Quantum Interference Effects on the Electronic Transmission Through Quantum Dot Molecules - Quantum Dot Molecules - page 262

8.2.6 Resonant Zeros of the Transmission Function

For the sake of simplicity, consider a nondegenerate energy eigenvalue E k of the
disconnected device with eigenfunction

| ψ k , which can be written as a linear

combination of the site orbitals | i , | ψ k = ∑ i c ki | i , with c ki = i | ψ k .Ifthestate

| ψ k has a non-vanishing weight at the connection sites l and r simultaneously, i.e.

c kl = 0 = c kr , then the spectral representation

c kl c kr
E

)= ∑ k l | ψ k ψ k | r = ∑ k

G lr ( E (8.17)

E − E k − E k

shows that E k is a pole of G lr because the term c kl c kr / (

−

E k ) E is present in the

expansion ( 8.17 ). Therefore, also the terms

c kl c kl
E

c kr c kr
E

and (8.18)

− −

E k E k

will be present in the spectral representation of G ll and G rr , respectively, and E k will
also be a pole of them. That is, the poles of G lr also become poles of G ll and G rr and
all three Green functions G ij diverge as G ij ( ,where R ij = c ki c kj E ) ≈ R ij / ( E − E k )

( i r ) is the residue of G ij at the simple pole E k . Therefore, the Green function

of the connected ring, ( 8.15 ), can be approximated as

, j = l ,

E k ) − 1 R lr ( E −

G lr ( E ) ≈

2

R lr )( E k ) − 2 E k ) − 1 1 − Γ ( R ll R rr − E − − i Γ ( R ll + R rr )( E −

R lr

=

) .

(8.19)

R lr )(

( −

E k ) − Γ

2

( − −

E k ) − 1

−

Γ ( R ll +

E R ll R rr E i R rr

R lr = |

2 2 2

Taking into account that R ll R rr − l | ψ k | | r | ψ k | −| l | ψ k ψ k | r | = 0, it

reduces to

R lr iR lr

E → E k
−→ G lr ≈

(8.20)

( E − E k ) − i Γ ( R ll + R rr ) Γ ( R ll + R rr )

which shows that G has a pole at E = E k + i Γ ( R ll + R rr ) and a finite transmission

4 R lr / (

2 . The pole E k acquires a finite width proportional to the

T lr = R ll + R rr )

coupling to the leads . In the particular case where the sites l and r are topologically

equivalent because of the symmetry of the system, R ll =

Γ

R lr so that perfect
transmission occurs. This is the case for the connection (1,3), because the sites l

R rr =

= 1

and r 3 are equivalent when their coupling strengths to their respective terminals

are the same. It should be noted that is not the case for the (1,2) connection.
On the other hand, if E

=

= E k is a pole of G ll or G rr , but not of G lr , the numerator

G lr ( of ( 8.15 ) is finite whilst its denominator diverges; therefore, T lr will show

an antiresonance at E

E k )

= E k . In other words, a finite transmission occurs when the