Page 33 - Photonics Essentials an introduction with experiments

Electrons and Photons

Electrons and Photons 27

Figure 2.10. In this sequence of calculations, we show how the periodic potential modifies
the energy–momentum relationship for a real three-dimensional semiconductor, GaAs. In
the first frame, you can clearly see the parabolic relationship between energy on the verti-
cal axis and momentum on the horizontal axis. In succeeding frames, we add the periodic
potential due to the actual atoms. This causes the crossings to separate. By the time we
arrive at GaAs, there is a band gap between the valence band and the conduction band.
Because the minimum of the conduction band and the maximum of the valence band oc-
cur at the same value of momentum, this is a direct energy gap. GaAs, InP, GaInAsP, and
GaN are examples of direct-gap semiconductors.

This splitting is shown symbolically in Fig. 2.9. In Eqs. 2.10 and 2.11,
we show how the splitting occurs in the real band structures of GaAs
and Ge. The crystalline potential is the direct expression of the atoms
that make up the material. So, the difference between direct band gap
and indirect band gap materials is a matter of chemistry.
The band gap expresses the difference in energy between an electron
in a bonding state and an electron in an antibonding state. In the anti-

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