Page 36 - Electrical Properties of Materials

Johnson noise 19

This is for one boundary. If there are two boundaries, each one of them will
have a surface wave. For a thick metal slab, the two surface waves do not know
about each other. However, for a thin slab the two surface waves interact. ∗ ‘Thin’ really means thin. At a
∗
Curve (c) in Fig. 1.8 splits into two branches, as may be seen in Fig. 1.11. wavelength of 360 nm the slab must be
thinner than 50 nm.
It may be shown that for the upper mode, less power is carried in the metal
than in the air. Since electron motion in the metal is responsible for losses,
the mode which propagates more in the air is less lossy. This mode is called
a long-range surface plasmon. How long is ‘long-range’? Not very long. At
optical frequencies, ‘long-range’ may mean 20 mm at best. On the other hand, v
with modern techniques, one can have a large number of various devices within
Light line
20 mm.
v p
1.8 Johnson noise
(b)
The electrons have so far behaved very reasonably. When no force acted upon v p
√ 2
them they stayed put. Under the effect of a force they moved. They obeyed
Newton’s law without qualm or hesitation. Unfortunately, this is not the whole (c)
truth. I may quote a very sophisticated and most profound theorem of phys-
ics the so-called fluctuation–dissipation theorem, which tells us that whenever (a)
there is dissipation there will be fluctuation as well. As we have already dis-
0 k
cussed there is bound to be dissipation. It occurs when an electron bumps into
a lattice atom. So there must also be fluctuation. There is no such thing as an Fig. 1.11
electron waiting patiently for some force to turn up and then it moves; they are Dispersion curves: (a) for a single
moving all the time. It makes good sense that when the temperature is finite metal–air boundary [same as curve
the electrons will jiggle. They will move a little bit to the left, a little bit to the (c) in Fig. 1.8]; (b), (c) for a thin
right, in fact a little bit in any conceivable direction. metal slab in air.
Is that good or bad? Definitely bad. For our communications we rely on
electronic signals. If the jiggling of the electrons interferes with that it’s bad.
The situation is the same as in direct verbal communications. If there is nearby
some source of unrequited sound waves (say a workshop where they repair
motorcycles) then the clarity of those verbal communications may suffer con-
siderably. When we rely on sound waves then we refer to the undesirable sound
waves as noise. As a generalization of this concept we may always refer to an
undesirable interfering agent as noise whatever the useful signal is. For ex-
ample if we gaze at the stars, then the random motion of the constituents of air
can be regarded as the source of noise that will make our star twinkle. When
it comes to electronic signals, then the noise is due to the random motion of
electrons. Resistors contain electrons hence they are a source of noise.
The first to investigate this effect was John B. Johnson, at Bell Labs in 1928.
Johnson carried out a careful set of experiments using resistors of different
values held at different temperatures, and a measurement system containing
a band-pass filter. His results showed that any resistor generates a random
series voltage whose power spectral density is flat, at least at low frequencies,
and proportional to both the resistance and the absolute temperature. Almost
immediately, his colleague Harry Nyquist provided a theoretical explanation.
We will avoid a detailed derivation and simply state the standard expression
for what is now known as Johnson noise. The root mean square (r.m.s.) voltage
generated by a resistance R at an absolute temperature T is
2
<V n > =4k B TRdf . (1.75)