Carrier multiplication in bulk and nanocrystalline semiconductors: Mechanism, efficiency, and interest for solar cells

Carrier multiplication (CM), the possibility to generate more than one exciton in a semiconductor quantum dot (QD) after absorption of a single photon has been intensely debated in recent years. Following on previous theoretical and experimental work, we report here that: (1) although the CM factor (i.e., number of generated photons per absorbed photon) at a given photon energy is higher in bulk than in QDs of the same material [Pijpers et al., Nature Phys. 5, 811 (2009)], the energy efficiency (the relative fraction of the photon energy that is transformed into excitons rather than heat) is higher in QDs; (2) for the same $\sim 1.2\text{eV}$ band gap, CM is more efficient in PbSe QDs than in bulk silicon; (3) nonetheless, the efficiency of solar cells based on PbSe QDs is not significantly enhanced by CM compared to a bulk silicon-based device.

Figure 1

DOS per atom versus the energy of the excited carrier for bulk PbSe (solid lines) and PbSe QDs with an energy gap of 1.2 eV (dashed lines). The thin lines represent the DOS of initial states and the thick lines correspond to the DOS of final states. The zero of energy corresponds to the top of the bulk valence band.

Figure 2

CM factor: Number of excitons generated by impact ionization after the absorption of a photon of energy $h\nu$. (a) Lines: simulations for bulk PbSe (gap ${\epsilon }_{\text{g}}=0.28\text{eV}$) and for QDs (${\epsilon }_{\text{g}}=0.6\text{eV}$, 6.9 nm diameter and ${\epsilon }_{\text{g}}=1.2\text{eV}$, 3.1 nm diameter). All the calculations were performed with ${\tau }_{\text{ph}}=0.5\text{ps}$. The symbols indicate the recent experimental results of Ref. 23 (◼) for bulk PbSe and those of Ref. 18 (●), Ref. 20 (◆), Ref. 19 (◻), Ref. 21 (▼), and Ref. 3 $(\times )$ for PbSe QDs. (b) Same data but plotted versus the energy-gap-normalized photon energy, $h\nu /{\epsilon }_{\text{g}}$.

Figure 3

CM factor: Number of excitons generated by impact ionization in PbSe QDs and bulk PbSe after the absorption of a photon of energy $h\nu =3.1\text{eV}$ represented as a function of the energy gap ${\epsilon }_{\text{g}}$ of the system (${\epsilon }_{\text{g}}=0.28\text{eV}$ for bulk PbSe). Lines: simulations for different values of the intraband relaxation lifetime ${\tau }_{\text{ph}}$ (0.1, 0.2, 0.5, and 1 ps). The symbols indicate the recent experimental results of Ref. 23 (◼) for bulk PbSe and those of Ref. 18 (●), Ref. 19 (◻), Ref. 20 (◆), and Ref. 21 (▼) for PbSe QDs.

Figure 4

Energy efficiency of the CM versus photon energy $h\nu$. Thick solid lines: simulations $\left({\tau }_{\text{ph}}=0.5\text{ps}\right)$ for bulk PbSe $\left({\epsilon }_{\text{g}}=0.28\text{eV}\right)$ and for two QDs (${\epsilon }_{\text{g}}=0.6\text{eV}$ and ${\epsilon }_{\text{g}}=1.2\text{eV}$). Thin dashed lines: same but assuming that there is no impact ionization, i.e., when excited carriers can only relax by emission of phonons. Black squares: experimental results of Ref. 23 for bulk PbSe.

Figure 5

CM factor: Number of excitons generated by impact ionization after the absorption of a photon of energy $h\nu$. Solid line: simulations for bulk Si $\left(\text{gap}=1.2\text{eV}\right)$. Crosses: experiments of Ref. 31. Dashed line: simulations for a PbSe QD characterized by a diameter of 3.1 nm and a gap of 1.2 eV. All the calculations are performed with ${\tau }_{\text{ph}}=0.5\text{ps}$.