BJT Physics: Injection Efficiency, Base Transport & the Gummel Plot

Inside the bipolar transistor. Learn NPN/PNP structures and doping profiles, transistor action (emitter injection, base diffusion, collector collection), minority-carrier distributions in each region, emitter injection efficiency γ, base transport factor α_T, current gain α and β from physical parameters, the role of emitter bandgap narrowing, base-width modulation and the Early effect, collector multiplication, high-injection and the Kirk effect, the Gummel plot and β vs current, and temperature dependence — with Python computing γ, α_T, β, I_S, Early voltage and Kirk current from doping and geometry.

A bipolar transistor is two PN junctions sharing a very thin base. The forward-biased emitter–base junction injects minority carriers into the base; because the base is much thinner than a diffusion length, almost all of them reach the reverse-biased collector junction and are swept out. A small base current controls a large collector current.

Every BJT parameter — gain, Early voltage, speed, maximum current — follows from doping levels and dimensions, which is what makes device design possible.

An analogy

Think of a waterfall with a narrow ledge (base): water (electrons) pushed over the first step (emitter junction) crosses the ledge so quickly that only a few drops soak in (recombination/base current) before plunging down the waterfall (collector junction).

Transistor action

In forward-active mode the emitter–base junction is forward biased and the collector–base junction reverse biased. The emitter injects electrons (NPN) into the base; they diffuse across and are collected. Since I_C depends on V_BE exponentially and hardly on V_CB, the collector behaves like a current source controlled by the base–emitter voltage.

Designing for gain

Levers on β:

Early and Kirk effects

Increasing V_CB widens the collector depletion region, narrowing the neutral base: I_C rises with V_CE (finite output resistance r_o = V_A/I_C). Heavier base doping or wider base raises V_A but lowers β or speed. At high current density, injected electrons neutralise the collector depletion charge; when their density approaches N_C, the effective base extends into the collector (base push-out/Kirk effect), sharply increasing transit time and reducing β.

Gummel plot and breakdown

Plotting log I_C and log I_B versus V_BE (at V_CB = 0) reveals ideality, base-recombination current at low bias (n ≈ 2), the ideal β region and high-injection roll-off. Breakdown: BV_CBO (junction avalanche) is higher than BV_CEO ≈ BV_CBO/β^(1/n) because avalanche-generated holes act as base current and are amplified. Punch-through occurs if the base depletes completely.

Formulas

Worked example: Base transport

Given: W_B 0.5 µm, L_B 20 µm

Find: α_T.

Answer: 0.99969

Worked example: Gain from α

Given: α = 0.995

Find: β.

Answer: 199

Worked example: Output resistance

Given: V_A 60 V, I_C 1 mA

Find: r_o.

Answer: 60 kΩ

NPN carrier flow

   Emitter n⁺ (10¹⁹)   │ Base p (10¹⁷, W_B 0.1 µm) │  Collector n (10¹⁶)
   e⁻ ═══ injected ════►│══ diffusion ════════════════►│═══ swept by field ══► I_C
   ◄── holes back-injected (I_B1: limits γ)
                         │  recombination (I_B2: limits α_T)
   minority profile in base: n(0) = n_p0 e^(V_BE/V_T) → ~0 at collector edge (linear)
   I_E = I_C + I_B

Collector current is the electron diffusion current across the base, set by the slope of the (nearly linear) minority-carrier profile. Base current has two parts: holes back-injected into the emitter and electrons recombining in the base. Heavy emitter doping and a thin, lightly doped base maximise β.

Gain, saturation current, Early voltage and Kirk current

import math
# BJT physics: emitter injection efficiency, base transport, beta from doping, Early voltage, Kirk current
q, eps, VT, ni = 1.602e-19, 11.7 * 8.854e-14, 0.02585, 9.65e9
NE, NB, NC = 1e19, 1e17, 1e16             # emitter, base, collector doping
WE, WB = 0.2e-4, 0.1e-4                    # neutral widths (cm)
DE, DB = 1.5, 20.0                         # hole diff. in emitter, electron diff. in base
Ln = math.sqrt(DB * 1e-6)                  # base electron lifetime 1 us
gamma = 1 / (1 + (DE * NB * WB) / (DB * NE * WE))
alphaT = 1 - WB**2 / (2 * Ln**2)
alpha = gamma * alphaT
print(f"injection efficiency gamma = {gamma:.5f}, base transport alphaT = {alphaT:.7f}")
print(f"alpha = {alpha:.5f} -> beta = alpha/(1-alpha) = {alpha/(1-alpha):.0f} (simple estimate DB*NE*WE/(DE*NB*WB) = {DB*NE*WE/(DE*NB*WB):.0f})")
dEg = 0.09                                 # bandgap narrowing in the 1e19 emitter raises its ni^2 by exp(dEg/VT)
print(f"with emitter bandgap narrowing ({dEg*1e3:.0f} meV): beta ~ {DB*NE*WE/(DE*NB*WB)/math.exp(dEg/VT):.0f} (realistic range)")
A = 1e-6                                   # 10 um x 10 um emitter
Is = q * A * ni**2 * DB / (NB * WB)
print(f"IS = {Is:.2e} A -> IC at VBE 0.7 V = {Is*math.exp(0.7/VT)*1e3:.2f} mA")
Cjc_per_area = eps / math.sqrt(2 * eps * (0.8 + 5) / q * (1 / NB + 1 / NC))
VA = q * NB * WB / Cjc_per_area
print(f"Early voltage ~ qNB*WB/Cjc = {VA:.0f} V")
Jk = q * NC * 1e7
print(f"Kirk effect onset J ~ q*NC*vsat = {Jk:.0f} A/cm^2 -> IC ~ {Jk*A*1e3:.1f} mA for this emitter")
# injection efficiency gamma = 0.99963, base transport alphaT = 0.9999975
# alpha = 0.99962 -> beta = alpha/(1-alpha) = 2649 (simple estimate DB*NE*WE/(DE*NB*WB) = 2667)
# with emitter bandgap narrowing (90 meV): beta ~ 82 (realistic range)
# IS = 2.98e-16 A -> IC at VBE 0.7 V = 0.17 mA
# Early voltage ~ qNB*WB/Cjc = 14 V
# Kirk effect onset J ~ q*NC*vsat = 16020 A/cm^2 -> IC ~ 16.0 mA for this emitter

With a 0.1 µm base, base transport is essentially perfect (α_T = 0.9999975), so β is set by injection efficiency. The naive doping ratio predicts β ≈ 2700; bandgap narrowing in the heavily doped emitter raises its effective n_i² ~30× and brings β to a realistic ~80. The thin, moderately doped base gives a low Early voltage (~14 V) — a trade-off with speed. Above ~16 mA the Kirk effect pushes the base into the collector and β and f_T collapse.

Common mistakes

Applications

Key takeaways

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