Many-body perturbation theory calculations on circular quantum dots

The possibility to use perturbation theory to systematically improve calculations on circular quantum dots is investigated. A few different starting points, including Hartree-Fock, are tested and the importance of correlation is discussed. Quantum dots with up to 12 electrons are treated and the effects of an external magnetic field are examined. The sums over excited states are carried out with a complete finite radial basis set obtained through the use of $B$ splines. The calculated addition energy spectra are compared with experiments and the implications for the filling sequence of the third shell are discussed in detail.

Figure 1

(Color online) Second-order perturbation theory correction to the energy as function of $\max \left(n\right)$ (squares) and $\max \left(\mid m_{\ell }\mid \right)$ (circles) in the second sum of Eq. (13) for the two electron dot with the confinement strength $\hslash \omega =6\mathrm{meV}$. Note that the sum over $m_{\ell }$ converges faster than the sum over $n$.

Figure 2

(Color online) The quotient between the calculated energies (of the respective method) and the CI energy as functions of the confinement strength $\hslash \omega$ for the ground state in the two electron dot. In (a) the results from HF, a reduced exchange HF with $\alpha =0.7$ and second-order MBPT using wave functions from the respective method are plotted. In (b) the results from LDA calculations (with two different alphas) + first- and second-order MBPT are plotted. For reference the results from calculations where we have used the one electron wave functions as a starting point for the perturbation expansion (taking the whole electron-electron interaction as the perturbation) have been plotted in both (a) and (b).

Figure 3

(Color online) $E∕E_0$ for the two electron dot and for different methods as functions of $l_0∕a_B^{*}$, where $l_0=\sqrt{\hslash ∕\left(m^{*}\omega \right)}$ is the characteristic dot length, $E_0=\hslash \omega$ is the single particle energy, and $a_B^{*}$ is the effective Bohr radius. Small values of $l_0∕a_B^{*}$ correspond to stronger confinement and therefore a faster expected convergence rate of a perturbation expansion.

Figure 4

(Color online) Comparison between our HF and second-order MBPT results for the six electron dot in the ground state with $M_L^{\mathit{TOT}}=0$ and $S_z^{\mathit{TOT}}=0$, with the LSDA and CI calculations by Reimann et al. (Ref. 16). The second-order MBPT calculations include the full sum over the complete radial basis set (corresponding to all $n$ values) and with $\max \left(\mid m_{\ell }\mid \right)=1,2,3,\dots ,30$ for the two strongest potentials. For clarity only the curves with $\max \left(\mid m_{\ell }\mid \right)=1$, 4, 6, and 30 have been labeled. The HF and the second-order MBPT with $\max \left(\mid m_{\ell }\mid \right)=30$ curves are plotted for all potential strengths calculated by Reimann et al. Moreover, the values of $⟨S^2⟩$ for the HF and the second-order MBPT with $\max \left(\mid m_{\ell }\mid \right)=30$ have been plotted in the figure.

Figure 5

(Color online) The chemical potentials for $N=1,2,\dots ,6$ as functions of the external magnetic field according to HF (dashed curve) and second-order MBPT (full curve) calculations for the potential strength $\hslash \omega =5\mathrm{meV}$. Note the big difference between the two different models regarding the behavior of the chemical potentials when the magnetic field varies.

Figure 6

(Color online) The ground state addition energy spectra for dots with $\hslash \omega =5\mathrm{meV}$ (a) and $\hslash \omega =7\mathrm{meV}$ (b). The squares (circles) represent the addition energy spectra according to HF (second-order MBPT). It is clear that the second-order MBPT spectra imply closer resemblances to the experimental picture in Tarucha (Ref. 7) than the HF spectrum.

Figure 7

(Color online) Ground state (a) and selected excited state (b)–(e) addition energy spectra for $\hslash \omega =5\mathrm{meV}$ according to second-order MBPT. The notation $({\sum }_{i=7}^N{n}^i,\mid {\sum }_{i=7}^N{m}_{\ell }^i\mid ,S)$ to label the states is used. Note the big differences between the different spectra. For example, the ground state spectrum (a) has peaks at $N=8,10$ while spectrum (b) has a peak at $N=8$ and the rest have a peak at $N=9$. That is, even if the spin is maximized at the half filled shell $(N=9)$ there is not always a peak there as seen in subfigure (a) and (b). Subfigure (e) resembles the experimental results of Ref. 3 best with dips at $N=7$ and 10 and a peak at $N=9$. Moreover, combining the addition energies for $N=6,7,8$ of sequence (c) or (d) with the addition energies for $N=10,11,12$ of sequence (e) would give a spectrum that closely resembles the experimental situation in Ref. 7 with dips at $N=8$ and 10 and a peak at $N=9$.

Figure 8

(Color online) Ground state (a) and selected excited state (b)–(e) addition energy spectra for $\hslash \omega =7\mathrm{meV}$ according to second-order MBPT. The notation $({\sum }_{i=7}^N{n}^i,\mid {\sum }_{i=7}^N{m}_{\ell }^i\mid ,S)$ to label the states is used. Note the big differences between the different spectra. Note also that all spectra have peaks at $N=9$. Even though the ground state spectra for $N=7$, 8, and 9 resemble the experimental results of Ref. 7, the dip at $N=11$ is uncharacteristic when compared with the experimental results of Refs. 7, 3. Subfigure (b) resembles the experimental result in Ref. 3 the most with a peak at $N=9$ and dips at $N=7$ and 10 while subfigure (c) resembles the experimental results of Ref. 7 the most with a peak at $N=9$ and dips at $N=8$ and 10.

Figure 9

(Color online) $⟨S^2⟩$ according to Hartree-Fock and second-order MBPT calculations as functions of the potential strength for the seven electron ground and excited state.