# Electrochemical potential, chemical potential and Fermi levels. Ashcroft is confusing

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

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