Dark and bright excitonic states in nitride quantum dots

Formation of excitonic states in quantum dots, of nitride based III-V semiconductors GaN and AlN, including Coulombic interaction, exchange interaction, and dielectric effects, are investigated. Dark exciton formation is found to occur for both GaN quantum dots (QD’s) with wurtzite structure having positive crystal field splitting and GaN and AlN QD’s with zinc-blende structure having zero crystal field splitting. The transition from dark to bright exciton occurs between radii range $20–40\mathrm{\AA }$ depending on the amount of dielectric mismatch between the dot and the surroundings. In wurtzite AlN QD’s with negative crystal field splitting, the splitting between the dark and bright excitonic states is very small and vanishes at about $15\mathrm{\AA }$.

Figure 1

The energy levels in units of ${ϵ}_0=({\gamma }_1∕2m_o){(\hslash ∕R)}^2$ for w-GaN QD’s as a function $R$ in (a) absence and (b) presence of e-h Coulomb interaction (without exchange). The energy reference is taken as the top of the valence band with the spin orbit effects included. Eigenvalues in the absence $(ϵ=1)$ as well as presence $(ϵ=5)$ of dielectric mismatch effects are also shown in (b). The states are labeled as $S_x\downarrow$, $P_x\uparrow$, etc., where capital letters represent the dominant $\mathbf{L}$ present and the subscripts for $X$-, $Y$-like or $Z$-like states, arrow indicating the spin state. (c) Eigenvalues including e-h Coulomb and exchange interactions. The states are labeled with $F_z$ with superscripts standing for upper and lower state. The midpoint of the exciton energies with $M=\pm \frac{3}{2}$ and $M=\pm \frac{1}{2}$ is taken as the reference energy (zero of energy).

Figure 2

The energy levels in units of ${ϵ}_o$ for z-GaN QD’s as a function of dot radius $R$ in the (a) absence and (b) presence of e-h Coulomb interaction and dielectric mismatch effects (without exchange). The energy reference is taken as the top of the valence band without the spin orbit effects included. The states are labeled as $S_{3∕2}$, etc. where the subscripts represent the total angular momentum. (c) Eigenvalues including the e-h Coulomb and exchange interactions.

Figure 3

The energy levels in units of ${ϵ}_0$ for w-AlN QD’s in the (a) absence and (b) presence of e-h Coulomb interaction and dielectric mismatch effects.

Figure 4

The energy difference in meV between bright excitonic state and dark excitonic state for w-GaN. (a) Energy difference of $S$ and $P$ states with $M=\frac{3}{2}$ and $\frac{1}{2}$, respectively. (b) The singlet triplet splitting in the absence $ϵ={ϵ}_{\text{dot}}∕{ϵ}_{\text{surrounding}}=1$ and presence $ϵ=5$ of dielectric mismatch effects.

Figure 5

The energy difference in meV between bright excitonic state and dark excitonic state for z-GaN. (a) Energy differences of $S$ and $P$ states. (b) Singlet triplet splitting in meV in the absence and presence of dielectric mismatch effects.

Figure 6

The energy levels in units of meV for w-AlN QD’s after the inclusion of exchange interaction. To show the splitting between the ground state and the first excited state, the ground state $0^L$ has been shown as dashed line and the excited states as the solid lines. Some of the excited states have been shown in the inset to clearly show the above splitting.

Figure 7

The excitation-deexcitation processes taking place at the band edges.