Question

Consider a system of degenerate electrons at a low temperature in thermal equilibrium under the simultaneous influence of a density gradient and an electric field. (a) How is the chemical potential $\mu$ related to the electrostatic potential $\phi(x)$ and the Fermi energy $E_{\mathrm{F}}$ for such a system? (b) How does $E_{\mathrm{F}}$ depend on the electron density $n$ ? (c) From the condition for $\mu$ under thermal equilibrium and the considerations in (a) and (b), derive a relation between the electrical conductivity $\sigma$, the diffusion coefficient $D$ and the density of states at the Fermi surface for such a system.

   Consider a system of degenerate electrons at a low temperature in thermal equilibrium under the simultaneous influence of a density gradient and an electric field.
(a) How is the chemical potential $\mu$ related to the electrostatic potential $\phi(x)$ and the Fermi energy $E_{\mathrm{F}}$ for such a system?
(b) How does $E_{\mathrm{F}}$ depend on the electron density $n$ ?
(c) From the condition for $\mu$ under thermal equilibrium and the considerations in (a) and (b), derive a relation between the electrical conductivity $\sigma$, the diffusion coefficient $D$ and the density of states at the Fermi surface for such a system.

Problems and Solutions on Thermodynamics and Statistical Mechanics

U.S.T. of China… 1st Edition

Chapter 2, Problem 206

Instant Answer

Step 1

In a system of degenerate electrons, the chemical potential $\mu$ can be expressed as: \[ \mu = E_{\mathrm{F}} - e\phi(x) \] where $e$ is the elementary charge. This equation indicates that the chemical potential is influenced by the electrostatic potential, which Show more…

Show all steps

Thanks for your feedback!

Consider a system of degenerate electrons at a low temperature in thermal equilibrium under the simultaneous influence of a density gradient and an electric field. (a) How is the chemical potential $\mu$ related to the electrostatic potential $\phi(x)$ and the Fermi energy $E_{\mathrm{F}}$ for such a system? (b) How does $E_{\mathrm{F}}$ depend on the electron density $n$ ? (c) From the condition for $\mu$ under thermal equilibrium and the considerations in (a) and (b), derive a relation between the electrical conductivity $\sigma$, the diffusion coefficient $D$ and the density of states at the Fermi surface for such a system.

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback