\[f(E) = \frac{1}{e^{(E - E_F)/k_\mathrm BT} + 1}\]
含义Meaning
费米子（电子）占据能量 E 的量子态的概率。T = 0 时为阶跃函数，费米能以下全满、以上全空；有限温度下只在 \(E_F\) 附近约 \(k_\mathrm BT\) 范围内被“抹平”。
The probability that a fermion state at energy E is occupied: a step at T = 0, smeared over ~k_BT around E_F at finite temperature.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(E_F\) | 费米能级（化学势）Fermi level (chemical potential) | eV |
| \(f(E)\) | 占据概率occupation probability | 1 |
推导Derivation
泡利原理：每个量子态最多一个电子（含自旋）。
Pauli: at most one electron per spin-state.
巨正则系综中单个态的配分函数 \(1 + e^{-(E-\mu)/k_\mathrm BT}\)，平均占据数为其对数导数。
In the grand canonical ensemble a single state has \(\Xi = 1 + e^{-(E-\mu)/kT}\); its mean occupancy is the result.
\(E - E_F \gg k_\mathrm BT\) 时退化为玻尔兹曼因子 \(e^{-(E-E_F)/k_\mathrm BT}\)（非简并半导体）。
For \(E - E_F \gg kT\) it reduces to the Boltzmann factor.
适用条件与局限Assumptions & limitations
只适用于平衡态；光照、注入载流子时用准费米能级描述。
Equilibrium only; use quasi-Fermi levels under illumination or injection.