shot noise in mesoscopic conductors - Low Temperature Laboratory

22 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
and s _ (E) is the element of the scattering matrix relating bK (E) toa( (E). Note that Eq. (34) is
independent of the coordinate z along the lead.
2.3.2. Average current
Before we proceed in the next subsection with the calculation of current}current correlations, it
is instructive to derive the average current from Eq. (34). For a system at thermal equilibrium
the quantum statistical average of the product of an electron creation operator and annihilation
operator of a Fermi gas is
a( (E)a( (E)" (E!E) f (E) . (36)
Using Eqs. (34) and (36) and taking into account the unitarity of the scattering matrix s, we obtain
I "
e
2 dE Tr[t(E)t(E)][ f (E)!f (E)] . (37)
Here the matrix t is the o!-diagonal block of the scattering matrix (30), t "s . In the
_
zero-temperature limit and for a small applied voltage, Eq. (37) gives a conductance
G" e
2 Tr[t(E )t(E )] . (38)
Eq. (38) establishes the relation between the scattering matrix evaluated at the Fermi energy and
the conductance. It is a basis invariant expression. The matrix tt can be diagonalized; it has a real
set of eigenvalues (transmission probabilities) ¹ (E) (not to be confused with temperature), each
of them assumes a value between zero and one. In the basis of eigen-channels we have instead of
Eq. (37)
I "
e
2
dE ¹ (E)[ f (E)!f (E)] . (39)
and thus for the conductance
G" e
2 ¹ . (40)
Eq. (40) is known as a multi-channel generalization of the Landauer formula. Still another version
of this result expresses the conductance in terms of the transmission probabilities ¹ "s
for carriers incident in channel n in the left lead L and transmitted into channel m in the right lead
R. In this basis the Hamiltonians of the left and right lead (the reservoirs) are diagonal and the
conductance is given by
G" e
2 ¹ . (41)
We refer to this basis as the natural basis. We remark already here that, independently of the choice
of basis, the conductance can be expressed in terms of transmission probabilities only. This is not
case for the shot noise to be discussed subsequently. Thus the scattering matrix rather then
transmission probabilities represents the fundamental object governing the kinetics of carriers.