Confinement effects in PbSe quantum wells and nanocrystals

The effect of quantum confinement in PbSe quantum wells and dots is studied using tight binding calculations. Compared to zinc-blende semiconductors, unusual physical properties are predicted for rock salt PbSe nanostructures. The energy gap increases as the inverse of the size both for wells and dots. For PbSe nanocrystals, the luminescence lifetime, the confinement energy, and the intraband optical properties are in good agreement with experiments. The high quantum yield observed experimentally can be explained by the absence of surface dangling bonds in these systems. The origin of the second peak measured in the absorption spectra is discussed, whereas S-P interband transitions exhibit very small oscillator strength. The full frequency-dependent dielectric function $ϵ\left(\omega \right)$ is calculated for PbSe quantum wells. Its imaginary part ${ϵ}_2\left(\omega \right)$ is strongly anisotropic and shows large variations with respect to its bulk value even far from the gap region.

Figure 1

(a) Band structure of bulk PbSe. The zero of energy corresponds to the top of the valence band. (b) Same but with $E_{pp\sigma }\left(\mathrm{ac}\right)=2.5\mathrm{eV}$, $E_p^{\mathrm{c}}-E_p^{\mathrm{a}}=1.5\mathrm{eV}$ and the spin-orbit coupling keeping its bulk value, all other interatomic terms of the Hamiltonian being set to zero.

Figure 2

Imaginary part of the frequency-dependent dielectric function ${ϵ}_2\left(\omega \right)$ for bulk PbSe calculated in tight binding (solid line) compared to the experimental value at $300\mathrm{K}$ of Ref. 40 (dashed line) and to the calculated joint density of states (dotted line). $E_1$, $E_2$, and $E_3$ indicate the main transitions at critical points of the band structure.

Figure 3

Lines: band structure of a well containing six PbSe planes. Shaded region: projection of the bulk band structure on the surface reciprocal space.

Figure 4

Squares confinement energy in quantum wells as a function of size. Line: fit of the energy gap.

Figure 5

The energy dispersion in bulk PbSe is mostly linear except very close to the gap region, where it becomes parabolic. The confinement energy behaves as $1∕L$, where $L$ is the size of the nanostructure. $k_0$ is the wave vector at the $L$ point ofthe Brillouin zone.

Figure 6

Imaginary part of the frequency-dependent dielectric function ${ϵ}_2\left(\omega \right)$ of quantum wells made of 2 (a), 4 (b), and 20 (c) PbSe (001) planes. The solid lines are for a polarization vector e parallel to the planes {${ϵ}_{2\Vert }\left(\omega \right)$, along [100]} and the dashed lines for $\mathbf{e}$ perpendicular {${ϵ}_{2\perp }\left(\omega \right)$, along [001]}. The dotted lines represent ${ϵ}_2\left(\omega \right)$ calculated for bulk PbSe.

Figure 7

The electronic structure of bulk PbSe can be described by linear chains of $p$ orbitals with only $pp\sigma$ interactions.

Figure 8

Band structure of bulk PbSe in the [001] direction starting from the critical point $\Sigma$ along the [110] axis (along the dashed line in the Brillouin zone).

Figure 9

Calculated confinement energy in spherical (×) and cubic (+) PbSe nanocrystals as function of size (diameter $D$ for a sphere, side length $L$ for a cube). Cubes and spheres with the same volume are at the same abscissa $[L=D{(\pi ∕6)}^{1∕3}]$. Continuous line: fit of the results. Dotted line: $\mathbf{k}\bullet \mathbf{p}$ results of Ref. 3. Experimental values (first excitonic transition): Ref. 10 (∎), Ref. 3 (●), Ref. 12 (▴), and Ref. 9 (◆).

Figure 10

(a) Optical absorption spectrum of PbSe nanocrystals, each transition line being broadened by a Gaussian $(\text{width}=35\mathrm{meV})$. Continuous line: theory, sphere $(\text{diameter}=7.3\mathrm{nm})$. Dashed line: theory, cube (side $(\text{length}=7.2\mathrm{nm}$). Dotted line: experiments of Ref. 9 (average $(\text{diameter}=7\mathrm{nm}$). (b) Same, but the theory shows the joint density of states; i.e., replacing the optical matrix element by a constant. Note that for the sake of comparison, the lower energy scale of the experimental spectrum is slightly shifted with respect to the upper one of theoretical spectra $(\text{shift}=0.1\mathrm{eV})$.

Figure 11

Splittings of the energy levels in the conduction band (full symbols) and in the valence band (empty symbols) of spherical PbSe nanocrystals as a function of their diameter. The splittings are defined with respect to the lowest unoccupied level and to the highest occupied level, respectively. The levels have either a twofold (circle) or a fourfold (square) degeneracy.

Figure 12

Evolution with size of the radiative lifetime in spherical (◻, ∎) and cubic (▵, ▴) nanocrystals. Empty symbols correspond to lifetimes calculated with ${ϵ}_{\text{in}}=23$ and ${ϵ}_{\text{out}}=23$, filled ones to those calculated with ${ϵ}_{\text{in}}=23$ and ${ϵ}_{\text{out}}=2.1$. The cross is the measured lifetime of Ref. 10.

Figure 13

Room temperature intraband optical absorption in the conduction (dotted line) and valence (dashed line) band of spherical nanocrystals charged with one extra electron or one extra hole, respectively (upper curves: $\text{diameter}=4.9\mathrm{nm}$; lower $\text{ones}=7.3\mathrm{nm}$). Continuous line: sum of electron and hole contributions.

Figure 14

Evolution with diameter $D$ of the energy of the main intraband transitions in the conduction band (+) and in the valence band (×) of spherical nanocrystals compared to the experimental values (∎) of Ref. 10, the size being deduced from the energy of the first exciton peak using Eq. (4). Straight line: fit of the electron intraband transition energy ($E$ in eV, $D$ in nanometer): $E=1∕\left(1.05D\right)$.