Page 62 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 49

Another peculiarity of mesoscopic TLs is the nature of their energy
spectrum when formed into a ring in the presence of a magnetic flux φ . In
this case, Schrödinger’s becomes,

− = 2 {D − D } ψ = εψ , (26)
2 q 2 L e q e q
e

where D and D are discrete derivatives that remain covariant in the
e q e q
presence of the magnetic flux φ and are defined by Li [61] as,

− iq e φ
φ
iq e
~
− q e φ Q − e = − q e φ e = − Q + ˆ
D ≡ e = ; D ≡ e = . (27)
q q
q e q e
e e
Applying the Hamiltonian in (26) to the eigenstate p > , the energy
eigenvalues are obtained as,
·
ε ( φp, ) = 2 = sin 2 § q e ( −φp )¸ , (28)
¨
q 2 © =2 ¹
e
where φ is the magnetic flux threading the TL. Thus, (28) implies that when
the discrete nature of charge is at play, the TL energy becomes a periodic
2=
function of p or φ , with maximum amplitude and nulls occurring
q 2
e
whenever p =φ + n= q . Furthermore, it has also been shown that the TL
e
current is given by,

φ
¨
¸ , (29)
I () = = sin § q e φ ·
q L © = ¹
e
which implies that it becomes an oscillatory function of the magnetic flux.
Since no applied forcing function was assumed, (29) leads to the important
observation [62] that a TL in the discrete charge regime will, in the presence
of a magnetic flux, exhibit persistent currents [59]. These are currents
without dissipation, such as the atomic orbital currents that elicit orbital
magnetism.