434 BELL SYSTEM TECIIXICM. JOVRXAL 



time / wliicli is to Ix' used in E(j. (10). Al high fre(}uencies, the charge at 

 the interface need not he in phase with the applied voltage. Tf the frequency 

 is low enough so that the charges maintain their equilibrium values during 

 the course of a cycle, Q will be in phase with V and the parallel capacitance 

 for unit area is simply: 



C = dQ/dV. (12) 



The barrier layer may be represented by this caj)acitance in parallel with 

 the d-c. differential resistance, R. 



Both R and C may depend on the d-c. bias current flowing. Variations of 

 R and C with frequency at moderate frequencies may result from large scale 

 nonuniformities of the barrier such that the patch fields extend over a large 

 fraction of the thickness of the barrier layer or from charge relaxation times 

 associated with acceptors, donors or trapped carriers. At low frequencies, 

 drift of ions may be involved. 



Attempts which have been made to determine the variation of resistivity 

 in the barrier layer from impedance data are invalid. It is not correct to 

 tike the impedance of an element of thickness dx to be 



dx/\a{x) + C/'co/v. Vtt)] 



and integrate over dx to obtain the impedance of the layer. This procedure 

 omits terms arising from diffusion and changes of concentration in time. 

 It is possible to obtain an integral of Eq. (1') if both sides are divided by 

 nen. Integrating over x from .v = to .v = .vi, and using the boundary condi- 

 tions (3a), (3b) and (5), we get 



K. = f lO + i^Mi^'V/"'^') ,,,. (13) 



Jo ne^L 



which means that the integral of the conduction current over the conduc- 

 tivity gives the applied voltage. This is consistent with the representation 

 of the barrier by a resistance and capacitance in parallel. 



Acknowledgments 



The author is indebted to W. Shockley, \\'. II. Hrattain, and P. Debye 

 for stimulating discussions and suggestions. 



