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BELL SYSTEM TECHNICAL JOURNAL 



Neglecting space charge, Poisson's equation becomes simply the con- 

 dition of electrical neutrality: 



(4) 



{p — pit) — (« — «o) = 0, 



assuming substantially complete ionization of donors and acceptors. 

 Similarly, equations (3) become 



(5) 



L - Jn = J; Ip + In = I. 



A\'ith electrical neutrality, the two continuity equations merge into one: 

 Since derivatives of p equal the corresponding ones of n, 



div Jp = - [p/r,, - ^o] - — 

 (6) 



= div Jn = — [«/t„ — gn] — 



dn 

 dT 



The neutrality condition in conjunction with the two equations obtained 

 by substituting for Jp and J,, from the diffusion equations in (6) thus pro- 

 vide three equations for the determination of p, n, and E or T^. 

 It is instructive to rewrite equations (6) in accordance with 



(7) 



div Jp = s 

 = div J„ = s 

 djp • i 



gradp 

 grad n, 



s = 



dx 



dp 

 dx 



i + 



dy 



dp 

 d~yj 



j + 



'd]p ■ k 

 dz 



dp 

 dz 



k, 



where i, j, and k are unit vectors in the directions of the respective axes. 

 The velocity s, which is given as well by the expression for electrons an- 

 alogous to that written for holes, may be defined alternatively as follows: 

 Suppose, for definiteness, that the second-order system of equations (4) 

 and (6) have been solved, so that the concentrations and flow densities 

 are known in terms of the cartesian coordinates .v, y, and z, and the time t. 

 The ;v-component of s is then the partial derivative with respect to p of 

 the x'-component of Jp in which x has been replaced by the proper func- 

 tion of p, y, z, and /, and similarly for the other components. Thus, with 

 s a known function, p or n may be considered to satisfy the first-order 

 partial differential equation obtained by substituting from (7) in (6), from 

 which it is evident that s is the velocity with which concentration transi- 

 ents are propagated^^. This velocity, which is here called the differential 



^* The identification of s as tiiis propagation velocity follows the example of C. Herring, 

 in whose method for solving the transient constant-current ])rol)lem in one dimension 

 the velocity depends in a known manner on concentration only, through the neglect of 

 diffusion, so that the general solution of the differential equation in which thus neither 

 independent variable x nor I occurs explicitly may be obtained; cf. reference 12, pp. 412 ff. 



