[반도체공학] 7.2. pn Admittance - Forward Bias

2. pn Admittance - Forward Bias

1. Forward-Bias Admittance

• $C=C_J+C_D$ : Depletion Capacitance + Diffusion Capacitance
  • $C_J$ : Depletion Width가 감소하면 $C_J$는 증가

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• Forward Bias
  • Forward Bias가 인가되면 Minority Carrier 증가 : Minority Carrier의 응답은 Majority만큼 빠르지 않기 때문에, 전압 인가에 따른 위상차 존재
  • Forward Bias 인가 시 Minority Carrier에 의한 커패시턴스 성분 생성 (Diffusion Capacitance)
  • 위상차 성분 일부는 Conductance에 영향
  • $i(t)=(G+wC^{out})v_0coswt-wC^{in}v_0sinwt$

• $p^+n$ (p농도 > n농도) 접합의 경우
  • 고주파 Bias 인가 시 도핑 농도가 Bias를 중심으로 진동하는(위상차가 나타나는) 형태로 나타나게 됨

2. Admittance Relationships

• small signal에 의한 Diffusion Equation
  • $\frac{\partial\Delta p_n(x,t)}{\partial t}=D_P\frac{\partial^2\Delta p_n(x,t)}{\partial x^2}-\frac{\Delta p_n(x,t)}{\tau_p}$
• $\Delta p_n(x,t)=\overline{\Delta p_n(x)}+\tilde{p_n}(x,w)e^{jwt}$ 가정
  • $\overline{\Delta p_n(x)}$ : 도핑 농도 Bias
  • $\tilde{p_n}(x,w)e^{jwt}$ : AC 전압에 의한 진폭

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• 가정한 해를 대입

  • $\frac{\partial\Delta p_n(x,t)}{\partial t}=jw\tilde{p_n}(x,w)e^{jwt}$
    $=D_P\frac{d^2\overline{\Delta p_n(x)}}{d x^2}+D_P\frac{d^2 \tilde{p_n}(x,w)e^{jwt}}{d x^2}-\frac{\overline{\Delta p_n(x)}}{\tau_p}-\frac{\tilde{p_n}(x,w)e^{jwt}}{\tau_p}$

  • DC : $D_P\frac{d^2\overline{\Delta p_n(x)}}{d x^2}-\frac{\overline{\Delta p_n(x)}}{\tau_p}=0$

  • AC : $jw\tilde{p_n}(x,w) = D_P\frac{d^2 \tilde{p_n}(x,w)}{d x^2}-\frac{\tilde{p_n}(x,w)}{\tau_p}$
    $\rarr D_P\frac{d^2 \tilde{p_n}(x,w)}{d x^2}-(jw+\frac{1}{\tau_p})\tilde{p_n}(x,w)=0$

    • $D_P\frac{d^2 \tilde{p_n}(x,w)}{d x^2}-\frac{\tilde{p_n}(x,w)}{\tau_p/(1+jw\tau_p)}$

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• Boundary Condition
  • $\Delta p_n(x=\infty)=0$
  • $\Delta p_n(x=x_n)$ : Depletion Edge
    $=p_{n0}[e^{(V_A+v_a)/V_T}-1]\simeq p_{n0}[e^{V_A/V_T}(1+\frac{v_a}{V_T})-1]$
    $=\overline{\Delta p_n}(x=x_n)+\tilde{p_n}(x=x_n,w)$
  • $\overline{\Delta p_n}(x=x_n)=p_{n0}[e^{V_A/V_T}-1]\simeq\frac{n_i^2}{N_D}[e^{V_A/V_T}-1]$
  • $\tilde{p_n}(x=x_n,w)\simeq\frac{n_i^2}{N_D}e^{V_A/V_T}\frac{v_a(w)}{V_T}$

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• Solution : Diffusion Current
  • DC : $\overline{\Delta p_n}(x)=p_{n0}[e^{V_A/V_T}-1]e^{-x/\sqrt{(D_P\tau_P)}}$
    • $I_{diff}=\frac{qAD_P}{L_P}p_{n0}[e^{V_A/V_T}-1]$
      $=qA\sqrt{\frac{D_P}{\tau_P}}p_{n0}[e^{V_A/V_T}-1]=I_0[e^{V_A/V_T}-1]$
  • AC : $\tilde{p_n}(x=x_n,w)=p_{n0}e^{V_A/V_T}\frac{v_a(w)}{V_T}e^{-x/\sqrt{D_P\tau_P/(1+jw\tau_p)}}$
    • $I_{diff}(w)=qA\sqrt{\frac{D_P(1+jw\tau_P)}{\tau_P}}p_{n0}e^{V_A/V_T}[V_a(w)/V_T]$
      $=I_0\sqrt{1+jw\tau_p}e^{V_A/V_T}\frac{V_A(w)}{V_T}$
      $=G_0\sqrt{1+jw\tau_p}V_A(w)$
    • $G_0=\frac{I_0}{V_T}e^{V_A/V_T}$

3. Forward-Bias Diffusion Admittance

• Small-signal Conducance : $Y_D=G_0\sqrt{1+jw\tau_p}$
  • $\sqrt{1+jw\tau_p}=x+jy$일 때,
    $x^2-y^2=1,\ 2xy=w\tau_p$
  • $x=[\frac{1+\sqrt{1+w^2\tau_p^2}}{2}]^{1/2},\ y=[\frac{\sqrt{1+w^2\tau_p^2}-1}{2}]^{1/2}$
• Diffusion Admittance : $Y_D=G_D+jwC_D$
  • $G_D=G_0[\frac{1+\sqrt{1+w^2\tau_p^2}}{2}]^{1/2}=\frac{G_0}{\sqrt{2}}[1+\sqrt{1+w^2\tau_p^2}]^{1/2}$
  • $C_D=\frac{C_0}{w}[\frac{\sqrt{w^2\tau_p^2-1}}{2}]^{1/2}=\frac{G_0}{\sqrt{2}w}[\sqrt{1+w^2\tau_p^2}-1]^{1/2}$

• 주파수가 일정 이하 ($w\tau_p<1$)일 때는 G, C가 일정하지만, 증가함에 따라 Diffusion에 의한 G는 증가, C는 감소