Al_0.6Ga_0.4As x-ray avalanche photodiodes for spectroscopy

For both Al_0.6Ga_0.4As p⁺-i-n⁺ photodiodes, device capacitance was measured under dark conditions and at a temperature of 20 °C (293 K), as functions of forward and reverse applied bias. A Keithley 6487 Voltage Source/Picoammeter and an HP 4275A LCR Meter (50 mV rms signal magnitude; 1 MHz frequency) were used to bias and measure the capacitance of the detectors respectively. The Al_0.6Ga_0.4As photodiodes were installed inside a custom test enclosure and placed inside a TAS Micro MT environmental chamber for temperature control. The environmental chamber was set to 20 °C (293 K) and left for 1 h before measuring to ensure thermal equilibrium; the temperature of the devices was also monitored by a thermocouple which was appropriately positioned. The test enclosure was first purged with dry N₂, then sealed. The environmental chamber was continually purged with dry N₂ such that a dry environment (<5% relative humidity) could be maintained, thus eliminating any humidity related effects [7]. The capacitance as functions of applied forward (figure 1(a)) and reverse (figure 1(b)) bias for the 200 µm diameter and 400 µm diameter devices is shown in figure 1.

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The maximum forward (1 V) and reverse (40 V) biases were selected to maintain device current at relatively low levels (<1 nA) to ensure there was no damage to the devices during the characterisation procedure. For both devices, the measured capacitance increased as a function of applied forward bias; increasing from 13.13 pF ± 0.02 pF at 0 V to 19.79 pF ± 0.03 pF at 1 V for the 200 µm device, and 46.56 pF ± 0.08 pF at 0 V to 71.32 pF ± 0.11 pF at 1 V for the 400 µm device. Conversely, the devices' capacitances decreased as functions of applied reverse bias; decreasing from 13.13 pF ± 0.02 pF at 0 V to 3.96 pF ± 0.01 pF at 40 V for the 200 µm device, and 46.55 pF ± 0.08 pF at 0 V to 12.51 pF ± 0.06 pF at 40 V for the 400 µm device. The stated uncertainties included those associated with a single measurement as well as those associated with disconnecting and reconnecting the measured devices to the test enclosure.

The measured capacitance, C_M, included both the diode capacitance, C_D, and the package capacitance, C_P, since the devices were measured after packaging. C_P was removed by assuming a constant capacitance density as a function of device area. At each applied bias, the capacitance density of the 200 µm diameter device and the 400 µm diameter device was compared, and the empty package capacitance calculated. A mean average empty package capacitance (1.29 pF ± 0.19 pF) was calculated for C_P and subsequently subtracted from C_M of each diode. The diode capacitance densities of each device are shown in figure 2.

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Since the depletion layer capacitance, C_DL, under reverse bias primarily defined the diode capacitance, the depletion width, W, can be calculated using

$$\begin{equation}W = \frac{{{{{\varepsilon }}_0}\varepsilon A}}{{{C_{DL}}}},\end{equation} \tag{ 1 }$$

where is the relative permittivity of the material (11.196 for Al_0.6Ga_0.4As [29]), A is the device area, and ₀ is the permittivity of free space [30]. The depletion width of the 200 µm device increased from 0.26 µm ± 0.02 µm at 0 V to 1.16 µm ± 0.08 µm at 40 V. Similarly, the depletion width of the 400 µm device increased from 0.28 µm ± 0.02 µm at 0 V to 1.11 µm ± 0.02 µm at 40 V. The stated depletion width uncertainty includes the uncertainty associated with the depletion layer capacitance and the uncertainty associated with the Debye length of the Al_0.6Ga_0.4As p⁺-i-n⁺ photodiodes (0.02 µm) [31]. The results indicated that a further increase in depletion width would be expected if the applied reverse bias was increased beyond 40 V, indicating that both detectors were not fully depleted at 40 V reverse bias. This was consistent with the device structure, which has a 2 µm i layer (see table 1). The depletion width has been plotted in figure 3(a), as a function of reverse bias; the implied quantum detection efficiency, assuming that the photodiode active region was confined to the depleted region of the i layer, has been plotted in figure 3(b).

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At the maximum reverse bias (40 V), the Al_0.6Ga_0.4As x-ray p⁺-i-n⁺ photodiodes quantum detection efficiency was calculated to be 0.07 at 5.9 keV and 0.05 at 6.49 keV for both the 200 µm and 400 µm device. However, since it has been shown that electrons generated in the p⁺ region (within 0.16 µm of the p⁺-i interface) of Al_0.8Ga_0.2As x-ray photodiodes contribute to detected signals [32], and that at least some charge carriers generated in the non-depleted sections of the i layer are likely to contribute to the collected charge, the calculated quantum detection efficiency should be viewed as a conservative assumption. It should be noted that the reported quantum efficiency was a result of a thin structure due to the early prototype nature of the devices. Real world devices would require thicker structures for use in low flux environments to increase the proportion of photons they detect.

The i layer carrier concentration, N, was calculated using the general nonuniform distributions equation

$$\begin{equation}\frac{{{\rm{d}}\left( {1/{C_{DL}}^2} \right)}}{{{\rm{d}}{V_R}}} = \frac{2}{{{\rm{q}}{{{\varepsilon }}_0}{{\varepsilon }}N}},\end{equation} \tag{ 2 }$$

where q is the elementary charge and all other symbols have been previously defined [30]; it was found to be 4.0 × 10¹⁶ cm⁻³ for both devices. The carrier concentration for the Al_0.6Ga_0.4As detectors as a function of distance below the p⁺-i junction, is shown in figure 4. Variation in the apparent carrier concentration between the 200 µm diameter device and the 400 µm diameter device was within the uncertainty of the measurements.

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The dark currents of both devices were measured as functions of applied forward and reverse bias using a Keithley 6487 Voltage Source/Picoammeter; the procedure followed was as per the capacitance measurements. Figure 5(a) presents the measured forward current as a function of applied forward bias and figure 5(b) presents the measured reverse leakage current as a function of applied reverse bias for both the 200 µm and 400 µm devices.

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The leakage current was measured to be 3.7 pA ± 0.4 pA and 6.3 pA ± 0.4 pA for the 200 µm and 400 µm devices respectively, at the maximum applied reverse bias (40 V). The uncertainties associated with the current measurements were dominated by the uncertainty associated with the Keithley 6487 Voltage Source/Picoammeter. Assuming the electric field strength, E_f, was uniform and across only the depleted region (implying E_f = 345 kV cm⁻¹ for the 200 µm device and E_f = 361 kV cm⁻¹ for the 400 µm device, at 40 V) it was expected that the photodiodes were operating in the avalanche regime.

The leakage current density, J_R, of the devices was calculated using the measured leakage current and is presented in figure 6. The leakage current density was calculated to be 11.9 nA cm⁻² ± 1.3 nA cm⁻² for the 200 µm device and 5.0 nA cm⁻² ± 0.3 nA cm⁻² for the 400 µm device at the maximum reverse bias (V_R = 40 V; E_f = 345 kV cm⁻¹ for the 200 µm device and E_f = 361 kV cm⁻¹ for the 400 µm device (assuming E_f was uniform and across only the depleted region)). The difference in measured leakage current density between the two devices suggested that the leakage current did not scale with junction area. This was attributed to a non-negligible surface leakage current component, possibly due to the devices being unpassivated, in addition to possible differences in contact deposition and wire bonding [12].

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For Al_xGa_1-xAs (x > 0) x-ray photodiodes, the best (lowest) leakage current density previously reported at room temperature was 2.2 nA cm⁻², at E_f = 100 kV cm⁻¹ [33]. At the same E_f (equivalent to 5 V reverse bias for the current detectors, assuming E_f was uniform and across only the depleted region), the Al_0.6Ga_0.4As devices had leakage current densities of 0.3 nA cm⁻² ± 1.3 nA cm⁻² for the 200 µm device and 0.1 nA cm⁻² ± 0.3 nA cm⁻² for the 400 µm diameter device, thus the present detectors have much lower leakage current densities. The reported leakage current densities were also lower than recently studied GaAs devices: at 20 °C (293 K) and E_f = 50 kV cm⁻¹, a 200 µm diameter GaAs device with a 10 µm i layer was measured to have a 6 nA cm⁻² ±1 nA cm⁻² leakage current density [34]. At the same E_f (equivalent to 2 V applied reverse bias for the present devices), the Al_0.6Ga_0.4As devices had leakage current densities of 0.1 nA cm⁻² ±1.3 nA cm⁻² for the 200 µm device and 1.5 pA cm⁻² ±0.3 nA cm⁻² for the 400 µm device. As previously mentioned, the difference in measured leakage current density between the two investigated devices was attributed to a non-negligible surface leakage current component, in addition to possible differences in contact deposition and wire bonding [12].

The Al_0.6Ga_0.4As p⁺-i-n⁺ photodiodes were connected to a low-noise charge-sensitive custom-made preamplifier, similar in design to that of [35], each in turn. The output of the preamplifier was then connected to a shaping amplifier (Ortec 572A) whose output was digitised by a multi-channel analyser (Ortec 927). An ⁵⁵Fe x-ray (Mn Kα = 5.9 keV; Mn Kβ = 6.49 keV) source (≈157 MBq) was positioned ≈4 mm above each Al_0.6Ga_0.4As detector in turn. The resultant spectrometers S₂₀₀ (employing the 200 µm diameter photodiode) and S₄₀₀ (employing the 400 µm diameter photodiode) were characterised. Each spectrometer was installed inside a TAS Micro MT environmental chamber to maintain an operating environment temperature of 20 °C (293 K). The environmental chamber was left for 1 h at this temperature to allow thermal equilibrium to be reached. To monitor temperature equilibrium between the environmental chamber and the spectrometer, a thermocouple was positioned close to the spectrometer. In order to reduce humidity related effects, the environmental chamber was purged with N₂ (<5% relative humidity) throughout the measurements.

The shaping amplifier shaping time constant, τ, was kept at 2 µs throughout the measurements such that direct comparisons could be made between spectra. This shaping time was the optimal shaping time available from the Ortec 572 A shaping amplifier (i.e. that which minimised the combination of the white series and white parallel noise contributions). The shaping time was selected for this purpose rather than for optimising the maximum count rate of the system. Since the two detectors had different active areas, the live time limits of each spectrum was set differently: 1500 s for S₂₀₀ spectra; 800 s for S₄₀₀ spectra. For S₂₀₀, spectra were accumulated with the detector operated at reverse biases of 0 V to 40 V in 2 V steps. For S₄₀₀, spectra were accumulated with the detector operated at reverse biases of 30 V to 40 V in 2 V steps, since at lower reverse biases (<30 V), the photopeak could not be deconvolved from the so called zero energy noise peak.

The ⁵⁵Fe x-ray spectra obtained with each spectrometer can be seen in figure 7. For clarity, not all spectra obtained are plotted; instead, a number of reverse biases have been selected to show the change in spectroscopic response. The form of response is consistent with an avalanche photodiode; this is further exemplified by plotting the change in main (largest) photopeak centroid position (corrected for changes in zero energy noise peak position) as a function of applied detector reverse bias, figure 8. Al_0.8Ga_0.2As avalanche x-ray photodiodes have been reported previously [25, 32, 36–38], but this is the first report of Al_0.6Ga_0.4 As avalanche x-ray photodiodes. The secondary peak observed on the left hand side of the primary peak is discussed in section 4.2.

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Gaussian fitting was applied to the main photopeak assuming that it was composed of 5.9 keV (Mn Kα) and 6.49 keV (Mn Kβ) x-rays from the ⁵⁵Fe x-ray source; the relative emission ratio of these x-rays (Mn Kα = 0.879; Mn Kβ = 0.122 [39]), and the relative detector quantum efficiency at these energies (see figure 3), were taken into account.

Energy calibration of each spectrum was achieved by using the zero energy noise peak and fitted 5.9 keV peak positions, assuming a linear variation of detected charge with energy. The energy resolution (FWHM at 5.9 keV) was determined for all spectra. Examples of the spectra with the Gaussians fitted are shown in figure 9 for S₂₀₀ with detector applied biases of 10 V (figure 9(a)) and 38 V (figure 9(b)). Figure 10 shows the determined FWHM at 5.9 keV of the S₂₀₀ and S₄₀₀ spectrometers as functions of applied detector reverse bias.

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The measured energy resolution (FWHM at 5.9 keV) of the presently reported devices was better than any previously reported Al_xGa_1-xAs x-ray photodiode at room temperature. For non-avalanche mode Al_xGa_1-xAs x-ray detectors, the best energy resolution previously reported was 760 eV FWHM at 5.9 keV at 20 °C (293 K) [23], using square, 200 µm by 200 µm, 3 µm i layer, Al_0.2Ga_0.8As p⁺-i-n⁺ x-ray photodiodes. For avalanche mode Al_xGa_1-xAs x-ray detectors, excluding separate absorption and multiplication region (SAM) avalanche photodiodes, the best previously reported energy resolution was 1.21 keV FWHM at 5.9 keV at room temperature [25], using an Al_0.8Ga_0.2As p⁺-p^–n⁺ circular device, 200 µm in diameter. Similar energy resolutions have been reported for GaAs-Al_0.8Ga_0.2As SAM avalanche photodiodes (1.08 keV FWHM at 5.9 keV at room temperature [37]). The presently reported energy resolution was also better than has been reported in recent studies of other wide bandgap materials, such as Al_0.52In_0.48P; non-avalanche mode, 200 µm diameter, 2 µm i layer, Al_0.52In_0.48P p⁺-i-n⁺ circular mesa x-ray photodiodes were reported to have an energy resolution of 930 eV FWHM at 5.9 keV [12]. Avalanche mode, 200 µm diameter, Al_0.52In_0.48P p⁺-i-p^–n⁺ circular mesa x-ray photodiodes were reported to have an energy resolution of 682 eV FWHM at 5.9 keV [11]. At optimal operating conditions, and at 20 °C (293 K), the presently reported spectrometers had a measured energy resolution of 630 eV ± 40 eV (at V_R = 38 V) and 730 eV ± 50 eV (at V_R = 40 V) FWHM at 5.9 keV, for S₂₀₀ and S₄₀₀ respectively. It should be noted that the reported energy resolutions, when compared to those measured for state-of-the-art Si detectors and GaAs detectors, are still relatively modest. For example, a GaAs 5 × 5 diode array (40 µm i layer), when connected to ultra-low-noise front end electronics, was reported to have an energy resolution of 266 eV FWHM at 5.9 keV at room temperature [4]. A Silicon Drift Detector (SDD) coupled to ultra-low-noise CMOS readout electronics was reported to have an energy resolution of 141 eV FWHM at 5.9 keV at room temperature [40]. A Si depleted p channel field effect transistor (DEPFET) detector was reported to have an energy resolution of 134 eV FWHM at 5.9 keV at room temperature [41].

The minimum energy cut-off was determined to be ≈2 keV at the optimal operating conditions for both S₂₀₀ and S₄₀₀. As a comparison, a minimum energy cut-off of 1.5 keV and 170 eV was estimated at room temperature using a GaAs 5 × 5 diode array [4] and Silicon Drift Detector (SDD) [40] respectively, when coupled to ultra-low-noise readout electronics. It should be noted that, aside from the quality of the detector, the use of low noise readout electronics plays a significant part in improving (reducing) the energy resolution and minimum energy cut-off of an x-ray spectrometer.

As seen in figure 7, at high detector reverse bias (⩾34 V for S₂₀₀; ⩾36 V for S₄₀₀) a secondary peak was present at the low energy side of the main photopeak. The separation between the secondary and main peak increased as the detector reverse bias was increased. Figure 11 shows how the positions of the main and secondary peaks change as functions of applied detector reverse bias for both spectrometers. A third peak, close to the low energy threshold, was also present in spectra obtained with S₂₀₀ at detector reverse biases ⩾38 V, and with S₄₀₀ at a detector reverse bias of 40 V. The third peak was hypothesised to be from Al Kα (1.49 keV [42]) x-rays from detector self-fluorescence; these x-rays became detectable due to improvement in the low energy x-ray performance of the spectrometers at high detector reverse biases as a consequence of the avalanche multiplication.

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Peaks similar to the secondary peak seen here have also been recorded with SAM APDs made from GaAs-Al_0.8Ga_0.2As [37] and Al_0.52In_0.48P [11]. In those cases, the secondary peaks were attributed to holes, created in the n region(s) of the detectors below the avalanche layer, receiving the maximum possible hole initiated avalanche multiplication as they benefitted from the full width of the avalanche region. However, other experimental and theoretical work on Al_0.8Ga_0.2As p⁺-p^–n⁺ x-ray APDs indicated that pure hole initiated multiplication of this type played no significant part in spectrum formation due to loss of the holes (e.g. by recombination) before those charge carriers could reach the avalanche region [25, 32].

Furthermore, in the case of the SAM APDs described in [11, 37], the main peak in each spectrum was formed by electrons. These were created by photons absorbed in a low doped absorption region and subsequently transported to a relatively thin avalanche region; they underwent maximum pure electron multiplication since they benefitted from the whole of the width of the avalanche region. This is a different mechanism of formation of main peak than was present in the p⁺-p^–n⁺ APDs [32, 38], where the main peak was formed from both electrons and holes which were created within the p^- layer of the detector. This p^- layer was also the avalanche region of that device and consequently charge carriers created there received a mixed multiplication, which would normally be dependent on the photon absorption position within the layer. However, the doping profile in the particular case of the previously reported p⁺-p^–n⁺ device was said to be such as to compensate for this position dependence [25, 32, 38]; a simulated spectrum showing the morphology which would have been expected without this special doping profile was presented as figure 3 in [32].

Those earlier reported p⁺-p^–n⁺ APDs did show an additional peak but this was at the right hand (high energy) side of the main peak. This additional peak was shown to be a consequence of electrons generated by photons absorbed within the p⁺ region of the detector diffusing towards the p⁻ layer and subsequently receiving the maximum pure electron initiated multiplication. It should be noted that, in that case, the only electrons that reached the p⁻ layer were those created within 0.16 µm of the p⁺−p⁻ junction. Thus, in that case, the 0.16 µm of material close to the p⁺−p⁻ junction acted analogously to the absorption region in the SAM APDs. In the case of the present devices, which are p⁺−i−n⁺ APDs rather than SAM APDs, the absorption and multiplication regions are not separate. Consequently, a spectrum morphology as per figure 3 of [32] was expected to be obtained, assuming there was no contribution from holes created in the device's n⁺ layer or the non-depleted portion of its i layer. If there was a contribution from holes created in those regions, the spectrum expected to be accumulated with the present Al_0.6Ga_0.4As devices would be similar to figure 3 of [32] but with a further additional peak (akin to that from the p⁺−p^–n⁺ device's p⁺ layer) but at the low energy side of the primary (p^- layer) peak.

However, the morphologies of the spectra obtained with the present devices appear to be more similar to those obtained with the SAM APDs than from the p⁺-p^–n⁺ APD spectra and earlier modelling. A superficial similarity between the current spectra (e.g. Figure 9(b)) and figure 3 of [32] is noted, but in the present case the depletion region is thicker (1.16 µm ± 0.08 µm for the 200 µm device and 1.11 µm ± 0.02 µm for the 400 µm device at V_R = 40 V) than the p⁺ region (0.5 µm), and both are relatively thin. Consequently, even if the whole of the p⁺ was active, the number of counts from the depletion region should be much greater than the number from the p⁺ region. However, the spectrum shape does not indicate this: the number of counts within the saddle between the main and secondary peaks is relatively small compared with the number of counts in the main peak. Consequently, the origin of the spectra morphologies obtained with the present devices is currently unknown.

As shown by figures 7, 8 and 11, the spectroscopic response of the 200 µm diameter device and 400 µm diameter device changed as a function of detector reverse bias in a manner consistent with an avalanche photodiode. Before determining the apparent multiplication factors, it should be noted that through extensive characterisation, the charge output of the low-noise charge-sensitive custom-made preamplifier used in this work has been found to be sensitive to changes in capacitance at its input (⩾0.2 pF); where a reduction in capacitance at the input (e.g. reduction in detector capacitance) caused an increase in output voltage. Since the capacitance of the 200 µm diameter device and 400 µm diameter device decreased as a function of detector reverse bias within the investigated range (see figure 1), the change in spectroscopic response due to the change in capacitance must be understood before the apparent gain from avalanche multiplication can be calculated.

Since the non-avalanche photopeak of the 400 µm diameter device could not be separated from the so called zero energy noise peak, only the 200 µm diameter device was considered. The 200 µm diameter device was coupled to the same preamplifier as used in section 4.1. A Berkeley Nucleonics Corporation model BH-1 tail pulse generator was coupled to the preamplifier test signal input, to quantify the change in apparent conversion gain of the preamplifier resulting from a change in applied bias (and consequently a change in capacitance) of the connected detector. Spectra were acquired whilst operating the detector at reverse biases from 10 V (assumed to be operating in non-avalanche mode) to 40 V in 10 V steps at a shaping time, τ, and live time, of 2 µs and 300 s, respectively, at 20 °C (293 K). The experiment was performed at four different pulse generator amplitudes (test signal input charges) in order to ensure there was no unexpected variation of preamplifier response as a function of pulse generator amplitude; no variation in this regard was detected.

The position of the pulser peak (corrected for any changes in zero energy noise peak position) was found to increase by 17% ± 1% as V_R increased from 10 V to 40 V. Previous investigations of the custom-made preamplifier found no appreciable change in spectral response as a function of detector current for currents ⩽6 pA. Consequently, the change in pulser peak position reported here was attributed solely to the change in detector capacitance. The mean change in conversion factor per unit capacitance was measured to be 6.7% pF⁻¹ ± 0.4% pF⁻¹ (rms deviance). The change in conversion factor due to change in capacitance was subtracted from the measured peak positions (see figure 11) and has been plotted in figure 12.

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The apparent multiplication factor, M, was calculated for the spectrometer S₂₀₀ by calculating the ratio between the fitted primary 5.9 keV peak position at each bias and the fitted primary 5.9 keV peak position at unity gain (M = 1, observed at an applied detector reverse bias of 10 V, given that the fitted primary 5.9 keV peak position did not change beyond the measurement uncertainty until an applied detector reverse bias of 14 V). The same procedure was used for calculating the secondary 5.9 keV peak apparent multiplication factor, assuming unity gain at 10 V applied reverse bias. The apparent multiplication factor for both peaks is shown in figure 13.

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Apparent multiplication factors of 5.20 and 3.43 were measured at an applied detector reverse bias of 40 V for the primary and secondary ⁵⁵Fe x-ray photopeaks, respectively. The tertiary peak mentioned in section 4.2, appeared to have the same avalanche multiplication as the secondary peak, assuming that the third peak is indeed a result of Al Kα x-ray fluorescence. The multiplication factors were larger than expected; at a detector reverse bias of 40 V, and assuming that the electric field strength was uniform and across only the depleted region (see figure 3(a)), a maximum pure electron multiplication factor, M_e, of 1.22, and a maximum pure hole multiplication factor, M_h, of 1.18 were calculated given the material's accepted impact ionization coefficients [43]. Given this, the measured relative positions of the peaks were considered in an attempt to establish if there was an alternative 'effective' field strength that would explain the results and maintain the accepted impact ionization coefficients. Two approaches were considered: one starting from the position of the primary peak; and one starting from the position of the secondary peak.

The first approach calculated the expected multiplication of the secondary peak given the apparent multiplication of the primary peak. Thus, it was considered that the primary peak may be a consequence of maximum pure electron multiplication; if this was the case then perhaps the secondary peak was from either maximum pure hole multiplication or mixed multiplication. Given the material's accepted impact ionization coefficients [43], if the primary peak's apparent multiplication factor of 5.20 was a consequence of maximum pure electron multiplication, then the secondary peak's multiplication, M, would be 4.66 if it was a result of maximum pure hole multiplication, or 4.66 < M < 5.20 if it was a consequence of mixed multiplication. Since the measured multiplication factor of the secondary peak was 3.43, neither explanation in the first case fits the measurements.

The second approach calculated the expected multiplication of the primary peak given the apparent multiplication of the secondary peak. Thus, correspondingly with the first case, if the secondary peak's measured multiplication (3.43) was a consequence of maximum pure hole multiplication then, given the accepted impact ionization coefficients, a primary peak multiplication of 3.80 was expected if it was a consequence of maximum pure electron multiplication, whereas 5.20 was measured experimentally.

Since both approaches indicated that the measured multiplication factors could not be explained by the accepted ratio between the electron and hole impact ionization coefficients, it was considered that it may be informative to calculate the apparent impact ionization coefficients implied by the experimental multiplication factor measurements. With the assumption that the primary peak's multiplication was a result of maximum pure electron multiplication and the secondary peak's multiplication was a result of maximum pure hole multiplication, the apparent impact ionization coefficients were calculated and are presented in figure 14. Since the secondary peak was only clearly resolved at reverse biases ⩾32 V, this is the minimum reverse bias for which the apparent impact ionization coefficients could be determined. The values were calculated by assuming the McIntyre local model [44], such that

$$\begin{equation}\alpha = \frac{1}{W}\left( {\frac{{{M_e} - 1}}{{{M_e} - {M_h}}}} \right){{ln}}\left( {\frac{{{M_e}}}{{{M_h}}}} \right),\end{equation} \tag{ 3 }$$

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where α is the electron impact ionization coefficient, M_e is the maximum pure electron multiplication factor, M_h is the maximum pure hole multiplication factor, and W is the depletion width, and

$$\begin{equation}\beta = \frac{1}{W}\left( {\frac{{{M_h} - 1}}{{{M_h} - {M_e}}}} \right){{ln}}\left( {\frac{{{M_h}}}{{{M_e}}}} \right),\end{equation} \tag{ 4 }$$

where β is the hole impact ionization coefficient [25].

The apparent electron and hole impact ionization coefficients were substantially greater than the generally accepted values reported by Plimmer et al [43]. For example, at a detector reverse bias of 40 V (E_f = 345 kV cm⁻¹), apparent ionisation coefficients α = 8513 and β = 4930 were measured, whereas the accepted ionisation coefficients at this field strength are α = 930 and β = 742 [43]. The origin of this discrepancy is currently unknown.