Module 6 : LIGHT EMITTING DIODE (LED)
Lecture : LED - I
 

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When electrons from the conduction band drops to a hole in the valence band, the transition is vertical in the $ E-k$diagram as there is no change in the momentum. The electron hole pair combines to give energy in the form of heat (called non-radiative transition ) or light ( radiative transition ) depending on the band-gap energy. The wavelength of the emitted photon is given by $ \lambda = hc/E$. If the energy is measured in eV, the wavelength is approximately given by the formula

$\displaystyle \lambda \ \ {\rm (in nm)} = \frac{1242}{E\ \ {\rm (in eV)}}$

Semiconductors like Ge and Si are indirect bandgap material in the sense that the bottom of the conduction band is shifted from $ k=0$to $ k=k_0$, so that the energy is given by

$\displaystyle E_n = \Delta + \frac{\hbar^2(k-k_0)^2}{2m_n^\ast}$

As both energy and momentum must be conserved in the process of transition, when an electron recombines with a hole in the valence band, the difference in the momentum must be transferred to a third entity, e.g., the lattice.

 

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  As $ \hbar k$is momentum, conservation of momentum implies conservation of wave vector $ k$. Photon carries very little momentum. For instance, the value of k for a photon with $ \lambda=600 $nm is $ 10^7$m$ ^{-1}$. The wavevector of electron in a lattice is in the range $ -\pi/a$to$ +\pi/a$. Taking the lattice spacing to be 5Å, the k value for the electron is
$ \simeq 10^{10}$m $ ^{-1}$. Thus while photon takes most of the energy in a radiative transition, it carries negligible momentum. Thus recombination in indirect gap semiconductors is essentially a three body process. A three body process is much less probable to occur than a two body process.
GaAs, direct bandgap material emits in infrared (870-900 nm). GaP has andirect band gap of 2.2 EV. Both As and P belong to Group III of the periodic table. It is possible to make alloys having composition GaAs$ _{1-x}$P$ _x$ in which phosphorus atoms ocuupy As sites substitutionally as the lattice constants of GaAs and GaP are nearly equal.
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