Interplay between electron spin and orbital pseudospin in double quantum dots

We investigate theoretically spin and orbital pseudospin magnetic properties of a molecular orbital in parabolic and elliptic double quantum dots (DQD’s). In our many-body calculation we include intradot and interdot electron-electron interactions, in addition to the intradot exchange interaction of $p$ orbitals. We find for parabolic DQD’s that, except for the half or completely filled molecular orbital, spins in different dots are ferromagnetically coupled while orbital pseudospins are antiferromagnetically coupled. For elliptic DQD’s spins and pseudospins are either ferromagnetically or antiferromagnetically coupled, depending on the number of electrons in the molecular orbital. We have determined orbital pseudospin quantum numbers for the ground states of elliptic DQD’s. An experiment is suggested to test the interplay between orbital pseudospin and spin magnetism.

Figure 1

(a) Basis set $B^1(N_p=1)$. ↑ (↓) is the electron with up (down) spin. The upper (lower) half circle represents a state in the $p$ orbital with $z$ component of angular momentum $\alpha =1$ $(-1)$. Not all the basis states are shown. Other states can be obtained by flipping all spins, by altering the signs of all angular momenta, or by doing both. (b) One of the eigenstates for parabolic dots is shown as an example. This state has the quantum numbers $\sigma =1$ and $\alpha =1$ and is given by a linear combination of the basis states in the set $B^1\left(1\right)$.

Figure 2

Some basis states in the set $B^1(N_p=2)$. States with two electrons in a single dot are not included in this basis set since we are interested only in the low-energy states.

Figure 3

Some basis states in the three sets from $B^1(N_p=3)$ to $B^3(N_p=3)$.

Figure 4

Some basis states in the five sets from $B^1(N_p=4)$ to $B^5(N_p=4)$.

Figure 5

The low-lying energy states for $N_p=1$ are shown. Left and right horizontal bars indicate energy levels of ${\mathcal{H}}_0$ and ${\mathcal{H}}_0+V_{\mathrm{hop}}$. The numbers on the bars indicate the number of degeneracies. (a) The energy spectrum and states are shown. Using the particle-hole transformation, we can obtain the energy spectrum for $N_p=7$ from that of $N_p=1$; see the text. (b) One of the ground states is depicted.

Figure 6

The low-lying energy states for $N_p=2$ are shown: (a) the energy spectrum and states and (b) two of the ground states are depicted.

Figure 7

The low-lying energy states for $N_p=3$ are shown: (a) the energy spectrum and states and (b) two of the ground states are depicted.

Figure 8

The low-lying energy states for $N_p=4$ are shown: (a) the energy spectrum and states and (b) the ground state are depicted.

Figure 9

(a) The low-lying energy spectrum for $N_p=1$ is depicted for an elliptic DQD. The left, middle, and right horizontal bars indicate energy levels of ${\mathcal{H}}_0$, ${\mathcal{H}}_0+V_{\text{elliptic}}$, and ${\mathcal{H}}_{\mathrm{tot}}={\mathcal{H}}_0+V_{\text{elliptic}}+V_{\mathrm{hop}}$. (b) One of the ground states for ${\mathcal{H}}_{\mathrm{tot}}$ is depicted.

Figure 10

(a) The low-lying energy spectrum for $N_p=2$ of an elliptic DQD. (b) The ground state is depicted.

Figure 11

(a) The low-lying energy spectrum for $N_p=3$ of an elliptic DQD. (b) Two ground states are depicted.

Figure 12

(a) The low-lying energy spectrum for $N_p=4$ of an elliptic DQD. (b) The ground state is depicted.