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MECHANISMS OF SCATTERING

So far we have talked about collisions and scattering in more or less loose terms, without specifying the source of the scattering, or equivalently, with what exactly the electrons collide. From early undergraduate years, the student learns that electrons collide with impurities and the lattice vibrations. Here we want to allocate a few lines to clarify a few points. In chapter 2, we proved that the eigenstates of electrons in a periodic lattice are Bloch functions, i.e. modulated plane waves, each eigenstate corresponding to an energy E„(k), where к is a wavevector and n is a band index. The wavevector к acts like the macroscopic momentum p of an electron, i.e. the two are related by p = bk. Furthermore, £ is given by the time derivative of k. In the 1 dimension we are considering here

So when a field is applied, the electrons will move to higher and higher к vectors, that is to higher and higher momenta in accordance with 2.41. It seems that nothing will stop this process of acceleration. Obviously, the acceleration will be stopped, as assumed in the previous section, but what will stop the validity of 2.41 or equivalently create collisions? The answer is: anything that will lift or destroy the periodicity of the crystalline potential which gives rise to the Bloch functions and consequently the validity of 2.41. These are:

a. lattice vibrations

b. impurities

c. crystal defects (like vacant sites)

d. interfaces

Whereas b), c), d) are obvious deviations from periodicity, as far as a) we note that as the atoms of the crystal vibrate they move away from their ideal static positions and therefore vibrations indeed constitute deviations from periodicity. The frequency of the electron collisions with the vibrations will depend on the exact type of these vibrations. An introduction into the formal theory of lattice vibrations is given in Appendix В where the reader will find a complete classical theory. The interactions of the electrons with the perturbing potentials caused by a, b, c, d above constitute a major portion of solid state physics. Luckily we do not need this theory itself to formulate a theory of electronic conduction, we only need the results, i.e. the time rate of collisions which are given by the relaxation times x.

Each scattering mechanism has its own relaxation time, so how come we have used a simple relaxation time in our formulation of mobility? The following argument shows how this is possible. If x, is the relaxation time of the ith scattering mechanism, then 1 / x, is the probability of scattering per unit time of the ith mechanism. The total probability of scattering per unit time x by any mechanism is then the sum of the individual ones assuming that each mechanism is independent of the others. Then

However, each of the x, exhibits a different temperature dependence. When the temperature rises, it is reasonable to suggest that the amplitude of the lattice vibrations increase, irrespective of their type, so we expect the collision of electrons with these vibrations to become more frequent. A lengthy calculation shows that the relaxation time for scattering with the lattice vibrations (or phonons) is

Conversely, the collisions of electrons with charged impurities are expected to become less frequent as the temperature rises because the electrons have a higher energy at a higher temperature and hence are expected to overcome the attraction or repulsion of charged impurities. A simpler calculation this time shows

If we add the two mechanisms appropriately, the total relaxation time x then exhibits a maximum at a certain temperature as figure 3.4 shows.

 
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