Two-electron lateral quantum-dot molecules in a magnetic field

Laterally coupled quantum dot molecules are studied using exact diagonalization techniques. We examine the two-electron singlet-triplet energy difference as a function of magnetic field strength and investigate the magnetization and vortex formation of two- and four-minima lateral quantum dot molecules. Special attention is paid to the analysis of how the distorted symmetry affects the properties of quantum-dot molecules.

Figure 1

(Color online) Confinement potential of square-symmetric $(L_x=L_y=5\mathrm{nm})$ four-minima quantum dot molecule.

Figure 2

(Color online) Relative error in the energy of a parabolic two-electron QD at $B=0$ as a function of the basis size $n_x=n_y$ from 3 to 8, which corresponds to around 40–2000 many-body basis functions in the expansion. The relative angular momentum state of electrons is denoted with $m$.

Figure 3

One-body density (dotted line), two-body spin-singlet state (dashed line), and two-body spin-triplet state (solid line) along $x$ axis. One of the electrons is fixed at the most probable position in the $x$ axis $\left(x^{*}\right)$ and the conditional density is plotted for the other electron $\mathbf{(}{\mid {\psi }_c\left(\mathbf{r}\right)\mid }^2={\mid {\Psi }_S[(x,y),(x^{*},0)]\mid }^2∕{\mid {\Psi }_S[(-x^{*},0),(x^{*},0)]\mid }^2\mathbf{)}$. The peaks on the left-hand side also indicate the most probable position, therefore by reflecting the peak position to the right-hand side of the $x$ axis one can perceive the position of the fixed electron. (a) and (b) show the densities of a parabolic QD at two different magnetic field values ($B=1$ and $8\mathrm{T}$), (c) and (d) represent a two-minima QDM with $L_x=20\mathrm{nm}$. The confinement potential, $V_{\mathrm{c}}$, is plotted with gray color on each figure.

Figure 4

(Color online) Ten lowest energy levels of the parabolic QD ($L_x=0$, $L_y=0$) as a function of magnetic field of (a) noninteracting two-body, (b) two-body singlet state $(S=0)$, and (c) two-body triplet state $(S=1)$. Some of the states in (a) are degenerate. (d) Three lowest energy levels of singlet (dashed line) and triplet (solid line) as a function of magnetic field up to $B=6\mathrm{T}$ for the parabolic QD $(L_x=L_y=0)$. In (b) the first singlet ground state corresponds to angular momentum $m=0$ which changes to $m=2$ at $B\approx 2.7\mathrm{T}$. In (c) the first triplet ground state is $m=1$ and it changes to $m=3$ at $B\approx 5.8\mathrm{T}$. In (d) the first ground state equals to $m=0$ singlet, then the ground state is $m=1$ triplet followed by $m=2$ singlet and $m=3$ triplet, where the latter is not visible as a ground state in (d). Zeeman energy is included in the triplet energies ($E_Z=-2\times 12.7B\left[T\right]\mu \mathrm{eV}$ in GaAs).

Figure 5

(Color online) Triplet-singlet energy difference in (a) and magnetization in (b) for the parabolic QD $(L_x=L_y=0)$. The lower curve in (a) shows the singlet-triplet splitting with Zeeman energy included. The smooth curve in (b) represents the magnetization of two noninteracting electrons and the other curve shows the magnetization for two interacting electrons. At low magnetic field values, the ground state is the angular momentum $m=0$ singlet, which shows as positive values in the triplet-singlet energy difference of (a) and as a smooth curve in (b). When the first transition from $m=0$ singlet to $m=1$ triplet occurs triplet-singlet energy difference changes its sign from positive to negative and there appears a peak in magnetization. Change of $m=1$ triplet to $m=2$ singlet and from $m=2$ singlet to $m=3$ triplet appear in the same way as peaks in magnetization and changes of sign in triplet-singlet splitting. Magnetization is given in the units of effective Bohr magnetons ${\mu }_B^{*}=e\hslash ∕2m^{*}$ (${\mu }_B^{*}=0.87\mathrm{meV}∕\mathrm{T}$ for GaAs).

Figure 6

(Color online) (a)–(e) Contours of conditional density ${\mid {\psi }_c(x,y)\mid }^2$ and phase of the conditional wave function ${\theta }_c(x,y)$ in grayscale for parabolic QD $(L_x=L_y=0)$. (White equals ${\theta }_c=0$ and darkest gray ${\theta }_c=2\pi$). Magnetic field value and the spin type are plotted on top of each figure. The plus sign (+) indicates the position of the fixed electron and small circles indicate the positions of the vortices. In (f) contours of total electron density of the three-vortex triplet state are plotted in the background and positions of the vortices are solved when the fixed electron is in three different positions. The fixed electron is marked with the plus sign and vortices with circles. The most probable position is marked with a star, on the left-hand side for clarity.

Figure 7

Total electron density of the ground state at different magnetic field values for the parabolic $(L_x=L_y=0)$ QD. In the left column are densities in $1∕{\mathrm{nm}}^2$ and on the right column are contours of the densities. Dark regions in the contours mark high density and brighter regions low density. Notice that in the densities the scale in the $x$ and $y$ axes is changing and the height of the peak is increasing with magnetic field. The range of axes in the contour plots is kept fixed. The magnetic field value and the spin type of the ground state are plotted above each subfigure.

Figure 8

(Color online) Most probable position in nm of singlet $(S=0)$ and triplet $(S=1)$ states for the parabolic $(L_x=L_y=0\mathrm{nm})$ QDM. The magnetic field region where singlet is a ground state is marked with gray background color.

Figure 9

Triplet-singlet energy difference $(\Delta E=E^{\uparrow \uparrow }-E^{\uparrow \downarrow })$ as a function of magnetic field in the two-minima quantum dot molecule. The energy difference is plotted as a function of dot-dot separation and magnetic field in (a) without Zeeman energy and in (c) with the Zeeman energy included $(\Delta E=E^{\uparrow \uparrow }+E_Z-E^{\uparrow \downarrow })$. In (b) and (d) the ground state regions of the singlet and triplet states are plotted as function of $B$ and $L$, without and with Zeeman energy, respectively.

Figure 10

(Color online) Energy levels of $L_x=5$, $L_y=0$ two-minima QDM. See Fig. 4 for details.

Figure 11

(Color online) Triplet-singlet energy difference (a) and magnetization (b) for $L_x=5$, $L_y=0$ two-minima QDM. See Fig. 5 for details.

Figure 12

(Color online) (a)–(e) Contours of conditional densities ${\mid {\psi }_c(x,y)\mid }^2$ and phase of the conditional wave function ${\theta }_c(x,y)$ in grayscale for $L_x=5$, $L_y=0$ two-minima QDM. See Fig. 6 for details. (f) Contours of total electron density of the three-vortex triplet state are plotted in the background and positions of the vortices with the fixed electron in three different positions.

Figure 13

(Color online) Most probable position in nm of singlet $(S=0)$ and triplet $(S=1)$ states for the two-minima ($L_x=5$, $L_y=0\mathrm{nm}$) QDM. Singlet ground state magnetic field region is marked with gray background color.

Figure 14

Density of the ground state at different magnetic field values for the two-minima ($L_x=5$, $L_y=0\mathrm{nm}$) QDM. See Fig. 7 for details.

Figure 15

Triplet-singlet energy difference $(\Delta E=E^{\uparrow \uparrow }-E^{\uparrow \downarrow })$ as a function of magnetic field in the square-symmetric $(L=L_x=L_y)$ four-minima quantum dot molecule. See Fig. 9 for details.

Figure 16

(Color online) Energy levels of $L_x=L_y=5\mathrm{nm}$ four-minima QDM. See Fig. 4 for details.

Figure 17

(Color online) Triplet-singlet energy difference in (a) and magnetization in (b) for $L_x=5=L_y=5$ QDM. See Fig. 5 for details.

Figure 18

(Color online) (a)–(e) Contours of conditional densities ${\mid {\psi }_c(x,y)\mid }^2$ and phase of the conditional wave function ${\theta }_c(x,y)$ in grayscale for the $L_x=5$, $L_y=5$ four-minima QDM. See Fig. 6 for details. (f) Contours of total electron density of the three-vortex triplet state are plotted in the background and positions of the vortices with the fixed electron in two different positions.

Figure 19

(Color online) Most probable position in nm of singlet $(S=0)$ and triplet $(S=1)$ states for the four-minima $L_x=L_y=5\mathrm{nm}$ QDM. Singlet ground state magnetic field region is marked with gray background color.

Figure 20

Density of the ground state at different magnetic field values for the four-minima $L_x=L_y=5\mathrm{nm}$ QDM. See Fig. 7 for details.

Figure 21

Triplet-singlet energy difference $(\Delta E=E^{\uparrow \uparrow }-E^{\uparrow \downarrow })$ as a function of magnetic field in the rectangular-symmetric four-minima quantum dot molecule. $L_x$ is fixed to $5\mathrm{nm}$ and $L_y$ is varied from $0\text{to}20\mathrm{nm}$. Therefore at $L_y=0$ we have $L_x=5\mathrm{nm}$ double dot and at $L_y=5\mathrm{nm}$ it is the square-symmetric four-minima QDM. The energy difference is plotted as a function of $L_y$ and magnetic field in (a) without Zeeman energy and in (c) with the Zeeman energy included $(\Delta E=E^{\uparrow \uparrow }+E_Z-E^{\uparrow \downarrow })$. In (b) and (d) the ground state regions of the singlet and triplet states are plotted as a function of $B$ and $L_y$, without and with Zeeman energy, respectively.

Figure 22

(Color online) Energy levels of the $L_x=5$, $L_y=10\mathrm{nm}$ rectangular four-minima QDM. See Fig. 4 for details.

Figure 23

(Color online) Triplet-singlet energy difference (a) and magnetization (b) in the $L_x=5$, $L_y=10\mathrm{nm}$ rectangular four-minima QDM. Magnetization is given in the units of effective Bohr magnetons ${\mu }_B^{*}=e\hslash ∕2m^{*}$.

Figure 24

(Color online) (a)–(e) Contours of conditional densities ${\mid {\psi }_c(x,y)\mid }^2$ and phase of the conditional wave function ${\theta }_c(x,y)$ in grayscale for the $L_x=5$, $L_y=10$ four-minima QDM. See Fig. 6 for details. (f) Contours of total electron density of the three-vortex triplet state are plotted in the background and positions of the vortices with the fixed electron in three different positions.

Figure 25

Density of the ground state at different magnetic field values for the $L_x=5$, $L_y=10\mathrm{nm}$ rectangular four-minima QDM. See Fig. 7 for details.

Figure 26

(Color online) Most probable position in nm of singlet $(S=0)$ and triplet $(S=1)$ states for the rectangular-symmetric four-minima ($L_x=5$, $L_y=10\mathrm{nm}$) QDM. The singlet ground state (magnetic field) region is marked with gray background color.

Figure 27

(Color online) Three lowest singlet and triplet energy levels and magnetization for all studied quantum dot confinements.