Single-electron shell occupation and effective $g$ factor in few-electron nanowire quantum dots

Nanowire double quantum dots occupied by an even number of electrons are investigated in the context of energy level structure revealed by electric dipole spin resonance measurements. We use a numerically exact configuration interaction approach up to six electrons for systems tuned to a Pauli spin blockade regime. We point out the differences between the spectra of systems with two and a greater number of electrons. For two electrons the unequal length of the dots results in a different effective $g$ factor in the dots as observed by the recent experiments. For an increased number of electrons the $g-\text{factor}$ difference between the dots appears already for symmetric systems and it is greatly amplified when the dots are of unequal length. We find that the energy splitting defining the resonant electric dipole spin frequency can be quite precisely described by the two electrons involved in the Pauli blockade with the lower-energy occupied states forming a frozen core.

Figure 1

Energy spectra of a two-electron double dot system. (a) Symmetric system with the dots width equal to 100 nm. (b) Asymmetric configuration: width of the left dot is 150 nm and the width of the right one is 50 nm. Inset in (a) shows the energy spectrum including the ground state (2,0) singlet. The insets in (b) depict the spin densities with the color curves presented in arbitrary units. The black contours show the confinement potential.

Figure 2

Energy difference (the red-dashed curve) between the energy level of $T_{+}$ state and the degenerated energy level from Fig. 1 or two levels that correspond to states with opposite spin configurations from Fig. 1 (black curves).

Figure 3

(a) and (e) Single-electron charge densities and the potential profile. (b) and (f) Energy spectrum: The colors of the curves corresponds to the states from (a) and (e). (c) and (g) Energy levels from (b) and (f) shifted to compare the Zeeman splittings. (d) and (h) Sum of the single-electron energies as they enter into the configuration interaction approach.

Figure 4

$g_1^{*}/g_2^{*}$ versus the length of the dots for (a) two electrons, (b) four electrons, and (c) six electrons.

Figure 5

(a) Energy levels of the four-electron symmetric double dot in a (3,1) occupation regime depicted with black-solid curves. The red-dashed curves are energy levels of a two-electron system with excluded $s_{l,\uparrow }$ and $s_{l,\downarrow }$ single-electron orbitals that form a singlet closed shell. (b) Energy spectrum for an asymmetric system with $l_1=150$ nm and $l_2=50$ nm.

Figure 6

(a)–(c) Charge densities of single-electron states and the potential profile for symmetric quantum dots with $V_b=-14.21$ meV. (d) Single-electron energy spectrum.

Figure 7

(a) Six-electron energy spectrum for a symmetric system with $V_b=-30.125$ meV. Black curves correspond to the exact six-electron calculation. Red-dashed curves are obtained in the two-electron calculation with the basis excluding the four lowest energy single-electron spin orbitals. (b) Single-electron energy spectrum.