Fluctuations in the collected charge in integrating photoconductive detectors under small and large signals: the variance problem

Abstract Charge collection efficiency (CE) η 0 under small signal conditions, corresponding to a uniform field in the detector medium, has been widely used in evaluating the performance of photoconductive detectors. The present paper answers the question, ‘ What is the variance of the collected charge in an integrating detector as a function of photoinjection level and what are the errors if we continue to use the small signal equations? ’ The variance <?CDATA $\sigma _0^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> in η 0 under small signals has been theoretically derived in the literature and has been a key factor in the detective quantum efficiency modeling of integrating detectors based on various semiconductors. <?CDATA $\sigma _0^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> is a noise source and can degrade the detector performance under incomplete charge collection. The statistical variance <?CDATA $\sigma _0^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> in the CE η 0 , under small signals and the variance <?CDATA $\sigma _r^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> in the CE η r under an arbitrary injection level r (injected charge divided by charge on the electrodes) have been studied using the Monte Carlo simulation model developed in this work to evaluate the difference between <?CDATA $\sigma _r^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> and <?CDATA $\sigma _0^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> from small to large signals. Initial injection of electron and hole pairs and their subsequent transport and trapping in the presence of an electric field, which is calculated from the Poisson equation, is used to calculate the photocurrent. Each injected carrier is tracked as it moves in the semiconductor until it is either trapped or reaches the collection electrode. Trapped carriers do not contribute to the photocurrent but continue to contribute to the field through the Poisson equation. The instantaneous photocurrent i ph ( t ) is calculated from the drift of the free carriers through the Shockley–Ramo theorem. i ph ( t ) is integrated over the duration of the photocurrent to calculate the total collected charge and hence the CE η r . The variance <?CDATA $\sigma _r^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> in η r is found from multiple simulations of η r . The η r and <?CDATA $\sigma _r^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> have been generated over varying charge injection ratios r , the electron and hole ranges μτ , mean photoinjection depths δ and drift mobility ratios b . At full injection, the deviation <?CDATA $\Delta \sigma _r^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> of the CE variance <?CDATA $\sigma _r^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> from the uniform field case <?CDATA $\sigma _0^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> ( <?CDATA ${\text{i}}.{\te

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