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.
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.
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
- 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.
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.