Double quantum dot photoluminescence mediated by incoherent reversible energy transport

We present a theoretical study of the stationary photoluminescence of two, direct-gap, semiconductor nanocrystals, taking into account electronic excitation energy-transfer processes due to electrostatic interaction. The results obtained here allow for the incoherent reversible energy transport that occurs when the intraband relaxation rate in a quantum dot acceptor is comparable to, or less than, the energy-transfer rate. We investigate the secondary emission of two different electronic level schemes that can be realized experimentally, obtain analytical expressions for the luminescence differential cross section, and perform an analysis of its spectrum. It is shown that when excitation is not in resonance with the levels involved in energy transfer, the energy transfer is more efficient.

Figure 1

Mutual orientation of vector $\mathbf{r}$ directed from the center of the acceptor to the center of the donor and vectors ${\mathbf{r}}_{vc}^{\left(D\right)}$ and ${\mathbf{r}}_{cv}^{\left(A\right)}$.

Figure 2

(Color online) Dependencies of energy-transfer matrix element on intersurface distance $R=r-R_D-R_A$ for different interband transitions in a QD acceptor. In this calculation, we use a fixed donor radius $R_D=2\text{nm}$ and consider resonant energy transfer from the lowest-energy state of the donor. Acceptor radii are obtained from the resonance condition ${\omega }_{iD}\left(R_D\right)={\omega }_{jA}\left(R_A\right)$. (a) Dipole-allowed transitions. Numbers denote energy transfer with creation of the following electron-hole pairs in the acceptor: 1—$c110;v110$ $\left(R_A=2.86\text{nm}\right)$, 2—$c120;v120$ $\left(R_A=3.67\text{nm}\right)$, 3—$c200;v200$ $\left(R_A=4\text{nm}\right)$, 4—$c210;v210$ $\left(R_A=4.92\text{nm}\right)$, and 5—$c220;v220$ $\left(R_A=5.79\text{nm}\right)$. (b) Dipole-forbidden transitions: 1—$c100;v110$ $\left(R_A=2.09\text{nm}\right)$, 2—$c110;v100$ $\left(R_A=2.80\text{nm}\right)$, 3—$c120;v110$ $\left(R_A=2.94\text{nm}\right)$, and 4—$c110;v120$ $\left(R_A=3.61\text{nm}\right)$.

Figure 3

(Color online) 2-3 scheme of double quantum dot for the allowed optical transitions in QD acceptor. ${\zeta }_{0D,1D}$, ${\zeta }_{0A,1A}$, ${\zeta }_{0A,2A}$ are the interband relaxation rates for the QD donor and QD acceptor. $W_{1D}$, $W_{1A}$, $W_{2A}$ are the spontaneous light emission rates for the QD donor and QD acceptor. ${\zeta }_{1A,2A}$ is the intraband relaxation rate for the QD acceptor. ${\gamma }_{DA}$ is the energy-transfer rate. ${\omega }_{1D,R}$, ${\omega }_{1A,R}$, ${\omega }_{2A,R}$ are the spontaneous light emission frequencies. ${\omega }_L$ is the incident light frequency. $|0D⟩$, $|0A⟩$ and $|1D⟩$, $|1A⟩$ are the ground and lowest-energy states of the QD donor and QD acceptor. $|2A⟩$ is the high-energy state of the QD acceptor.

Figure 4

(Color online) 2-3 scheme of double quantum dot for forbidden optical transitions in QD acceptor. Symbols are the same as in Fig. 3.

Figure 5

(Color online) 3-3 scheme of double quantum dot for the allowed optical transitions in the QD acceptor. ${\zeta }_{0D,2D}$ is the interband relaxation rate for the QD donor. $W_{2D}$ is the spontaneous light emission rate for the QD donor. ${\zeta }_{1D,2D}$ is the intraband relaxation rate for the QD donor. ${\omega }_{2D,R}$ is the spontaneous light emission frequency. $|2D⟩$ is the high-energy state of the QD donor. Other symbols are the same as in Fig. 3.

Figure 6

(Color online) 3-3 scheme of double quantum dot for the forbidden optical transitions in the QD acceptor. Symbols are the same as in Figs. 3, 5.

Figure 7

(Color online) Methods of separation of photoluminescence signal from the secondary emission spectrum of donor QD ($T=4\text{K}$, $R_D=2\text{nm}$, and ${\Gamma }_F=2\times {10}^{10}{\text{s}}^{-1}$). (a) Elimination of narrow scattering signal from the wide luminescence contour $\left({\gamma }_{i\alpha ,0\alpha }⪢{\Gamma }_F\right)$ with a roughened spectral resolution in case of resonant excitation, $R=3\text{nm}$. Solid line depicts the calculated spectrum of donor, Eq. (34), round point positions have been obtained with step of calculation larger than ${\Gamma }_F$. (b) Separation of resonant scattering and luminescence signals by using nonresonant excitation. Here detuning energy $\hslash {\Delta }_L=-1\text{meV}$, $R=10\text{nm}$, and ${\gamma }_{0D,1D}\sim {\Gamma }_F$. The luminescence component is defined by the solid and the resonant scattering component by the dashed line.

Figure 8

(Color online) Dependencies of maxima of LDCS on intersurface distance for dipole-allowed energy transfer $\left\{c100;v100\right\}D\to \left\{c110;v110\right\}A$. The donor relaxation rate is ${\gamma }_{1D,1D}^{\left(0\right)}={10}^8{\text{s}}^{-1}$. (a) $T=300\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$, (b) $T=300\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$, (c) $T=300\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$, (d) $T=90\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$, (e) $T=90\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$, and (f) $T=90\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$.

Figure 9

(Color online) Dependencies of the maxima of the LDCS on intersurface distance for dipole-forbidden energy transfer $\left\{c100;v100\right\}D\to \left\{c100;v110\right\}A$. The donor relaxation rate is ${\gamma }_{1D,1D}^{\left(0\right)}={10}^8{\text{s}}^{-1}$. (a) $T=300\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$, (b) $T=300\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$, (c) $T=300\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$, (d) $T=30\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$, (e) $T=30\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$, and (f) $T=30\text{K}$, ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$.

Figure 10

(Color online) Ratios of LDCSs calculated for the approximation of incoherent reversible $d{\sigma }_{\beta }$ and irreversible $d{\sigma }_{\beta }^{\prime }$ energy transport as functions of the intersurface distance at room temperature. (a) ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$ and (b) ${\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$.

Figure 11

(Color online) LDCS maximum as a function of intersurface distance for dipole-dipole energy transfer $\left\{c100;v100\right\}D\to \left\{c110;v110\right\}A$. The external excitation is in resonance with the electron-hole pair $\left\{c110;v110\right\}D$. (a) $T=300\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$; (b) $T=300\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$; (c) $T=300\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$; (d) $T=90\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$; (e) $T=90\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$; and (f) $T=90\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$.

Figure 12

(Color online) LDCS maximum as a function of intersurface distance for dipole-forbidden energy transfer $\left\{c100;v100\right\}D\to \left\{c100;v110\right\}A$, scheme 3-3, ${\gamma }_{1D,1D}^{\left(0\right)}={10}^8{\text{s}}^{-1}$. (a) $T=300\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$; (b) $T=30\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$; (c) $T=300\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$; (d) $T=30\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{12}{\text{s}}^{-1}$; (e) $T=30\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$; and (f) $T=30\text{K}$, ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$.

Figure 13

(Color online) Ratios of LDCSs calculated for the approximation of incoherent reversible $d{\sigma }_{\beta }$ and irreversible $d{\sigma }_{\beta }^{\prime }$ energy transport as functions of the intersurface distance at room temperature. (a) ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^{10}{\text{s}}^{-1}$ and (b) ${\gamma }_{2D,2D}^{\left(0\right)}={\gamma }_{2A,2A}^{\left(0\right)}=3\times {10}^8{\text{s}}^{-1}$.