Appl. Phys. 22, 185-187 (1980) Appl ied Physics

�9 by Springer-Verlag 1980

Analysis of Resistance Fluctuations Independent of Thermal Voltage Noise

H. Stoll

Max-Planck-Institut fiir Metallforschung, Institut fiir Physik, D-7000 Stuttgart 80, Fed. Rep. Germany

Received 8 February 1980

Abstract. New methods for thermal-equilibrium investigations of atomic vacancies or of dislocation movements in metals require measurements of minute resistance fluctuations. The accuracy of such measurements is limited by thermal noise from the resistor. The present paper proposes a method for analysing resistance fluctuations independent of voltage noise.

PACS: 06, 07, 61.70

By analysing electrical resistance fluctuations, Celasco et al. [1] were able to measure properties of atomic vacancies in aluminum in thermal equilibrium. The same technique was later used by Bertotti et al. [-2, 3] in a study of dislocations during plastic deformation. Recently, Venkataraman and Balakrishnan [4, 5] have shown that the analysis of resistance fluctuations ("Resistance Fluctuation Spectroscopy") can be ap- plied to a study of many other physical properties, provided that very small resistance fluctuations can be detected. Usually a resistance fluctuation, AR(t), is determined by applying a constant current I 0 to the resistor and measuring the fluctuation in voltage across the re- sistor, i.e. I o �9 AR(t) (Fig. 1). The thermal voltage noise V(t) of the resistor [which may be much larger than I o �9 AR(t)] interferes with this measurement. Therefore, V(t) is usually determined in a second measurement without current (I o = 0), and the first measurement is corrected accordingly. However, V(t) may be different in the two measurements (e.g. due to a temperature change of the resistor after switching off the constant current) which limits the accuracy of AR(t) for small resistance fluctuations. Methods [6-9] have been published which suggest the replacement of the dc current I o by an ac current Io.cOscot. In that way a low-frequency resistance fluctuation is shifted to frequencies near to the carrier frequency co. This can be helpful in eliminating a 1I f

noise of the amplifier. But if the spectrum of V(t) is frequency independent (e.g., thermal noise) this meth- od also requires the substracting of two measure- ments, one with and one without current. The present paper proposes a method for determining AR(t) independent of V(t) by a single measurement. The basic idea is to utilize two carrier frequencies which are modulated simultaneously by the resistance fluctuation and demodulated in a special manner.

Analysis

The arrangement for the measurement is shown in Fig. 2. The outputs of the two oscillators

Ul( t ) = U 0 cos((/) 1 t + q)l) (1)

and

U2(t) = Uo c~ + (P2) (2)

constant- current s o ulroce t R , A R (t)

V(t)

AU(t)= AR(t).Io. V(t)

Fig. 1. Usual circuit for measurement of resistance fluctuation, AR(t). The output A U(t) is also influenced by a voltage fluctuation V(t) (e.g. by thermal noise of the resistor)

0340-3793/80/0022/0185/$01.00

186 H. Stoll

oscillator 01

surT

oscillator 02

V[t) cror

Uo co5 (w2t +~o 2)

" " I '''''r I "

] 2 channel Fourier - anQlyser

amplifier

Fig. 2. Circuit for measurement of a resistance fluctuation AR(t) independent of a voltage fluctuation V(t)

amplitude ~ amplitude modulated carrier amplitude modulated carrier

I \ / ~ ' ~ | tower sideband ~ upper $1deband ~ upper skJeband speclrum

Fig. 3. Frequency spectrum of the two carrier frequencies behind the Wheatstone bridge [modulated by the resistance fluctuation AR(t)]. A broadband voltage fluctuation V(t) interferes with the signal throughout the entire frequency range

are summed in network N 1. (Pl and ~0 z are the (random) phases of the oscillators at the onset (t = 0) of each measuring cycle. The signal [Ul ( t )+ U2(t)] is applied to a Wheatstone bridge. If one resistor R+AR(t) of the bridge is fluc- tuating, both carrier frequencies suffer an amplitude modulation. (If all four resistors R are allowed to fluctuate, the modulation is further increased. To simplify calculations, only the fluctuations of one resistor will be considered and AR~R is assumed.) The signal at the output of the bridge is

U(t)=�89 lTc~ + ~--~ AR(t)]" Uo cos(0)l t + (Pl)

+ Ic~ + ~---R AR(t)]" Uo cos(0)zt + q)2)+ V(t)} �9

(3)

The parameter c~ describes a possible asymmetry of the configuration. V(t) is the thermal noise of the resistors or another stationary random voltage fluctuation. The spectrum of V(t) can extend above 0)t and 0)z.

The signal spectrum behind the bridge is shown in Fig. 3. In order to allow room for the sidebands, the frequencies 0)1 and 0) 2 should be chosen higher than the maximum frequency of the resistance fluctuation, COma x (i.e., the maximum frequency with measurable amplitude), and their difference 0)2-0) 1 , should be higher than twice that maximum fluctuation fre- quency. The signal is separated into two parts by a crossover network, N z (with crossover frequency be- tween 0)1 and 0)2), and both components are separately amplified by a factor G. Two double-balanced mixers mix (multiply) them with the original frequencies. A high-pass filter suppresses the dc part of the signal, which is present for ~ 40, and a low-pass filter sup- presses frequencies higher than 0)re,x, which are pro- duced in the mixer. The signal is analysed using a two- channel Fourier analyser, which determines the cross spectrum (or the cross correlation) between the two output channels. In order to show that the cross spectrum (or cross correlation) agrees with the spectrum (auto corre- lation) of the resistance fluctuation AR(t), and is inde- pendent of V(t), when averaged over a sufficiently large number of cycles of the Fourier analyser, we consider separately the influence of V(t) and AR(t) on the cross spectrum (cross correlation).

a) The outputs of the double-balanced mixers are channel 1 :

�89 [~+ 2~ AR(t)]'Uoc~ +(P~) } �9 cos(0)lt + ~ot), (4)

Resistance Fluctuations Independent of Thermal Voltage Noise 187

channel 2:

�89 [~+ ~--~ AR(t))'Uocos(cozt +~o2)}

�9 cos(o~2t + ~o2). (5)

[In general V(t) will be altered by the crossover network N2, yielding Vl(t ) and V2(t). ] The phases qo t and ~02 are r andom for every cycle of the Fourier analyser. When summing over many cy- cles, the averaged cross spectrum (cross correlation) becomes independent of V(t): The cross correlation function C(z, q01, cp2 ) due to V(t) is

G z / C('c, cpl, q)2) = --s 1] \

�9 V2(t) c~ + (P2)~ �9 (6) After averaging over the cycles it becomes

G2/ ] 2~ c(~) = -4- _(, v~(t + ~ ) ~ ,!v cosCo~(t + ~3 + ~o 1] d~o~

-~0

�9 V2(t)~2fcos(o)2t+q)2)dcP2~). (7) 0 ~ It

= 0 Thus C(z)=0, since the averaging over t and ~o may be interchanged. It can be also shown that the cross correlat ion between the term due to V(t) and the other output terms averages to zero. This proves that the cross spectrum, which is the Fourier t ransform of C(z), is independent of V(t). b) The outputs of the double-balanced mixers due to the resistance fluctuations AR(t) are channel 1 :

1 2 �89 (COxt+~o 0

_ G. ~Jo AR(t) + (dc + higher frequencies), (8) 8R

channel 2:

1 2 �89 ((D2 t "k- ~02)

_ G ' U o

8R - - - A R(t) + (dc + higher frequencies). (9)

After the dc and the higher-frequency components are removed by the filters, the cross spectrum (cross correlation) between channel 1 and channel 2 gives the spectrum (auto correlation) of the resistance fluc- tuat ion AR(t). As demonstra ted above, the result is not influenced by the voltage noise V(t).

Acknowledgements. The author would like to thank Prof. P. Mazzetti, F. Fiorillo and the other members of the Istituto Elettrotecnico Nazionale for their kind hospitality and stimulating discussions during his stay at Torino, Dr. G. Venkataraman for providing him with the manuscript [4], and Prof. A. Seeger for suggesting this investigation and for his continuous interest in this work.

R e f e r e n c e s

1. M.Celasco, F.Fiorillo, P.Mazzetti: Phys. Rev. Lett. 36, 3842 (1976)

2. G.Bertotti, M.Celasco, F.Fiorillo, P.Mazzetti: Scripta Metall. 12, 943-948 (1978)

3. G.Bertotti, M.Celasco, F.Fiorillo, P.Mazzetti: J. Appl. Phys. 50, 6948-6955 (1979)

4. G.Venkataraman: Lectures delivered at the Banaras Hindu University during Nov. 1978

5. G.Venkataraman, V.Balakrishnan: Mater. Sci. Bull. 1, 59-62 (1979)

6. C.S.Bull, S.M.Bozic: Br. J. Appl. Phys. 18, 883-895 (1967) 7. J.H.J. Lorteije, A.M.H. Hoppenbrouwers : Philips Res. Rep. 26,

29-39 (1971) 8. H.Bittel, N.Miiller: Appl. Phys. 5, 283-284 (1974) 9. D.Wolf (ed.): Noise in Physical Systems, Springer Series in

Electrophysics, Vol. 2 (Springer, Berlin, Heidelberg, New York 1978)