BACKGROUND

The power spectral density, or ``first spectrum'' ( ), of a signal x(t) is given by the Wiener-Khinchin theorem:

where is the standard autocorrelation function and a second-order noise statistic. In the discrete case of data sampled by a digital instrument, x(t) is limited to a bandpass bounded above and below by and , respectively, and is measured over a finite total time interval . The frequency components in the signal's power spectrum are expressed by expanding x(t) in a Fourier series

 

and then applying the convolution theorem:

 

For comparison of noise among measurements at different applied current levels, it is more convenient to use the first spectrum normalized by the squared d.c. value, denoted by . However, the information in alone is insufficient to characterize non-Gaussian contributions to x(t).

Restle et al. pioneered a computationally accessible approach to emphasize correlations within the power spectrum of a noise signal. [7, 8, 9] Their method involves storing a series of measurements of binned into frequency octaves so that a trace of each octave's evolution in time is obtained. The octave variances and interoctave linear correlation coefficients for the temporal evolution of the power are then calculated according to the standard statistical definitions and compared to values expected for Gaussian fluctuations. Following the nomenclature of Ref. [11], the nth octave's normalized variance is quantified as

where an octave at an instant in time t is defined as

with one of the lowest frequencies in , and the octave fluctuations are given by . The expectation values are calculated over a series of N successive power spectra acquired at time intervals . Gaussian noise is shown to lead to ,[7, 11] and calculated variances in excess of this value indicate non-Gaussian behavior. [7] The interoctave correlation coefficient between two octaves and is

 

For a Gaussian system, fluctuations in distinct bins are uncorrelated and therefore when the average is taken over all possible octave pairings. Ensemble averages of correlation coefficients calculated in most studies of a-Si:H thus far have shown decidedly non-Gaussian behavior. [4, 5, 12, 14, 16] When the octave variance does not dramatically exceed the Gaussian value, however, this framework provides little new information.

The ``second spectrum'' was also introduced by Restle et al. [8] as another technique for understanding how noise power redistributes among octaves as a system evolves in time. The second spectrum is obtained for each octave by taking the power spectrum of the series of power samplings in and is intended to show slow dynamical behavior of the fluctuation mechanisms within a narrow frequency band. Features in the second spectrum reveal interactions among local fluctuators so that the underlying kinetics of a system may be understood. This approach has also been applied to conductance fluctuations in a-Si:H, although more recent analyses have focused on instead. [12, 17] Furthermore, earlier papers adopted normalization conventions which did not account for the bandwidth dependence of the noise consistently, so that comparisons among reports are difficult.

A refinement of this framework appears in the work by Seidler and Solin where analysis reveals that estimating higher order correlations with optimal sensitivity requires measurements in a bandpass extending over more than two octaves. [15] Additionally, the Seidler and Solin analysis accounts quantitatively for the effect of finite measurement bandwidth and distinguishes the relative importance of phase and amplitude correlations. This approach builds upon the pioneering work by Beck and Spruit [18] who reexamined the variance of Johnson noise as studied in the experiments of Voss and Clarke [19] and derived a criterion for the detection limit of 1/f noise above the white background value. A full discussion of the new formulation of the second spectrum is given in Ref. [15], but the salient points necessary to understand its applicability are summarized here as follows. First, the two-point autocorrelation is calculated for the amplitude-squared signal because the signal variance is sensitive to slow energy fluctuations over long time scales. [19] A Fourier expansion of the bandpass limited signal may be taken in the same way as in Eq. (1), leading to a second-order correlation analogous to that in Eq. (2) which is defined to be the second spectrum, denoted by :

 

where and are the complex coefficients from the Fourier analysis of . Alternatively, may be expressed as the product of two independently indexed Fourier expansions of x(t) as given in Eq. (1). Restricting the summation to low beat frequencies and simplifying algebraically, the second spectrum is approximately given in terms of the original signal's Fourier components as

 

which is a fourth-order statistic in x(t). A physical way of interpreting this quantity is that it gauges how closely pairs of fluctuations with beat frequency f are correlated throughout the bandpass studied, indicating power leakage between frequencies or modulation by other fluctuators.

To compare the magnitudes of the fourth-order correlations expressed in Eq. (5) with those expected for purely Gaussian fluctuations, is divided by the square of the averaged total power over unit time in the bandwidth limited to obtain the normalized second spectrum, :

Gaussian fluctuations are expected to have stationary second spectra, and so in the cases of the Johnson and Gaussian 1/f noise backgrounds the expectation values of may be used:

for the Johnson noise background. The purely Gaussian 1/f noise background also asymptotes to a general expression dependent only upon bandwidth in the limit of low frequency:

The sensitivity of to higher-order correlations is found to depend upon both the width and the upper frequency of the bandpass in which the noise spectrum is studied. Specifically, the optimal ratio of bandpass end frequencies is found to be , and should be maximized for minimal Gaussian background signal. Non-Gaussian effects are then evident from deviations of from the relevant background for the bandpass studied. Additionally, the contributions to non-Gaussian behavior due respectively to amplitude and to phase correlations are also analyzable by separately randomizing the phase and amplitude components in the power spectrum of the initial signal and then calculating the second spectrum of the modified signal. We now apply these results to study non-Gaussian phenomena in a-Si:H.

A new method of acquiring data and calculating the second spectrum is particularly welcome at this juncture when conflicting reports have appeared regarding the character of conductance fluctuations in a-Si:H. Reports of highly non-Gaussian noise indicated by strong interoctave correlations according to Eq. (3) have been interpreted as evidence that transport through a-Si:H is dominated by a small number of current microfilaments. [4, 5, 12, 14, 16] Initial work by Johanson et al. [3] reported the observation of two-level fluctuations reminiscent of RTSN in their system above a current density threshold of but did not investigate correlations or higher-order statistics. Johanson et al. have more recently studied correlations between time-evolved amplitudes at discrete frequencies of similar to Eq. (3) and found the noise to be purely Gaussian. [13]

The change in the conductance noise amplitude and correlations after exposing samples to intense light also differs between studies. The dark conductivity of a-Si:H decreases after prolonged exposure to light with photon energy comparable to the semiconductor bandgap. [20] This phenomenon is currently understood to result from the conversion of 4-fold coordinated Si atoms into 3-fold coordinated Si atoms each with an unbonded orbital. Such unbonded orbitals quickly accept an electron and move the Fermi level toward the middle of the bandgap, thereby reducing the conductivity. If noise is a probe of the microstructural dependence of local current flow, then changes in the noise statistics of the dark conductivity after light exposure should yield information about the validity of the current microfilament picture. Fan and Kakalios found that noise changed from being very strongly non-Gaussian to having relatively low interoctave correlations in the dark conductivity following light exposure,[14] but Johanson et al. observed only Gaussian noise both before and after exposure. [13] A more sensitive indicator of the higher-order correlations together with consideration of other plausible mechanisms for the observed noise may provide new insights regarding the origins of these discrepancies if similar experiments are performed using the new second spectrum analysis technique summarized above.