Spatial ordering of charge and spin in quasi-one-dimensional Wigner molecules

Few-electron systems confined in quasi-one-dimensional quantum dots are studied by the configuration interaction approach. We consider the parity symmetry of states forming Wigner molecules in large quantum dots and find that for the spin-polarized Wigner molecules it strictly depends on the number of electrons. We investigate the spatial spin ordering in the inner coordinates of the quantum system and conclude that for small dots it has a short-range character and results mainly from the Pauli exclusion principle while the Wigner crystallization in large dots is accompanied by spin ordering over the entire length of the dot.

Figure 1

(a) Lowest energy levels multiplied by the dot length for $N=2$. Numbers close to the curves denote the total spin quantum number of the corresponding states and signs $+$, $-$ stand for even and odd parity symmetry. (b), (c), (d) Charge density of $0+$, $1-$, $1+$, and $0-$ states plotted with solid, dotted, dashed, and dash-dotted lines for $d=50$, 100, and $200\mathrm{nm}$, respectively.

Figure 2

(a) Lowest energy levels multiplied by the dot length for $N=3$. (b), (c), (d) Charge density of $1∕2-$, $1∕2+$, and $3∕2-$ states plotted with solid, dashed, and dotted lines for $d=50$, 150, and $200\mathrm{nm}$, respectively. In (d) the charge density of the $3∕2+$ state is shown by the dash-dotted curve.

Figure 3

(a) Four-electron energy levels multiplied by the dot length. (b), (c), (d) Charge density of $0+$, $1-$, $1+$, and $2+$ four-electron states plotted with solid, dash-dotted, dotted, and dashed lines for $d=100$, $200$, and $300\mathrm{nm}$, respectively. In (d) the charge densities of $2-$ and $0+$ states are marked with crosses and dots, respectively.

Figure 4

(a) Five-electron energy levels multiplied by the dot length. Even (odd) parity levels are plotted with solid (dotted) lines. (b), (c) Charge density of $1∕2+$, $1∕2-$, $3∕2+$, $3∕2-$, and $5∕2+$ states plotted with solid, dash-dotted, dotted, dashed, and dash-double-dot lines for $d=100$ and $200\mathrm{nm}$, respectively. In (d) the charge density of the $1∕2+$, $3∕2-$, and $5∕2-$ state is shown by solid, dashed, and dotted lines, respectively [charge densities of $1∕2-$, $3∕2+$, and $5∕2+$ are almost identical with the $1∕2+$ and $3∕2-$ charge densities are therefore omitted in (d) for the sake of clarity].

Figure 5

Illustration to the proof that for $N=2M+1$ or $N=2M$ electrons the parity of the spin-polarized state which exhibits Wigner localization is accordant with the parity of $M$.

Figure 6

(a) Lowest energy levels for $N=2$ as functions of the length of the well with a central attractive cavity [Eq. (18)]. Numbers close to the curves denote the total spin quantum number of the corresponding states and signs $+$, $-$ stand for even and odd parity symmetry, respectively. (b), (c), (d) Charge density of $0+$, $1-$, $1+$, and $0-$ states plotted with solid, dotted, dashed, and dash-dot curves for $d=50$, 100, and $200\mathrm{nm}$, respectively.

Figure 7

(a) Lowest energy levels for $N=3$ as function of the length of the well with a central attractive cavity [Eq. (18)]. Numbers close to the curves denote the total spin quantum number of the corresponding states and signs $+$, $-$ stand for even and odd parity symmetry. (b),(c),(d) Charge density of $0+$, $1-$, $1+$, and $0-$ states plotted with solid, dotted, dashed, and dash-dot curves for $d=50$, 100, and $200\mathrm{nm}$, respectively.

Figure 8

$\mathrm{PCFs}$ for four electrons in state $0+$ [(a),(b)] and state $1-$ with $S_z=\hslash$ (c) for $d=100$ (a) and $300\mathrm{nm}$ [(b),(c)]. One of the electrons is fixed and its position is marked by a thin vertical line. Solid curves show the spin-independent $\mathrm{PCF}$, dashed (dotted) curves show the opposite (same) spin $\mathrm{PCF}$.

Figure 9

Spin-up (solid lines) and spin-down (dashed lines) densities for the ground-state 5-electron system $1∕2+$ with $S_z=\hslash ∕2$ for different system sizes.

Figure 10

$\mathrm{PCFs}$ for five electrons in the state $1∕2+$ for $d=100$ (a), 200 (b), and $300\mathrm{nm}$ (c). The position of the fixed electron is marked by a thin vertical line. Solid lines show the spin-independent $\mathrm{PCF}$, dashed (dotted) lines show the opposite (same) spin $\mathrm{PCF}$.