10. Hamiltonian of the system:
H H 0 + HT ,
=
=H0 ∑
i=S ,I ,D
H i + H env
∑ (l
H i = + eVi )cl†,i cl ,i ,(i =
l
S , D)
=HI ∑α
α cα cα + Q 2 / 2CΣ
†
H env = ∑ ωα bα bα
†
α
H T = H T 2 , H Ti = H i− ,
HT 1 + H i+ +
=H1+ ∑
= ( H i+ )†
Tα p cα (t )c p (t )e − iϕ1 ( t ) , H i−
α,p
†
10
11. Inelastic cotunneling
γ ∝ ∫ d ε1 ∫ d ε 2 Γ1 (ε1 + eV1 ) Re[ D(ε1 , ε 2 )] Γ 2 (ε 2 + eV2 )
Γi (eV ) ∝ ∫ d ε1 ∫ d ε 2 f (ε1 )[1 − f (ε 2 )] P(ε1 − ε 2 + eV )
P( E ) ∝ ∫ dt exp( J (t ) + iEt / )
d ω Re[ Z (ω )] β ω
J (t ) ∫ ω RK [coth 2 (cos(ωt ) − 1) − isin(ωt )]
11
46. Bogoliubov-de Gennes approach
1 e
= ∑ ∫ d r Ψσ (r )[
H BCS 3
( ∇ − A(r )) 2 + U (r ) − µ ]Ψσ (r )
†
σ 2m i c
+ ∫ d 3 r [∆(r ) Ψ † (r )Ψ † (r ) + ∆* (r )Ψ ↓ (r )Ψ ↑ (r )]
↑ ↓
∆(r ) = (r ) Ψ ↓ (r )Ψ ↑ (r )
−g
{Ψσ (r ), Ψσ ' (r ')}+ δσ ,σ 'δ (r − r ')
†
=
46
47. ∆( x) = ∆ Θ( x); U ( x) = U 0δ ( x); Z = mU 0 / 2 k F
A Z=0 A Z=1
B
B
E/∆ E/∆
A: Probability of Andreev reflection
B: Probability of ordinary reflection
Blonder, Tinkham, and Klapwijk: Phys. Rev. B 25, 4515 (1982)
47
55. Summary
• We have developed theories of electron transport in:
semiconducting QD’s, metallic, superconducting islands,
and carbon nanotubes taking into account charging as well
as the effects of the electromagnetic environment.
• Applications include most accurate SET devices and their
use in metrology, computing and sensing.
55
