Mobility

Electrons in doped semiconductor that is thermalised at room temperature have a thermal energy which allows them to wander in any direction. Under thermal equilibrium, elementary statistical mechanics tells us that the average kinetic energy associated with each electron, is 1/2 k _B T , per degree of freedom, and since the electron is free to wander in 3 dimensions the total kinetic energy is given by

(1)

The electron travels in a straight line until it path is influenced by another atom in the lattice, impurity atom or other scattering mechanism. The average distance which the electron travels before being scattered is known as the mean-free path and the average time between collisions is imaginatively called the mean-free time t _c . As you might expect, it depends on the material but typical magnitudes are 10 ^-5 cm and a few picoseconds, respectively.

If an electric field E is applied across the semiconductor, the free electrons will experience a force F =- qE . In the opposite direction of the field (since the electron has negative charge). Now the additional component of the electric field is imposed on the random motion of the electron causing an overall drift in the opposite direction to the electric field.

By equating the momentum gained by the electron during its mean free flight to the momentum lost in a collision we can obtain the drift velocity.

(2)

(3)

We call the ratio of the drift velocity to the applied electric field the mobility and it has the units (cm ² V ^-1 s ^-1 ). A similar argument applies to holes in the valence band, with the result that the mobility is given by (5) where we have used m _h ^* as the effective mass of the hole.

(4)

(5)

Table 1. Electron and Hole mobilities for various semiconductor materials

Looking at equations (4) and (5) we can see that mobility is directly effected by the mean free time for electrons and holes which is determined by the various scattering mechanisms. The most important are lattice scattering and impurity scattering.

Lattice scattering results from the thermal motion of the lattice atoms at temperatures above T= 0 K. The agitation of the atoms cause variations in the potential resulting in the emission of phonons which transfer energy between the lattice and the free carriers. Lattice scattering becomes dominant at higher temperatures because lattice vibrations increase with increasing temperature. A full theoretical analysis is shows that the mobility due to lattice scattering varies as T ^-3/2 .

Impurity scattering results from the ionised donor and acceptor impurities. A passing charge carrier will be deflected due to the Coulomb force between it and the ion. The probability of impurity scattering depends on the doping density and the proportion of those atoms that are ionised. Impurity scatting become less significant with temperature since above a certain temperature, the impurity atoms will have ionised and also the charge carriers are moving faster and interact with the impurity for a shorter time. The variation mobility due to impurity scattering decreases as T ^3/2 / N _T , where N _T is the total impurity concentration.

In summary, the total scattering time is the sum of two scatting times, the lattice scatting time t _L and the impurity scattering time t _I . Using (4) we can simply obtain (7),

(6)

(7)

References

[1] Ilegems, M., Montgomery, H.C., J. Phys. Chem. Solids 34 (1972) 885.
[2] Ilegems, M., Dingle, R., J. Appl. Phys. 44 (1973) 4234.
[3] Crouch, R.K., Debnam, W.J., Fripp, A.L., J. Mater. Sci. 13 (1978) 2358.
[4] S. M. Sze, Semiconductor Devices Physics and Technology.