Solving Confusion: Ashcroft & Electrochemical/Chemical Potential

[Fernsanz]

Hello.

There is no agreement on the meaning of terms electrochemical potential and chemical potential (see for example http://web.mit.edu/6.730/www/ST04/Lectures/Lecture26.pdf"). While proper definitions would call chemical potential to

$$\mu\equiv\left(\frac{\partial U}{\partial n}\right)_{neutral}$$​

-i.e., the variation in energy if the mass were not charged- and electrochemical potential or Fermi level to

$$\overline{\mu}\equiv\left(\frac{\partial U}{\partial n}\right)_{charged}=\left(\frac{\partial U}{\partial n}\right)_{neutral} +q\phi \equiv F_n$$​

-i.e, the actual variation in energy taking into account the mass is charged, which is the only measurable observable- Ashcroft (p. 593) seem to be totally misleading because he uses the letter $$\mu$$ to refer to the electrochemical potential and then defines an electrochemical potential as $$\mu_e=\mu - e\phi$$.

Equilibrium condition (deduced from the fundamental thermodynamic relation in energetic form)

$$dU=TdS - pdV+ \mu dn + Fz\phi dn=TdS - pdV + \overline{\mu} dn$$​

imposes that electrochemical potential should be constant along the semiconductor, so the actual picture is the electrochemical potential being constant along the semiconductor dimension and the valence and conduction energies bending wherever electric field exists.

Has anyone else observed -suffered- this misleading (wrong?) point in Aschroft?

Thanks

 

[eljose79]

for bringing up this issue. I can understand your confusion and frustration with the different definitions and usage of the terms electrochemical potential and chemical potential. It is important for scientists to have a clear understanding and agreement on the definitions and usage of terms in order to communicate effectively and accurately.

After reviewing the article you shared, it seems that the main source of confusion lies in the use of the letter \mu to refer to both the chemical potential and the electrochemical potential. As you mentioned, Ashcroft defines an electrochemical potential as \mu_e=\mu - e\phi, which can be misleading since \mu is typically used to refer to the chemical potential. This can lead to confusion and misinterpretation of the equations and concepts.

In order to avoid this confusion, it is important to clearly define and distinguish between the two terms. As you stated, the chemical potential \mu refers to the variation in energy if the mass were not charged, while the electrochemical potential or Fermi level \overline{\mu} takes into account the fact that the mass is charged. Therefore, it would be more accurate to use \overline{\mu} instead of \mu_e to refer to the electrochemical potential.

In conclusion, I agree with you that the use of the letter \mu to refer to both the chemical potential and the electrochemical potential can be misleading and should be clarified in order to avoid confusion. Thank you for bringing this issue to our attention and let's continue to strive for clear and accurate communication in the scientific community.