半导体物理—半导体中载流子统计分布公式总结（精华版）

李子一个爱学习

2024年10月25日 00:01

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共8篇

说明：本总结公式符号与刘恩科《半导体物理学（第8版）》一致。

一、状态密度

导带底附近状态密度 $g_c(E)%3D%5Cfrac%7BdZ%7D%7BdE%7D%3D%5Cfrac%7BV%7D%7B2%5Cpi%5E2%7D%5Cfrac%7B(2m_n%5E*)%5E%5Cfrac%7B3%7D%7B2%7D%7D%7B%5Chbar%5E3%7D(E-E_c)%5E%5Cfrac%7B1%7D%7B2%7D$

价带顶附近状态密度 $g_v(E)%3D%5Cfrac%7BdZ%7D%7BdE%7D%3D%5Cfrac%7BV%7D%7B2%5Cpi%5E2%7D%5Cfrac%7B(2m_p%5E*)%5E%5Cfrac%7B3%7D%7B2%7D%7D%7B%5Chbar%5E3%7D(E_v-E)%5E%5Cfrac%7B1%7D%7B2%7D$

二、Fermi 分布函数和 Boltzmann 分布函数

Fermi分布函数 $f(E)%3D%5Cfrac%7B1%7D%7B1%2Be%5E%5Cfrac%7BE-E_F%7D%7Bk_0T%7D%7D%3D%5Cfrac%7B1%7D%7B1%2B%5Cexp%0A(%5Cfrac%7BE-E_F%7D%7Bk_0T%7D)%7D$

Boltzmann分布函数 $f(E)%3D%5Cfrac%7B1%7D%7Be%5E%5Cfrac%7BE-E_F%7D%7Bk_0T%7D%7D%3D%5Cexp(-%5Cfrac%7BE-E_F%7D%7Bk_0T%7D)$

三、本征半导体

导带中的电子浓度

$n_0%3D2(%5Cfrac%7Bm_n%5E*k_0T%7D%7B2%5Cpi%5Chbar%5E2%7D)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Cexp(-%5Cfrac%7BE_c-E_F%7D%7Bk_0T%7D)$

$N_c%3D2(%5Cfrac%7Bm_n%5E*k_0T%7D%7B2%5Cpi%5Chbar%5E2%7D)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D$

价带中的空穴浓度

$p_0%3D2(%5Cfrac%7Bm_p%5E*k_0T%7D%7B2%5Cpi%5Chbar%5E2%7D)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Cexp(%5Cfrac%7BE_v-E_F%7D%7Bk_0T%7D)$

$N_v%3D2(%5Cfrac%7Bm_p%5E*k_0T%7D%7B2%5Cpi%5Chbar%5E2%7D)%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D$

本征半导体的费米能级

$E_i%3DE_F%3D%5Cfrac%7BE_c%2BE_v%7D%7B2%7D%2B%5Cfrac%7Bk_0T%7D%7B2%7D%5Cln%7B%5Cfrac%7BN_v%7D%7BN_c%7D%7D%3D%5Cfrac%7BE_c%2BE_v%7D%7B2%7D%2B%5Cfrac%7B3k_0T%7D%7B4%7D%5Cln%7B%5Cfrac%7Bm_p%5E*%7D%7Bm_n%5E*%7D%7D$

本征载流子浓度

$n_i%3Dn_0%3Dp_0%3D(N_cN_v)%5E%5Cfrac%7B1%7D%7B2%7D%5Cexp(-%5Cfrac%7BEg%7D%7B2k_0T%7D)%3D(N_cN_v)%5E%5Cfrac%7B1%7D%7B2%7D%5Cexp(-%5Cfrac%7BE_c-E_v%7D%7B2k_0T%7D)$

$n_i%3D%5B%5Cfrac%7B2(2%5Cpi%20k_0T)%5E%5Cfrac%7B3%7D%7B2%7D(m_p%5E*m_n%5E*)%5E%5Cfrac%7B3%7D%7B4%7D%7D%7Bh%5E3%7D%5D%5Cexp(-%5Cfrac%7BE_c-E_v%7D%7B2k_0T%7D)%3D%5B%5Cfrac%7B2(2%5Cpi%20k_0T)%5E%5Cfrac%7B3%7D%7B2%7D(m_p%5E*m_n%5E*)%5E%5Cfrac%7B3%7D%7B4%7D%7D%7Bh%5E3%7D%5D%5Cexp(-%5Cfrac%7BE_g%7D%7B2k_0T%7D)$

$n_0p_0%3Dn_i%5E2$

四、杂质能级上的电子和空穴

电子占据施主能级的概率

$f_D(E)%3D%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B1%7D%7Bg_D%7D%5Cexp(%5Cfrac%7BE_D-E_F%7D%7Bk_0T%7D)%7D$

空穴占据受主能级的概率

$f_A(E)%3D%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7B1%7D%7Bg_A%7D%5Cexp(%5Cfrac%7BE_F-E_A%7D%7Bk_0T%7D)%7D$

施主能级上电子浓度

$n_D%3DN_Df_D(E)%3D%5Cfrac%7BN_D%7D%7B1%2B%5Cfrac%7B1%7D%7Bg_D%7D%5Cexp(%5Cfrac%7BE_D-E_F%7D%7Bk_0T%7D)%7D$

施主能级电离浓度

$n_D%5E%2B%3DN_D-n_D%3D%5Cfrac%7BN_D%7D%7B1%2Bg_D%5Cexp(-%5Cfrac%7BE_D-E_F%7D%7Bk_0T%7D)%7D$

受主能级上空穴浓度

$p_A%3DN_Af_A(E)%3D%5Cfrac%7BN_A%7D%7B1%2B%5Cfrac%7B1%7D%7Bg_A%7D%5Cexp(%5Cfrac%7BE_F-E_A%7D%7Bk_0T%7D)%7D$

受主能级电离浓度

$p_A%5E%2B%3DN_A-p_A%3D%5Cfrac%7BN_A%7D%7B1%2Bg_A%5Cexp(-%5Cfrac%7BE_F-E_A%7D%7Bk_0T%7D)%7D$

五、n型半导体的载流子浓度

低温弱电离区：（电中性方程 $n_0%3Dn_D%5E%2B$ ）

$E_F%3D%5Cfrac%7BE_C%2BE_D%7D%7B2%7D%2B(%5Cfrac%7Bk_0T%7D%7B2%7D)%5Cln(%5Cfrac%7BN_D%7D%7B2N_c%7D)$

$n_0%3D(%5Cfrac%7BN_cN_D%7D%7B2%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5Cexp(-%5Cfrac%7BE_c-E_D%7D%7B2k_0T%7D)%3D(%5Cfrac%7BN_cN_D%7D%7B2%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5Cexp(-%5Cfrac%7B%5CDelta%20E_D%7D%7B2k_0T%7D)$

强电离区：（电中性方程 $n_D%5E%2B%3DN_D$ ）

$E_F%3DE_C%2B%7Bk_0T%7D%5Cln(%5Cfrac%7BN_D%7D%7BN_c%7D)$

$n_D%3DD_-%20N_D$

$D_-%3D(%5Cfrac%7B2N_D%7D%7BN_c%7D)%5Cexp(%5Cfrac%7B%5CDelta%20E_D%7D%7Bk_0T%7D)$

少子 $p_%7Bn0%7D%3D%5Cfrac%7Bn_i%5E2%7D%7BN_D%7D$

过渡区：（电中性方程 $n_0%3DN_D%2Bp_0$ ）

$E_F%3DE_i%2Bk_0T%20%5Ctext%7Barsh%7D(%5Cfrac%7BN_D%7D%7B2n_i%7D)$

$n_0%3D%5Cfrac%7BN_D%2B(N_D%5E2%2B4n_i%5E2)%5E%5Cfrac%7B1%7D%7B2%7D%7D%7B2%7D%3D%5Cfrac%7BN_D%7D%7B2%7D%5B1%2B(1%2B%5Cfrac%7B4n_i%5E2%7D%7BN_D%5E2%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5D$

$p_0%3D%5Cfrac%7Bn_i%5E2%7D%7Bn_0%7D%3D%5Cfrac%7B2n_i%5E2%7D%7BN_D%7D%5B1%2B(1%2B%5Cfrac%7B4n_i%5E2%7D%7BN_D%5E2%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5D%5E%7B-1%7D$

六、p型半导体的载流子浓度

低温弱电离区：（电中性方程 $p_0%3Dp_A%5E-$ ）

$E_F%3D%5Cfrac%7BE_v%2BE_A%7D%7B2%7D-(%5Cfrac%7Bk_0T%7D%7B2%7D)%5Cln(%5Cfrac%7BN_A%7D%7B4N_v%7D)$

$p_0%3D(%5Cfrac%7BN_AN_v%7D%7B4%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5Cexp(-%5Cfrac%7BE_A-E_v%7D%7B2k_0T%7D)%3D(%5Cfrac%7BN_AN_v%7D%7B4%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5Cexp(-%5Cfrac%7B%5CDelta%20E_A%7D%7B2k_0T%7D)$

强电离区：（电中性方程 $p_A%5E-%3DN_A$ ）

$E_F%3DE_v-%7Bk_0T%7D%5Cln(%5Cfrac%7BN_A%7D%7BN_v%7D)$

$p_A%3DD_%2BN_A$

$D_%2B%3D(%5Cfrac%7B4N_A%7D%7BN_v%7D)%5Cexp(%5Cfrac%7B%5CDelta%20E_A%7D%7Bk_0T%7D)$

少子 $n_%7Bp0%7D%3D%5Cfrac%7Bn_i%5E2%7D%7BN_A%7D$

过渡区：（电中性方程 $p_0%3DN_A%2Bn_0$ ）

$E_F%3DE_i-k_0T%20%5Ctext%7Barsh%7D(%5Cfrac%7BN_A%7D%7B2n_i%7D)$

$p_0%3D%5Cfrac%7BN_A%2B(N_A%5E2%2B4n_i%5E2)%5E%5Cfrac%7B1%7D%7B2%7D%7D%7B2%7D%3D%5Cfrac%7BN_A%7D%7B2%7D%5B1%2B(1%2B%5Cfrac%7B4n_i%5E2%7D%7BN_A%5E2%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5D$

$n_0%3D%5Cfrac%7Bn_i%5E2%7D%7Bp_0%7D%3D%5Cfrac%7B2n_i%5E2%7D%7BN_A%7D%5B1%2B(1%2B%5Cfrac%7B4n_i%5E2%7D%7BN_A%5E2%7D)%5E%5Cfrac%7B1%7D%7B2%7D%5D%5E%7B-1%7D$