Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

38 J.M. Elzerman et al.
1.6 Spin Configurations in Few-Electron Quantum Dots
The fact that electrons carry spin determines the electronic states of the quantum
dot. In the simplest case – a dot containing just a single electron – spin
leads to a splitting of all orbitals into Zeeman doublets, with the ground state
corresponding to the electron spin pointing up (↑), and the excited state to
the spin pointing down (↓). The difference between the corresponding energy
levels E↑ and E↓ is given by the Zeeman energy, ∆EZ = gµBB, whichis
approximately 25 µeV/T in GaAs.
For two electrons in a quantum dot, the situation is more complicated. For
a Hamiltonian without explicit spin-dependent terms, the two-electron state
is the product of the orbital and spin state. Since electrons are fermions, the
total two-electron state has to be anti-symmetric under exchange of the two
particles. Therefore, if the orbital part is symmetric, the spin state must be
anti-symmetric, and if the spin part is anti-symmetric, the orbital state must
be symmetric. The anti-symmetric two-spin state is the so-called spin singlet
(S):
| ↑↓〉 − | ↓↑〉
S = √ (3)
2
which has total spin S = 0. The symmetric two-spin states are the so-called
spin triplets (T+, T0 and T−):
T+ = | ↑↑〉T0 =
| ↑↓〉 + | ↓↑〉
√ T− = | ↓↓〉 (4)
2
which have total spin S = 1 and a quantum number ms (corresponding to the
spin z-component) of 1, 0, and −1, respectively. In a finite magnetic field, the
three triplet states are split by the Zeeman splitting, ∆EZ.
Even at zero magnetic field, the energy of the two-electron system depends
on its spin configuration, through the requirement of anti-symmetry of the
total state. If we consider just the two lowest orbitals, ε0 and ε1, then there
are six possibilities to fill these with two electrons (Fig. 9). At zero magnetic
field [28], the two-electron ground state is always the spin singlet (Fig. 9a),
and the lowest excited states are always the three spin triplets (Fig. 9b–d).
The energy gain of T0 with respect to the excited spin singlet S1 (Fig. 9e)
is known as the exchange energy, J. It essentially results from the fact that
electrons in the triplet states tend to avoid each other, reducing their mutual
Coulomb energy. As the Coulomb interaction is very strong, the exchange
energy can be quite large (a few 100 µeV) [29].
The energy difference between T0 and the lowest singlet S, the “singlettriplet
energy” EST, is thus considerably smaller than ε1 − ε0. In fact, besides
the gain in exchange energy for the triplet states, there is also a gain in the
direct Coulomb energy, related to the different occupation of the orbitals [29].
For a magnetic field above a few Tesla (perpendicular to the 2DEG plane),
EST can even become negative, leading to a singlet-triplet transition of the
two-electron ground state [30].