In-plane magnetic-field-induced Wigner crystallization in a two-electron quantum dot

The orbital effects of the in-plane magnetic field on a two-electron harmonic quantum dot are studied using a variational method. For flat quantum dots the singlet-triplet transitions appearing in a perpendicular magnetic field are absent in a magnetic field oriented parallel to the plane of confinement. Instead, a degeneracy of orbital energies for symmetric and antisymmetric states at high in-plane magnetic field is observed. This degeneracy is due to the formation of Wigner molecules in the laboratory frame of reference with charge islands elongated along the direction of the magnetic field and localized within the plane perpendicular to it.

Figure 1

Ground-state wave function of the relative Hamiltonian calculated for the parameters used in Table . Solid line shows the exact wave function, the dashed (dotted) line is the wave function obtained by the variational method for $M=6\left(M=22\right)$. Inset: comparison of the exact radial probability density (solid line) with the dependence obtained variationally with $M=22$ (open circles).

Figure 2

Energy eigenvalues of the relative Hamiltonian (3) calculated with respect to the lowest Landau level for a spherical quantum dot with $\hslash \omega =\hslash {\omega }_z=3\mathrm{meV}$ (Zeeman effect neglected). The solid and dotted lines show the singlet and triplet energy levels, respectively. In parentheses the parity of the corresponding eigenstates in the $x$ direction and within the ($y$, $z$) plane is given. The numbers indicate the $x$ component of the angular momentum in $\hslash$ units. The inset shows the lowest singlet probability density integrated over the $x$ direction for magnetic fields $B=0$, 12, and 18 T.

Figure 3

Energy eigenvalues of the relative Hamiltonian calculated with respect to the lowest Landau level for a flat quantum dot with $\hslash \omega =3\mathrm{meV}$ and $\hslash {\omega }_z=12\mathrm{meV}$. The solid and dotted lines show the singlet and triplet energy levels, respectively. In parentheses the parity of the corresponding eigenstates in the $x$ direction and within the ($y$,$z$) plane is given.

Figure 4

Triplet-singlet energy difference (without the spin Zeeman effect) as function of the magnetic field for $\hslash \omega =3\mathrm{meV}$ and various confinement energies in the $z$ direction. The thick dashed line shows the spin Zeeman splitting between the triplet and the singlet states. The inset shows the low-magnetic-field region.

Figure 5

Contour plots at the left side of the figure shows the relative motion probability density for the lowest singlet state integrated over the $x$ direction (left panel) and over the $z$ coordinate (right panel) for $\hslash \omega =3\mathrm{meV}$, $\hslash {\omega }_z=12\mathrm{meV}$, and different magnetic fields. Larger values of density are marked with darker colors. At the right side of the figure we show the surface at which the probability density takes one-fifth of its maximum value.

Figure 6

The same as Fig. 5 but now for the lowest triplet state.

Figure 7

Probability densities of the ground state of the center of mass (CM) and relative (rel) Hamiltonians, and the charge density of the noninteracting and interacting system of two electrons integrated over the $z$ direction as a function of the radial coordinate $\rho =\sqrt{x^2+y^2}$ for $\hslash \omega =3\mathrm{meV}$, $\hslash {\omega }_z=12\mathrm{meV}$, and $B=0\mathrm{T}$. The two-electron charge densities have been divided by 2.

Figure 8

Contour plots at the left side of the figure show the two-electron probability densities $\left[n\left(\mathbf{r}\right)\right]$ integrated over the $x$ (left panel) and $z$ coordinates (right panel) for $\hslash \omega =3\mathrm{meV}$, $\hslash {\omega }_z=12\mathrm{meV}$. The surface plots at the right side of the figure show the surface at which the charge density takes one-fifth of its maximum value.

Figure 9

(Color online) Triplet-singlet energy difference as a function of the magnetic field and vertical confinement energy for $\hslash \omega =3\mathrm{meV}$. Blue (white and red) regions correspond to the triplet (singlet) ground state for the spin Zeeman effect neglected. The dashed line shows the values of $B$ and $\hslash {\omega }_z$ for which the triplet-singlet energy difference is equal to 0.1 meV. Color scale is given at the right-hand side of the figure.