SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

10.5. MESOSCOPIC STRUCTURES 501

In small structures where phase information is retained we will see that conductance has quan-
tized behavior. The conductance is

G=

δI

δV

(10.5.1)

Also
δI=δn.e.vk (10.5.2)

where the carrier velocity is

vk=

1



δE

δk

(10.5.3)

IfδVis the potential change we have

δn=

dn

dE

(eδV) (10.5.4)

Using these equations we get for a small change in current

δI=

e^2



δn

δ

δV (10.5.5)

The conductance is then

G=

e^2



δn

δk

(10.5.6)

Now for a 1-dimensional case the number of electron states available perk-state is

dn

dk

=

1

π

(10.5.7)

so that (including the spin degeneracy factor of 2)

G=

2 e^2

h

(10.5.8)

The expression shows that the fundamental unit of conductance is 2 e^2 /h. By using Landauer
formalism where electrons are treated as incident, transmitted and reflected waves it can be
shown that there is a remarkable universality in the magnitude of the fluctuations independent
of the sample size, dimensionality and extent of disorder, provided the disorder is weak and the
temperature is low (a few Kelvin). Such universal conductance fluctuations have been measured
in a vast range of experiments involving magnetic field and Fermi level position (voltage).
In figure 10.10 we show experimental results of Wees, et al., carried out on a GaAs/AlGaAs
MODFET at low temperatures. As shown, a pair of contacts are used to create a short channel of
the high mobility region, and conductance is measured. The gates form a 1-dimensional channel
in which the Fermi level and thus the electron wavefunctions can be altered. As can be seen from
the figure 10.10, there are quantized steps in the conductance.
Transistors based on the mesoscopic effects described here are called single electron transis-
tors. They promise low power operation although they require low temperatures. In figure 10.11
we show an SEM image of a single electron transistor where the conductance fluctuations dis-
cussed here have been observed.