Magnetic-field dependence of hole levels in self-assembled $\mathrm{InGaAs}$ quantum dots

Recent magnetotransport experiments of holes in $\mathrm{InGaAs}$ quantum dots [D. Reuter, P. Kailuweit, A. D. Wieck, U. Zeitler, O. Wibbelhoff, C. Meier, A. Lorke, and J. C. Maan, Phys. Rev. Lett. 94, 026808 (2005)] are interpreted by employing a multiband $\mathbf{k}\bullet \mathbf{p}$ Hamiltonian, which considers the interaction between heavy hole and light hole subbands explicitly. No need of invoking an incomplete energy shell filling is required within this model. The crucial role we ascribe to the heavy hole-light hole interaction is further supported by one-band local-spin-density functional calculations, which show that Coulomb interactions do not induce any incomplete hole shell filling and therefore cannot account for the experimental magnetic field dispersion.

Figure 1

Addition energy versus number of holes in a QD with $R=16\mathrm{nm}$ and $h=4\mathrm{nm}$ (solid line) and in a QD with $R=20$ nm and $h=6$ nm (dashed line). The insets show the occupation of the Kohn-Sham orbitals for each $N$, the arrows representing the sign of the spin.

Figure 2

Single-particle, four-band energy levels versus magnetic field for a QD with $R=16\mathrm{nm}$ and $h=4\mathrm{nm}$. Solid (dashed) lines are used for the energy levels calculated without (with) spin Zeeman splitting.

Figure 3

Absolute value of the magnetic field-induced energy shift for charging peaks of 1 to 6 holes versus the magnetic field. Solid lines represent the average slope of each pair of peaks.