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Lasers 115

Si Si Si
Si Si Si Si Si Si

+‒ +‒ +‒
(a) (b) (c)

Figure 3.19 Electron and hole current in Si.

+ + + +
.... .... .... ....

Atom 1 Atom 2 Atom 3 Atom n

Figure 3.20 Electron in a periodic Coloumb potential.

be attracted toward the nucleus of atom 2. When it is in the vicinity of atom 2, there is a chance that it will
be attracted toward atom 3 or atom 1. An alectron can hop on and off from atom to atom as if it were a free
classical particle, but with the following difference. A free non-relativistic particle can acquire any amount of
energy and the energy states are continuous. However, for an electron in periodic potential, there is a range of
energy states that are forbidden and this is called the energy band gap. The existence of a band gap can only
be explained by quantum mechanics. At very low temperature, electrons have energy states corresponding
to valence bands. As the temperature increases, electrons occupy energy states corresponding to conduction
bands, but are not allowed to occupy any energy states that are within the band gap, as shown in Fig. 3.21.
For a free electron, the energy increases quadratically with k as given by Eq. (3.66),
2 2
ℏ k
E = , (3.92)
2m
0
where m is the rest mass of an electron. Differentiating Eq. (3.92) twice, we find
0
ℏ 2
m = . (3.93)
0 2 2
d E∕dk
The dotted line in Fig. 3.21 shows the plot of energy as a function of the wavenumber, k, for the free electron.
For an electron in a pure semiconductor crystal, the plot of energy vs. wavenumber is shown as a solid line.
We define the effective mass of an electron in the periodic potential as
ℏ 2
m (k)= (3.94)
eff 2 2
d E∕dk
in the allowed range of energy states, in analogy with the case of free electrons. The effective mass can be
larger or smaller than the rest mass, depending on the nature of the periodic potential. For example, for GaAs,