442 BELL SYSTEM TEL II MCA L JOIRSAL 



On the other hand, if /./; » /„ , a large change in impurity concentration 

 occurs near .v = without compensating electron and hole densities oc- 

 curring. Mathematically, we find that (2.11) can be expressed in the form 



'^ = _L (_v + sinhw) (2.16) 



and 



y = -v 2A„, K = Ln/lLa (2.17) 



In Appendix VII, it is verified that the api)ropriate solution for iv « 1 is that 

 giving local neutrality, u = sinh"^ y; while for A' » 1, there is space charge 

 as described below. 



For Li) » La , or A' » 1, there is a space charge layer in which Xd — Xa 

 is uncompensated. To a first approximation, we can neglect the electron and 

 hole space charge in the layer and obtain, by integrating twice, 



^= -^-^Va..v, (2.18) 



where we have chosen the zero of potential as the value at .v = 0, a condi- 

 tion required by the symmetry between +.v and — .v of (2.14). Although 

 the potential rise is steep in the layer, dxp/dx should be small at the point 

 x„i where the neutral ;;-type material begins. As an approximation Ave set 



,]\P/(ix = at .V - .v,„ : 



d\l/ lirqax] 



+ as = 0; (2.19) 



ax K 



this leads to a value for a-i which may be inserted in (2.18) to evaluate x}/ 



at .v„, : 



where n,„ = aXm is the density of electrons required to neutralize X,i — .Ya = 

 ax„, at the edge of the space-charge layer. This value of n„, must corre- 

 spond to that associated with xj/m by (2.2) 



We thus have two equations relating \p,n and //,„ and the j^arameter "a 

 To solve them we i)lot \n\l/„, versus In 11,,, as shown in Fig. 3. On this figure the 

 relationship 



_ iirq Urn 

 6k a- 



3 

 = 3.18 X 10~' ^' volts for Ge 



3 

 = 4.83 X 10"' ^ volts for Si (2.22) 



I'PnJl^-T^ (2.21) 



ii^ " 



