F. N. HOOGE and A. M. H. HOPPENBROUWERS Philips Research Laboratories, N. V. Philips Gloeila~penfabrieken, Eindhoven, Nederland

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1 Physica 42 (1969) 33 l North-Holland Publishing Co., Amsterdam AMPLITUDE DISTRIBUTION OF l/f NOISE F. N. HOOGE and A. M. H. HOPPENBROUWERS Philips Research Laboratories, N. V. Philips Gloeila~penfabrieken, Eindhoven, Nederland Received 26 September 1968 synopsis The amplitude distribution of l/f noise was investigated by displaying the noise on an oscilloscope, and scanning the picture with a movable slit. The distribution curve is gaussian; its variance does not fluctuate with time. 1. Introduction. Many attempts to establish a theory of I/f noise have been made. Because of the rather unsatisfactory results of linear models it is tempting to study nonlinear models. One of the questions that arises then is whether the amplitude distribution is gaussian or not. A review article on this problem of nonlinear models and distribution function has been written by Van Kampeni). His conclusion is that a linear model always leads to a gaussian distribution, and that nonlinear models under certain conditions also lead to gaussian distributions. Non-gaussian distributions arise from nonlinear models that do not fulfil these special conditions. If for a certain type of noise a non-gaussian distribution is found experimentally the properties of the distribution function give some information on the model for that noise. But for the I/f noise we found a gaussian distribution, from which one cannot infer whether the model for 1 /f noise must be linear or nonlinear. When we started our investigations there was only one publication on the amplitude distribution of 1/f noises). In that paper Bell reported a gaussian distribution. We followed his method in principle, but we obtained a much higher accuracy. Our investigations led to the same result, the l/f noise is gaussian. During our investigations Brophya) published a paper on the same subject. There are two reasons why we still think it worthwhile to present our results. 1. Our accuracy being higher, we could follow the distribution curve to large amplitudes with a probability of 0.3% of the probability of the top, whereas Brophy s smallest probability is 5%. This point is of importance since it is only at the foot of the curve that possible deviations from a gaussian distribution will show up. 331

2 332 F. N. HOOGE AND A. M. H. HOPPENBROUWERS 2. Brophy observed that at any moment the distribution curves were gaussian, but the gaussian curves measured at different times had different variances. The variances fluctuated considerably with time. Our results show quite the opposite: the variance is constant, even if averaged over smaller measuring periods than Brophy used. 2. Measurement of the amfilitude distribution. l/f noise was generated by passing a current through an evaporated CdSe layer. The amplified noise was displayed on a cathode-ray oscilloscope. The picture on the screen was projected onto a vertical plane with a horizontal slit. The slit was slowly moved vertically by a motor, scanning in this way the picture on the scope from top to bottom in half an hour, and integrating at each moment the light over the whole slit at a certain noise-voltage level. The light passing the slit was detected by a photomultiplier. The signal from the photomultiplier was measured by a DC microvoltmeter, connected to a recorder P (A ) PO YiqIY--:;,,,,i I E 0.4 v* to-' 5 2 t + lo-i 5 2 1O-3 -+i I 4c (av)'(orb. units) --C Fig. 1. Amplitude distribution of l/f noise. 1 Av unit = 0.29 volt. ((AU) 2> from power spectrum and bandwidth.

3 AMPLITUDE DISTRIBUTION OF l/f NOISE r-4 N N

4 334 F. N. HOOGE AND A. M. H. HOPPENBROUWERS l (AV) PO (A@(arb. units) ----c Fig. 4. Amplitude distribution of thermal noise. lhv unit = 1.14 volt. ((Av) 2> from power spectrum and bandwidth. The light intensity is proportional to the probability of that amplitude that corresponds to the position of the slit. The vertical distance of the recorded curves is proportional to the proba- bility for an amplitude. The horizontal distance is proportional to the time during which the slit has moved, hence it is proportional to the noise ampli- tude itself. The horizontal scale depends on the sensitivity of the scope, the reduction of the camera, and the speeds of the slit and the recorder. This horizontaldi stance is expressed by Av in figs. 1, 2, 3 and 4, where Av = 0 corresponds to the top of the curve. One Av unit is 1.14 volt for high-current experiments, and it is 0.29 volt for low-current experiments, as could easily be verified, since the 1 cm marks on the oscilloscope screen showed up clearly in our recorded curves. For measurements of thermal noise wire-wound resistors of 104 IR were used. Thermal noise is known to be gaussian. It was used in our experiments to show that the measuring equipment did not introduce experimental errors. We did not use it for calibrating the equipment.

5 AMPLITUDE DISTRIBUTION OF l/f NOISE 335 Without any correction other than subtraction of a zero level, we found experimental curves that were gaussian, both for thermal and for 1 /f noise. This was found by replotting the recorded curves as log P(Av)/P(O) vers%s (Av)s. Our highest accuracy was obtained by comparing two distribution curves, one of l/f noise and one of thermal noise, that had about the same width <(Av)s). Both curves were measured under the same experimental conditions, such as bandwidth and gain of the amplifiers, sensitivity of the oscilloscope, etc. Neither here did we find a deviation from a purely gaussian distribution for the l/f noise. The results are shown in figs. 1, 2, 3 and 4. The values for <(Av)s> from the distribution curves agree reasonably well with the values calculated from the power spectra and the bandwidths of the amplifiers (see table I). The relatively high values from the power spectra for 1 /f noise and the correct values for white noise suggest that the low frequencies do not fully contribute to the noise signal displayed on the scope. For thermal noise <(Av)s> is in agreement with the resistance of the wirewound resistor used. Table I Variance in volt 2 * l/f noise thermal noise from power spectrum from distribution curve from power spectrum from distribution curve * The figures are for amplified noise at the input of the scope. 3. Experimental details of the measurement of the amfilitude distribution. The I/f noise was generated by passing a current (max ma) through an evaporated CdSe layer. The spectrum was exactly l/f. For i = 0.28 ma the l/f noise at 100 khz was a factor of 500 higher than the white thermal noise of this sample. The noise was also studied with smaller currents. With the smallest current the 1 /f noise at 100 khz was still a factor of 30 higher than the thermal noise. For the amplification of the noise a Philips amplifier PM 6045 was used with a gain of 100 and a bandwidth from 1 Hz to 100 khz, and a Keithley AC amplifier model 103 with a gain of 1000 and a bandwidth from 100 Hz to 100 khz. The amplified noise was displayed on a Philips universal laboratory oscilloscope PM 3330, using a time base of 5 ms/cm and a repetition of 6 per second. Since the circular form of the oscilloscope introduces errors at large amplitudes which are screened off at the sides, only the central part

6 336 F. N. HOOGE AND A. M. H. HOPPENBROUWERS of the screen was used with a width of 8 cm, which means that with the time base we used the noise is exposed each time for 40 ms. Facing the oscilloscope screen there was an Exakta camera with a movable horizontal slit of 125 tn. in the plane where normally the sensitive film is placed. The screen was projected onto this plane with a reduction of about l/4. The light passing through a slit passed through a ground glass plate, and reached a photomultiplier tube (Philips XP 1003). The signal from the photomultiplier was measured by a DC microvoltmeter PM 2440 with a ca- pacitor of 40 FF at the input. The PM 2440 was connected to a recorder. The recordings obtained in this way were used for a further analysis. Usually the top of the recorded distribution curve was at about 80% of the full recorder sensitivity scale. In order to study the feet of the curves more closely the of the PM 2440 was set 3 and 10 times higher than for the re- cording of the central part of the distribution curve. The resolving power of our system is 0.05 cm on the screen of the scope or 0.1 Av unit. This was found by scanning a horizontal line with the same intensity and width as the lines we used for our measurements. This resolving power is in agreement with the slit width of 125 l_~ and the reduction of l/4. 4. Experimental proof that the variance is constant. According to Brophy the variance of the gaussian amplitude distributions of l/f noise (and of white noise) fluctuate considerably with time. As is shown in his figures 7 and 8 it is not exceptional to find a factor of 2 between variances. If this were true then the probability for any given large amplitude would fluctuate enormously. A numerical example may serve as an illustration. The probability curve will be written as For a chosen value of <(Av)~> there is an amplitude Av for which P(Av)/P(O) = Compare this probability 0.20 with the probability that the same amplitude Av would have if ((Av)~) became 2 x <(Av)s> or (+) x <(Av)s>. These values are 0.45 and Such large fluctuations in the probability for a chosen amplitude are easy to detect. Our experiments showed that such fluctuations do not exist. That slow fluctuations, of the order of minutes, do not exist, follows from the already described experiments in which the distribution curves were determined. The recorded amplitude curves are smooth curves. This means that neighbouring points on the slope of the curve belong to distribution curves with the same variance. The measuring time for one point is obviously long enough to give one average value for the variance. For this measuring

7 AMPLITUDE DISTRIBUTION OF l/f NOISE 337 time we may take the time in which the slit moves over its own width. This time is 11 s. There are no extremely slow variations in the variance, because the recorded curves are perfectly symmetrical. The scanning time of the complete curve is 30 minutes. The points in figs 1, 2, 3 and 4 are from both slopes of the recorded curve. In order to detect possible fluctuations in much smaller times (40 ms) the following experiment was done. The same apparatus as for the study of the amplitude distribution was used. Now the slit was not moved but kept fixed, first on the top of the distribution curve, where the intensity was measured and where only small fluctuations in the intensity were observed. Then the slit was kept fixed at a position on the slope where about 20% of Fi g. 5. Sequences of 40 ms pulses, each the result of the average probability for a fixed amplitude. Top : l/f noise Bottom : thermal noise Left: zero amplitude Right : finite fixed amplitude All pictures have the same time and voltage scale. Time base 125 ms/cm.

8 338 F. N. HOOGE AND A. M. H. HOPPENBROUWERS the top intensity was measured. Here we did not observe the large fluctu- ations that could be expected if Brophy were right. l/f- and white noise were studied. For a good comparison with Brophy s experiment we used in this part of our investigation a bandwidth of the Keithley amplifier from 10 Hz to 10 khz. The signal from the photomulti- plier was displayed on a second oscilloscope, from which we took the photo- graphs presented in fig. 5. During 40 ms a series of light pulses came from the first scope on which the noise was displayed. These pulses arose from the intersections of the slit with one single noise trace. The 40 ms corresponded to the screen width of 8 cm and the time base of 5 ms/cm. There were 6 of such series of pulses per second. The second scope had a time base of 125 ms/cm. Thus on the 10 cm wide screen of the second scope there were 8 blocks in one sweep. Each block was 40 ms wide, and its height gave the average intensity 40 ms. during this A 3 PF capacitor at the input of the second scope smoothed out the indi- vidual pulses in a block. Each picture of fig. 5 gives the result of one sweep of the second scope. These pictures show that even 40 ms is long enough to give a reasonable average. We observed these pictures for several minutes, without seeing serious fluctuations in the average height. The fluctuations that were observed can be accounted for by considering the low number of data points in a measurement of 40 ms. This can be demonstrated by an analysis of the pictures of fig. 5. For the 1 /f noise one finds : Top P(0) = (7.2 f 0.7) div. = 7.2 div. & lo%, Slope P(Av) = (1.2 f 0.4) div. = 1.2 div. & 30%. The ratio of the relative errors is a factor of 3. From the numbers of data points, which are 6 times higher on the top than at the chosen point on the - slope, the expected ratio is J6 = 2.5. Thus the ratio of the relative errors is in agreement with the ratio of data points. We can also make an estimate for the errors themselves. According to Rice*) the number of zeros per second is given by where w(f) is the spectral density of the power spectrum of the noise. For 1 /f noise with frequencies between 10 Hz and 10 khz this formula gives Na = 5000 per s. In 40 ms there will be 200 zeros, which is the number of data points on the top of the probability curve. The experimental error inferred from this number is:d200 = 14 or 7%. This compares well with the 10%~welfoundIfrom:theIpicture:oflfig. 5.

9 AMPLITUDE DISTRIBUTION OF l/f NOISE 339 For the white noise we found: Top P(0) = (6.1 f 0.2) div. = 6.1 div. f 3%, Slope P(Av) = (1.6 &- 0.1) div. = 1.6 div. & 6%. The probabilities and hence the numbers of data points have a ratio of 4. The expected ratio in the errors will be J4 = 2, the same value as found experimentally. The number of zeros for white noise between 10 Hz and 10 khz is according to Rice s formula per s. In 40 ms there will be 440 data points, giving an error of 4440 = 22 or 5%, which agrees with the experimental values of 3%. Thus we reach the conclusion that the observed fluctuations in the average intensity can be accounted for by considering the low number of data points. The variance of the gaussian distribution curve for l/f noise does not fluctuate with time. REFERENCES 1) van Kampen, N. G., Chapter 5 in Fluctuation phenomena in solids. Edited by R. E. Burgess (Pure and applied Physics 19, Academic Press, New York - London, 1965). 2) Bell, D. A., Proc. Phys. Sot. (London) B68 (1955) ) Brophy, J. J., Phys. Rev. 166 (1968) ) Rice, S. O., Bell System techn. J. 24 (1945) 46.