Quantum confinement of nanocrystals within amorphous matrices

Nanocrystals encapsulated within an amorphous matrix are computationally analyzed to quantify the degree to which the matrix modifies the nature of their quantum-confinement power—i.e., the relationship between nanocrystal size and the gap between valence- and conduction-band edges. A special geometry allows exactly the same amorphous matrix to be applied to nanocrystals of increasing size to precisely quantify changes in confinement without the noise typically associated with encapsulating structures that are different for each nanocrystal. The results both explain and quantify the degree to which amorphous matrices redshift the character of quantum confinement. The character of this confinement depends on both the type of encapsulating material and the separation distance between the nanocrystals within it. Surprisingly, the analysis also identifies a critical nanocrystal threshold below which quantum confinement is not possible—a feature unique to amorphous encapsulation. Although applied to silicon nanocrystals within an amorphous silicon matrix, the methodology can be used to accurately analyze the confinement softening of other amorphous systems as well.

Figure 1

Confinement power $\alpha$ of a finite square well based on the difference between the lowest two energy levels. $d_0$ is a characteristic length.

Figure 2

Si NCs of increasing size ($d$) are encapsulated within a cloned a-Si:H matrix with Si atoms (tan) and H atoms (magenta) that is periodic in all directions. For this particular a-Si structure, the hydrogenation procedure primarily affects the region to the right of the NC. Four of the ten structures considered are shown. HOMO isosurfaces (green where in a-Si:H and red where in NC) make clear that there is a minimum crystal size required for the HOMO to be localized within the crystal.

Figure 3

Emergence of confined states with increasing NC size. Conduction and valence energies are referenced so that the band edges have zero energy for the largest NC size. The solid curves passing through the highest occupied and lowest unoccupied orbital (HOMO and LUMO) energies were generated using Eqs. (1) and (2). Gray dots represent calculated eigenstates of the combined a-S:H/NC system.

Figure 4

(a) Kronig-Penney model with an isolated well for a-Si:H and multiple wells representing the NC. Matrix HOMOs plotted for each of the NCs are essentially identical (green) with a confined HOMO (red) emerging for NCs with at least eight wells. (b) Confined states appear in the NC at higher energy than the matrix HOMO as size increases. The color scheme of the orbitals (top) and valence energies (bottom) is consistent.

Figure 5

Effect of boundary conditions on HOMO energy for a-Si:H/NC. The size of the amorphous matrix, L, is held fixed, while the length of the NC, $d$, is increased. (a) Example of “hard” boundaries (H-terminated, magenta). Si atoms are green (a-Si) and tan (NC). Gray lines indicate periodicity. The associated HOMO isosurface is shown in blue. (b) HOMO energy vs NC size for periodic and hard boundaries.

Figure 6

(a) HOMOs for a two-level model as a function of domain size with hard and periodic boundary conditions. The size of the amorphous matrix, L, is held fixed, while the length of the NC, $d$, is increased. Gray lines indicate the extent of the NC in each case. Periodic wave functions (red) bend more with increasing length, while wave functions associated with hard boundaries (blue) bend less. (b) Corresponding HOMO energies.

Figure 7

Comparison of energy gaps for free-standing and encapsulated Si NCs. The crystal are otherwise identical and the matrix is the same for all encapsulated NCs. The confinement effect is softened by the surrounding a-Si matrix.