118 BELL SYSTEM TECHNICAL JOURNAL 



from single contacts and have obtained results in agreement with the CP 

 empirical formula down to frequencies of the order of 10""^ — 10~- cps. 



Significant theoretical work upon this problem has not been attempted 

 until recently. G. G. Macfarlane" has advanced a theory based upon a 

 non-linear mechanism containing one degree of freedom which seems to be 

 in agreement with the CP law. W. Miller^- has worked out a general theory 

 of noise in crystal rectifiers. His theory is linear, contains essentially an 

 infinite number of degrees of freedom, and is equivalent in many respects 

 to the theory discussed in this paper; however, he has not succeeded in ob- 

 taining agreement with the experimental data on crystal rectifiers (which 

 satisfy approximately the CP law) for any of the specific models he used. 



The purpose of this paper is the calculation of the spectral density of the 

 fluctuations of the electrical resistance when it is linearly coupled to a diffus- 

 ing medium (particles or heat), or, mathematically speaking, is equal to a 

 linear function of the concentration deviations of this diffusing medium. 

 This diffusing medium undergoes thermally excited fluctuations and thereby 

 causes fluctuations in the resistance. The motive behind this investigation 

 was the understanding of the frequency dependence of contact noise dis- 

 cussed in the following paragraphs, but at the present time it is apparent 

 that this treatment in addition may apply to rectifying crystals, thin films, 

 transistors, etc. The quantitative details of the coupling between the resist- 

 ance and the diffusing medium are not considered here; in consequence of 

 which, this work can hardly pretend to give a complete explanation of con- 

 tact noise. However, important results are given concerning the relation 

 between the spectral density of the resistance, on one hand, and the geom- 

 etry of the coupling and the dimensionality of the diffusion field on the 

 other. 



Now let us consider the CP empirical formula in detail. Let R be the 

 average resistances^ of the contact (we will henceforth consider only contacts 

 and will regard a granular resistance as a contact assemblage) and let Ri 

 (/) be the instantaneous deviation from the average. By theorems 1-3 of 

 Appendix I, we can express the m.s. value of R\ as a sum of the m.s. values 

 of Ri in each frequency interval as follows : 



R\ = f 5(co) do:, (1.1) 



•'0 



" G. G. Macfarlarie, Proc. Phys. Soc. 59, Pi. 3, 366-374 (1947). 



'^ To ho. published. 



"The resistance of a contact is composed of two jiarls: the "gap resistance" and the 

 "spreading resistance." The term "gip resistance" is seU'-explanatory. The "spreading 

 resistance" is the resistance involved in driving the electric current through the body of 

 the contact material along paths converging near the area of lowest gap resistance. The 

 measured contact resistance is the sum of these two parts. In some of the particular 

 physical models considered in Section 5, fi is taken to be the gap resistance necessitating 

 ad hoc arguments relating gap resistance and total resistance. 



