shot noise in mesoscopic conductors - Low Temperature Laboratory

Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 37
energies E . In many situations, however, the Lorentz tails of ¹(E) far from the resonances are
not important, and one can write
/4 ¹(E)" ¹
(E!E )#/4
.
Despite the fact that the transparencies of both barriers are low, we see that the total transmission
coe$cient shows sharp peaks around resonant energies. This e!ect is a consequence of constructive
interference and is known as resonant tunneling. The transmission coe$cient at the top of a peak
equals ¹;
for a symmetric resonance " the transmission is ideal, ¹"1.
This
dependence may be probed by applying a gate voltage. The gate voltage moves the positions of
the resonant levels, and the conductance exhibits peaks around each resonance.
In the linear regime the shot noise is determined by the transmission coe$cient evaluated at the
Fermi level, and is thus an oscillating function of the gate voltage, vanishing almost completely
between the peaks. The Fano factor (60) at the top of each peak is equal to F"( ! )/.
It vanishes for a symmetric barrier. For a resonance with ¹'1/2
the Fano factor reaches
a maximum each time when the transmission probability passes through ¹"1/2; for a resonance
with ¹(1/2
the shot noise is maximal at resonance.
One-dimensional problem, non-linear regime. For arbitrary voltage, direct evaluation of the expressions
(39) and (61) gives an average current,
I" e
, (76)
and a zero-temperature shot noise,
S" 2e ( # )
. (77)
Here N is the number of resonant levels in the energy strip e< between the chemical potentials of
the left and right reservoirs. Eqs. (76) and (77) are only valid when this number is well de"ned } the
energy di!erence between any resonant level and the chemical potential of any reservoir must be
much greater than . Under this condition both the current and the shot noise are independent of
the applied voltage. The dependence of both the current and the shot noise on the bias voltage < is
thus a set of plateaus, the height of each plateau being proportional to the number of resonant
levels through which transmission is possible. Outside this regime, when one of the resonant levels
is close to the chemical potential of left and/or right reservoir, a smooth transition with a width of
order from one plateau to the next occurs.
Consider now for a moment, a structure with a single resonance. If the applied voltage is large
enough, such that the resonance is between the Fermi level of the source contact and that of the
sink contact, the Fano factor is
F"(# )/ . (78)
We assume that the resonances are well separated, ;/2mw.
For discussion of experimental realizations, see below.