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6 Nonequilibrium Excess Carriers in Semiconductors

6.1 Carrier Generation and Recombination

  • generation is the process whereby electrons and holes are created
  • recombination is the process whereby electrons and holes are annihilated.

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6.1.1 The Semiconductor in Equilibrium

  • \(G_{n0},G_{p0}\) thermal-generation rates of electrons and holes
\[ G_{n0}=G_{p0} \]
  • \(R_{n0},R_{p0}\) recombination rates of electrons and holes
\[ R_{n0}=R_{p0} \]

对于热平衡状态而言

\[ G_{n0}=G_{p0}=R_{n0}=R_{p0} \]

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  • \(g_n',g_p'\) 过剩电子/空穴产生率，一定有\(g_n'=g_p'\)

电子跃迁到导带以后，空穴浓度会高于热平衡时的值

  • \(\delta n,\delta p\) 过剩电子空穴浓度
\[ \begin{align*} n&=n_0+\delta n\\ p&=p_0+\delta p \end{align*} \]

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电子落回价带

\[ R_n'=R_p' \]

直接的带间复合是自发的行为。因此相对于时间是一个常数，且与电子和空穴的浓度成比例

\[ \frac{dn(t)}{dt}=\alpha_r\left[n_i^2-n(t)p(t)\right] \]
  • \(n(t)=n_0+\delta n(t)\)
  • \(p(t)=p_0+\delta p(t)\)
  • \(\alpha_rn_i^2\)热平衡状态的生成率，\(n_i^2=n_0p_0\)
\[ \frac{d(\delta n(t))}{dt}=-\alpha_r\delta n(t)[(n_0+p_0)+\delta n(t)] \]

low-level injection 小注入条件

Low-level injection puts limits on the magnitude of the excess carrier concentration compared with the thermal-equilibrium carrier concentrations. In an extrinsic n-type material, we generally have $n_0>>p_0 $and, in an extrinsic p-type material, we generally have \(p_0>>n_0\). Low-level injection means that the excess carrier concentration is much less than the thermal-equilibrium majority carrier concentration. Conversely, high-level injection occurs when the excess carrier concentration becomes comparable to or greater than the thermal-equilibrium majority carrier concentrations.

考虑 p type semi (\(\delta n(t)<<p_0\)，\(p_0>>n_0\))

\[ \frac{d(\delta n(t))}{dt}=-\alpha_r\delta n(t)p_0 \]
\[ \delta n(t)=\delta n(0)e^{-\alpha_rp_0t}=\delta n(0)e^{-t/\tau_{n0}} \]

过剩少数载流子衰减的情况

  • \(\tau_{n0}\)过剩少数载流子寿命 \(\tau_{n0}=(\alpha_rp_0)^{-1}\)
\[ R_n'=\frac{-d(\delta n(t))}{dt}=+\alpha_r p_0\delta n(t)=\frac{\delta n(t)}{\tau_{n0}}=R_p' \]

同理 对于n type

\[ R_n'=\frac{\delta n(t)}{\tau_{p0}}=R_p' \]
  • \(\tau_{p0}=(\alpha_rn_0)^{-1}\)

6.2 Characteristics of excess carriers

ambipolar transport

the excess electrons and holes do not move independently of each other, but they diffuse and drift with the same effective diffusion coeffi cient and with the same effective mobility.

6.2.1 Continuity Equations

\[ F_{px}^+(x+dx)=F_{px}^+(x)+\frac{\partial F_{px}^+}{\partial x}\cdot x \]

以下仅讨论一维下的情况

\[ \frac{\partial p}{\partial t}dxdydz=[F^+_{px}(x)-F^+_{px}(x+dx)]dydz=-\frac{\partial F_{px}^+}{\partial x}dx dydz \]
\[ \frac{\partial p}{\partial t}dxdydz=-\frac{\partial F_p^+}{\partial x}dxdydz+g_pdxdydz-\frac{p}{\tau_{pt}}dxdydz \]
空穴的连续性方程
\[ \frac{\partial p}{\partial t}=-\frac{\partial F_p^+}{\partial x}+g_p-\frac{p}{\tau_{pt}} \]

同理：

\[ \frac{\partial n}{\partial t}=-\frac{\partial F_n^-}{\partial x}+g_n-\frac{n}{\tau_{nt}} \]

6.2.2 Time-Dependent Diffusion Equations

\[ J_p=e\mu_ppE-eD_p\frac{\partial P}{\partial x} \]
\[ J_n=e\mu_nnE+eD_n\frac{\partial n}{\partial x} \]

同除e得到\(F_p^+\)和\(F_n^-\)

\[ \frac{\partial p}{\partial t}=-\mu_p\frac{\partial(pE)}{\partial x}+D_p\frac{\partial^2p}{\partial x^2}+g_p-\frac{p}{\tau_{pt}} \]
\[ \frac{\partial n}{\partial t}=\mu_n\frac{\partial(nE)}{\partial x}+D_n\frac{\partial^2p}{\partial x^2}+g_n-\frac{n}{\tau_{nt}} \]

总之一通操作后得到：

\[ \frac{\partial (\delta p)}{\partial t}=-\mu_p(E\frac{\partial(\delta p)}{\partial x}+p\frac{\partial E}{\partial x})+D_p\frac{\partial^2p}{\partial x^2}+g_p-\frac{p}{\tau_{pt}} \]
\[ \frac{\partial (\delta n)}{\partial t}=\mu_n(E\frac{\partial(\delta n)}{\partial x}+n\frac{\partial E}{\partial x})+D_n\frac{\partial^2n}{\partial x^2}+g_n-\frac{n}{\tau_{nt}} \]

6.3 Ambipolar Transport

\[ E=E_{app}+E_{int} \]

ambipolar transport equation

\[ D'\frac{\partial^2(\delta n)}{\partial x^2}+\mu'E\frac{\partial(\delta n)}{\delta x}+g-R=\frac{\partial (\delta n)}{\partial t} \]
  • \(D'=\frac{\mu_nnD_p+\mu_ppD_n}{\mu_nn+\mu_pp}\)，根据爱因斯坦关系式，\(D'=\frac{D_nD_p(n+p)}{D_nn+D_pp}\)
  • \(\mu'=\frac{\mu_n\mu_p(p-n)}{\mu_nn+\mu_pp}\)

小注入模型

  • n type: \(D'=D_p\), \(\mu'=-\mu_p\), \(\tau_{pt}=\tau_p\)
  • p type: \(D'=D_n\), \(\mu'=\mu_n\), \(\tau_{nt}=\tau_n\)

原式中：

\[ g-R=g_n-R_n=(G_{n0}+g_n')-(R_{n0}+R_n') \]
\[ \begin{gather*} \because G_{n0}=R_{n0}\\ \therefore g-R=g'_n-R'_n=g_n'-\frac{\delta n}{\tau_n} \end{gather*} \]

as to holes

\[ g-R=g_p'-R_p'=g_p'-\frac{\delta p}{\tau_p} \]

过剩电子的产生率必须等于过剩空穴的产生率

\[ g_n'=g_p'=g' \]
小注入p型半导体双极运输方程
\[ D_n\frac{\partial^2(\delta n)}{\partial x^2}+\mu_nE\frac{\partial (\delta n)}{\partial x}+g'-\frac{\delta n}{\tau_{n0}}=\frac{\partial(\delta n)}{\partial t} \]

同理，小注入n型半导体双极运输方程

\[ D_p\frac{\partial^2(\delta p)}{\partial x^2}-\mu_pE\frac{\partial (\delta p)}{\partial x}+g'-\frac{\delta p}{\tau_{p0}}=\frac{\partial(\delta p)}{\partial t} \]

6.4 Quasi-Fermi Energy Level

\[ n_0+\delta n=n_i\exp(\frac{E_{Fn}-E_{Fi}}{kT}) \]
\[ p_0+\delta p=n_i\exp(\frac{E_{Fi}-E_{Fp}}{kT}) \]