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

Low-Temperature Conduction of a Quantum Dot 115
Each term in the sum here corresponds to a process during which an electron
or a hole is created virtually on the level n in the dot, cf. (42). For Gα ≪ e 2 /h
and ES ≪ δE the main contribution to the sum in (53) comes from singlyoccupied
energy levels in the dot. A dot with spin S has 2S such levels near the
Fermi level (hereafter we assign indexes n = −S,...,n = S to these levels),
each carrying a spin S/2S, and contributing
J n λn
αα ′ = t
EC
∗ αntα ′ n , λn =2/S, |n| ≤S (54)
to the exchange amplitude in (51). This yields
�
Jαα ′ ≈
EC
n
|n|≤S
J n αα ′ . (55)
It follows from (53) and (54) that
tr ˆ J = 1 � � 2
λn |tLn| + |t 2 Rn| � . (56)
By restricting the sum over n here to |n| ≤S, asin(55), and taking into
account that all λn in (54) are positive, we find J1 + J2 > 0. Similarly, from
�
λmλn|D 2 � �
tLm tRm
mn| , Dmn = det
(57)
det ˆ J = 1
2E 2 C
m,n
tLn tRn
and (54) and (55) follows that J1J2 > 0forS>1/2. Indeed, in this case
the sum in (57) contains at least one contribution with m �= n; all such
contributions are positive. Thus, both exchange constants J1,2 > 0ifthe
dot’s spin S exceeds 1/2 [30]. The pecularities of the Kondo effect in quantum
dots with large spin are discussed in [30, 50].
Here we concentrate on the most common situation of S =1/2 onthe
dot [3]. The ground state of such dot has only one singly-occupied energy
level (n = 0), so that det ˆ J ≈ 0, see (55) and (57). Accordingly, one of the
exchange constants vanishes,
J2 ≈ 0 , (58)
while the remaining one, J1 = trˆ J, is positive. Equation (58) resulted, of
course, from the approximation made in (55). For the model (15) the leading
correction to (55) originates in the co-tunneling processes with an intermediate
state containing an extra electron (or an extra hole) on one of the empty
(doubly-occupied) levels. Such contribution arises because the spin on the level
n is not conserved by the Hamiltonian (15), unlike the corresponding occupation
number. Straightforward calculation [49] yields the partial amplitude in
the form of (54), but with
λn = − 2ECES
, n �= 0.
(EC + |ɛn|) 2