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 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
*θ*_{min} which is taken as limit for
the integration.

The description of shielded potentials was given by Yukawa and bears his name.
He obtained a potential with an exponential decay that is described by a
screening radius *r _{s}*.

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 the following scattering cross section:

With this expression, the term for the relaxation time becomes integrable.

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 ξ 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 ξ 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].

[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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