Tight-binding model for semiconductor quantum dots with a wurtzite crystal structure: From one-particle properties to Coulomb correlations and optical spectra

In this work we investigate the electronic and optical properties of self-assembled quantum dots by means of a tight-binding model. Coulomb and dipole matrix elements are calculated from the one-particle wave functions which fully include the atomistic wurtzite structure of the low-dimensional heterostructures and serve as an input for the calculation of optical spectra. For the investigated $\mathrm{In}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ material system, the optical selection rules are found to be strongly affected by band-mixing effects for the localized valence band states. Within this framework, excitonic absorption and emission spectra are analyzed for different sizes of the investigated lens-shaped quantum dots, including the influence of the intrinsic and strain-induced electrostatic field of the wurtzite structure. A dark exciton ground state for small quantum dots is found.

Figure 1

Tight-binding band structure for bulk InN and GaN. The corresponding tight-binding parameters for each material are given in Table .

Figure 2

(Color online) Illustration of the finite-size supercell in which the QD geometry is modeled. The investigated lens-shaped InN QDs are sitting atop of an InN wetting layer (WL) and have circular symmetry around the $z$ axis. The InN QD-WL system is embedded in a GaN matrix.

Figure 3

Contour plot of the electrostatic potential energy $V_p$. A slice in the $x\text{−}z$ plane through the center of the QD is depicted. The result is shown for the smallest QD but is qualitatively the same for all the investigated QD sizes.

Figure 4

(Color online) The QD geometry is shown for the largest QD from atop. The structure is visualized and isosurfaces of the probability density for the three lowest electron (left) and hole (right) states with built-in field are included for 10% (blue) and 50% (red) of the maximum value. The atomistic structure and the $C_{3v}$ symmetry of the wurtzite crystal becomes most apparent for the hole states. The corresponding energies $\left(E_{1,2,3}^{e,h}\right)$ of electron and hole states, measured from the valence band maximum of bulk GaN, and the atomic orbital character for each wave function are given. The dominant contributions are highlighted.

Figure 5

(Color online) Planar-integrated probability density $P_{\text{planar}}$ for the electron (left) and the hole (right) ground state, without (solid lines) and with (dashed lines) built-in field for the smallest (top) and the largest (bottom) QD.

Figure 6

(a) The first five electron and 10 hole one-particle energy levels are shown for the three investigated QD sizes with increasing diameter from left to right (the dashed lines are included as a guide to the eye). The results are shown with the influence of the built-in field. As a consequence of the stronger quantum confinement of the carriers inside the QD structure for decreasing size, a clear blueshift of the effective gap energy is observed. (b) The energy splitting $\Delta E_{s,p}^h=E_s^h-E_p^h$ between the $s$ and the $p$ shell for the holes is depicted. By definition the splitting is positive for the smallest QD and changes sign with increasing QD diameter $d$, where the twofold degenerate $p$ shell constitutes the hole ground states (the dashed lines are included as a guide to the eye).

Figure 7

(Color online) For the intermediate QD size the influence of the built-in field on the hole-state ordering is demonstrated. Without the electrostatic field the hole ground state is built by a single $s$-like state (solid line), whereas with the influence of the field the twofold degenerate $p$ state (dashed line) constitutes the ground state for the holes. The dotted lines are included as a guide to the eye.

Figure 8

(Color online) (a) Excitonic absorption for the small QD with (solid line) and without (dashed line) the influence of the built-in field. (b) Same as (a) but for the large QD. (c) Excitonic ground state emission for the large QD with the influence of the built-in field. Without the built-in field, the ground state emission vanishes and is consequently not shown.

Figure 9

Coulomb matrix elements calculated with a cutoff radius $R_0$ with $\mid \mathbf{R}-{\mathbf{R}}^{\prime }\mid <R_0$ in the sum of Eq. (A1). Results are shown for the direct electron-electron (solid line), hole-hole (dashed line), and electron-hole (dashed-dotted line) Coulomb matrix elements for the ground state wave functions without the influence of the built-in field. (a) Small QD. (b) Large QD.

Figure 10

(Color online) Overlap matrix $S_{\alpha \beta }^{0{\mathbf{R}}^{\prime }}$ for a nitrogen atom to the atomic orbitals centered at the atom sites ${\mathbf{R}}^{\prime }$ in the neighborhood up to third-nearest neighbors. The results are calculated with the original Slater orbitals. The overlap for the four different nitrogen orbitals $\alpha ∊\{p_x,p_y,p_z,s\}$ is stacked in vertical direction and is visualized by the color coding. Contributions from different neighbors and their different orbitals $\beta$ are given on the horizontal axis with increasing distance $\mid {\mathbf{R}}^{\prime }\mid$ to the center nitrogen atom from left to right.

Figure 11

(Color online) Same as Fig. 10 but calculated with the orthogonalized Slater orbitals.

Figure 12

(Color online) Nitrogen basis orbitals (Slater orbitals) without (top) and with (bottom) orthogonalization up to third-nearest neighbors. The depicted orbitals are from left to right: $p_x$, $p_y$, $p_z$, and $s$. The viewpoint is on the $z$ axis. Two isosurfaces of the probability density are included with 20% (inner surface) and 0.001% (outer surface) of the maximum value, respectively.