460 bELL SYSTEM TECnNlCAL JOURNAL 



ance. For cjiTp small, the real term Gp is simply conductance per cm^ 

 of a layer Lp cm thick with hole conduction corresponding to the density 

 pn -\- to ; it is also the differenliil conductance obtained by differentiating 

 (4.11) in respect to tq . For the case of zero bias this establishes the result 

 quoted in Section 1 that the voltage drop is due to hole flow in the /^-region 

 where the hole conductivity is low. 



In this section we have treated Tp as arising from body recombination. 

 In a sample whose y and z dimensions are comparable to Lp or L„ , surface 

 recombination may play a dominant role. However, as we show in Appendix 

 V, the theory given here may still apply provided appropriate values for 

 Tp and T„ are used. 



4.3 D-C. Formulae 



The total direct hole current flowing in at Xa is /^o plus the current re- 

 quired to recombine with electrons in the />-region. This latter current is, of 

 course, equal to the electron current flowing into the ^-region. This electron 

 current, denoted by /,o or I,.o{xtp), is obtained by the same procedure 

 as that leading to (4.11) for /^o except that bD replaces D and the subscripts 

 of L and t are now n. Combining the two currents leads to the total direct 

 current: 



/o = / 



pO 



+ /no = iqL) (j^ + ^"^ ie'-'""' - 1) (4.13) 



for the direct current per unit area for applied potential tq . The algebraic 

 signs are such that / > corresponds to current from the /^-region to the 

 w-region in the specimen; tq > corresponds to a plus potential applied 

 to the p-end. The ratio of hole current to electron current across the transi- 

 tion re^iion is 



/pO _ Pn Ln _ pp \/bD7 



(4.14) 



7„o Lp blip biin -s/Dtp 



P . /bitn _ . /^ 

 ^n 'V pp V <Xn 



2 



where we have used the relationships Unpn = npPp = Ui from (2.2) and 

 TpUn — T„pp = 1/'' from (3.2) and (3.3). These results can be summarized 

 by saying that the current flows principally into the material of higher re- 



'2 For convenience we repeat the definitions here: q ^ magnitude of electronic charge; 

 D ^ dilTusion constant for holes; />„ antl «„ ^ thermal ecjuihurium value of p and n, as- 

 sumed constant throughout </ region (.r > .Vv,,); n,. and />p = similar values for x < Xtp', 

 Lp^ dilTusion length ^^y/ Dt^ for holes in »-rcgion; r^ ^ lifetime of hole in «-region be- 

 fore recombination; b = electron moijilily/hole mobilily; L,. and t„ similar in (|uantities 

 for electrons in /^-region; cr„ = qiihn,, and Op = quPp are the conductivities of the two regions. 



