Universality in the energy spectrum of medium-sized quantum dots

In a two-dimensional parabolic quantum dot charged with $N$ electrons, Thomas-Fermi theory states that the ground-state energy satisfies the following nontrivial relation: $E_{\text{gs}}/\left(\hslash \omega \right)\approx N^{3/2}{f}_{\text{gs}}\left(N^{1/4}\beta \right)$, where the coupling constant, $\beta$, is the ratio between Coulomb and oscillator $\left(\hslash \omega \right)$ characteristic energies and $f_{\text{gs}}$ is a universal function. We perform extensive configuration-interaction calculations in order to verify that the exact energies of relatively large quantum dots approximately satisfy the above relation. In addition, we show that the number of energy levels for intraband and interband (excitonic and biexcitonic) excitations of the dot follows a simple exponential dependence on the excitation energy whose exponent, $1/\Theta$, satisfies also an approximate scaling relation á la Thomas-Fermi, $\Theta /\left(\hslash \omega \right)\approx N^{-\gamma }g\left(N^{1/4}\beta \right)$. We provide an analytic expression for $f_{\text{gs}}$ based on two-point Padé approximants and two-parameter fits for the $g$ functions.

Figure 1

(Color online) Different contributions to the ground-state wave function entering the configuration-interaction calculation.

Figure 2

(Color online) Scaling of the ground-state energy in medium-sized dots. The large-$N$ expression for the Padé estimate [Eq. (6)] is shown as a solid line.

Figure 3

(Color online) (a) The number of excited states in the 42-electron quantum dot as a function of the excitation energy. The confinement strength is $\hslash \omega =6\text{meV}$. (b) The excitation gap to the first excited state as a function of the scaled variable $z=N^{1/4}\beta$. (c) The temperature parameter, $\Theta$, in scaled variables. Fits in (b) and (c) correspond to Eqs. (8, 10).

Figure 4

(Color online) (a) The interband (excitonic) excitations in the quantum dot with 42 electrons and $\hslash \omega =18\text{meV}$. In the $x$ axis the reference energy is the first excitonic state. (b) Scaling of the temperature parameter for the lowest-energy excitonic states.

Figure 5

(Color online) Structure of the biexcitonic wave function in the configuration-interaction calculations.

Figure 6

(Color online) (a) The interband (biexcitonic) excitations in the quantum dot with 42 electrons and $\hslash \omega =6\text{meV}$. In the $x$ axis the reference energy is the first biexcitonic state. (b) Scaling of the temperature parameter for the biexcitonic excitations.