Odd formula for calculating chemical potential

[duggles]

Hi all,

I've been looking through some code for semi-classical transport characteristics of solids and one of the features it can do is artificially "dope" the system by adding in charge carriers. However it seems a bit unstable so I had a peek to see what they actually did to change the system to accommodate the extra holes/electrons. It seems that they shift the chemical potential based on the difference between the number of electrons in the pure system and the number of electrons in the system with carriers. However they divide this difference by a factor which is completely throwing me. It is the "integral" (sum, since it's a computer) of the DOS multiplied by the derivative of the Fermi-Dirac distribution over the energy range of the DOS. I've never come across a formula for how to shift the chemical potential using the derivative of the FD distribution before (only that basic one which uses the log of the quotient of the carrier concentrations), so I am hoping someone can point in the direction of a book/journal/website that explains what they're doing.

Thanks

 

[eljose79]

for the interesting question about artificially doping a system with charge carriers. From what you've described, it sounds like the code is using a method called the "chemical potential shift" technique to adjust the chemical potential in the system. This technique is commonly used in simulations of electronic transport in solids.

The purpose of artificially doping a system is to mimic the effects of introducing impurities or defects, which can significantly alter the electronic properties of a material. By adding extra charge carriers, the code is essentially creating a non-equilibrium state in the system, which can be useful for studying the effects of doping on transport properties.

To shift the chemical potential, the code is using the derivative of the Fermi-Dirac (FD) distribution, which describes the distribution of electrons among energy levels in a system at thermal equilibrium. This derivative is multiplied by the density of states (DOS), which represents the number of available energy states for electrons in the system.

The integral (or sum) of this product over the energy range of the DOS is then divided by a factor, which is likely a normalization constant to ensure the correct magnitude of the shift. This formula may seem unfamiliar, but it is based on statistical mechanics and is commonly used in simulations of electronic transport.

If you would like to learn more about this technique and its application in simulations of electronic transport, I recommend checking out some textbooks or research papers on the topic. Some good resources to start with include "Electronic Transport in Mesoscopic Systems" by Supriyo Datta and "Introduction to the Theory of Thermal Neutron Scattering" by G.L. Squires.

I hope this helps to answer your question and provides some resources for further exploration. Good luck with your research!