404 BELL. SYSTEM TECIIMC A L JOURNAL 



hole current densities on the two sides must ditTer by 7, ; if a part oi jc is 

 due to withdrawal of electrons, then the electronic current will have a cor- 

 responding discontinuity. If /„ is positive, i.e., flows from left to right in the 

 specimen, the current can be assumed to be practically entirely electronic 

 over most of the range from -(/ to 0; i.e., as .v becomes negative the hole 

 current must rapidly approach zero and the electron current must rapidly 

 api)roach7„ . In fact, if diffusion is ignored the electron and hole currents 

 must have these limiting values for any negative .v. 



The preceding discussion and the mathematics to follow have been 

 couched in purely one-dimensional language, i.e., have been formulated as if 

 the electron and hole densities were functions of x alone, independent of y 

 and a, and as if the semiconductor extended to infinity in the y- and s-direc- 

 tions. However, it is easy to see at each stage that practically the same 

 equations can be written for transport of holes along a narrow filament whose 

 thickness is small compared with the linear scale of the phenomena along its 

 length, even when the density of holes is not uniform over the cross-section 

 of the filament. If the density of holes is uniform over the cross-section, all 

 the equations will of course hold as written. However, recent work- has 

 suggested that holes recombine with electrons so rapidly at the surface that 

 the density of holes may be much smaller near the surface than in the center 

 of the cross-section. In such case all the equations of this memorandum must 

 be interpreted as applying to the mean value, nh{x), of the density of holes, 

 «fc(.r, y, s), averaged over the cross-section of the filament; also, the rate of 

 recombination of holes and electrons must be set equal to some function of 

 fih , as yet not reliably known, instead of to a constant times the product of 

 electron and hole densities. This will of course alter most of the quantitative 

 predictions of Section 4, but will not require any change in the method of 

 calculation. 



2. FORMUL.\TIOX .\XD SOLUTIOX OF THE PROBLEM WITH XeGLECT OF 



Diffusion* and Recombix.\tiox 



For this case the electron and hole currents can each be equated to the 

 product of tield strength E by particle density 11 by mobility m- and the 

 continuity equations are 



dl dx 



dl dx 



-U. Sulil and \V. Shockley, paper Qll prcscnlcd at the Washington Meeting of the 

 .\merican Physical Society, .April 29, 1Q49; see also Shockley, Pearson, Sparks and Hrat- 

 tain, reference 1. 



