# SATURATION OF DRIFT VELOCITY

Extensive mobility experiments have shown that the velocity does not increase continuously with an increasing electric field *£* but actually saturates at a constant value *v ^{sal}* as shown in figure 4.7 (ID model). The low field part of the curve is discussed in section 4.4 where the whole electron distribution

*f(k*

_{x}) shifts rigidly along the

*k*axis (along which

_{x}*£*is applied) by an amount proportional to the electric field

*£.*But at high

*£*it seems as if the electric field is unable to change

*f(k).*We have to look and find out what particular assumptions that we have made in sections 4.3 and 4.4 are no longer valid and no longer give a drift velocity proportional to

*£.*

Before we do that, we present the physical processes that lead to the saturation of velocity. As the electron is accelerated and its energy increases, it interacts strongly with the lattice and gives up the energy acquired by the field to the lattice in the form of optical

phonons, a mode of lattice vibrations(see Appendix B). It should then be obvious that the collisions of the electrons with the lattice are no longer elastic and the assumption of the equality of the microscopic transition rates *P(k,k')* and *P(k',k)* (see equation 4.9) is no longer valid. In short, from a certain value of the electric field onwards, the extra kinetic energy of the electron is given up to the lattice and the electron velocity stays constant. The description of this phenomenon through the Boltzmann equation is beyond the scope of this book. It must be emphasized that when the collisions are inelastic even the relaxation time approximation is no longer valid. We note, however, that one can define a mean time between inelastic collisions, and we are certainly going to make use of this concept in the next chapter, but one cannot use it to approximate the collisions term in the Boltzmann equation. An empirical relation which reproduces the above discussed effect is

where (Д is the low-field mobility.