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

Low-Temperature Conduction of a Quantum Dot 101
Hint = EC ˆ N 2 − ES ˆ S 2 + Λ (2/β − 1) ˆ T † ˆ T (10)
of the interaction part of the Hamiltonian of the dot. Here
ˆN = �
d † nsdns ,
�
S ˆ = d † σss
ns
′
2 dns ′ �
T ˆ =
ns
nss ′
d
n
†
n↑d† n↓ (11)
are the operators of the total number of electrons in the dot, of the dot’s
spin, and the “pair creation” operator corresponding to the interaction in the
Cooper channel.
Thefirsttermin(10) represents the electrostatic energy. In the conventional
equivalent circuit picture, see Fig. 1, the charging energy EC is related
to the total capacitance C of the dot, EC = e2 /2C. For a mesoscopic
(kF L ≫ 1) conductor, the charging energy is large compared to the mean
level spacing δE. Indeed, using the estimates C ∼ κL and (3) and (5), we find
√
EC/δE ∼ L/a0 ∼ rs N. (12)
Except an exotic case of an extremely weak interaction, this ratio is large
for N ≫ 1; for the smallest quantum dots formed in GaAs heterostructures,
EC/δE ∼ 10 [3]. Note that (4), (6), and (12) imply that
ET /EC ∼ 1/rs � 1 ,
which justifies the use of RMT for the description of single-particle states with
energies |ɛn| � EC, relevant for Coulomb blockade.
GL
L dot
R
CL
GR
CR
Cg
VL Vg VR
Fig. 1. Equivalent circuit for a quantum dot connected to two leads by tunnel
junctions and capacitively coupled to the gate electrode. The total capacitance of
the dot C = CL + CR + Cg
The second term in (10) describes the intra-dot exchange interaction, with
the exchange energy ES given by
�
ES = dr dr ′ U(r − r ′ )F 2 (|r − r ′ |) (13)
In the case of a long-range interaction the potential U here should properly
account for the screening [20]. For rs ≪ 1 the exchange energy can be estimated
with logarithmic accuracy by substituting U(r) =(e 2 /κr)θ(a0 −r) into