Derivation of TSC and PITS Equations

We wish to solve, under various conditions, the following differential equations for the electron and hole concentrations  

We suppose that only electrons are being emitted in appreciable quantities, and n p at all times. Then, if a constant heating rate, a = dT/dt is imposed upon the sample, Eq. (C7) becomes 

A glance at the temperature dependence of eni, as shown in Eq. (31), quickly convinces one that Eq. (CIO) cannot be solved analytically. However, progress can be made by moving the exp( - T/ax ) factor inside the integral. Then 

Now if r 1 is large enough, specifically, if t 1 e h then the exp[(T — T)/ ] term will dominate the temperature dependence and, in fact, will keep the integrand small until T approaches T. Thus, we can evaluate the slowly varying part, eni(T )exp[ —Q(T )] at T = T and pull it out of the integral. That is, 

Here the last term will be negligible if 1 s, since usually a l°K/s. Also, we will be able to nelgect the last term in Eq. (CIO) [i.e., exp(— 7/ )] in the spirit of the present approximations. The final result is therefore 

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