Let's show how the **base transport factor** \(\alpha_r\) can be simplified to: \[ \alpha_r = 1 - \frac{W^2}{2L_p^2} \] where: - \(W\) = base width (width of the neutral base region) - \(L_p\) = diffusion length of holes in the base (for an NPN, holes are minority carriers in the base) --- ## **Step 1: Definition of Base Transport Factor** The **base transport factor** (\(\alpha_r\)) is defined as the fraction of injected minority carriers (holes for NPN, electrons for PNP) that diffuse across the base and reach the collector-base junction **without recombining** in the base. For a uniformly doped base, and neglecting surface recombination, the transport factor is given by: \[ \alpha_r = \frac{\text{number of carriers reaching the collector}}{\text{number injected from emitter}} \] --- ## **Step 2: Carrier Diffusion Equation** The minority carrier concentration (for holes in the NPN base) satisfies the diffusion equation: \[ \frac{d^2p(x)}{dx^2} = \frac{p(x)}{L_p^2} \] General solution (assuming steady state, low-level injection): \[ p(x) = A \sinh\left(\frac{W - x}{L_p}\right) + B \sinh\left(\frac{x}{L_p}\right) \] But we usually set up boundary conditions: - At \(x=0\) (emitter-base junction), \(p(0) = p_{E}\) - At \(x=W\) (collector-base junction), \(p(W) = 0\) (assuming perfect collector, all holes swept away) The **hole current density** at the collector-base edge (\(x = W\)) is: \[ J_p(W) = -q D_p \left.\frac{dp}{dx}\right|_{x=W} \] --- ## **Step 3: Ratio of Collector to Emitter Current** The **fraction of holes injected from the emitter that reach the collector** (i.e., transport factor) is: \[ \alpha_r = \frac{J_p(W)}{J_p(0)} \] From the solution to the diffusion equation, the **current density at \(x\)** is: \[ J_p(x) = -q D_p \frac{dp}{dx} \] \[ \frac{dp}{dx} = \frac{p_{E}}{\sinh(W/L_p)} \cdot \frac{1}{L_p} \cosh\left(\frac{W-x}{L_p}\right) \] At \(x=0\): \[ J_p(0) = -q D_p \frac{p_{E}}{L_p} \frac{\cosh(W/L_p)}{\sinh(W/L_p)} \] At \(x=W\): \[ J_p(W) = -q D_p \frac{p_{E}}{L_p} \frac{1}{\sinh(W/L_p)} \cdot \cosh(0) = -q D_p \frac{p_{E}}{L_p} \frac{1}{\sinh(W/L_p)} \] But, the current at the collector side is **in the opposite direction** (positive value), so we take the absolute value. The **ratio** is: \[ \alpha_r = \frac{J_p(W)}{J_p(0)} = \frac{\cosh(0)}{\cosh(W/L_p)} = \frac{1}{\cosh(W/L_p)} \] Actually, more commonly, the **emitter current** is: \[ J_{pE} = \frac{q D_p p_{E}}{L_p} \frac{1}{\sinh(W/L_p)} \] and the **collector current** is: \[ J_{pC} = \frac{q D_p p_{E}}{L_p} \frac{\exp(W/L_p)}{\sinh(W/L_p)} \] But let's use the more general form: \[ \alpha_r = \frac{J_p(W)}{J_p(0)} = \frac{\cosh(W/L_p) - 1}{\sinh(W/L_p)} \div \frac{\cosh(W/L_p)}{\sinh(W/L_p)} = \frac{\cosh(W/L_p) - 1}{\cosh(W/L_p)} \] But, **according to standard textbooks (e.g., S. M. Sze, *Physics of Semiconductor Devices*, equation 6.9)**, the transport factor can be approximated by: \[ \alpha_r = \text{sech}(W/L_p) = \frac{1}{\cosh(W/L_p)} \] --- ## **Step 4: Taylor Series Expansion for Small \(W/L_p\)** When the base is **narrow compared to the diffusion length** (\(W \ll L_p\)), expand \(\cosh(W/L_p)\): \[ \cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots \] Let \(x = W/L_p\): \[ \cosh(W/L_p) \approx 1 + \frac{1}{2} \left(\frac{W}{L_p}\right)^2 \] So: \[ \alpha_r = \frac{1}{\cosh(W/L_p)} \approx \frac{1}{1 + \frac{1}{2}\left(\frac{W}{L_p}\right)^2} \] Now for small \(x\), \(1/(1+y) \approx 1 - y\): \[ \alpha_r \approx 1 - \frac{1}{2}\left(\frac{W}{L_p}\right)^2 \] Or, \[ \alpha_r \approx 1 - \frac{W^2}{2L_p^2} \] --- ## **Step 5: Conclusion** Thus, the base transport factor for a narrow base (\(W \ll L_p\)) is: \[ \boxed{ \alpha_r \approx 1 - \frac{W^2}{2L_p^2} } \] --- ### **References** - S. M. Sze, *Physics of Semiconductor Devices*, 2nd Edition, Eq. 6.9 and discussion - Streetman & Banerjee, *Solid State Electronic Devices*, Eq. 7.19 --- **Summary:** By expanding the hyperbolic cosine for small \(W/L_p\), the base transport factor simplifies to \(\alpha_r \approx 1 - \frac{W^2}{2L_p^2}\) for a narrow base.