Effects of thickness in quantum dots in strong magnetic fields

We study the effects of thickness on the ground states of two-dimensional quantum dots in high magnetic fields. To be specific, we assume the thickness to be small so that only the lowest state in the corresponding direction is occupied, which however leads to a modification of the effective interaction between the electrons. We find the ground-state phase diagram and demonstrate the emergence of different phases as the thickness is accounted for. Finally, the wave-functional form and vortex structure of different phases are analyzed.

Figure 1

Likely configurations in six-electron quantum dots.

Figure 2

The ground-state angular momentum phase diagram for (a) five and (b) six electrons as a function of the thickness parameter $d$ and the magnetic field $B$. The vertical lines indicate the regular angular momentum difference $\Delta M$ between adjacent phases. For six electrons, the angular momentum difference changes depending on the thickness except for phases A and B, as well as for the states with $\nu =1$, $\frac{1}{3}$, and $\frac{1}{5}$.

Figure 3

Same as Fig. 2 but for (a) seven and (b) eight electrons.

Figure 4

The addition energy $\mu \left(N\right)=E_{\text{gs}}\left(N\right)-E_{\text{gs}}\left(N-1\right)$ for $N=8$ and $d=0\left(\text{uppermost}\right),0.1l,0.2l,\dots ,1.9l$. The black dots mark the ground-state transitions of the $N=8$ quantum dot for integer $d/l$.

Figure 5

The complex phase of the conditional wave functions for the six-electron quantum dot, in which the electron coordinates are fixed according to the most likely configuration indicated by $+$, and the uppermost electron is used as the probe electron. In the first row, the thickness parameter $d=l$ and the angular momenta are $M=M_{\text{MDD}}+5k$ with $k=2$, 3, 4, 5, and 6. In the second row, the thickness parameter $d=3l$ and the angular momenta are $M=M_{\text{MDD}}+6k$ with $k=1$, 2, 3, 4, and 5.