Collective excitations in charged nanocrystals and in close-packed arrays of charged nanocrystals

We present self-consistent tight binding calculations of the dielectric response of charged semiconductor nanocrystals. We predict collective effects such as the oscillation in opposite phase of the additional electron and of the plasma of valence electrons near the resonance. In arrays of charged nanocrystals, the coupling between infrared light and collective excitations is shown to be strong, the real part of the dielectric constant becoming negative in some part of the spectrum. Transverse and longitudinal excitation modes obeying the Lyddane-Sachs-Teller equation are predicted.

Figure 1

Polarizability (real part) of a nanocrystal $(R=1.69\mathrm{nm})$ charged with one electron calculated from Eq. (2) using ${\omega }_{\mathrm{sp}}=0.46\mathrm{eV}$ (vertical line), ${ϵ}_b=8.12$, and $\eta =5\mathrm{meV}$. Continuous line: total polarizability of the sphere $\left[{\alpha }_{\mathrm{tot}}\left(\omega \right)\right]$. Dashed line: response of the valence electrons $\left[{\alpha }_{\mathrm{val}}\left(\omega \right)\right]$.

Figure 2

Polarizability (real part) of a CdSe nanocrystal $(R=1.69\mathrm{nm})$ charged with one electron calculated in RPA $(\eta =10\mathrm{meV})$. Continuous line: total polarizability of the sphere. Dashed line: response of the valence electrons only. The vertical line indicates the single-particle transition $(\omega =0.38\mathrm{eV})$. (a) Nanocrystals charged with $n=1$ electron; (b) $n=2$.

Figure 3

Dielectric constant ${ϵ}_M$ [$\mathrm{Re}\left({ϵ}_M\right)$: straight line, $\mathrm{Im}\left({ϵ}_M\right)$: dashed line, $-\mathrm{Im}(1∕{ϵ}_M)$: dotted line] of a cubic array of CdSe nanocrystals ($R=1.69\mathrm{nm}$, $a=3.66\mathrm{nm}$, $\eta =10\mathrm{meV}$). (a) Nanocrystals charged with $n=1$ electron; (b) $n=2$. The vertical line indicates the single-particle transition $(\omega =0.38\mathrm{eV})$.

Figure 4

Dielectric constant ${ϵ}_M$ [$\mathrm{Re}\left({ϵ}_M\right)$: straight line, $\mathrm{Im}\left({ϵ}_M\right)$: dashed line, $-\mathrm{Im}(1∕{ϵ}_M)$: dotted line] of a cubic array of InAs nanocrystals ($R=1.69\mathrm{nm}$, $a=3.65\mathrm{nm}$, $\eta =10\mathrm{meV}$) charged with two electrons $(n=2)$. The vertical line indicates the single-particle transition $(\omega =0.46\mathrm{eV})$.

Figure 5

$S\to P$ transition energy in CdSe nanocrystals as function of size. Experiments (Ref. 22): +. Theory: single-particle energy ${\omega }_{\mathrm{sp}}$ (line), same but including the self-energy correction ${\Sigma }_p-{\Sigma }_s$ (dashed line), RPA peak energy (∎).

Figure 6

Band structure of bulk CdSe calculated in tight binding (continuous lines) compared to the one calculated in local density approximation (LDA) (dashed line) including a rigid shift of the conduction band to fit the experimental band gap. The LDA bands at about $-8\mathrm{eV}$ are $d$ bands which obviously cannot be reproduced in the $s{p}^3$ TB model.