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Ionized impurity scattering

Ionized impurity scattering is a very important process for highly doped semiconductors. One of the first mobility models for this case was proposed by Conwell and Weisskopf [1]. They used the scattering cross section of an electron in the Coulomb field of an ionized impurity for an iterative solution of the Boltzmann transport equation. This yields a modified velocity distribution which is then used to calcultate the current density in dependence of the electric field.

Scattering in the Coulomb potential

The figure below illustrates the scattering event in a Coulomb potential for a point charge. Basically it is a gradual deviation of the charge carrier trajectory by coulombic interaction with an ionized impurity.
The theory of scattering of a charge in a Coulomb field was worked out by Rutherford for the case of a particles scattered at a thin gold foil. He obtained a formula for the probability of scattering into an angle q, called differential cross section:

Here, Z is the charge of the scattering center and E is the energy of the scattered particle. The distance between the axis of the incident electron and the scattering center is called impact parameter a and enters into the angle q. The scattering event is rotationally invariant about this axis and also independent of the charge of the scattering center as illustrated in the figure figure below.

This property is particularly important for compensated semiconductors; as the carriers are scattered at all ionized impurities, donors and acceptors alike, the number of scattering centers may be much higher than the net carrier density.

The collision integral

The next step in the determination of the mobility is the calculation of the collision integral. Unfortunately, the integral over the differential cross section of Rutherford is divergent because of the long range of the Coulomb potential.

Truncation

Conwell and Weiskopf suggested to truncate the potential at a value which corresponds to half the distance between two impurities. Intuitively this seems a valid approach because the deflection from a straight path will be governed by the closer of two given impurities rather than those further away.

The average distance between two impurites ist just the cube root of the impurity density, and half of this value is then taken as the maximum for scattering parameter a. Correspondingly, there is a minimum scattering angle qmin which is taken as limit for the integration.

Shielding

An different argument was was suggested by Brooks and Herring [2]. Their discussuion makes use of an effect that is very common in the treatment of plasmas; free charge carriers are able to accumulte in the vicinity of attractive potentials, or they move away from repellent ones. Effectively this results in a redistribution of the charge density and shields the range of the potential.

The description takes the form of the well known Yukawa potential and involves a screening radius rs.

For dilute electron gases like in non-degenerate semiconductors the screening radius is given by the expression of Debye and Hückel. For highly doped semiconductors we must use the result of Thomas and Fermi.

The Yukawa potential results in a modified scattering cross section, and the collision term is integrable.

The current density integral

After the evaluation of the collision integrals there is one more step to take, we must calculate another integral for the current density. For both cases we must integrate over a fraction with a log-term in the denominator. Usually this term is evaluated only at the maximum of the remaining integrand because the log is a slowly varying function.

The resulting mobilities of these two approaches have the same pre-factor which shows a temperature dependence with an exponent of 3/2.

It is multiplied by (truncation)

or (shielding)

where the quantity x is given by:

The considerations have also been applied to degenerate semiconductors, the result for the truncated potential was given by Shockley [3], the screened case was considered by Dingle [4]. Note that this result is independent of the temperature.

multiplied by (truncation):

In case of shielding the factor involving the logarithm is the of the same form as above, but x is now given by:

Numerical differences between the different results of truncation and screening were discussed by Dingle [4], an attempt to reconcile the theories by third body interactions was proposed by Ridley [5]. Experimental data about ionized impurity scattering in doped ZnO can be found in a publication of Ellmer [6].


References

[1] E. Conwell and V. F. Weisskopf, Phys. Rev. 77, 388, (1950)
[2] H. Brooks, Adv. Elect. Electron Phys. 7, 85, (1955)
[3] W. Schockley, Electrons and Holes in Semiconductors (1950)
[4] R. B. Dingle, Phil. Mag. 47, 831, (1982)
[5] B. K. Ridley, J. Phys. C: Solid State Phys. 10, 1589 (1977)
[6] K. Ellmer, J. Phys. D: Appl. Phys. 34, 3097 (2001)
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