THEORY OF THE SWEPT INTRINSIC STRUCTURE 1265 



tion E increases so s = I/E must decrease. Thus the correct solution 

 must break away from the cubic near the junction. 



Instability of the Solution 



Equation (5.1) has two Hmiting forms and makes a rather abrupt 

 transition between them. Over most of the intrinsic region, the quantity 

 in brackets [£'s — /] = [E{E^ — x^ -\- A) — I] ahnost vanishes. It differs 

 from zero just enough that when multiplied by the very large factor 

 £^ ;^ 10*^ it gives the correct second derivative of E. In Section III we 

 derived an upper limit on the small deviation bE from the cubic required 

 to satisfy the differential equation. If E deviates from the cubic by more 

 than this small amount, then the second derivative of E becomes too 

 large. This increases the deviation from the cubic, which further in- 

 creases the second derivative and so on. E and s rapidly approach 

 infinity in a short distance. This, of course, is the reciuired behavior at 

 the junction. The rapid increase in s makes it possible for s to approach 

 P/2ni . 



In Section III we showed that there is a solution to the differential 

 equation that lies within a small interval bE from the cubic. Suppose we 

 try to solve (5.1) graphically or on a machine starting at x = 0. There 

 are two boundary conditions: By symmetry dE/dx = at x = 0. We 

 choose for E(0) a value somewhere in the interval 8E(0). The resulting 

 solution will not long remain in the interval 8E(x). In fact there is only 

 one choice of E(0) for which the solution remains close to the cubic 

 from .T = to a; = oc . For any other E(0) the solution would remain 

 close to the cubic for a certain distance and then abruptly become un- 

 stable and both E and s approach infinity. £"(0) must be so chosen that 

 the solution becomes unstable and E and s become large at the junction. 

 However it is impractical to set E(0) on a machine with sufficient ac- 

 curacy to insure that the solution will remain stable for a reasonable 

 distance. A more practical procedure is to find a solution which holds 

 near the junction and joins the cubic to a solution in the adjacent 

 extrinsic region. 



Zero Bias 



It may be helpful to approach the junction solution by reviewing the 

 simple case of an IP junction under zero bias. Both charge and particle 

 flow vanish. The vanishing of particle flow means that in the intrinsic 

 region E' — s is constant, (2.7). Since E = and s = 1 in the normal 

 intrinsic material, it follows that E^ — s = 1. With 7 = the equation 



