4 Coulomb blockade

70 4 Coulomb blockade
Assume now, that the basis states |α〉 are the states with definite spin
quantum number σα. It means, that only one spin component of the wave
function, namely ψα(σα) is nonzero, and ψα(¯σα) = 0. In this case the only
nonzero matrix elements are those with σα = σγ and σβ = σδ, theyare
Vαβ,γδ = dr dr ′ ψ ∗ α(r)ψ ∗ β(r ′ )V (|r − r ′ |)ψγ(r)ψδ(r ′ ). (4.12)
In the case of delocalized basis states ψα(r), the main matrix elements
are those with α = γ and β = δ, because the wave functions of two different
states with the same spin are orthogonal in real space and their contribution
is small. It is also true for the systems with localized wave functions ψα(r),
when the overlap between two different states is weak. In these cases it is
enough to replace the interacting part by the Anderson-Hubbard Hamiltonian,
describing only density-density interaction
ˆHAH = 1
Uαβ ˆnαˆnβ. (4.13)
2
α=β
with the Hubbard interaction defined as
Uαβ = dr
dr ′ |ψα(r)| 2 |ψβ(r ′ )| 2 V (|r − r ′ |). (4.14)
In the limit of a single-level quantum dot (which is, however, a two-level
system because of spin degeneration) we get the Anderson impurity model
(AIM)
ˆHAIM =
σ=↑↓
ɛσd † σdσ + U ˆn↑ˆn↓. (4.15)
The other important limit is the constant interaction model (CIM), which
is valid when many levels interact with similar energies, so that approximately,
assuming Uαβ = U for any states α and β
ˆHAH = 1
Uαβ ˆnαˆnβ ≈
2
α=β
U
2
ˆnα
2
α
− U
ˆn
2
α
2
α
= U ˆ N( ˆ N − 1)
.
2
(4.16)
where we used ˆn 2 =ˆn, forlargeNit is equivalent to (4.2).
Thus, the CIM reproduces the charging energy considered above, and the
Hamiltonian of an isolated system is
ˆHCIM =
αβ
˜ɛαβd † αdβ + E(N). (4.17)
Note, that the equilibrium compensating charge density can be easily introduced
into the AH Hamiltonian
ˆHAH = 1
Uαβ (ˆnα − ¯nα)(ˆnβ − ¯nβ) , (4.18)
2
α=β
and at N ≫ 1 we obtain the electrostatic energy (4.2) with C = U/e 2 .