4.9.4 On the Subject of Debye Length Limitation

Since no analytical approximations are required in the numerical   solution of Poisson's equation, the method has a better resolution than analytically based profile extraction methods [109][49]. Indeed some of the classical Debye length limitations associated with C-V extraction techniques stem directly from these approximation and are not inherent to C-V profiling per se. To validate the previous statement, the numerical experiments of [49] were repeated. First the C-V characteristics of a MOS capacitor was simulated using two profiles:

• Profile A is a step doping profile with a sharp transition:
  

  

• Profile B is a step doping profile with a graded transition over an extrinsic Debye length:
  

  

  where = is the extrinsic Debye length at (); is the Bolzmann constant, is the absolute temperature, is the dielectric permittivity of silicon, and is the doping concentration.

In Fig. 4.18, the two CV characteristics simulated using the profiles A and B are plotted. As seen, there is clearly a difference between the two characteristics that could be used to reconstruct the original profiles.

 
Figure 4.18: Comparison of CV characteristics of profiles A (solid) and B (dashed).  

The two step profiles were parameterized using the length of the transition region and the two levels of doping (, , ). Using nonlinear least squares parameters extraction, the values of the step profile parameters were then extracted from the capacitance data.

In Fig. 4.19 and Fig. 4.20, profiles A and B are shown with the analytically extracted profiles using (4.6) and (4.7), as well as the profiles extracted using the inverse modeling method from the same data. As originally reported in [49], both analytically extracted profiles are a poor approximation to the original step profiles and are indistinguishable. On the other hand, it is clear that the original profiles can be reconstructed using the inverse method.

 
Figure 4.19: Comparison of Profile A (symbols) and the extracted profiles using the analytical approach (dashed) and the inverse modeling method (solid).  

 
Figure 4.20: Comparison of Profile B (symbols) and the extracted profiles using the analytical approach (dashed) and the inverse modeling method (solid).