Tight-binding calculation of the optical absorption cross section of spherical and ellipsoidal silicon nanocrystals

Electronic and optical properties of silicon nanocrystals are calculated and discussed within a semiempirical tight-binding approach, which allows to study systems composed of thousands of atoms. Oscillator strengths, frequency-dependent optical absorption cross sections, and static dielectric constants are investigated for both spherical and ellipsoidal nanocrystals, with the aim of pointing out their size- and shape-dependent features. We show that the anisotropy of the optical functions follows the nanocrystal shape, and a comparison is discussed between very elongated structures and quantum wires.

Figure 1

(Color online) Imaginary part of the bulk silicon dielectric function. The empty circles are experimental data (Ref. 42); the solid and the dot-dashed lines are the TB curves calculated using, respectively, Tserbak (Ref. 16) and Niquet (Ref. 15) parameters; the dashed line refers to a local empirical pseudopotential calculation performed with the Chelikowsky-Cohen (Ref. 41) form factors.

Figure 2

Extinction coefficient for a Si nanocrystal having the diameter of about $1.8\mathrm{nm}$, obtained from absorption measurements (Ref. 50) (dashed line), compared to the ${\mathrm{Si}}_{159}{\mathrm{H}}_{124}$ $(d=1.83\mathrm{nm})$ extinction coefficient, calculated within our TB method (solid line). The arrow labels the first calculated transition energy. Our theoretical data have been rescaled in such a way that the two curves have the same height of the main peak.

Figure 3

Absorption cross section for a set of spherical nanocrystals. A $0.1\mathrm{eV}$ Gaussian broadening has been used. The solid arrows show the first dipole-allowed transition (HOMO-LUMO gap); the dashed arrows point to the absorption gap, as defined in the text. For the first two structures, the first allowed transition and the absorption gap have the same value. Calculations have been performed for structures with increasing diameter.

Figure 4

(Color online) $k$-space projection of the spherical nanocrystals eigenstates on the bulk Si band complex. The curves centered on $\Gamma$ and $X$ are, respectively, the projections of the HOMO and LUMO nanocrystal states. The normalization has been done in such a way that the HOMO states projection is 1 at $\Gamma$.

Figure 5

(Color online) Oscillator strengths (left) and electron-hole radiative recombination times (right) for the lowest-energy transitions, calculated for a set of spherical silicon nanocrystals. The results shown in figure refer to the following nanocrystals: ${\mathrm{Si}}_{29}{\mathrm{H}}_{36}$ $(d=1\mathrm{nm})$, squares; ${\mathrm{Si}}_{87}{\mathrm{H}}_{76}$ $(d=1.5\mathrm{nm})$, circles; ${\mathrm{Si}}_{191}{\mathrm{H}}_{148}$ $(d=1.9\mathrm{nm})$, triangles-up; ${\mathrm{Si}}_{465}{\mathrm{H}}_{252}$ $(d=2.6\mathrm{nm})$, triangles-right; ${\mathrm{Si}}_{705}{\mathrm{H}}_{300}$ $(d=3\mathrm{nm})$, diamonds.

Figure 6

Calculated spherical nanocrystal static dielectric constant. The circles correspond to our TB results (the line is only a guide for the eyes). The solid line is the pseudopotential result from Ref. 34, while the dot-dashed line is the TB calculation of the screening average dielectric constant of Ref. 54. The data from Ref. 34 have been taken without any rescaling (see the text).

Figure 7

(Color online) Absorption cross section for a set of Si ellipsoids with $a=1\mathrm{nm}$, as a function of the aspect ratio $\chi$. The solid and the dashed curves refer, respectively, to the perpendicular and the parallel polarization. The dotted arrow is the first transition energy, which is almost insensitive to the polarization. The solid and dashed arrows point, respectively, to the perpendicular and parallel absorption thresholds. A $0.1\mathrm{eV}$ Gaussian broadening has been used.

Figure 8

Comparison between an ellipsoid with $a=1\mathrm{nm}$, $\chi =3$ (dashed lines), and a [001] cylindrical quantum wire having a radius $r=1\mathrm{nm}$ (solid lines). The absorption cross section is shown for both the perpendicular (upper panel) and the parallel (lower panel) polarization. The arrows point to the first dipole-allowed transition, whereas the lines point to the absorption threshold.

Figure 9

(Color online) Degree of linear polarization $\rho$ for the ellipsoid frequency-dependent absorption cross section, defined in Eq. (12), plotted for $a=1\mathrm{nm}$ and different values of the aspect ratio: $\chi =0.5$, solid line; $\chi =0.6$, double dot-dashed line; $\chi =1.8$, dot-dashed line; $\chi =3.0$, dashed line.

Figure 10

(Color online) Left panel: Principal components of the static dielectric tensor for a set of ellipsoids with a fixed semiaxis $a=1\mathrm{nm}$. The circles and the squares are, respectively, the perpendicular and the parallel component with respect to the nanocrystal symmetry axis. Right panel: The static dielectric tensor degree of linear polarization. The lines are guides for the eyes.