Direct and Indirect Semiconductors

• In the typical quantitative calculation of band structures, wave function of the single electron traveling through a perfectly periodic lattice is supposed to be in form of a plane wave moving in the x-direction having the propagation constant k, also called as wave vector.
• In the quantum mechanics, the electron momentum can be given as 
• The space dependent wave function for electron is 

                          (2.1)

here the function    modulates the wave function in accordance to the periodicity of the lattice.

• Allowed values of energy, while plotted as the function of k, gives E-k diagram.
• As the periodicity of most lattices is different in the various directions, the E-k diagram is complex surface, which is to be visualized in the three dimensions.

Figure- Direct and indirect transition of the electrons from conduction band to valence band:

(a) Direct - with accompanying photon emission, (b) indirect by means of defect level.

• In the direct band gap semiconductor the minima of the conduction band and the maxima of valence band take place at the same value of k        an electron making smallest energy transition from conduction band to the valence band can do so without change in k (and, momentum).
• In the indirect band gap semiconductor the minima of conduction band and maxima of the valence band take place for different values of k, therefore, the smallest energy transition for an electron requires a change in the momentum.
• Electron  falling  from  the conduction  band  to  an  empty  state  in the valence band   recombination.

• Recombination probability for the direct band gap semiconductors is quite higher than that for the indirect band gap semiconductors.
• Direct band gap semiconductors give up energy released during this transition (=E_g) in the form of light used for the optoelectronic applications (such as LEDs and LASERs).
• Recombination in the indirect band gap semiconductors occurs through some defect states within band gap, and energy is released in the form of heat given to the lattice.

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