Excitons in core-only, core-shell and core-crown CdSe nanoplatelets: Interplay between in-plane electron-hole correlation, spatial confinement, and dielectric confinement

Using semianalytical models we calculate the energy, effective Bohr radius, and radiative lifetime of neutral excitons confined in CdSe colloidal nanoplatelets (NPLs). The excitonic properties are largely governed by the electron-hole in-plane correlation, which in NPLs is enhanced by the quasi-two-dimensional motion and the dielectric mismatch with the organic environment. In NPLs with lateral size $L\gtrsim 20$ nm the exciton behavior is essentially that in a quantum well, with super-radiance leading to exciton lifetimes of 1 ps or less, only limited by the NPL area. However, for $L<20$ nm excitons enter an intermediate confinement regime, hence departing from the quantum well behavior. In heterostructured NPLs, a different response is observed for core-shell and core-crown configurations. In the former, the strong vertical confinement limits separation of electrons and holes even for type-II band alignment. The exciton behavior is then similar to that in core-only NPL, albeit with weakened dielectric effects. In the latter, charge separation is also inefficient if band alignment is quasi-type-II (e.g., in CdSe/CdS), because electron-hole interaction drives both carriers into the core. However, it becomes very efficient for type-II alignment, for which we predict exciton lifetimes reaching microseconds.

Figure 1

(a) Schematic of the core-only NPL under study. (b) Exciton emission energy in square and rectangular CdSe NPLs with different size, shape, and thickness. Square NPLs have lateral dimensions $(L_x,L_y)=(L,L)$, and rectangular ones $(L,15)$ nm. Two thicknesses are considered, $L_z=1.4$ nm ($\sim 4.5\mathrm{ML}$) and $L_z=1.7$ nm ($\sim 5.5\mathrm{ML}$). The effect of lateral confinement is felt under $L=15$ nm. (c) Binding energy of the square NPL with 5.5 ML thickness. Solid line: Fully correlated exciton. Dashed line: Perturbational estimate, neglecting correlation. Dotted line: Same but neglecting dielectric confinement. (d) Effective Bohr radius for the NPLs of (a). Values lie between the 3D and 2D limit. The latter is reached at $L\approx 20$ nm.

Figure 2

Exciton radiative lifetime in CdSe NPLs with different lateral size, shape, and thickness. The lifetime is approximately proportional to ${\left(a_B^{*}\right)}^2/\left(L_x{L}_y\right)$, as shown by dotted-dashed lines. Inset: Lifetime of a 4.5 ML square NPL neglecting electron-hole correlation. Notice they are orders-of-magnitude longer, and the size dependence is weaker.

Figure 3

(a) Energy splitting of excited exciton states (${\mathrm{\Phi }}^{ps},{\mathrm{\Phi }}^{sp},{\mathrm{\Phi }}^{pp}$) with respect to the ground state (${\mathrm{\Phi }}^{ss}$). Solid (dashed) lines are used for optically bright (dark) states. (b) Ratio between radiative recombination rates of the ground (${\mathrm{\Phi }}^{ss}$) and first bright (${\mathrm{\Phi }}^{pp}$) exciton states. Inset: Thermally averaged recombination rate. The NPL dimensions are $(L_x,L_y,L_z)=(30,30,1.4)$ nm.

Figure 4

(a) Schematic of a core-shell NPL and band-offset potential along $z$ direction. (b) Exciton lifetime including (solid line) and excluding (dashed line) Coulomb correlation, as a function of the $cbo$. The insets show the electron and hole charge density along the growth direction of the NPL for the exact solution. As the electron and hole become separated, the exciton lifetime increases, but it remains super-radiant.

Figure 5

Effect of electron-hole separation on dielectric confinement energetic terms. Dashed lines: Self-energy repulsion. Dotted lines: Electron-hole Coulomb attraction enhancement due to dielectric mismatch. Solid line: Net effect on exciton emission energy.

Figure 6

(a) Schematic of a core-crown NPL. (b) Exciton lifetime in quasi-type-II (solid line) and type-II (dotted line) core-crown NPLs as a function of the core size. The insets show the electron charge density, with dark/light areas indicating low/high density. The electron stays in the core for quasi-type-II band alignment, but it delocalizes into the shell for type-II ones. $L_{\text{crown}}=2L_{\text{core}}$.

Figure 7

Integration domain defined by the limits given in Eqs. (A4, A5, A6, A7). The circled numbers label the three subdomains involved in Eq. (A8).