Ground-state densities and pair correlation functions in parabolic quantum dots

We present an extensive comparative study of ground-state densities and pair distribution functions for electrons confined in two-dimensional parabolic quantum dots over a broad range of coupling strength and electron number. We first use spin-density-functional theory to determine spin densities that are compared with diffusion Monte Carlo (DMC) data. This accurate knowledge of one-body properties is then used to construct and test a local approximation for the electron-pair correlations. We find very satisfactory agreement between this local scheme and the available DMC data, and provide a detailed picture of two-body correlations in a coupling-strength regime preceding the formation of Wigner-like electron ordering.

Figure 1

Density profile $n\left(r\right)$ (in units of ${\ell }_0^{-2}$) as a function of $r∕{\ell }_0$ for a paramagnetic QD with $N=6$,12,20, and 30 electrons at $\lambda =1.89$. The results of the LSDA and of the HF are compared with the DMC data of Ref. 13. The dash-dotted lines are for noninteracting electrons.

Figure 2

Density profile $n\left(r\right)$ (in units of ${\ell }_0^{-2}$) as a function of $r∕{\ell }_0$ for a partially spin-polarized QD with $N=9$ electrons at $\lambda =1.89$. The symbols are as in Fig. 1.

Figure 3

Local spin polarization $\zeta \left(r\right)$ as a function of $r∕{\ell }_0$ for a partially spin-polarized QD with $N=9$ electrons at $\lambda =1.89$. The symbols are as in Fig. 1. The inset shows the difference between $\zeta \left(r\right)$ calculated with two different parametrizations for ${\epsilon }_{\mathrm{xc}}^{\mathrm{hom}}(n,\zeta )$ (see text).

Figure 4

Spin-summed PDF $g(r,0)$ as a function of $r∕{\ell }_0$ for a partially spin-polarized QD with $N=9$ at $\lambda =1.89$. The results of the LSDA and ASDA are compared with the DMC data of Ref. 13. The inset shows $r_s^{\mathrm{LSDA}}\left(r\right)$ (dash-dotted line) and $r_s^{\mathrm{ASDA}}(r,0)$ (solid line) as functions of $r∕{\ell }_0$ (see text).

Figure 5

Top panel: definition of $x$ and $x^{\prime }$ appearing on the horizontal axis of Fig. 6 ($\mathbf{r}=x\widehat{\mathbf{x}}$ and ${\mathbf{r}}^{\prime }=x^{\prime }\widehat{\mathbf{x}}$). Bottom panel: definition of the azimuthal angle $\theta$ giving the abscissa in Fig. 11.

Figure 6

Spin-summed PDF $g(x,x^{\prime })$ as a function of $x∕{\ell }_0$ and $x^{\prime }∕{\ell }_0$ for a partially spin-polarized QD with $N=9$ electrons at $\lambda =1.89$. The bottom panel shows a contour plot of $g(x,x^{\prime })$: the thin solid lines limit the bulk central region $(\mid x\mid ,\mid x^{\prime }\mid )\lesssim 3{\ell }_0$ which is shown in the top panel.

Figure 7

Parallel-spin PDF $g_{\sigma \sigma }(r,0)$ as a function of $r∕{\ell }_0$ for a partially spin-polarized QD with $N=9$ electrons at $\lambda =1.89$. The results of the Hartree-Fock approximation are compared with the DMC data of Ref. 13.

Figure 8

Probability density (in units of ${\ell }_0^{-1}$) as a function of $r∕{\ell }_0$ for a QD with $N=6$ electrons at varying $\lambda$: profiles for the paramagnetic state (top) and for the ferromagnetic state (bottom). The inset shows the height $\Delta$ of the minimum (in units of ${\ell }_0^{-1}$) as a function of $\lambda$.

Figure 9

Spin-summed PDF $g(r,0)$ as a function of $r∕{\ell }_0$ for a QD with $N=6$ electrons in its ground state at various $\lambda$. The curves for $\lambda =1.89$ and 3.54 refer to the paramagnetic state, while those for $\lambda =10$ and 12 refer to the ferromagnetic state. The inset shows $r_s^{\mathrm{ASDA}}(r,0)$.

Figure 10

Total conditioned probability density $2\pi rP(r\mid r^{\prime }=0)$ as a function of $r∕{\ell }_0$ for the ground state of a QD with $N=6$ electrons. The curves for $\lambda =1.89$ and 3.54 refer to the paramagnetic state, while those for $\lambda =10$ and 12 refer to the ferromagnetic state.

Figure 11

Azimuthal plot of the spin-summed PDF as a function of the angle $\theta$ defined in Fig. 5, for the ground state of a QD with $N=6$ electrons. The reference electron is located at $\theta =0$. The curves for $\lambda =1.89$ and 3.54 refer to the paramagnetic state, while those for $\lambda =10$ and 12 refer to the ferromagnetic state. The inset shows the preferred first-neighbor distance $d\left(\lambda \right)$ along the circle at $r=r_{\max }$, in units of $a_B^{\star }$.

Figure 12

Rotationally broken HF one-body density (in units of ${\ell }_0^{-2}$) for a QD with $N=6$ electrons at $\lambda =3.18$, in the paramagnetic (upper panel) and ferromagnetic (lower panel) case. At this value of $\lambda$ the HF approximation predicts the ferromagnetic state to lie at lower energy.