Carrier multiplication in semiconductor nanocrystals via intraband optical transitions involving virtual biexciton states

We propose and analyze a physical mechanism for photogeneration of multiexcitons by single photons (carrier multiplication) in semiconductor nanocrystals, which involves intraband optical transitions within the manifold of biexciton states. In this mechanism, a virtual biexciton is generated from nanocrystal vacuum by the Coulomb interaction between two valence-band electrons, which results in their transfer to the conduction band. The virtual biexciton is then converted into a real, energy-conserving biexciton by photon absorption on an intraband optical transition. The proposed mechanism is inactive in bulk semiconductors as momentum conservation suppresses intraband transitions. However, it becomes highly efficient in the case of zero-dimensional nanocrystals, where quantum confinement results in relaxation of momentum conservation, which is accompanied by the development of strong intraband absorption. Our calculations show that the efficiency of the carrier multiplication channel mediated by intraband optical transitions can be comparable to or even greater than that for impact-ionization-like processes mediated by interband transitions.

Figure 1

(Color online) Direct photogeneration of a biexciton from vacuum, ∣0⟩, via intermediate single-exciton (on the left) or intermediate biexciton (on the right) states. The final biexciton state $\mid x_1{x}_2⟩$ is in resonance with the absorbed photon, while intermediate (virtual) single-exciton $\mid x^{\prime }⟩$ and biexciton $\mid {x^{\prime }}_1{x^{\prime }}_2⟩$ states can be off resonance (energy conservation is not required for virtual processes). The order of processes involving electron-photon and carrier-carrier Coulomb interactions for the virtual biexciton channel is reversed with respect to that for the virtual exciton channel. $H_{\mathrm{II}}$ and $H_{xx}$ are the components of the Coulomb-interaction operator responsible for impact-ionization and biexciton generation from vacuum, respectively. $H_{e\nu }^{\mathit{\text{inter}}}$ and $H_{e\nu }^{\mathit{\text{intra}}}$ are the operators of the electron-photon interaction responsible for the interband and intraband transitions, respectively.

Figure 2

(Color online) Spectral dependence of quantum efficiencies measured for bulk PbS (data from Ref. 13) as well as PbS and PbSe NCs (from Ref. 6).

Figure 3

(Color online) (a) Calculated degeneracy factors for four low-energy $1l$ biexcitons (solid circles) and corresponding near-resonant single-exciton states (open squares) along with their ratio (solid diamonds) shown as a function of energy in access of $2E_g$. Solid lines are fits to ${(E-2E_g)}^m$, with $m=4$ $\left(g_{xx}\right)$ and 3.7 $(g_{xx}∕g_x)$; the dashed line is a guide for the eye. (b) Comparison of calculated (solid circles) and measured (open squares; from Ref. 6) values of $f=W_{2,xx}∕W$.

Figure 4

(Color online) Spectral dependence of the absorption coefficient $\left({\alpha }_0\right)$ of PbSe NCs of three different mean radii (indicated in the figure). Arrows show the onset for CM calculated from $\hslash {\omega }_{\mathrm{CM}}=2.85E_g$.