Analytical Solutions of Transient Drift-Diffusion in P-N Junction Pixel Sensors

Abstract

Radiation detection in applications ranging from high energy physics to medical imaging rely on solid state detectors, often hybrid pixel detectors with (1) reverse biased p-n junction pixel sensors and (2) readout ASICs, attached by flipchip-bonding. Transient signals characteristics are important in, e.g., matching ASIC and sensor design, modeling and optimizing detector parameters and describing timing and charge sharing properties. Currently analytical forms of transient signals are available for only a few limited cases (e.g., drift or diffusion) or for the steady state (which is not relevant for high energy radiation detection). Tools are available for (relatively slow) numerical evaluation of the transient charge transport. We present here the first analytical solutions of partial differential equations describing drift-diffusion-recombination charge transport in planar p-n junction sensors in a variety of conditions: (1) undepleted, (2) fully depleted, (3) taking into account the gradual velocity saturation, and (4) overdepleted. We deduce the Green's functions which can be applied to any detection problem through simple convolution with the initial conditions. We compare the analytical solutions with Monte Carlo simulations and industry standard simulations (Synopsys Sentaurus), demonstrating good agreement. Using the analytical equations enables fast modeling of the influence of various detector parameters on tracking, imaging and timing performance, describing performance and enabling optimizations for different applications. Finally, we illustrate this model with applications in 3D+T (x,y,z,time) photon tracking and 4D+T (x,y,θ,φ,time) relativistic charged particle tracking.

Performance of ePix10K, a high dynamic range, gain auto-ranging pixel detector for FELs

“…ePix10K cameras are built around hybrid pixel detector modules obtained by flip-chip bonding 4 ePix10K readout ASICs [14,19] to a typical Si sensor (500 µm thick, n-type, resistivity 10 kΩ cm; other options possible) [20]. For increased radiation hardness, the ASIC balconies are protected with high-Z strips [21].…”

Dead-time correction for spectroscopic photon-counting pixel detectors

“…A sensor (typically a reverse biased p-n junction) detects a photon which is converted in electron-hole pairs. Depending on the sensor, holes (usually, for n-type silicon sensors) or electrons drift toward the pixel flip-chip bonding pads [36]. The charge is collected, amplified, and converted to a voltage in the charge preamplifier of each pixel [3].…”

Hammerhead, an ultrahigh resolution ePix camera for wavelength-dispersive spectrometers

“…Sensors in hybrid pixel detectors are typically reverse-biased p-n junctions; after photon detection and conversion to electron-hole charge clouds, the charge drift can be described by the partial differential equations describing charge drift, diffusion and recombination [11]. We derive from first principles the factors that influence the size of the charge clouds, which in turn are important for optimizing detector parameters for subpixel resolution.…”

“…(1) which for the sensor described in this paper is ∼85 V (calculations and measurements in good agreement). Using the convention in [11], with coordinate x denoting the depth of the sensor measured from the origin of real or virtual transition from fully depleted to undepleted region, the front and back planes of the fully depleted sensors are at…”