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THE BOLTZMANN EQUATION

When an equilibrium is established the probability of finding an electron with energy E depends only on E and is given by the Fermi-Dirac distribution. This may seem like a tautology but what it means is that if we have two states with two distinct wavevec- tors k *k2 with E(kt)= E(k2), the probability of occupation of the states is the same, that is, it does not depend on the wavevector or crystal momentum. This is not the case when non-equilibrium processes are present. In fact, the Boltzmann equation is the equation that governs the probability f(k,r,t) at non-equilibrium of finding an electron at time t with wavevector к inside an infinitesimal volume around r. This classical description is founded quantum mechanically on equations 4.1 and 4.2 of the previous section. For electrons to occupy such simple Bloch wavepackets, the system under investigation must be macroscopically large (of the order of fractions of microns) in each dimension. If one or more dimensions reach the nanometer scale, then additional complications arise which we will deal with in the next section. The requirement that the system be large is not the only one which prohibits the use of the Boltzmann equation. A more demanding requirement is that any applied fields or potentials must vary slowly—the wavelength of any applied electric field £{r) must be large compared to the width of the wavepackets representing the electrons. Figure 4.1 gives a pictorial representation of this statement. The necessity of this restriction comes from the fact that the notion of point-like particles acted on by the local field at point r is no longer valid. Then one has to abandon the Boltzmann equation altogether as we will do in the next chapter.

The probability or distribution function f(k,r,t) can change in 3 different ways: a) an external electric field can change the wavevector к of any electron cf. equation 2.41, b) an

Requirements for a particle picture to be valid

FIGURE 4.1 Requirements for a particle picture to be valid: the extent of a wavepacket representing a particle must be much smaller than the wavelength of the potential energy distribution.

initially non-uniform distribution may produce diffusion and change r in f(k,r,t), and c) collisions abruptly change the wavevectors of the electrons. Therefore, we can write

Before we continue, a small point on notation. As we will need both the gradient in real space and in к space, we will differentiate between them by denoting them as V,- and V* respectively. We now assume that the distribution of carriers in the neighbourhood of r at time t is equal to the distribution in the neighbourhood of r — vt at time 0. Then

Then the second term in equation 4.3 becomes In a similar manner to equation 4.4 we have

where к denotes the time derivative of k. We then get for the first term in 4.3

where, in equation 4.7 we have used equation 2.41 indicating the rate of change of the crystal momentum ftk is equal to the electric force exerted on the crystal. Note that the charge q may be either negative or positive so far. We have written F = qS above but in fact the magnetic as well as the electric field produce a rate of change of the crystal momentum. So we generalize equation 4.7 to

The rate of change of/due to collisions is more complicated than the first two terms in 4.3. A collision, say, transfers electrons from state к to k'. The state at k' must be empty for an electron to fall in and, likewise, the state at к must be occupied. Therefore, the rate of transitions depends on the occupation probabilities of the individual states. Furthermore, there is also a reverse transition from k' to к and all possible pairs (k,k') must be considered to account for the rate of change of/due to collisions. Using for the moment the abbreviation

we have

where in 4.9 Pk# is the probability of transition from k' to к if k' is known to be completely occupied and к completely empty. Pkk- and Pk-k are equal in equilibrium but they are also equal if the transitions from k' to к and vice versa are elastic. We limit ourselves for the moment to elastic collisions. Putting expression 4.5,4.8, and 4.9 in equation 4.3 will result in the time dependent Boltzmann equation, an integrodifferential equation which can be solved in a limited number of cases that are, however, of paramount importance. The

Boltzmann equation then reads after substituting for v(k) = —VkE(k)

h

Note that in equation 4.10 we have omitted the products fk *fk- in 4.9, which cancel out.

 
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