Spin droplets in confined quantum Hall systems

Two-dimensional semiconductor quantum dots are studied in the filling-factor range $2<\nu <3$. We find both theoretical and experimental evidence of a collective many-body phenomenon, where a fraction of the trapped electrons form an incompressible spin droplet on the highest occupied Landau level. The phenomenon occurs only when the number of electrons in the quantum dot is larger than $\sim 30$. We find the onset of the spin-droplet regime at $\nu =5∕2$. This proposes a finite-geometry alternative to the Moore-Read-type Pfaffian state of the bulk two-dimensional electron gas. Hence, the spin-droplet formation may be related to the observed fragility of the $\nu =5∕2$ quantum Hall state in narrow quantum point contacts.

Figure 1

(Color online) Total energies of spin states of the $N=48$ quantum dot $(\hslash {\omega }_0=2\mathrm{meV})$ calculated with the quantum Monte Carlo method in the spin-droplet regime. The linewidths denote the statistical error in the calculations. The table shows the density-functional-theory result for the maximum number of electrons in the spin droplet, $N_{\mathrm{SD}}$, as a function of $N$, as well as the total-energy decrease at $\nu =5∕2$ due to the spin polarization.

Figure 2

(Color online) Electron densities of a 60-electron quantum dot calculated with the density-functional method at $\nu =2$ (a), at an intermediate state between $\nu =2$ and $\nu =5∕2$ (b), and at $\nu =5∕2$ (c). The red (solid) and blue (transparent) regions mark the electron densities in the spin-compensated lowest (0LL) and the spin-polarized second-lowest Landau levels (1LL), respectively. The series of figures thus demonstrates the formation and growth of the spin droplet in the second lowest Landau level.

Figure 3

(Color online) Chemical potentials calculated with density-functional theory (a) and measured from a vertical (b) and lateral (c) quantum-dot devices for various electron numbers. Both experiments show the emergence of the spin-droplet signals in the peak position data when $N\gtrsim 30$ in agreement with the theoretical result. Data for the vertical device in (b) is courtesy of L. Kouwenhoven, Delft (Ref. 18).

Figure 4

(Color online) Chemical potential of two samples of lateral 45-electron dots obtained using spin-blockade spectroscopy and the corresponding theoretical result from density-functional calculations. Both samples show a clear spin-droplet plateau at $5∕2\ge \nu >2$ in agreement with the theory. The high; medium; low; and very-low-amplitude peaks in theory correspond to transport involving the 0LL spin $\uparrow$, 0LL spin $\downarrow$, 1LL spin $\uparrow$, and 1LL spin $\downarrow$ states, respectively. The numbers correspond to the $z$ components of the ground-state total angular momenta $\left(L\right)$ and spins $\left(S\right)$ for the high- and medium-amplitude states.