Concise text companions to the PNG study sheets. Key relations are typeset with MathJax for quick review.
Based on summary-1-1.png and summary-1-2.png.
Based on summary-2-1.png and summary-2-2.png.
Based on summary-3-1.png and summary-3-2.png.
Based on summary-4-1.png and summary-4-2.png.
Based on table-3-1-1.png and table-3-1-2.png.
| Quantity | Relationship / Formula | Key constants and notes |
|---|---|---|
| Intrinsic carrier concentration \(n_i\) | \(n_i = B T^{3/2} e^{-E_g / (2 k T)}\) | \(B = 7.3 \times 10^{15}\, \text{cm}^{-3}\text{K}^{-3/2}\), \(E_g = 1.12\, \text{eV}\), \(k = 8.62 \times 10^{-5}\, \text{eV/K}\); at \(T = 300\,\text{K}\), \(n_i \approx 1.5 \times 10^{10}\, \text{cm}^{-3}\). |
| Diffusion current densities | \(J_p = -q D_p \frac{dp}{dx},\; J_n = q D_n \frac{dn}{dx}\) | \(q = 1.60 \times 10^{-19}\,\text{C}\); \(D_p = 12\, \text{cm}^2/\text{s}\), \(D_n = 34\, \text{cm}^2/\text{s}\) for intrinsic Si at 300 K. |
| Drift current density | \(J_{\text{drift}} = q (\mu_p p + \mu_n n) E\) | Mobilities \(\mu_p \approx 480\) and \(\mu_n \approx 1350\, \text{cm}^2/(\text{V}\cdot\text{s})\); both decrease as doping increases. |
| Resistivity \(\rho\) | \(\rho = 1 / [ q (\mu_p p + \mu_n n) ]\) | Higher carrier density or mobility reduces resistivity. |
| Mobility-diffusivity link | \(\frac{D_p}{\mu_p} = \frac{D_n}{\mu_n} = V_T\) | Thermal voltage \(V_T = kT/q \approx 25.9\,\text{mV}\) at 300 K. |
| n-type carrier concentrations | \(n_{n0} \approx N_D,\; p_{n0} = n_i^2 / N_D\) | Majority carriers follow dopant density; minority carriers are suppressed by \(n_i^2\) scaling. |
| p-type carrier concentrations | \(p_{p0} \approx N_A,\; n_{p0} = n_i^2 / N_A\) | Analogous relationships for p-type regions. |
| Built-in junction voltage | \(V_0 = V_T \ln \left( \frac{N_A N_D}{n_i^2} \right)\) | Increases with heavier doping on either side. |
| Forward diffusion currents | \(I = I_p + I_n\) with \(I_p = A q n_i^2 \frac{D_p}{L_p N_D} (e^{V/V_T} - 1)\), \(I_n = A q n_i^2 \frac{D_n}{L_n N_A} (e^{V/V_T} - 1)\) | Device area \(A\); diffusion lengths \(L_p, L_n = 1\,\mu\text{m} \text{ to } 100\,\mu\text{m}\). |
| Saturation current | \(I_S = A q n_i^2 \left( \frac{D_p}{L_p N_D} + \frac{D_n}{L_n N_A} \right)\) | Sets the scale for pn-junction forward conduction. |
| Diode I-V law | \(I = I_S (e^{V/V_T} - 1)\) | Valid for moderate forward biases before high-level injection. |
| Minority-carrier lifetimes | \(\tau_p = L_p^2 / D_p,\; \tau_n = L_n^2 / D_n\) | Typical ranges \(\tau_p, \tau_n = 1\,\text{ns}\) to \(10^4\,\text{ns}\). |
| Minority charge storage | \(Q_p = \tau_p I_p,\; Q_n = \tau_n I_n,\; Q = Q_p + Q_n = \tau_T I\) | Total stored charge scales with current and lifetimes. |
| Zero-bias depletion capacitance | \(C_{j0} = A \sqrt{ \frac{\varepsilon_s q}{2} \frac{N_A N_D}{N_A + N_D} \frac{1}{V_0} }\) | \(\varepsilon_s\) is the silicon permittivity; area \(A\) scales capacitance. |
| Junction capacitance | \(C_j = C_{j0} (1 + V_R/V_0)^{-m}\) | Grading coefficient \(m\) ranges from \(1/3\) to \(1/2\); reverse bias reduces capacitance. |
| Diffusion capacitance | \(C_d = (\tau_T / V_T) I\) | Dominant under forward bias when stored charge is large. |
Based on table-1-1.png.
Based on table-3-2.png.
| Metric | Low-Pass (LP) | High-Pass (HP) |
|---|---|---|
| Transfer function \(T(s)\) | \(\displaystyle \frac{K}{1 + s/\omega_0}\) | \(\displaystyle \frac{K s}{s + \omega_0}\) |
| Frequency-domain \(T(j\omega)\) | \(\displaystyle \frac{K}{1 + j(\omega/\omega_0)}\) | \(\displaystyle \frac{K j(\omega/\omega_0)}{1 + j(\omega/\omega_0)}\) |
| Magnitude \(|T(j\omega)|\) | \(\displaystyle \frac{|K|}{\sqrt{1 + (\omega/\omega_0)^2}}\) | \(\displaystyle \frac{|K|}{\sqrt{1 + (\omega_0/\omega)^2}}\) |
| Phase \(\angle T(j\omega)\) | \(-\tan^{-1}(\omega/\omega_0)\) | \(+\tan^{-1}(\omega_0/\omega)\) |
| Transmission at \(\omega = 0\) | \(K\) | 0 |
| Transmission at \(\omega = \infty\) | 0 | \(K\) |
| 3 dB frequency | \(\omega_0 = 1/\tau\) where \(\tau = CR\) or \(L/R\) | |
Based on table-4-1.png.