On the experimental determination of the Fano factor in Si at soft X-ray wavelengths

Abstract

We have investigated the experimental determination of the Fano factor in silicon using a low-noise CCD detector. We have tested the hypothesis of Fraser et al. (Nucl. Instr. and Meth. A 350 (1994) 365) that the distribution of secondary electrons generated by low-energy X-ray interactions is not normally distributed, leading to an asymmetry in the electron number distribution with energy. This in turn, leads to systematically low values of the Fano factor when derived using a traditional analysis of the energy resolution function. Based on measurements taken at the Daresbury Synchrotron Radiation Source (SRS) we find that monoenergetic energy-loss distributions are indeed non-Gaussian to a similar degree predicted by Fraser et al. (1994). The Fano factors determined from a probability analysis of the pulse height data are typically (0.155±0.002) which are significantly different from the value of (0.143±0.001) derived from a Gaussian decomposition of the energy resolution function. The results are in excellent agreement with the theoretical values of ∼0.158 derived from the photoionization theory of Fraser et al. (1994). Lastly, we show that for practical detection systems, failure to correct for a finite energy threshold can also lead to an underestimate in the derived value of F—by as much as ∼5% in the present case.

Introduction

In a silicon X-ray detector, only about one-tenth of the total energy deposition results in the creation of electron–hole pairs, the remainder is lost to lattice vibrations, phonons and plasmons. Consequently, unlike scintillation detection systems, the limiting spectroscopic resolution is no longer a simple function of total energy deposition. In fact, until recently the only realistic descriptions were semi-empirical, based on the so-called Fano factor [1]. This was originally introduced to quantify the departure of the observed statistical fluctuations in the number of charge carriers in a gas from that expected from pure Poisson statistics. It is generally expressed as

$$\text{F=}\text{σ}{}_{\mathrm{<mtext>exp</mtext>}}{}^2\text{var}{}_{\mathrm{<mtext>poisson</mtext>}}$$

where σ_exp² is the experimentally observed variance and var_poisson is the Poissonian variance (equal to the mean number of events, N). Obviously, for purely Poisson statistics, σ_exp² is equal to the variance and F=1.

Historically, theoretical values of F in semiconductors have tended to disagree with experimental values derived from a Gaussian decomposition of the FWHM energy resolution. Until recently, this discrepancy had been attributed to the idealised nature of the calculation and specifically the uncertainties in interaction cross-sections and the assumed energy partition. However, as calculations have improved, it has become apparent that this may not be the case. In fact, Fraser et al. [2] (hereafter F94) have shown that the differences may actually arise from the departure from normality of the distribution of secondary electrons. The exact reason for this is unclear at this time.

To test the hypothesis of F94, it is necessary to make high precision energy-loss measurements at several discrete energies. In semiconductors, such measurements have only recently become possible at soft X-ray wavelengths for two reasons. First, the spectroscopic resolution of Charge Coupled Devices (CCDs) is now sufficiently high that Fano noise dominates the energy response above ∼1 keV—making it possible to isolate it from other sources of noise. Secondly, experiments [3] at the Daresbury Synchrotron Radiation Source (SRS) have demonstrated that the synchrotron can be operated at extremely low ring currents (∼0.5 μA as opposed to ∼300 mA). After monochromation, this allows CCDs to single photon count a stable source with an acceptably high-count rate per eV when compared to other X-ray sources (e.g., radioisotopes, fluorescent target or electron bombardment sources). Synchrotron sources are ideal to test the hypothesis of F94, since the intrinsic energy resolution obtainable at most beamlines is typically a fraction of an eV and most importantly, the spectra are free from the normal background contamination associated with conventional X-ray sources. Such contamination tends to skew energy-loss spectra, making measurements of small non-Gaussian components in the pulse-height spectrum extremely difficult.