Universality in the Energy Spectrum of Medium-Sized Quantum Dots

TL;DR

This paper demonstrates that the energy spectrum of medium-sized quantum dots exhibits universal scaling laws, verified through extensive calculations, and provides analytic expressions for these universal functions.

Contribution

It introduces and verifies universal scaling relations for quantum dot energies and excitations, supported by extensive computational data and analytical approximations.

Findings

Ground-state energy scales with N^{3/2} and a universal function of N^{1/4}β.

Number of energy levels follows an exponential dependence on excitation energy.

Analytic expressions for the universal functions are provided.

Abstract

In a two-dimensional parabolic quantum dot charged with $N$ electrons, Thomas-Fermi theory states that the ground-state energy satisfies the following non-trivial relation: $E_{gs}/(\hbar\omega)\approx N^{3/2} f_{gs}(N^{1/4}\beta)$, where the coupling constant, $\beta$, is the ratio between Coulomb and oscillator ($\hbar\omega$) characteristic energies, and $f_{gs}$ 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 {\it a la} Thomas-Fermi, $\Theta/(\hbar\omega)\approx N^{-\gamma} g(N^{1/4}\beta)$. We provide an analytic expression for $f_{gs}$, based on two-point Pad\'e approximants, and two-parameter fits for the $g$ functions.