Branch : Computer Science and Engineering
Subject : Fundamental of Electronic Devices
The Fermi Dirac distribution
The fermi dirac distribution:
- In applying the Fermi-Dirac distribution to semiconductors, we recall that f(E) is the probability of occupancy of an available state at E.
- Thus if there is no available state at E (e.g., in the band gap of a semiconductor), there is no possibility of finding an electron there.
- We can best visualize the relation between f(E) and the band structure by turning the f(E) vs. E diagram on its side so that the E scale corresponds to the energies of the band diagram (Fig. given below).
- For intrinsic material we know that the concentration of holes in the valence band is equal to the concentration of electrons in the conduction band.
- Therefore, the Fermi level EF must lie at the middle of the band gap in intrinsic material.
- Since f(E) is symmetrical about EF, the electron probability "tail" of f(E) extending into the conduction band of Fig.(a) is symmetrical with the hole probability tail [1 - f(E)] ' in the valence band.
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The distribution function has values within the band gap between Ev and Ec, but there are no energy states available, and no electron occupancy
results from f(E) in this range. - The tails in f(E) are exaggerated in Fig. for illustrative purposes. Actually, the probability values at Ev and Ec are quite small for intrinsic material at reasonable temperatures.
- For example, in Si at 300 K, ni = Pi = 1010 cm-3, whereas the densities of available states at Ev and Ec are on the order of 1019 cm-3.
- Thus the probability of occupancy f(E) for an individual state in the conduction band and the hole probability [1 - f(E)] for a state in the valence band are quite small.
- Because of the relatively large density of states in each band, small changes in f(E) can result in significant changes in carrier concentration.
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In n-type material there is a high concentration of electrons in the conduction band compared with the hole concentration in the valence band (recall
Fig. a). - Thus in n-type material the distribution function f(E) must lie above its intrinsic position on the energy scale (Fig. b).
- Since f(E) retains its shape for a particular temperature, the larger concentration of electrons at Ec in n-type material implies a correspondingly smaller hole concentration at Ev.
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We notice that the value of f(E) for each energy level in the conduction band (and therefore the total electron concentration n0) increases as EF moves
closer to Ec . - Thus the energy difference (Ec - EF) gives a measure of n.
- For p-type material the Fermi level lies near the valence band (Fig. c) such that the [1 - f(E)] tail below Ev is larger than the f(E) tail above Ec.
- The value of (EF - Ev) indicates how strongly p-type the material is.
- It is usually inconvenient to draw f(E) vs. E on every energy band diagram to indicate the electron and hole distributions.
- Therefore, it is common practice merely to indicate the position of EF in band diagrams.
- This is sufficient information, since for a particular temperature the position of EF implies the distributions.