Two electrons in a strongly coupled double quantum dot: From an artificial helium atom to a hydrogen molecule

We study the formation of molecular states in a two-electron quantum dot as a function of the barrier potential dividing the dot. The increasing barrier potential drives the two electron system from an artificial helium atom to an artificial hydrogen molecule. To study this strongly coupled regime, we introduce variational wave functions which describe accurately two electrons in a single dot, and then study their mixing induced by the barrier. The evolution of the singlet-triplet gap with the barrier potential and with an external magnetic field is analyzed.

Figure 1

Comparison of the relative motion energies in a parabolic dot with ${\omega }_0=\frac{2}{3}\mathrm{Ry}$ calculated variationally and by the Numerov method. Solid lines: variational $m=0,1,-1,2,-2,3,-3$ states (increasing order at ${\omega }_c=0.1\mathrm{Ry}$). Dashed line: numerical $m=0$ state. Other numerical states coincide with variational in this figure.

Figure 2

The profile of the double dot potential with ${\omega }_0=\frac{2}{3}\mathrm{Ry}$, $V_0=1\mathrm{Ry}$, $\Delta =0.5a_B$.

Figure 3

The lowest lying singlet and triplet energies in a double dot with ${\omega }_0=\frac{2}{3}\mathrm{Ry}$, $V_0=1\mathrm{Ry}$, $\Delta =0.5a_B$. Dashed lines: ${\tilde{E}}_0$—singlet, ${\tilde{E}}_{\pm 1}$—triplet. Solid lines: ${\tilde{E}}_{0,\pm 2}$—singlet, ${\tilde{E}}_{\pm 1,\pm 3}$—triplet. Dotted lines: numerical singlet and triplet energies calculated by the configuration-interaction method. (In increasing order of energy at ${\omega }_c=0$.)

Figure 4

(Color online) The two-electron density of the lowest singlet calculated by diagonalization using $m=0,\pm 2$ states at ${\omega }_c=0\mathrm{Ry}$ (upper), $1\mathrm{Ry}$ (lower).

Figure 5

(Color online) The two-electron density of the lowest singlet calculated by the configuration-interaction method, at ${\omega }_c=0\mathrm{Ry}$ (upper), $1\mathrm{Ry}$ (lower).

Figure 6

(Color online) The two-electron density of the lowest triplet calculated by diagonalization using $m=\pm 1,\pm 3$ states at ${\omega }_c=0\mathrm{Ry}$ (upper), $1\mathrm{Ry}$ (lower).

Figure 7

(Color online) The two-electron density of the lowest triplet calculated by the configuration-interaction method at ${\omega }_c=0\mathrm{Ry}$ (upper), $1\mathrm{Ry}$ (lower).

Figure 8

The singlet-triplet gap as a function of the magnetic field in a single dot with ${\omega }_0=\frac{2}{3}\mathrm{Ry}$. Variational calculations.

Figure 9

The singlet-triplet gap as a function of the magnetic field in a double quantum dot with ${\omega }_0=\frac{2}{3}\mathrm{Ry}$, $V_0=1\mathrm{Ry}$, $\Delta =0.5a_B$. Solid line: variational results. Dotted line: numerical calculations by the configuration-interaction method.

Figure 10

The energy of the first excited state in a single parabolic dot with ${\omega }_0=\frac{2}{3}\mathrm{Ry}$. Solid line: variational calculation. Dotted line: numerical calculations by the Numerov method.

Figure 11

(Color online) The two-electron density at ${\omega }_c=0\mathrm{Ry}$ calculated using $m=0$, $m=0^{*}$, $m=2$, $m=-2$ states.