1244 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



II. BASIC RELATIONS 



The problem can be stated in terms of the hole density p, the electron 

 density Ji, and the electric field E and their derivatives. Let the distance 

 .T be measured in the direction from N to P. The field will be taken as 

 positive when a hole tends to drift in the -{-x direction. The field in- 

 creases in going in the -fx direction when the space charge is positive. 

 Poisson's equation for intrinsic material is 



f = a(p - «) (2.1) 



where the constant a has the dimensions of volt cm and is given in terms 

 of the electronic charge q and the dielectric constant k by 



■iirq 



a = — - 



K 



For germanium a = 1.17 X 10~ volt cm. 



The hole and electron flow densities Jp and J„ are^ 



Jp = nEv - d'^ = ^,Je -—^Inv 

 ax \ q dx 



Jn = —hi nEn + D -^ j = —bun 



E -\ -Inn 



q dx 



(2.2) 



where n and D = n kT/q are the hole mobility and diffusion constant re- 

 spectively, k is Boltzmann's constant (8.63 X 10~^ cv per °C) and T is 

 the absolute temperature. The ratio b of electron mobility to hole mo- 

 bility we take to be unity. This makes the problem symmetrical in n and 

 p and consequently easier to understand. Section \T will extend the re- 

 sults to the general case of arbitrary b. 



Charge and Particle Flow 



For some purposes it helps to express the flow not in terms of Jp and 

 Jn but rather in terms of the current density / and the flow density 

 J = Jp -\- Jn oi particles, or carriers. The current density / = q{Jp — 

 Jn). Each carrier, hole or electron, gives a positive contribution to 

 J if it goes in the +.r direction and a negative contribution if it goes in 

 the —X direction. In other words, J is the net flow of carriers regardless 

 of their charge sign. The current / is constant throughout the intrinsic 



^ See, for example, Electrons and Holes in Semiconductors, by W. Shockley. 

 D. Van Nostrand Co., New York, 1950. 



