Intrinsic Carrier Density - Temperature Dependence [+ Code]

For this short tutorial the requirements are the basic knowledge of Solid State Physics, in particular the electronic band structure.

Skipping the theoretical part (people interested in it can write their questions on the comments section or by private message), the intrinsic carrier concentration (or density) is given by the following formula:

Neglecting T dependence of Nc, Nv and Eg

In the first plot the dependency of Nc, NV and Eg on temperature was neglected. Three curves are plotted for three different imaginary band gaps.

#define constants and energy gap list
Egs = [0.2, 0.5, 5]
kB = 8.6173303e-5 #eV/K

Lower band gap should give higher value of intrinsic carrier density.

Considering Energy Band Gap dependency on T

The previous formula has to be corrected because now we include the temperature dependence of the band gap.

The latter is given by the following empirical model:

This model shows us that increasing temperature leads to smaller energy gap. Therefore we should observe an higher intrinsic carrier density. Moreover this effect is more evident on curves with a little Eg(0) as the blue line shows. Indeed at the highest temperature we analyzed we could observe a value almost five times greater when we include the temperature dependence of the band gap.

Considering Nc, Nv dependency on T

In the previous cases we consider approximated values of Nc and Nv. Now it is taken into account the temperature dependency and more precise value of conduction/ valance band states density.

This dependency should further increase the intrinsic carrier density.

All the plots and code are available on the following link.

Code (open the file with Google Colab)