We have already said that the electrons in a solid behave almost as if they are free although they have a different mass ( m* ). This implies the electrons accelerate without impedence, this does not happen. This lecture will cover the frictional force that electrons feel.
13.2 Momentum Relaxation
In reality the perfect lattice assumed in Bloch's Theorem is tainted by:
thermal vibration (aka Phonons) which slightly displace the atoms from their lattice sites, the displacement from these mean position increases as the temperature increases
impurities (eg. donor/accepter atoms) are charged positive and negative and so the electrons are scattered
Figure 13.1 - Electron Scattering
the scattering is random and reduces the average electron momentum to zero in the momentum scattering time ( t ).
13.3 Carrier Mobility
The mean electron velocity in an electric field is
VD=
eE
m*
t
where VD is the drift velocity, eE/m* is the electron acceleration and t is the time between collisions. Hence
VDºµE
where µ is known as the mobility factor and is µ=et/m* .
We can obtain a current density
J=neVD=
æ ç ç è
ne2t
m*
ö ÷ ÷ ø
EºsE
where s is the conductivity and is the reciprical of the resistivity ( sº1/r ).
This is the bulk version of Ohm's Law. A good conductor has a low m* , high mobile carrier concentrations and a long t .
metals:
n is almost constant with temperature and so conductivity is determined by the temperature variation of µ . As the temperature increases µ falls due to a increased level of phonon scattering, therefore the conductivity falls
semiconductors/insulators:
ni is proportional to an exponential function
niµ e
-
Eg
kT
so the temperature dependence of the free carrier concentration easily outweighs the decrease in µ and so the conductivity increases as the temperature also increases
13.4 Diffusion Currents
Like anything else mobile carriers diffuse by a random walk process. The carriers diffuse along a decreasing concentration gradient to give a diffusion current
J=-qD.(Ñ n)
where J is the direction of the decreasing concentration gradient, q is the charge on the carrier +|e| for holes and -|e| for electrons. D is the diffusion coefficient and Ñ n is the concentration gradient.
