shot noise in mesoscopic conductors - Low Temperature Laboratory

Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 127
short-ranged impurities. They found that the Fano factor in d"3 crosses over monotonically from
F"0.38 ("! )toF"0 ("1). There is no shot noise in this model for '1.
This example clearly demonstrates that Fermi statistics are not necessary to suppress shot noise,
in accordance with general expectations.
6.5.2. Ballistic non-degenerate conductors
Bulashenko et al. [264] address the noise in charge-limited ballistic conductors. They consider
a two-terminal semiconductor sample with heavily doped contacts. Carriers in the semiconductor
exist only due to injection from the contacts which thus determine the potential distribution inside
the sample. The self-consistent "eld determines a barrier at which carriers are either completely
re#ected or completely transmitted (no tunneling). This system is thus a close analog of the
charge-limited shot noise in vacuum tubes.
It is easy to adapt the Boltzmann}Langevin formulation to this problem: Since carrier motion
inside the conductor is determined by the Vlasov equation (collisionless Boltzmann equation),
(R #v R #eE R ) f (x, p, t)"0 , (238)
the distribution function f (x, p, t) at any point inside the sample is determined by the distribution
function at the surface of the sample. The only source of noise arises from the random injection of
carriers at the contacts. Thus the boundary conditions are
f (0, p, t) "f #f ( p, t) ,
(239)
f (¸, p, t) "f #f ( p, t) .
The stochastic forces f are zero on average, and their correlation is
f ( p, t)f ( p, t)J (p !p )(t!t) f (p ) , (240)
where f (p ) is the Maxwell distribution function restricted to p '0("¸) orp (0("R).
Note that in the degenerate case we would have to write an extra factor 1!f on the right-hand
side of Eq. (240), thus ensuring that there is no noise at zero temperature. We have checked already
in Section 2 that this statement is correct. On the contrary, in the non-degenerate case f ;1, and
thus (1!f )&1. The results for this regime do not of course allow an extrapolation to k ¹"0.
The crossover between the shot noise behavior in degenerate and non-degenerate conductors was
investigated by Gonzàlez et al. [265] using Monte Carlo simulations.
Eq. (238) is coupled to the Poisson equation,
!dE /dx"4 f (x, p, t) . (241)
p
Solving the resulting equations, Bulashenko et al. [264] found that interactions, at least in a certain
parameter range, suppress shot noise below the Poisson value. The suppression may be arbitrarily
strong in long (but still ballistic) samples. The results are in good agreement with previous
numerical studies of shot noise in the same system by GonzaH lez et al. [266,268] and Bulashenko
et al. [267].
The crossover between the ballistic and di!usive behavior of the non-degenerate Fermi gas was
numerically studied by GonzaH lez et al. [268,259]. Analytical results on this crossover are presently
unavailable.