p-n JUNCTIONS IN SEMICONDUCTORS 475 



APPENDIX I 



A Theorem on Junction Resistance 



We shall here prove that the junction resistance is never less than the value 

 obtained by integrating the local resistivity l/qii(P + bn). This is accom- 

 plished by analyzing the following equation which we shall discuss before 

 giving the derivation: 



75^ = i r (i + ^P) dx + qg r {^, - ^rdie'^'^-'-'^'-Ddx, 



9M J^a \ P on J Jxa 



the meaning of the symbols being that shown in Fig. 5. This expression is 

 valid even if large disturbances in p and n from their equilibrium values 

 occur. The second integral is positive since the integrand is never negative. 

 It may be very large if (^p — (pn » kT/q in some regions. If, in the first in- 

 tegral, we consider that Ip and In may be varied subject to the restraint 

 Ip-^ In = I, we may readily prove that the first integrand takes on a mini- 

 mum value when 



T P^ AT ^'^^ 



Iv = — T— ^ and /„ = 



p -\- bn p -{- bn 



For this minimum condition, the first integral becomes 



/ 



lb 



dx/qix{p + In) = I Re 



where i?o is simply the integrated local resistivity. If / does divide in this 

 way, the second integral is zero, a result which we can see as follows: 



Ip = — qnp dipp/dx 



In — ~ qiibn difn/dx 



d(pp/dx _ Ip/p 

 d(pn/dx In/bn 



Hence, if the current divides in the ratio of p to bn, then d<pp = d<pn and. 

 since ipp = tpn at Xa ,<Pp^(Pn everywhere and the second integral vanishes. 



In general, of course, the conditions governmg recombination prevent 

 current division in the ratio p:bn and then 8ip/I > Fq . 



The equation discussed above is derived as follows: We suppose that 



SJ'p(-Vo) = ^n{Xa) = ^a 

 <Pp(Xb) = <Pn{Xb) = <Pb 



