Why does silicon have an indirect gap?
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Suppose they are different, it is called an indirect gap. The band gap is called 'direct' if the crystal momentum of electrons and hole is the same in both the conduction band and the valence band; an electron can directly emit a photon.
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As a result of such considerations, gallium arsenide and other direct band gap semiconductors are used to make optical devices such as LEDs and semiconductor lasers, whereas silicon, which is an indirect band gap semiconductor, is not.
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