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

40 J.M. Elzerman et al.
a
T-
T0 T +
S
∆EZ ∆EZ EST b
Energy
N =1 N =2
∆E Z
Electrochemical potential
/
/
↑↔T0
↓↔T-
↑↔T+
↓↔T0
↑↔S
↓↔S
N=1↔2 E ST
∆E Z
∆E Z
Fig. 10. One- and two-electron states and transitions at finite magnetic field. (a)Energy
diagram for a fixed gate voltage. By changing the gate voltage, the one-electron
states (below the dashed line) shift up or down relative to the two-electron states
(above the dashed line). The six transitions that are allowed (i.e. not spin-blocked)
are indicated by vertical arrows. (b) Electrochemical potentials for the transitions
between one- and two-electron states. The six transitions in (a) correspond to only
four different electrochemical potentials. By changing the gate voltage, the whole
ladder of levels is shifted up or down
tunnelling between the dots, with tunnelling matrix element t, ε0 (the “bonding
state”) and ε1 (the “anti-bonding state”) are split by an energy 2t. By
filling the two states with two electrons, we again get a spin singlet ground
state and a triplet first excited state (at zero field). However, the singlet
ground state is not purely S (Fig. 9a), but also contains a small admixture of
the excited singlet S2 (Fig. 9f). The admixture of S2 depends on the competition
between inter-dot tunnelling and the Coulomb repulsion, and serves to
lower the Coulomb energy by reducing the double occupancy of the dots [33].
If we focus only on the singlet ground state and the triplet first excited
states, then we can describe the two spins S1 and S2 by the Heisenberg
Hamiltonian, H = JS1 · S2. Due to this mapping procedure, J is now defined
as the energy difference between the triplet state T0 and the singlet ground
state, which depends on the details of the double dot orbital states. From a
Hund-Mulliken calculation [34], J is approximately given by 4t 2 /U +V , where
U is the on-site charging energy and V includes the effect of the long-range
Coulomb interaction. By changing the overlap of the wavefunctions of the
two electrons, we can change t and therefore J. Thus, control of the interdot
tunnel barrier would allow us to perform operations such as swapping or
entangling two spins.