Study of charge sharing effect and energy resolution of the Timepix hybrid detector based on gallium arsenide compensated with chromium

For the mathematical modeling of the beam sweep the initial cloud center was placed 15 μm next to the cathode inside the active material, most likely position for low energy photons verified in previous simulations with the MCNPX code system [4] reported in [5,6]. The cloud diameter (2.5 μm) was established based on the calculation of the generated photoelectrons range [7].The sensor anode is segmented forming a 256 x 256 matrix, with 45 x 45 µm² pixels and 10 µm gap.

The cloud induced charge on the target pixels(Q) is a detector characteristic that leads to the magnitude influence of the charge sharing effect in the detector response. In order to obtain the induced by the carriers cloud charge during its drift and diffusion movement was used the Shockley-Ramo theorem [3] (Eq.1)

where q is the punctual carrier charge value and $\Delta {\phi }_0$ is the weighting potential difference at the initial cloud position and at the detector anode location, value obtained with the employment of the ARCHIMEDES 2.0.1code system [8].

As radiation sources were employed the X-rays generated by a VEPP-3 synchrotron of the Budker Institute of Nuclear Physics [9], and by an X-ray tube with different materials for obtaining the corresponding fluorescence photons (table 1) provided by the Dzheliepov Laboratory of Nuclear Problems (DLNP) of the JINR.

The fine collimated synchrotron beam was used to irradiate point zones of independent pixels, as well as to perform the line sweep between two continuous pixels. The X-ray tube with a reflection configuration allowed to irradiate larger areas including several pixels at the same time.

This detector presents fourth possible operation modes, but is only used the “Single photon” mode for convenience. This approach delivers the number of times that signal exceeds the threshold level (THL) while the shutter is still open. This number of times by THLis the relation experimentally known as the integral pulse height spectrum [3].

The method employed [10] to experimentally study the charge sharing effect and obtain the detector energy resolution consists in fitting the observed points of the integral pulse height spectrum to a general function F(x). Then gets the correspondent differential pulse height spectrum f(x) and the different sub functions that compound it, each one describing a particular effect to take into account.

The function that describe the charge sharing effect is one from a function family called “repetitive integrals of the complementary error function” [11], numerically calculated by:

where $\Gamma$ is the Gamma function, j the iterative variable, a the integration order and $i^{a}Erfc$ is theorder a complementary error function integral. Was concluded that the order 2 complementary error function integral is the adequate function to describe the charge sharing effect contribution.

Other of these sub functions is the known “complementary error function” [12]:

where u is the integration variable. $Erfc\left(x\right)$ describe appropriately the counts behavior without considering the charge sharing effect corresponding function. Finally, a linear function is considered, which slope d is the background counts, and a constant g with fitting purposes only.

The applied general function to fit the experimental points is then:

Where b is the charge sharing effect contribution corresponding parameter, c the energy peak corresponding parameter, $\mu$ the THL mean value and $\sigma$ the corresponding standard deviation.

Once fitted the function, is only needed derive F(x) to get the differential pulse height spectrum f(x):

This is the standard procedure applied to three cases, chosen under the base that charge sharing effect is manifested differently in each one, such cases are:

After been obtained the corresponding $\mu$ and $\sigma$ parameters, is performed the calibration curve. Then the energy resolution is found to be [3]:

where R is the energy resolution, FWHM is the full width at half maximum (≈ 2.35σ) and E _x is the peak energy.

Figure 1 presents the induced charge modeling during the beam sweep from a pixel center to the center of the next one. The secondary electron cloud diameter and the interaction position inside the detector active volume are relevant elements to analyze in order to study the induced charge and the charge sharing effect in sensitive pixels.

The electron cloud diameter influence and the interaction position contribution to the simulated induced charge are shown in the fiigure 1 (a) and figure 1 (b) respectively. The figure 1 (a) simulated initial clouds are positioned at a constant initial distance of 885 μm between their center and the detector anode surface. The interaction positions shown in figure 1 (b) are represented by the distance between the interaction position and the anode surface.

The behavior found in the mathematical modeling is similar to the modeling presented in [13] and the experimental results seen in [14]. In both cases, although the detector material is different (Si and CdZnTe respectively), a behavior characterized by a plateau on the pixels central zone is verified, and a fall that extends to the gap center, where the signal intensity decreases to 45 - 50% of the recorded maximum. These two articles and other similar results such as [15] confirms the adequate correspondence of the used model with the already performed experiments.

The results presented in figure 1 (a) shows that dependence between the cloud sizes of charge carriers generated at the time of the interaction of the photons with the GaAs:Cr does not significantly influence the induced charge values in the pixels, as long as the number of generated charges remains constant (10³ electrons).

On the other hand, in the figure 1 (b) is verified that, within the studied limits, the depth at which the photon interaction occurs is not significant to the induced charge in the pixels. This is a direct consequence of the well-known "small pixel effect" [3]. At a distance lower than 100 μm for this 900 μm detector, a strong influence on the results is expected, since it is the area where the weighting potential starts its abrupt growth [6].

An example of the applied full procedure to get the detector energy resolution and study the charge sharing effect contribution in the integral and differential pulse height spectrum is presented in the next figure, for the irradiation to a 230x230 pixels arrangement with 15,691 keV X rays. The differential pulse height spectrum of figure 2 (c) represents functions that characterizes the total counts (red), the charge sharing effect (green), the energy peak gauss distribution (blue) and the background (black).

For free fitting parameters are consider $\mu$ , $\sigma$ , b, c, d and g. figure 2 (c) and (d)y axis values are different because in the first one is shown the derivative of F(x) respect to $\frac{x-\mu }{\sqrt{2}\bullet \sigma }$ , while in the second one is represented the derivative with respect to x of the experimental points shown in figure 2 (a).

The gauss distribution parameters ( $\mu$ , $\sigma$ ) are used to find the detector energy resolution for the incoming X rays energies considered in table 1. Meanwhile the charge sharing effect contribution function is used to study this effect, through comparison with the remaining two cases.

The same procedure is applied to the collimated beam directed to a pixel center case with 27,9 incoming keV X rays (figure 3) and the collimated beam directed to the center of the gap between two neighboring pixels case with 18 keV incoming X rays (figure 4).

For both neighboring pixels, the gap irradiation case procedure, showed through figure 4, presents the counts vs. THL experimental dependence with their corresponding F(x) fitting function (figure 4 (a) and (c) respectively), and the derivative fitting function with its components (figure 4 (b) and (d) respectively).

In order to compare the charge sharing effect contributions for the several analyzed configurations is defined the quotient $\eta$ :

The attenuation of the energy peak gauss distribution by the charge sharing effect is exposed with the coefficient $\eta$ . A comparison between the observed cases is presented in table 2.

The higher charge sharing effect contribution is presented in the pixel arrangement irradiation case, followed by the irradiation directed to gap between two neighboring pixels case, and finally the irradiation directed to a pixel center, facts verified by figures 2, 3, 4 and table 2.

The charge sharing effect contribution depends on the irradiation beam transversal section, its target zone, and others factors mentioned previously. This effect contribution is larger as larger the beam transversal section is and involved pixels are. The induced charge registered in the gap irradiation case is lower than the corresponding incoming radiation energy reported [16], reason why is excluded this case in following analysis.

This detector calibration, for the pixel arrangement irradiation case and the irradiation directed to a pixel center case, seen in figure 5 (a), is defined as the ratio between the incoming X rays energies and their corresponding THL peak value, this last one assigned by the analog-digital converser (DAC) [9].

Once developed the detector calibration is used Eq. 6 to obtain the detector energy resolution, presented in figure (b).

Each pixel possesses its own electronics and therefore its own electronic noise, which means that each pixel has its own THL different from the others [17], this is reflected in the calibration curve by the two different linear fitting functions shown in figure (a).

figure (b) reveals that energy resolution is improved for lower incoming X rays energies values in both cases, and affected negatively by the charge sharing effect, whose contribution is greater as bigger the collimated beam transversal section is.