Coulomb-Interaction-Induced Incomplete Shell Filling in the Hole System of InAs Quantum Dots

We have studied the hole charging spectra of self-assembled InAs quantum dots in perpendicular magnetic fields by capacitance-voltage spectroscopy. From the magnetic-field dependence of the individual peaks we conclude that the $s$-like ground state is completely filled with two holes but that the fourfold degenerate $p$ shell is only half filled with two holes before the filling of the $d$ shell starts. The resulting six-hole ground state is highly polarized. This incomplete shell filling can be explained by the large influence of the Coulomb interaction in this system.

Figure 1

Capacitance-voltage spectra for the hole system in InAs quantum dots for various magnetic fields. The traces are offset vertically for clarity, and the lines are guides to the eye to indicate the shifts with the magnetic field.

Figure 2

The energetic position of the charging peaks (symbols) from Fig. 1 as function of the magnetic field. The peaks are identified by the same labels as in Fig. 1. $N$ is the total number of holes per QD. The nature of the states with respect to their angular momentum is indicated at the individual graphs. The insets show the occupation of the single-particle levels for each charging peak before and after the level crossing. The arrows represent $J_z=3/2$ and $J_z=-3/2$, respectively. The assignment of the squares to different angular momentum states is given in the upper left corner of the figure. The solid lines represent fits, using a harmonic oscillator model as outlined in the text.

Figure 3

The absolute value of the change in energy $|E(B)-E(0\mathrm{T})|$ for charging peaks 1 to 6 (labels according to Fig. 1), for fields up to 15 T. Three classes of slopes can clearly be distinguished, corresponding to different angular momenta: Peaks 1 and 2 show almost no shift ($L_z=0$), whereas peaks 5 and 6 have twice the slope ($L_z=\pm 2$) as peaks 3 and 4 ($L_z=\pm 1$).