1.1139587

ac method for measuring lowfrequency resistance fluctuation spectraJohn H. Scofield

Citation: Review of Scientific Instruments 58, 985 (1987); doi: 10.1063/1.1139587 View online: http://dx.doi.org/10.1063/1.1139587 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/58/6?ver=pdfcov Published by the AIP Publishing

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ac method for measuring low .. frequency resistance fluctuation spectra John H. Scofield AT&T Bell Laboratories, Holmdel, New Jersey 07733 (Received 24 November 1986; accepted for publication 19 February 1987)

An ac technique is described for measuring low-frequency resistance fluctuation spectra with improved sensitivity over de methods achieved by avoiding preamplifier 1/fnoise. The technique, easily implemented with decade resistors and a lock-in amplifier, allows the current noise oflow-resistance (r< 10 kfl) specimens to be measured to frequencies below 1 mHz. Use of a center-tapped, four-probe specimen geometry allows discrimination between specimen and contact noise and eliminates noise due to bath temperature variations. The technique is demonstrated in use to determine the dependence of the 1/fnoise ofCr films on film area. Measurements with simultaneous direct and alternating currents provide means to study the noise of nonlinear devices and frequency-dependent conductors.

INTRODUCTiON A. Background

Fluctuations about equilibrium are generally interesting in that they convey information about system dynamics near equilibrium. Resistance fluctuations reflect the dynamics of the physical systems to which the conduction processes are coupled. For instance, resistance fluctuations ofSn films just above their superconducting transition reflect the dynamics of heat flow. I It has been suggested2 and verified3 that resis-tance fluctuations of thin metal films may probe the kinetics of chemisorption. Others have used resistance fluctuation spectra of Al films to probe vacancy creation/annihilation4 and dislocation dynamics.s More recently, resistance fluctu-ations ofNb films have been shown to reflect the dynamics of hydrogen diffusion. 6

Moreover, a wide variety of conductors exhibit a phe-nomenon known as "1/f noise," i.e., low-frequency resis-tance fluctuations .5r(t) having power spectra Sr ( f) 0:: 1/ fa, a = 1.7 The apparent decrease of the relative noise magni-tude is, (f)jr with increasing conductor size makes it diffi-cult to study the phenomenon in bulk materials, and accen-tuates the potential practical problem of 1/f noise in submicron conductors. In metals the 1/f-resistance fluctu-ations are so low in level that they have been measured only for thin films U

B. Four probe de noise measurement The standard dc four-probe method for measuring resis-

tance fluctuations is illustrated in Fig. 1.8 9 17 A constant current 1= 10 = EoIR, established by a stable battery Eo in series with a large, stable ballast resistance R = Rl + Rz flows through the current leads of a fluctuating specimen resistancer(t) = = I (r> is removed by a dc blocking capacitor C (i.e., a high-pass fil-ter) so that fluctuations (jv(t) = v(t) - (v) in the voltage drop v across the specimen may be amplified and measured. The effect of fluctuations in the current contact resistances SI andS2 (i.e., contact noise) is reduced by choosing R II'> 1. For sufficiently large R Ir (see Appendix A) the power spec-trum Sv (j;Io) of the voltage oVen at the preamplifier out-put is

Sv((;lo)::"GZ[S~(/) +16S,(/)], (1) where S, (I) is the power spectrum of orU) =01'1(1) + Dr2 (t) , S~(/) is the power spectrum of the

background noise (amplifier + thermal noise), and G is the preamplifier voltage gain. The excess noise S v (I; 1)-S v (1;0) should be measured for at least two different values of the ballast resistance R to determine whether the current is sufficiently constant to eliminate contact noise. Indepen-dence of calculated S, (j) on the ratio R Ir is proof that the contact noise has been eliminated. 8

The above four-probe dc method has several limitations, especially when applied to low resistance conductors (e.g., metal films). First, measurements are limited to frequencies above the RA C knee of the blocking capacitor C and amplifi-er input impedance RA Second, fluctuations DE in the bat-tery voltage show up directly in the measured noise, prohib-iting the use of power supplies. Third, the sensitivity is limited by S ~ (f), which at low frequencies is dominated by preamplifier 111 noise, which greatly exceeds the intrinsic thermal noise. While an impedance matching transformer may be used to reduce this latter problem its frequency re-sponse will then typically limit measurements to frequencies above 1 Hz.18 And finally, one must always worry about thermoelectric effects. electromigration, and resistance fluc-tuations associated with bath temperature fluctuations.

3 1 6 5ro G 4 21-' 0-----1

FIG. 1. Circuit for the usual four-probe noise measurement. The six-probe device is the noise specimen of Fig. 2. The ballast resistance R is split into two equal pieces R 1 and R2 for comparison with the five-probe bridge noise measurement.

986 Rev. Sci. Instrum., Vol. 58, No.6, June 1987

(a)

L

1 ( b)

3 !1 7)1 1 " ~ -

r 1 ~

6 ~o ~ 5 ~ r.. "". 'V

-

~ r 2

4 ~2 7)2 2 ~ ......

'"" vv 'v

-

r '= r 1 + r z

FIG. 2. Detail of the multiprobe noise specimen. (a) Geometry and (b) equivalent circuit. The usual four-probe resistance l''''''rJ + r2 (with I'J ""rz ) of width w and length L is obtained by ignoring the center contacts (5 and 6).

I. MEASUREMENT IMPROVEMENTS A. de bridge

The use of a bridge circuit to remove the average voltage drop I (r> avoids the first two limitations mentioned above. 17 There are a variety of ways to place a four-probe resistor in a Wheatstone bridge. In each case contact noise may be eli-minated only at the expense of increasing the input imped-ance (seen by the preamplifier) and thus the background noise. This problem may be avoided by using a symmetric four-probe resistor with center tap as shown in Fig. 3.19 When balanced, the bridge error signal is insensitive to fluc-tuations DE in the supply voltage or, equivalently, to fluctu-ations in the center contact resistance S. As with the stan-dard four-probe measurement large ballast resistors R j and

Resistance fluctuation spectra 986


R., "0

3 1

6 5

4 :2

II 2I

FIG. 3. Circuit for the bridge measurements. The six-probe device is the noise specimen of Fig. 2.

R;t reduce the effect of fluctuations in the current contact resistances 51 and 52' In the limit Rj/rj 1 (j = 1,2) the input impedance to the preamplifier is I" (thus the back-ground noise S ~ ( f) is the same as with the standard four-probe method) and the measured voltage spectrum is

Sv(f;Ioh",G 2 [ S~(f) +nS r- (I)], (2) where S ; (f) is the power spectral density of the difference 8r_(t)=8r2 (t) -8r1 (t). This has the added advantage of rendering the measurement insensitive to bath temperature fluctuations. If (Br /j1' 1> = 0 then S r- (I) = Sr ( f) and the measured spectrum is the same as before. If 8rz and or! are correlated, as is the case for noise associated with hydrogen diffusion in Nb films then S r- (1) and Sr ( f) may differ significantly. 6

B. Background and preamplifier noise While the bridge arrangement improves upon the stan-

dard four-probe method, the sensitivity is still limited at low frequencies by preamplifier noise. 18 This is best illustrated with a specific example. Consider the background noise of a r = lOO-n resistor in a dc bridge arrangement measured with a Princeton applied research (PAR) model 113 low-noise preamplifier. The intrinsic thermal noise, 4kTr = 1.6x 10-- 18 V 2/Hz, is indicated by the horizontal solid line in Fig. 4. The actual measured noise (corrected for a G = 104 ) is plotted as curve (a) in the same figure. At high frequencies the background noise exceeds the thermal noise by more than 15 dB, and only gets worse at lower frequen-cies. Since the 100-0 sample resistance is poorly matched to the lOO-MH input impedance of the preamplifier, an imped-ance matching transformer may be used to give (in princi-ple) noise-free voltage gain before the preamplifer. 18

Figure 5 summarizes the noise added by a PAR 113 preamplifier for various input impedances and frequen-cies.20 The noise figure (NF) of the amplifier is a measure of the amount of noise (referred to the input) added by the amplifier over and above the thermal noise of the impedance at its input at room temperature; NF( J,r) = lOdB logw( SeC/)/4kTor], where To=290 K. Thehor-izontal dashed line for a source resistance of 100 n repre-

987 Rev. Scl.lnstrum., Vol. 58, No.6, June 1987

;> :.;.;.; ' ...... ;-.:-7.;;;.:-: ............ ' :.:.:.:.:.;.:.: 0; -.-.-.;.-, ....... .

10-12 (100m ++ ... (a) N 10-13 :r: I"~ +"'+ ... , , ++

N 10-14 ' ... " ++ > TYPICAL '.:++ x

(\! 10- 15 NOISE FIGURES ~, iO (!) \ ::: '10- 16 (b} -

(1M.n) " '. IC) o 10-17 ' ...... '0 .... (e) " f

" .... _- 'eo

...... ...

"" '.

, ...... 10-18

"t: >

" (I) 10-19

10-3 10-2 10-1 10 101 f (Hz)

FIG. 4. Background noise measurements with a 100-0 load resistance using a PAR 113 preamplifier. Curves (a) and (b) are the measured de back-ground noise; (a) is measured with the preamplifier directly while (b) is measured with a PAR 190 impedance matching transformer and preampli-fier. The two dashed curves represent the expected background noise for these measurements (based on manufacturer's "typical" noise figure con-tours}for an input impedanceoflOO n (direct coupled) and 1 Mfl. (trans-former coupled). Curve (c) is measured atthe output of the lock -in amplifi-er (reference at 65 Hz) and illustrates the background noise using the ac technique.

sents the typical amplifier noise for this dc measurement; these same data give the upper dashed curve in Fig. 4. The dashed curve and measured background noise [curve (a)] disagree simply because the noise figure contours of Fig. 5 are not actually measured from the preamplifier used, but rather are "typical." The PAR 113 NF contours show a min-

FIG. 5. Typical noise figure contours for the PAR model 113 low-noise preamplifier. The horizontal dashed lines represent expected preamplifier noise levels when used directly (100 fl.) and with a 100: 1 impedance matching transformer (1 M!!) with a load resistance uf lOOn (see text) .

Resistance fluctuation spectra 987


imum (called the "eye") near a source resistance re = 1 MH and a frequency of 100 Hz. An impedance matching trans-former with turns ratio (NzINI )2 = re1r may be used to shift the operation of the PAR 113 with a 100-rt bridge to the 1-MH horizontal line (see Fig. 5); these data give the lower dashed curve in Fig. 4. The measured background noise us-ing a PAR 190 (100: 1) low-noise transformer before the preamp (total gain = 104N21Nl = 106 ) is plotted as curve (b) in Fig. 4. Note that the transformer adds some noise of its own, much of which may be eliminated by magnetic shielding, vibration isolation, and cooling the transformer. 2 1 Properly impedance matched, the background noise ap-proaches the thermal noise between 1 and 100 Hz. Below 100 mHz the frequency response of the transformer (not shown) drops to zero so that it is no longer useful. The roll-off above 100 Hz is also due to the frequency response of the trans" former. A flatter frequency response may be obtained by slightly mismatching the impedances of the bridge and PAR 113, but at the expense of additional preamplifier noise. The trade-off is usually worth it for room-temperature measure-ments, but may not be if the sample is cold.

C. ac bridge with phase-sensitive detection Impedance matching allows the preamplifier to be used

at its optimum input impedance. However, to measure resis-tance fluctuations down to zero frequency the preamp must still be used far away from its optimum frequency. While cross-correlation techniques can reduce preamplifier noise their use below 1 Hz is limited.22 An ac technique overcomes this difficulty by shifting the frequency at which the pream-plier is used from zero (dc) up to the carrier frequency fo. II With an alternating current 1 = i(t) = io sin(21T fot) the re-sistance fluctuations modulate the carrier to produce noise sidebands. The preamplifier will contribute little noise in a bandwidth I:::.f if 10 is chosen within the eye of its NF con-tours. After amplification the carrier is demodulated to re-trieve the desired low-frequency resistance fluctuations. The point is that an ac current allows the preamplifier to be used near its optimum frequency. The lock-in amplifier was de-signed to perform precisely this function.

Let the voltage source in Fig. 3 be E (t) = eo sin (21T lot). For a sufficiently low carrier frequency 10 capacitance effects may be neglected so that the bridge error signal is approxi-mately 8r _ (t)io sin (21Tlot) , where io=eofCRj + rj ) (j = 1,2). The bridge error signal is inserted into the signal input of a lock-in amplifier with detection at a phase angle 15 with respect to the bridge current. A block diagram of the setup is shown in Fig. 6. For frequencies I

will be mixed to de by the higher harmonics of the PSD square wave (see Appendix B).

II. IMPLEMENTATION OF FIVEMPROBE ac METHOD A. Instrumentation

The original realization of this technique used a parallel combination of 1.000 and 10.000 kfl, noninductively wound, SoW, 2-ppmiC stable PRC (Precision Resistor Corporation) resistors mounted in large aluminum heat sinks for baUast resistor R I . Ballast resistor R2 was similarly formed but with a GenRad mode11433 decade resistor pro-viding the (nominally) lO-kH shunt. The entire bridge was enclosed in an aluminum box surrounded with 2 in. styro-foam on all sides. Subsequent adaptations of the technique have used two rack-mounted GenRad 1433 decade resistors for RI and R z, trading stability for versatility. To achieve 1-ppm bridge balance at/o = 700 Hz it was necessary to shunt Rz with NPO trimmer capacitors of a few hundred pF.

Bridge current was supplied either by the lock-in itself or by a Hewlett-Packard (HP) 3325A frequency synthesiz-er, typically operated at 700 Hz, In some cases a Kepco oper-ational power supply was used after the synthesizer to pro-vide a very low impedance voltage source. The bridge error signal (v 2 - v1 ) was fed into either a PAR 118 (l00 H

--,-------.---r-l T " 10K 1

I

1 J I

f (Hz)

FIG. 7. Noise measurements in the presence of bath-temperature fluctu-ations. The solid lines are guides to the eye. (MC) Excess noise from a car-bon resistor ( = r J ) and a commercial metal film resistor ( = r2 ) at T = 10 K (see text). (Ce) Exeess noise from two matched carbon resistors mea-sured under similar conditions.

as has been reported for continuous metal films. 10 The ac technique has been used to measure room-tem-

perature 1/fnoise from more than 60 continuous metal film specimens from 23 depositions of Ag, AI, Au, Cr, Cu, Ni, Mo, and W. 26 At sufficiently low current levels the excess noise [Sv ({;i)-Sv ((;O)] ex i6. At higher current levels stronger current dependences were sometimes observed ac-companied by instabilities in the bridge balance. They are thought to be associated with joule heating and may be con-veniently recognized and avoided by attention to bridge sta-bility. With isolated exceptions the 1/1 noise of specimens prepared from films of a given deposition were found to be reproducible to within about 30%.27 Moreover, with similar precision the 1I/noise spectraS r- (I) of specimens (differ-ing only in lengthL and width w) are found to vary inversely with their area, A = Lw. This is illustrated by the excess noise offour specimens prepared on the same substrate from a (120 20) -nm-thick Cr film of resistivity p = (115 20) f..tn cm. The size-normalized, relative resis-tance fluctuation spectra, NaS; (/)/?, for these four specimens are plotted in Fig. 8, each with a different symbol. (Na is the number of atoms in the specimen volume 0, = Lwh.) The figure clearly shows that, at all frequencies, the normalized spectra differ by no more than 50% while the areas (Lw) vary by a factor of 200. The inset of Fig. 8 shows a log-log plot of the relative noise level [IS r- (/)/?}f= 1 Hz vs Na A least-squares fit to the four points gives a slope of - 1.05, not significantly different from a - 1.0 slope. The same behavior has always been ob-served for other sets of specimens with the area commonly varying by a factor of 16. These results are so consistent that checking for an A _.( dependence serves as a useful test to eliminate noise due to extraneous sources like bath-tempera-


No 10 11 14 102 r---'--- 10 10- 13

p=115/-l(lcm N -1.05 h=120nm ;::. a N

ffOl~ .... 10 ~ .--... ::=... ., 0-N :r: "' ... 10-2 I I I 110-17 "- w '--"" D 0.7 ~f-1.141 u') o 1.5 80 " 1 Z 10-4 x 4.3 300 9.3 600 ~

(Hz)

FIG. 8. Log-log plots of the size-normalized relative resistivity fluctuation spectra of four Cr specimens fabricated from the same film. The symbols, lengths L (inllm), and widths w (in{tm) are tabulated in the figure. The figure inset is a log-log plot ofthe relative noise level [is, (f)/r2]f~ \H,. vs specimen size (Na ). The distance between either voltage probe and the cen-ter contact is L /2. In practice 1', "",1'2 lind R, =R2; the specimen resistance r""'Y j + '2'

ture fluctuations. It should be noted, however, that these data are in no way evidence for a bulk noise origin since both the surface area and volume vary together. Evidence distin-guishing bulk from surface origins is obtained only by vary-ing the film thickness (Le., the surface to volume ratio).28 The w -I dependence here places an upper limit of about 1 f.lm on the correlation length of the local resistivity fluctu-ations in the frequency range considered. We have not expe-rienced the factor of2-1 0 irreproducibility reported by other laboratories. 9,29

The III noise of a metal film was never found to depend on the carrier frequency 10 nor was significant excess noise observed for 8 = 90. The carrier frequency was systemati-cally varied between 80 Hz and 5 kHz for one gold film; measurements agreed with each other, and also with the re-sults of a de five-probe bridge measurement. Results for an Al specimen were typical of out-of-phase noise measure-ments. In this case the power spectrum of the out-of-phase excess noise was less than 1/ 100th the level of that of the in-phase excess noise.

Some lock-in amplifiers (e.g., PAR 124A) maybe oper-ated in a fiat frequency response mode and the amplified signal monitored both at the input and output of the phase-sensitive detector (i.e., before and after demodulation). With a direct bridge current If) the PSD-input may be spec-trum analyzed to obtain the dc five-probe measurement of Eq. (2). With a bridge current 10 + io sin(21Tji + 0) the spectrum of Eg. (2) is available at the PSD input while the spectrum of Eq. (3) is available at the PSD output. These two signals have been used to cross correlate the 1I/noise and 1/ Ll/noise l2 of two matched 50-n carbon resistors (sim-ilar to those of Fig. 7). With a SOD-Hz carrier frequency 10 the measured coherence3o r (f) between the two signals reaches a peak value r( 5 Hz) = 0.9975 falling either side to Resistance fluctuation spectra 990


values of y2(256 Hz) = 0.975 and y(2 Hz) = 0.984. The decreased coherence at the low-frequency end is expected due to the presence of preamplifier 1// noise and the fact that the lock-in preamplifier is ac coupled to its input. y( f) is also expected to decrease at higher frequencies where the two signals are dominated by thermal noise. The results are completely consistent with those reported by Jones and Francis.l:l

IV. DISCUSSION Several groups have reported observations of unrecti-

fied 1// noise from resistors passing an alternating cur-rent31- 34 while others have detected noise sidebands about harmonics of the carrier?5 More recent efforts have failed to confirm these reports.36 Such effects suggest nonlinear mechanisms; a simple model involving diodes and resistors has been suggested.37 As mentioned earlier, all previ.ous ac noise measurements have used only two probes and have been subject to contact noise. It is easy to imagine that some rectification would take place at the contact interface between two materials. Such effects have not been observed with five-probe ac measurements on metal films.

The Cornell group has performed measurements on tunnel junctions with slightly nonlinear I-V characteris-tics.38 Preliminary results suggest that complicated 10 depen-dences of S v (f;Io) measured with a direct current 10 resolve into simpler io and 10 dependences of S v (/J) determined by ac measurements. The junctions show noise sidebands about the carrier's second harmonic, easily measured with the "2/0" reference mode of the lock-in.

In principle, this ac technique may be used to study the excess noise of any resistance. However, preamplifier 11/ noise tends to be most problematic for low resistance sam-ples. Moreover, stray capacitances complicate analysis for high resistances. The circuit acts like a simple ac resistance bridge as long as 21T'/0-r~ L With r= 100 n the bridge tends to become unstable for fo > 10kHz; dc noise measurements are preferred for r> 10 kn.

Proper impedance matching and choice of carrier fre-quency allows resistance fluctuations to be measured down to I-mHz frequency with preamplifier noise minimized. With the PAR 116 preamplifier this amounts to no more than a O.OS-dB NF at 1 kHz, or 1 % of the room-temperature thermal noise of the sample resistor. In principle this can be accomplished with a PAR 190 low-noise transformer for 0.1 ft

the limit as (r)IR -0, the current 1-> (1), and 8uI -> lor, i.e., the current does not fluctuate and there is no contact noise.) In addition to the above current noise (SUI)' Nyquist (or thermal) noise 8uN contributes to 8u. The preamplifier also injects noise t)v A (as referenced to its input) so that the voltage 8 V at the preamplifier output is

t)V = G(6uo + 8uI ), where G is the preamplifier voltage gain and 8uo = OUN + t)uA is the "background noise."

Assuming (8d)R') = 0, the power spectral density S v (/;1) of 8 V(t) for a current fis

Sy(f;l) =G2{S~(f) + (1)2[S,(/) + < (r)IR ')2SR',(f)J),

where S~ (f), Sr (/), and SR" (/) are the power spectra of oUo (t), ore t) , and oR I (t). The background noise spectra S ~ (f) = G - 2 S V (};1 = 0) are measured with zero current; current noise spectra are subsequently obtained from S y (/;1) by subtracting background. To distinguish Sr (f) from S R' , (f) it is necessary to determine the current noise spectra for two or more ratios ( (r) I R ').8

For comparison with the bridge circuit it is useful to define Ozj =R jl(R j + (1') [orj - rj )IR j)85j ] (j = 1,2) so that 8uI = 1(&1 + &2) =It)z. With these defi-nitions the power spectrum S v (/;1) of the measured voltage 8 VU) is just

Sv(f;l)/G 2 = S~ (j) + 12Sz (f), whereSz (f) is the spectrum of 8z(t). For sufficiently large ballast resistors (i.e., as (r)IR-.O) 8z--or and Sz (f) ->Sr (/).

The stability of the battery Eo and ballast resistors R J and R2 may be determined by replacing the specimen with equivalent noise-free resistors (e.g., wire-wound) and mea-suring S v (/;1), both for zero current and for the maximum current to be used, Imax. The necessary stability in E is usual-ly not achieved by power supplies. We have found the switch

I odd

contacts of most decade resistors to be too noisy for use with continuous metal films.

APPENDIX B: SPECTRAL DENSITY MEASUREMENTS WITH A LOCK~IN AMPLIFIER

Recall that in the five~probe bridge measurement, speci-men resistance fluctuations (or2 and or j ) modulate the bridge current to produce a bridge error signal &(t) =1(t)or _ (I), where 8r _ = orz - brIo Here we calcu-late the power spectral density Sv (U)) of the output ofa lock-in amplifier for an input signal

ou(t) = ovo(t) + ioor _ (t) sin(21T fot + 0). The lock-in consists of an ac amplifier, a phase-sensitive de-tector (PSD), and a de amplifier (see Fig. 6). The ac and dc amplifier stages are characterized by complex transfer func-tions HA (U) and Hv (w). The PSD stage multiplies the time-domain output of the ac amplifier by a square wave

odd e(t) = (4ec/1T) L (lln)sin[n(U)ot + 8)]

n~ I

offrequency fo = U)0/(21T) , amplitude eo. and fixed phase 8 with respect to the carrier. The Fourier transform of the output voltage 8 V(w) = fe

Hans introduce less than I-dB error for 1 UJo the output of the lock-in's own low-pass filter is fed into a Unigon digital filter with a 120-dB/octave roUoff.

'M. B. Ketchen and J. Clarke, Phys. Rev. B 17, 114 (1978). 2G. S. De and H. SuM, Surf. Sci. 95, 67 (\980). 3M. R. Shanabarger, J. Wilcox, and H. G. Nelson, J. Vac. Sci. Technol. 20, 898 (1982).

4H. Celasco, F. Fiorillo, and P. Mazzetti, Phys. Rev. Lett. 36,38 (1976). 'G. Bertotti, H. CeJasco, F. Fiorillo, and P. Mazzetti, J. App!. Phys. 50, 6948 (1979); also see G. Bertotti and F. Fiorillo, in Noise ill Physical Sys-tems and 11/ naive, edited by M. Savelli, G. Lecoy, and J. P. Nougier (North-Holland, Amsterdam, 19I.H), p. 339.

"J. H. Scofield and W. W. Webb, Phys. Rev. Lett. 54, 353 (1985). 7M. Nelkin, in Chaos and Statl:5tical Mechanics, Springer Series in Synerge-tics, edited by Y. Kuramoto (Springer, Berlin, 1984).

"C. Leemann, M. J. Skove, and E. P. Stillwell, Solid State Commun. 35, 97 (1980).

"1'. Dutta and P. M. Horn, Rev. Mod. Phys. 53,497 (198!). IOJ. W. Eberhard and P. M. Horn, Phys. Rev. Lett. 39, 643 (1977); also

Phy. Rev. B 111, 6681 (1978). liJ. H. J. Lorteije and A. M. H. Hoppenbrouwers, Philips Res. Rep. 26, 29

(1971 ). 12H. Sutcliffe, Electron. Lett. 7, 160 (1971). uB. K. Jones and J. D, Francis, J. Phys. D 8,1172 (1975). "B. K. Jones and J. D. Francis, J. Phys. D 8,1937 (1975). 15R. Rzemien, C. K. Iddings, and W. F. Love, Rev. Sci. lnstrum. SO, 488

(1979). 16H, Stoll, App!. Phys. 22,185 (1980). I7S. Demolder, H. Vandendriessche, and A. Van Calster, J. Phys. E 13, 1323

(1980).


..'...-:.;N.. ... .:. . ;....'.:..,: . :~ . .:..; . 'l'; ...... 7,> - '.-. ........ .-; . :'. ; ': - .. - ;' ...... ~ -.- -,"

ISS. Letzter and N. Webster, IEEE Spectrum 7,67 (1970). lOR. F. Voss and J. Clarke, Phys. Rev. B 13, 556 (1976). 2"These noise figure contours were reproduced (with permission) from the

operation and service manual of the Princeton Applied Research model 113 low-noise preamplifier.

21D. E. Prober, Rev. Sci.lnstrum. 45,849 (1974). 22Significant reduction of preamp noise by cross correlation is time consum-

ing. For instance, to reduce amplifier noise at 0.1 Hz by 20 dB (factor of 102 ) requires averaging of N =- 10" time records, each 20 s long; total time = 6 h. See Ref. 17 for a description of the cross-correlation technique.

23Conductivity fluctuations generally imply both resistive (8 = 0) and re-at,tive (8 = 90') components due to Kramers-Kronig relations. Here we consider only conductors with a negligible imaginary component of the conductivity. See Ref. 15 for discussion of this.

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1.1139587

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