# 费米能

## 三维形式的推导

${\displaystyle \psi =A\sin \left({\frac {n_{x}\pi x}{L}}\right)\sin \left({\frac {n_{y}\pi y}{L}}\right)\sin \left({\frac {n_{z}\pi z}{L}}\right)\,}$

A为波函数的归一化常数
nx、ny、nz正整数

${\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right)\,}$

${\displaystyle N={\frac {1}{8}}\times 2\times {\frac {4}{3}}\pi n_{F}^{3}\,}$

${\displaystyle n_{F}=\left({\frac {3N}{\pi }}\right)^{1/3}}$

 ${\displaystyle E_{F}\,}$ ${\displaystyle ={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}n_{F}^{2}}$ ${\displaystyle ={\frac {\hbar ^{2}\pi ^{2}}{2mL^{2}}}\left({\frac {3N}{\pi }}\right)^{2/3}}$

 ${\displaystyle E_{F}={\frac {\hbar ^{2}}{2m}}\left({\frac {3\pi ^{2}N}{V}}\right)^{2/3}\,}$

${\displaystyle \hbar }$约化普朗克常数${\displaystyle =1.055\times 10^{-34}\ {\mbox{J}}\cdot {\mbox{s}}}$
m为粒子质量

## 典型的费米能

### 白矮星

${\displaystyle E_{F}={\frac {\hbar ^{2}}{2m_{e}}}\left({\frac {3\pi ^{2}(10^{36})}{1\ \mathrm {m} ^{3}}}\right)^{2/3}\approx 3\times 10^{5}\ \mathrm {eV} \,}$

### 原子核

${\displaystyle R=\left(1.5\times 10^{-15}\mathrm {m} \right)\times A^{1/3}}$

${\displaystyle n={\frac {A}{{\begin{matrix}{\frac {4}{3}}\end{matrix}}\pi R^{3}}}\approx 7\times 10^{43}\ \mathrm {m} ^{-3}}$

${\displaystyle E_{F}={\frac {\hbar ^{2}}{2m_{p}}}\left({\frac {3\pi ^{2}(7\times 10^{43})}{1\ \mathrm {m} ^{3}}}\right)^{2/3}\approx 33\times 10^{6}\ \mathrm {eV} =33\ \mathrm {MeV} }$

## 费米能级

${\displaystyle E_{av}={\frac {3}{5}}E_{F}}$

${\displaystyle p_{F}={\sqrt {2m_{e}E_{F}}}}$

${\displaystyle V_{F}={\sqrt {\frac {2E_{F}}{m_{e}}}}}$

${\displaystyle T_{F}={\frac {E_{F}}{k_{B}}}}$

## 自由电子气

${\displaystyle \mu =E_{F}\left[1-{\frac {\pi ^{2}}{12}}\left({\frac {k_{B}T}{E_{F}}}\right)^{2}+{\frac {\pi ^{4}}{80}}\left({\frac {k_{B}T}{E_{F}}}\right)^{4}+\cdots \right]}$

## 注释

1. ^ Sze, Simon M. Physics of Semiconductor Devices（2nd ed.）. John Wiley and Sons（WIE）. 1981. ISBN 978-0-471-05661-4.
2. ^ The use of the term "Fermi energy" as synonymous with Fermi level (a.k.a. electrochemical potential) is widespread in semiconductor physics. For example: Electronics (fundamentals And Applications)页面存档备份，存于互联网档案馆） by D. Chattopadhyay, Semiconductor Physics and Applications页面存档备份，存于互联网档案馆） by Balkanski and Wallis.
3. ^ 在参考文献（1）中，绝对零度下的费米能用${\displaystyle E_{F}^{0}}$表示，其他温度下的费米能或化学势用${\displaystyle E_{F}}$表示，与本条目中的符号有差异，敬请留意。