Temperature dependence of the optical properties of silicon nanocrystals

Silicon nanocrystals (SiNCs) have been under active investigation in the last decades and have been considered as a promising candidate for many optoelectronic applications including highly efficient solar cells. Some of the fundamental properties of interest in these nanostructures is the temperature dependence of their optical absorption onset and how this is controlled by different passivation regimes. In the present work we employ first-principles calculations in conjunction with the special displacement method to study the temperature dependence of the band gap renormalization of freestanding hydrogen-terminated, and oxidized SiNCs, as well as matrix-embedded SiNCs in amorphous silica, and we obtain good agreement with experimental photoluminescence data. We also provide strong evidence that the electron-phonon interplay at the surface of the nanocrystal is suppressed by oxidation and the surrounding amorphous matrix. For the matrix-embedded SiNCs, we show a high correlation between the temperature dependence of the band gap and the Si-Si strained bonds. This result emphasizes the immanent relationship of electron-phonon coupling and thermal structural distortions. We also demonstrate that, apart from quantum confinement, Si-Si strained bonds are the major cause of zero-phonon quasidirect transitions in matrix-embedded SiNCs. As a final point, we clarify that, unlike optical absorption in bulk Si, phonon-assisted electronic transitions play a secondary role in SiNCs.

Figure 1

Structures of (a) H-terminated freestanding (FS) and (b) matrix-embedded (ME) SiNCs with diameter $d=2$ nm. Si atoms in the nanocrystals and passivating H atoms are shown in green and white. Matrix Si and O atoms are colored blue and gray, respectively.

Figure 2

(a) Band gap of silicon nanocrystals (SiNCs) as a function of average diameter $d$. Calculated band gaps of H-terminated (${\mathrm{Si}}_n{\mathrm{H}}_m$), oxidized SiNCs (${\mathrm{Si}}_n{\mathrm{OH}}_m$), and matrix-embedded $({\mathrm{Si}}_n/\mathrm{a}\text{−}{\mathrm{SiO}}_2)$ SiNCs are shown as red discs, green discs, and blue squares, respectively. The solid curves are fits of the form $E_0+a/d^b$ to the calculated band gaps of H-terminated SiNCs (red) and oxidized SiNCs (green) [67]. The dashed curves represent vertically shifted theoretical fits such that $E_0$ corresponds to the GW corrected band gap of bulk Si [70]. Experimental data of matrix-embedded SiNCs in a-${\mathrm{SiO}}_2$ (black squares) and freestanding SiNCs with oxidized surface (black discs) are from Refs. [22] and [65]. (b) Decomposition of the electronic density of states (EDOS) of the matrix-embedded SiNC into contributions from Si core atoms (black solid line), passivated surface Si atoms (black dashed dotted line), O matrix atoms (gray solid line), and Si matrix atoms (black dashed line). The Fermi level is set at the middle of the gap. A Gaussian broadening of 70 meV was used.

Figure 3

(a) Temperature dependence of the band gap renormalization of freestanding (FS) and matrix-embedded (ME) SiNCs up to 350 K. Calculated band gaps using the ZG displacement [26] for H-terminated (${\mathrm{Si}}_{217}{\mathrm{H}}_{150}$), oxidized (${\mathrm{Si}}_{217}{\mathrm{O}}_7{\mathrm{H}}_{136}$) and matrix-embedded (${\mathrm{Si}}_{215}/\mathrm{a}\text{−}{\mathrm{SiO}}_2$) SiNCs are shown as red discs, green discs and blue squares, respectively. The diameter of the nanocrystals employed for our calculations is 2 nm. Magenta squares represent the band gaps of matrix-embedded SiNCs including also the effect of thermal expansion (TE). Orange diamonds represent data of bulk Si reported in Ref. [27]. Experimental data of oxidized freestanding SiNCs (black discs) is from Ref. [23] and of matrix-embedded SiNCs in a-${\mathrm{SiO}}_2$ (black squares) is from Refs. [20, 21, 22, 73]. All curves represent fits to Eq. (11). (b) Variation of the zero-point renormalization of the matrix-embedded SiNC with the number of deterministic ZG configurations. The horizontal dashed line indicates the zero-point renormalization of 86 meV calculated using only one antithetic pair of ZG configurations. (c) Band gap renormalization versus volume expansion of H-terminated SiNCs (red discs) and matrix-embedded SiNCs in a-${\mathrm{SiO}}_2$ (blue squares). The vertical dashed line indicates the volume expansion at 300 K. (d) Comparison of temperature-dependent band gaps of the H-terminated SiNC (top panel) and matrix-embedded SiNCs in a-${\mathrm{SiO}}_2$ (bottom panel) calculated using the SDM (solid lines) and finite differences (dashed lines). The shaded area can be taken as the uncertainty of the band gap renormalization calculated for matrix-embedded SiNCs in a-${\mathrm{SiO}}_2$. Experimental data of matrix-embedded SiNCs is as for (a).

Figure 4

Phonon density of states (PDOS) of the H-terminated freestanding SiNC (red), oxidized freestanding SiNC (green), matrix-embedded SiNC in a-${\mathrm{SiO}}_2$ (blue), and bulk Si (black). The shaded areas define the frequency range of the vibrational modes associated mainly with displacements of surface (light gray) and core atoms (dark gray). Percentages in round [square] brackets show the contribution to the vibrational modes associated with displacements of the indicated group of atoms in the freestanding [matrix-embedded] SiNC.

Figure 5

Temperature dependence of the Eliashberg spectral function up to 300 K versus phonon frequency calculated for the freestanding fully H-terminated SiNC, the freestanding oxidized SiNC, and the matrix-embedded SiNC in a-${\mathrm{SiO}}_2$ from top to bottom, respectively. Thin arrows indicate the direction of increasing temperature and thick arrows highlight the suppression of electron-phonon coupling (EPC) after oxidation, and after placing the nanocrystal inside the matrix.

Figure 6

(a) Top panel: Decomposition of the imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$ of the matrix-embedded SiNC (blue solid line) into site contributions from nanocrystal (black solid line) and matrix (black dashed line) atoms. Bottom panel: Decomposition of the imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$ of the embedded nanocrystal (black solid line) into site contributions from suboxides: ${\mathrm{Si}}^0$ (gray solid line), ${\mathrm{Si}}^1$ (gray dashed line) and ${\mathrm{Si}}^2+{\mathrm{Si}}^3$ (green solid line). A Gaussian broadening of 50 meV was used in all plots. (b) Si-Si bond length in freestanding (red) and matrix-embedded (blue) SiNCs as a function of the radial distance from the center of the nanocrystal. The variation of Si-Si bond length in the matrix-embedded SiNC after applying ZG displacements is also shown for $T=0\mathrm{K}$ (gray) and $T=300\mathrm{K}$ (black). The horizontal black line indicates the average Si-Si bond length (2.375 Å) in the matrix-embedded SiNC with the nuclei clamped at their equilibrium positions. (c) Band gap renormalization versus Si-Si bond length variance [Eq. (12)] calculated for matrix-embedded SiNCs (blue squares) that correspond to equilibrium and ZG geometries for four temperatures. The blue line represents the linear regression [Eq. (13)] to the data with slope ${\partial }^2{E}_{\mathrm{g}}/\partial \mathrm{\Delta }{l}^2=$ $-33$ $\mathrm{meV}/{10}^{-3}$ ${\AA }^2$. Data for bulk Si (orange diamonds) are included for comparison.

Figure 7

Tauc plots ${\left[{\epsilon }_2{\omega }^2\right]}^{1/2}$ (black spectra) and ${\left[{\epsilon }_2{\omega }^2\right]}^2$ (blue spectra) of the matrix-embedded SiNC, $d=2$ nm, for indirect and direct transitions. The inset shows the Tauc plots close to the absorption onset. The dashed and solid lines represent spectra calculated with the nuclei at their relaxed equilibrium positions (zero-phonon) and for $T=300\mathrm{K}$ (phonon-assisted), respectively. The thin red lines represent the corresponding linear fits in the range of photon energies 1.3–2.0 eV and 3.4–4.0 eV for ${\left[{\epsilon }_2{\omega }^2\right]}^{1/2}$ and ${\left[{\epsilon }_2{\omega }^2\right]}^2$, respectively. The dashed-dotted line in the inset represents the zero-phonon spectrum of the freestanding SiNC, $d=2$ nm, calculated with the nuclei at their relaxed QC positions. A Gaussian broadening of 50 meV was used for all spectra.