含义Meaning
未掺杂半导体中热激发产生的电子与空穴浓度，强烈依赖带隙与温度。质量作用定律 \(np = n_i^2\) 对掺杂半导体同样成立。
Thermally generated electron and hole densities in an undoped semiconductor, strongly dependent on gap and temperature. The mass-action law \(np = n_i^2\) also holds when doped.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(n, p\) | 电子、空穴浓度electron and hole concentrations | cm⁻³ |
| \(N_c, N_v\) | 导带、价带有效态密度effective densities of states | cm⁻³ |
| \(m^*\) | 有效质量effective mass | kg |
推导Derivation
导带电子 \(n = \int g_c(E)f(E)dE\)，非简并时用玻尔兹曼近似：\(n = N_ce^{-(E_c - E_F)/k_\mathrm BT}\)。
Integrate density of states × occupation: \(n = N_ce^{-(E_c - E_F)/kT}\).
同理 \(p = N_ve^{-(E_F - E_v)/k_\mathrm BT}\)。
Similarly \(p = N_ve^{-(E_F - E_v)/kT}\).
相乘消去 \(E_F\)：\(np = N_cN_ve^{-E_g/k_\mathrm BT} = n_i^2\)。
Multiplying eliminates E_F.
适用条件与局限Assumptions & limitations
非简并近似（E_F 离带边几个 kT 以上）；重掺杂或窄带隙材料需用费米积分。
Non-degenerate approximation; heavy doping needs Fermi integrals.
例题Worked example
Si 在 300 K：n_i ≈ 1.0×10¹⁰ cm⁻³；掺 10¹⁶ cm⁻³ 磷后 n ≈ 10¹⁶，p = n_i²/n ≈ 10⁴ cm⁻³。
Si at 300 K: n_i ≈ 10¹⁰ cm⁻³; with 10¹⁶ cm⁻³ P, p ≈ 10⁴ cm⁻³.