shot noise in mesoscopic conductors - Low Temperature Laboratory

24 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
The current is taken to be positive if it #ows from the reservoir towards the mesoscopic structure.
For the average current in the two-terminal geometry, we have I #I "0. We emphasize that
current conservation must hold not only on the average but at each instant of time. In particular,
current conservation must also hold for the #uctuation spectra which we discuss subsequently. In
general, for time-dependent currents, we have to consider not only contacts which permit carrier
exchange with the conductor, but also other nearby metallic structures, for instance gates, against
which the conductor can be polarized. The requirement that the results are gauge invariant means
in this context, that no current arises if voltages at all reservoirs are simultaneously shifted by the
same value (and no temperature gradient is applied). For the average currents (see Eqs. (39), (47))
both properties are a direct consequence of the unitarity of the scattering matrix.
For the conductance matrix G current conservation and gauge invariance require that the
elements of this matrix in each row and in each column add up to zero,
G " G "0 . (48)
Note that for the two terminal case this implies G,G "G "!G "!G . In the two
terminal case, it is thus su$cient to evaluate one conductance to determine the conductance
matrix. In multi-probe samples the number of elements one has to determine to "nd the conductance
matrix is given by the constraints (48) and by the fact that the conductance matrix is
a susceptibility and obeys the Onsager}Casimir symmetries
G (B)"G (!B) .
In the scattering approach the Onsager}Casimir symmetries are again a direct consequence of the
reciprocity symmetry of the scattering matrix under "eld reversal.
In the stationary case, the current conservation and the gauge invariance of the results are
a direct consequence of the unitarity of the scattering matrix. In general, for non-linear and
non-stationary problems, current conservation and gauge invariance are not automatically
ful"lled. Indeed, in ac-transport a direct calculation of average particle currents does not yield
a current conserving theory. Only the introduction of displacement currents, determined by the
long-range Coulomb interaction, leads to a theory which satis"es these basic requirements. We will
discuss these issues for noise problems in Section 3.
2.4. General expressions for noise
We are concerned with #uctuations of the current away from their average value. We thus
introduce the operators IK (t),IK (t)!I .Wede"ne the correlation function S (t!t) of the
current in contact and the current in contact as
S (t!t),IK (t)IK (t)#IK (t)IK (t) . (49)
Note that several de"nitions, di!ering by numerical factors, can be found in the literature. The one we use
corresponds to the general de"nition of time-dependent #uctuations found in Ref. [11]. We de"ne the Fourier transform
with the coe$cient 2 in front of it, then our normalization yields the equilibrium (Nyquist}Johnson) noise S"4k ¹G
and is in accordance with Ref. [1], see below. The standard de"nition of Fourier transform would yield the
Nyquist}Johnson noise S"2k ¹G. Ref. [9] de"nes the spectral function which is multiplied by the width of the
frequency interval where noise is measured.