Week 2: Doping density and Function difference

Introduction

As we had got some basic properties of the material of CMOS,  this week we will go further in this week based on these outcomes to research in the doping density and the function difference.

Doping Density of the Silicon Substrate

Doping is the intentional introduction of impurities into an intrinsic semiconductor for the purpose of adding more electrons or holes. 

To get the results, we firstly following the advices from academic advisor, and find the equation for the density as usual. Initially, we have: 

Since C_min, the minimum capacitance, which is given as the series contribution of oxide and semiconductor.
We have C_ox = C_max, and:

By combining these equations, the equation should be:

However, the power -2 was omitted mistakenly, it takes disastrous consequences which lead the results is more than 20 times worse than the correct one. In addition, after figuring out the formal part of the equation (without the logarithmic part), we realized that this equation is a transcendental equation, which can not be solved in an ordinary way. 

Moreover, we faced another predicament since the vast majority of the websites with the function of calculation and calculator can not give complete solutions, what can we obtained was a single solution, while we need are both of those solutions. 

After failed many times in different method, we try to convert the logarithmic part to an exponential function, and finally we re-enter the equation and get the second solution. N_D = 2.6771⨉10²¹. 

Function Difference Assuming a Gold (Au) gate

Figure 1: The Fermi Level

In solid-state physics, the work function is the minimum dynamic work needed to remove an electron from a solid to a point in the vacuum.

The function difference can be indicated as:

Where the equation of 𝜙_F has been given in the previous sections:

Then, 𝜙_MS = 0.7899 V

Summary

The process in this week are not smooth, even we can be blocked by a trouble for hours and nowhere with solving equations and handling data. But it is better than the first week, since our knowledge about this topic have been increased by the previous research. 

Once the key ideal are understand and the critical trouble are overcame, the rest of the process can be straightforward, just plugging parameters what we have then the results will come out.