Formation of spin droplet at $\nu =\frac{5}{2}$ in an asymmetric quantum dot under quantum Hall conditions

In this work, a quantum dot that is defined asymmetrically by electrostatic means induced on a GaAs/AlGaAs heterostructure is investigated to unravel the effect of geometric constraints on the formation of spin droplets under quantized Hall conditions. The incompressibility of the excited $\nu =\frac{5}{2}$ state is explored by solving the Schrödinger equation within spin density functional theory, where the confinement potential is obtained self-consistently utilizing the Thomas-Fermi approximation. Our numerical investigations show that the spatial distribution of the $\nu =2$ incompressible strips and electron occupation in the second lowest Landau level considerably differ from the results of the laterally symmetric quantum dots. Our findings yield two important consequences: first, the incompressibility of the intriguing $\nu =\frac{5}{2}$ state is strongly affected by the asymmetry, and second, since the Aharonov-Bohm interference patterns depend on the velocity of the particles, asymmetry yields an additional parameter to adjust the oscillation period, which imposes a boundary condition dependency in observing quasiparticle phases.

Figure 1

Sketch of the layer sequence of GaAs/AlGaAs heterostructure, which has the dimensions of $2550\times 2550\times 784$ $\mathrm{n}{\mathrm{m}}^3$. The heterostructure is grown on a thick GaAs substrate, where 2DEG is formed at the interface of the GaAs/AlGaAs denoted by the green region. Silicon donor layers (dark brown regions), distributed homogeneously on the $xy$ plane, provide electrons both to the surface and to the 2DEG. The light gray and black regions on the surface are shown as the etched and the metallic gate regions, respectively.

Figure 2

The confinement potential for electrons at the interface of the GaAs/AlGaAs heterostructure obtained with self-consistent electronic calculations. Inset graph shows the symmetric confinement potential (when applied $-2.3$ V) more clearly.

Figure 3

A FPI device. Current flows along the counterpropagating edge states (shown by yellow lines). Tunneling occurs in the two narrow constrictions, when the edge channels are at a sufficient proximity.

Figure 4

The slopes of the confinement potential of the sample give the electron velocity as a function of the position.

Figure 5

Spin-up, spin-down, and total electron density of quantum Hall state in a QD includes 30 electrons calculated with the SDFT at $\nu =\frac{5}{2}$ for various gate voltages.

Figure 6

LDOS for different electron numbers and filling factors.

Figure 7

Chemical potentials compared with symmetric confining potential for various $N$-electron symmetric quantum dots as a function of the magnetic field $B$. The dotted ellipse denotes the spin-droplet regime.

Figure 8

Chemical potentials compared with asymmetric confining potential for various $N$-electron symmetric quantum dots as a function of the magnetic field $B$. The dotted ellipse denotes the spin-droplet regime.

Figure 9

Spin-up, spin-down, and total electron density of quantum Hall state in a QD that includes 48 electrons calculated with the SDFT at $\nu =\frac{5}{2}$ for various gate voltages.

Figure 10

Spin-up, spin-down, and total electron density of quantum Hall state in a QD that includes 54 electrons calculated with the SDFT at $\nu =\frac{5}{2}$ for various gate voltages.

Figure 11

Spin-up, spin-down, and total electron density of quantum Hall state in a QD that includes 60 electrons calculated with the SDFT at $\nu =\frac{5}{2}$ for various gate voltages.