652 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



and is the same as that due to an infinite source of strength 



Z) «^(^, Vr , Zr) 



r 



uniformly distributed over the plane x-^. 



But the density w due to such a source will be a function of x and ^ 

 only, and will be the solution of the one-dimensional diffusion equation. 

 Hence for a geometry approaching that of figure 1 sufficiently closely, 

 the problem is one-dimensional. 



The Evaluation of r 



We now have to write down the one-dimensional diffusion equation 

 in the presence of a magnetic field along Oy , which combines with the 

 drift velocity of the carriers so as to force them towards one of the 

 surfaces x = dz a.li Fx is the effective field arising in this manner, and 

 D is the diffusion coefficient, the equation is 



which expresses the fact that the diffusion current —Dq dw/dx plus the 

 drift current qtiFxW must be constant since the carrier density cannot 

 build up indefinitely, n is the mobility of the minority carriers: At„ for 

 electrons, /Up for holes. As is shown elsewhere^ the effective field F^ is 

 given by 



Fx = (Bn + Bp) Ez = BE^ , 



where Ez is the biasing field causing the drift current, Bn , Bp are the Hall 

 angles for electrons and holes, respectively. If /x„ , Hp are the electronic 

 and hole-mobilities, and if the magnetic field is not too large, 



B = Bn + Bp= 10-\fjLn + tJip)H, 



where ff is in oersteds, and the mobilities are in cm^/ volt-second, and 6 

 is in radians. (Strictly speaking, the diffusion current is not in the direc- 

 tion of the density gradient when a magnetic field is present.'* As the 

 result mixed derivatives 



dxdy 



occur in the diffusion equation. But in the reduction to one dimension 

 these terms integrate out. All that remains is a small correction to 7), 



