Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

3.2 Introduction to quantum dots and various derivations 61
to the leads where the number of electrons in the dot is a well defined integer
and the discrete nature of charge determines the single electron transport through
the system in a mostly classical way. A current through the QD is given by a
sequence of tunnelling events of single electrons. The Coulomb repulsion between
electrons leads to a considerable energy cost when adding an extra electron charge
to the QD. Current can only flow if this energy is supplied by external voltages – a
phenomenon known as Coulomb blockade [126]. The energy threshold which is to
be overcome when adding an electron to a QD that already contains N electrons is
the addition energy E add := µ(N +1)−µ(N) where µ(N) is the chemical potential
of the QD with N electrons. Exact calculation of E add is a formidable task, since
it requires the solution of the full many body problem of a few dozen electrons.
An estimate for the addition energy is given by the constant interaction (CI)
model [125, 126] which assumes that the Coulomb interaction between the electrons
in the dot is independent of N. Thus, the addition energy is approximated by
E add = e 2 /C + ∆E, where ∆E is the energy difference between subsequent singleelectron
states (calculated e.g. in the Fock–Darwin framework) and the Coulomb
interactions are represented as a single capacitance C, which leads to the classical
charging energy E c = e 2 /C for a charge e. This approximation for E add requires
that the Coulomb interaction does not mix up the single electron wavefunctions. A
condition which is generally fulfilled in vertical GaAs dots where the confinement
induced single electron energy scale is comparable or larger than the interaction
energies, ∆E ≥ E c [108].
Despite its simplicity, the CI model is remarkably successful in providing a
basic understanding of experiments on few-electron QDs [108]. The implication
that when adding electrons to a QD they will successively occupy consecutive
single-particle levels – while the energy cost of E c has to be paid for each electron
– is sufficient to explain the shell structure which is found in the addition energy
spectra [125]. For zero magnetic field the two-dimensional oscillator states at
E = (M + 1)ω 0 with M = 2n + |l| are (M + 1)-fold degenerate, see Fig. 3.4.
Therefore, in the CI model, each single-particle level can take up to 2(M + 1)
electrons due to spin degeneracy. This leads to the magic numbers 2,6,12,... in
the shell structure. Many magnetic field effects in QD spectra can be explained
by the crossover from degenerate two-dimensional harmonic oscillator states for
B = 0 to the lowest Landau level in the high field limit. The CI model can be
extended to include effects of exchange interaction such as Hund’s rule, which
obliges the dot shells to be filled with electrons of parallel spin first [108].
Not included in the CI model are effects which violate the assumption of constant
interaction. In the presence of a magnetic field the characteristic length ˜l
decreases for larger B, indicating that the confinement becomes stronger. Thus,
two electrons which occupy the same state will be pushed closer together, significantly
increasing the Coulomb interaction. Therefore, at some magnetic fields it