shot noise in mesoscopic conductors - Low Temperature Laboratory

34 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
Fig. 6. Geometry of the quantum point contact in the hard-wall model (a) and the e!ective potential for one-dimensional
motion (b).
Fig. 7. Conductance (upper plot) and shot noise (lower plot) as functions of the gate voltage, as measured by Reznikov
et al. [42]. Di!erent curves correspond to "ve di!erent bias voltages. Copyright 1995 by the American Physical Society.
(Fig. 6a). If the distance between the walls d(x) (width of the contact) is changing slowly in
comparison with the wavelength, transverse and longitudinal motion can be approximately
separated. The problem is then e!ectively reduced to one-dimensional motion in the adiabatic
potential ;(x)"n/2md(x), which depends on the width pro"le and the transverse channel
number n. Changing the gate voltage leads to the modi"cation of the potential pro"le. Theoretically
it is easier to "x the geometry of the sample, i.e. the form of the potential, and vary the Fermi
energy in the channel E (Fig. 6b). The external potential is smooth, and therefore may be treated
semi-classically. This means that the channels with n(k d / (here k ,(2mE ), and d is
the minimal width of the contact) are open and transparent, ¹ "1, while the others are closed,
¹ "0. The conductance (40) is proportional to the number of open channels and therefore
exhibits plateaus as a function of the gate voltage. At the plateaus, shot noise is equal to zero, since
all the channels are either open or closed. The semi-classical description fails when the Fermi
energy lies close to the top of the potential in one of the transverse channels. Then the transmission
coe$cient for this channel increases from zero to one due to quantum tunneling through the
barrier and quantum re#ection at the barrier. The transition from one plateau to the next is
associated with a spike in the shot noise as we will now discuss.
A more realistic description of the quantum point contact takes into account that the potential in
the transverse direction y is smooth [34]. The constriction can then be thought of as a bottleneck
with an electrostatic potential of the form of a saddle. Quite generally, the potential can be
expanded in the directions away from the center of the constriction,