Is the drift current of minority carriers negligible in the continuity equation?

Question

When deriving the ambipolar transport equations for an extrinsic semiconductor under low injection, I have found certain discrepances between different references.

For instance, in Neamen's Semiconductor Physics and Devices (see section 6.3.2 for reference), he constructs said equations from the continuity equations, by supposing that $n_0, p_0$ (the thermal equilibrium concentrations) are constant in space and assuming that the recombination and generation rates of holes and electrons are the same. Lastly, after arriving to a differential equation with non-constant coefficients he applies the hypothesis of an extrinsic semiconductor under low injection. In this limit, the coefficients turn constant and equal to the ones corresponding to the minority carrier. For instance, in the case of a p-type semiconductor under low injection, he arrives to

$$ D_n \frac{\partial^2\left(\delta n\right)}{\partial x^2} + \mu_n \varepsilon \frac{\partial\left(\delta n\right)}{\partial x} + g - \frac{\delta n}{\tau_{n}} = \frac{\partial\left(\delta n \right)}{\partial t} $$

Where $\varepsilon$ stands for the electric field on the semiconductor, and $\tau_n$ for the electron mean lifetime.

However, Streetman (in section 4.4.3 of Solid State Electronic Devices) approximates one step further and assumes that the drift current of minority carriers is negligible in front of its diffusion current, since the former is proportional to the concentration itself whereas the latter is proportional to the gradient of the concentration. He arrives to the following equation, (taking $g=0$)

$$ D_n \frac{\partial^2\left(\delta n\right)}{\partial x^2} - \frac{\delta n}{\tau_{n}} = \frac{\partial\left(\delta n \right)}{\partial t} $$

(hence, the same as before but with one term missing and with $g=0$). What I don't understand is why this is a valid assumption: when you introduce the drift term in the continuity equation as Neamen does, since the divergence of the current appears you will eventually obtain a term (that comes from this drift term) in the continuity equation that depends on the gradient of the concentration (while you will get a second order derivative term from the diffusion component of current), which we have stated to be non-negligible.

So, is Streetman's approximation correct? Under what circumstances?

Answer 1

The E-term in the minority diffusion equation is, in general, not negligible. In a homogenous semiconductor under low-injection conditions and zero applied field you can neglect it. This corresponds to the simplest models of the pn-junctions and the bipolar transistor. In modern bipolar transistors, however, a graded doping and/or bandgap in the base are used to introduce a built-in electric field. This accelerates the transport of minority carriers through the base and thus the transistor speed. Also, high-injection conditions lead to an electric field in the base which accelerates the transport of minority carriers through the base.

PS: Streetman seems to argue only for the neglect of the E-field induced minority drift current compared to the much larger drift current of the majority carriers while the minority diffusion current for high concentration gradients could be comparable to it. This applies only in situations where the total (electron and hole) current is considered. Even in the explanation of the Haynes-Shockley experiment, where the E-field drift current is essential, he uses the E-field free minority diffusion equation.