A general relation between resistance fluctuations from enthalpy fluctuations andresistance response to current changesMichael B. Weissman Citation: Applied Physics Letters 32, 193 (1978); doi: 10.1063/1.89986 View online: http://dx.doi.org/10.1063/1.89986 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/32/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin-current noise from fluctuation relations AIP Conf. Proc. 1566, 363 (2013); 10.1063/1.4848436 High current in general relativity AIP Conf. Proc. 453, 498 (1998); 10.1063/1.57111 The relation between the changes of thermodynamic quantities at the glass transition and the longwavelengthfluctuations of glass structure J. Chem. Phys. 79, 4463 (1983); 10.1063/1.446332 Noise from equilibrium enthalpy fluctuations in resistors J. Appl. Phys. 52, 3095 (1981); 10.1063/1.329171 A Note on the Relation between Entropy and Enthalpy of Solution J. Chem. Phys. 15, 875 (1947); 10.1063/1.1746367

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A general relation between resistance fluctuations from enthalpy fluctuations and resistance response to current changes

Michael B. Weissman

Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 (Received 12 September 1977; accepted for publication 16 November 1977)

The spectral shape and magnitude of spontaneous resistance fluctuations due to enthalpy fluctuations are shown to be determined by the resistance response to current impulses for a broad class of resistors, including any resistor containing only one type of conducting material, regardless of the resistor geometry. Implications for the design and interpretation of experiments are discussed.

PACS numbers: 72.70.+m, 05.40.+j. 73.60.Dt

A series of experiments by Clarke and co-workers have provided strong evidence that the '1/f" current or voltage noise observed in thin metal film resistors when a voltage is applied arises from resistance fluctuations caused by spontaneous fluctuations in local "tempera­ture", i. e" enthalpy density, 1-4 This same mechanism has been invoked to account for noise in one ionic sys­tem. 5 While both the magnitude and frequency spectrum of the noise in the ionic resistor fit the model of dif­fusing thermodynamic fluctuations, 5 the frequency spectrum of the noise in metal films remains problem­atic. 4 In this note I derive a rather general relation (suggested by one given for more restricted conditions by Voss and Clarke4) relating the autocorrelation func­tion of the thermal resistance fluctuations to the re­sponse of the resistor to current impulses, Implications for the interpretations of experimental results are discussed,

For simplicity we may consider the case of fixed voltage (V) and fluctuating current (1) and resistance (R), (Since we are interested only in determining the fluctuations in R, an exactly equivalent analysis with identical results could be performed by assuming con­stant I rather than constant V 0 ) The power dissipation is given by P= IV, so that OP / P= - oR/R, for small fluctuations, Fluctuations in P are particularly conve­nient to analyze, since they may be expressed directly as an integral of fluctuations in the power dissipation density p(r), As I have argued previously, 6 the fluctua­tion weighting function in resistors, i. e" the function which gives the effect of a local fluctuation in some parameter on the net resistance, is proportional to p(r).

For noise from enthalpy fluctuations, since

op(r, t) _ oa(r, t) _ a lna(r) o1z(r, t) p(r) - ---a(r) - ---ar-~

we have

oR(t) R

oP(t) -p-=

(1)

where a(r) is the electrical conductivity, c p(r) is the heat capacity per unit volume, T is the absolute tem­perature, and JBolz(r, t) d 3r is the enthalpy fluctuation within any arbitrary volume B. When, as is ordinarily

the case, illna/aT and cp are constants determined by the conducting material over the region in which p(r) *0, Eq, (1) reduces to

oR(t) 1 a Ina f :3 ~= -PCp ----aT Ol!(r,t}p(r)d r, (2)

The noise autocorrelation function is then given by

G(T) =(OR(t)~~(t + T))

= (p;J2 e :~ar<ff p(r)p(r' )l5li(r, t)oh(r' ,t + T)d 3r' d 3r) ,

(3)

Standard thermodynamic theory 7 gives the magnitude of the enthalpy fluctuations: (UBI51z(r,t)d 3r)2) =kcpBT2, where k is Boltzmann's constant, Familiar linear fluctuation theory8 then allows us to express G(T) in terms of the Green's function for the evolution of enthal­py perturbations, c(r, r', T):

G(T)=(il ~arCk~22)ffp(r)p(r')c(r, r', T)d 3r' d 3r, p

At time T after a brief pulse of current produces Joule heating till, the enthalpy perturbation density is given by Ah(r', T) = tillJp(r)c(r, r', T)d3r/P, since Alz(r,O)=AJ!p(r)/p, Then from Eq, (2)

~T) = t-!¥f)(a :~a)ffp(r)p(r')c(r, r', T)d>r'd 3r,

(4)

(5)

Combining Eqs, (4) and (5), one obtains

G( )=_(AR(T)\T2alna T RAJ! ( aT' (6)

Since it is sometimes convenient experimentally to use step-function current changes rather than pulses, 4 it is helpful to have the equivalent relation in terms of the response :5.R(T) to step-function current changes:

G(T)= _(JAR(T) (RAP)"I\T2 a Ina (7) aT J aT'

The magnitude and average time course of the sponta­neous noise are then completely determined by easily measurable macroscopic quantities, as long as (a Ina/ a T)(Cpt1 is constant over the region of non -negligible p(r),

193 Appl. Phys. Lett. 32(3), 1 February 1978 0003-6951178/3203-193$00.50 © 1978 American Institute of Physics 193

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At this point a comparison with the related prediction of Voss and Clarke may be helpful. By assuming that the fluctuation weighting function is uniform within a volume Wand zero elsewhere and that the fluctuations decay by uniform diffusion, they obtain (in our notation)4

G( ) =(.:ill(T)XkT2Xillna)2 T .:ill (0) cvW aT .

(8)

For this special case, this formula is equivalent to Eq. (6), except for a minor distinction between Cv and cpo

The special case of Ref. 4 is not, however, typical of metal film resistors, in which there are regions of low power dissipation outside the narrow "neck" and of high power dissipation near sharp corners. Also, as Voss and Clarke point out, 4 such resistors are normal­ly surrounded by materials with drastically different thermal properties, producing non -uniform thermal diffusion.

Use of Eq. (7) of this paper does not require an esti­mate of W nor even a measurement of cP' but only one of I:J..P, which is, of course, easily obtained from P = IV. 9 We have dispensed with the assumption of uni­form diffusion in this derivation, allowing application of the theorem to resistors on thermally inhomogeneous substrates and resistors with non- negligible electro­phoretic heat transporL The main new physical content of this relation, however, comes from the proportional­ity of the weighting function to the local power dissipa­tion density 0

Unfortunately, the more general relationship between G(T) and .:ill(T) only heightens the apparent contradiction between the predictions of standard thermodynamics and the observed noise spectra in metal films. 4 In order for the data shown in Fig, 10 of ReL 4 to fit this theory, it would be necessary for the estimate of c v W in that paper to be an order of magnitude too small and for some extraneous noise source which in many respects behaves like enthalpy fluctuations to contribute to the low-frequency noise. Voss and Clarke suggest changing the prediction by introducing an assumption that the free energy of a Fourier component of a thermal fluc­tuation is proportional to the square of its wave vector. 4

In addition to the thermodynamic implausibility of this assumption, such a postulate would seriously contradict the experimental results, since it may be shown to lead to a noise magnitude proportional to 11'-1/3 rather than to the observed W- I dependence. 1,4 [It should be noted that the relation between .:ill (T) and G (T) proposed by Voss and Clarke for such spatially correlated fluctua-

194 Appl. Phys. Lett., Vol. 32, No.3, 1 February 1978

tions cannot be derived under the more general condi­tions assumed in this work, since the derivation re­quires the assumption of uniform diffusion. 1

Even more disturbing results have been obtained by Eberhard and Horn, 10 who observed noise in metal film resistors as a function of temperature. Near 100'K, the noise (above Johnson noise) in Cu films was more than two orders of magnitude less than predicted by the formula of Clarke and Voss. I Several other anomalous effects were observed, including a dependence of the noise magnitude on I a Ina/a T I of the wrong sign. 10

Dutta, Eberhard, and Horn have recently confirmed that the resistance fluctuations in the region of 10 Hz from Cu and Ag films are much smaller than Eq. (8) predicts over a wide temperature range. II

No resolution to these difficulties is apparent. Never­theless, the unambiguous adequately general relation­ship between the thermodynamically predicted noise and the experimentally accessible response to current jumps may be useful in determining rigorously if there are cases in which theory and experiment actually dis­agree. In particular, this theory shows that it would be more useful to measure I:J..P than to estimate Wand measure or estimate cp in comparisons of noise with current jump responses,

This work was supported by a post-doctoral fellow­ship from the National Science Foundation. I thank J.W. Eberhard, P,M. Horn, and P. Dutta for sending me preprints of their work on the temperature depen­dence of 1/! noise,

lJ. Clarke and R. F. Voss, Phys. Rev. Lett. 33, 24 (1974). 2J. Clarke and T. Y. HSiang, Phys. Rev. Lett. 34, 1217 (1975).

3p. F. Voss and J. Clarke, Phys. Rev. Lett. 36,42 (1976). 4R.F. VossandJ. Clarke, Phys. Rev. B13, 556(1976). 5M. Weissman and G. Feher, J. Chern. Phys. 63, 586 (1975). ('M.B. Weissman, Phys. Fev. Lett. 35, 689 (1975). 1J. W. Gibbs, Elementary Principles ill Sta tistical Mechanics (Yale University, New Haven, 1902), pp. 68-86.

8K. M. Van Vliet and J. R. Fassett, in Fluctuation Phenomena in Solids, edited by R.E. Burgess (Academic, New York, 1965), PP. 267-354.

9Equation (4) may be used to obtain an effective noise volume W~p2/Jp2(r) d3r for any noise source for which the only spatially varying factor in the fluctuation weighting function is p(r).

IOJ.W. Eberhard and P.M. Horn, Phys. Rev. Lett. 39, 643 (1977).

lip. Dutta, J.W. Eberhard, and P.M. Horn (unpublished).

Michael B. Weissman 194

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