Spin-orbit splitting and critical point energy at $\Gamma$ and $L$ points of cubic CdTe nanoparticles: Effect of size and nonspherical shape

This study reports the observation of changes in critical point energy at $\Gamma$ and $L$ points and an increase in spin-orbit splitting energy of cubic CdTe nanoparticles in comparison to the bulk single-crystal value. ${\epsilon }_1$ and ${\epsilon }_2$ spectra of CdTe nanoparticles, accurately derived from spectroscopic ellipsometry measurements on CdTe nanoparticles dispersed in ${\mathrm{SiO}}_2$ films, show $E_0$, $E_0+{\Delta }_0$, $E_1$, and $E_1+{\Delta }_1$ critical points at 1.75, 2.74, 3.29, and 3.83 eV, respectively. Glancing-angle x-ray diffraction and high-resolution transmission electron microscopy investigations confirm cubic CdTe nanoparticles with good crystallinity and a nonspherical shape. Variable magnitudes of size-induced stress along the minor and major axes cause dissimilar shifts of heavy-hole and split-off bands resulting in a positive contribution to spin-orbit splitting energy, in addition to the small size.

Figure 1

GAXRD spectra of (curve a) as-deposited (A0) and vacuum annealed samples (curve b) A2, (curve c) A4, and (curve d) A6.

Figure 2

Four-layer optical model used to simulate the optical constants. EMA $(\mathrm{CdTe}+{\mathrm{SiO}}_2)$ layer is a dummy layer with a zero thickness, which provides variable optical functions to the EMA layer. The maximum thickness variation will be $\pm 0.5\mathrm{nm}$. In the absence of the dummy layer, the model serves as three-layer model.

Figure 3

Curve with open squares (◻) represents the simulated values of ${\epsilon }_2$ corresponding to CdTe nanoparticles. These simulated values provide a match between the measured (엯) and fitted (line) values of ${\epsilon }_2$ spectra corresponding to CdTe nanoparticles dispersed in ${\mathrm{SiO}}_2$ samples.

Figure 4

${\epsilon }_2$ spectra of samples A0, A2, A4, A6, and bulk CdTe.

Figure 5

Fitted ${\epsilon }_1$ and ${\epsilon }_2$ spectra of CdTe nanoparticles (NP) (solid lines) and bulk (dashed lines).

Figure 6

Simulated ${\epsilon }_2^{″}$ spectra of CdTe nanoparticles showing the position of critical point transitions. Vertical bars denote critical point transitions in bulk CdTe (Ref. 13). Arrows indicate the values corresponding to CdTe nanoparticles.

Figure 7

Nonlinear best-fit curves for second derivatives of ${\epsilon }_1$ and ${\epsilon }_2$ spectra. $E_0$ and $E_0+{\Delta }_0$ are the critical point transitions at the $\Gamma$ point and $E_1$ and $E_1+{\Delta }_1$ are the critical point transitions at the $L$ point.

Figure 8

(a) TEM image showing CdTe nanoparticles dispersed in ${\mathrm{SiO}}_2$ sample. (b), (c) Typical HRTEM images showing the nonspherical shape of CdTe nanoparticles. Nanoparticle dimensions along major and minor axes are also shown. Plane (220) is oriented at $\sim 40°$ with respect to major axis. Major $\left(x\right)$ and minor $\left(y\right)$ axes have dimensions of 10–11 and 7–9 nm and the average diameter ${\left(y^{2}x\right)}^{1∕3}$ of the nonspherical nanoparticles is $\sim 8.5\mathrm{nm}$.

Figure 9

Schematic diagram showing the comparison of the band structure of bulk CdTe (solid curves) and CdTe nanoparticles (dotted curves). Heavy-hole, light-hole, and split-off hole subbands are denoted as $A$, $B$, and $C$ bands, respectively. For clarity, the conduction band is denoted as the $D$ band. Based on the result of the present study, the critical point transitions $E_0$ and $E_0+{\Delta }_0$, SO splitting ${\Delta }_0$, shifts in conduction band $\left(\Delta E_D\right)$, heavy hole $\left(\Delta E_A\right)$, and split-off subbands $\left(\Delta E_C\right)$ are also shown.