4 Coulomb blockade

4.1 Electron-electron interaction in nanosystems 69
the energy spectrum E(q, λ) of the isolated molecule, where q is the charge
state, and λ is the eigenstate in this charge state. Finally, for nanosystems we
understand the charging energy as the minimal energy required to add one
electron to the neutral system, and the level spacing as the typical energy
difference between two levels of the system in the same charge state.
The consideration in this chapter is devoted mostly to the case of the dense
energy spectrum ∆ɛ ≪ EC, when the directness of the energy levels can be
neglected or taken into account within the simplified models.
It is reasonable to note here, that the last very important energy scale is
the temperature T . Both discrete energy spectrum and charging effects can
be observed only if the temperature is lower, than the corresponding energy
scales
T ≪ ∆ɛ, T ≪ EC. (4.6)
4.1.3 Anderson-Hubbard and constant-interaction models
To take into account both discrete energy levels of a system and the electronelectron
interaction, it is convenient to start from the general Hamiltonian
ˆH =
αβ
˜ɛαβd † αdβ + 1
2
αβγδ
Vαβ,γδd † αd †
β dγdδ. (4.7)
The first term of this Hamiltonian is a free-particle discrete-level model (2.8)
with ˜ɛαβ including electrical potentials. And the second term describes all
possible interactions between electrons and is equivalent to the real-space
Hamiltonian
ˆHee = 1
2
dξ
where ˆ ψ(ξ) are field operators
dξ ′ ˆ ψ † (ξ) ˆ ψ † (ξ ′ )V (ξ,ξ ′ ) ˆ ψ(ξ ′ ) ˆ ψ(ξ), (4.8)
ˆψ(ξ) =
ψα(ξ)dα, (4.9)
α
ψα(ξ) are the basis single-particle functions, we remind, that spin quantum
numbers are included in α, and spin indices are included in ξ ≡ r,σ as variables.
The matrix elements are defined as
Vαβ,γδ = dξ dξ ′ ψ ∗ α(ξ)ψ ∗ β(ξ ′ )V (ξ,ξ ′ )ψγ(ξ)ψδ(ξ ′ ). (4.10)
For pair Coulomb interaction V (|r|) the matrix elements are
Vαβ,γδ =
dr
σσ ′
dr ′ ψ ∗ α(r,σ)ψ ∗ β(r ′ ,σ ′ )V (|r − r ′ |)ψγ(r,σ)ψδ(r ′ ,σ ′ ).
(4.11)