In a textbook about semicoductor physics, I came across a passage about deriving

Question

In a textbook about semicoductor physics, I came across a passage about deriving the carrier concentration at thermal equilibrium in semiconductors I didn't quite grasp:

The recombination R(T,n,p) rate depends on the ionization energy, temperature T, and carrier concentrations n, p. As the recombination rate must be zero if one of the carrier concentrations n, p is zero, it follows for the first non-vanishing term of a Taylor expansion:
R(T,n,p)=r(T)np

The Taylor expansion, as I know it from the mathematics class, is defined only for functions of one variable. Besides that, there is always a point involved about which the expansion is carried out. Strictly speaking, R is a function of three variables. And no point is given. I guess this is the kind of inaccuracy practiced among physicists in order not to bloat their argumentation, and generally understood -- among physicists.

So, how does an expansion of a function of three variables work and which point should I assume here?

Explanation / Answer

The argument is not a mathematical Taylor expansion, there is an implicit physical argument here which is very well known but not given.

The rate of a reaction (like recombination) at low density should be the product of the densities, with a coefficient that is independent of the density. This happens to be a mathematical Taylor expansion in the densities. But zero is a special point for an expansion, because there are natural power-law change of variables, and you can't determine which power is right without knowing the physics. For example, the period of a pendulum goes to zero as the length of the pendulum goess to zero, does this mean that the period must be proportional to L? Of course not. It's the square-root of L.
The correct argument

When you have two objects which are moving randomly and have to find each other, the probability each A object finding a B object per unit time is proportional to the density of B objects. If you have twice as many B objects per unit volume, it takes half as long to find one, just from independent search statistics. Similarly, the rate is doubled if you double the concentration of A objects. So the leading term is the so-called mass-action term, the product of the concentrations.

At higher concentrations, the A's and B's start to have excluded volume effects--- it is slightly easier for A to find B because the mass-action assumes indepedent motion of the different B's while the real B's only can travel in the volume not excluded by other B's. There are also corrections to the diffusion rate fron interactions.

The corrections are negligible when the gas is dilute compared to the physical cross-section of interaction, and this limit defines the domain of applicability of mass action dynamics. The argument that the molecules are in the mass-action limit is what is used here, and the Taylor expansion phrasing is suboptimal.

But the result is that the physics of recombination is by the first term in a taylor expansion in the two densities, so the calculation that follows is correct, even though the physical argument is wrong. This is typical of textbooks, and you should probably throw this one away

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