Anomalous quantum confinement of the longitudinal optical phonon mode in PbSe quantum dots

We have investigated the diameter dependence of the Raman spectra of lead selenide nanocrystals. The first-order Raman peak at about 136 cm${}^{-1}$ and its second-order overtone at twice this wavenumber move up in energy with decreasing nanocrystal diameter. This anomalous behavior is interpreted in terms of quantum confinement of the longitudinal optical (LO) phonon whose frequency displays a minimum at Γ in the dispersion of bulk PbSe. We perform ab initio calculations of the phonons of PbSe slabs with up to 15 layers. The LO mode perpendicular to the slab shifts indeed upwards with decreasing layer thickness, thus validating the interpretation of the anomalous radius dependence of the Raman spectra in terms of quantum confinement.

Figure 1

(a) Normalized absorbance spectra of PbSe nanocrystals for a range of mean diameter between 7.1 nm (bottom) and 4.5 nm (top). The optical transitions are labelled α, β, γ, and $E_1$. Inset: TEM image of a monolayer of PbSe nanocrystals with a mean diameter of 6.0 nm. (b) and (c) Raman spectra of PbSe nanocrystals with a mean diameter of 6.8 nm for two optical excitation wavelengths.

Figure 2

(a) Raman spectra of a 5.5-nm PbSe nanocrystal film for different laser powers. The experiment was performed at 300 K. (b) Raman spectra of a 5.5-nm PbSe nanocrystal film for different sample temperatures. The laser power was set to 1.0 mW. (c) Raman wavenumbers of the 1-LO peak as a function of the laser power for the 5.5-nm PbSe nanocrystal film and a bulk PbSe sample. (d) Raman wavenumbers of the 1-LO phonon mode as a function of the temperature for the 5.5-nm PbSe nanocrystal film.

Figure 3

(a) Comparison of Raman spectra for a 5.2-nm PbSe nanocrystal film measured in an air-free environment (red) and air exposed (black), both excited at 5 mW. (b) Spatial fluctuations in the position of the 1-LO peak of 5.5-nm PbSe nanocrystal films. The laser power was set to 1.0 mW in (b).

Figure 4

Raman shift of the 1-LO peak as a function of the PbSe nanostructure size. Open circles: calculated frequencies of the 1-LO peak for PbSe (100) slabs with a width varying from 6 layers up to 15 layers. Open squares: 1-LO frequencies for PbSe(100) slabs according to the quantum-confinement model (extracted from the dispersion in Fig. 5). Filled squares: measured Raman frequencies for films of PbSe NCswith different sizes. Inset: Raman wavenumbers of the 2-LO peak as a function of the PbSe nanocrystal size. The right panel shows the vibrational pattern of the quantum confined LO mode (with polarization perpendicular to the slab) of a 12-layer slab. The double column of Pb (gray) and Se (orange) atoms represents the motif of the elementary unit cell of the slab.

Figure 5

Calculated dispersion of the LO phonon branch between Γ and $X$ (Ref. 2). The phonon dispersion has been upscaled by 6$\%$ in order to match the experimental value for the LO phonon at Γ. The red circles mark the wave vectors and frequencies of the perpendicular LO mode of an $n$-layer slab according to the zone-folding model (see text). For comparison with Fig. 4: the 6-layer slab has a width of 3.45 nm, the 14-layer slab has a width of 8 nm.

Figure 6

Raman spectra of (a) a 5.5-nm PbSe nanocrystal film and (b) a bulk PbSe sample measured for different laser powers.