440 BELL SYSTEM TECDNICAL JOURNAL 



where the mobility n and diffusion constant D for holes are related by Ein- 

 stein's equation 



^i = qD/kT (2.7) 



and b is the ratio of electron mobiUty to hole mobility.'' 



Under equilibrium conditions (pp ^ ipn = <p where ^ is independent of 

 position. Under those conditions, Ip and /„ are both zero according to equa- 

 tions (2.5) and (2.6). The electrostatic potential \p, however, will not in 

 general be constant and there will be unbalanced charge densities throughout 

 the semiconductor. We shall consider the nature of the conditions which 

 determine \}/ for a general case and will later treat in detail the behavior of 

 4/ for p-n junctions. 



For equilibrium conditions, there is no loss in generality in setting <p 

 arbitrarily equal to zero. The charge density expression (2.3) may then be 

 rewritten as 



p = Pd — Pi sinh ti (2.8) 



where 



u = q^/kT, Pi = Imq, Pd = q(Nd - No) (2.9) 



In equation (2.8) pd and u and, consequently, p may be functions of position. 

 The potential \p must satisfy Poisson's equation which leads to the equation 



VV = -47rp//c (2.10) 



where /c is the dielectric constant, (2.10) can be rewritten as 



V% = '^-^(smhu-'j) (2.11) 



klK \ Pi/ 



What this equation requires in physical terms is that the electrostatic po- 

 tential produces through (2.8) just such a total charge density p that this 

 charge density, when used in Poisson's Equation (2.10), in turn produces 

 xf/. It seems intuitively evident that the equation for u will always have a 

 physically meaningful solution; no matter how the charge density pd due 

 to the impurities varies with position, the holes and electrons should be 

 able to distribute themselves so that equilibrium is produced. For a 

 one-dimensional case, it is not difficult to prove that a unique solution exists 

 for «(x) for any pd{x) (Appendix VII). 



* We prefer b in comparison to c for this ratio since c for the speed of light also occurs in 

 formulae involving b. 



