# 密度梯度理论简介——半导体器件仿真

2019年 11月 27日

### 静电和电荷载流子守恒

(1)

\nabla \cdot (\epsilon \mathbf{E})=\rho \qquad \mathbf{E}= – \nabla V

(2)

\rho=q(p-n+N_d^{+}-N_a^{-})

(3)

\frac{\partial n} {\partial t}&=&\frac{-\nabla\cdot\mathbf{J}_n}{-q}-Re_n+Ge_n
\frac{\partial p}{\partial t}&=&\frac{-\nabla\cdot\mathbf{J}_p}{+q}-Re_p+Ge_p

(4)

\mathbf{J}n &=& q~n~\mu_n \nabla E{fn} + q~n ((E_c-E_{fn})\mu_n + Q_n) \nabla T /T
\mathbf{J}p &=& -q~p~\mu_p \nabla E{fp} – q~p ((E_v-E_{fp})\mu_p + Q_p) \nabla T /T

(5)

E_c = -V-\chi \qquad E_v = E_c – E_g

### 漂移扩散和密度梯度理论的状态方程

(6)

n&=&N_c F_{1/2}(\frac{E_{fn}-E_c}{k_B T/q})
p&=&N_v F_{1/2}(\frac{-E_{fp}+E_v}{k_B T/q})

(7)

n&=&N_c F_{1/2}(\frac{E_{fn}-E_c+V^{DG}_n}{k_B T/q})
p&=&N_v F_{1/2}(\frac{-E_{fp}+E_v+V^{DG}_p}{k_B T/q})

(8)

\nabla \cdot \left( \mathbf{b}_n \nabla \sqrt{n} \right) &\equiv& \frac{\sqrt{n}}{2} V^{DG}_n
\nabla \cdot \left( \mathbf{b}_p \nabla \sqrt{p} \right) &\equiv& \frac{\sqrt{p}}{2} V^{DG}_p

(9)

\mathbf{b}_n &=&\frac{\hbar^2}{12 q} \left[\mathbf{m}_n\right]^{-1}
\mathbf{b}_p &=&\frac{\hbar^2}{12 q} \left[\mathbf{m}_p\right]^{-1}

### 求解策略

(10)

n&\equiv& exp(\frac{\phi_n}{k_B T/q})
p&\equiv& exp(\frac{\phi_p}{k_B T/q})

(11)

V^{DG}_n &=& + E_c – E_{fn} + k_B T/q \left[ log(F_{1/2})\right]^{-1} \left( \frac{\phi_n} {k_B T/q} – log(N_c) \right)
V^{DG}_p &=& -E_v + E_{fp} + k_B T/q \left[ log(F_{1/2})\right]^{-1} \left( \frac{\phi_p}{k_B T/q}- log(N_v) \right)

### 重组率

(12)

r_e &=& n~C_n~N_t (1-f_t) (1-e^{\frac{E_{ft}-E_{fn}}{k_B T/q}})
r_h &=& p~C_p~N_t~f_t (1-e^{\frac{E_{fp}-E_{ft}}{k_B T/q}})

(13)

f_t = \frac{1}{1+\frac{1} {g_D}~e^{\frac{E_t-E_{ft}}{k_B T/q}}} \qquad \mbox{或等同} \qquad \frac{1-f_t}{f_t} = \frac{1}{g_D}~e^{\frac{E_t-E_{ft}}{k_B T/q}}

(14)

R_n=R_p=\frac{n~p-n_{eq}^{DG}~p_{eq}^{DG}}{\tau_p(n+n_1)+\tau_n(p+p1)}

(15)

(16)

n_1 &\equiv& n~e^{\frac{E_{t}-E_{fn}}{k_B T/q}}
p_1 &\equiv& p~e^{\frac{E_{fp}-E_{t}}{k_B T/q}}

### Slotboom变量的边界条件

(17)

\mathbf{n} \cdot (\mathbf{b}_n \nabla \sqrt{n}) = 0 \qquad \mathbf{n} \cdot (\mathbf{b}_p \nabla \sqrt{p}) = 0

(18)

\mathbf{n}\cdot (\mathbf{b}_n \nabla \sqrt{n}) = -\frac{b_{n,ox}}{d_n}\sqrt{n}

(19)

### 参考文献

1. M.G. Ancona, “Density-gradient theory: a macroscopic approach to quantum confinement and tunneling in semiconductor devices,” Journal of Computational Electronics, vol. 10, p. 65, 2011.
2. M.G. Ancona, Z. Yu, R.W. Dutton, P.J. Vande Voorde, M. Cao, and D. Vook, “Density-Gradient Analysis of MOS Tunneling,” IEEE Transactions On Electron Devices, p. 2310, Vol. 47, No. 12, December 2000.
3. M.G. Ancona, D. Yergeau, Z. Yu, and B.A. Biegel, “On Ohmic Boundary Conditions for Density-Gradient Theory”, Journal of Computational Electronics 1: 103–107, 2002.
4. S. Jin, Y.J. Park, and H.S. Min, “Simulation of Quantum Effects in the Nano-scale Semiconductor Device,” Journal of Semiconductor Technology and Science, vol. 4, no. 1, p. 32, 2004.