Maximum density hole droplets of an antidot in strong magnetic fields

We investigate a quantum antidot in the integer quantum Hall regime (the filling factor is two) by using a Hartree-Fock approach and by transforming the electron antidot into a system which confines holes via an electron-hole transformation. We find that its ground state is the maximum density droplet of holes in certain parameter ranges. The competition between electron-electron interactions and the confinement potential governs the properties of the hole droplet such as its spin configuration. The ground-state transitions between the droplets with different spin configurations occur as magnetic field varies. For a bell-shape antidot containing about 300 holes, the features of the transitions are in good agreement with the predictions of a recently proposed capacitive interaction model for antidots as well as recent experimental observations. We show this agreement by obtaining the parameters of the capacitive interaction model from the Hartree-Fock results. An inverse parabolic antidot is also studied. Its ground-state transitions, however, display different magnetic-field dependence from that of a bell-shaped antidot. Our study demonstrates that the shape of antidot potential affects its physical properties significantly.

Figure 1

A quantum Hall antidot with localized antidot states and extended edge channels. The localized states are weakly coupled to the extended channels, as indicated by dashed and dotted lines. The small arrows show the spin direction of the states. The local filling factor around the antidot is two.

Figure 2

Schematic diagram of particle densities, as a function of distance from antidot center, in a hole MDD where $N_{\uparrow }$ spin-up holes occupy single-particle states with angular momentum $m=0,1,\dots ,N_{\uparrow }-1$ and $N_{\downarrow }$ spin-down holes with $m=0,1,\dots ,N_{\downarrow }-1$. Single-particle hole states with spin $\sigma$ are empty for $m>N_{\sigma }-1$. Note that a single-particle state with smaller $m$ is located at a smaller distance from the center. The changes in electron and hole densities around the antidot are also shown when a spin-down MDD transition $\mid N_{\uparrow },N_{\downarrow }⟩\to \mid N_{\uparrow },N_{\downarrow }+1⟩$ occurs. In this process, a spin-down electron tunnels out of the antidot (indicated as thin arrows) and thus the total hole spin decreases.

Figure 3

Same diagram as in Fig. 2, but for the spin-flip transition $\mid N_{\uparrow },N_{\downarrow }⟩\to \mid N_{\uparrow }+1,N_{\downarrow }-1⟩$. In this process, a cotunneling event (see thin arrows) takes place, where a spin-up electron moves out of the antidot while a spin-down electron moves in, and thus the total electron (hole) spin decreases (increases).

Figure 4

Chemical potential, ${\mu }_N\equiv E_{N+1}-E_N$, versus magnetic field $B$ for a bell-shaped antidot potential. Different values of $N=N_{\uparrow }+N_{\downarrow }∊[368,384]$ are used. The horizontal dotted line represents hole Fermi energy, which can be rather different from electron Fermi energy. Diamonds represent the spin-flip transition $\mid N_{\uparrow },N_{\downarrow }⟩\to \mid N_{\uparrow }+1,N_{\downarrow }-1⟩$ as $B$ increases, while vertical jumps show the spin-down transition $\mid N_{\uparrow },N_{\downarrow }⟩\to \mid N_{\uparrow },N_{\downarrow }+1⟩$. Following the zigzag solid line, the MDD ground state evolves as $\mid N_{\uparrow },N_{\downarrow }⟩=\mid 175,193⟩\to \mid 175,194⟩\to \mid 175,195⟩\Rightarrow \mid 176,194⟩\to \mid 176,195⟩\to \mid 176,196⟩\Rightarrow \mid 177,195⟩\to \mid 177,196⟩\to \mid 177,197⟩\Rightarrow \mid 178,196⟩\to \mid 178,197⟩\to \mid 178,198⟩$ with increasing $B$. Here $\to$ and $\Rightarrow$ indicate, respectively, the spin-down and spin-flip transitions. The parameters of this antidot can be found in Sec. 5B.

Figure 5

The boundary (critical magnetic fields) of stable and unstable MDDs of a bell-shaped antidot. The following parameters are used: $\hslash \Omega =1.5\mathrm{meV}$, $C∕\left(2{\ell }^2\right)=2100\sqrt{B\left(\mathrm{Tesla}\right)}\times e^2∕\left(ϵ\ell \right)$, and $m_t=118$.

Figure 6

The renormalized Hartree-Fock single hole energies ${\epsilon }_{m\sigma }^{\mathrm{HF}}$ of a MDD are plotted for both spin up (solid) and down (dashed line). This MDD is stable as ${\epsilon }_{m\sigma }^{\mathrm{HF}}⩽{\epsilon }_{N_{\sigma }-1,\sigma }^{\mathrm{HF}}$ for all $m⩽N_{\sigma }-1$ and for both spins. We choose parameters $\hslash \Omega =1.5\mathrm{meV}$, $C∕\left(2{\ell }^2\right)=2100\sqrt{B\left(\mathrm{Tesla}\right)}\times e^2∕\left(ϵ\ell \right)$, $m_t=118$, $N=355$, and $B=2.13\mathrm{T}$. The last occupied states $\left(m=N_{\sigma }-1\right)$ are marked with vertical lines. Note that $N_{\uparrow }=152$ and $N_{\downarrow }=203$.

Figure 7

Same curves as in Fig. 6 but at $B=2.205\mathrm{T}$. This MDD is unstable since ${\epsilon }_{m\sigma }^{\mathrm{HF}}>{\epsilon }_{N_{\sigma }-1,\sigma }^{\mathrm{HF}}$ for $m\approx 125$. Note that $N_{\uparrow }=153$ and $N_{\downarrow }=202$.

Figure 8

Same curves as in Fig. 4 but for a weaker bell-shaped potential. Twenty one different values of $N∊[350,370]$ are used. Following the topmost zigzag solid line, for example, the MDD ground state evolves as $\mid N_{\uparrow },N_{\downarrow }⟩=\mid 128,233⟩\to \mid 128,234⟩\Rightarrow \mid 129,233⟩\to \mid 129,234⟩\to \mid 129,235⟩\Rightarrow \mid 130,234⟩\to \mid 130,235⟩\to \mid 130,236⟩\Rightarrow \mid 131,235⟩\to \mid 131,236⟩\to \mid 131,237⟩\Rightarrow \mid 132,236⟩$ with increasing $B$. Similarly, in the fourth zigzag solid line, $\mid N_{\uparrow },N_{\downarrow }⟩$ evolves as $\mid 127,231⟩\to \mid 127,232⟩\Rightarrow \mid 128,231⟩\to \mid 128,232⟩\to \mid 128,233⟩\Rightarrow \mid 129,232⟩\to \mid 129,233⟩\to \mid 129,234⟩\Rightarrow \mid 130,233⟩$. Note that both of $N_{\uparrow }$ and $N_{\downarrow }$ increase with $B$.

Figure 9

Energy differences of the ground and first excited states of the antidot studied in Fig. 8 as a function of $B$. We choose the topmost Fermi level shown in Fig. 8 so that the transition of MDD ground states (their spin configurations are marked in this figure) follows the top zigzag line of Fig. 8. At a cusp in $E_1\left(B\right)-E_0\left(B\right)$, the spin-down transition $\mid N_{\uparrow },N_{\downarrow }⟩\to \mid N_{\uparrow },N_{\downarrow }+1⟩$ occurs. The spin-flip transitions $\mid N_{\uparrow },N_{\downarrow }⟩\to \mid N_{\uparrow }+1,N_{\downarrow }-1⟩$ are also found around $B=\mathrm{1.213\hspace{0.17em}75},1.2425,\mathrm{1.271\hspace{0.17em}25}$, and $1.3\mathrm{T}$. A domain boundary between two MDD ground states is indicated by the numbers $(N_{\uparrow },N_{\downarrow })$. The state right to the domain boundary has $N_{\uparrow }$ and $N_{\downarrow }$ numbers of spin-up and -down holes, respectively.

Figure 10

The evolution trajectory (solid line) of $\left(\delta q_{\uparrow }\right(B),\delta q_{\downarrow }(B\left)\right)$ for the bell-shaped antidot potential, while the MDD ground state evolves following the fourth zigzag solid line in Fig. 8. The conditions of the spin-down [Eq. (13)] and spin-flip [Eq. (14)] transitions define the cell boundaries (dashed and dotted lines, respectively). The trajectory is determined by Eq. (16), thus it evolves parallel to the line of $\delta q_{\downarrow }=(a_{\downarrow }∕a_{\uparrow })\delta q_{\uparrow }$. When the trajectory collides with a cell boundary, it jumps following vertical (spin-down) or diagonal dashed-dot lines (spin-flip transition). The trajectory starts at $B=1.205\mathrm{T}$ and ends at $1.245\mathrm{T}$, and the arrows represent the direction of evolution with increasing $B$. The parameters $\left(\alpha =1,a_{\downarrow }∕a_{\uparrow }=1.55,b_{\uparrow }=85.33,b_{\downarrow }=165.79\right)$ are determined from the result in Fig. 8 (see text).

Figure 11

Magnetic field dependence of the chemical potential for an inverse parabolic antidot with $\hslash \Omega =1.5\mathrm{meV}$. Chemical potentials are shown for two different values of hole Fermi energies. Squares indicate spin-flip transitions, while vertical jumps show spin-up or spin-down transitions. In the spin-flip transitions, the occupation number configuration changes from the lower to the upper panels of Fig. 3 as $B$ increases. For the Fermi level drawn by the dashed horizontal line, $\mid N_{\uparrow },N_{\downarrow }⟩$ evolves as $\mid 66,67⟩\Rightarrow \mid 65,68⟩\to \mid 66,68⟩\Rightarrow \mid 65,69⟩\Rightarrow \mid 64,70⟩\Rightarrow \mid 63,71⟩\Rightarrow \mid 62,72⟩$ as $B$ increases from $1.4\mathrm{T}$. Here $\Rightarrow$ indicates the spin-flip transition, while $\to$ shows the spin-up or -down transition. For the Fermi level drawn by the solid horizontal line, $\mid N_{\uparrow },N_{\downarrow }⟩$ evolves as $\mid 66,67⟩\Rightarrow \mid 65,68⟩\to \mid 65,69⟩\Rightarrow \mid 64,70⟩\Rightarrow \mid 63,71⟩\Rightarrow \mid 62,72⟩$. The main different result between the upper and lower Fermi energies is that the vertical jumps represent the spin-up and -down transitions, respectively.