PRINCIPLES OF TRANSISTOR ACTION 271 



effect of recombination, leads to a non-linear differential equation which 

 must be solved by numerical methods. A simple solution can be obtained, 

 however, if it is assumed that all of the forward current consists of holes 

 and if recombination is neglected. 



The electron current then vanishes everywhere, and the electric field is 

 such as to produce a conduction current of electrons which just cancels the 

 current from diffusion, giving 



WeF = -(kT/e) gradwe. (4.18) 



This equation may be integrated to give the relation between the electro- 

 static potential, V, and ««, 



V = (kT/e) log (ue/neo). (4.19) 



The constant of integration has been chosen so that F = o when «, is 

 equal to the normal electron concentration Ueo- The equation may be 

 solved for He to give: 



He = tieo exp{eV/kT). (4.20) 



If trapping is neglected, electrical neutrality requires that 



ne = ilk + WeO. (4.21) 



Using this relation, and taking n-eo a constant, we can express field F in 

 terms of »/, 



F = - {kT/e{nn + n^)) grad iih (4.22) 



The hole current density, h, is the sum of a conduction current resulting 

 from the field F and a diffusion current: 



ih = nhCtikF — kTnh grad tih (4.23) 



Using Eq. (4.22), we may write this in the form 



ih = —kTnh({2nh + neo)/(nh + iieo)) grad rih (4.24) 



The current density can be written 



ih = - grad lA, (4.25) 



where 



\l/ = kTnh{1nh — neo log ((nh + neo)/neo)) (4.26) 



Since ih satisfies a conservation equation, 



div ih = o, (4.27) 



^ satisfies Laplace's equation. 



