Dear Venkatesh, I presume that it primarily relies on the doping element. I'd like to know what is the doping element you use, so that it's quite easy to seek answer for your context.

With increasing doping concentration above a certain critical value, Mott critical concentration Nc given by Nc1/3*aH=0.25 where aH is the fundamental hydrogenic-like Bohr Radius, the ionization energy of the doping element Ei starts decreasing as a result of increasing interactions (correlation interactions) between doping impurities leading to a gaussian-like broadening of the impurity level which is no more a discrete level.

This boadening of the impurity level is responsible for the reduction of the energy Ed or Ea between the impurity (donor or acceptor) level and allowed (conduction or valence) bands, hence the ionization energy Ei.

Increasing furhter doping concentration N* well above the critical value Nc results in a vanishing Ei value, and the material becomes metallic like: i.e. experiences a Semiconductor to metal transition.

The fact that N* is often well above Nc is due to the fact that in adition to correlation interactions, there is onset of disorder effects (Anderson-like) also triggered by increasing impurity concentrations above a critical value.

In real situations both correlation interactions (Mott like) and disorder effects (Anderson like) are responsible for decreasing ionization energy of doping impurities.

Best regards.

Professor A. Kadri

Dinesh Kumar Madheswaran Sir, it does not matter what the element is, as doping increases, the effect of incomplete ionization is seen even at higher temperatures. For example, assuming a constant Ea for Boron, at Na=10^18/cm^3, all dopants are not completely ionized. This can be calculated by calculating the position of fermi level using the charge balance equation and including the effect of incomplete ionization. And using that to back-calculate the percentage of ionized dopants. The higher the value of Ea, the more we see the effect of incomplete ionization. A recent paper where you can see this effect is

A. Beckers, F. Jazaeri, A. Grill, S. Narasimhamoorthy, B. Parvais and C. Enz, "Physical Model of Low-Temperature to Cryogenic Threshold Voltage in MOSFETs," in IEEE Journal of the Electron Devices Society, vol. 8, pp. 780-788, 2020, doi: 10.1109/JEDS.2020.2989629.

Here, in figure 5, we can clearly see the difference in bulk fermi potential when we assume complete ionization and when we assume incomplete ionization, even for T=300K for Na=10^18/cm^3.

Abderrahmane Kadri Sir, while what you said is true, the effect of incomplete ionization can be seen even without taking the interaction of doped atoms with each other as well as the crystal into account. Also, what you have said implies that dopant ionization energy effectively decreases in such a case which means dopant ionization percentage must increase. I believe that would happen at a much higher doping than the range in which I'm working.

Venkatesh Jayanti

I think I have to make the point clearer, because in real life, the behavior of impurities in a semiconductor can be neither so simple, nor so obvious:

- Yes, you are right if we consider the basic model for impurities in a semiconductor in the dilute regime where the impurities are too far apart from each other to interact sufficiently (i.e. Ni or =Nc, then the impurity energy level is strongly disturbed and instead of a Dirac distribution of the impurity density of states as is the case in the dilute regime, now we end with a Gaussian distribution of the impurity density of states centered around a new ''impurity energy level'' (corresponding to an ''apparent ionization energy'') which shifts slowly with increasing impurity concentration towards respective allowed band (conduction band for donors and valence band for acceptors).

Hence, a statistical (Gaussian) reduction of the apparent ionization energy of the impurity level.

However, because of impurity interactions and Gaussian-like distribution of impurity density of states (not all the impurities are at the same energy: some are nearer to the allowed band than the center of the Gaussian and others are farther from the allowed band) this Ei reduction does not induce a linear increase of free carriers (as is the case in the dilute regime) but a saturating increase (i.e. smaller than the expected value which otherwise would have lied on the perfect line giving n=f(ND) or p=f(NA) variations in a log-log plot) of free carrier ionization from impurity levels.

In other words, the ionization process from Gaussian-like impurity levels is modulated (dumped) by the Gaussian distribution of energies within the impurity levels.

Moreover, in this ''strong doping regime'' (Ni> or =Nc) in addition to the Gaussian-like impurity energy distribution, impurity correlation effects induce a new energy level called the mobilty threshold EC which separates states ''able to conduct'' (called delocalized states) from states which are ''not able to conduct'' (called strongly localized states).

This localization of impurity states can come from two possible different processes:

- The first such process is ''electron-electron correlation interactions'' described by the model worked out by N.F. Mott (Nobel Prize in Physics). Here, the impurities are so close (Nc1/3*aH)< or =0.25) that they are strongly overlapping and thus strongly interacting. An accurate quantum mechanical treatment due to Mott shows that some of the impurities are no more able to contribute in the conduction because of electron-electron interactions giving rise to the Gaussian-like distribution and a mobilty edge Ec.

Here, we therefore end up with much lower free carrier densities (n=f(ND) and p=f(NA) than expected when ignoring electron-electron correlation interactions, despite the fact the apparent ionization energy decreases as discussed above.

- The second such processes is related to ''disorder effects'' which are triggered in a semiconductor by an increasing random distribution of impurities in the crystal as the impurity concentration increases above a critical value also well above the dilute regime.

Here, thanks to the model introduced by W.P. Anderson (Nobel Prize in Physics), we know that random distribution of impurities do induce potential fluctuations giving rise in turn to a Gaussian-like distribution of impurity density of states.

Here also because of impurity potential fluctuations, some of the impurities are no more able to contribute to the conduction, those lying below a special energy level called also the ''mobility edge Ec'' . Ec is here separating delocalized impurity states with E>Ec (not affected by disorder) from strongly localized impurity states with E