Digital measurement of resistance fluctuationsJ. S. Moon, Arshia F. Mohamedulla, and Norman O. Birge Citation: Review of Scientific Instruments 63, 4327 (1992); doi: 10.1063/1.1143732 View online: http://dx.doi.org/10.1063/1.1143732 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/63/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High speed digital holography for density and fluctuation measurements (invited)a) Rev. Sci. Instrum. 81, 10E527 (2010); 10.1063/1.3492423 Measurement errors in the mean and fluctuation velocities of spherical grains from a computer analysis ofdigital images Rev. Sci. Instrum. 75, 811 (2004); 10.1063/1.1666989 Surface diffusion measurements by digitized autocorrelation of field emission current fluctuations Rev. Sci. Instrum. 65, 3707 (1994); 10.1063/1.1144496 A method for doping fluctuations measurement in high resistivity silicon J. Appl. Phys. 71, 3593 (1992); 10.1063/1.351389 ac method for measuring lowfrequency resistance fluctuation spectra Rev. Sci. Instrum. 58, 985 (1987); 10.1063/1.1139587

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Digital measurement of resistance fluctuations J. S. Moon, Arshia F. Mohamedulla,a) and Norman 0. Birge Department of Physics and Astronomy and The Center for Fundamental Materials Research, Michigan State University, East Lansing, Michigan 48824

(Received 16 March 1992; accepted for publication 23 June 1992)

We describe a digital technique for measuring the spectral density of resistance fluctuations with simultaneous background subtraction. The technique implements either of two equivalent ac bridge techniques that rely on extremely good orthogonality between the two channels of a dual-phase lock-in amplifier. We use a* digital signal processor (DSP) chip and a personal computer in place of both the lock-in amplifier and spectrum analyzer. The digital system can measure sample l/f noise that is 100 times smaller than the experimental background noise; this performance equals or surpasses that of any analog system. The system is low cost, very flexible, and can function as a stand-alone digital lock-in amplifier or low-frequency spectrum analyzer.

1. INTRODUCTION

A wide variety of electrically conducting materials ex- hibit resistance fluctuations with an approximately l/f power spectrum.’ The standard procedures for measuring l/f noise involve passing a current through the sample, thereby converting resistance noise to voltage noise by Ohm’s Law. The chief difficulty in such measurements is that the noise from the sample may be hidden by back- ground noise-either Johnson noise or preamplifier noise. Of course the signal from the sample can be increased simply by increasing the measurement current, but this is often unacceptable due to Joule heating. Several methods have been used to reduce the background noise. ac bridge techniques* reduce the preamplifier noise contribution by moving the signal frequency away from the l/f noise tail of the amplifier. These techniques also allow the use of step-up transformers to increase the sample-to-preamp noise ratio for low-impedance samples. Cooling these transformers further reduces the background noise.3 If the background is still not negligible, it can be determined in a second measurement without current and then subtracted from the first total noise measurement. This procedure has several drawbacks. First, if the measurement current heats the sample, then the sample Johnson noise will be different in the two measurements. Second, the background may have long-term drifts due to, for example, changing lead resistances as the cryogen level drops in the sample cry- ostat. The Appendix discusses a third possible drawback of the zero-current background measurement.

The above difficulties can be surmounted with tech- niques that measure noise and background simultaneously. Several such techniques have been discussed in the litera- ture. The two-amplifier cross-correlation technique4 rejects the preamplifier contribution to the background, but does not reject the sample Johnson noise. The double-frequency ac method5’6 suppresses all background noise by correlat- ing the outputs of two lock-in amplifiers operating at dif- ferent frequencies. While this method can be very power-

‘kb-rent address: Department of Electrical Engineering, Michigan State University, East Lansing, MI 48824.

ful, it is often unsatisfactory because the bridge circuit must be in balance at both frequencies.

There are two methods that subtract background noise using the ac bridge method with a single drive frequency. First, one can simultaneously measure the in-phase (0”) and quadrature (90”) signals from the bridge using a dual- phase lock-in amplifier. The power spectrum of the former contains both sample resistance noise and background, while the power spectrum of the latter contains only back- ground.2 Subtracting the two spectra yields the sample noise alone. Unfortunately, commercial lock-in amplifiers tend to have large phase noise, so this technique is limited in practice. An elegant method was recently developed by Verbruggen et al.,’ who showed that the background is eliminated by correlating the two orthogonal outputs of a dual-phase lock-in amplifier set to phases of plus and mi- nus 45” with respect to the bridge current. Mathematically, these two methods are equivalent (see Appendix), so their relative performance depends only on the limitations of the physical implementation. Like the O”-90” subtraction method, the 45” cross-correlation method requires ex- tremely good orthogonality and low phase noise in the lock-in amplifier, which is not achievable with commercial instruments.* Verbruggen et al. circumvented this problem by building homemade edge-triggered analog lock-in am- plifiers.

An alternative way to achieve the extreme phase sta- bility and orthogonality required for the two techniques described above is to use a completely digital measurement system. There are several examples in the literature of dig- ital lock-in amplifiers with excellent phase stability and orthogonality.’ In addition to these requirements, a digital noise measurement system demands a large real-time cal- culation bandwidth, since it acts as both lock-in amplifier and spectrum analyzer. The recent development of digital signal processor (DSP) integrated circuits make it possible to implement such a system using only a DSP board with a small amount of memory, a digital-to-analog interface, and a personal computer.

We have developed a noise measurement system based on the Motorola DSP56001 digital signal processor. We obtained a DSP system board, an analog interface board,

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FIG. 1. Schematic diagram of noise measurement system. The samole forms the lower two arms of the Wheatstone bridge, while adjustable ballast resist&.

and a computer interface board, free of charge from Mo- torola through their DSP University Support Program.” Our only hardware cost was for a 386-based personal com- puter, which has many other uses in the lab. The only other expense was the cost of software development.”

Our digital noise measurement system has excellent background rejection, due to the extremely low phase noise and excellent orthogonality of the digital lock-in amplifier component. The system can also function as a stand-alone dual-phase lock-in amplifier or as a low-frequency spec- trum analyzer. We describe only the noise measurement system in this article because it best demonstrates the ex- cellent performance of the system.

II. PRINCIPLE OF OPERATION

The principle of operation is described in Refs. 2 and 7. We outline the main points here, and leave the mathemat- ical details to the Appendix. In the ac bridge technique, a five-terminal sample and two ballast resistors form a Wheatstone bridge, which is driven with a sinusoidal volt- age. In the limit that the ballast resistors are large com- pared to the sample resistance, the current through the bridge does not fluctuate. The resistance tIuctuations of the sample modulate the current to produce voltage noise side- bands which are detected by a low-noise differential pre- amplifier. In addition to the bridge error signal, the signal at the output of the preamplifier, V(t), contains Johnson noise and preamplifier noise

V(t) =n(t) +6r(f)io COS(W,~I), (1)

where n(t) is the background ‘noise and Sr( t) is the fluc- tuating resistance difference bet.ween the two sample arms of the bridge. As shown in the Appendix, if V(t) is de- modulated by a cosine wave in phase with the driving cur- rent, and then low-pass filtered, the resulting signal con- tains both the low-frequency -Ructuations of Sr and the tluctuations of n(t) near tic. If V(f) is demodulated with a sine wave (90” out of phase with the drive), the resultant signal contains only the background contribution. A sub- traction of the two power spectra leaves only the sample

4328 Rev. Sci. Instrum., Vol. 63, No. 10, October 1992

the upper arms are

noise contribution. In the alternative method of Ref. 7, the demodulating signals have phase shifts of plus and minus 45” with respect to the drive current. The Appendix shows that the real part of the cross-spectral density of the two signals produces the same result as the subtraction of the 0” and 90” spectra.

III. IMPLEMENTATION

Figure 1 shows our noise measurement system using the Motorola DSP boards and computer. The digital signal processor and several kilobytes of memory lie on one board. A second board contains 16-bit D/A and A/D con- verters, which communicate with the first board via a rib- bon cable. A third board lies inside the PC, and commu- nicates with the DSP board through a second ribbon cable. While the DSP board obtains its power from the PC, we supply the analog interface board with external power to reduce noise pickup from the PC.

The D/A and A/D converters operate from a single clock, assuring their synchronization. The analog board comes with three internal clocks, for sample rates of 44.1, 48, and 100 kHz, respectively. We provided an external clock to lower the sample rate to 14.4 kHz. This rate is still much greater than the typical bridge excitation frequencies of several hundred Hz, and allows more DSP clock cycles between each D/A cycle. (The DSP chip runs at 20.48 MHz.) We needed the extra DSP clock cycles to perform all of the necessary mixing and digital filtering of the signal in real time, without missing any D/A cycles.

A sine wave generated by the D/A converter excites the bridge. A reconstruction filter on the board reduces the harmonic distortion of the output sine wave. By choosing the frequency of the excitation signal to be commensurate with the D/A clock frequency, we obtain a sine wave with total harmonic distortion (THD) 140 dB below the car- rier. For a sine wave of arbitrary frequency, THD is 60-70 dB below the carrier.

The difference signal from the bridge is amplified by a low-noise preamp, such as the Stanford SR560 or the PAR 116.” The output of the preamp is low-pass filtered to

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decimation LOW pass filter

-ia

-

FIG. 2. Block diagram of digital signal processing system. The mixing, digital filtering, and decimation are performed in real time by the Motorola DSP

prevent aliasing, then digitized by the A/D converter. A single-pole RC filter is sufficient to prevent aliasing with the sigma-delta-type A/D converter.i3 Since the Motorola A/D converter is supplied with dc power levels of 0 and 5 V, the A/D inputs are dc biased to 2.5 V. Hence the signal must either be ac coupled or level shifted between the preamp and the A/D converter. While the former solution is simpler, it imposes a low-frequency roll-off on the input signal. Since we also use our DSP system as a stand-alone lock-in amplifier or spectrum analyzer with very low- frequency signals, we constructed a simple differential-to- differential level shifter from two standard op-amp circuits.

Figure 2 shows the block diagram for the digital pro- cessing of the input signal. First the phase of the signal is shifted to compensate for the overall phase shift of the experiment. The signal is digitally mixed (multiplied) by two orthogonal sine waves chosen to be either at 0” and 90” or at *445” with respect to the reference. The mixed signals are digitally filteredI and the sampling rate is decimated to reduce the number of points stored in memory on the DSP board and later Fourier transformed by the PC. The pro- cess of filtering and decimation is represented by the fol- lowing equation:

y(m) = 2 h(k)x(mn--k), k=l

(2)

where n is the decimation ratio, h(k) are the filter impulse response coefficients, N is the number of filter taps, and x and y are the input and output data streams, respectively.

4329 Rev. Sci. Instrum., Vol. 63, No. 10, October 1992

An advantage of digital filtering is that it is accurate to extremely low cut-off frequencies where analog filtering becomes awkward.

The digital filter must satisfy three criteria for our sys- tem. First, the passband must be flat, so as not to introduce errors into the final low-frequency noise data. Second, the transition band should be narrow, to minimize the number of stored data points. Insisting on too narrow a transition band, however, requires an excessive number of filter taps, and hence longer computing time at each step of the real- time data acquisition. Third, the stopband must have suf- ficient attenuation to eliminate aliasing of high-frequency background noise. We demanded at least 100 dB of atten- uation in the stopband, primarily to suppress 60-Hz har- monics that inevitably enter into ac noise measurements.

We perform the digital filtering in three nested stages to reduce the number of filter taps in any one stage.15 The advantage of a multistage digital filter is that the initial stages may have wide transition bands, as long as aliased signals do not appear in the final passband of the complete three-stage filter. Only the final stage need have a sharp transition band to minimize the number of unusable points in the power spectrum. We designed three Chebychev equiripple FIR filters14 with maximally flat passbands, us- ing the Monarch software package. These filters have 75, 51, and 40 taps, respectively. Transfer functions of the three filters we used are shown in Fig. 3. When used with a 10-5-2 decimation scheme, these filters provide over lOO-

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0

-25

-50

-75

-100

-125

0.00 0.02 0.04 0.06 0.08 0.10 0.12

10-15

10-16

10-17

10-18

Normalized Frequency Frequency (Hz)

FIG. 3. Transfer functions of the three FIR digital filters preceding the three decimations. The frequency scale is referred to the original sample rate of 14.4 kHz before decimation. The sample rate is reduced by factors of 10, 5, and 2 in successive decimation stages.

FIG. 4. Noise measurements of a pair of 480-R carbon resistors. The background (Cl) was measured with the 90” method, and the sample noise (0) was measured with the 45” cross-correlation method.

dB attenuation of signals above the Nyquist frequency of the final (decimated) sample rate.

To acquire noise data, the system runs until the mem- ory on the DSP board is full (2048 points per channel), at which time the data are sent to the personal computer. The PC restarts the DSP system, and simultaneously analyzes the data from the previous run. The PC performs fast Fou- rier transforms (FFT) of each data channel to get the l/f noise spectrum by cross correlation, or it calculates two power spectra and subtracts the results in the 0”90” method. Our program for real-time digital signal process- ing on the Motorola DSP56001 is written in assembly lan- guage, while the computer interface program is written in c,‘6

A minor limitation of the current system is due to the presence of only one processor on the DSP board. Because the DSP56001 controls all functions on the board, the out- put of the sine wave by the D/A ceases during communi- cation between the DSP board and the PC. To eliminate transients during the subsequent start-up, we run the D/A for several hundred sine periods before starting data acqui- sition.

IV. PERFORMANCE

To test the performance of our system, we measured the l/f noise of a pair of 480-a carbon resisters at room temperature. To measure the overall phase shift in the Wheatstone bridge, we first unbalance the bridge and mea- sure the large error signal. The phase shift is very small for a low-resistance sample. We then balance the bridge and measure the l/f noise with the DSP system. The measure- ments in Fig. 4 are obtained by averaging 128 runs ( z-35 minutes) with an excitation frequency of 281.25 Hz. The background noise was measured with a 90” phase difference

between modulation and demodulation.‘7 The sample noise spectrum was measured using the 45” cross-correlation technique, and is reduced by a factor of 2 with respect to the background (see Appendix). The slope of the sample noise spectrum is -0.972 rtO.018. Our system suppresses the background noise by a factor of 100. The limit is set by statistical errors due to the finite measurement time. The performance of this system equals that of the homemade electronics of Ref. 7, and substantially out-performs com- mercial analog lock-in amplifiers.“*

ACKNOWLEDGMENTS

We acknowledge helpful discussions with R. Koch, I, Scofield, and A. Verbruggen. This work was supported in part by the National Science Foundation through grant DMR-9023458.

APPENDIX: TWO-PHASE NOISE MEASUREMENTS AND BACKGROUND SUBTRACTION

First we show that the 45” cross-correlation method introduced in Ref. 7 is mathematically equivalent to the O-90” subtraction method discussed in Ref. 2. The esti- mated power spectrum S( f ) of a signal X(t), measured over a finite length of time, is proportional to the modulus squared of the Fourier Transform (or Fourier Series) of that signal.

s(s)=; Imd12, (31

where w=Zn-f and

s

T X(0) = X( t)e’“*dt.

0 14)

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Let V(t) be the signal from the Wheatstone bridge, after amplification. V(t) is given by Eq. ( 1) in the text, where the noise term n(t) encompasses both the Johnson noise from the bridge and the noise from the preamp, referred to its input. (We are ignoring the gain of the preamp for this discussion.) In the V-90” technique, the mixers (or de- modulators) are in phase and out of phase with the bridge current. If X( t) and Y(t) are the outputs of the two mix- ers, then X(t) = V( t)cos w,+ and Y(t) = V( t)sin mot. We show below that the sample noise is obtained by subtract- ing the 90” power spectrum from the 0” spectrum,

S(w) ==; [ IX(w) 12- I Y(w) 19. (5)

In the 45” cross-correlation method, the mixers are set to phases of plus and minus 45” with respect to the bridge current. If X’(t) and Y’(t) are the output signals from these mixers, then X’(t)=[X(t)-Y(t)]/v? and Y’(t) = [X( t) + Y( t)]/v”Z. With this method, the sample noise is obtained from the real part X’(t) and Y’(f):

Re[X’(w) Y’*(w)]

=i Re{[X(o) - Y(w)

of the cross spectral density of

I tx*(@) - r*(u) 11

=f [ IX(o) I*- I Y(w) [*I. (6)

The cross-correlation method gives exactly l/2 the result of the V-90” subtraction method.

Next we consider in detail the different contributions to the measured power spectrum given by Eq. ( 5). First we analyze the signal, X(t), from the 0” mixer. From Eq. ( 1 ), we have

X(t)=[n(t)+Sr(t)i~cos(wot>]cos(w~~).

The Fourier transform of X( t) is

(7)

X(0) =; [2Sr(o)+Sr(w+24 +&(o--24]

+; [n(@+oo) Sn(Q+@o) I (8)

=a+b+c+d+e. We have labeled the five terms by the letters u-e for dis- cussion. We draw attention to terms b and c, which arise from the sample noise at frequencies of 2wo. In most sit- uations, the sample noise spectrum is roughly l/f, so these extra terms are negligible if w<wo. These terms could be- come important, however, if one tries to extend the fre- quency range of the measurement to 0~:00/2 or greater, and if the sample noise spectrum falls off less rapidly than l/f?

The power, or ( X(w) ( *, contains five modulus squared term, lal*,Ibl*, etc., plus ten cross terms, ab*, a*, etc. and their complex conjugates. If the five amplitudes a-e are uncorrelated, then all of the cross terms average to zero over long times. The averaged power spectrum then con-

tains only the squared modulus terms: the sample noise at w and w * 2oo, and the background noise at w f o,,.

We now perform the same calculation for the signal from the 90” mixer, Y(t) . The result is

Y(w) = -i(b-c+d-e), (9)

where the terms b-e have the same meaning as above. The power, or ( Y(w) I 2, contains the four squared modulii of b-e, and six cross terms of various signs and their complex conjugates. When we subtract ( Y(w) I * from I X(w) ) *, we are left with 1 a ] * and those cross terms that did not cancel due to a sign change in Y(o) relative to X(w). Since the cross terms average to zero, the subtraction yields only the noise signal from the sample:

IX(w)12-lY(w)l*=)u(*=(i~4)16r(w)(*. (10)

Note that a background measurement taken with zero drive current would not include the terms I b I 2 and ( c I *, hence those contributions would appear in the final result of such a measurement. With the 90” background measure- ment, those terms are properly subtracted.

‘For a review, see M. B. Weissman, Rev. Mod. Phys. 60, 537 (1988). *J. H. Scofield, Rev. Sci. Instrum. 58, 985 (1987). 3 D. E. Prober, Rev. Sci. Instrum. 45, 849 (1974). 4A. Van der Ziel, Noise (Prentice Hall, Englewood Cliffs, 1970); see also

S. Demolder, M. Vandendriessche, and A. Van Calster, J. Phys. E 13, 1323 (1980).

‘M. B. Weissman, Ph. D. Thesis, University of California, San Diego (1976).

6H. Stoll, Appl. Phys. 22, 185 (1980). ‘A. H. Verbruggen, H. Stoll, K. Heeck, and R. H. Koch, Appl. Phys. A 48, 233 (1989).

* R. Koch (private communication). 9P. A. Probst and B. Collet, Rev. Sci. Instrum. 56, 466 (1985); N. 0.

Birge and S. R. Nagel, Rev. Sci. Instrum. 58, 1464 (1987); P. K. Dixon and L. Wu, Rev. Sci. Instrum. 60, 3329 (1989). For a history of digital lock-in amplifiers, see L. G. Rubin, Rev. Sci. Instrum. 59, 514 (1988).

“For information about the Motorola DSP University Support Program, write to: Motorola Inc., Attention Bob Bergeler, M/D OE314, 6501 Wm. Cannon Drive West, Austin, TX 787358598.

‘I We are willing to provide software to seriously interested parties. Write to Prof. N. Birge at Michigan State University.

121t is convenient to use a preamplifier inside a commercial lock-in am- plifier, so that the bridge can be balanced using that instrument. Alter- natively, the digital system can be used in lock-in amplifier mode to balance the bridge.

13The sigma-delta A/D converter samples the incoming signal at a rate 128 times faster than the nominal data conversion rate, then averages groups of 128 points to get the output. Due to the high sample rate, the single-pole RC filter is adequate to prevent aliasing during digitization of the input. See Horowitz and Hill, The Arr of Electronics (Cambridge University Press, Cambridge, 1980), p. 423 for an elementary discus- sion of sigma-delta A/D conversion.

14For an introduction to digital filtering, see the standard text by Oppen- heim and Schafer, Digitd Signal Processing (Prentice-Hall, Englewood Cliffs, 1975); A brief introduction to the subject is given in L. R. Rab- iner et al., IEEE Trans. Audio Electroacoust. AU-20, 322 (1972).

“For a discussion of multistage digital filtering, see R. E. Crochiere, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, 1983).

160f great help in programming the digital filtering on the DSP was the book by M. El-Sharkawy, Real Time Digital Signal Processing Applica-

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tions with Motorola’s DSP56000 Fumily (Prentice-Hall, New Jersey, 18As this project neared completion, Stanford Research Systems started 1990). For programming the PC, we used the FFT and graphics rou- advertising a completely digital lock-in amplifier, the SRMO, based on tines from “Science and Engineering Graphics” sold by Quinn-Curtis DSP technoIogy. Since that instrument is not yet being shipped, we do Software. not know if its performance equals that of our DSP system.

“We could further reduce the background in Fig. 4 by using the cooled- 191n circumstances where the noise at frequency of2oc is not neghble transformer technique mentioned earlier (Ref. 3)-but that is unnec- compared to the noise at w, it would be advisable to use a dc bridge essary for the demonstration of this technique. technique rather than the ac technique discussed in this articIe.

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