Page 38 - Electrical Properties of Materials

Heat 21

Although this term has reduced as well, we must add to it the noise due
to the additional resistor. If this is done, we obtain a total noise power of
2
2
P n =(V 2 + V )/{4R L (1 + R/2R L ) . Assuming that everything is at the same
nS n
2
temperature, the average value of V is 4k B TRdf . As a result, the SNR at the
n
2
load is now P S /P n = V /{4k B TR L df (1 + R/R L )}. Clearly, the SNR has got
S
worse, by a factor 1 + R/R L , known as the noise factor F. All intermediate cir-
cuits, whether passive or active, have a noise factor. Often this is expressed in
decibels, as the noise figure NF =10 log (F). The better the circuit, the closer
10
F will be to unity, and the closer NF will be to zero. In this case, R should
clearly be made as small as possible. The effect of any additive noise should
be minimized, by placing amplifiers at the front end of the system.
Is that the end of the story for noise? Unfortunately no, because there are
other physical mechanisms that create it. We will discuss these later.
1.9 Heat
When the aim is to unravel the electrical properties of materials, should we
take a detour and discuss heat? In general, no, we should not do that but when
the two subjects overlap a little digression is permissible. I want to talk here
first about the relationship between the electrical conductivity and heat con-
ductivity, and then point out some discrepancies suggesting that something is
seriously wrong with our model.
We have already discussed electrical conductivity. Heat conductivity is the
same kind of thing but involves heat. An easy but rather unpleasant way of
learning about it is to touch a piece of metal in freezing weather. The heat
from your finger is immediately conducted away and you may get frostbite.
Now back to that relationship. Denoting heat conductivity by κ, it was claimed
around the middle of the nineteenth century that for metals

κ
= C WF T, (1.76)
σ
where C WF , the so-called Wiedemann–Franz constant, was empirically de-
rived. It was taken as

–8
–1
–2
C WF =2.31 × 10 WS K . (1.77)
How well is the Wiedemann–Franz law satisfied? Very well, as Table 1.2
shows. Can it be derived from our model in which our electrons bounce about
in the solid? Yes, that is what Drude did in about 1900. Let us follow what he
did.
At equilibrium, the average energy of an electron (eqn 1.1) is E = 3/2 k B T.
The specific heat C V is defined as the change in the average energy per unit
volume with temperature

3
dE
C V = N e = N e k B . (1.78)
dT 2
Let us now consider heat flow, assuming that all the heat is carried by the
electrons. We shall take a one-dimensional model in which the electrons move