Energy transfer between semiconductor nanocrystals: Validity of Förster’s theory

We theoretically study the energy transfer between semiconductor nanocrystals and we calculate the rate of this process in the case of direct-gap (InAs) and indirect-gap (Si) semiconductors. The calculations are based on the tight-binding method that enables us to determine the transfer rate, including electronic structure effects, dielectric screening, and multipolar Coulomb interactions using a microscopic approach. In the case of direct-gap semiconductor nanocrystals, we show that the energy transfer arises only from dipole-dipole interactions in agreement with experimental observations. We obtain that the transfer rate is well described by Förster’s theory, and we provide an analytical expression in which the Förster rate is determined not only by the spectral overlap between the emission and absorption spectra of the nanocrystals but also by a factor that simulates the dielectric effects on the Coulomb interactions. In the case of Si nanocrystals, the situation is different because dipolar transitions are weak due to the indirect gap of Si. For this reason, we show that multipolar terms dominate when the distance between the nanocrystals is small and surface effects play an important role in the screening. We predict that the energy transfer between Si nanocrystals by a no-phonon process is possible only when the dots are almost in close contact.

Figure 1

Many-particle electronic configurations of the donor $\left(D\right)$ and acceptor $\left(A\right)$ nanocrystals in the (i) initial and (f) final states of the system, before and after the energy transfer.

Figure 2

Energy transfer time (+, $1∕\Gamma$; ×, $1∕{\Gamma }^{\left(0\right)}$) between two identical InAs nanocrystals $(\text{radius}=1.52\mathrm{nm})$ in vacuum versus the interdot distance $R$. $\Gamma$ is calculated using the screened Coulomb interaction, and ${\Gamma }^{\left(0\right)}$ using the bare one. The solid (dotted) curve shows that $1∕\Gamma$ $(1∕{\Gamma }^{\left(0\right)})$ varies like $R^6$.

Figure 3

Ratio $(\Gamma ∕{\Gamma }^{\left(0\right)})∕{\eta }^2$ calculated for InAs nanocrystals in vacuum versus the radius $R_A$ of the acceptor. The figure shows that $\Gamma ∕{\Gamma }^{\left(0\right)}$ is close to ${\eta }^2$, in average. The radii of the donors are $0.58\mathrm{nm}$ (+), $0.68\mathrm{nm}$ (×), $0.83\mathrm{nm}$ $(*)$, $0.99\mathrm{nm}$ (◻), $1.22\mathrm{nm}$ (∎), $1.52\mathrm{nm}$ (☉), $1.67\mathrm{nm}$ (•), and $1.82\mathrm{nm}$ (▵). $\eta$ is calculated from Eqs. (15, 20) using $\delta =0.324\mathrm{nm}$, ${\epsilon }_{\mathrm{out}}=1$, and ${\epsilon }_{\mathrm{in}}^{\text{bulk}}=9.21$.

Figure 4

Calculated energy transfer time $1∕\Gamma$ for InAs nanocrystals in vacuum plotted versus the radius $R_A$ of the acceptor (interdot distance $R=10\mathrm{nm}$). The radii of the donors are $0.58\mathrm{nm}$ (+), $0.68\mathrm{nm}$ (×), $0.83\mathrm{nm}$ $(*)$, $0.99\mathrm{nm}$ (◻), $1.22\mathrm{nm}$ (∎), $1.52\mathrm{nm}$ (☉), $1.67\mathrm{nm}$ (•), $1.82\mathrm{nm}$ (▵). The lines connect the values $1∕{\Gamma }_F$ calculated for the same nanocrystal sizes using Eq. (13) based on Förster’s theory $({\epsilon }_{\mathrm{out}}=1)$.

Figure 5

Lifetime for the energy transfer between two InAs nanocrystals (radius of the $\text{donor}=1.22\mathrm{nm}$) in vacuum versus the interdot distance $R$. Radii of the acceptors: $2.07\mathrm{nm}$ (solid curve), $1.75\mathrm{nm}$ (dashed curve), and $1.25\mathrm{nm}$ (dotted curve). The horizontal line denotes the radiative lifetime ${\tau }_{\exp }$ of the donor in vacuum.

Figure 6

Energy transfer time (+, $1∕\Gamma$; ×, $1∕{\Gamma }^{\left(0\right)}$) between two identical Si nanocrystals $(\text{radius}=1.36\mathrm{nm})$ in vacuum versus the interdot distance $R$. $\Gamma$ is calculated using the screened Coulomb interaction, and ${\Gamma }^{\left(0\right)}$ using the bare one. The solid (dotted) curve shows that $1∕\Gamma$ $(1∕{\Gamma }^{\left(0\right)})$ varies like $R^6$ at large distance.

Figure 7

Energy transfer time $1∕\Gamma$ calculated for Si nanocrystals in vacuum and plotted versus the radius $R_A$ of the acceptor (interdot distance $R=10\mathrm{nm}$). Radii of the donors: $0.52\mathrm{nm}$ (×), $0.61\mathrm{nm}$ (+), $0.75\mathrm{nm}$ $(*)$, $0.84\mathrm{nm}$ (◻), $1.09\mathrm{nm}$ (∎), $1.36\mathrm{nm}$ (☉), $1.50\mathrm{nm}$ (•), and $1.63\mathrm{nm}$ (▵).

Figure 8

Energy transfer time $1∕{\Gamma }^{\left(0\right)}$ calculated using the unscreened Coulomb interaction for Si nanocrystals in vacuum and plotted versus the radius $R_A$ of the acceptor (interdot distance $R=10\mathrm{nm}$). Radii of the donors: $0.52\mathrm{nm}$ (×), $0.61\mathrm{nm}$ (+), $0.75\mathrm{nm}$ $(*)$, $0.84\mathrm{nm}$ (◻), $1.09\mathrm{nm}$ (∎), $1.36\mathrm{nm}$ (☉), $1.50\mathrm{nm}$ (•), and $1.63\mathrm{nm}$ (▵). The lines connect the values $1∕{\Gamma }_F^{\left(0\right)}$ calculated for the same nanocrystal sizes using Eq. (13) with $\eta =1$.

Figure 9

Lifetime for the energy transfer between two Si nanocrystals (diameter of the $\text{donor}=2.7\mathrm{nm}$) in vacuum versus the interdot distance $R$. Diameters of the acceptors: $2.7\mathrm{nm}$ (+), $3.3\mathrm{nm}$ (×), and $3.9\mathrm{nm}$ (∎). The horizontal line denotes the radiative lifetime ${\tau }_{\exp }$ of the donor in vacuum.