p-n JUNCTIONS IN SEMICONDUCTORS 453 



cordance with the scheme used in Fig. 2. Part (b) shows the current dis- 

 tribution for a forward current I from left to right and (c) shows 

 the corresponding potential distribution and values of ^„ and <pp , the total 

 applied iwtential being Sip. Recombination prevents the hole current from 

 penetrating far into the u-region, the dept h of pe netration being described 

 by the diffusion length Lp = VD^p = VDp„/g, where pn is the hole con- 

 centration in the /^-region. The electron current similarly is limited by 

 /.„ = s/bDTn = x^bDiip/g. (Diffusion lengths are evaluated for pirticular 

 models of the junction in Section 4.) Far from the junction, therefore, the 

 hole and electron concentrations have their normal values and consequently 

 <Pp = <Pn and ipp — \[/ has its normal value. This accounts for the equal dis- 

 placement dip for all three curves at x = Xa . The curves for tpp and ipn have 

 a continuous downward trend which produces the currents 



Ip = -qy.p~ and /„ = -qbixn^. (3.7) 



The area between the <pp and (p^ curves has a special significance: This differ- 

 ence is related to the excess rate of recombination and the integral of this 

 rate over the entire specimen must be sufhcient to absorb the hole current 

 Ip = I entering at Xa so that the entire current at Xi, is carried by electrons. 

 In terms of cpp — <pn and equation (3>.6) we obtain 



I = Ip{Xa) - Ip{Xb) = -dip 



= gq\ {e'^'^p-^'^^"'' - 1) dx. 



•I Xa 



From (3.8) we conclude that if g is increased indefinitely for a specified 

 current /, then cpp — <pn must approach zero. For this case, in which the 

 rate of recombination and generation is very high, (pp = ipn and 



I = Ip-\- I„= -qtJi{p + bn) dipp/dx (3.9) 



and 



5^ = - f ' diPp = I f dx/qix(p -f bn) = IRo, (3.10) 



where Ro is simply the integral of the local resistivity corresponding to 

 densities p and n. For smaller values oi g, I does not divide in the ratio p:bn 

 and cpp 7^ <pn and dip > IRo . 



We shall next give an approximate treatment for the case in which b^pj 

 (J for junction), the value of ipp - <Pn at x = 0, is an appreciable fraction of 



»" A general proof that 5^ > IRo is given in Appendix I. 



