Advanced Classical Theory of Conduction
THE NEED FOR A BETTER CLASSICAL THEORY OF CONDUCTION
In section 2.6 we proved that under the action of an electric field £, an electron accelerates as if it were a classical particle, i.e. its acceleration given by e£ divided by the effective mass m in 1 dimension and in 3 dimensions by e£ with each component divided by the corresponding diagonal element of the effective mass tensor m*j (see equation 2.49b). In chapter 3, on the other hand, we have seen that the motion of an electron under the action of £ is not that of perpetual acceleration and because of the presence of collisions a steady velocity appears for the ensemble of electrons that we call drift velocity. We gave an elementary proof of that in section 3.2. The proof ignored the existence of к space and Bloch functions since the treatment there was intentionally elementary. A qualitative picture of the application of £ and collisions in terms of к space was described in figures 3.2a and 3.2b, which portray a situation where more electrons travel against £ with higher velocities than in the opposite direction. However that was not proven. It is time now to acquire a deeper understanding by studying the Boltzmann equation which allows electric fields, diffusion, and collisions to be examined simultaneously in a rigorous manner within the framework of к space.
We saw in section 2.5 that when an electric field £ is applied to a crystalline solid the wavefunctions are no longer Bloch waves, which fill homogeneously the whole solid, but wavepackets, i.e. localized in space states, see equation 2.37. Hence the notion of acceleration and collision for particles has meaning because it is impossible to reconcile the notion of collision at some point in space with a wavefunction for electrons that extends from one end of the crystal to the other. We now rewrite equation 2.37 more analytically in terms of the components of a Bloch wave.
where d}k = dkxdkydkz and A(/c) is sharply peaked around a given k = k' with kk' being the momentum of that wavepacket if the bands are parabolic. The function u(k,r) is the periodic part of the Bloch function. As we have seen in section 2.5, such a wavepacket is also sharply localized in space around a maximum point r= rmax. Furthermore, it is easy to prove (see problem 4.1) that the group velocity of this particular wavepacketis in an arbitrary band is
where by the symbol V* we mean the gradient in к space. From the properties of wavepackets presented above, we conclude that for the rest of this chapter we can forget the wave properties of electrons and assume that they can be treated as classical particles with their position given by the point of maximum in 4.1 (or the point of average value) and their velocity by 4.2 above. The validity of this approximation is examined below.