40 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS
general is
dp
dt
=Fext+FintHowever this is not very useful for a meaningful description of the electron because it includes
the internal forces on the electron. We need a description which doesnot include the evaluation
of the internal forces. Using a wavepacket description of electrons as with any wave phenomena
it is the wave group velocity that represents the propagation of wave energy. In the case of
a particle wave the group velocity represents the particle velocity. The group velocity of this
wavepacket is
vg=dω
dk(2.3.13)
whereωis the frequency associated with the electron of energyE; in quantum mechanics,
ω=E/:
vg =1
dE
dk
=1
∇kE(k)If we have an electric fieldEpresent, the work done on the electron during a time intervalδtis
δE=−eE·vgδt (2.3.14)We may also write, in general
δE =(
dE
dk)
δk= vg·δk (2.3.15)Comparing the two equations forδE,weget
δk=−eE
δtgiving us the relation
dk
dt=−eE (2.3.16)In general, we may write
dk
dt=Fext (2.3.17)The termkresponds to theexternalforcesasifitisthemomentumoftheelectron,although,
ascanbeseenbycomparingthetrueNewtonsequationofmotion,itisclearthatkcontains
theeffectsoftheinternalcrystalpotentialsandisthereforenotthetrueelectronmomentum.
The quantitykis called the crystal momentum. We can, for all practical purposes, treat the
electrons as if they are free and obey the effective Newtons equation of motion. This physical
picture is summarized in figure 2.7.