Interrelation of structural and electronic properties in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}∕\mathrm{Ga}\mathrm{N}$ quantum dots using an eight-band $\mathbf{k}\bullet \mathbf{p}$ model

We present an eight-band $\mathbf{k}\bullet \mathbf{p}$-model for the calculation of the electronic structure of wurtzite semiconductor quantum dots (QDs) and its application to indium gallium nitride $\left({\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}\right)$ QDs formed by composition fluctuations in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ layers. The eight-band $\mathbf{k}\bullet \mathbf{p}$-model accounts for strain effects, piezoelectricity and pyroelectricity, and spin-orbit and crystal-field splitting. Exciton binding energies are calculated using the self-consistent Hartree method. Using this model, we studied the electronic properties of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ QDs and their dependence on structural properties, i.e., their chemical composition, height, and lateral diameter. We found a dominant influence of the built-in piezoelectric and pyroelectric fields, causing a spatial separation of the bound electron and hole states and a redshift of the exciton transition energies. The single-particle energies as well as the exciton energies depend heavily on the composition and geometry of the QDs.

Figure 1

Schematic drawing of the model InGaN QDs used in the calculation. Ellipsoids with a height of $d_z$ and lateral diameter of $d_x=d_y$. The indium concentration inside the QDs is modeled with a linear gradient from the maximum indium fraction $x_c$ at the center of the QDs to $x=0.1$ at their border. The QDs are embedded in an ${\mathrm{In}}_{0.1}{\mathrm{Ga}}_{0.9}\mathrm{N}$ layer with a height of $2\mathrm{nm}$. The layer itself is embedded in a matrix of pure GaN.

Figure 2

Linescan of the built-in electric potential of the largest QD—D4 in Table —along the [0001] direction through the QD’s center. The solid line shows the overall potential. The dashed line and the dotted line show the contributions of the piezoelectric polarization and the pyroelectric polarization. The dotted ellipse indicates the position of the QD.

Figure 3

$\pm 230\mathrm{mV}$ isosurfaces of the overall built-in electric potential of QD D4 (see Table ). The surface of the QD is shown in the middle in light gray. The positive isosurface is located atop the QD, making this area attractive for electrons. The negative one is found beneath it; this area becomes attractive for holes.

Figure 4

Line scan of the local band-edge profile of QD D4 (see Table ) along the [0001] direction through the QD’s center. (a) Without electric potentials. (b) Including electric potentials. The dashed ellipses indicate the position of the QD.

Figure 5

Bound single-particle states for QD D4 ($d_z=2\mathrm{nm}$, $d_{x∕y}=6.4\mathrm{nm}$, $x_c=0.5$): electron ground state $\left(e0\right)$, hole ground state $\left(h0\right)$, and the first two excited states for both ($e1∕e2$ and $h1∕h2$). The picture shows the 65% isosurfaces of the propability density distribution ${\mid \Psi \mid }^2$ in the top and side view. Each state is twofold spin-degenerate. The energies of the single-particle states are listed in Table .

Figure 6

Single-particle electron and hole state energies—with respect to the VB edge of unstrained GaN—for all three calculated series. (a) Variation of lateral diameter; (b) variation of height; (c) variation of indium concentration.

Figure 7

Transition (upper panel) and binding energies (lower panel) of the $\mid A⟩$ excitons for all three calculated series. (a) Variation of lateral diameter; (b) variation of height; (c) variation of indium concentration. The binding energies of $\mid B⟩$ excitons are identical to the $\mid A⟩$-exciton binding energies $(\pm 0.6\mathrm{meV})$.