Structural properties of electrons in quantum dots in high magnetic fields: Crystalline character of cusp states and excitation spectra

The crystalline or liquid character of the downward cusp states in $N$-electron parabolic quantum dots at high magnetic fields is investigated using conditional probability distributions obtained from exact diagonalization. These states are of crystalline character for fractional fillings covering both low and high values, unlike the liquid Jastrow-Laughlin wave functions, but in remarkable agreement with the rotating-Wigner-molecule ones [Phys. Rev. B 66, 115315 (2002)]. The crystalline arrangement consists of concentric polygonal rings that rotate independently of each other, with the electrons on each ring rotating coherently. We show that the rotation stabilizes the Wigner molecule relative to the static one defined by the broken-symmetry unrestricted-Hartree-Fock solution. We discuss the nonrigid behavior of the rotating Wigner molecule and pertinent features of the excitation spectrum, including the occurrence of a gap between the ground and first-excited states that underlies the incompressibility of the system. This leads us to conjecture that the rotating crystal (and not the static one) remains the relevant ground state for low fractional fillings even at the thermodynamic limit.

Figure 1

(Color online) Conditional probability distributions at high $B$ for $N=6$ electrons with $L=135$ ($\nu =\frac{1}{9}$, left column) and $L=105$ ($\nu =\frac{1}{7}$, right column). Top row: RWM case. Middle row: The case of exact diagonalization. Bottom row: The Jastrow-Laughlin case. It is apparent that the exact diagonalization and RWM wave functions have a pronouned crystalline character, corresponding to the (1,5) polygonal configuration of the rotating Wigner molecule. In contrast, the Jastrow-Laughlin wave functions fail to capture this crystalline character, exhibiting a rather “liquid” character. The observation point (identified by a solid dot) was placed at the maximum of the outer ring of the radial electron density (Ref. 4, 5) of the EXD wave function, namely at $r_0=7.318l_B$ for $L=135$ and $r_0=6.442l_B$ for $L=105$. Here, $l_B={(\hslash c∕eB)}^{1∕2}$. The EXD Coulomb interaction energies (in the lowest Landau level) are 1.6305 and 1.8533 $e^2∕\kappa l_B$ for $L=135$ and $L=105$, respectively. The errors relative to the corresponding EXD energies and the overlaps of the trial functions with the EXD ones are (i) for $L=135$, RWM: 0.34%, 0.860; JL: 0.50%, 0.665 and (ii) for $L=105$, RWM: 0.48%, 0.850; JL: 0.46%, 0.710.

Figure 2

(Color online) Conditional probability distributions at high $B$ for $N=6$ electrons and $L=75$ ($\nu =\frac{1}{5}$, left column) and for $N=7$ electrons and $L=105$ (again $\nu =\frac{1}{5}$, right column). Top row: RWM case. Middle row: The case of exact diagonalization. Bottom row: The Jastrow-Laughlin case. The exact diagonalization and RWM wave functions have a pronouned crystalline character, corresponding to the (1,5) polygonal configuration of the RWM for $N=6$, and to the (1,6) polygonal configuration for $N=7$. In contrast, the Jastrow-Laughlin wave functions exhibit a characteristic liquid profile that depends smoothly on the number $N$ of electrons. The observation point (identified by a solid dot) is located at $r_0=5.431l_B$ for $N=6$ and $L=75$ and $r_0=5.883l_B$ for $N=7$ and $L=105$. The EXD Coulomb interaction energies (lowest Landau level) are 2.2018 and 2.9144 $e^2∕\kappa l_B$ for $N=6$, $L=75$ and $N=7$, $L=105$, respectively. The errors relative to the corresponding EXD energies and the overlaps of the trial functions with the EXD ones are (i) for $N=6$, $L=75$, RWM: 0.85%, 0.817; JL: 0.32%, 0.837 and (ii) for $N=7$, $L=105$, RWM: 0.59%, 0.842; JL: 0.55%, 0.754.

Figure 3

(Color online) Conditional probability distributions at high $B$ for $N=7$ electrons and $L=69$ ($\nu =\frac{7}{23}=0.304>\frac{1}{5}$, top row), $L=57$ ($1>\nu =\frac{7}{19}=0.368>\frac{1}{3}$, middle row), and $L=45$ ($1>\nu =\frac{7}{15}=0.467>\frac{1}{3}$, bottom row). RWM case: Left column. The case of exact diagonalization is depicted in the right column. Even for these low magic angular momenta (high fractional fillings), both the exact-diagonalization and RWM wave functions have a pronouned crystalline character [corresponding to the (1,6) polygonal configuration of the RWM for $N=7$ electrons]. The observation point (identified by a solid dot) is located at $r_0=4.752l_B$ for $L=69$, $r_0=4.278l_B$ for $L=57$, and $r_0=3.776l_B$ for $L=45$.

Figure 4

(Color online) CPD’s at high $B$ for $N=7$ and $L=63$ $(\nu =\frac{1}{3})$. Top: RWM case. Middle: EXD case. Bottom: JL case. Unlike the JL CPD (which is liquid), the CPD’s for the exact-diagonalization and RWM wave functions exhibit a well developed crystalline character [corresponding to the (1,6) polygonal configuration of the RWM for $N=7$ electrons]. The observation point (identified by a solid dot) is located at $r_0=4.568l_B$.

Figure 5

(Color online) Additional CPD’s at high $B$. RWM results: Left column. Results from exact diagonalization are depicted on the right column. Top row: $N=8$ electrons and $L=91$ $(\frac{1}{5}<\nu =\frac{4}{13}=0.308<\frac{1}{3})$. Two bottom rows: $N=9$ electrons and $L=101$ $(\frac{1}{3}<\nu =\frac{36}{101}=0.356<1$, see text for explanation). Even for these low magic angular momenta (high fractional fillings), both the exact-diagonalization and RWM wave functions have a pronouned crystalline character [corresponding to the (1,7) and (2,7) polygonal configurations of the RWM for $N=8$ and 9 electrons]. The observation point (identified by a solid dot) is located at $r_0=5.105l_B$ for $N=8$, $L=91$, and $r_0=5.218l_B$ (outer) and $r_0=1.662l_B$ (inner) for $N=9$, $L=101$.

Figure 6

Stabilization energies $\Delta E_{\mathrm{gs}}^{\text{gain}}$ for $N=6$ (dashed curve) and $N=7$ (solid curve) fully polarized electrons in a parabolic QD as a function of $B$. The troughs associated with the major fractional fillings ($\frac{1}{3}$, $\frac{1}{5}$, and $\frac{1}{7}$) and the corresponding ground-state angular momenta [see Eq. (1)] are indicated with arrows. We have extended the calculations up to $B=120\mathrm{T}$ (not shown), and verified that $\Delta E_{\mathrm{gs}}^{\text{gain}}$ remains negative while its absolute value vanishes as $B\to \infty$. The choice of parameters is $\hslash {\omega }_0=3\mathrm{meV}$ (parabolic confinement), $m*=0.067m_e$ (electron effective mass), and $\kappa =12.9$ (dielectric constant).

Figure 7

Low-energy part of the spectrum of the parabolic QD whose parameters are the same as those in Fig. 6, calculated as a function of the angular momentum $L$ through exact diagonalization for $N=7$ electrons with a magnetic field $B=18.8\mathrm{T}$. We show here the spectrum in the interval $95⩽L⩽115$ (in the neighborhood of $\nu =\frac{1}{5}$). The magic angular momentum values corresponding to cusp states are marked (99, 105, and 111), and they are seen to be separated from the rest of the spectrum. For the given value of $B$, the global energy minimum (ground state) occurs for $L_{\mathrm{gs}}=105$, and the gap $\Delta$ to the first excited state $(L=99)$ is indicated. The lowest energies for the different $L$’s in the plotted range are connected by a dashed line, as a guide to the eye. The zero of energy corresponds to $7\hslash \Omega$, where $\Omega ={({\omega }_0^2+{\omega }_c^2∕4)}^{1∕2}$ and ${\omega }_c=eB∕(m*c)$. The horizontal arrow denotes the energy of the Laughlin quasihole at $L=112$. It is seen that the Laughlin quasihole is not the lowest excited state.

Figure 8

Top: RWM projected energies [see Eq. (8)] calculated as a function of the magnetic field $B$ for $N=7$ electrons in a parabolic QD with the same parameters as those used in Figs. 6, 7. Each of the parabolalike curves (made partly of a solid and partly of a dashed line) corresponds to the marked value of the angular momentum—i.e., for the range of magnetic fields shown here, $L_{\mathrm{gs}}=57\left(\frac{7}{19}\right)$, $63\left(\frac{1}{3}\right)$, $69\left(\frac{7}{23}\right)$, $75\left(\frac{7}{25}\right)$, and $81\left(\frac{7}{27}\right)$, with the corresponding value of the fractional filling $\nu =N(N-1)∕\left(2L_{\mathrm{gs}}\right)$ given in parentheses. The solid lines denote the ground-state energies, and the dashed lines give the values of the first-excited-state energies. Note that the gap between the ground and the first excited state, $\Delta =E^1-E^{\mathrm{gs}}$, oscillates as a function of $B$. The arrows denote the values of $B$ for which the gap vanishes, occuring between the fractional fillings. The zero of energy corresponds to $7\hslash \Omega +E_{\mathrm{cl}}^{\mathrm{st}}$, where $\Omega ={({\omega }_0^2+{\omega }_c^2∕4)}^{1∕2}$ [with ${\omega }_c=eB∕(m*c)$] and $E_{\mathrm{cl}}^{\mathrm{st}}$ is the classical energy of the static Wigner molecule (see Ref. 17). Bottom: The gap $\Delta$ plotted versus $B$.