RESULTS

In the above setup, data was acquired and rapidly analyzed in terms of , octave variance and correlations , and . This facilitates comparisons of the relative sensitivities of each method to different properties of noise and emphasizes the complementarity of the analysis techniques. Examples of noise in the strongly non-Gaussian limit and weakly non-Gaussian fluctuations both upon the initial annealing and after five months of aging the sample were collected and investigated.

In order to obtain an example of fluctuations in the limit of extremely non-Gaussian behavior, we measured both samples prior to annealing. We have previously observed that both the amplitude and non-Gaussian characteristics of a sample's noise decrease substantially after annealing. The striking effect of annealing on noise production might suggest a relationship of the noise power to the comparatively high density of defect states in the as-deposited material; on the other hand, large non-Gaussian fluctuations superimposed on the more common flicker noise could be ascribed to adsorbates on the sample surface influencing the local conductivity. [21] One trace of points measured from sample A at an applied current density of 0.9 A/cm showed significant non-Gaussian behavior which is comparable in magnitude to the most dramatic switching noise reported in annealed samples in other studies. The acquisition frequency for the data in this set was 33 kHz for a maximum power spectral frequency of 12.8 kHz; the sample and preamplifier setup is depicted in Fig. 1(a). Spurts of RTSN amid long spells of flicker noise are evident, with one such instance depicted in Fig. 2(a). The power spectral density for this section of the trace appears in the inset to Fig. 2(b) and has the Lorentzian form expected for two-state switching signals with a corner frequency of 200 Hz. This Lorentzian form is not characteristic of the entire data set, however. The average of for 24 data blocks from a period of flicker noise immediately following the RTSN appears in Fig. 2(b). In contrast to the Lorentzian spectrum in the inset, this trace closely follows with spectral slope . Such abrupt alternation between switching and flicker noise is a familiar aspect of noise attributed to hydrogen motion in some annealed a-Si:H films, [16] although in an unannealed film similar behavior might reflect artifacts due to adsorbates. The average for all 245 segments contains both types of behavior and clearly deviates from the 1/f noise spectrum by an order of magnitude in the low frequency range because of the Lorentzian segments which enhance the power in that portion of the spectrum. This is a significantly greater difference than that observed between noise traces acquired from sample B on different dates following the initial anneal and between subsequent annealing cycles, which led to a factor of two change in the noise power at most.

Both and the interoctave correlations confirmed non-Gaussian behavior in a quantitative fashion beyond the visual observation of RTSN. While this is not surprising in itself, it demonstrates the utility of the second spectrum to quantify non-Gaussian noise processes. We will be able to take advantage of this method's precision and sensitivity when comparing the present signals which have dramatic features to more subtly correlated noise. First, the nearest neighbor octave correlation coefficients calculated according to Eq. (3) are plotted in Fig. 3 with the average value given by . Correlation coefficients for octave separations greater than one are not represented here, as in all cases studied here the averages of such correlations uniformly decreased with increasing separation. Although recent reports in the literature have focused on the statistics of ,[12] inspection of the dependence of upon octave position can yield further information about the regions of the spectrum that are most correlated, as weakly non-Gaussian modulations, unlike RTSN, influence different octaves to dramatically varying degrees. The values of shown here resemble those given in Ref. [9] for a strongly non-Gaussian system as well as previously reported on other a-Si:H samples after annealing. [12, 16, 17] was studied for the bandpass between 3 kHz and 12.8 kHz close to the suggested optimal bandpass end frequency ratio and range where the Gaussian background is minimized. is plotted in Fig. 2(c) together with a post-anneal Johnson noise trace from sample B to illustrate the dramatic increase in fourth-order correlations at the lowest frequencies in an extremely non-Gaussian system. This type of behavior was more the exception than the rule for this sample, however, as reports of post-anneal measurements will next show. Interoctave correlations measured on sample B immediately before the initial sample anneal were not substantially above Gaussian levels.

A clear example of weakly non-Gaussian behavior was observed following the initial annealing of sample B. Sample B was chosen for focused and controlled measurements because its linear dimensions were comparable to samples investigated previously. [4, 5, 12, 16, 17] Upon annealing the sample, the noise power decreased by about an order of magnitude even at an increased current density of 2.85 A/cm so that the signal was dominated by preamplifier input noise above about 2 kHz in the configuration shown in Fig. 1(a). This limitation was avoided by changing to the configuration of Fig. 1(b) so that the current preamplifier could be set to a more sensitive scale with less input noise. Reducing the sampling frequency to 4 kHz further ensured that the noise signal would be undistorted by the Johnson floor. In this arrangement ranges between 1 Hz and 1.6 kHz, enabling direct comparison with many of the previously published second spectral and correlation data for a-Si:H. [4, 12, 16, 17]

A section of a time trace after annealing is shown in Fig. 4(a) and appears relatively featureless compared to that in Fig. 2(a) at the same density of time sample points. Measurements of the power spectral density at room temperature consistently yielded 1/f behavior of nearly two orders of magnitude lower power than was seen prior to annealing, as can be seen in the trace in Fig. 4(b) with the slope . However, the inset to Fig. 4(b) depicting (perfectly 1/f noise yields a horizontal spectrum in this format) emphasizes that a subtle spectral feature is indeed present centered about f = 200 Hz which was not present in a continuous data run of the same length acquired immediately prior to this one. This is an example of what has been referred to as ``spectral wandering'', [8, 10, 17] in which noise power redistributes to different regions of the spectrum over time. Also, both spectra show some superimposed spikes due to odd harmonics of 60 Hz which intermittently appeared during some data runs. These did not affect the salient broadband noise characteristics, however, and have been omitted from the plots for clarity.

 

 

Table 1: Octave variances for the signals acquired at 4 kHz both before and after aging

Interoctave correlation coefficients and were calculated for both data sets. The nearest neighbor values for both traces are plotted in Fig. 3 and the corresponding octave variances are presented in Table 1. The variances are all weakly non-Gaussian and increase in value towards the higher frequency octaves for both traces, although this statistic is seen to be greater for the second trace which exhibits the spectral feature. This trend is amplified in the plot of the which reveals the nearest neighbor octave correlations of the second data set to be moderately correlated especially between the higher frequency octaves which contain the spectral feature, yielding . On the other hand, the first data set is practically Gaussian in its weak correlations, with . [8] This contrast between Gaussian and non-Gaussian behavior in the correlation coefficients is emphasized by the plots of in Fig. 4(c), calculated for a bandpass extending from 320 Hz to 1.6 kHz. While the first data set shows an increase in the fourth-order correlations of about a factor of five at the lowest frequencies, the later data set shows an increase of over two orders of magnitude above the background. The for this later trace were not quite so strongly non-Gaussian as for the measurements on sample A discussed earlier. Yet, displays dramatic higher-order correlations at small frequency differentials. The narrowband peaks in the first spectrum due to power line pickup cannot account for the increase in in the latter trace, as identical peaks are seen in the plot of (inset to Fig. 4(b)) for the earlier trace which shows subtler, non-monotonic low frequency behavior in .

Furthermore, randomizing the phases of the Fourier coefficients, , and recalculating removes all traces of non-Gaussian behavior. As pointed out by Seidler and Solin, such observations indicate that phase correlations rather than amplitude correlations are responsible for the observed non-Gaussian behavior. The data for the second data set, 4NG, were analyzed in this way and are plotted in the inset to Fig. 4(c) for the phase-randomized and amplitude-whitened analyses; only the latter situation shows higher order correlations. The narrowband peaks in the first spectrum due to power line pickup cannot account for the increase in in the latter trace, as identical peaks are seen in the plot of (inset to Fig. 4(b)) for the earlier trace which shows subtler, non-monotonic low frequency behavior in .

An investigation into the dependence of the noise amplitude on current in Channel B after the initial annealing revealed that at 310 K, which is close to the commonly observed relation for linear noise, . Such behavior is seen in other systems as well as in certain a-Si:H specimens. [3, 10] Some previous reports on a-Si:H, however, indicate nonlinearity in the current dependence to be a unique signature of noise in this material. [4, 14, 22] Linear noise is generally understood to indicate that the fluctuations probed result from fluctuations intrinsic to the measured sample and not from effects induced by the applied signal. In many cases in which the noise is linear, the empirically derived Hooge parameter

in which is the number of free carriers, provides a rough quantitative measure of the magnitude of the noise. Given the sample's conductivity and estimated carrier mobility , we estimate at 310 K. This is almost three orders of magnitude lower than estimated in Ref. [22] and an order of magnitude lower than that found in Ref. [3] for another sample with linear noise. Other reports indicate that this quantity can vary between and 1 with both temperature and hydrogen content.[2]

Finally, weakly non-Gaussian behavior was observed in sample B after five months of aging. Sample B was set aside and allowed to age in a jar dried with desiccant for five months. The sample was then annealed at 435 K for 30 minutes, cooled to 310 K at a rate of 6 K/min, and measured again in an atmosphere of dry nitrogen with a flow rate of 0.6 l/min. The current density increased to 3.96 A/cm at the same bias of 24.1 V which had been applied to sample B after its very first annealing five months earlier. As for the earlier measurements on sample B, data runs were acquired at a frequency of 4 kHz and analyzed in the bandpass extending from 320 Hz to 1.6 kHz. The unnormalized power spectral density also decreased by about a factor of two, leading to an order of magnitude drop in the normalized quantity . Because the noise power was seen to depend upon the square of the current rather closely, the large change in over time suggests that some type of relaxation may have occurred in the material. Despite this, the usual 1/f behavior is seen in the main portion of Fig. 5(a) with a slope of , and two octavally binned traces of acquired within an hour of each other in the same data run are plotted in the inset to the figure. The difference in the spectral features is even subtler than in Fig. 4(b), but the divergence between the nearly Gaussian and weakly non-Gaussian behavior is again suggested by the values depicted in Fig. 3 and octave variances listed in Table 1. The former trace exhibits , which is very close to being Gaussian, and the latter trace shows an only slightly more correlated signal, with . The strongest indicator of the difference between these traces with nearly identical power spectral densities again comes from the behavior of the second spectra, seen in Fig. 5(b). Although the first trace is already slightly non-Gaussian with significant fourth-order correlations appearing at beat frequencies as high as 0.1 Hz, the later trace manifests interfrequency correlations extending nearly two orders of magnitude farther out in extent. The inset to Fig. 5(b) also shows the phase and amplitude correlations in for the 5NG trace, again demonstrating that higher order correlations only come from the phases of the Fourier components. Similar transitions between Gaussian behavior evolving into non-Gaussian behavior were seen on different days on the aged sample B, indicating that weakly non-Gaussian behavior is not a static property but rather waxes and wanes over time.