THEORY OF THE SWEPT INTRINSIC STRUCTURE 1281 



breaks down. First we find where the carrier concentration is in error by 

 finding the bias at which the minimum drift current as calculated from 

 qn(n + p)E becomes equal to the total current, as found from the excess 

 of generation over recombination in the intrinsic region. We then go on 

 to find where the error in carrier concentration gives a sufficient error in 

 space charge to affect the calculation of electric field. As we shall see, the 

 zero-current approximation gives too low a carrier concentration in the 

 interior of the intrinsic region. This will lead to serious errors in the field 

 distribution only if the space charge of the carriers is important. When 

 the bias is sufficiently high or the intrinsic region sufficiently narrow 

 that the intrinsic region is swept so clean that the carrier space charge is, 

 in fact, negligible, it will not matter that the calculated carrier density 

 is too low, even by orders of magnitude. In such cases, the electric field is 

 constant throughout most of the intrinsic region. 



In the following we shall, for simplicity, take 6=1 and assume that 

 the extrinsic regions are ecjually doped so that the problem is symmetri- 

 cal. 



Carrier Density 



We now find where, on the zero current assumption, the drift current 

 becomes equal to the total current. This involves knowing only the 

 carrier concentrations and the field Ei in the center of the intrinsic 

 region, where the drift current qyi,{n -f- p)Ei is a minimum. By symmetry 

 n and y are equal here and n = p = 7ii exp ( — qVa/2kT) where Va is 

 the applied bias. The minimum field Ei is given by the total voltage 

 drop V and the field penetration parameter rj, which is the ratio of the 

 minimum field to the average field. Thus r) = 2LEi/V where 2L is the 

 width of the intrinsic regions. The difference between V and Va is the 

 built-in voltage {2kT/q)/ln{N /rii) where N is the majority carrier 

 concentration in the extrinsic regions. We now have for the drift current 

 in the center of the intrinsic region 



,.(,. + p)£ = , ,0(1;)^ exp (-1^) (Al) 



We next find the total current from the excess of generation over re- 

 combination in the intrinsic region. From the zero current assumption, 

 np = Ui exp ( — qVa/kT) is constant throughout the intrinsic region. 

 Hence g — r is constant. So the current / = q(g — r)2L = qL{ni — np)/ 

 TUi is 



qLrii 



1 — exp 



(A2) 



