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VELOCITY OF ELECTRONS IN SOLIDS

In the previous section we made a distinction between, on the one hand, electrons and holes in donor and acceptor states respectively and, on the other hand, electrons and holes in the conduction and valence bands respectively. The latter are in Bloch states and can conduct. From the discussion so far, it should be clear that Bloch states are modulated plane waves running from one end of the crystal to the other. The question then arises how can these states represent particles that are accelerated, then scattered and accelerated again (according to a simple picture of conduction), i.e. how can the electrons have a localized nature but extend all over the crystal. We will delve into this question in chapter 4, but for the moment it suffices to say the following. When an electric field is applied to a solid, the wavefunctions of the electrons are no longer Bloch states, Ч'щд, but are wavepackets made out of Bloch states which are localized in space. Each wavepacket is sharply peaked around a particular Bloch state with wavevector k. Mathematically, these wavepackets can be written 'F(r) as

where the A{k) are peaked around a given к as shown in figure 2.15a. BZ stands for the Brillouin zone. The space variation of 'F(r) is shown in figure 2.15b. Both figures give a 1-dimensional representation of these variables at an instant of time.

Now the motion of a wavepacket is determined by its group velocity which from the theory of simple optics is

But in quantum mechanics, we have the relationship

Weighting factors of the Bloch waves in a wavepacket 4(x,t = 0) (a) and |'В(л:,0)|~ itself (b)

FIGURE 2.15 Weighting factors of the Bloch waves in a wavepacket 4/(x,t = 0) (a) and |'В(л:,0)|~ itself (b).

Hence the motion of a wavepacket, each composed of wavefunctions Tm )( with an energy Em(k), where m is the band index, is given in one dimension by the group velocity

and in 3 dimensions is given by

where V* is the grad in к space.

We can immediately verify an important conclusion we drew previously. A completely filled band cannot exhibit a current. The current density / is given by

where the summation over к is over the allowed к values in the Brillouin zone. However an examination of figure 2.8 shows that the BZ of the FCC lattice is symmetric around zero, i.e. for every allowed к there is always an allowed -k, with V-kE(-k) = -VkE{k). This is a property of all BZs. We have already encountered this property in the E(k) expression for the simple linear chain of the hydrogenic atoms, equation 2.31. Therefore, the sum in equation 2.40 above is zero. It is worth noting that while donor atoms fill an empty conduction band with electrons, thereby increasing its conductivity, the acceptors do exactly the same thing, i.e. increase the conductivity by partially emptying the valence band, however odd it may seem at first glance.

Another point we would like to discuss is the notion of “crystal momentum”. The quantity fik plays the role of momentum as far as the external forces are concerned, i.e. we can write

There is a formal proof of equation 2.41, which we wil not give, but the reader may be satisfied by noting that, on intuitive grounds, Fdt is the impulse in time dt and hence this quantity must give the corresponding change in momentum d(tlk). We note however that in 2.41 the electrons must be considered as wavepackets as already described and the к refers to the wavevector of the dominant Bloch component. It must be stressed though, that the Bloch functions are not eigenfunctions of the momentum operator and hence hk would not be a result of a momentum measurement in a semiconductor. Equation 2.41 merely relates the external force applied to a semiconductor to the rate of change of the к vector of an electron.

 
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