Quantum theory of the charge-stability diagram of semiconductor double-quantum-dot systems

We complete our recently introduced theoretical framework treating the double-quantum-dot system with a generalized form of Hubbard model. The effects of all quantum parameters involved in our model on the charge-stability diagram are discussed in detail. A general formulation of the microscopic theory is presented and, truncating at one orbital per site, we study the implication of different choices of the model confinement potential on the Hubbard parameters as well as the charge-stability diagram. We calculate the charge-stability diagram keeping three orbitals per site and find that the effect of additional higher-lying orbitals on the subspace with lowest-energy orbitals only can be regarded as a small renormalization of Hubbard parameters, thereby justifying our practice of keeping only the lowest orbital in all other calculations. The role of the harmonic-oscillator frequency in the implementation of the Gaussian model potential is discussed, and the effect of an external magnetic field is identified to be similar to choosing a more localized electron wave function in microscopic calculations. The full matrix form of the Hamiltonian, including all possible exchange terms and several peculiar charge-stability diagrams due to unphysical parameters, is presented in the Appendices, thus emphasizing the critical importance of a reliable microscopic model in obtaining the system parameters defining the Hamiltonian.

Figure 1

Charge-stability diagram calculated at $U=6.1$ meV, $U_{12}=2.5$ meV, and $t=J_e=J_p=J_t=0$.

Figure 2

Charge-stability diagram calculated at $U_1=5$ meV, $U_2=7$ meV, $U_{12}=2.5$ meV, and $t=J_e=J_p=J_t=0$.

Figure 3

Charge-stability diagram calculated at $t=0.6$ meV, $U=6.1$ meV, $U_{12}=2.5$ meV, and $J_e=J_p=J_t=0$.

Figure 4

Charge-stability diagram calculated at $J_e=1$ meV and (a) $J_p=0$, (b) $J_p=1$ meV. $U=6.1$ meV, $U_{12}=2.5$ meV, and $t=J_t=0$.

Figure 5

Charge-stability diagram calculated at $U=6.1$ meV, $U_{12}=2.5$ meV, $t=0.6$ meV, and $J_e=J_p=0.2$ meV. $J_t=0$.

Figure 6

Charge-stability diagram calculated at (a) $J_t=0$, (b) $J_t=0.5$ meV, and $U=6.1$ meV, $U_{12}=2.5$ meV, $t=0.6$ meV, and $J_e=J_p=0.1$ meV.

Figure 7

(Color online) Profiles of biquadratic (black solid line) and Gaussian (red dashed line) model potentials under two different values of detuning energy $\varepsilon ={\mu }_2-{\mu }_1$. For the biquadratic potential, $\hslash {\omega }_0=3.956$ meV, $a=33$ nm. The parameters of the Gaussian potential are derived from the biquadratic one. The gray dotted lines denote the energy reference values of the chemical potential.

Figure 8

(Color online) Charge-stability diagrams calculated from biquadratic [panels (a) and (b)] and Gaussian [panels (c) and (d)] model potential different interdot distances. The parameters for the biquadratic model are $\hslash {\omega }_0=3.956$ meV, $a=33$ nm [panels (a) and (c)], and $a=28$ nm [panels (b) and (d)]. The white crosses [indicating the points at which the comparison of Hubbard parameters (Fig. 9) is conducted] are located at ${\mu }_1={\mu }_2=9.743$ meV for panels (a) and (c), and at ${\mu }_1={\mu }_2=10.095$ meV for panels (b) and (d).

Figure 9

(Color online) Parameters in the generalized Hubbard model as functions of half of the interdot distance $a$, calculated at the white crosses shown in Fig. 8, using both biquadratic and Gaussian potentials.

Figure 10

(Color online) Charge-stability diagrams calculated from the Gaussian potential using (a) $s$ orbitals only and (b) $s$ and $p$ orbitals. The parameters corresponding to the biquadratic potential are $a=30$ nm and $\hslash {\omega }_0=3.956$ meV. Panel (c) shows the full charge-stability diagram including all 49 components, while panel (b) shows the stability diagram in the two-electron regime. Panel (b) is a replica of the area enclosed by the white frame in the lower left corner of panel (c).

Figure 11

(Color online) Effective on-site Coulomb interaction [panel (a)] and intersite Coulomb interaction [panel (b)] of the results shown in Fig. 10 as a function of the detuning energy $\varepsilon ={\mu }_2-{\mu }_1$ calculated keeping $s$ orbitals only (black solid line) and $s$ and $p$ orbitals (red dashed line). The traces where calculations take place are shown as white diagonal lines in Figs. 10a and 10b. As $\varepsilon$ is swept from the top left corner to the bottom right corner, the dominating ground state is changed from (2,0) to (1,1), and then to (0,2). The black dotted lines and red dashed-dotted lines denote the phase boundaries (where the probabilities of states cross) in the case of $s$ orbitals only and $s$ and $p$ orbitals, respectively.

Figure 12

(Color online) Charge-stability diagrams calculated from Gaussian potential with $a=30$ nm and $\hslash {\omega }_0=3.956$ meV. (a) $B=0$, the zero-point energies of Fock-Darwin states are chosen according to the microscopic confinement potential at ${\mu }_1={\mu }_2=9.938$ meV (indicated by the white cross) with $\hslash {\omega }_{0G}=6.487$ meV. (b) $B=0$, the zero-point energies of Fock-Darwin states are $\hslash {\omega }_{0G}=9.731$ meV. (c) $B=0$, the harmonic-oscillator frequencies of the Fock-Darwin states are chosen according to the microscopic potentials for each given ${\mu }_1$ and ${\mu }_2$. (d) $B=6$ T, the zero-point energies of Fock-Darwin states are $\hslash {\omega }_{0G}=6.487$ meV.

Figure 13

Illustration of the connections between different components discussed in this paper as well as in Refs. 50 and 51.

Figure 14

Charge-stability diagram calculated at $U=6.1$ meV, $U_{12}=7$ meV, and $t=J_e=J_p=J_t=0$.

Figure 15

Charge-stability diagram calculated at $U=6.1$ meV, $U_{12}=2.5$ meV, (a) $J_e=2.5\mathrm{meV}=U_{12}$, (b) $J_e=4\mathrm{meV}>U_{12}$. $t=J_p=J_t=0$.