1282 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



Equating this to the drift current (Al) in the center of the intrinsic 

 region gives 



The error in carrier concentration is less for narrower intrinsic regions 

 and lower biases. Thus (A3) gives a curve of L versus Va such that the 

 zero current solution gives a good approximation to carrier concentration 

 for points in the VaL plane lying well below the curve. As expected, for 

 zero bias, the solution is good for any value of L. However, for a bias of \ 

 several kT/q, the solution for carrier concentration breaks down unless ^ 

 L is a very small fraction of a diffusion length. 



Carrier Space Charge. 



In Prim's analysis the carrier space charge is so low throughout most 

 of the intrinsic region that the field remains approximately constant 

 and equal to Ei . However there must be enough carriers present that 

 the drift currents of holes and electrons can remove the carriers as fast 

 as they are generated. In this section we ask where the space charge of 

 the necessary carriers becomes large enough that its effect on the field 

 can no longer be neglected. Let i^E be the change in field due to the 

 space charge in the intrinsic region (not counting the high field regions 

 near the junctions). Unless LE is small compared to Ei the neglect of 

 carrier space charge will not be justified. We shall find the ratio of AE 

 to Ei . 



If the field is to be approximately constant, then the hole and electron 

 concentrations can easily be found from the hole and electrons currents. 

 We shall deal with applied biases of at least a few kT/q, for which 

 recombination is negligible and the total current I = qg2L = qUiLlr. 

 Since g — r = g\s> constant, the hole and electron currents are linear in 

 X and, for constant field, are proportional to the hole and electron con- 

 centrations respectively. Thus the net space charge of the moving 

 carriers q{'p — n) is proportional to x and varies from zero in the center i 

 of the intrinsic region to qp near the IP junction, where n is small 

 compared to p and the current flows by hole drift, so / = q^ipEi . Thus 

 the maximum charge is I/iiEi and the total positive charge of the car- 

 riers on the P side of the center is IL/2iJ.Ei . This gives a drop in field 



„ _ alL _ arii kT L 

 " 2qiJ.Ei ~ 'YqEiL} 



