Even-Odd Effect in Spontaneously Coherent Bilayer Quantum Hall Droplets

Using exact diagonalization in the disk geometry we predict a novel even-odd effect in the Coulomb-blockade spectra of vertically coupled double quantum dots under an external magnetic field. The even-odd effect in the tunneling conductance is a direct manifestation of spontaneous interlayer phase coherence, and is similar to the even-odd resonance in the Cooper pair box problem in mesoscopic superconducting grains. Coherent fluctuations in the number of Cooper pairs in superconductors are analogous to the fluctuations in the relative number difference between the two layers in quantum Hall droplets.

Figure 1

Coulomb interaction energy as a function of $S_z$ which is half the relative electron number difference between different layers. The layer separation is chosen to be $d/a=1$ where we define $a=l_B(1+4{\omega }_0^2/{\omega }_c^2{)}^{-1/4}$, the magnetic length $l_B=\sqrt{\hslash c/eB}$, and the cyclotron frequency ${\omega }_c=eB/m^{*}c$. ${\omega }_0$ is the frequency of the confining potential. The state with angular momentum $M_z=21$, is the maximum density droplet state for $N=7$.

Figure 2

Energy spectrum as a function of total angular momentum, $M_z$, in the Hilbert space of $S_z=1/2$. Note that the energy in the graph is a sum of the Coulomb interaction energy and the confining potential energy: $E=V_{\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{l}}+\gamma M_z$ where $\gamma =\frac{\hslash }{2}(\sqrt{{\omega }_c^2+4{\omega }_0^2}-{\omega }_c)$. In the graph, we have chosen $\gamma =0.1187e^2/ϵa$ which gives us the largest possible gap. The arrows indicate the three lowest energies.

Figure 3

The lowest three energy gaps as a function of interlayer separation $d/a$. The lowest three excitations are categorized as follows: (i) $\Delta M_z=+1$ (the edge excitation), (ii) $\Delta M_z=0$, and (iii) $\Delta M_z=-1$ (the internal excitation). The energy gap is given by $\Delta E=\Delta V_{\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{l}}+\gamma \Delta M_Z$ where $\gamma$ is defined as in Fig. 2.

Figure 4

Interlayer coherence in the limit of zero tunneling as a function of interlayer separation $d/a$. $|4,3⟩$ represents the lowest energy state with 4 (3) electrons in the top (bottom) layer for a 7 electron system, which can be alternatively denoted by $|S_z=+1/2⟩$. $|3,4⟩$ ($|S_z=-1/2⟩$) is similarly defined.

Figure 5

Schematic diagram illustrating the even-odd effect in bilayer quantum Hall dots. The total energy including the charging energy is plotted as a function of $C{V}_g/e$ where $V_g$ is the gate voltage and $C$ is the capacitance between a lead and one of the two dots. $⟨N⟩$ is the average number of electrons inside double quantum dots. Figure 5a depicts the situation where there is no interlayer coherence. $|+1/2⟩$ ($|-1/2⟩$) represents the degenerate set of low energy states with $S_z=1/2$ ($-1/2$). It is shown in Fig. 5b that, due to the interlayer coherence, the odd $N$ system acquires an energy splitting between the symmetric ($|{\varphi }_{+}⟩$) and antisymmetric ($|{\varphi }_{-}⟩$) superposition of $|+1/2⟩$ and $|-1/2⟩$, which is $2{\Delta }_x$.