shot noise in mesoscopic conductors - Low Temperature Laboratory

38 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
It varies between 1/2 (symmetric barrier) and 1 (very asymmetric barrier). Expression (78) was
obtained by Chen and Ting [48] using a non-equilibrium Green's functions technique, and
independently in Ref. [19] using the scattering approach. It was con"rmed in Monte Carlo
simulations performed by Reklaitis and Reggiani [49,50]. If the width of the resonance is
comparable to the applied voltage, Eq. (78) has to be supplemented by correction terms due to the
Lorentz tails of the Breit}Wigner formula, as found in Ref. [19] and later by Averin [51].
It is also worthwhile to point out that our quantum-mechanical derivation assumes that the
electron preserves full quantum coherence during the tunneling process (coherent tunneling model).
Another limiting case occurs when the electron completely loses phase coherence once it is inside
the well (sequential tunneling model). This latter situation can be described both classically (usually,
by means of a master equation) and quantum-mechanically (e.g., by connecting to the well one or
several "ctitious voltage probes which serve as `dephasinga leads). These issues are addressed in
Section 5, where we show that the result for the Fano factor, Eq. (78), remains independent of
whether we deal with a coherent process or a fully incoherent process. The Fano factor Eq. (78) is
thus insensitive to dephasing.
Quantum wells. The double-barrier problem is also relevant for quantum wells, which are two- or
three-dimensional structures, consisting of two planar (linear in two dimensions) potential
barriers. Of interest is transport in the direction perpendicular to the barriers (across the quantum
well, axis z). These systems have drawn attention already in the 1970s, when resonant tunneling was
investigated both theoretically [52] and experimentally [53].
If the area of the barriers (in the plane xy) A is very large, the summation over the transverse
channels in Eqs. (39), (61) can be replaced by integration, and we obtain for the average current
I" e A
2
dE dE ¹(E ) f (E #E )!f (E #E )
and the shot noise
S" e A
dE dE ¹(E )[1!¹(E )] f (E #E )!f (E #E ) ,
with "m/2 the density of states of the two-dimensional electron gas (per spin). The key point
is that the transmission coe$cient depends only on the energy of the longitudinal motion E ,
and thus is given by the solution of the one-dimensional double-barrier problem, discussed above.
Denoting "E #e