The doping profile of a NPN BJT is given in the figure below. Estimate the emitter injection efficiency
𝛾
assuming the minority carrier diffusion coefficient for the base and emitter are the same.

[ans]

Question

The doping profile of a NPN BJT is given in the figure below. Estimate the emitter injection efficiency 
𝛾
 assuming the minority carrier diffusion coefficient for the base and emitter are the same.

[ans]

BlackTom AI · Accepted Answer

The problem asks to estimate the emitter injection efficiency γ for a NPN BJT, given a doping profile and the assumption that the minority carrier diffusion coefficients for base and emitter are the same. To reason through this, we consider how injection efficiency is typically defined and what factors influence it in a simplified, one-dimensional, abrupt- junction picture with equal diffusion coefficients.

First, recall that emitter injection efficiency is a measure of how effectively carriers injected from the emitter into the base contribute to forward-active operation, relative to the minority-carrier flow that would recombine or inject in the opposite direction. In many textbook treatments, the forward-emission injection efficiency (often denoted γ or α_E) depends on how heavily the emitter is doped compared with the base, and on the diffusion properties that govern minority-carrier transport.

Second, with the assumption that the minority carrier diffusion coefficients are equal in base and emitter, the transport aspects simplify: the ratio of minority-carrier concentrations and the effective diffusion lengths enter the expression for the currents but are constrained by the same diffusivity in both regions. In this limit, the dominant factor controlling γ becomes the relative doping levels across the emitter-base junction. A heavily doped emitter relative to the base tends to push more of the injection toward the base and reduces back-injection, increasing γ.

Third, to form a concrete estimate, one typically looks at the order-of-magnitude relationship between the emitter and base doping levels (N_D in the emitter and N_A in the base, for an NPN transistor). If the emitter is much more heavily doped than the base, the emitter injection efficiency increases; conversely, if the base doping is not negligible, the efficiency is reduced. The figure provided shows a very heavy emitter-side doping concentration (on the order of 10^21 cm^-3) compared with a much lower base doping (around 10^18 cm^-3), with the base region spanning a few micrometers, and a distinctly lighter collector-side doping in the collector region. This large emitter-to-base doping ratio is the key to achieving a high emitter injection efficiency.

Fourth, qualitative analysis of the geometry and doping suggests a high γ value, since the emitter is heavily doped while the base is relatively lightly doped, reducing holes or electrons returning into the emitter side and favoring forward injection into the base. When the base is relatively lightly doped (N_A around 10^18) and the emitter is substantially more doped (N_D around 10^21 in the emitter region for an NPN device, as implied by the color-coded bars in the figure), the ratio of forward to reverse injections becomes favorable, leading to γ well above 0.9 in typical dopant-distribution scenarios with equal diffusion coefficients.

Fifth, consider dimensional and material-physics caveats: the exact numerical value depends on the precise geometry (base width, emitter width), boundary conditions, and the exact diffusion lengths (which are influenced by mobilities and recombination lifetimes). Since the problem statement asks for an estimate and provides the assumption of equal diffusion coefficients, it is reasonable to treat the dominant contribution as the emitter/base doping contrast, yielding a high efficiency value in the vicinity of about 0.9 or higher. The reported numerical answer in the provided solution is 0.91, which aligns with this qualitative expectation given the strong emitter doping relative to the base.

Sixth, to summarize the logic in a compact way: with a very heavily doped emitter (ND) compared to the base (NA) and with identical minority-carrier diffusion coefficients in the two regions, the forward injection from emitter-to-base dominates, pushing γ toward a high value. Minor deviations from the ideal, such as a nonzero base doping or slight transport differences, would slightly reduce γ from its maximum, but the overall trend and the numerical estimate around 0.9–0.92 are consistent with the depicted doping profile, where the emitter is far more doped than the base and the diffusion properties are matched.

Note: If you’re solving similar problems, one useful approach is to write the expression for emitter injection efficiency γ in terms of the emitter and base minority-carrier concentrations and diffusion parameters, then substitute the relative dopant concentrations and set D_E ≈ D_B to simplify. The exact form of the expression may vary depending on the level of approximation (simplified one-dimensional model vs. full Ebers–Moll treatment), but the qualitative dependence on the emitter/base doping contrast remains robust.

EESM5200 (L1) Part I: Topic 13 - BJT Physical Structure and Switching

The doping profile of a NPN BJT is given in the figure below. Estimate the emitter injection efficiency 𝛾 assuming the minority carrier diffusion coefficient for the base and emitter are the same. [ans]

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