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Ionisation

The estimation of the ionisation energy from the hydrogen analogy in the previous chapter was illustrative, but not very accurate. We can determine the ionisation energy of dopants by the temperature dependence of the carrier density. Assume a n-type semiconductor with a donor density N_D. The density of occupied donor states n_D is given by the an expression very similar to the Fermi-Dirac distribution:

Here, E_D is the "depth" of the donor, and β a factor for describing how many equivalent band minima are available at the conduction band minimum (also called band degeneracy). At low temperatures all donor electrons are bound to their donor atoms, they are "frozen". The Fermi level is at its intrinsic position. As we increase the temperature, more and more donors are ionised. The increased electron density also raises the Fermi level which in turn controls how many donors are ionised and so forth. Charge neutrality requires that the number of excited and bound electrons add up to the number of donors:

n + n_D = N_D

For the calculation of the actual electron density we combine the equation of charge neutrality and the relation between electron density and Fermi level of the previous section. Then, we eliminate n_D as well as the explicite dependence on the Fermi level. This yields a second order equation for the dependence of the electron densitiy n on the temperature T. A graphical representation in Arrhenius form is shown below, assuming a bandgap of 1.2 eV, a donor depth of 100 meV, and a donor density of 10¹⁵ 1/cm³.

In the useful range between 200 and 500 K all donors are ionised, we can assume n = N_D. For lower temperatures the electron density freezes out with an activation energy of half the donor depth. Thus, Hall measurements at different temperatures can be used to find the activation energy for donor ionisation. For higher temperatures the electron density is dominated by the intrinsic carrier density n=sqrt(n_i²).

Compensation

In less ideal semiconductors both, donors and acceptors, will be present simultaneously. With a donor density N_D and an acceptor density N_A we observe only a net dopant concentration in the equation for charge neutrality.

n + n_D = N_D- N_A

The degree of compensation K is defined as the ratio of N_A and N_D. The activation energy plot becomes a little more difficult. For moderate compensation there are two regions with different slope as shown in the figure. More often compensation levels of 90 or 99% percent are found which substantially complicates the interpretation.

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