Coulomb-Interaction-Induced Incomplete Shell Filling in the Hole System of InAs Quantum Dots

Abstract

We have studied the hole charging spectra of self-assembled InAs quantum dots in perpendicular magnetic fields by capacitance-voltage spectroscopy. From the magnetic field dependence of the individual peaks we conclude that the s-like ground state is completely filled with two holes but that the fourfold degenerate p-shell is only half filled with two holes before the filling of the d-shell starts. The resulting six-hole ground state is highly polarized. This incomplete shell filling can be explained by the large influence of the Coulomb interaction in this system.

Multiple charging ofInAs∕GaAsquantum dots by electrons or holes: Addition energies and ground-state configurations

“…Since the spin index "σ" is not an actual electronic spin, but rather an index for two Kramers degenerate states, in principle the exchange integrals K iσ,jσ ′ are not simply diagonal in σ, σ ′ , as has been widely used in the dot charging literature. 3,11,15,16 However, adopting the literature approximation, K iσ,jσ ′ =K i,j δ σσ ′ and considering the s and two p orbitals, ǫ p1 < ǫ p2 and assuming the particle filling order follows Hund's rule, as shown in Fig.1, the total energies for N =1, 2, 3, 4 electrons in the Hartree-Fock approximation are…”

“…is the correlation energy correction to the addition energy calculated in HF. Since the HF equations are used by many experimentalists to deduce Coulomb energies, 3,8,11 we review it here. In the Hartree-Fock approximation, where the effect of correlations is neglected but the direct Coulomb and exchange interactions are retained, simple expressions can be derived for the addition energies.…”

“…Recently, it became possible to load and measure electrons 4,8 and holes 8,11,13 into much smaller, epitaxially grown self-assembled dots of InGaAs/GaAs, where electron addition energies are about 10 -60 meV, 4,8 and hole addition energies are between 10 -30 meV. 11,13 More interestingly, while electrons still follow the Aufbau principle, recent hole charging experiment 11 show that holes have unusual charging patterns that defy the Aufbau principle and Hund's rule.…”

“…This is afforded either by injecting from a tip of a scanning tunneling microscope, 1 or by various gate structures. 2,3,4,5,6,7,8,9,10,11 Since the energy scale of both single-particle levels and Coulomb interactions in quantum dots (QDs) (10 −4 -10 −2 Ry) are a few order of magnitudes smaller than those of the real atoms ( ∼ 1 Ry), dots can be loaded by as many as six 4 to ten 12 electrons in colloidal 12 and self-assembled 4,8,9 dots having confining dimension of ≤ 50Å, and up to hundreds of electrons in larger 500Å electrostatically confined dots. 2,3,7 The "charging energy" µ(N ) is the energy needed to add a carrier to the dot that is already loaded by N − 1 carriers,…”

“…It indeed has been recently shown 18 that hole ground state configurations predicted by the 2D-EMA model are qualitatively different from those measured by hole charging experiments. 11,13 An atomistic pseudopotential description of electronic structure effects can be used instead of 2D-EMA to calculate charging energies. 18,19,20,21,22 Here, we show that such an atomistic theory correctly reproduces the many-particle configurations as well as addition spectra for carriers in self-assembled quantum dots.…”