Force field potentials for the vibrational properties of II-VI semiconductor nanostructures

We derive force field potentials for CdS, CdSe, CdTe, ZnS, ZnSe, and ZnTe using the vibrational and elastic properties of the bulk materials based on density functional theory calculations and experiment. The potentials have bond-stretching (first- and second-nearest neighbor), bond-bending, as well as long-range Coulomb interactions. The bulk vibrational properties are very well reproduced for all materials considered. The vibrational properties of nanoclusters are compared to the results of large-scale density functional theory calculations with up to 1000 atoms. We obtain a very good agreement for all the core vibrational modes and an overall reasonable agreement over the entire spectrum. However, the low-frequency acoustic-type modes with surface character tend to be too soft in the force field calculations. We demonstrate the capability of the force field to obtain vibrational properties of realistic colloidal quantum dots with sizes up to 60 Å diameter.

Figure 1

(a) Two-body 1NN interaction as a function of the bond length and (b) three-body term as a function of the angle, assuming in Eq. (4) that the bond lengths $r_{ik}$ and $r_{ij}$ are the ideal bulk values.

Figure 2

Phonon dispersion calculated using the empirical force field model (black solid lines) and via DFPT (red circles) for (a) CdS, (b) CdSe, (c) CdTe, (d) ZnS, (e) ZnSe, and (f) ZnTe.

Figure 3

Comparison of the phonon dispersion (left) and vibrational DOS (right) obtained from the empirical force field (fitted to zinc-blende structure) calculations (black lines) and from DFT (thin red lines) for Wurtzite CdSe.

Figure 4

Comparison of the vibrational DOS obtained from the empirical force field model calculations (black lines) and from DFT (thin red lines) for (a) ${\mathrm{Cd}}_{321}{\mathrm{S}}_{312}$, (b) ${\mathrm{Cd}}_{321}{\mathrm{Se}}_{312}$, and (c) ${\mathrm{Cd}}_{321}{\mathrm{Te}}_{312}$ NCs in the zinc-blende crystal structure.

Figure 5

Vibrational DOS of a ${\mathrm{Cd}}_{225}{\mathrm{Se}}_{240}$ NC resolved into contributions from the core atoms (black lines) and the surface atoms (thin red lines), along with the corresponding bulk phonon DOS (dashed blue lines), all calculated from empirical force field potential calculations.

Figure 6

(a) Bulk vibrational DOS for CdSe and CdS. (b) Vibrational DOS of a ${\mathrm{Cd}}_{79}{\mathrm{Se}}_{68}\text{−}{\mathrm{Cd}}_{242}{\mathrm{S}}_{244}$ core/shell NC with a radius of 9.5 Å for the core part and 15.0 Å for the whole cluster. The total DOS (blue dashed line) is separated into contributions from the core part (black line) and from the shell part (red line). (c) DFT calculation for the structure in (b).

Figure 7

Vibrational DOS of NCs (thick black lines) with corresponding bulk phonon DOS (thin red lines) calculated using our empirical force field potentials for (a)–(c) CdS, (d)–(f) CdSe, and (g)–(i) CdTe NCs with diameters between 40 and 60 Å.

Figure 8

Vibrational DOS of NCs (thick black lines) with corresponding bulk phonon DOS (thin red lines) calculated using our empirical force field potentials for (a)–(c) ZnS, (d)–(f) ZnSe, and (g)–(i) ZnTe NCs with diameters between 40 and 60 Å.

Figure 9

Total vibrational DOS of NCs (dashed lines), contribution of the core atoms (black liness), and contribution of surface atoms (red lines) to the vibrational DOS for (a) ${\mathrm{Cd}}_{952}{\mathrm{S}}_{959}$, (b) ${\mathrm{Cd}}_{952}{\mathrm{Se}}_{959}$, (c) ${\mathrm{Cd}}_{952}{\mathrm{Te}}_{959}$, (d) ${\mathrm{Zn}}_{952}{\mathrm{S}}_{959}$, (e) ${\mathrm{Zb}}_{952}{\mathrm{Se}}_{959}$, (f) ${\mathrm{Zn}}_{952}{\mathrm{Te}}_{959}$.