Most of the device modeling for nanoscale MOSFET where degenerate doping of the substrate is a must, assumes that although there is incomplete ionization at lower substrate temperature than 300 K but at room temperature, doping ionization is 100%. This modeling practice has provided erroneous results for MOSFET device performance simulation where substrate doping is essential modeling parameter. I am attaching the following papers for reference. I would like the readers to concentrate on Figure 6 of the first paper attached. Here with increased doping level from 10^17/cm^3 to 3x10^18/cm^3, the ionization ratio gradually decreases to to about >0.75 and then rises again to unity close to 10^20/cm^3. Since the source and drain regions of all nanoscale MOSFET are doped of the order of 10^20/cm^3 to 10^21/cm^3, it looks like the source dopants are more or less ionized close to 100% but the substrate for example junctionless FET where very high degenerate doping level of the order of high 10^18/cm^3 is required, Figure 6 of the first paper attached shows that ionization will be in the vicinity of 0.8 or 80% even at room temperature. Band gap narrowing which is dependent on degenerate doping levels, need to be remodeled where the continuous variation of dopant ionization values for high degenerate doping levels need to be properly estimated through analytical means that replicate Figure 6 for n-type substrate (first attached paper) or Figure 5 for p-type substrate (second attached paper). Similar ionization curve for p-substrate can be found from the second attached paper. Intrinsic carrier concentration for silicon at T = 300 K rises from its actually quoted value when due to degenerate doping, energy band gap narrowing takes place. Model equations are available to quantify this narrowed band gap with respect to substrate doping values, albeit assuming complete ionization. But as we see < 10^20/cm^3 doping levels, ionization ratio is between 0.8 and 0.9 or 80%-90% and this feature needs to be included in an analytical derivation format just like Figure 6 of the first paper attached or Figure 5 of the second paper attached. Once the ionization is precisely defined, the different band gap narrowing models can be computed to extract intrinsic carrier concentration value for silicon at T = 300 K when it sharply increases from its quoted value. Not accounting for this incomplete ionization of degenerately doped substrate value, most of the modeling calculations including high doping value dependent intrinsic carrier concentration at T = 300 K for Silicon FET will be error prone. I am still not sure Silvaco modeling library provides the users the analytical equations that precisely provide the total ionization at T = 300 K for degenerate donor or acceptor dopants as per Figure 6 of first attached paper or Figure 5 of the second attached paper.
I initiate this discussion related conversation here based on the above contents shared with Research gate community.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
Dear Dr. Nabil,
welcome,
When the doping is relatively low, the Fermi level will lie nearer to the intrinsic level and away from the doping level, therefore the occupation probability of the doping level will be nearly zero for the donor and one for the acceptor pointing out full ionization. So, the use of Boltzmann statistics is justified and the material is nondegenerate. When the doping increases and the Fermilevels approaches the bandgap edges, one has to use the Fermi statistics. Using Statistics and the neutrality equations for a given doping concentration will the fraction of the unionized doping concentration spontaneously. So, normally in solving the semiconductor equation one assumes Fermi statistics. So, the incomplete ionization at low temperature or high doping concentration is taken into consideration by utilizing Fermi statistics instead of Boltzmann statistics.
Best wishes
Dr. Zekry:
Best regards. The Fermi statistics shows the reduced free carriers only when the Ec-Ef or Ef-Ev difference is smaller than 3 kT. The carriers just get reduced but still it does not provide the complete modeling outcome of the Figure 6 of the first attached paper of mine in this discussion, where the ionization ratio gradually declines in line with Fermi statistics you are pointing about but the ionization ratio reaches a minima of about 0.8 and then reverses trend to full ionization again at very high 10^21/cm^3 doping levels. The single use of Fermi-Dirac distribution function to determine free carrier concentration does not replicate this picture ( ionization increasing up again, please look at the Figure 6 of the attached first paper closely and look how the brilliant authors attempted to model the experimental findings of ionization at T = 300 K for n-type substrate when degeneracy is extended up to 10^21/cm^3). It would have been more a requirement that a sequence of analytical model equations can be derived for this purpose which may be lacking in prominent Silvaco device modeling software.
Sincerely,
Dr. Nabil Shovon Ashraf.
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