Spontaneous surface plasmon polariton decay of band-edge excitons in quantum dots near a metal surface

Surface plasmon polaritons (SPPs) due to their subwavelength nature could significantly modify electronic transition behaviors in various optoelectronic systems. Here, using a model system with a spherical quantum dot (QD) close to a flat metal surface, we show that the conventional forbidden optical transitions in a QD could be largely enabled by the spontaneous SPP decay. The electronic states of the QD are approximated by a Bloch state combined with wave functions in a spherical potential well, which provides multiple hole states with mixed electronic multipoles. Moreover, the SPP is quantized by using a canonical quantization scheme followed by a Green's function approach to introduce its dissipation. In particular, we find that when the SPP dissipation is included, the spontaneous decay of the corresponding QD exciton is dominant by the transition into the off-resonance mode of SPPs with large momenta. Also, we have studied the dependence of spontaneous decay rates on the size and crystal orientation of a QD, the distance between the QD and metal surface, and the linewidth of SPPs. Some useful scaling relations have been revealed, and the multipole transitions are found to be comparable with the dipole transition under specific system parameters. These findings have important implications for our understanding of the electronic transition at a metal near field and might prove instrumental for the future design of plasmonic and QD devices.

Figure 1

The sketch of the system with a spherical QD above a metal surface in vacuum. We consider two different coordinates: one is the coordinate $xyz$ fixed by the metal surface, and the other is the coordinate $XYZ$ depicted by the red dashed lines attach to the crystal symmetry axes.

Figure 2

The value of ${\left|A(\omega ,\mathit{k})\right|}^2$ is mapped as the energy $\omega$ and momentum $k$ in (a), where the dashed lines indicate ${\left|A(\omega ,\mathit{k})\right|}^2$ at $\omega /{\omega }_p=0.5,0.55,0.6,\text{and}0.65$ corresponding to the solid curves with the same color in (b).

Figure 3

The schematic plot of the electronic states in the spherical QD. The blue solid line indicates the electron state. The red thick (thin) line indicates the $M=\pm \frac{3}{2}$ ($\pm \frac{1}{2}$) hole states, and the solid (dashed) line indicates the odd (even) state. The black solid (dashed) arrows represent the dipole allowed (forbidden) transitions.

Figure 4

The numerical results of the $k$ distribution of $\partial A_s/\partial k$ in Eq. (55) for $s=0,2,4$, and 6, corresponding to the lines in black, red, green, and blue, respectively. The corresponding parameters are ${\omega }_0=0.5{\omega }_p$, $\mathrm{\Gamma }=0.05{\omega }_p$, $a=0.5z_0$, and $z_0=30$ nm. The dashed vertical line in the upper zoomed-in view indicates $k\simeq 0.612$, which gives the resonant energy ${\omega }_k=0.5{\omega }_p$.

Figure 5

The numerical results of $A_s$ with $s=0,2,4$, and 6 corresponding to the lines in black, red, green, and blue, respectively. (a) $\mathrm{\Gamma }$ dependence of $A_s$ with ${\omega }_0=0.5{\omega }_p$, $z_0=15$ nm, $a=0.8z_0$. (b) ${\omega }_0$ dependence of $A_s$ with $\mathrm{\Gamma }=0.2{\omega }_p$, $z_0=15$ nm, $a=0.8z_0$. (c) $z_0$ dependence of $A_s$ with ${\omega }_0=0.5{\omega }_p$, $\mathrm{\Gamma }=0.2{\omega }_p$, $a=0.7z_0$. (d) $a/z_0$ dependence of $A_s$ with ${\omega }_0=0.5{\omega }_p$, $\mathrm{\Gamma }=0.2{\omega }_p$, $z_0=20$ nm.

Figure 6

The numerical results of ${\eta }_{3/2}^o$, ${\eta }_{1/2}^o$, ${\eta }_{3/2}^e$, and ${\eta }_{1/2}^e$ corresponding to the solid lines in red, green, blue, and purple, respectively. The numerical results of ${\eta }_{3/2}$ and ${\eta }_{1/2}$ corresponding to the dashed lines in red and green, respectively, indicate the dipole-only transition of the two bright excitons. The thick and thin lines represent the results for the QD with its crystal $Z$ axis parallel ($\parallel$) ($\theta =\pi /2$) and perpendicular ($\perp$) ($\theta =0$) to metal surface, respectively. (a) $\mathrm{\Gamma }$ dependence of $\eta$ with ${\omega }_0=0.5{\omega }_p$, $z_0=20$ nm, $a=0.8z_0$. (b) $\mathrm{\Gamma }$ dependence of $\eta$ with ${\omega }_0=0.68{\omega }_p$, $z_0=20$ nm, $a=0.8z_0$. (c) ${\omega }_0$ dependence of $\eta$ with $\mathrm{\Gamma }=0.2{\omega }_p$, $z_0=20$ nm, $a=0.8z_0$. (d) $a/z_0$ dependence of $\eta$ with ${\omega }_0=0.68{\omega }_p$, $\mathrm{\Gamma }=0.2{\omega }_p$, $z_0=20$ nm.

Figure 7

The dispersion of the SPP with a lossy dielectric function at $\gamma =0.05{\omega }_p$. (a) The dispersion with a real frequency $\omega$. The blue solid (red dashed) line indicates the real (imaginary) part of $k$. (b) The dispersion with a real wave vector $k$. The blue solid (red dashed) line indicates the real (imaginary) part of $\omega$. The black solid line indicates the difference between the dispersion line with and without the loss.

Figure 8

The values of $J_i$ and $J_i/J_0$ as a function of $\mu$.