf{(}\alpha (E)/R^3\mathbf{)}$] of 3.2 nm Si NCs (${\mathrm{Si}}_{573}$) doped with 0, 2, 10, 28, 58, 100, and 140 P atoms. The corresponding doping levels are also indicated. The dotted lines show the spectra obtained by using the single-particle response instead of the self-consistent one, i.e., using $f_{\mathrm{lf}}^2{\chi }^0$ instead of ${\chi }^{\mathrm{RPA}}$. (b) Energy position of the high (circle) and low (square) energy Gaussian peak obtained from the decomposition of the absorption spectrum of a 3.2 nm Si NC containing $\ge 10\mathrm{P}$ atoms. The symbol size is proportional to the amplitude of the Gaussian peak. Solid line: $\hslash {\omega }_{\mathrm{sp}}$ predicted by Eq. (1) with $m_e^{\mathrm{Si}}=0.3m_0$, ${\epsilon }_b^{\mathrm{Si}}=11.7$, and ${\epsilon }_m=1$. Dashed line: Same but with ${\epsilon }_b$ replaced by ${\epsilon }^{\mathrm{Si}}(R)=7.7$ [26].

Figure 4

Absorption spectrum [$\mathrm{Im}\mathbf{(}\alpha (E)/R^3\mathbf{)}$] of a 3.2 nm Si NC containing 58 Si dangling bonds (corresponding concentration of 10.1%) calculated in the TB-RPA (solid line) or TB-IPA (dashed line).