COLLISIONS AND DRIFT MOBILITY

In section 2.5, we made it clear that a full band cannot exhibit any conductivity because the sum of all the velocities is zero, cf. equation 2.40 and discussion therein. This statement simply states that in a full band there are always as many velocities in a direction as in the opposite. This statement holds true also in a non-full band under the absence of an external field. An external electric field £ can change this balance so that more electrons flow in the opposite direction to £ than in the same direction and the mechanisms to do this are the collisions. (Note that due to the negative charge of the electron, the force is opposite to £.) What happens is that the collisions in the direction of £ become more frequent. Note that the individual velocity of any electron is still given in the presence of a field by V* [£(&)], equal to for a parabolic band. The electric field does not change the individual velocities, only their distribution.

Hence, current is a property of an ensemble of electrons; £ changes the distribution of the velocities in к-space: more electrons are flowing with к vectors opposite to the direction of £ than electrons with к vectors in the same direction as £, see figures 3.2a and 3.2b. The average velocity of the excess electrons in the opposite direction to £ is called drift velocity. Although figures 3.2a and 3.2b refer to a 1-dimensional picture of a crystalline solid, they have all the physical ingredients that we will need in this chapter. In the next chapter where more advanced physical models will be used, a 3-dimensional picture will be employed. It is with this aim that we have used vector notation so far here.

The distribution of velocities shown in figure 3.2b is time independent but derives from a highly dynamic situation: electrons are constantly accelerated and scattered by collisions to lower energies. An electron with momentum of, for example, hk in 1 dimension is accelerated

Distribution of velocities in a conduction band (a) before and (b) after an electric field is applied to a solid

FIGURE 3.2 Distribution of velocities in a conduction band (a) before and (b) after an electric field is applied to a solid.

by the electric field £ to hk2 > bkx and then scattered by collision to a state with momentum bk} < hk2. As a result of all these processes, one gets the static picture of figure 3.2b with a net collective velocity. In section 2.6, we showed that under the assumptions of a parabolic band and a slowly varying field, an electron’s motion can be considered as classical with the effective mass substituted for the real mass. Hence we will consider a set of N classical particles, each with an effective mass m* and derive the current density / of this system in terms of the electric field £ and the properties of the system, such as the conductivity a and mobility u defined by / = and a = nex. We emphasize that what follows is the lowest order of an approximate theory of charge transport in semiconductors valid only under the conditions we have described above. More advanced theories will follow.

For such a system of classical charged particles, each with charge -e, the total momentum increase (dP)el in time dt due to the electric field is

At the same time, the system loses momentum due to collisions. The simplest approximation we can make is that each particle loses by collision all the kinetic energy it has gained so that, in our 1-dimensional model, the variation of velocity with time of a single particle will be the one shown in figure 3.3. In this figure the change in time of the velocity of two electrons is shown for clarity, but we can imagine many more electrons all colliding randomly with the lattice at a random frequency that is sharply peaked around a mean value which can be written as 1/x where x is the so-called relaxation time and can be thought of as roughly the mean time between collisions with the lattice. At the moment the reader should interpret the word “lattice” very loosely, since we have not specified with what exactly the electrons collide. We leave this critical issue for the next section.

Time evolution of the velocities of two particles under the action of an electric field and of collisions

FIGURE 3.3 Time evolution of the velocities of two particles under the action of an electric field and of collisions.

So if the system loses all the momentum it has gained in time x, in time dt it loses Therefore, in a time independent state we must have

But the drift velocity, as explained so far, is the mean of the extra momentum per particle divided by the (effective) mass. Therefore

where {p) is the average additional momentum per electron. By the word ‘additional, we mean the additional momentum gained by the system due to the electric field.

Using equation 3.6, the above equation can be written as

and hence the mobility p defined from the relation vdr = p£ is given by

The above classical model can be applied to both electrons in the conduction band and holes in the valence band of a semiconductor. Using the superscripts n and p for electrons and holes respectively we have

and for the corresponding mobilities

 
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