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

102 M. Pustilnik and L.I. Glazman
(13) (here we took into account that the screening length in two dimensions
coincides with the Bohr radius a0), which yields
ES ∼ rs ln (1/rS) δE ≪ δE . (14)
The estimate (14) is valid only for rs ≪ 1. However, the ratio ES/δE remains
small for experimentally relevant 2 value rs ∼ 1 as long as the Stoner criterion
for the absence of itinerant magnetism [23] is satisfied. This guarantees the
absence of a macroscopic (proportional to N) magnetization of a dot in the
ground state [19].
The third term in (10), representing interaction in the Cooper channel,
is renormalized by higher-order corrections arising due to virtual transitions
to states outside the energy strip of the width ET about the Fermi level.
For attractive interaction (Λ 0),
in which case Λ is very small,
δE
Λ ∼
ln(ɛF /ET )
∼ δE
ln N
≪ δE .
This estimate accounts for the logarithmic renormalization of Λ when the
high-energy cutoff is reduced from the Fermi energy ɛF down to the Thouless
energy ET [20]. In addition, if the time-reversal symmetry is lifted (β =2)
then the third term in (10) is zero to start with. Accordingly, hereinafter we
neglect this term altogether by setting Λ =0.
Obviously, the interaction part of the Hamiltonian (10), is invariant with
respect to a change of the basis of single-particle states φi(r). Picking up the
basis in which the first term in (1) is diagonal, we arrive at the universal
Hamiltonian [19, 20],
Hdot = �
ɛnd † � �2 nsdns + EC
ˆN − N0 − ES ˆ S 2 . (15)
ns
We included in (15) the effect of the capacitive coupling to the gate electrode:
the dimensionless parameter N0 is proportional to the gate voltage,
N0 = CgVg/e ,
where Cg is the capacitance between the dot and the gate, see Fig. 1. The
relative magnitude of fully off-diagonal interaction terms in (1) (corresponding
2 For GaAs (m ∗ ≈ 0.07me, κ ≈ 13) the effective Bohr radius a0 ≈ 10 nm, whereas
a typical density of the two-dimensional electron gas, n ∼ 10 11 cm −2 [3], corresponds
to kF = √ 2πn ∼ 10 6 cm −1 . This gives kF a0 ∼ 1.