Question 7.3: Extend Eq. (7–20a) to include the effects of nonunity emitte...
Extend Eq. (7–20a) to include the effects of nonunity emitter injection efficiency (\gamma < 1). Derive Eq. (7–25) for g. Assume that the emitter region is long compared with an electron diffusion length.
I_{E} \simeq qA\frac{D_{p} }{L_{p} } \Delta p_{E}\coth \frac{W_{b} }{L_{p} } (7–20a)
\gamma =\left[1+\frac{L^{n}_{p}n_{n}\mu ^{p}_{n} }{L^{p}_{n}P_{p}\mu ^{n}_{p}} \tan h\frac{W_{b} }{L^{n}_{p} } \right]^{-1}\simeq \left[1+\frac{W_{b}n_{n}\mu ^{p}_{n} }{L^{p}_{n}P_{p}\mu ^{n}_{p}}\right]^{-1} (7–25)
Equation (7–20a) is actually I_{Ep}.
I_{En} =\frac{qAD^{p}_{n} }{L^{p}_{n} }n_{p} e^{qV_{EB}/kT}for V_{EB} \gg kT>q
Thus, the total emitter current is
I_{E} =I_{Ep} +I_{En} =qA\left[\frac{D^{n}_{p} }{L^{n}_{p} } p_{n}\coth \frac{W_{b} }{L^{n}_{p} } +\frac{D^{n}_{n} }{L^{p}_{n} } n_{p}\right] e^{qV_{EB}/kT}\gamma =\frac{I_{Ep} }{I_{E} } =\left[1+\frac{I_{En} }{I_{Ep} }\right]^{-1}=\left[1+\frac{\frac{D^{p}_{n} }{L^{p}_{n} } n_{p}}{\frac{D^{n}_{p} }{L^{n}_{p} }P_{n} }\tanh \frac{W_{b} }{L^{n}_{p} } \right] ^{-1}
Using \frac{n_{p} }{P_{n} } =\frac{n_{n} }{P_{p} } , \frac{D^{p}_{n} }{D^{n}_{p} } =\frac{\mu ^{p}_{n} }{\mu ^{n}_{p} } and ,\frac{D}{\mu }=\frac{kT}{q}yields
\gamma =\left[1+\frac{L^{n}_{p}n_{n} \mu ^{p}_{n} }{L^{p}_{n}P_{p} \mu ^{n}_{p}}\tanh \frac{W_{b} }{L^{n}_{p} } \right]^{-1}Since we have assumed that the emitter is long compared with the electron diffusion length in the emitter, the long-diode expression applies to I_{En} . On the other hand, for a very narrow emitter width W_{e}, L^{n}_{ p} would be replaced by W_{e} , corresponding to the narrow-diode expression.