Singlet-triplet transition in double quantum dots in two-dimensional topological insulators

We study two-electron states confined in two coupled quantum dots formed by a short-range potential in a two-dimensional topological insulator. It is shown that there is a fairly wide range of the system parameters where the ground state is a tripletlike state formed by a superposition of two spin-polarized states. Outside this range, the ground state is a singlet. A transition between the singlet and triplet states can be realized by changing the potential of the quantum dots. The effect is caused by a significant change in the energies of the Coulomb repulsion and the exchange interaction of electrons due to the presence of the pseudospin components of the wave function when the band spectrum is inverted.

Figure 1

Bound-state energy as a function of the quantum well potential for two interwell distances ($d=4$ and $d=5$) in the topological phase. Thin black lines depict the single-well energy spectrum. The parameters used in the calculations are $a=2,\mathrm{\Lambda }=3$.

Figure 2

Bound-state energy of the symmetric ${\varepsilon }_s$ and antisymmetric ${\varepsilon }_a$ states as a function of the interwell distance in the topological phase. Red and blue lines show, respectively, the symmetric and asymmetric electron- and holelike states. Numerical parameters are chosen as $v=15,\mathrm{\Lambda }=3$, and $a=2$.

Figure 3

Evolution of the two-electron spectrum as the $e\text{−}e$ interaction amplitude is changed. Lines 1, 2, 3, and 6 depict the energy of unpolarized states ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2,{\mathrm{\Psi }}_3$, and ${\mathrm{\Psi }}_6$. Line T depicts polarized states ${\mathrm{\Psi }}_4$ and ${\mathrm{\Psi }}_5$. Calculations are carried out for the topological phase at $d=6,v=15,\mathrm{\Lambda }=3$, and $a=2$. The energies of the one-particle states are ${\varepsilon }_s=-0.5142$ and ${\varepsilon }_a=-0.4786$.

Figure 4

Evolution of the two-electron spectrum as the $e\text{−}e$ interaction amplitude is changed for a variety of the well potentials. The low-lying singlet ${\mathcal{E}}_1$ (line 1), polarized triplets ${\mathcal{E}}_T$ (line T), and unpolarized tripletlike state ${\mathcal{E}}_3$ (line 3) are shown. The numerical data are as follows: (a) $v=15,{\varepsilon }_s=-0.5142$, and ${\varepsilon }_a=-0.4786$; (b) $v=18,{\varepsilon }_s=-0.2803$, and ${\varepsilon }_a=-0.2473$; and (c) $v=25,{\varepsilon }_s=0.0515$, and ${\varepsilon }_a=0.0849$. Other parameters are $d=6,\mathrm{\Lambda }=3$, and $a=2$.

Figure 5

Critical potential of the quantum wells $v_c$ as a function of the $e\text{−}e$ interaction potential $u$ for a variety of interwell distances $d$. In the region $v<v_c$ the ground state is formed by the triplet states. The parameters used in the calculations are $a=2,\mathrm{\Lambda }=3$.

Figure 6

Evolution of the two-electron spectrum as the $e\text{−}e$ interaction amplitude is changed for two values of the well potential in the topologically trivial case. The low-lying singlet ${\mathcal{E}}_1$ (line 1), polarized triplets ${\mathcal{E}}_T$ (line T), and unpolarized tripletlike state ${\mathcal{E}}_3$ (line 3) are shown. The numerical data are as follows: (a) $v=8,{\varepsilon }_s=-0.698$, and ${\varepsilon }_a=-0.83$ and (b) $v=12,{\varepsilon }_s=0.248$, and ${\varepsilon }_a=0.182$. Other parameters are $d=6,\mathrm{\Lambda }=3$, and $a=2$.