SHOT NOISE IN DIODES 595 



A third fundamental relation is Poisson's equation which becomes in 

 the parallel plane case under consideration 



In the a-region the total charge density is made up of three classes of 

 electrons, namely 



1. Those destined to pass the potential minimum and arrive at the 



anode. 



2. Those moving away from the cathode but which will not travel as 



far as the minimum point. 



3. Those returning to the cathode. 



Corresponding to each class of electrons, there is an associated 

 current, pu, so that each of the three densities p\, pz or pa, may be 

 expressed by a relation of the form, 



P,=^. (4) 



When it is remembered that the potential and velocity at a given 

 value of X are uniquely related through (2), then it is easy to see th-~t 

 the total density for a given plane in the a-region is given by 



,„ = ,r^Wd„,+2<.r'^(^rf«,. (5) 



where the first term represents the contribution of electrons in class 1 

 above, while the second term represents the contribution of electrons in 

 classes 2 and 3. The contribution of class 3 is equal to that of class 2. 

 The lower integration limit v of the second term of (5) represents the 

 initial velocity of an electron which would just arrive at the value of x 

 under consideration before coming to rest and starting back toward the 

 cathode and the limit u/ in both terms represents the initial velocity of 

 an electron which" comes to rest just at the potential minimum. 

 Thus, from (2) 



V = 



^^V and «/=V^r. (6) 



In the jS-region there is only one class of electrons, so that the 

 density is more simply expressed. Thus, 



J'* 00 



""^du,. (7) 



The value of p in (5) and (7) may each be expressed in terms of 

 d^V/dx' by the use of (3), and the integration of these two Poisson's 

 relations for the common boundary condition that the electric force is 



