Increasing impact ionization rates in Si nanoparticles through surface engineering: A density functional study

We propose design pathways to improve the efficiency of multiple exciton generation in nanoparticle-based solar cells by carrying out ab initio calculations of impact ionization (II) rates in semiconducting nanoparticles (NPs). In NPs with unreconstructed surfaces, quantum confinement has two competing effects: it enhances the effective Coulomb interaction and thus the II rates, but it also blue-shifts the gap, which tends to reduce the II rates. The competition of these effects determines the utility of NP-based solar cells. We report that surface reconstruction of NPs can tip the balance towards the enhancement of II by creating a substantial density of states at lower energies. Our results suggest that manipulating the surfaces of NPs, e.g., by engineering the ligands and embedding structures, may lead to an efficient multiexciton generation within the solar spectrum.

Figure 1

Schematic representation of an impact ionization process. (a) If a photon with energy ($\hslash \omega$) two times larger than the band gap ($E_{\mathrm{gap}}$) is absorbed (b) then the generated hot carriers may decay to (c) biexcitons by exciting an additional electron-hole pair via Coulomb scattering. The physical interactions involved in the processes are also depicted: dipole ($er$) for the absorption process and screened Coulomb interaction ($W$) for impact ionization.

Figure 2

Ball-and-stick geometries of the nanoparticles considered in our work. NP with nonreconstructed surfaces: (a) Si${}_{35}$H${}_{36}$, (b) Si${}_{66}$H${}_{64}$, (d) Si${}_{78}$H${}_{64}$, (e) Si${}_{124}$H${}_{96}$, (f) Si${}_{220}$H${}_{144}$, and with reconstructed ones: (c) Si${}_{66}$H${}_{40}$, and (g) Si${}_{220}$H${}_{120}$. Representative long Si-Si bonds arising from surface reconstruction are shown in gray. Yellow (white) spheres represent silicon (hydrogen) atoms.

Figure 3

Size dependence of the single-particle-averaged impact ionization rate for nonreconstructed clusters: (a) rate plotted on an absolute energy scale; (b) rate plotted on a relative energy scale, the energy being normalized with the gap of the corresponding nanoparticle. The color code for the lines in (b) is the same as in (a). Note that while in (a) the largest $d=2.0$ nm nanoparticle has the highest impact ionization rate, the opposite is true in (b) where the smallest $d=1.1$ nm nanoparticle wins.

Figure 4

(a) Impact ionization rate for the $d=1.2$ nm and $d=2.0$ nm nanoparticle with and without surface reconstruction. (b) Enhancement of the electron and hole trion density of states (TDOS) due to surface reconstruction for the $d=2$ nm nanoparticle. (c) Enhancement of the effective matrix element ($W_{\mathrm{eff}}$, see text) due to surface reconstruction for the $d=2$ nm nanoparticle. The color code for the markers in (c) is the same as in (b).

Figure 5

(a) Hole and electron contributions [Eq. (2)] to the averaged II rate for the $d=1.1$ nm nanoparticle. Blue solid and red dotted lines represent electron and hole impact ionization rates, respectively. (b) Comparison of the effective matrix element and inverse participation ratio for the same nanoparticle. Blue crosses, red circles denote electron and hole quantities. Double-headed arrows show a gap of approximately $3\times E_g$ in $W_{\mathrm{eff}}$, IPR.

Figure 6

(a) Averaged impact ionization and Auger recombination rate for the $d=1.2$ nm nanoparticle plotted on a relative energy scale. (b) Comparison of the number of exciton and biexcitons corresponding to a given photon energy.

Figure 7

Comparison of single-particle-averaged impact ionization rates obtained with or without including oscillator strengths. (a) $d=1.1$ nm and $d=1.2$ nm NPs, (b) $d=1.4$ and $d=1.6$ nm NPs, (c) nonreconstructed (Si${}_{66}$H${}_{64}$) and reconstructed (Si${}_{66}$H${}_{40}$) $d=1.2$ nm NPs, (d) nonreconstructed (Si${}_{220}$H${}_{144}$) and reconstructed (Si${}_{220}$H${}_{120}$) $d=2.0$ nm NPs. Legends in the subfigures: “osc” and “ari” denote oscillator strength weighted and arithmetic average, respectively.

Figure 8

Comparison of the contribution of electrons and holes to the single-particle-averaged impact ionization rates obtained with or without including oscillator strengths. (a) $d=1.1$ nm NP without oscillator strength, (b) $d=2.0$ nm NP without oscillator strength, (c) $d=1.1$ nm NP with oscillator strength, (d) $d=2.0$ nm NP with oscillator strength.

Figure 9

Dependence of averaged impact ionization rates on the width of the window function ($\Delta$) used to represent the energy conservation rule in Eq. (2): (a) $d=1.2$ nm NP (b) $d=1.6$ nm NP.

Figure 10

(a) Comparison of the effective matrix element and inverse participation ratio for the $d=1.1$ nm nanoparticle. (b) Comparison of the single-particle rate and trion density of states (TDOS) for the $d=1.1$ nm nanoparticle. (c), (d) show the same for the $d=2$ nm nanoparticle. Double-headed arrows show a gap of approximately $3\times E_g$ in $\Gamma$, $W_{\mathrm{eff}}$, TDOS. For (a) and (c) blue crosses, red circles denote electron and hole quantities, while for (b) and (d) blue crosses, red circles represent $\Gamma$, TDOS, respectively.