P型半导体载流子密度推导

p_a^-=N_A(1-f_A(E))=\frac{N_A}{1+g_A(E)e^{-\frac{E_F-E_A}{k_0T}}}

取g_A(E)=4

p_a^-=N_A(1-f_A(E))=\frac{N_A}{1+4e^{-\frac{E_F-E_A}{k_0T}}}

有

n_0p_0=n_i^2,p_0=n_0+p_a^-

其中

n_0=N_C\cdot e^{-\frac{E_C-E_F}{k_0T}},p_0=N_V\cdot e^{-\frac{E_F-E_V}{k_0T}}

进而可以得到

N_V\cdot e^{-\frac{E_F-E_V}{k_0T}}=N_C\cdot e^{-\frac{E_C-E_F}{k_0T}}+\frac{N_A}{1+4e^{-\frac{E_F-E_A}{k_0T}}}

可以直接求得解析解为:

k_0 T ln \left(\frac{\sqrt[3]{\sqrt{4 \left(12 N_c e^{\frac{E_c}{k_0 T}} \left(N_A e^{\frac{E_D}{k_0 T}}-4 N_v e^{\frac{E_v}{k_0 T}}\right)-N_c^2 e^{\frac{2 E_D}{k_0 T}}\right){}^3+\left(36 N_A N_c^2 e^{\frac{E_c}{k_0 T}+\frac{2 E_D}{k_0 T}}+288 N_c^2 N_v e^{\frac{E_c}{k_0 T}+\frac{E_D}{k_0 T}+\frac{E_v}{k_0 T}}-2 N_c^3 e^{\frac{3 E_D}{k_0 T}}\right){}^2}+36 N_A N_c^2 e^{\frac{E_c}{k_0 T}+\frac{2 E_D}{k_0 T}}+288 N_c^2 N_v e^{\frac{E_c}{k_0 T}+\frac{E_D}{k_0 T}+\frac{E_v}{k_0 T}}-2 N_c^3 e^{\frac{3 E_D}{k_0 T}}}}{12 \sqrt[3]{2} N_c}-\frac{12 N_c e^{\frac{E_c}{k_0 T}} \left(N_A e^{\frac{E_D}{k_0 T}}-4 N_v e^{\frac{E_v}{k_0 T}}\right)-N_c^2 e^{\frac{2 E_D}{k_0 T}}}{6\ 2^{2/3} N_c \sqrt[3]{\sqrt{4 \left(12 N_c e^{\frac{E_c}{k_0 T}} \left(N_A e^{\frac{E_D}{k_0 T}}-4 N_v e^{\frac{E_v}{k_0 T}}\right)-N_c^2 e^{\frac{2 E_D}{k_0 T}}\right){}^3+\left(36 N_A N_c^2 e^{\frac{E_c}{k_0 T}+\frac{2 E_D}{k_0 T}}+288 N_c^2 N_v e^{\frac{E_c}{k_0 T}+\frac{E_D}{k_0 T}+\frac{E_v}{k_0 T}}-2 N_c^3 e^{\frac{3 E_D}{k_0 T}}\right){}^2}+36 N_A N_c^2 e^{\frac{E_c}{k_0 T}+\frac{2 E_D}{k_0 T}}+288 N_c^2 N_v e^{\frac{E_c}{k_0 T}+\frac{E_D}{k_0 T}+\frac{E_v}{k_0 T}}-2 N_c^3 e^{\frac{3 E_D}{k_0 T}}}}-\frac{1}{12} e^{\frac{E_D}{k_0 T}}\right)

当然我们不可能直接使用上面的式子
使用Si中多子浓度与温度的关系曲线可以得到三个工作温区,分别为

• 电离温区
  • 弱电离温区
  • 中等电离温区
  • 强电离温区
• 非本征温区
• 本征温区
  

  电离温区

  弱电离温区

  在弱电离温区内,本征激发可以忽略,主要的载流子来源于电离的杂质,此时有:

  p_0\approx p_a^-=\frac{N_A}{1+4e^{-\frac{E_F-E_A}{k_0T}}}=N_V\cdot e^{-\frac{E_F-E_V}{k_0T}}

  N_A>>p_a^-\to e^{-\frac{E_F-E_A}{k_0T}}>>1\to\frac{N_A}{4e^{-\frac{E_F-E_A}{k_0T}}}=N_V\cdot e^{-\frac{E_F-E_V}{k_0T}}

  E_F=\frac{1}{2} \left(-k_0 T ln \left(\frac{N_A}{4 N_V}\right)+E_A+E_V\right)

  进而可以得到

  p_0=\frac{1}{2}\sqrt{N_AN_V} e^{\frac{E_V-E_A}{2 k_0 T}}

  中等电离温区

  此时,需要直接按照

  p_0\approx p_a^-=\frac{N_A}{1+4e^{-\frac{E_F-E_A}{k_0T}}}=N_V\cdot e^{-\frac{E_F-E_V}{k_0T}}

  进行求解

  E_F=E_A-k_0 Tln(\frac{1}{8}[\sqrt{1+16\frac{N_A}{N_V}e^{\frac{E_A-E_V}{k_0T}}}-1])

  进而有

  p_0=\frac{N_A e^{\frac{E_A-E_V}{k_0T}}}{\frac{1}{8}[\sqrt{1+16\frac{N_A}{N_V}e^{\frac{E_A-E_V}{k_0T}}}-1]}

  强电离温区

  此时,大部分杂质全部电离,有

  p_0\approx N_A\Rightarrow N_A=N_V\cdot e^{-\frac{E_F-E_V}{k_0 T}}

  E_F=E_V+k_0 T \left(ln \left(\frac{N_V}{N_A}\right)\right)

  进而有:

  p_0=N_A

  非本征区

  此时的激发包括杂质激发和部分的本征热激发,杂质提供的载流子几乎已经全部激发

  p_0=N_A+n_0,n_0 p_0=n_i^2

  解得

  p_0=\frac{1}{2} \left(\sqrt{N_A^2+4 n_i^2}+N_A\right),n_0=\frac{1}{2} \left(\sqrt{N_A^2+4 n_i^2}-N_A\right)

  进而我们可以求得$E_F$

  E_F=E_i+k_0 T \left(ln \left(\frac{2 n_i}{\sqrt{N_A^2+4 n_i^2}+N_A}\right)\right)

  本征温区

  此时,本征激发的效果逐渐明显,杂质激发已经弱于本征激发

  n_0\approx p_0=n_i=\sqrt{N_C N_V}e^{-\frac{E_C-E_V}{k_0 T}}

  需要注意的是,此处出现了禁带宽度

  E_g=E_C-E_V

  随着温度的变化,禁带宽度可以由下式给出

  E_g(T)=E_g(0K)-\frac{\alpha T^2}{\beta+T}

  此时

  E_F\approx E_i