THEORY OF THE SWEPT INTRINSIC STRUCTURE 1251 



Combining this with (2.7) gives 



^(^-,) = JL (32) 



where L/ = Z)r is the diffusion length in intrinsic material. 



Equation (3.2) can be immediately integrated. There are two con- 

 'stants of integration, one of which can be made to vanish by choosing 

 [a; = at the center of the intrinsic region, where the first derivatives of 

 E and s vanish. {E is a minimum here and s a maximum). The solution 

 obtained by two integrations is 



r -^=(~] -A (3.3) 



As we shall see later, the constant A is determined by the voltage drop 



j across the unit. 



I The exact procedure now would be to substitute s from (3.3) into (2.6). 



The resulting second order differential equation could, in principle, 

 Ithen be solved for E versus x. The exact solution, however, would be 

 i quite difficult. We shall discuss it in Section V. Here we make the assump- 

 'tion that throughout the intrinsic region the charge flow is mainly by 

 'drift, so that we can neglect the diffusion term in (2.6) and take / = 



asE, as discussed in Section IL Later in this section we find an upper 



limit on the error due to this assumption and show how the cubic can be 

 i corrected to take account of the diffusion term. 

 1 Putting s = I/(tE in (3.3) gives a cubic equation 



1 for E/Ei as a function of x/2Li . This equation contains two parameters 



I / and A . A determines the voltage and / is determined by the length 



i 2L of the intrinsic region. The relation is as follows: Let the applied 



[ voltage drop across each junction be at least a few kT/q. Then the 



minority carrier currents from the extrinsic regions will have reached 



their saturation values. Call Is the sum of the hole current from the 



A^ region and the electron current from the P region. Is comes from pairs 



generated in the extrinsic regions near the junctions. 7s can be made 



arbitrarily small by making the N and P regions sufficiently highly 



doped (provided the diffusion length in the extrinsic material does not 



decrease with doping faster than the majority carrier concentration in- 



