Tight-binding model for semiconductor nanostructures

An empirical $s_c{p}_a^3$ tight-binding (TB) model is applied to the investigation of electronic states in semiconductor quantum dots. A basis set of three $p$ orbitals at the anions and one $s$ orbital at the cations is chosen. Matrix elements up to the second nearest neighbors and the spin-orbit coupling are included in our TB model. The parametrization is chosen so that the effective masses, the spin-orbit splitting and the gap energy of the bulk CdSe and ZnSe are reproduced. Within this reduced $s_c{p}_a^3$ TB basis the valence $\left(p\right)$ bands are excellently reproduced and the conduction $\left(s\right)$ band is well reproduced close to the $\Gamma$ point, i.e., near to the band gap. In terms of this model much larger systems can be described than within a (more realistic) $s{p}^3{s}^{*}$ basis. The quantum dot is modeled by using the (bulk) TB parameters for the particular material at those sites occupied by atoms of this material. Within this TB model we study pyramidal-shaped CdSe quantum dots embedded in a ZnSe matrix and free spherical CdSe quantum dots (nanocrystals). Strain effects are included by using an appropriate model strain field. Within the TB model, the strain effects can be artificially switched off to investigate the influence of strain on the bound electronic states and, in particular, their spatial orientation. The theoretical results for spherical nanocrystals are compared with data from tunneling spectroscopy and optical experiments. Furthermore the influence of the spin-orbit coupling is investigated.

Figure 1

Tight-binding band structures for CdSe (a) and ZnSe (b).

Figure 2

Schematic visualization of the pyramidal CdSe QD buried in a ZnSe matrix. The wetting layer has a thickness of one lattice constant $\left(1a\right)$ of bulk ZnSe. The pyramidal QD has a base length $b$ of ten times the ZnSe lattice constant $(b=10a)$.

Figure 3

Strain distribution in and around the embedded pyramidal CdSe QD with a base length of $b=10a$. The WL at the base of the QD is $1a$ thick. The whole structure is buried in a ZnSe matrix. Line scans along the [001] direction through the WL outside the dot (a) and inside the dot through the tip of the pyramid (b) are displayed. The diagonal elements of the strain tensor $\mathbf{ϵ}$ are shown as solid $\left({ϵ}_{zz}\right)$ and dashed-dotted lines $({ϵ}_{xx}={ϵ}_{yy})$.

Figure 4

Electron and hole energies for embedded pyramidal $\mathrm{Cd}\mathrm{Se}∕\mathrm{Zn}\mathrm{Se}$ QDs on a WL of thickness $1a$ (roughly 4 ML) for different base lengths of the dot without strain (a) and with strain effects taken in to account (b). The ground state energies for electrons $\left({\mathrm{WL}}_{e_1}\right)$ and holes $\left({\mathrm{WL}}_{h_1}\right)$ for the WL (of thickness $1a$) alone are also displayed.

Figure 5

Isosurfaces of the squared electron wave functions with and without strain for the embedded $b=10a$ pyramidal QD. The light and dark surfaces correspond to 0.1 and 0.5 of the maximum probability density, respectively.

Figure 6

Isosurfaces plots of the squared hole wave functions with and without strain for the embedded $b=10a$ pyramidal QD. See the caption of Fig. 5 for more details.

Figure 7

First five electron and hole state energies for the embedded pyramidal CdSe QD with $b=10a$ and a 1 ML thick WL. On the left-hand side the results for the unstrained EQD are shown while on the right-hand side the results for the strained EQD are displayed. The zero of the energy scale is the bulk ZnSe valence-band maximum (VBM). The energies are compared with the conduction-band minimum (CBM) of the bulk ZnSe.

Figure 8

Isosurfaces (at 30% of the maximum probability density) of the squared electron and hole wave functions of spherical CdSe nanocrystals of diameter $d=4.85\mathrm{nm}$ for the five lowest states.

Figure 9

Energy gap $E_{\text{gap}}^{\mathrm{nano}}$ as a function of the nanocrystal diameter $d$. Compared are the results from our TB model with (TB) and without (TB-NO SO) spin-orbit coupling, a STM (STM, Ref. 68) and an optical measurement (optical, Ref. 68).

Figure 10

Electron (a) and hole (b) energies as a function of the nanocrystal diameter $d$. For electrons $(e_1–e_5)$ and holes $(h_1–h_5)$ the first five eigenvalues are displayed. Each state is twofold degenerated.

Figure 11

Splitting ${\Delta }_{e_1,e_2}=E_{e_2}-E_{e_1}$ between the lowest two electronic states as a function of the nanocrystal diameter $d$. The results from our TB model, with (TB) and without (TB-NO SO) spin-orbit coupling, and from a STM measurement (STM, Ref. 68) are displayed. Besides this, results from infrared spectroscopy (IR, Ref. 73) and optical methods (optical, Ref. 68) are shown.